Introduction to Photonics ——— Draft

Technical University of Clausthal WS 2000/2001 – WS 2004/2005

W. Lücke

Preface

Photonics deals with controlled production, evolution, and detection of (mainly coherent) light (photons) within the ‘optical’ region of the electromagnetic spectrum (vacuum wavelength (λ_{0}) from 50 nm to 500 μm or photon energies (ℏω) from 25 to 0.002 eV). With the advent of the laser non-linear properties of optical media have become essential.

...

This draft should be considered as just the germ of a more complete introduction to photonics.

Recommended Literature: (Loudon, 2000; Mandel and Wolf, 1995; Yariv, 1997; Vogel et al., 2001; Shen, 1984; Bachor, 1998; Saleh and Teich, 1991)

4

Contents

1 Fundamentals 9

1.1 Classical Electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . 9 1.1.1 Maxwell Equations . . . . . . . . . . . . . . . . . . . . . . . 9 1.1.2 Energy and Energy Flux of the Field . . . . . . . . . . . . . . 11 1.1.3 Frequency Analysis . . . . . . . . . . . . . . . . . . . . . . . . 12

1.2 Quantized Electromagnetic Radiation . . . . . . . . . . . . . . . . . . 13 1.2.1 Quantum Aspects of Light . . . . . . . . . . . . . . . . . . . . 14 1.2.2 Field Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.2.3 Field Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.2.4 ‘Classical’ States of Light . . . . . . . . . . . . . . . . . . . . 24

2 Linear Optical Media 35

2.1 General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.1.1 A Simple Model . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.1.2 Exploiting Homogeneity . . . . . . . . . . . . . . . . . . . . . 37 2.1.3 Radiation off a Given Source . . . . . . . . . . . . . . . . . . . 40

2.2 Monochromatic Waves of Exponential Type . . . . . . . . . . . . . . 42 2.2.1 Basic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.2.2 Simple Reflection and Refraction . . . . . . . . . . . . . . . . 44 2.2.3 Birefringent Media (Crystal Optics) . . . . . . . . . . . . . . . 51

2.3 Monochromatic Light Rays . . . . . . . . . . . . . . . . . . . . . . . . 57 2.4 Group-Velocity Dispersion . . . . . . . . . . . . . . . . . . . . . . . . 59

3 Nonlinear Optical Media 61

3.1 General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.1.1 Functionals of the Electric Field . . . . . . . . . . . . . . . . . 62 3.1.2 Various Electric Polarization Effect . . . . . . . . . . . . . . . 63 3.1.3 Perturbative Solution of Maxwell’s Equations . . . . . . . . 63

3.2 Classical Nonlinear Optical Effects . . . . . . . . . . . . . . . . . . . 65 3.2.1 Phase Matching . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.2.2 Three-Wave Mixing . . . . . . . . . . . . . . . . . . . . . . . . 70 3.2.3 Four-Wave Mixing . . . . . . . . . . . . . . . . . . . . . . . . 76

3.3 Quantum Aspects of Three-Wave Mixing . . . . . . . . . . . . . . . . 77 3.3.1 Quantized Radiation Inside Linear Media . . . . . . . . . . . . 77 3.3.2 Nonlinear Perturbation . . . . . . . . . . . . . . . . . . . . . . 79 3.3.3 Type-II Spontaneous Down Conversion . . . . . . . . . . . . . 81

4 Photodetection 83

4.1 Simple Detector Models . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.1.1 Single Localized Detectors . . . . . . . . . . . . . . . . . . . . 83 4.1.2 Independent Detectors . . . . . . . . . . . . . . . . . . . . . . 85

4.2 Quantum Theory of Coherence . . . . . . . . . . . . . . . . . . . . . 86 4.2.1 Correlation Functions in General . . . . . . . . . . . . . . . . 86 4.2.2 Correlation Functions of Single-Mode States . . . . . . . . . . 88

4.3 Simple Interference Experiments . . . . . . . . . . . . . . . . . . . . . 95 4.3.1 Young’s Double-Slit Experiment . . . . . . . . . . . . . . . . 95 4.3.2 Hanbury-Brown-Twiss Effect . . . . . . . . . . . . . . . . 95

5 Applications of Non-Classical Light 99 5.1 Criteria for Non-Classical Light . . . . . . . . . . . . . . . . . . . . . 99 5.2 The Action of Beam Splitters on Photons . . . . . . . . . . . . . . . . 102

5.2.1 General Action of Linear Optical Devices . . . . . . . . . . . . 102 5.2.2 Classical Description of Lossless Beam Splitters . . . . . . . . 103 5.2.3 Transformation of Photon Modes . . . . . . . . . . . . . . . . 105 5.2.4 Mach-Zehnder Interferometer . . . . . . . . . . . . . . . . . 109

5.3 Applications to Quantum Information Processing . . . . . . . . . . . 111 5.3.1 Single-Photon States and Quantum Cryptography . . . . . . . 111 5.3.2 Entangled Pairs of Photons . . . . . . . . . . . . . . . . . . . 113 5.3.3 Quantum Teleportation . . . . . . . . . . . . . . . . . . . . . 116

6 Coupling of Quantum Systems 127

6.1 Closed Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 6.1.1 States and Observables in the Heisenberg Picture . . . . . . . 127 6.1.2 Time Evolution and Projective Measurements . . . . . . . . . 130 6.1.3 Time Dependent Perturbation Theory . . . . . . . . . . . . . 132

6.2 Bipartite Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 6.2.1 Composition of Distinguishable Systems . . . . . . . . . . . . 133 6.2.2 Partial States . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 6.2.3 Composition of Indistinguishable Systems . . . . . . . . . . . 135

6.3 Open Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

7 Radiative Transitions 139

7.1 Single Atoms in General . . . . . . . . . . . . . . . . . . . . . . . . . 139 7.1.1 Naive Interaction Picture . . . . . . . . . . . . . . . . . . . . . 139 7.1.2 Electric Field Representation . . . . . . . . . . . . . . . . . . 142 7.1.3 First Order Perturbation Theory . . . . . . . . . . . . . . . . 144

7.2 Photoelectric Detection of Light . . . . . . . . . . . . . . . . . . . . . 147 7.2.1 Ionisation of One-Electron Atoms . . . . . . . . . . . . . . . . 147 7.2.2 Simple Photodetectors . . . . . . . . . . . . . . . . . . . . . . 150 8 Few-Level Atomic Systems 153

8.1 The Exterior Electric Field Representation . . . . . . . . . . . . . . . 153 8.2 2-Level Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

8.2.1 Interpretation of the Bloch Vector . . . . . . . . . . . . . . . 157 8.2.2 Rotating Wave Approximation . . . . . . . . . . . . . . . . . . 158

8.3 3-Level Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 8.3.1 Population Trapping . . . . . . . . . . . . . . . . . . . . . . . 161 8.3.2 Electromagnetically Induced Transparency . . . . . . . . . . . 163 8.3.3 Change of Linear Susceptibility . . . . . . . . . . . . . . . . . 165

9 Cavity QED 171

9.1 The Jaynes-Cummings Model . . . . . . . . . . . . . . . . . . . . . 171 9.1.1 Quantized Radiation Field for Selected Modes . . . . . . . . . 171 9.1.2 Interaction of the Two-Level System with the Cavity Field . . 172

A Appendix 179 A.1 Functional Taylor Expansion . . . . . . . . . . . . . . . . . . . . . . . 179 A.2 Elementary Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . 181

A.2.1 Scalar Particles in One Dimension . . . . . . . . . . . . . . . . 181 A.2.2 Energy Eigenstates of the Harmonic Oscillator . . . . . . . . . 183 A.2.3 Coherent States . . . . . . . . . . . . . . . . . . . . . . . . . . 186

A.3 Construction of Field Operators . . . . . . . . . . . . . . . . . . . . . 187 A.4 Dyson Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

Bibliography 191

Index 204 8 CONTENTS

Chapter 1

Fundamentals

1.1 Classical Electrodynamics

1.1.1 Maxwell Equations

The classical electromagnetic field E(x, t) , B(x, t) is defined (implicitly) by the force density

f(x, t) = ρ_{test}(x, t) E(x, t) + ȷ_{test}(x, t) × B(x, t)  (1.1)

that it exerts on the carriers of test 4-current densities (c ρ_{test}(x, t), ȷ_{test}(x, t)); see, e.g., Sections 3.3.3 and 4.1.3 of (Lücke, rel). The field generated by a 4-current density (c ρ(x, t), ȷ(x, t)) fulfilling the continuity equation

ρ̇(x, t) + div ȷ(x, t) = 0  (1.2)

has to obey Maxwell’s equations^{1}

curl B(x, t) = μ_{0}(ϵ_{0} \frac{∂}{∂t}E(x, t) + ȷ(x, t)) ,  (1.3)
curl E(x, t) = −\frac{∂}{∂t}B(x, t) ,
div E(x, t) = \frac{1}{ϵ_{0}} ρ(x, t) ,
div B(x, t) = 0

(we use SI units; see Appendix A.3.3 of (Lücke, edyn)). More precisely, in the classical case, it is the physical solution

E(x, t) = −grad Φ_{ret}(x, t) − \frac{∂}{∂t}A_{ret}(x, t) , B(x, t) = curl A_{ret}(x, t)

Draft, November 5, 2011

1Note that μ_{0} ϵ_{0} = c^{−2} , where

ϵ_{0} ≈ 8, 85 · 10^{−12} \frac{As}{Nm}

and c denotes the velocity of light in vacuum (≈ 3 · 10^{8} m/s).

10 CHAPTER 1. FUNDAMENTALS

given by the retarded electromagnetic Potentials

Φ_{ret}(x, t) = \frac{1}{ϵ_{0}} ∫ \frac{ρ (x′, t − \frac{|x−x′|}{c})}{4π |x − x′|} dV_{x′} ,

A_{ret}(x, t) = μ_{0} ∫ \frac{ȷ (x′, t − \frac{|x−x′|}{c})}{4π |x − x′|} dV_{x′} .

(1.4)

In typical applications the 4-current density is approximated as

ρ(x, t) = ρ_{ex}(x, t) − div P(x, t) ,
ȷ(x, t) = ȷ_{ex}(x, t) + \frac{∂}{∂t} P(x, t) ,

(1.5)

with some known 4-current density (c ρ_{ex}(x, t), ȷ_{ex}(x, t)) of excess charges and some generalized polarization P(x, t) .

Remarks:

1. For every choice of ρ_{ex}(x, t), ȷ_{ex}(x, t) obeying the continuity equation there is^{2} a time-dependent vector field P(x, t) fulfilling (1.5).

2. Under suitable conditions we have

\frac{∂}{∂t} P(x, t) = \frac{∂}{∂t} P_{el}(x, t) + \frac{1}{μ_{0}} rot M(x, t) ,  (1.6)

(see, e.g., Chapter 4 of (Lücke, edyn)), where P_{el}(x, t) = D(x, t) − ϵ_{0} E(x, t) denotes the electric dipole density and M(x, t) = B(x, t) − μ_{0} H(x, t) the magnetic dipole density.^{3}

Then Maxwell’s equation become equivalent to

curl B(x, t) = μ_{0} \frac{∂}{∂t} (ϵ_{0} E(x, t) + P(x, t)) + μ_{0} ȷ_{ex}(x, t) , (1.7)
curl E(x, t) = −\frac{∂}{∂t}B(x, t) ,  (1.8)
div (ϵ_{0} E(x, t) + P(x, t)) = +ρ_{ex}(x, t) ,  (1.9)
div B(x, t) = 0 .  (1.10)

Draft, November 5, 2011

2See (Lücke, 1995), in this connection. 3Contrary to a wrong argument given in (Landau und Lifschitz, 1967, § 60) the interpretation

of M(x, t) does not imply \frac{∂}{∂t} P_{el}(x, t) ≈ 0 for ȷ_{ex}(x, t) = 0 . For higher dipole moments see (Bloembergen, 1996b, p. 63).

1.1. CLASSICAL ELECTRODYNAMICS 11

1.1.2 Energy and Energy Flux of the Field

For arbitrary (sufficiently well-behaved) regions G we have ∫_{G} μ_{0} ȷ(x, t) · E(x, t) dV_{x}

=_{(1.3)} ∫_{G}( \underbrace{E(x, t) · (∇_{x} × B(x, t))}_{=B(x,t)·(^{∇_{x} × E(x, t)}_{{= \atop (1.8)} − \frac{∂}{∂t} B(r,t)})−∇_{x}·(E(x,t)×B(x,t))} −ϵ_{0} μ_{0} E(x, t) · \frac{∂}{∂t} E(x, t)) dV_{x}

= − ∫_{G} (ϵ_{0} μ_{0} E(x, t) · \frac{∂}{∂t}E(x, t) + B(x, t) · \frac{∂}{∂t}B(x, t) + ∇_{x} · (E(x, t) × B(x, t))) dV_{x} .

This implies

\frac{1}{2} \frac{∂}{∂t} (ϵ_{0} |E(x, t)|^{2} + \frac{1}{μ_{0}} |B(x, t)|^{2})

= −ȷ(x, t) · E(x, t) − ∇_{x} · \frac{E(x, t) × B(x, t)}{μ_{0}}

(1.11)

and therefore, by Gauß’ theorem, the Poynting theorem^{4}

\frac{d}{dt} ∫_{G} \frac{1}{2} (ϵ_{0} |E(x, t)|^{2} + \frac{1}{μ_{0}} |B(x, t)|^{2}) dV_{x}

= − ∫_{G} ȷ(x, t) · E(x, t) dV_{x} − ∫_{∂G} \frac{E(x, t) × B(x, t)}{μ_{0}} · dS_{x} .

(1.12)

also imply

∫_{G} (E(x, t) · \frac{∂}{∂t} D(x, t) + H(x, t) \frac{∂}{∂t} B(x, t)) dV_{x}

(1.13)

where

D(x, t) =^{def} ϵ_{0} E(x, t) + P(x, t) , H(x, t) =^{def} \frac{B(x, t) − M(x, t)}{μ_{0}} .

Draft, November 5, 2011

4See also (Kinsler et al., 2009).

Remark: If (1.6) holds then (1.12) and Gauß’s theorem, because of

div (E(x, t) × M(x, t)) = M(x, t) · curl E(x, t) − E(x, t) · curl M(x, t)

(1.8) −M(x, t) · \frac{∂}{∂t} B(x, t) − E(x, t) · curl M(x, t) , = − ∫_{G} ȷ_{ex}(x, t) · E(x, t) dV_{x} − ∫_{∂G}(E(x, t) × H(x, t)) · dS_{x} ,

Since, by (1.1), ∫_{G} ȷ(x, t) · E(x, t) dV_{x} is to be interpreted as the rate of the work

done by the electromagnetic field on the charges, (1.12) suggests the interpretation

S(x, t) =^{def} \frac{1}{μ_{0}} E(x, t) × B(x, t) = \{ \begin{matrix}engergy current density of the \\ plain electromagnetic field \end{matrix}

(1.14)

of the (free) Poynting vector S(x, t) and

H_{0}(x, t) =^{def} \frac{1}{2} (ϵ_{0} |E(x, t)|^{2} + \frac{1}{μ_{0}} |B(x, t)|^{2})

= \{ \begin{matrix}energy density of the \\ plain electromagnetic field . \end{matrix}

(1.15)

Assuming (1.5) and defining^{5}

U(x, t) =^{def} \{ \begin{matrix}work done by the field \\ on the non-excess charges . \end{matrix}

we get

\frac{∂}{∂t} U(x, t) =_{(1.1)} E(x, t) · \frac{∂}{∂t} P(x, t) .  (1.16)

Remark: Recall that the power performed by a force F(t) on a masspoint with trajectory x(t) is F(t) · ẋ(t) . Correspondingly, the power performed by the angular momentum M(t) on a rigid body rotating with circular angular velocity ω(t) is M · ω(t) . Likewise, the power performed by an electrical field E(t) on an electrical

dipole P(t) is E(t) · Ṗ(t) — not \frac{d}{dt} (E(t) · P(t)) .

Therefore, (1.11) and (1.5) imply

\frac{∂}{∂t}(H_{0}(x, t) + U(x, t)) = −ȷ_{ex}(x, t) · E(x, t) − ∇_{x} · \frac{E(x, t) × B(x, t)}{μ_{0}} .

Consequently:

ȷ_{ex} = 0 ⟹ \frac{d}{dt} ∫ (H_{0}(x, t) + U(x, t))dV_{x} = 0 .  (1.17)

1.1.3 Frequency Analysis

Using the Fourier transform

f̃(x, ω) =^{def} \frac{1}{\sqrt{2π}} ∫ f(x, t) e^{iω t} dt

Draft, November 5, 2011

5Concerning the classical field energy see (Landau und Lifschitz, 1967, Chapt. IX, § 61).

1.2. QUANTIZED ELECTROMAGNETIC RADIATION 13

resp.

f̃(x, ω) =^{def} \frac{1}{\sqrt{2π}} ∫ f(x, t) e^{iω t} dt

we may write (1.7)–(1.10) in the equivalent form

\frac{1}{μ_{0}} curl B̃(x, ω) = −i ω (ϵ_{0} Ẽ(x, ω) + P̃(x, ω)) + j̃_{ex}(x, ω) , (1.18)
curl Ẽ(x, ω) = +i ω B̃(x, ω) ,  (1.19)
div (ϵ_{0} Ẽ(x, ω) + P̃(x, ω)) = +ρ̃_{ex}(x, ω) ,  (1.20)
div B̃(x, ω) = 0 .  (1.21)

Remark: If the Fourier-integrals do not exist in the ordinary sense — as, e.g., for monochromatic waves — they have to be interpreted in the distributional sense; see Exercise 33 of (Lücke, qft) or Section 3.1.1 of (Lücke, ftm).

The continuity equation for ρ_{ex}, ȷ_{ex} implies

ρ̃_{ex}(x, ω) = \frac{1}{iω} div j̃_{ex}(x, ω) ∀ ω ≠ 0  (1.22)

and, consequently, equivalence of the equations (1.18)–(1.21) to

B̃(x, ω) = \frac{1}{i ω} curl Ẽ(x, ω) ∀ ω ≠ 0

and

curl (curl Ẽ(x, ω)) = (\frac{ω}{c} )^{2} (Ẽ(x, ω) + \frac{1}{ϵ_{0}} P̃(x, ω)) + i ω μ_{0} j̃_{ex}(x, ω)

(1.23)

for ω ≠ 0 .

Of course, the physical fields have to be real. For the electric field, e.g., this means

E(x, t) = (E(x, t))^{∗} ,

i.e.

Ẽ(x, −ω) = (Ẽ(x, ω))^{∗} .  (1.24)

Therefore, it is sufficient to determine Ẽ(x, ω) for ω ≥ 0 .

1.2 Quantized Electromagnetic Radiation

Of course, the classical electromagnetic theory should be quantized.^{6} A rigorous relativistic quantum theory describing the interaction of radiation with matter (nonperturbative QED) does not yet exist. Fortunately, as we will see, for many important applications the classical macroscopic Maxwell theory supplies sufficient information concerning the influence of linear optical devices on photons.^{7}

Draft, November 5, 2011

6For a detailed discussion why a semiclassical theory (treating only ‘particles’ quantum mechanically) plus vacuum fluctuations is insufficient see, e.g., (Scully and Zubairy, 1999).

7The notion ‘photon’ was introduced in (Lewis, 1926). See (Lamb, Jr., 1995) for a historical review.

1.2.1 Quantum Aspects of Light

Nowadays (almost) monochromatic electromagnetic radiation of angular frequency ω is considered to be composed of quanta (photons) having (total) energy ℏ ω each (and zero rest mass). This leads to additional information on optical processes.

To give an example, consider second harmonic generation which has to be considered as formation of quanta having energy 2 ℏ ω out of pairs of quanta having energy ℏ ω each.^{8} Reversibility of quantum dynamics predicts also the reverse process, called spontaneous down conversion: instantaneous formation of pairs of photons out of single photons. Contrary to the quantized theory^{9} classical optics does not give any hint at (essentially) simultaneous creation of both partners of each pair, as confirmed by experiment.^{10}

More generally, quantum optics describes multi-photon processes mediated by matter. These processes are called nonlinear if what happens to a photon depends in an essential way on the presence and properties of other optical photons.^{11} This kind of nonlinearity is essential for optical implementations of modern quantum information processing.

1.2.2 Field Operators

In the so-called radiation gauge a classical electromagnetic vacuum field E(x, t) , B(x, t) (of sufficiently rapid decrease at spatial infinity) may be represented in the form

B(x, t) = curl A(x, t) , E(x, t) = −\frac{∂}{∂t}A(x, t)

(1.25)

with the vector potential

A(x, t) = \frac{1}{4π} ∫ \frac{curl B(x′, t)}{|x − x′|} dV_{x′}  (1.26)

fulfilling the Coulomb condition

div A(x, t) = 0 ,  (1.27)

Remark: Equations (1.25) follow from (1.26) by curl curl = grad div − Δ and the Poisson equation Δ_{x} \frac{1}{|x − x′|} = −4π δ (x − x′) ; see, e.g., equations 4.87 and 4.63 of (Lücke, ein). The Coulomb condition follows for (1.26) by div curl = 0 ; see, e.g., Equation 4.89 of (Lücke, ein).

Draft, November 5, 2011

8Momentum conservation holds for the momenta inside the medium, corresponding to phase (-velocity) matching from the classical point of view.

9See (Mandel and Wolf, 1995, Sect. 22.4.2).

10See (Mandel and Wolf, 1995, Sect. 22.4.7).

11This type of nonlinearity does not contradict the fundamental linearity of quantum mechanical time evolution!

In this gauge the free Maxwell equations are equivalent to the vectorial wave equation

□ A(x, t) = 0 ,  □ =^{def} Δ − (\frac{1}{c} \frac{∂}{∂t})^{2} .  (1.28)

If no boundary conditions are imposed, every (sufficiently well-behaved, real) solution of (1.28) fulfilling the Coulomb condition is of the form

A(x, t) =^{def} A^{(+)}(x, t) + (A^{(+)}(x, t))^{∗}  (1.29)

with a complex vector potential ^{12}

A^{(+)}(x, t) = (2π)^{−3/2}\sqrt{μ_{0}ℏc} ∫ (∑_{j=1}^{2} ϵ_{j}(k) a_{j}(k)) e^{−i(c|k|t−k·x)} \frac{dV_{k}}{\sqrt{2 |k|}} , (1.30)

where the a_{j}(k) are (sufficiently well-behaved) complex-valued functions and the ϵ_{j}(k) are vector-valued functions forming a

right handed orthonormal basis \{ϵ_{1}(k), ϵ_{2}(k), \frac{k}{|k|}\} of ℝ^{3}

for every k ≠ 0 .

In the Heisenberg picture the observables Ê(x, t) and B̂(x, t) should obey the same linear relations as their classical counter parts E(x, t) and B(x, t) . This can be achieved replacing the complex-valued functions a_{j}(k) by suitable operator-valued functions â_{j}(k) :

Â^{(+)}(x, t) =^{def} (2π)^{−3/2}\sqrt{μ_{0}ℏc} ∫ (∑_{j=1}^{2} ϵ_{j}(k) â_{j}(k)) e^{−i(c|k|t−k·x)} \frac{dV_{k}}{\sqrt{2 |k|}} , (1.31)

B̂^{(+)}(x, t) =^{def} curl Â^{(+)}(x, t)  (1.32)
= (2π)^{−3/2}\sqrt{μ_{0}ℏc} ∑_{j=1}^{2} ∫ ik × ϵ_{j}(k) â_{j}(k) e^{−i(c|k|t−k·x)} \frac{dV_{k}}{\sqrt{2 |k|}} ,

Ê^{(+)}(x, t) =^{def} −\frac{∂}{∂t} Â^{(+)}(x, t)  (1.33)
= (2π)^{−3/2}\sqrt{μ_{0}ℏc} ∑_{j=1}^{2} ∫ ic |k| ϵ_{j}(k) â_{j}(k) e^{−i(c|k|t−k·x)} \frac{dV_{k}}{\sqrt{2 |k|}} ,

Draft, November 5, 2011

12The factor 1/\sqrt{2 |k|} under the integral will be necessary for Lorentz covariance of the quantized field tensor. The factor \sqrt{μ_{0}ℏc} is chosen in view of (1.57).

Â(x, t) =^{def} Â^{(+)}(x, t) + (Â^{(+)}(x, t))^{†} ,  (1.34)
B̂(x, t) =^{def} B̂^{(+)}(x, t) + (B̂^{(+)}(x, t))^{†} ,  (1.35)
Ê(x, t) =^{def} Ê^{(+)}(x, t) + (Ê^{(+)}(x, t))^{†} .  (1.36)

If the operators â_{j}(k) , (â_{j}(k))^{†} fulfill the canonical commutation relations^{13}

[â_{j}(k), â_{j′}(k′)]_{−} = 0 , [â_{j}(k), (â_{j′}(k′))^{†}]_{−} = δ_{jj′} δ(k − k′)  (1.37)

on a suitable dense subspace D_{0} , of the Hilbert space H_{field} , containing a cyclic normalized vacuum state vector Ω characterized by^{14}

â_{j}(k) Ω = 0 ,  (1.38)

then the observables B̂^{(+)}(x, t) , Ê^{(+)}(x, t) fulfill all reasonable physical postulates.

Remarks:

1. These properties fix the quantized theory up to unitary equivalence (see Sect. 4.1.1 of (Lücke, qft) and (Fredenhagen, 2001, Sect. III.4)). This also implies, by the way, that all irreducible realizations of (1.37)/(1.38) are unitarily equivalent.

2. Actually, the (â_{j}(k))^{†} have to be considered as operator-valued distributions (generalized functions):

∫ â_{j}^{†}(k) φ^{∗}(k) dk ⊂ (∫ â_{j}(k) φ(k) dk)^{†} ∀ φ ∈ S(ℝ^{3}) .

See, e.g., Section 2.1.3 of (Lücke, qft) for details.

Especially, the field observables fulfill the quantized free Maxwell equations:

curl B̂(x, t) = μ_{0} ϵ_{0} \frac{∂}{∂t} Ê(x, t) ,  (1.39)
curl Ê(x, t) = −\frac{∂}{∂t} B̂(x, t) ,  (1.40)
div Ê(x, t) = 0 ,  (1.41)
div B̂(x, t) = 0 .  (1.42)

Draft, November 5, 2011

13We simply write z for z ˆ1 , z ∈ ℂ, as long as this does not cause any confusion. For a possible realization of (1.37)/(1.38) see A.3.

14Because of (1.38) the â_{j}(k) are called annihilation operators (see (Mizrahi and Dodonov, 2002) in this connection). Here, ‘cyclic’ means that the linear span of all vectors of the form

\underbrace{∫ f(k_{1}, …, k_{n}) (â_{j_{1}}(k_{1}))^{†} ⋯ (â_{j_{n}}(k_{1}))^{†} dV_{k_{1}} … dV_{k_{n}}}_{=^{def} 1 for n=0} Ω , n ∈ ℤ_{+} , j_{1}, …, j_{n} ∈ \{1, 2\} ,

with square integrable f’s is dense in H_{field} .

1.2. QUANTIZED ELECTROMAGNETIC RADIATION 17 However, the canonical commutation relations also imply

\begin{matrix}[(Â^{j_{1}})^{(+)}(x_{1}, t_{1}) , ((Â^{j_{2}})^{(+)}(x_{2}, t_{2}))^{†}] & − \\ = i μ_{0} ℏ c (δ_{j_{1}j_{2}} − \frac{\frac{∂}{∂x_{1}^{j_{1}}} \frac{∂}{∂x_{1}^{j_{2}}}}{Δ_{x_{1}}} ) Δ_{0}^{(+)}(x_{1} − x_{2}, t_{1} − t_{2}) \end{matrix}  (1.43)

(in the distributional sense), where

Δ_{0}^{(+)}(x, t) =^{def} −i (2π)^{−3} ∫ e^{i k·x} e^{−i c|k|t} \frac{dV_{k}}{2 |k|}  (1.44)
= −i (2π)^{−3} ∫_{k^{0}>0} δ (k^{0} k^{0} − k · k) e^{−i(k^{0}ct−k·x)} dV_{k} dk^{0} .

Outline of proof for (1.43): Since

[∑_{j=1}^{2} ϵ_{j}^{j_{1}}(k) â_{j}(k) , ∑_{j′=1}^{2} ϵ_{j′}^{j_{2}}(k′)(â_{j′}(k′))^{†}]_{−}
= ∑_{j,j′=1}^{2} ϵ_{j}^{j_{1}}(k) ϵ_{j′}^{j_{2}}(k′) [â_{j}(k) , (â_{j′}(k′))^{†}]_{−}
= ∑_{j=1}^{2} ϵ_{j}^{j_{1}}(k) ϵ_{j}^{j_{2}}(k′) δ (k − k′) ,

(1.31) implies

[(Â^{j_{1}})^{(+)}(x_{1}, t_{1}) , ((Â^{j_{2}})^{(+)}(x_{2}, t_{2}))^{†}]_{−}

= (2π)^{−3} μ_{o} ℏ c ∫ \underbrace{∑_{j=1}^{2} ϵ_{j}^{j_{1}}(k) ϵ_{j}^{j_{2}}(k′)}_{=δ_{j_{1}j_{2}} − \frac{k^{j_{1}} k^{j_{2}}}{|k|^{2}}} e^{−i(c|k|(t_{1}−t_{2})−k·(x_{1}−x_{2}))} \frac{dV_{k}}{2 |k|} .

(1.43) implies

\begin{matrix}[Â^{j_{1}}(x_{1}, t_{1}) , Â^{j_{2}}(x_{2}, t_{2})] & − \\ = i μ_{0} ℏ c (δ_{j_{1}j_{2}} − ^{\frac{\frac{∂}{∂x{j_{1} \atop 1}} \frac{∂}{∂x{j_{2} \atop 1}}}{Δ_{x_{1}}}} ) Δ_{0}(x_{1} − x_{2}, t_{1} − t_{2}) , \end{matrix}  (1.45)

where^{15}

Δ_{0}(x, t) =^{def} Δ_{0}^{(+)}(x, t) − Δ_{0}^{(+)}(−x, −t) .  (1.46)

Draft, November 5, 2011

15Note that Δ_{0}(x, t) , being an anti-symmetric Lorentz invariant distribution, vanishes for c |t| ≥ |x|. However, the operator on the r.h.s of (1.45) is non-local.

Therefore, we have

⟨Ω | Ê(x, t) · Ê(x′, t′) Ω⟩ ≠ 0  (1.47)

in spite of^{16}

⟨Ω | Ê(x, t) Ω⟩ = 0 .

Since Ê(x, t) is the observable of the electric field,^{17} field, (1.47) shows the presence of vacuum fluctuations^{18} (quantum noise). (1.45) also implies^{19}

[Â^{j_{1}}(x_{1}, t) , Â^{j_{2}}(x_{2}, t)]_{−} = 0 ,

[Â^{j_{1}}(x_{1}, t) , B̂^{j_{2}}(x_{2}, t)]_{−} = 0  (1.48)

[B̂^{j_{1}}(x_{1}, t) , B̂^{j_{2}}(x_{2}, t)]_{−} = 0 ,

[Ê^{j_{1}}(x_{1}, t) , Ê^{j_{2}}(x_{2}, t)]_{−} = 0

and

[Â^{j_{1}}(x_{1}, t) , Ê^{j_{2}}(x_{2}, t)]_{−} = −i \frac{ℏ}{ϵ_{0}} δ_{⊥}^{j_{1}j_{2}}(x_{1} − x_{2}) ,  (1.49)

[B̂(x_{1}, t) , Ê^{j_{2}}(x_{2}, t)]_{−} = i \frac{ℏ}{ϵ_{0}} e_{j_{2}} × ∇_{x_{1}}δ(x_{1} − x_{2}) ,

where

δ_{⊥}^{j_{1}j_{2}}(x) =^{def} (2π)^{−3} ∫ (δ_{j_{1}j_{2}} − \frac{k^{j_{1}}k^{j_{2}}}{|k|^{2}} )e^{i k·x} dV_{k}  (1.50)

fulfills

∑_{j,j′=1}^{3} ∫ δ_{⊥}^{jj′}(x − x′) F^{j}(x′) dV_{x′} e_{j′} = F_{⊥}(x)

=^{def} F(x) + grad ∫ \frac{div F(x′)}{4π |x − x′|} dV_{x′} (1.51)

for sufficiently well-behaved F(x) .

Draft, November 5, 2011

16Also for Johnson noise, the electric current in a simple circuit resulting from fluctuating electromagnetic forces due to thermal motion of the charges, one has zero mean but nonzero mean square. Ω being an eigenvector of the (non-hermitean!) operator Ê^{(+)}(x, t) does not mean that the positive-frequency part of the electric field has a definite value in the vacuum state.

17I.e. ⟨Φ | Ê(x, t) Φ⟩ is the expectation value for the electric field in the quantum state Φ , for all Φ ∈ D_{0} .

18These provide, e.g., a simple explanation for the spontaneous transitions of exited atoms into lower energy states; see (Milonni, 1994, Chapter 3) for some further effects. Also spontaneous down-conversion can be explained as a result of vacuum fluctuations.

19The first three equations hold because of Δ_{0}(x, 0) = 0 . The last one follows from

(\frac{∂}{∂t})^{2} Δ_{0}(x, t)_{|_{t=0}} = 0 .

Remark: Note that F_{⊥}(x) is just the transversal part of

F(x) = curl ∫ \frac{curl F(x′)}{4π |x − x′|} dV_{x} − grad ∫ \frac{div F(x′)}{4π |x − x′|} dV_{x}

(see, e.g., Section 4.5.3 of (Lücke, ein)).

Let us finally note that one may prove relativistic covariance of the theory in the sense that there is a unitary representation^{20} Û(a, Λ) of the restricted Poincaré group P_{+}^{↑} fulfilling

Û(a, Λ)Ω = 0

and

Û^{−1}(a, Λ)F̂^{μν} (Λx + a) Û(a, Λ) = ∑_{α,β=0}^{3} Λ_{α}^{μ}Λ_{β}^{ν}F̂^{αβ}(x)

(see, e.g., Chapter 4 of (Lücke, qft) for details), where^{21}

(F̂^{μν}(x)) =^{def} ( \begin{matrix}0 & −\frac{1}{c} Ê^{1}(x, t) & −\frac{1}{c} Ê^{2}(x, t) & −\frac{1}{c} Ê^{3}(x, t) \\ + \frac{1}{c} Ê^{1}(x, t) & 0 & −B̂^{3}(x, t) & +B̂^{2}(x, t) \\ + \frac{1}{c} Ê^{2}(x, t) & +B̂^{3}(x, t) & 0 & −B̂^{1}(x, t) \\ + \frac{1}{c} Ê^{3}(x, t) & −B̂^{2}(x, t) & +B̂^{1}(x, t) & 0 \end{matrix} )

and

x =^{def} (x^{0}, …, x^{3}) , x^{0} =^{def} c t .

This together with the above commutation relations also shows that the field measurements at space-like separated space-time points are compatible:

[F̂^{μν}(x), F^{μ′ν′}(x′)]_{−} = 0 for c |t − t′| > |x − x′| (Einstein causality) .

1.2.3 Field Modes

The free Hamiltonian Ĥ_{field} , characterized as the self-adjoint operator fulfilling the commutation relation

\frac{i}{ℏ}[Ĥ_{field}, Â^{(+)}(x, t)]_{−} = \frac{∂}{∂t} Â^{(+)}(x, t)  (1.52)

Draft, November 5, 2011

20This means that the Û(a, Λ) are unitary operators fulfilling

Û(a_{1}, Λ_{1})Û(a_{2}, Λ_{2}) = Û(a_{1} + Λ_{1}a_{2}, Λ_{1}Λ_{2}) ∀ (a_{1}, Λ_{1}), (a_{2}Λ_{2}) ∈ P_{+}^{↑} .

21Note that F̂^{μν} is the quantized version of \frac{1}{μ_{0}} H^{μν} , as introduced in 1.1.1.

on D_{0} ⊂ DĤ_{field} and

Ĥ_{field} Ω = 0 ,

is given by^{22}

Ĥ_{field} = ∫ ℏ c |k| n̂(k) dV_{k} , n̂(k) =^{def} ∑_{j=1}^{2} (â_{j}(k))^{†} â_{j}(k) ,  (1.53)

where n̂(k) is interpreted as the observable (in the Heisenberg picture) for the k-space density of the number of photons:

∫_{O} ⟨Φ | n̂(k) Φ⟩ dV_{k} = \{ \begin{matrix}expectation value for the \\ number of photons in the state Φ \\ with momentum p = ℏ k ∈ ℏ O . \end{matrix}  (1.54)

This is consistent in the sense that, for every open subset O of the k-space, the

t-independent observable ∫_{O} n̂(k) dV_{k} commutes^{23} with Ĥ_{field} , has only nonnegative

integer eigenvalues,^{24} and

P̂_{0} =^{def} ∫ ℏ k n̂(k) dV_{k}  (1.55)

is the momentum observable^{25} as uniquely characterized (on D_{0}) by

\frac{i}{ℏ}[P̂_{0}, Â^{j}(x, t)]_{−} = −∇_{x}Â^{j}(x, t)  (1.56)

and

P̂_{0}Ω = 0

According to (1.54),

N̂ =^{def} ∫ n̂(k) dV_{k}

has to be interpreted as the observable for the total number of photons and the subspace H_{field}^{(n)} of n-photon state vectors coincides with the closed linear span of the

Draft, November 5, 2011

22Note that (1.37) implies [n̂(k′), â_{j}(k)]_{−} = −â_{j}(k) δ(k − k′) . This, actually, is the heuristic motivation for (1.37). See (Green, 1953), in this connection.

23Note that (1.37) implies [(â_{j}(k))^{†}â_{j}(k) , (â_{j′}(k′))^{†}â_{j′}(k′)]_{−} = 0 .

24Since (1.37) implies

((â_{j}(k))^{†}â_{j}(k)) (â_{j_{1}}(k_{1}))^{†} ⋯ (â_{j_{n}}(k_{n}))^{†}

= (â_{j_{1}}(k_{1}))^{†} ⋯ (â_{j_{n}}(k_{n}))^{†} (((â_{j}(k))^{†}â_{j}(k)) + ∑_{ν=1}^{n} δ_{jj_{ν}} δ(k − k_{ν})) ,

(recall the derivation of (A.26)).

25This observable is t-independent in the Heisenberg picture, since it commutes with Ĥ_{field} .

set of all vectors of the form^{26}

∫ f(k_{1}, …, k_{n}) (â_{j_{1}}(k_{1}))^{†} ⋯ (â_{j_{n}}(k_{1}))^{†} dV_{k_{1}} … dV_{k_{n}} Ω

with square integrable f’s.

As already indicated in Footnote 12, the normalization factor on the r.h.s. of (1.31) has been chosen to yield^{27}

Ĥ_{field} = \frac{1}{2} ∫ (ϵ_{0} :Ê(x, t) · Ê(x, t): + \frac{1}{μ_{0}} :B̂(x, t) · B̂(x, t):) dV_{x} ,
c^{2} P̂_{0} = \frac{1}{μ}_{0} ∫ :Ê(x, t) × B̂(x, t): dV_{x} ,

(1.57)

where : : means normal ordering, i.e. in all products the creation operators

â_{j}^{†}(k) =^{def} (â_{j}(k))^{†} have to be placed — irrespective of the initial ordering — to the left of the annihilation operators â_{j}(k) .

A dense subset of H_{field}^{(n)} is given by the linear span of the set of all (i.g. not yet normalized) vectors of the form

â_{1}^{†} ⋯ â_{n}^{†} Ω

with modes â_{1}, … â_{n}, i.e. operators of the form^{28}

â_{ν} = ∑_{j=1}^{2} ∫ â_{j}(k)(f_{ν}^{j}(k))^{∗}dV_{k} ,  (1.58)
[â_{ν}, â_{ν}^{†}]_{−} = 1 .  (1.59)

Here

⟨Ω | A^{(+)}(x, t) â_{ν}^{†} Ω⟩ = (2π)^{− \frac{3}{2}} \sqrt{μ_{0} ℏ c} ∫ ∑_{j=1}^{2} ϵ_{j}(k) f_{ν}^{j}(k) e^{−i(c|k|t−k·x)} \frac{dV_{k}}{\sqrt{2 |k|}}

(1.60)

Draft, November 5, 2011

26Recall Footnote 14.

27Recall Footnote 1. Using the commutation relations for the components of Â(x, t) , Ê(x, t) and B̂(x, t) one can verify (1.52) and (1.56) also directly for (1.57). For the connection between the energy current density and the momentum density of the electromagnetic field see (Landau und Lifschitz, 1967, Footnote 1 on Page 284).

28Equation (1.59) is equivalent to

∑_{j=1}^{2} ∫ |f_{ν}^{j}(k)|^{2} dV_{k} = 1 .

can be interpreted as wave function of the single photon^{29} in the (pure) state ˆ= â_{ν}^{†} Ω . Since

⟨Ω | â_{j}(k) ∑_{j=1}^{2} â_{j}^{†}(k) f^{j}(k) dV_{k} Ω⟩ = ⟨Ω | ∑_{j=1}^{2}[â_{j′}(k′) , â_{j}^{†}(k)]_{−} f^{j}(k) dV_{k} Ω⟩
= f^{j′}(k′) ,

for every ϕ^{(1)} ∈ H_{field}^{(1)} there is exactly one mode â with

ϕ^{(1)} = â^{†} Ω .

Motivated by

⟨â^{†} Ω | ĉ^{†} Ω⟩ = ⟨Ω | â ĉ^{†} Ω⟩
= ⟨Ω | [â , ĉ^{†}]_{−} Ω⟩
= [â , ĉ^{†}]_{−} ,

modes â, ĉ are said to be orthogonal if

[â, ĉ^{†}]_{−} = 0 .

A family \left\{{ \atop â_{ν}}\right\}_{ν∈ℕ} of pairwise orthogonal modes is said to be maximal, if \{â_{ν}^{†} Ω : ν ∈ ℕ\} is a maximal orthonormal system (MONS) of H_{field}^{(1)} . For every maximal family \left\{{ \atop â_{ν}}\right\}_{ν∈ℕ} of pairwise orthogonal modes we have

â_{j}(k) = ∑_{ν=1}^{∞} ⟨Ω | â_{j}(k) â_{ν}^{†} Ω⟩ â_{ν} ;

since

â_{j}^{†}(k) Ω = ∑_{ν=1}^{∞} |â_{ν}^{†} Ω⟩⟨â_{ν}^{†} Ω| â_{j}^{†}(k) Ω

= ∑_{ν=1}^{∞} (⟨Ω | â_{j}(k) â_{ν}^{†} Ω⟩)^{∗} â_{ν}^{†} Ω  (1.61)

and modes â are uniquely characterized by the corresponding 1-photon state vectors â^{†} Ω . Therefore, the field operators â_{j}(k) may be replaced by the countable set of ordinary operators â_{ν} and every normalized 1-photon state vector Ψ^{(1)} may be written in the form

Ψ^{(1)} = ∑_{ν=1}^{∞} λ_{ν} â_{ν}^{†} Ω , ∑_{ν=1}^{∞} |λ_{ν}|^{2} = 1 ,

where the complex coefficients λ are the probability amplitudes for the ‘modes’ â_{ν} , i.e.:

Draft, November 5, 2011

29If ∑_{j=1}^{2} ϵ_{j}(k) f_{ν}^{j}(k) ∝ ϵ_{1}(k) + i ϵ_{2}(k) resp. ∑_{j=1}^{2} ϵ_{j}(k) f_{ν}^{j}(k) ∝ ϵ_{1}(k) − i ϵ_{2}(k) the photon is said to have positive resp. negative helicity.

For every ν_{0} ∈ ℕ

|λ_{ν_{0}}| = |⟨â_{ν}^{†} Ω | Ψ^{(1)}⟩|^{2}

is the probability for finding a photon, randomly chosen from an ensemble (characterized by) Ψ^{(1)} , in the mode â_{ν}^{†} if an ideal test for being in just one one the modes â_{1}, â_{2}, … is performed.^{30}

Obviously, for ν ∈ ℕ,

n̂_{ν} =^{def} â_{ν}^{†} â_{ν}

is the observable for the number of photons in mode â_{ν} and for the observable N̂ for the total number of photons we get

N̂ = ∑_{ν=1}^{n} n̂_{ν} .

Warning: According to the simple theory of photodetection considered in 7.2.2 the states corresponding to elements of H_{field}^{(n)} are, in fact, nfold localized: The maximal number of photodetectors which may fire simultaneously when testing such a state is n.

Typical for nonlinear quantum optics:

b̂_{ν}^{†} ω ⟼ ĉ_{ν}^{†} Ω ∀ ν  ≠⇒  b̂_{1}^{†} ⋯ b̂_{n}^{†} Ω ⟼ ĉ_{1}^{†} ⋯ ĉ_{n}^{†} Ω ∀ n .

Such ‘nonlinear’ behavior is needed, e.g., for optical implementations of the CNOT gate acting according to

H ⊗ H ⟼ H ⊗ H ,
H ⊗ V ⟼ H ⊗ V ,
V ⊗ V ⟼ V ⊗ H ,
V ⊗ H ⟼ V ⊗ V .

Finally, let us derive some useful relations for modes â :

A simple consequence of the commutation relations (1.37) and (1.59) is

(â^{†}â) â = â (â^{†}â − 1) , (â^{†}â) â^{†} = â^{†} (â^{†}â + 1)  (1.62)

which, by iteration, implies

(â^{†}â) \binom{}{â}^{n} = \binom{}{â}^{n} (â^{†}â − n) , (â^{†}â) (â^{†})^{n} = (â^{†})^{n} (â^{†}â + n)

Draft, November 5, 2011

30Typical for quantum mechanics is, that such tests are considered even if

Ψ^{(1)} = â^{†} Ω ,  â ≠ â_{ν} ∀ ν ∈ ℕ .

for all n ∈ ℕ. This, in turn, implies^{31}

(â^{†})^{n} â^{n} = (â^{†}â) (â^{†}â − 1) ⋯ (â^{†}â − (n − 1)) ∀ n ∈ ℕ  (1.63)

and

(â^{†}â) (â^{†})^{n} Ω = n (â^{†})^{n} Ω ∀ n ∈ ℕ .  (1.64)

1.2.4 ‘Classical’ States of Light

They are important,^{32} not only because they are one of the quantum mechanical states whose properties most closely resemble those of a classical electromagnetic wave, but also because a single-mode laser operated well above threshold generates a coherent state excitation...

(Loudon, 2000, p. 190)

Radiation off Classical Currents

The expectation values for the (no longer free) electromagnetic field associated with a classical 4-current (c ρ(x, t), ȷ(x, t)) in the vacuum should fulfill the corresponding Maxwell equations:

curl ⟨B(x, t)⟩ = μ_{0}(ϵ_{0} \frac{∂}{∂t} ⟨E(x, t)⟩ + ȷ(x, t)) ,  (1.65)
curl ⟨E(x, t)⟩ = −\frac{∂}{∂t} ⟨B(x, t)⟩ ,  (1.66)
div ⟨E(x, t)⟩ = \frac{1}{ϵ_{0}} ρ(x, t) ,  (1.67)
div ⟨B(x, t)⟩ = 0 .  (1.68)

In the interaction picture the expectation values (for pure states) are given by^{33}

⟨B(x, t)⟩ = ⟨Ψ_{t}^{I} | B̂(x, t) Ψ_{t}^{I}⟩ ,
⟨E(x, t)⟩ = ⟨Ψ_{t}^{I} | (Ê(x, t) − grad ∫ \frac{ρ(x′, t)}{4 π ϵ_{0} |x − x′|} dV_{x′}) Ψ_{t}^{I}⟩

(1.69)

and, therefore, the equations (1.65)–(1.68) are guaranteed by the t-dependence

iℏ\frac{d}{dt} Ψ_{t}^{I} = Ĥ_{int}(t) Ψ_{t}^{I} ,  (1.70)

of the normalized state vector Ψ_{t}^{I} , where

Ĥ_{int}(t) =^{def} − ∫ ȷ(x, t) · Â(x, t) dV_{x} .  (1.71)

Draft, November 5, 2011

31Compare with (A.31).

32See also (Klauder, 2010).

33Note that, in the presence of charges, Ê(x, t) is only the transversal part of the observable for the field strength in the interaction picture (see also Footnote 2 of Chapter 7).

1.2. QUANTIZED ELECTROMAGNETIC RADIATION 25

Outline of proof for (1.65)–(1.68): While (1.68) resp. (1.67) follows directly from (1.69) and (1.42) resp. (1.41), (1.66) follows according to

curl ⟨E(x, t)⟩ =_{(1.69)} curl ⟨Ψ_{t}^{I} | Ê(x, t) Ψ_{t}^{I}⟩
=_{(1.40)} ⟨Ψ_{t}^{I} | \frac{∂}{∂t} B̂(x, t) Ψ_{t}^{I}⟩
=_{(1.70)} \frac{∂}{∂t} ⟨Ψ_{t}^{I} | B̂(x, t) Ψ_{t}^{I}⟩ − ⟨Ψ_{t}^{I} | \underbrace{\frac{i}{ℏ} [Ĥ_{int}(t), B̂(x, t)]_{−}}_{=_{(1.71),(1.48)} 0} Ψ_{t}^{I}⟩ .

(1.65), finally, follows from

\frac{1}{μ_{0} ϵ_{0}} curl ⟨B(x, t)⟩ =_{(1.39)} ⟨Ψ_{t}^{I} | \frac{∂}{∂t} Ê(x, t) Ψ_{t}^{I}⟩
=_{(1.70)} \frac{∂}{∂t} ⟨Ψ_{t}^{I} | Ê(x, t) Ψ_{t}^{I}⟩ − ⟨Ψ_{t}^{I} | \frac{i}{ℏ} [Ĥ_{int}(t), Ê(x, t)]_{−} Ψ_{t}^{I}⟩

and

−\frac{i}{ℏ} [Ĥ_{int}(t), Ê(x, t)]_{−}  =_{(1.71)}  −\frac{i}{ℏ} ∫ ȷ(x′, t) · [Â_{0}(x′, t), Ê(x, t)]_{−} dV_{x′}
=_{(1.49),(1.51)} \frac{1}{ϵ} ȷ(x) + grad ∫ \frac{div ȷ(x′)}{4π ϵ_{0} |x − x′|} dV_{x}
=_{cont. eq.}  \frac{1}{ϵ} ȷ(x, t) − \frac{∂}{∂t} grad ∫ \frac{ρ(x′, t)}{4 π ϵ_{0} |x − x′|} dV_{x′} .

If

\begin{matrix}ρ(x, t) & = & 0 \\ ȷ(x, t) & = & 0 \end{matrix} \} ∀ t < 0  (1.72)

then the retarded solution of (1.70), characterized by

Ψ_{t}^{I} = Ω ∀ t < 0 ,

is^{34}

Ψ_{t}^{I} = e^{λ(t)} eÂ_{t} , Â_{t} =^{def} −\frac{i}{ℏ} ∫_{−∞}^{t} Ĥ_{int}(t′) dt′  (1.73)

(for sufficiently well-behaved ρ, ȷ), where

λ(t) =^{def} \frac{1}{2} ∫_{−∞}^{t} \frac{d}{dt′} [Â_{t′}, Â_{t}]_{−} dt′ ∈ ℂ .

Remark: We assume, without proof, that this solution is unique. For the expectation values this is a simple consequence of Poynting’s theorem; see, e.g., Section 5.1.2 of (Lücke, edyn).

Draft, November 5, 2011

34For more complicated interactions the Dyson series (see A.4) has to be evaluated.

That this is a solution can be shown by application of the Baker-Hausdorff formula^{35}

[Â, [Â, B̂]_{−}]_{−} = [B̂, [Â, B̂]_{−}]_{−} = 0 ⟹ e^{Â+B̂} = e^{− \frac{1}{2} [Â,B̂]_{−}} eÂ eB̂ .  (1.74)

Outline of proof for ‘ (1.73) ⟹ (1.70)’:

\frac{d}{dt} eÂ_{t} Ω

= lim_{Δt→+0} \frac{eÂ_{t+Δt}e^{−Â_{t}} − 1}{Δt} eÂ_{t} Ω

=_{(1.74)} lim_{Δt→+0} \frac{e^{Â_{t+Δt}−Â_{t}}e^{− \frac{1}{2}[Â_{t+Δt},Â_{t}]_{−}} − 1}{Δt} eÂ_{t} Ω

= lim_{Δt→+0} (\frac{e^{Â_{t+Δt}−Â_{t}} − 1}{Δt} e^{− \frac{1}{2}[Â_{t+Δt},Â_{t}]_{−}} + \frac{e^{− \frac{1}{2}[Â_{t+Δt},Â_{t}]_{−}} − 1}{Δt}) eÂ_{t} Ω

=_{(1.73)} (−\frac{i}{ℏ} Ĥ_{int} − λ′(t)) eÂ_{t} Ω .

Since

\frac{i}{ℏ} ∫_{−∞}^{t} Ĥ_{int}(t′) dt′ = â_{ȷ}(t) − (a_{ȷ}(t))^{†} ,  (1.75)

where

a_{ȷ}(t) =^{def} ∑_{j=1,2} ∫ (g^{j}(k, t))^{∗} â_{j}(k) dV_{k} ,

and

g^{j}(k, t) =^{def} \frac{i}{ℏ} \sqrt{\frac{μ_{0}ℏc}{2 |k|}} ϵ_{j}(k) · ∫_{−∞}^{t} ((2π)^{− \frac{3}{2}} ∫ ȷ (x, t′) e^{−i k·x} dV_{x}) e^{+i c|k|t′} dt′ ,

Draft, November 5, 2011

35For operators in finite dimensional vector spaces (1.74) may be proved as follows: Since e^{λÂ}B̂ e^{−λ Â} and e^{ad^{λÂ}} B̂ (adĈ D̂ =^{def} [Ĉ, D̂]_{−}) fulfill the same first order differential equation and initial condition (for λ = 0), the Campbell-Hausdorff formula

eÂ B̂ e^{−Â} = e^{adÂ} B̂

holds for arbitrary Â , B̂. Therefore, also

f_{1}(λ) =^{def} e^{λ(Â+B̂)+ \frac{λ^{2}}{2} [Â,B̂]_{−}}

and

f_{2}(λ) =^{def} e^{λ Â} e^{λ B̂}

fulfill the same first order differential equation and initial condition (for λ = 0) and hence f_{1} = f_{2} if the l.h.s. of (1.74) holds. For details concerning the case of unbounded operators Â, B̂ see (Fröhlich,

(1.73) can be written in the form

Ψ_{t}^{I} = e^{λ(t)} e^{(aȷ(t))^{†}−âȷ(t)} Ω .  (1.76)

Proof of (1.75): Using the general definition

G(k, t) =^{def} (2π)^{− \frac{3}{2}} ∫ G(x, t)e^{−i k·x} dV_{x}

we get

Ĥ_{int} =_{(1.71)} − ∫ ȷ_{cr}(−k, t) · Â(k, t) dV_{k}

and

Â(k, t) =_{(1.31),(1.34)} \sqrt{\frac{μ_{0}ℏc}{2 |k|}} ∑_{j=1,2} (ϵ_{j}(k) â_{j}(k) e^{−i c|k|t} + ϵ_{j}(−k)(â_{j}(−k))^{†} e^{+i c|k|t}) .

Since

ȷ(x, t) = (ȷ(x, t))^{∗} ⟹ ȷ(−k, t) = (ȷ(+k, t))^{∗} ,

this implies (1.75).

Remarks:

1. The state of the electromagnetic field depends only on the timedependent vector field ȷ(x, t) . This is no surprise since the latter — together with the continuity equation and (1.72) — fixes the time-dependent scalar field ρ(x, t) .

2. Note that for time intervals without interaction^{36} the interaction picture coincides with the free Heisenberg picture.

3. This is why â_{ȷ}(t) does not change during time intervals over which ȷ(x, t) vanishes.

Single Modes

Obviously, the state (1.76) generated from the vacuum by an exterior current is a coherent state,^{37} i.e. a pure state corresponding to a state vector χ of the form

χ ∝ D̂â(α) Ω , α ∈ ℂ ,  (1.77)

Draft, November 5, 2011

36For the present case this means that ȷ(x, t) = 0 over the corresponding time intervals.

37These states are called coherent since their analogs for the harmonic oscillator correspond to wave packets which do not spread but — apart form shifts of the expectation values — have the same probability distributions for position and momentum as the ground state, for all times; see A.2.3.

where â is a mode and

D̂â(α) =^{def} e^{α â^{†}−α^{∗} â} ∀ α ∈ ℂ .  (1.78)

Obviously, the operators D̂â(α) are unitary. Using the Baker-Hausdorff formula (1.74) we easily see that

D̂â(α) = e^{+ \frac{1}{2} |α|^{2}} e^{−α^{∗} â} e^{α â^{†}} = e^{− \frac{1}{2} |α|^{2}} e^{α â^{†}} Ω ∀ α ∈ ℂ .  (1.79)

The latter especially implies

χâ(α) =^{def} \underbrace{e^{− \frac{1}{2} |α|^{2}}}_{normalizing factor} e^{α â^{†}}Ω
= D̂â(α) Ω ∀ α ∈ ℂ .

(1.80)

Now, the relation

[â, e^{α â^{†}}]_{−} = α e^{α â^{†}} ,  (1.81)

corresponding to the formal rule

[Â, B̂]_{−} ∝ 1 ⟹ [Â, f(B̂)]_{−} = [Â, B̂]_{−} f′(B̂) ,  (1.82)

(compare Exercise E29a) of (Lücke, eine)) shows that

â χâ(α) = α χâ(α) ∀ α ∈ ℂ  (1.83)

More generally we have

\begin{aligned}â_{j}(k) D̂â(α) Ω & = & [â_{j}(k), D̂â(α)]_{−} Ω \\ & =_{(1.82)} & α [â_{j}(k), â^{†}]_{−} D̂â(α) Ω \end{aligned}

and

\begin{aligned}[â_{j}(k), â^{†}]_{−} & = & ⟨Ω | [â_{j}(k), â^{†}]_{−} Ω⟩ \\ & = & ⟨Ω | â_{j}(k) â^{†} Ω⟩ , \end{aligned}

hence

Â^{(+)}(x, t) χâ(α) = \underbrace{⟨Ω | Â^{(+)}(x, t) α â^{†} Ω⟩}_{= complex vector potential of the expectation value of the electromagnetic field in the state χâ(α) = α×wave function of a photon in the state â^{†} Ω} χâ(α) .  (1.84)

Thus:

In a coherent state χ the (vector-valued) ‘wave function’ of every photon coincides (up to normalization) with the complex vector potential ⟨χ | Â_{0}^{(+)}(x, t) χ⟩.

Remarks:

1. For (Fock-)states of the form

Ψ^{(n)} = \frac{1}{\sqrt{n!}} â^{†} Ω

the phase of the single-photon mode â is irrelevant. But it is crucial for coherent superpositions of such states with different values of n.

2. The unitary D̂â(α) a displacement operator in the sence that^{38}

[â, D̂â(α)]_{−} =_{(1.81)} α D̂â(α) , [â^{†}, D̂â(α)] =_{(1.81)} \overline{α} D̂â(α) . (1.85)

In coherent states χĉ(α) the so-called quadrature components

x̂â =^{def} \frac{â + â^{†}}{\sqrt{2}} , p̂â =^{def} \frac{â − â^{†}}{i\sqrt{2}}

of an arbitrary mode â have the same uncertainties as in the vacuum state and their product takes the minimum value allowed by the uncertainty relations^{39}

Δxâ Δpâ ≥ \frac{1}{2} |[x̂â, p̂â]_{−}| = \frac{1}{2} .

Outline of proof: With

λ =^{def} [â, α ĉ^{†}]_{−}

we have

⟨x̂ââ⟩ = ⟨χĉ(α) | x̂â χĉ(α)⟩
= \frac{1}{\sqrt{2}}(⟨χĉ(α) | â χĉ(α)⟩ + ⟨â χĉ(α) | χĉ(α)⟩)
= \frac{1}{\sqrt{2}} (λ + λ^{∗})

and

2 ⟨\binom{}{x̂â}^{2}⟩ = ⟨χĉ(α) | (\binom{}{â}^{2} + ((â^{†})^{2} + 2 â^{†} â + ˆ1)χĉ(α)⟩
= (λ^{∗})^{2} + λ^{2} + 2 λ^{∗} λ + 1
= (λ + λ^{∗})^{2} + 1 ,

Draft, November 5, 2011

38Note that (1.85) implies

â Ψ = α Ψ ⟹ â (D̂â(β) Ψ) = (α + β) (D̂â(β) Ψ) .

Actually:

D̂â(α) D̂â(β) =_{(1.74)} e^{^{\frac{1}{2}}(α \overline{β}−\overline{α} β)} D̂â(α + β) ∀ α, β ∈ ℂ .

39Siehe Footnote 14 of Appendix A.

30 CHAPTER 1. FUNDAMENTALS

hence

\begin{aligned}(Δxâ)^{2} & = & ⟨\binom{}{x̂â}^{2}⟩ − (\left⟨{ \atop x̂â}\right⟩)^{2} \\ & = & 1/2 . \end{aligned}

Similarly we get

(Δpâ)^{2} = 1/2 .

In this sense coherent states are ‘as classical as possible’.^{40}

Remarks:

1. For every t ∈ ℝ and arbitrary real-valued f^{1}(x), f^{1}(x), f^{1}(x) ∈

S(ℝ^{3}) there is a mode â with

∫ Ê(x, t) · f(x) dV_{x} ∝ x̂â .

2. If â is a mode then so is e^{i φ} â for every φ ∈ ℝ.

3. For every mode â we have p̂â = x̂_{−i â}. Let â be a mode and denote by Hâ the smallest closed subspace of H_{field} that

contains Ω and is invariant under â^{†}. Then^{41}

∑_{ν=0}^{∞} \frac{1}{ν!} |(â^{†})^{ν} Ω⟩⟨(â^{†})^{ν} Ω| = P̂_{Hâ} ,  (1.86)

holds as a consequence of

⟨(â^{†})^{ν} Ω | (â^{†})^{μ} Ω⟩ = ν! δ_{νμ} .  (1.87)

Proof of (1.87): The statement follows by successive application of

⟨â^{†}Ψ | (â^{†})^{μ} Ω⟩ = ⟨Ψ | [â , (â^{†})^{μ}]_{−} Ω⟩
=_{(1.82),(1.59)} μ ⟨Ψ | (â^{†})^{μ−1} Ω⟩ ∀ μ ∈ ℕ , Ψ ∈ H_{field} .

Draft, November 5, 2011

40States with Δxâ < 1/\sqrt{2} or Δpâ < 1/\sqrt{2} are called squeezed states (see (Scully and Zubairy, 1999, Sections 2.5–2.8) for a discussion of these states).

41 As usual, P̂_{Hâ} denotes the orthogonal projection onto Hâ.

1.2. QUANTIZED ELECTROMAGNETIC RADIATION 31 (1.86) implies^{42}

\frac{1}{π} ∫ |D̂â(x + iy) Ω⟩⟨D̂â(x + iy) Ω| dx dy = P̂_{Hâ} ,  (1.88)

i.e.^{43}

χ = \frac{1}{π} ∫ ⟨D̂â(x + iy) Ω | χ⟩ Dâ(x + iy) Ω dx dy ∀ χ ∈ Hâ .

Proof of (1.88): According to (1.86) and (1.87) we have to show

⟨(â^{†})^{ν} Ω | (∫ |D̂â(x + iy) Ω⟩⟨D̂â(x + iy) Ω| dx dy) (â^{†})^{μ} Ω⟩ = π ν! δ_{νμ} . This, however, may be shown as follows:

⟨(â^{†})^{ν} Ω | (∫ |D̂â(x + iy) Ω⟩⟨D̂â(x + iy) Ω| dx dy) (â^{†})^{μ} Ω⟩

= ∫ ⟨(â^{†})^{ν} Ω | D̂â(x + iy) Ω⟩ ⟨D̂â(x + iy) Ω | (â^{†})^{μ} Ω⟩ dx dy
=_{(1.79)} ∫ e^{−|x+iy|^{2}} ⟨(â^{†})^{ν} Ω | e^{(x+iy)^{â^{†}}} Ω⟩ ⟨e^{(x+iy)^{â^{†}}} Ω | (â^{†})^{μ} Ω⟩ dx dy
= ∫ e^{−|x+iy|^{2}} ∑_{ν′,μ′=0}^{∞} \frac{(x + iy)^{ν′} (x − iy)^{μ′}}{ν′! μ′!} ⟨(â^{†})^{ν} Ω | (â^{†})^{ν′} Ω⟩ ·
· ⟨(â^{†})^{μ′} Ω | (â^{†})^{μ} Ω⟩ dx dy
=_{(1.87)} ∫ (x + iy)^{ν} (x − iy)^{μ} e^{−|x+iy|^{2}} dx dy
= ∫_{0}^{∞} r^{ν+μ+1} e^{−r^{2}} dr ∫_{0}^{2π} e^{i(ν−μ)φ} dφ
= \frac{1}{2} ∫_{0}^{∞} ξ^{(ν+μ)/2} e^{−ξ} dξ ∫_{0}^{2π} e^{i(ν−μ)φ} dφ
= π ν! δ_{νμ} .

Draft, November 5, 2011

42The integral is to be understood in the weak sense:

⟨χ_{1}| (∫ |D̂â(x + iy) Ω⟩⟨D̂â(x + iy) Ω| dx dy) χ_{2}⟩

=^{def} ∫ ⟨χ_{1} |D̂â(x + iy) Ω⟩ ⟨D̂â(x + iy) Ω| χ_{2}⟩ dx dy ∀ χ_{1}, χ_{2} ∈ H_{field} .

43Note that, for every ϵ > 0 , already {D̂â(α) Ω : α ∈ ℂ , |α| < ϵ} is a complete set of states since, e.g.,

|(â^{†})^{n} Ω⟩ = ((\frac{d}{dα})^{n} e^{α â^{†}} Ω)_{|_{α=0}} ∀ n ∈ ℤ_{+}

and

|(â^{†})^{n} Ω⟩ = \frac{n!}{2π} e^{r^{2}/2} r^{−n} ∫ e^{−nφ} |D̂â (r e^{iφ}) Ω⟩ dφ ∀ n ∈ ℤ_{+} , r > 0 .

32 CHAPTER 1. FUNDAMENTALS By (1.88), every vector in Hâ way be written as a continuous superposition of coherent states |D̂â(x + iy) Ω⟩ , z ∈ ℂ. But, of course, this representation is not unique:^{44}

{|D̂â(x + iy) Ω⟩ : z ∈ ℂ} is an over-complete subset of Hâ. The states D̂â(α) Ω and D̂â(α) Ω are only approximately orthogonal for large |α − β| :

⟨D̂â(α) Ω | D̂â(β) Ω⟩ =_{(1.79)} e^{− \frac{1}{2}(|α|^{2}+|β|^{2})}⟨Ω | \underbrace{e^{α^{∗}â} e^{βâ^{†}} Ω}_{=_{(1.83)} e^{α^{∗}β} e^{βâ^{†}} Ω}⟩
= e^{− \frac{1}{2}(|α|^{2}+|β|^{2})+α^{∗}β} ⟨e^{β^{∗}â} Ω | Ω⟩
= e^{− \frac{1}{2}(|α|^{2}+|β|^{2})+α^{∗}β}  (1.89)

Multiple Modes More generally, let \left\{{ \atop â_{ν}}\right\}_{ν∈ℕ} be a maximal family of pairwise orthogonal modes and, for n ∈ ℕ, denote by H_{â_{1},...,â_{n}} the smallest closed subspace of H_{field} that contains Ω and is invariant under â_{1}^{†}, …, â_{n}^{†}. Then

s- lim_{n→∞} P̂_{H_{â_{1},...,ân}} = ˆ1_{H_{field}} .  (1.90)

Now also the vectors

Φ_{α_{1},...,α_{n}} =^{def} e^{− \frac{1}{2}(|α_{1}|^{2}+...|α_{n}|^{2})} e^{α_{1}â_{1}^{†}+...α_{n}â_{n}^{†}} Ω ∀ n ∈ ℕ , α_{1}, …, α_{n} ∈ ℂ  (1.91)

describe coherent states and have the inner products

⟨Φ_{α_{1},...,α_{n}} | Φ_{β_{1},...,β_{m}}⟩ = ∏_{ν=1}^{min\{n,m\}} e^{− \frac{1}{2}(|α_{ν}|^{2}+|β_{ν}|^{2})+α_{ν}^{∗}β_{ν}} .  (1.92)

Outline of proof for n = m :

e^{+(|α_{1}|^{2}+...+|β_{n}|^{2})} ⟨Φ_{α_{1},...,α_{n}} | Φ_{β_{1},...,β_{n}}⟩

= ⟨Ω | e^{α_{1}^{∗}â_{1}} e^{β_{1}â_{1}^{†}} … \underbrace{e^{α_{n}^{∗}â_{n}} e^{β_{n}â_{n}^{†}} Ω}_{=e^{α^{∗}_{n}βn} e^{αnâ^{†}_{n}} Ω}⟩
= e^{α_{n}^{∗}β_{n}}⟨\underbrace{e^{α_{n}^{∗}â_{n}} Ω}_{=Ω} | e^{α_{1}^{∗}â_{1}} e^{β_{1}â_{1}^{†}} … e^{α_{n−1}^{∗}â_{n−1}} e^{β_{n−1}â_{n−1}^{†}} Ω⟩
⋮
= ∏_{ν=1}^{n} e^{α_{ν}^{∗}β_{ν}} .

Draft, November 5, 2011

44A simple calculation (see (Mandel and Wolf, 1995, Section 11.6.1)) even shows that

∫ f(x^{2} + y^{2}) (x + iy)^{n} |D̂â(x + iy) Ω⟩ dx dy = 0

holds for every function f and every n ∈ ℕ for which the integral exists.

(1.92) implies

|⟨Φ_{α_{1},...,α_{n′}} | Φ_{β_{1},...,β_{n}}⟩|^{2} = ∏_{ν=1}^{n} e^{−|α_{ν}−β_{ν}|^{2}} for n′ ≥ n .  (1.93)

The corresponding generalization of (1.88) is

π^{−n} ∫ |Φ_{α_{1},...,α_{n}}⟩⟨Φ_{α_{1},...,α_{n}}| d^{2}α_{1} ⋯ d^{2}α_{n} = P̂_{H_{â_{1},...,ân}} ,  (1.94)

where

d^{2}(x + iy) =^{def} dx dy

and this gives a nice mode expansion:

(1.90) ⟹ ˆ1_{H_{field}} = s- lim_{n→∞} ∫ |Φ_{α_{1},...,α_{n}}⟩⟨Φ_{α_{1},...,α_{n}}| \frac{d^{2}α_{1}}{π} ⋯ \frac{d^{2}α_{n}}{π} . (1.95)

This expansion is extremely useful since^{45}

â_{j}(k) Φ_{α_{1},...,α_{n}} = [â_{j}(k) , α_{1}â_{1}^{†} + … α_{n}â_{n}^{†}]_{−} Φ_{α_{1},...,α_{n}} .  (1.96)

Now let ρ̂ be some density operator on H_{â_{1},...,â_{n}}. Then, thanks to (1.95), ρ̂ is uniquely determined by the matrix elements

f(α_{1}, …, β_{n}) =^{def} ⟨Φ_{α_{1},...,α_{n}} | ρ̂Φ_{β_{1},...,β_{n}}⟩ , n ∈ ℕ , α_{1}, …, β_{n} ∈ ℂ .

Since, by (1.91) ,

P (α_{1}, …, α_{n}, β_{1}, …, β_{n}) =^{def} e^{^{\frac{1}{2}} ∑{n \atop ν=1}(|α_{ν}|^{2}+|β_{ν}|^{2})}f(\overline{α_{1}}, …, \overline{α_{n}}, β_{1}, …, β_{n})

is a convergent power series of the variables α_{1}, …, α_{n}, β_{1}, …, β_{n} the following lemma shows that ρ̂ is already determined by the expectation values^{46}

⟨Φ_{α_{1},...,α_{n}} | ρ̂Φ_{α_{1},...,α_{n}}⟩ , n ∈ ℕ , α_{1}, …, α_{n} ∈ ℂ .

Lemma 1.2.1 Let n ∈ ℕ and let P (z_{1}, …, z_{2n}) be a power series of the complex variables z_{1}, …, z_{2n} that converges for all (z_{1}, …, z_{2n}) ∈ ℂ^{2n} and vanishes if

z_{ν} = \overline{z_{n+ν}} ∀ ν ∈ \{1, …, n\} .

Then P is identically zero.

Draft, November 5, 2011

45Recall (1.82) and (1.37).

46Note that Φ_{α_{1},...,α_{n}} = Φ_{α_{1},...,α_{n},0,...,0} .

Outline of proof: Obviously,

P̌(z_{1}, …, z_{2n}) =^{def} P (z_{1} − i z_{n+1}, …, z_{n} − i z_{n+n}, z_{1} + i z_{n+1}, …, z_{n} + i z_{n+n})

is a power series of the complex variables z_{1}, …, z_{2n} that vanishes on ℝ^{2n}. Therefore, it vanishes identically.

This suggests that ρ̂ may be represented in the form

ρ̂ = s- lim_{n→∞} ∫ ρ_{n}(α_{1}, …, α_{n}) |Φ_{α_{1},...,α_{n}}⟩⟨Φ_{α_{1},...,α_{n}}| \frac{d^{2}α_{1}}{π} ⋯ \frac{d^{2}α_{n}}{π}

with suitably chosen functions ρ_{n}(α_{1}, …, α_{n}) . Indeed, careful elaboration on this idea leads to the following theorem:

Theorem 1.2.2 For every density operator ρ̂ on H_{field} there is a sequence of functions φ_{N} ∈ S (ℝ^{2N}) — see, e.g. Definition 9.1.1 of (Lücke, eine) for the Definition of the Schwartz space S — such that

Tr (ρ̂B̂)

= lim_{N→∞} ∫ φ_{N}(ℜ(α_{1}), ℑ(α_{1}), …, ℜ(α_{N}), ℑ(α_{N})) ⟨Φ_{α_{1},...,α_{N}} | B̂ Φ_{α_{1},...,α_{N}} ⟩ \frac{dα_{1}}{π} ⋯ \frac{dα_{N}}{π}

(1.97)

is uniformly convergent for all B̂ ∈ L (H_{field}) with ∥B̂∥ ≤ 1 .

Proof: See (Klauder and Sudarshan, 1968, Section 8-4 B).

Strictly speaking (1.97) is established only for bounded operators B̂. Typically, however, this formula will be applied to experimentally realizable ρ with normally ordered polynomials (eventually of infinite degree) of radiation field operators substituted for B̂. The assertion that such applications are justified is called the optical equivalence theorem since for coherent states — thanks to (1.96) — the expectation values of normal ordered products of (positive- and negative-frequency) radiation field operators coincide with the corresponding products of expectation values of these operators.^{47}

Draft, November 5, 2011

47The latter implies the optical equivalence theorem for normally ordered bounded analytic functions of the â_{ν} . Limits of such functions are used in typical applications.

Linear Optical Media

2.1 General Considerations

In situations of practical interest a given medium never extends over all of ℝ. Consequently, nontrivial boundary conditions have to be taken into account and this spoils the use of spatial Fourier techniques. Then, instead of a Cauchy problem, one usually considers a scattering problem: For t → ∞ the electromagnetic field tends to a prescribed vacuum solution. This together with (1.7)–(1.10) and suitable constitutive equations determines the field for all times.

Usually, since the evolution of electromagnetic waves inside materials is very delicate, only very special types of (idealized) media and/or waveforms are discussed in the literature. This makes it very difficult to get an overview.

In their most general form the constitutive equations for linear media^{1} just specify P(x, t) as a linear functional of E(x, t) (recall Section 3.1.1) and B(x, t) . Of course, we are not able to establish linear optics in this general form. Rather we will consider simple models for special situations. However, in order to give at least some feeling for complications arising in more general situations, we derive Fresnel’s formulas ßindexFresnel@Fresnel’s formulas for (isotropic) dispersive media with absorption (on both sides of the boundary plane) and present the solutions of exponential type for Maxwell’s equations in nonisotropic media with absorption and dispersion.

2.1.1 A Simple Model

Let us consider (non-moving) linear optical media inside which the generalized polarization P(x, t) , and the induced macroscopic current density ȷ_{ind}(x, t) are given via t-dependent tensor fields^{2}

χ̌^{↔}(x, t) = (χ̌_{k}^{j}(x, t)) and σ̌^{↔}(x, t) = (σ̌_{k}^{j}(x, t))

Draft, November 5, 2011

1Linearity should be a good approximation for ‘normal’ electromagnetic fields.

2We use matrix multiplication: m^{↔} A =^{def} ∑_{j,k=1}^{3} m_{k}^{j} A^{k} e_{j} .

by^{3}

P(x, t) = \frac{ϵ_{0}}{\sqrt{2π}} ∫ χ̌^{↔}(x, t − t′) E(x, t′) dt′ ,  (2.1)

ȷ_{ind}(x, t) = \frac{1}{\sqrt{2π}} ∫ σ̌^{↔}(x, t − t′) E(x, t′) dt′

(2.2)

(see Appendix A of (Lücke, ein) for an outline of tensor calculus). Thinking of P and ȷ_{ind} as (initially^{4}) induced by E leads to the following causality condition:^{5}

χ̌^{↔}(x, t) = σ̌^{↔}(x, t) = 0 ∀ t < 0 .  (2.3)

Moreover, the inverse Fourier transforms χ_{k}^{j}(x, ω) , and σ_{k}^{j}(x, ω) — usually called “material constants” — should be ordinary functions, converging rapidly to 0 for ω → ±∞. Then also χ̌_{k}^{j}(x, t) and σ̌_{k}^{j}(x, t) are ordinary functions which should decrease exponentially for t → +∞ (damping).

Exercise 1 Show that the oscillation x(t) , induced by f(t) according to

\begin{matrix}ẍ(t) + 2ρ ẋ(t) + ω_{0}^{2} x(t) = f(t) , \\ x(t) = f(t) = 0 für t < t_{0} , \end{matrix}

is of the form

x(t) = \frac{1}{\sqrt{2π}} ∫ ř(t − t′) f(t′) dt′

where

r(ω) = \frac{1}{ω_{0}^{2} − ρ^{2} − (ω + iρ)^{2}}

can be analytically continued into an open neighborhood of the closed upper half plane. Moreover, show that

\begin{aligned}ř(t) & = & \frac{1}{\sqrt{2π}} ∫ r(−ω) e^{iωt} dω \\ & = & \frac{1}{\sqrt{ω_{0}^{2} − ρ^{2}}} e^{−ρt} sin (t \sqrt{ω_{0}^{2} − ρ^{2}}) θ(t) \end{aligned}

holds for weak damping, i.e. for 0 < ρ ≪ ω_{0} .

Draft, November 5, 2011

3This means — among other things — that we exclude ferroelectrics, ferromagnets, and nonlocal effects such as optical rotation in quartz (optical activity, see (Saleh and Teich, 1991, Eq. (6.4-2))) or anomalous skin effect, here. Media with (spatially) nonlocal response are discussed in (Agranovich and Ginzburg, 1984).

4Concerning the backreaction of P onto E see, e.g., (Mandel and Wolf, 1995, Sect. 16.3).

5Exact vanishing for t < 0 is not essential but quite convenient and reasonable for macroscopic considerations.

2.1.2 Exploiting Homogeneity

Let us assume that the medium is homogeneous, i.e. that

χ̌^{↔}(x, t) = χ̌^{↔}(t) , σ̌^{↔}(x, t) = σ̌^{↔}(t) .

Then the causality condition (2.3) implies that

χ_{l}^{k}(z) =^{def} \frac{1}{\sqrt{2π}} ∫ χ̌_{l}^{k}(t) e^{izt} dt (generalized susceptibility )

σ_{l}^{k}(z) =^{def} \frac{1}{\sqrt{2π}} ∫ σ̌_{l}^{k}(t) e^{izt} dt (specific conductivity )

are bounded holomorphic functions on a sufficiently small neighborhood of the closed upper half plane.

Remark: This together with (2.4), (2.5), and the causality condition allows for the derivation of dispersion relations (see Sect. 4.1 of (Lücke, ftm) providing valuable insight into the frequency dependence of the permittivity (see (Mills, 1998, Sect. 2.1)).

The same, consequently, holds for the (relative) permittivity^{6}

ϵ^{↔}(z) =^{def} 1^{↔} + χ^{↔}(z) .

Since χ̌_{l}^{k} and σ̌_{l}^{k} have to be real-valued, the so-called crossing relations

\begin{matrix}ϵ^{↔}(z) & = & ϵ^{↔}^{∗} (−z^{∗}) \\ σ^{↔}(z) & = & σ^{↔}^{∗} (−z^{∗}) \end{matrix} \} for ℑ(z) ≥ 0  (2.4)

have to be fulfilled^{7} according to Rule 7 for the Fourier transform^{8} and the principle of analytic continuation. The asymptotic behaviour along the real axis is:^{9}

\begin{matrix}ϵ^{↔}(ω) & → & 1^{↔} \\ σ^{↔}(ω) & → & 0^{↔} \end{matrix} \} for ω → ±∞ .  (2.5)

If (2.1) holds, we may use

ϵ_{0} Ẽ(x, ω) + P̃(x, ω) = ϵ_{0} σ^{↔}(ω) Ẽ(x, ω)  (2.6)

in (1.18) resp. (1.20) and get

curl B̃(x, ω) = \frac{−i ω}{c^{2}} ϵ^{↔}(ω) Ẽ(x, ω) + μ_{0} j̃_{ex}(x, ω)

(2.7)

Draft, November 5, 2011

6The inverse ϵ^{↔}^{−1}(z) is called impermeability.

7By definition, the matrix m^{↔}^{∗} results from m^{↔} by complex conjugation of the entries.

8See Appendix.

9Compare Exercise 1 and (Römer, 1994, Sect. 2.8).

resp.

div (ϵ_{0} σ^{↔}(ω) Ẽ(x, ω)) = +ρ̃_{ex}(x, ω) ,

(2.8)

If also ȷ_{ex} = ȷ_{ind} holds with ȷ_{ind} given by (2.2), then (2.7) becomes^{10}

curl B̃(x, ω) = \frac{−i ω}{c^{2}} ( ϵ^{↔}(ω) + i \frac{σ^{↔}(ω)}{ω ϵ_{0}}) Ẽ(x, ω) .  (2.9)

Remark: The material tensors ϵ^{↔}(ω), σ^{↔}(ω) enter Maxwell’s equations only via the combination

ϵ^{↔}_{c}(ω) =^{def} ϵ^{↔}(ω) + i \frac{σ^{↔}(ω)}{ω ϵ_{0}} ,

usually called complex dielectric ‘constant’ — even though, from the physical point of view,^{11} already ϵ(ω) should be complex-valued.^{12} From the purely mathematical point of view, if only ϵ^{↔}_{c}(ω) is given, ϵ_{l}^{k}(ω) and σ_{l}^{k}(ω) may be considered as real-valued (but not separately analytic). Similar remarks apply to the complex conductivity

σ^{↔}_{c}(ω) =^{def} σ^{↔}(ω) − i ω ϵ_{0} ϵ^{↔}(ω) (= −i ω ϵ_{0} ϵ^{↔}_{c}(ω)) .

Exercise 2 Given ω ∈ ℝ \ {0} and assuming (1.7)–(2.2) show that

\begin{matrix}0 ≠ ϵ^{↔}_{c}(ω) ∼ 1^{↔} \\ ȷ_{ex}(x, t) = ȷ_{ind}(x, t) \end{matrix} \} ⟹ ρ̃_{ex}(x, ω) = 0 .

Thanks to (1.19), the radiative Part of B(x, t) as as a function of time is already fixed by E(x, t) and its first order spatial derivatives at the same position in space. Therefore:

For radiation fields inside homogeneous linear media it is sufficient to determine E(x, t) .

Draft, November 5, 2011

10Since the 4-current ȷ_{cr}^{ν} =^{def} ȷ_{ex}^{ν} − ȷ_{ind}^{ν} creating the electromagnetic wave (see Section 2.1.3) is not included, (2.9) is — strictly speaking — only relevant for times t with ȷ_{cr}^{ν}(x, t′) = 0 ∀ t′ ≥ t.

11Compare Exercise 1. 12Thus allowing for a phase difference between Ẽ(x, ω) and D̃(x, ω) .

In the following we consider only media for which (2.1) holds. Then we have

curl (curl Ẽ(x, ω)) =_{(1.19)} i ω curl B̃(x, ω)
=_{(2.7)} \frac{ω^{2}}{c^{2}} ϵ^{↔}(ω) Ẽ(x, ω) + i ω μ_{0} j̃_{ex}(x, ω)
=_{(2.2)} \frac{ω^{2}}{c^{2}} ϵ^{↔}_{c}(ω) Ẽ(x, ω) + i ωμ_{0} j̃_{cr}(x, ω) ,  (2.10)

where

j̃_{cr}(x, ω) =^{def} j̃_{ex}(x, ω) − j̃_{ind}(x, ω) .  (2.11)

Thanks to

∇_{x} × (∇_{x} × Ẽ(x, ω)) = ∇_{x} (∇_{x} · Ẽ(x, ω)) − Δ_{x} Ẽ(x, ω) ,

(2.10) is equivalent to

(Δ_{x} + (\frac{ω}{c} )^{2} ϵ^{↔}_{c}(ω)) Ẽ(x, ω) = grad (div Ẽ(x, ω)) − i ω μ_{0} j̃_{cr}(x, ω) .

(2.12)

Therefore

B̃(x, ω) =_{(1.19)} \frac{curl Ẽ(x, ω)}{i ω} for ω ≠ 0  (2.13)

together with (2.12) implies (2.7). (1.21) is a direct consequence of (2.13) and (2.8), finally, serves as a definition for ρ̃_{ex} respecting the continuity equation. The latter implies

ρ̃_{ex}(x, ω) = ρ̃_{cr}(x, ω) + ρ̃_{ind}(x, ω) for ω ≠ 0 ,  (2.14)
where: ρ̃_{cr}(x, ω) =^{def} div (j̃_{cr}(x, ω)) /(iω) ,  (2.15)
ρ̃_{ind}(x, ω) =^{def} div(σ^{↔}(ω) Ẽ(x, ω))/(iω) .  (2.16)

We conclude:

Solving (1.18)–(1.21) for ω ≠ 0 is equivalent to solving (2.12) and determining B̃(x, ω) by (2.13).

Lemma 2.1.1 Let E(x, t) be a solution of (2.12) for j̃_{cr}(x, ω) = 0 with

supp E(x, t) ⊂ U_{R} (V_{+})

for some R > 0 . Then E(x, t) = 0 .

Proof: Use the tube theorem, the edge-of-the-wedge theorem, and (2.5).

2.1.3 Radiation off a Given Source

In this subsection we consider only isotropic^{13} homogeneous media, i.e. we assume

χ̌^{↔}(x, t) = χ̌(t) 1^{↔} , σ̌^{↔}(x, t) = σ̌(t) 1^{↔} .  (2.17)

Then, by (2.8), (2.12) becomes

(Δ_{x} + (\frac{ω}{c} )^{2}ϵ_{c}(ω)) Ẽ(x, ω) = \frac{1}{ϵ_{0} ϵ(ω)} grad ρ̃_{ex}(x, ω) − i ω μ_{0} j̃_{cr}(x, ω) . (2.18)

Defining

Ẽ_{ret}(x, ω) =^{def} −grad ∫ ρ̃_{cr}(x′, ω) \frac{e^{i \frac{ω}{c} N (ω)|x−x′|}}{4π ϵ_{0} ϵ_{c}(ω) |x − x′|} dV_{x′}

− μ_{0} ∫ (−iω) j̃_{cr}(x′, ω) \frac{e^{i \frac{ω}{c} N (ω)|x−x′|}}{4π |x − x′|} dV_{x′} ,

(2.19)

where

N (ω) =^{def} \sqrt{ϵ_{c}(ω)} , ℜ(N (ω)) ≥ 0 ,

and using (2.15) we get

div Ẽ_{ret}(x, ω) = \frac{ρ̃_{cr}(x, ω)}{ϵ_{0} ϵ_{c}(ω)}  (2.20)

and hence, by (2.14) and (2.16),

ρ̃_{cr}(x, ω)/ϵ_{c}(ω) = ρ̃_{ex}(x, ω)/ϵ(ω) .

Therefore Ẽ(x, ω) = Ẽ_{ret}(x, ω) is a solution of (2.18) (for ω ≠ 0). By (2.13) this implies

B̃_{ret}(x, ω) =^{def} curl ∫ μ_{0} j̃_{cr}(x′, ω) \frac{e^{i \frac{ω}{c} N (ω)|x−x′|}}{4π |x − x′|} dV_{x′} .  (2.21)

Fourier transforming (2.19) and (2.21) we get solutions

E(x, t) = −grad Φ(x, t) − \frac{∂}{∂t}A(x, t) , B(x, t) = curl A(x, t)  (2.22)

of Maxwell’s equations (1.7)–(1.10) for^{14}

ρ_{ex}(x, t) = − ∫_{−∞}^{t} (\frac{1}{\sqrt{2π}} ∫ σ̌(t′′ − t′) div E(x, t′) dt′) dt′′ + ρ_{cr}(x, t) ,

ȷ_{ex}(x, t) = \frac{1}{\sqrt{2π}} ∫ σ̌(t − t′) E(x, t′) dt′ + ȷ_{cr}(x, t)

(2.23)

Draft, November 5, 2011

13All crystals of the cubic system are isotropic. It would be interesting to generalize the results of this subsection for nonisotropic media.

14Note that (2.23) and the continuity equation for _{ex} imply the continuity equation for _{cr} .

with electromagnetic Potentials Φ , A of the form

Φ(x, t) = Φ_{ret}(x, t) =^{def} \frac{1}{\sqrt{2π}} ∫ ρ̃_{cr}(x′, ω) \frac{e^{−i \frac{ω}{c} (c t−N (ω)|x−x′|)}}{4π ϵ_{0} ϵ_{c}(ω) |x − x′|} dV_{x′} dω ,

A(x, t) = A_{ret}(x, t) =^{def} \frac{1}{\sqrt{2π}} ∫ μ_{0} j̃_{cr}(x′, ω) \frac{e^{−i \frac{ω}{c} (c t−N (ω)|x−x′|)}}{4π |x − x′|} dV_{x′} dω .

(2.24)

Using standard techniques one may prove (see (Borchers, 1990) and, for the main techniques, also Sect. 4.1 of (Lücke, ftm)) that

ρ_{cr}(x, t) , ȷ_{cr}(x, t) = 0 for (|x| , t) ∉ [0, R] × [0, ∞]
⟹ B_{ret}(x, t) = 0 for t ∉ [0, \frac{|x|−R}{c}] .

and, assuming^{15}

∫ \frac{1}{ϵ_{c}(ω)} e^{−iωt} dt = 0 ∀ t < 0 ,  (2.25)

(in addition to (2.3)) also

ρ_{cr}(x, t) , ȷ_{cr}(x, t) = 0 for (|x| , t) ∉ [0, R] × [0, ∞]
⟹ E_{ret}(x, t) = 0 for t ∉[0, \frac{|x|−R}{c}] .

(2.26)

holds. In other words:

The speed of a lightning inside the medium is strictly bounded by c.

All this becomes obvious for constant^{16} real ϵ_{c} . Then we have

Φ_{ret}(x, t) = \frac{ρ_{cr}(x′, t − \frac{n}{c} |x − x′|)}{4π ϵ_{0} ϵ_{c} |x − x′|} , A_{ret}(x, t) = μ_{0} \frac{ȷ_{cr}(x′, t − \frac{n}{c} |x − x′|)}{4π |x − x′|}

with n = N (ω) , showing that the speed of light in such a medium is c_{medium} = c/n. Taken together with the former argument we see that

ℝ ∋ ϵ_{c} constant ⟹ n ≥ 1 .  (2.27)

Remarks:

1. By Lemma 2.1.1, there is at most one retarded solution for given ρ_{cr} , ȷ_{cr} .

2. Therefore we consider the solution (2.22)/(2.24) as the physical one.

3. Then (2.20) implies

div Ẽ(x, ω) = 0  (2.28)

outside the (spatial) support of ρ_{cr}(x, ω) .

Draft, November 5, 2011

15This assumption may presumably be proved using (2.5).

16Strictly speaking, this is only compatible with (2.5) if ϵ = μ = 1 and σ = 0 .

42 CHAPTER 2. LINEAR OPTICAL MEDIA

2.2 Monochromatic Waves of Exponential Type

2.2.1 Basic Equations

In the following we consider only the case

ȷ_{cr} = 0 .

Then (2.12) becomes

(Δ_{x} + (\frac{ω}{c} )^{2} ϵ^{↔}_{c}(ω)) Ẽ(x, ω) = grad (div Ẽ(x, ω))  (2.29)

Let us look for monochromatic solutions of exponential type

Ẽ(x, ω′) = E_{s}(ω) exp (i \frac{ω}{c} N_{s}(ω) s · x) \sqrt{2π} δ(ω′ − ω) ,

(2.30)

i.e. for (damped) monochromatic plane waves^{17}

E(x, t) = E_{s}(ω) e^{−i \frac{ω}{c} (c t−N_{s}(ω) s·x)} ,

where

s , E_{s}(ω) ∈ ℂ^{3}

and s is a (complex) direction, i.e.^{18}

s · s = 1 .  (2.31)

Then (2.29) holds iff^{19}

ϵ^{↔}_{c}(ω) E_{s}(ω) = (N_{s}(ω))^{2} P^{↔}_{⊥s} E_{s}(ω) ,

where: P^{↔}_{⊥s} z =^{def} z − (s · z) s ∀ z ∈ ℂ^{3} .

(2.32)

Draft, November 5, 2011

17Note that, by (2.31), N (ω) s can only be real if (N_{s}(ω))^{2} is real. One may show (see (Mills, 1998, p. 12)) that ℜ (N_{s}(ω)) > 0 ⟹ ω ℑ (N_{s}(ω)) ≤ 0 . Actually, N_{s}(ω) may depend on the polarization (see 2.2.3).

18We use standard notation:

a · b =^{def} ∑_{j=1}^{3} a^{j} b^{j} , \overline{a^{1}e_{1} + a^{2}e_{2} + a^{3}e_{3}} =^{def} \overline{a^{1}}e_{1} + \overline{a^{2}}e_{2} + \overline{a^{3}}e_{3} , |a| =^{def} \sqrt{\overline{a} · a} ≥ 0

(a^{1}e_{1} + a^{2}e_{2} + a^{3}e_{3}) × (b^{1}e_{1} + b^{2}e_{2} + b^{3}e_{3}) =^{def} (a^{2}b^{3} − a^{3}b^{2}) e_{1} + (a^{3}b^{1} − a^{1}b^{3}) e_{2} + (a^{1}b^{2} − a^{2}b^{1}) e_{3} .

Note that the symmetric bilinear form a·b is an indefinite inner product on ℂ^{3} and that ℑ (s · s) = 0 ⟹ ℜ(s) · ℑ(s) = 0 . Modes with 0 ≠ e · s ∈ i ℝ for some e ∈ ℝ^{3} are called evanescent.

19(2.32) implies (2.8) for ρ_{ex} = ρ_{ind} according to (2.16). Note that, thanks to (2.31), P^{↔}_{⊥s} is a

projection operator: P^{↔}_{⊥s} P^{↔}_{⊥s} = P^{↔}_{⊥s} .

According to (2.13), the corresponding magnetic field is given by

B̃(x, ω) = μ_{0} \frac{N_{s}(ω)}{Z_{0}} s × Ẽ(x, ω)  (2.33)

for ω ≠ 0 , where

Z_{0} =^{def} μ_{0} c ≈ 377 Ω vacuum impedance .

Even though the object of physical interest is the real field^{20}

E(x, t) + \overline{E(x, t)} = 2 e^{− \frac{ω}{c} ℑ(N_{s}(ω) s)·x} (ℜ (E_{s}) cos(ω t − \frac{ω}{c} ℜ(N_{s}(ω) s) · x)

+ ℑ (E_{s}) sin(ω t − \frac{ω}{c} ℜ(N_{s}(ω) s) · x))

(2.34)

it is mathematically quite convenient to work with the complex field^{21} E(x, t) . For the corresponding Poynting vector S(x, t) we have

μ_{0} ⟨S(x, t)⟩ = 4 ⟨ℜ(E(x, t)) × ℜ(B(x, t))⟩

=_{(2.33)} μ_{0} \frac{2}{Z_{0}} e^{−2 \frac{ω}{c} ℑ(N_{s}(ω) s)·x} ℜ(\overline{E_{s}(ω)} × (N_{s}(ω) s × E_{s}(ω))) ,(2.35)

where ⟨ ⟩ means averaging over t. By (2.31) and

a × (b × c) = (a · c) b − (a · b) c ∀ a, b, c ∈ ℂ^{3}  (2.36)

this gives^{22}

⟨S(x, t)⟩ = \frac{2}{Z_{0}} e^{−2 \frac{ω}{c} ℑ(N_{s}(ω) s)·x} ℜ(|E_{s}(ω)|^{2} (N_{s}(ω) s)
−N_{s}(ω) (\overline{E_{s}(ω)} · s) E_{s}(ω)) .

(2.37)

For real s and ϵ^{↔}(ω) = ϵ(ω) 1^{↔} the polarization of the wave is given, according to

Draft, November 5, 2011

20Note that with (1.24) the crossing relations (2.4) imply that (2.29) also holds for ω replaced by −ω. Similar formulas hold for B(x, t) and B̃(x, ω) with B_{s} = μ_{0} \frac{N (ω)}{Z_{0}} s × E_{s} .

21See (Bloembergen, 1996b, Section 1–2) for the justification of this convention.

22Note that for isotropic media and real s the wave is propagating into the direction of ℜ(N_{s}(ω))s perpendicular to the planes of constant phase while the planes of constant amplitude are perpendicular to ℑ(N_{s}(ω))s.

polarization  Jones vector
linear  e^{iφ} ( \begin{matrix}cos α \\ sin α \end{matrix} )
right circular^{24}  \frac{e^{iφ}}{\sqrt{2}} ( \begin{matrix}1 \\ i \end{matrix} )
left circular  \frac{e^{iφ}}{\sqrt{2}} ( \begin{matrix}1 \\ −i \end{matrix} )
elliptic  else^{25}

Table 2.1: Polarization of monochromatic plane waves

Table 2.1, by the corresponding Jones vector^{23}

J_{s}(x, ω) =^{def} ( \begin{matrix}a_{s} · \frac{Ẽ(x, ω)}{|Ẽ(x, ω)|} \\ b_{s} · \frac{Ẽ(x, ω)}{|Ẽ(x, ω)|} \end{matrix} ) ,

relative to a right-handed orthonormal basis (a_{s}, b_{s}, s) of ℝ^{3} .

2.2.2 Simple Reflection and Refraction

For isotropic media, i.e. if (2.17) holds, the solution of (2.32) is obvious:

s · E_{s}(ω) = 0 , ±N_{s}(ω) = N (ω) =^{def} \sqrt{ϵ_{c}(ω)} , ℜ(N (ω)) ≥ 0 .

Let us now consider the following situation:^{26}

The region G_{±} =^{def} {x ∈ ℝ^{3} : ±x^{1} ≥ 0} is filled by some homogeneous isotropic linear optical medium with (generalized) permittivity ϵ^{↔}_{±}(ω) =

Draft, November 5, 2011

23See (Jones, 1941; Azzam and Bashara, 1987; Kliger et al., 1990). For actual measurement of polarization see (Gjurchinovski, 2002). .

24Here — contrary to the convention used, e.g., in (Reider, 1997, Sect. 1.5) — right circular means that the b_{s} component of E(x, t) lags the a_{s} component of E(x, t) by 90^{∘}. This way — as pointed out in (Jackson, 1975, Sect. 7.2) — right circular light corresponds to positive helicity, hence to right handed photons.

25Note that, for ω ≠ 0 : \frac{d}{dt} |ℜ (\frac{e^{−iω t}}{\sqrt{2}} ( \begin{matrix}1 \\ z \end{matrix} ))|^{2} = 0 ∀ t ⟺ z ∈ \{−i, +i\} .

26See also (Stratton, 1941, Ch. IX) for the special case that G_{−} is a perfect dielectric. For realizable refractive indices see (Skaar and Seip, 2006; Grigorenko, 2006). For generalization to curved boundaries see (Hentschel and Schomerus, 2002; Hentschel and Schomerus, 2006).

ϵ_{±}(ω) 1^{↔}, and conductivity σ^{↔}_{±}(ω) = σ_{±}(ω) 1^{↔}. Moreover, assume

(N_{+}(ω))^{2} ≠ (N_{−}(ω))^{2} ,  (2.38)

where

N_{±}(ω) =^{def} \sqrt{ϵ_{c}^{±}(ω)} , ℜ(N_{±}(ω)) ≥ 0 .

Then scattering theory suggests to look for modes of the following form:^{27}

• Inside G_{−} the electric field has the form E_{−}(x, t) = E_{s_{−}} e^{−i \frac{ω}{c}(c t − N_{−}(ω) s_{−}·x)} + E_{s′}_{−} e^{−i \frac{ω}{c}(c t − N_{−}(ω) s′_{−}·x)} .

• The electric field inside G_{+} has the form E_{+}(x, t) = E_{s_{+}} e^{−i \frac{ω}{c}(c t − N_{+}(ω) s_{+}·x)} .

• s_{−}, s′_{−}, s_{+}, E_{s_{−}}, E_{s′}_{−}, E_{s_{+}} ∈ ℂ^{3} \ {0} .

• (2.31) holds for s ∈ {s_{−}, s′_{−}, s_{+}}.

• s_{−} − P̂_{e_{1}} s_{−} is not isotropic^{28} and P̂_{e_{1}} s_{−} =^{def} (e_{1} · s_{−}) e_{1} ≠ 0 . For this Ansatz the boundary condition for the elctric field requires

e_{1} × (E_{s_{−}} e^{−i \frac{ω}{c} N_{−}(ω) s_{−}·x} + E_{s′}_{−} e^{−i \frac{ω}{c} N_{−}(ω) s′_{−}·x} − E_{s_{+}} e^{−i \frac{ω}{c} N_{+}(ω) s_{+}·x})_{|_{x^{1}=0}} = 0

for all x^{2}, x^{3} ∈ ℝ. This is equivalent to

e_{1} × (E_{s_{−}}(ω) + E_{s′}_{−}(ω) − E_{s_{+}}(ω)) = 0  (2.39)

and Snell’s law ^{29}

e_{1} × N_{−}(ω) s′_{−} = e_{1} × N_{−}(ω) s_{−} = e_{1} × N_{+}(ω) s_{+} .  (2.40)

Defining

c_{3} =^{def} \frac{s_{−} − P̂_{e_{1}} s_{−}}{\sqrt{(s_{−} − P̂_{e_{1}} s_{−}) · (s_{−} − P̂_{e_{1}} s_{−})}} , c_{2} =^{def} c_{3} × e_{1}

where

Draft, November 5, 2011 27For nonisotropic media the Ansatz has to be refined.

28A vector z ∈ ℂ^{3} is called isotropic iff z · z = 0 .

29Usually Snell’s law is given for the special case N_{±}(ω) = n_{±}(ω) ∈ ℝ , s′_{−}, s_{±} ∈ ℝ^{3} . Then:

n_{−}(ω) sin θ′_{−} = n_{−}(ω) sin θ_{−} = n_{+}(ω) sin θ_{+} , s_{±} = cos θ_{±} e_{1} + sin θ_{±} e_{3} ; s′_{−} = − cos θ′_{−} e_{1} + sin θ′_{−} e_{3} , e_{3} =^{def} c_{3} ∈ ℝ^{3} .

we get a basis {e_{1}, c_{2}, c_{3}} of ℂ^{3} that is orthonormal in the sense that

c_{j} · c_{k} = δ_{jk} , c_{1} =^{def} e_{1} .

For c_{3} introduced this way Snell’s law implies

s′_{−}, s_{−}, s_{+} ∈ Span {e_{1}, c_{3}} ⊂ ℂ^{3} .  (2.41)

Similarly to (2.39), assuming vanishing permeability, we get

e_{1} × (B_{s_{−}}(ω) + B_{s′}_{−}(ω) − B_{s_{+}}(ω)) = 0 .  (2.42)

By (2.33) the latter is equivalent to

0 = e_{1} × (N_{−}(ω) (s_{−} × E_{s_{−}}(ω) + s′_{−} × E_{s′}_{−}(ω)) − N_{+}(ω) s_{+} × E_{s_{+}}(ω)) . (2.43)

Thanks to (2.41) and (2.31) the splitting^{30}

E_{s}(ω) = E_{s}^{⊥}(ω) c_{2} + E_{s}^{∥}(ω) (c_{2} × s) for s = s_{±}, s′_{−} .  (2.44)

is possible iff (2.32) holds.

Proof of (2.44): Obviously, due to our assumptions concerning the material constants, (2.32) is equivalent to

s · E_{s}(ω) = 0 .  (2.45)

According to (2.41) and (2.31) {s , c_{2} , c_{2} × s} is a Basis^{31} of ℂ^{3} . Therefore, a splitting of the form

E_{s}(ω) = E_{s}^{⊥}(ω) c_{2} + E_{s}^{∥}(ω) (c_{2} × s) + E_{s}^{add}(ω) s

is possible. But, since s · c_{2} = 0 = s · (c_{2} × s) , E_{s}^{add}(ω) vanishes iff (2.45) holds.

With this splitting, because (2.36) and (2.41) imply

e_{1} × (s × (c_{2} × s)) = e_{1} × c_{2} , e_{1} × (c_{2} × s) = (e_{1} · s) c_{2} ,

(2.43) becomes equivalent to

N_{−}(ω) (E_{s_{−}}^{∥}(ω) + E_{s′_{−}}^{∥}(ω)) = N_{+}(ω) E_{s_{+}}^{∥}(ω) ,

N_{−}(ω) ((e_{1} · s_{−}) E_{s_{−}}^{⊥}(ω) + (e_{1} · s′_{−})E_{s′_{−}}^{⊥}(ω)) = N_{+}(ω) (e_{1} · s_{+}) E_{s_{+}}^{⊥}(ω) .

(2.46)

Similarly we see that (2.39) is equivalent to

E_{s_{−}}^{⊥}(ω) + E_{s′_{−}}^{⊥}(ω) = E_{s_{+}}^{⊥}(ω) ,
(e_{1} · s_{−}) E_{s_{−}}^{∥}(ω) + (e_{1} · s′_{−})E_{s′_{−}}^{∥}(ω) = (e_{1} · s_{+}) E_{s_{+}}^{∥}(ω) .

(2.47)

Draft, November 5, 2011

30The notions ^{⊥} and ^{∥} refer to the e_{1}-c_{3}-plane in ℂ^{3} , z_{1} ⊥ z_{2} meaning z_{1} · z_{2} = 0 .

31Note that (c_{2} × s) · (c_{2} × s) = (c_{2} · c_{2}) (s · s) − (c_{2} · s)^{2} = 1 .

2.2. MONOCHROMATIC WAVES OF EXPONENTIAL TYPE 47

Exercise 3 Using (2.31) and (2.41), show the following:

a) (2.47), (2.46), (2.38), and Snell’s law imply

e_{1} · s′_{−} = − e_{1} · s_{−} ,  (2.48)

if solutions with E_{s_{+}}^{∥}(ω) ≠ 0 as well as those with E_{s_{+}}^{⊥}(ω) ≠ 0 exist. b)

s′_{−} = (c_{3} · s_{−}) c_{3} − (e_{1} · s_{−}) e_{1} .  (2.49)

c)

s_{+} = \frac{N_{−}(ω)}{N_{+}(ω)} (c_{3} · s_{−}) c_{3} ± \sqrt{1 − (\frac{N_{−}(ω)}{N_{+}(ω)} (c_{3} · s_{−}))^{2}} e_{1} .  (2.50)

By (2.48), both (2.46) and (2.47) together are equivalent to Fresnel’s formulas^{32}

E_{s_{+}}^{∥}(ω) = \frac{2 N_{−}(ω) (e_{1} · s_{−})}{N_{−}(ω) (e_{1} · s_{+}) + N_{+}(ω) (e_{1} · s_{−})} E_{s_{−}}^{∥}(ω) , (2.51)

E_{s_{+}}^{⊥}(ω) = \frac{2 N_{−}(ω) (e_{1} · s_{−})}{N_{−}(ω) (e_{1} · s_{−}) + N_{+}(ω) (e_{1} · s_{+})} E_{s_{−}}^{⊥}(ω) , (2.52)

E_{s′_{−}}^{∥}(ω) = \frac{N_{+}(ω) (e_{1} · s_{−}) − N_{−}(ω) (e_{1} · s_{+})}{N_{+}(ω) (e_{1} · s_{−}) + N_{−}(ω) (e_{1} · s_{+})} E_{s_{−}}^{∥}(ω) , (2.53)

E_{s′_{−}}^{⊥}(ω) = \frac{N_{−}(ω) (e_{1} · s_{−}) − N_{+}(ω) (e_{1} · s_{+})}{N_{−}(ω) (e_{1} · s_{−}) + N_{+}(ω) (e_{1} · s_{+})} E_{s_{−}}^{⊥}(ω) . (2.54)

Exercise 4 Using Snell law, show the following:

a) (2.43) implies e_{1} · (ϵ_{c}^{−}(ω) E_{s_{−}}(ω) + ϵ_{c}^{−}(ω) E_{s′}_{−}(ω) − ϵ_{c}^{+}(ω) E_{s_{+}}(ω)) = 0 .

b) grad (c_{2} · E(x, t)) is continuous.

c) grad (c_{3} · E(x, t)) is not continuous, in general.

Draft, November 5, 2011

32We assume that the denominators on the r.h.s. of (2.51)–(2.54) do not vanish. For the evaluation of these formulas in the case of nonabsorbing media and s ∈ ℝ^{3} see, e.g., (Monzón and Sánchez-Soto, 2001).

Let us consider the case^{33}

ℜ (e_{1} · s_{±}) > 0 .  (2.55)

Then E_{s_{−}} e^{−i \frac{ω}{c}(c t − N_{−}(ω) s_{−}·x)} may be interpreted as incoming wave,^{34} reflected into G_{−} as E_{s′}_{−} e^{−i \frac{ω}{c}(c t − N_{−}(ω) s′_{−}·x)} and transmitted into G_{+} as E_{s_{+}} e^{−i \frac{ω}{c}(c t − N_{+}(ω) s_{+}·x)} . As a direct consequence of (2.37) we have

e_{1} · ⟨S_{s_{−}}(x, t)⟩_{|_{x^{1}=0}} = e_{1} · ⟨S_{s_{−}}^{⊥}(x, t)⟩_{|_{x^{1}=0}} + e_{1} · ⟨S_{s_{−}}^{∥}(x, t)⟩_{|_{x^{1}=0}} ,

where^{35}

S_{s_{−}}(x, t) =^{def} Poynting vector of E_{s_{±}} e^{−i \frac{ω}{c}(c t − N_{−}(ω) s_{−}·x)} ,
S_{s_{−}}^{⊥}(x, t) =^{def} Poyn�ting vector of E_{s_{−}}^{⊥}c_{2} e^{−i \frac{ω}{c}(c t − N_{−}(ω) s_{−}·x)} ,
S_{s_{−}}^{∥}(x, t) =^{def} Poynting vector of E_{s_{−}}^{∥}(c_{2} × s_{−}) e^{−i \frac{ω}{c}(c t − N_{−}(ω) s_{−}·x)} .

Since, similar relations hold for s′_{−} and s_{+} instead of s_{−}, we may analyze the energy flow for the ^{⊥}- and ^{∥}-modes separately. Let us define

\begin{matrix}R_{s_{−}}^{⊥}(ω) & =^{def} & − \frac{e_{1} · ⟨S_{s′_{−}}^{⊥}(x, t)⟩_{|_{x^{1}=0}}}{e_{1} · ⟨S_{s_{−}}^{⊥}(x, t)⟩_{|_{x^{1}=0}}} , \\ T_{s_{−}}^{⊥}(ω) & =^{def} & + \frac{e_{1} · ⟨S_{s_{+}}^{⊥}(x, t)⟩_{|_{x^{1}=0}}}{e_{1} · ⟨S_{s_{−}}^{⊥}(x, t)⟩_{|_{x^{1}=0}}} \end{matrix} \} for e_{1} · ⟨S_{s_{−}}^{⊥}(x, t)⟩_{|_{x^{1}=0}} ≠ 0

and similarly for ^{∥} instead of ^{⊥}. Then by (2.37) and Snell’s law (2.40) we have

R_{s_{−}}^{⊥}(ω)  =  − \frac{e_{1} · ℜ(N_{−}(ω)s′_{−})}{e_{1} · ℜ(N_{−}(ω)s_{−})} |E_{s′_{−}}^{⊥}(ω)|^{2} / |E_{s_{−}}^{⊥}(ω)|^{2}
=_{(2.48),(2.54)} |\frac{N_{−}(ω) (e_{1} · s_{−}) − N_{+}(ω) (e_{1} · s_{+})}{N_{−}(ω) (e_{1} · s_{−}) + N_{+}(ω) (e_{1} · s_{+})}|^{2}  (2.56)

and

T_{s_{−}}^{⊥}(ω) = + \frac{e_{1} · ℜ(N_{+}(ω)s_{+})}{e_{1} · ℜ(N_{−}(ω)s_{−})} |E_{s_{+}}^{⊥}(ω)|^{2} / |E_{s_{−}}^{⊥}(ω)|^{2}

=_{(2.53)} \frac{ℜ(N_{+}(ω) (e_{1} · s_{+}))}{ℜ(N_{−}(ω) (e_{1} · s_{−}))} |\frac{2 N_{−}(ω) (e_{1} · s_{−})}{N_{−}(ω) (e_{1} · s_{−}) + N_{+}(ω) (e_{1} · s_{+})}|^{2} .(2.57)

35Recall (2.44).

Draft, November 5, 2011 33For ℜ (e_{1} · s_{+}) < 0 we have incoming waves from both sides of the boundary.

34Actually we should form wave packets by integrating with suitable weights over s_{−} and ω.

Therefore^{36} R_{s_{−}}^{⊥}(ω) + T_{s_{−}}^{⊥}(ω) = 1

⟺ \frac{ℜ(N_{+}(ω) (e_{1} · s_{+}))}{ℜ(N_{−}(ω) (e_{1} · s_{−}))} = ℜ (\frac{N_{+}(ω) (e_{1} · s_{+})}{N_{−}(ω) (e_{1} · s_{−})})

⟺ \{N_{−}(ω) (e_{1} · s_{−}) , N_{+}(ω) (e_{1} · s_{+}) \overline{N_{−}(ω) (e_{1} · s_{−})}\} ∩ ℝ ≠ ∅ , i.e.:

In general, the usual interpretation of R_{s_{−}}^{⊥}(ω) as reflectance and T_{s_{−}}^{⊥}(ω)

as transmittance (of the ^{⊥}-mode) is not strictly justified. For the ^{∥}-mode the situation is more complicated, since, e.g.,^{37}

ℜ (N_{−}(ω) (\overline{E_{s_{−}}(ω)} · s_{−}) E_{s_{−}}(ω)) = i c_{2} · (\overline{s_{−}} × s_{−}) ℑ (N_{−}(ω) \overline{E_{s_{−}}^{∥}(ω)} E_{s_{−}}(ω)) . The latter, however, implies e_{1}·ℜ (N_{−}(ω) (\overline{E_{s_{−}}(ω)} · s_{−}) E_{s_{−}}(ω)) = i c_{2}·(\overline{s_{−}} × s_{−}) |E_{s_{−}}^{∥}(ω)|^{2} ℑ(N_{−}(ω) (c_{3} · s_{−})) .

(2.58)

and, therefore, similarly to (2.56) and (2.57) we get ℑ(N_{−}(ω) (c_{3} · s_{−})) (\overline{s_{−}} × s_{−}) = ℑ(N_{+}(ω) (c_{3} · s_{+})) (\overline{s_{+}} × s_{+}) = 0

⟹ R_{s_{−}}^{∥}(ω) = |\frac{N_{+}(ω) (e_{1} · s_{−}) − N_{−}(ω) (e_{1} · s_{+})}{N_{+}(ω) (e_{1} · s_{−}) + N_{−}(ω) (e_{1} · s_{+})}|^{2}  (2.59)

and ℑ(N_{−}(ω) (c_{3} · s_{−})) (\overline{s_{−}} × s_{−}) = ℑ(N_{+}(ω) (c_{3} · s_{+})) (\overline{s_{+}} × s_{+}) = 0

⟹ T_{s_{−}}^{∥}(ω) = \frac{ℜ(N_{+}(ω) (e_{1} · s_{+}))}{ℜ(N_{−}(ω) (e_{1} · s_{−}))} |\frac{2 N_{−}(ω) (e_{1} · s_{−})}{N_{−}(ω) (e_{1} · s_{+}) + N_{+}(ω) (e_{1} · s_{−})}|^{2} . (2.60)

36Note that

\frac{ℜ(z_{+})}{ℜ(z_{−})} = ℜ (\frac{z_{+}}{z_{−}} ) ⟺ \overline{z_{+}} z_{−} (\overline{z_{−}} − z_{−}) = z_{+} \overline{z_{−}} (\overline{z_{−}} − z_{−}) .

1991).

37Note that, e.g.,

Draft, November 5, 2011 For a discussion of the reflection and transmission in the case of normal incidence see (Lodenquai,

i c_{2} · (\overline{s_{−}} × s_{−}) = 2 α β for s_{−} = α e_{1} + i β c_{3} α, β ∈ ℝ .

Warning: Note that (2.56), (2.57), (2.59), and (2.60) hold only for ℜ(N_{−}(ω) (e_{1} · s_{−})) ≠ 0 .

A direct consequence of (2.57) is that ℜ(N_{+}(ω) (e_{1} · s_{+})) = 0 implies T_{s_{−}}^{⊥}(ω) = 0 , i.e. total reflection of the ^{⊥}-mode.^{38} If, in addition,

ℑ(N_{−}(ω) (c_{3} · s_{−})) (\overline{s_{−}} × s_{−}) = ℑ(N_{+}(ω) (c_{3} · s_{+})) (\overline{s_{+}} × s_{+}) = 0

holds then, by (2.60), also T_{s_{−}}^{∥}(ω) = 0 .

Strictly speaking, however, total reflection appears only since G_{+} has infinite diameter. If G_{+} is replaced by a layer of finite depth, then optical tunneling is possible, i.e. partial transmission of radiation under the conditions of otherwise total reflection. Then, however, there is a superposition of two evanescent modes inside G_{+} with non-vanishing e_{1}-component of the total Poynting vector.^{39}

Remarks:

1. Multiple total reflection may be used for building useful optical devices (e.g. wave guides).

2. In case of total reflection the evanescent wave inside G_{+} has higher spatial frequency (ˆ=higher resolution) in the direction of propagation than usual and may have considerable amplitudes near the boundary of the media. These modes are exploited by near-field scanning optical microscopy^{40} (NSOM), converting the evanescent modes into propagating ones.

3. Usually scattered light contains evanescent near-field components (Wolf and Nieto-Vesperinas, 1985).

4. Optical tunneling may be used for coupling wave guides.

Exercise 5 For real N_{±}(ω) = N_{±}(ω) = n_{±}(ω) and s_{−} ∈ ℝ^{3} show that R_{s_{−}}^{∥}(ω) vanishes iff

\frac{(e_{1} · s_{−})^{2}}{1 − (e_{1} · s_{−})^{2}} = (\frac{n_{−}(ω}{n_{+}(ω)})^{2} ,

Draft, November 5, 2011

38Usually the case imaginary.

N_{±}(ω) = n_{±}(ω) , n_{−}(ω) sin θ_{−} > n_{+}(ω) , s_{−} ∈ ℝ^{3} is considered, corresponding to total reflection since the e_{1}-component of (2.50) becomes purely

39And the results for incident monochromatic plane waves should not be applied too naively to almost monochromatic incident rays!

40See, e.g., (Ohtsu and Hori, 1999) and (Paesler and Moyer, 1996).

i.e. iff ∠e_{1}, s_{−} coincides with the so-called Brewster angle arctan \frac{n_{−}(ω)}{n_{+}(ω)} . More-

over, show that in this case

R_{s_{−}}^{∥}(ω) = 0 ⟺ s′_{−} · s_{+} = 0 .

2.2.3 Birefringent Media (Crystal Optics)

Generally, (2.32) is equivalent to

M^{↔}_{s}(ω) E_{s}(ω) = 0 ,  (2.61)

where

M^{↔}_{s}(ω) =^{def} (N_{s}(ω))^{2} (1^{↔} − ( \begin{matrix}s^{1}s^{1} & s^{1}s^{2} & s^{1}s^{3} \\ s^{2}s^{1} & s^{2}s^{2} & s^{2}s^{3} \\ s^{3}s^{1} & s^{3}s^{2} & s^{3}s^{3} \end{matrix} )) − ϵ^{↔}_{c}(ω) .

(2.61) can be fulfilled iff

det(M^{↔}_{s}(ω)) = 0 .  (2.62)

Therefore, we have to choose the complex refractive index N_{s}(ω) such that (2.62) is fulfilled and determine the (complex) direction of E_{s}(ω) from (2.32) for such N_{s}(ω) .

Reasonable physical assumptions imply that^{41}

a · ϵ^{↔}_{c}(ω) b = b · ϵ^{↔}_{c}(ω) a ∀ a, b ∈ ℂ^{3} .  (2.63)

Therefore, if N_{s}(ω) resp. N′_{s}(ω) are solutions of (2.62) and E_{s}(ω) resp. E′_{s}(ω) corresponding solutions of (2.32), we have

(N_{s}(ω))^{2} ≠ (N′_{s}(ω))^{2} ⟹ ( ϵ^{↔}_{c}(ω) E_{s}(ω)) · ϵ^{↔}_{c}(ω) E′_{s}(ω) = 0 .  (2.64)

Proof of (2.64):

(N′_{s}(ω))^{2} ( ϵ^{↔}_{c}(ω) E_{s}(ω)) · ϵ^{↔}_{c}(ω) E′_{s}(ω)

=_{(2.32)} (N_{s}(ω))^{2} (N′_{s}(ω))^{4} P^{↔}_{⊥s} E_{s}(ω) · P^{↔}_{⊥s} E′_{s}(ω)
= (N_{s}(ω))^{2} (N′_{s}(ω))^{4} E_{s}(ω) · P^{↔}_{⊥s} E′_{s}(ω)
=_{(2.32)} (N_{s}(ω) N′_{s}(ω))^{2} E_{s}(ω) · ϵ^{↔}_{c}(ω) E′_{s}(ω)
=_{(2.63)} (N_{s}(ω) N′_{s}(ω))^{2} E′_{s}(ω) · ϵ^{↔}_{c}(ω) E_{s}(ω)
=_{(2.32)} (N_{s}(ω))^{2} ( ϵ^{↔}_{c}(ω) E′_{s}(ω)) · ϵ^{↔}_{c}(ω) E_{s}(ω) .

Draft, November 5, 2011

41See (Römer, 1994, Ends of Sections 2.4 and 2.7).

Because of

a · b = b · a ∀ a, b ∈ ℂ^{3}

this implies

((N_{s}(ω))^{2} − (N′_{s}(ω))^{2}) ( ϵ^{↔}_{c}(ω) E_{s}(ω)) · ϵ^{↔}_{c}(ω) E′_{s}(ω) = 0

and hence (2.64).

Note that (2.64), (2.32), and (2.31) imply

(N_{s}(ω))^{2} ≠ (N′_{s}(ω))^{2} ⟹ E_{s}(ω) · E′_{s}(ω) = (s · E_{s}(ω)) (s · E′_{s}(ω)) . (2.65)

s is said to correspond to an optical axis for given ω, if there is a unique complex refractive index N_{s}(ω) = N_{s}^{ord}(ω) for which there are two independent solutions E_{s}(ω) of (2.32).

Lemma 2.2.1 s corresponds to an optical axis for given ω iff

s · e = 0 ⟹ e · ( ϵ^{↔}_{c}(ω))^{−1}e = (N_{s}^{ord}(ω))^{−2}  (2.66)

holds for all (complex) directions e .

Proof: Let s correspond to an optical axis and let E_{s}(ω), E′_{s}(ω) be independent solutions of (2.32) for N_{s}(ω) = N_{s}^{ord}(ω) . Then, because of

E_{s}(ω) = (N_{s}^{ord}(ω))^{2} ( ϵ^{↔}_{c}(ω))^{−1}P^{↔}_{⊥s} E_{s}(ω) ,  (2.67)

(and the corresponding equation for E′_{s}(ω)) also P^{↔}_{⊥s} E_{s}(ω) and P^{↔}_{⊥s} E′_{s}(ω) have to be independent. Moreover, (2.67) implies

(P^{↔}_{⊥s} E_{s}(ω)) = (N_{s}^{ord}(ω))^{2} I^{↔}_{s} (P^{↔}_{⊥s} E_{s}(ω))

for

I^{↔}_{s} =^{def} P^{↔}_{⊥s} ( ϵ^{↔}_{c}(ω))^{−1}P^{↔}_{⊥s} .

Therefore,

I^{↔}_{s} = \frac{1}{(N_{s}^{ord}(ω))^{2}} P^{↔}_{⊥s} .

Since

s · e = 0 ⟺ e = P^{↔}_{⊥s} e

and

a · (P^{↔}_{⊥s} b) = (P^{↔}_{⊥s} a) · b ∀ a, b ∈ ℂ^{3} ,

we conclude^{42} that (2.66) holds. Conversely, if (2.66) holds, then

E_{s}(ω) = ( ϵ^{↔}_{c}(ω))^{−1} z

is a solution of (2.32) for every z ∈ P^{↔}_{⊥s} ℂ^{3} . The latter shows that s corresponds to an optical axis.

Let us assume that ϵ^{↔}_{c}(ω) may be diagonalized:

ϵ^{↔}_{c}(ω) = ( \begin{matrix}ϵ_{c}^{1}(ω) & 0 & 0 \\ 0 & ϵ_{c}^{2}(ω) & 0 \\ 0 & 0 & ϵ_{c}^{3}(ω) \end{matrix} ) w.r.t. \{z_{1}(ω), z_{2}(ω), z_{3}(ω)\} ⊂ ℂ^{3} ,

z_{j}(ω) · z_{k}(ω) = δ_{jk} ∀ j, k ∈ {1, 2, 3} .

(2.68)

Remark: (2.68) is guaranteed by (2.63), for suitable z_{j}(ω) ∈ ℂ^{3}, if ϵ^{↔}_{c}(ω) does not have any isotropic eigenvector; see Lemma 7.4.13 of (Lücke, eine). Note, however, that there are media (monoclinic and triclinic crystals) for which the eigenvectors of ϵ^{↔}_{c}(ω) depend on ω.

If (2.68) holds with ϵ_{c}^{2}(ω) = ϵ_{c}^{3}(ω) , then ±z_{1}(ω) corresponds to an optical axis and

N_{z_{1}(ω)}^{ord}(ω) = \sqrt{ϵ_{c}^{2}(ω)} .  (2.69)

In this case, whenever s ≁ z_{1}(ω) ,

E_{s}(ω) = E_{s}^{ord}(ω) ∼ s × z_{1}(ω) ,  (2.70)

is a solution of (2.32) for

N_{s}(ω) = N_{z_{1}(ω)}^{ord}(ω) .

Note that, by (2.37),

s ∈ ℝ ⟹ ⟨S_{s}^{ord}(x, t)⟩ ∼ s .  (2.71)

To determine the second solution of (2.62) we may assume, without loss of generality, that

s = s^{1}(ω) z_{1}(ω) + s^{3}(ω) z_{3}(ω) .

Then, w.r.t. {z_{1}(ω), z_{2}(ω), z_{3}(ω)}, by (2.31), we have

M^{↔}(k_{0}) = (N_{s}(ω))^{2} ( \begin{matrix}+s^{3}(ω)s^{3}(ω) & 0 & −s^{1}(ω)s^{3}(ω) \\ 0 & 1 & 0 \\ − s^{3}(ω)s^{1}(ω) & 0 & +s^{1}(ω)s^{1}(ω) \end{matrix} ) − ϵ^{↔}_{c}(ω)

Draft, November 5, 2011

42Recall that P^{↔}_{⊥s} P^{↔}_{⊥s} = P^{↔}_{⊥s} .

54 CHAPTER 2. LINEAR OPTICAL MEDIA and hence (2.62) holds iff either N_{s}(ω) = N_{z_{1}(ω)}^{ord}(ω) or

0 = ((N_{s}(ω))^{2}s^{3}(ω)s^{3}(ω) − ϵ_{c}^{1}(ω))((N_{s}(ω))^{2}s^{1}(ω)s^{1}(ω) − ϵ_{c}^{3}(ω))
−(N_{s}(ω))^{4}s^{1}(ω)s^{1}(ω)s^{3}(ω)s^{3}(ω)
= ϵ_{c}^{1}(ω) ϵ_{c}^{3}(ω) (1 − \frac{(N_{s}(ω) s^{1}(ω))^{2}}{ϵ_{c}^{3}(ω)} − \frac{(N_{s}(ω) s^{3}(ω))^{2}}{ϵ_{c}^{1}(ω)} ) .

Therefore another solution E′_{s}(ω) of (2.32) exists for N_{s}(ω) = N_{s}^{exo}(ω) , the latter being implicitly defined by

(\frac{N_{s}^{exo}(ω) s^{1}(ω)}{N_{s}^{ord}(ω)})^{2} + (\frac{N_{s}^{exo}(ω) s^{2}(ω)}{\sqrt{ϵ_{c}^{1}(ω)}})^{2} + (\frac{N_{s}^{exo}(ω) s^{3}(ω)}{\sqrt{ϵ_{c}^{1}(ω)}})^{2} = 1 . (2.72)

In the nonisotropic case, (2.70) and (2.64) imply E_{s}^{exo}(ω) ∼ ( ϵ^{↔}_{c}(ω))^{−1}(s × (s × z_{1}(ω))) .

(2.73)

In agreement with (2.65) we have E_{s}^{exo}(ω) · E_{s}^{ord}(ω) = 0 , in the present case.^{43} Exercise 6 Show that

E_{s}^{ord}(ω) · ⟨S_{s}^{ord}(x, t)⟩ = E_{s}^{exo}(ω) · ⟨S_{s}^{exo}(x, t)⟩ = 0

and

⟨S_{s}^{exo}(x, t)⟩ ∈ Span {s, E_{s}^{exo}(ω)} ⊂ Span{s, z_{1}(ω)}

if z_{1}(ω) ≁ s ∈ ℝ^{3} and ϵ^{↔}_{c} = ( ϵ^{↔}_{c})^{†}.

Remarks:

1. For real directions e and real ϵ^{↔}_{c}(ω) we have

( ϵ^{↔}_{c}(ω))^{−1}e = \frac{1}{2} (∇_{x} (x · ( ϵ^{↔}_{c}(ω))^{−1}x))_{|_{x=e}}

and, therefore, ( ϵ^{↔}_{c}(ω))^{−1}e ∼ n(e) , where n(e) denotes the normal of the index ellipsoid

{x ∈ ℝ : (x · ( ϵ^{↔}_{c}(ω))^{−1}x) = 1}

at x ∼ e .

Draft, November 5, 2011

43Note that, in the present case, ( ϵ^{↔}_{c}(ω))^{−1} leaves the linear span of s and z_{1}(ω) invariant.

2. Since, therefore, s · E_{s}^{exo}(ω) ≠^{i.g.} 0 Exercise 6 and (2.71) show that

⟨S_{s}^{exo}(x, t)⟩ ≁^{i.g.} s ∼ ⟨S_{s}^{ord}(x, t)⟩;

i.e. the medium is birefringent.

Lemma 2.2.2 If (2.68) holds with ϵ_{c}^{1}(ω) ≠ ϵ_{c}^{2}(ω) and if s is a (complex) direction

corresponding to an optical axis for given ω then ∏_{j=1}^{3} s · z_{j}(ω) = 0 .

Outline of proof: Let s correspond to an optical axis. Then, by Lemma 2.2.1,

s · x = 0 ≠ x · x ⟹ ∑_{j=1}^{3}(x^{j})^{2}/ϵ_{c}^{j} = C x · x ∀x ∈ ℂ^{3}

holds for some C = C(ω) ∈ ℂ, where x^{j} =^{def} x · z_{j}(ω) for j = 1, 2, 3 . Now assume

∏_{j=1}^{3} s^{j} ≠ 0 . Then

s · x = 0 ⟺ (x^{1})^{2} = (\frac{s^{2}x^{2} + s^{3}x^{3}}{s^{1}} )^{2}

and, therefore, we have

\begin{matrix}(\frac{s^{2}x^{2} + s^{3}x^{3}}{s^{1}} )^{2} + (x^{2})^{2} + (x^{3})^{2} ≠ 0 \\ ⟹ \begin{matrix}\frac{1}{ϵ_{c}^{1}} (\frac{s^{2}x^{2} + s^{3}x^{3}}{s^{1}} )^{2} + \frac{1}{ϵ_{c}^{2}} (x^{2})^{2} + \frac{1}{ϵ_{c}^{3}} (x^{3})^{2} \\ = C ((\frac{s^{2}x^{2} + s^{3}x^{3}}{s^{1}} )^{2} + (x^{2})^{2} + (x^{3})^{2}) \end{matrix} \end{matrix}

for all x^{2}, x^{3} ∈ ℂ. Comparing the x^{2}x^{3}-terms on both sides of the latter equation shows that C = 1/ϵ_{c}^{1} . Similarly we conclude that C = 1/ϵ_{c}^{2} . Therefore, ∏_{j=1}^{3} s^{j} ≠ 0 cannot hold if ϵ_{c}^{1} ≠ ϵ_{c}^{2} .

Corollary 2.2.3 If (2.68) holds with ϵ_{c}^{1}(ω) ≠ ϵ_{c}^{2}(ω) and if s is a (complex) direction with s · z_{2}(ω) = 0, for given ω, then s corresponds to an optical axis if and only if ϵ_{c}^{1}(ω) ≠ ϵ_{c}^{3}(ω) and

s = ±\sqrt{1 − (s^{3})^{2}} z_{1}(ω) + s^{3} z_{3}(ω) , s^{3} = ±\sqrt{\frac{ϵ_{c}^{3}(ω) − ϵ_{c}^{2}(ω)}{ϵ_{c}^{3}(ω) − ϵ_{c}^{1}(ω)} \frac{ϵ_{c}^{1}(ω)}{ϵ_{c}^{2}(ω)}} .  (2.74)

Outline of proof: Let the direction s ⊥ z_{2}(ω) correspond to an optical axis. Then there are complex numbers C and s^{3} with

s = ±\sqrt{1 − (s^{1})^{2}} z_{1}(ω) + s^{3} z_{3}(ω)

and

s · s′ = 0 ⟹ s′ · ( ϵ^{↔}_{c}(ω))^{−1}s′ = C

for all directions s′ (by Lemma 2.2.1). Especially for

e =^{def} z_{2}(ω) , e′ =^{def} s^{3} z_{1}(ω) ∓ \sqrt{1 − (s^{1})^{2}} z_{3}(ω) ,

therefore, we have

1/ϵ_{c}^{2} = e · ( ϵ^{↔}_{c}(ω))^{−1}e
= e′ · ( ϵ^{↔}_{c}(ω))^{−1}e′
= \frac{(s^{3})^{2}}{ϵ_{c}^{1}} + \frac{1 − (s^{3})^{2}}{ϵ_{c}^{3}} .

This implies ϵ_{c}^{1} ≠ ϵ_{c}^{3} (since ϵ_{c}^{1} ≠ ϵ_{c}^{2} , by assumption) and (2.74). Conversely, since e · ( ϵ^{↔}_{c}(ω))^{−1}e′ = 0 , it is obvious (by Lemma 2.2.1) that s corresponds to an optical axis under these conditions.

Remarks:

1. For the case

ϵ_{c}^{1}(ω) ≠ ϵ_{c}^{2}(ω) = ϵ_{c}^{3}(ω) ,  (2.75)

Lemma 2.2.2 and Corollary 2.2.3 show that only ±z_{1}(ω) corresponds to an optical axis. Therefore media for which (2.68) and (2.75) hold are called uniaxial.

2. Media with

ϵ_{c}^{1}(ω) ≠ ϵ_{c}^{2}(ω) ≠ ϵ_{c}^{3}(ω) ≠ ϵ_{c}^{1}(ω)

are called biaxial. Actually, by Corollary 2.2.3, there are 6 optical axes,^{44} but only those corresponding to ‘sufficiently real’ directions are relevant for physical applications.

3. For biaxial media with

ϵ_{c}^{3}(ω) > ϵ_{c}^{2}(ω) > ϵ_{c}^{1}(ω) > 0  (2.76)

the real directions e corresponding to optical axes are given by Corollary 2.2.3 (for the choice of indices corresponding to (2.76)).

4. Crystals of the hexagonal, tetragonal and trigonal system are uniaxial and the optical axis coincides with the crystal axis of six-, four-, or three-fold symmetry. Crystals of the orthorhombic, monoclinic and triclinic system are optically biaxial.

Draft, November 5, 2011

44Note that the choice of indices in (2.68) is arbitrary

2.3 Monochromatic Light Rays^{45}

In this section we consider only isotropic media and — in order to be able to apply standard Fourier analysis (for tempered distributions) — only the case^{46}

N (ω) = n(ω) > 0 .

Then, by monochromatic light ray we mean a solution of Maxwell’s equations fulfilling the following two conditions for suitable ω and s ∈ ℝ^{3} with |s| = 1 :

1. E(x, t) = E(x) e^{−i ω t} (i.e. : Ẽ(x, ω′) = E(x) δ(ω′ − ω)).

2. |Ě(k)| ≪ 1 unless |k|^{2} − (s · k)^{2} ≪ |k|^{2} ,

where

Ě(k) =^{def} (2π)^{−3/2} ∫ E(x) e^{ix·k} dV_{x} .  (2.77)

Let us assume s = e_{3} and define

F(k^{1}, k^{2}; x^{3}) =^{def} e^{−i \frac{ω}{c} n(ω)x^{3}} \frac{1}{2π} ∫ E(x) e^{−i(x^{1}k^{1}+x^{2}k^{2})} dx^{1}dx^{2} .

Then the Helmholtz equation (2.29) gives^{47}

(− (k^{1})^{2} − (k^{2})^{2} + (\frac{∂}{∂x^{3}} )^{2} + (\frac{ω}{c} n(ω))^{2}) (e^{+i \frac{ω}{c} n(ω)x^{3}} F(k^{1}, k^{2}; x^{3})) = 0 .

Since^{48}

(\frac{∂}{∂x^{3}} )^{2} (e^{+i \frac{ω}{c} n(ω)x^{3}} F(k^{1}, k^{2}; x^{3}))

= e^{+i \frac{ω}{c} n(ω)x^{3}} (− (\frac{ω}{c} n(ω))^{2} + 2 i \frac{ω}{c} n(ω)\frac{∂}{∂x^{3}} + (\frac{∂}{∂x^{3}} )^{2}) F(k^{1}, k^{2}; x^{3})

≈ e^{+i \frac{ω}{c} n(ω)x^{3}} (− (\frac{ω}{c} n(ω))^{2} + 2 i \frac{ω}{c} n(ω)\frac{∂}{∂x^{3}} ) F(k^{1}, k^{2}; x^{3}) ,

we conclude:

((k^{1})^{2} + (k^{2})^{2} − 2 i \frac{ω}{c} n(ω)\frac{∂}{∂x^{3}} ) F(k^{1}, k^{2}; x^{3}) ≈ 0 .  (2.78)

Draft, November 5, 2011

45See also (Gomez-Reino et al., 2002).

46Recall (2.30).

47Recall that, in the isotropic case, (2.29) implies div Ẽ(x, ω) = 0 .

48Thanks to the second defining condition for light rays, we may neglect (\frac{∂}{∂x^{3}} )^{2} F(k^{1}, k^{2}; x^{3}) .

This justifies to approximate F(k^{1}, k^{2}; x^{3}) by the unique solution^{49}

F_{≈}(k^{1}, k^{2}; x^{3}) = e^{−i \frac{(k^{1})^{2}+(k^{2})^{2}}{2 \frac{ω}{c} n(ω)} x^{3}} F(k^{1}, k^{2}; 0)  (2.79)

of the initial-value problem

\begin{aligned}\frac{∂}{∂x^{3}} F_{≈}(k^{1}, k^{2}; x^{3}) & = & \frac{(k^{1})^{2} + (k^{2})^{2}}{2 i \frac{ω}{c}n(ω)} F_{≈}(k^{1}, k^{2}; x^{3}) , \\ F_{≈}(k^{1}, k^{2}; 0) & = & F(k^{1}, k^{2}; 0) = \frac{1}{2π} ∫ E(x^{1}, x^{2}, 0) e^{−i(x^{1}k^{1}+x^{2}k^{2})} dx^{1}dx^{2} . \end{aligned}

Because of

E(x) = \frac{1}{2π} ∫ e^{+i \frac{ω}{c} n(ω)x^{3}} F(k^{1}, k^{2}; x^{3}) e^{+i(x^{1}k^{1}+x^{2}k^{2})} dk^{1} dk^{2}  (2.80)

this means:

\begin{matrix}E(x) ≈ (2π)^{−2} ∫ (∫ E(x^{1}, x^{2}, 0) e^{−i(x^{1}k^{1}+x^{2}k^{2})} dx^{1}dx^{2}) · \\ · e^{−i \frac{(k^{1})^{2}+(k^{2})^{2}}{2 \frac{ω}{c} n(ω)} x^{3}} e^{+i(x^{1}k^{1}+x^{2}k^{2})} dk^{1}dk^{2} . \end{matrix} (2.81)

Hence:

For monochromatic light rays,^{50} E(x) can be calculated everywhere approximately from its values on any plane that is perpendicular to the propagation direction s .

Especially for (linearly polarized) Gaussian rays, characterized (up to spatial rotation and translation) by

E(x)_{|_{x^{3}=0}} = E_{0} exp (−\frac{(x^{1})^{2} + (x^{2})^{2}}{δ^{2}} ) , s = e_{3} ,

Draft, November 5, 2011

rays.

49We skip the detailed mathematical prove of uniqueness.

50We do not elaborate on which profiles E(x)_{|_{x^{3}=0}} are actually possible for monochromatic light

2.4. GROUP-VELOCITY DISPERSION 59 (2.81) gives

E(x) ≈ (2π)^{−2} E_{0} ∫ (∫ e^{−^{\frac{(x^{1})^{2}+(x^{2})^{2}}{δ^{2}}}} e^{−i(x^{1}k^{1}+x^{2}k^{2})} dx^{1}dx^{2}) ·
· e^{−i \frac{(k^{1})^{2}+(k^{2})^{2}}{2 \frac{ω}{c} n(ω)} x^{3}} e^{+i(x^{1}k^{1}+x^{2}k^{2})} dk^{1}dk^{2}
= \frac{δ^{2}}{4π} E_{0} ∫ e^{−((k^{1})^{2}+(k^{2})^{2})δ^{2}/4} e^{−i \frac{(k^{1})^{2}+(k^{2})^{2}}{2 \frac{ω}{c} n(ω)} x^{3}} e^{+i(x^{1}k^{1}+x^{2}k^{2})} dk^{1}dk^{2}
= \frac{δ^{2}}{4π} E_{0} ∏_{j=1}^{2} ∫ exp (−(k R(x^{3}))^{2} + i x^{j} k) dk
= \frac{δ^{2}}{4π} E_{0} ∏_{j=1}^{2} (e^{−( \frac{x^{j}}{2 R(x^{3})} )^{2}} ∫ exp (−(k R(x^{3}) − i \frac{x^{j}}{2 R(x^{3})})^{2}) dk)
= (\frac{δ}{2 R(x^{3})})^{2} exp (− (\frac{x^{1}}{2 R(x^{3})})^{2} − (\frac{x^{2}}{2 R(x^{3})})^{2}) ,

where

R(x^{3}) =^{def} \sqrt{(δ/2)^{2} + i \frac{x^{3}}{2 \frac{ω}{c}n(ω)}} , ℜ (R(x^{3})) > 0 ,

since

ℜ (R^{2}) > 0 < ℜ(R) ⟹ ∫_{−∞}^{+∞} e^{−(Rx+z_{0})^{2}} dx = \frac{\sqrt{π}}{R} ∀ R , z_{0} ∈ ℂ  (2.82)

(see, e.g., Sect. 2.1.3 of (Lücke, ftm) for a proof of (2.82)) and hence

\frac{1}{\sqrt{2π}} ∫ e^{−x^{2}} e^{−i x k} dx = \frac{1}{\sqrt{2}} e^{−k^{2}/4} ∀ k ∈ ℝ .  (2.83)

2.4 Group-Velocity Dispersion

See (Reider, 1997, Sect. 3.2.1).

60 CHAPTER 2. LINEAR OPTICAL MEDIA

Nonlinear Optical Media

3.1 General Considerations

If, in Exercise 1, the damped harmonic oscillator is replaced by an anharmonic one then its evolution depends nonlinearly on the driving force. As a consequence — even if the driving force is monochromatic — higher harmonics show up in the motion of the oscillator; mainly second order harmonics if the medium is not inversion symmetric^{1} (compare (Reider, 1997, Sect. 8.1)). Moreover, if the driving force is a superposition of two monochromatic ones, the motion of the oscillator contains harmonic components corresponding to the sum as well as to the difference of both frequencies.^{2} If the damped harmonic oscillator model is applied to the electric dipoles of an optical medium then the accelerated charge of the dipoles create a radiation field acting on the other charges in addition. Therefore, actually, the driving field is a superposition of the incident exterior field with the radiation emitted from the medium’s electric dipoles. Now, if the harmonic oscillator is replaced by a more realistic nonlinear one, then the radiation contributed by the medium itself (to the macroscopic electromagnetic field) contains higher harmonics of every monochromatic component of the incident (microscopic) field as well as components with frequencies corresponding to the sum or difference of any two frequencies present in the incident field. Usually, very strong electromagnetic fields — as provided by laser pulses of high intensity — or long optical path length’s in low-loss nonlinear optical media (optical fibers, see (Agrawal, 1999, Sect. 1.3.3)) — have to be applied in order to gain recognizable nonlinear effects. Then, however,

Draft, November 5, 2011

1Since, e.g., silicia glasses — the material of choice for low-loss optical fibers, formed by fusing SiO_{2} molecules — are inversion symmetric they do not normally exhibit second-order nonlinear effects.

2See (Mills, 1998, Sect. 3.1) for a perturbative calculation.

interesting phenomena show up several of which will be studied in this chapter.

3.1.1 Functionals of the Electric Field

Usually, especially in optical applications, P(x, t) is assumed to be a (sufficiently well-behaved) functional of E . Then^{3} we have the functional Taylor expansion^{4}

P_{j}(x, t) = ∑_{ν=0}^{∞} P_{j}^{(ν)}(x, t) ,  (3.1)

P_{j}^{(ν)}(x, t) = \frac{ϵ_{0}}{(2π)^{\frac{ν}{2}}} ∫ χ̌_{jk_{1}...k_{ν}}^{(ν)}(x, t; x′_{1}, t′_{1}; …; x′_{ν}, t′_{ν}) E^{k_{1}}(x′_{1}, t′_{1}) …
… E^{k_{ν}}(x′_{ν}, t′_{ν}) dV_{x′}_{1} dt′_{1} … dV_{x′}_{ν} dt′_{ν} ,

(3.2)

w.r.t. an arbitrary Basis {b_{1}, b_{2}, b_{3}} of ℝ, where^{5}

P_{j}(x, t) =^{def} b_{j} · P(x, t) , E^{k}(x′, t′) =^{def} b^{k} · E(x, t) .

Note that the coefficients χ̌^{(ν)} fulfill the symmetry condition^{6}

χ̌_{jk_{1}...k_{ν}}^{(ν)}(x, t; x′_{1}, t′_{1}; …; x′_{ν}, t′_{ν}) = χ̌_{jk_{π̌(1)}...k_{π̌(ν)}}^{(ν)}(x, t; x′_{π̌(1)}, t′_{π̌(1)}; …; x′_{π̌(ν)}, t′_{π̌(ν)}) (3.3)

for every permutation π̌ of (1, …, ν) . If, in addition, we assume homogeneity in space and time these coefficients have to be of the form

χ̌_{jk_{1}...k_{ν}}^{(ν)}(x, t; x′_{1}, t′_{1}; …; x′_{ν}, t′_{ν}) = χ̌_{jk_{1}...k_{ν}}^{(ν)}(x − x′_{1}, t − t′_{1}; …; x − x′_{ν}, t − t′_{ν}) .

Assuming, moreover, spatial nonlocality of the electric field contributions to be negligible we have

χ̌_{jk_{1}...k_{ν}}^{(ν)}(x, t; x′_{1}, t′_{1}; …; x′_{ν}, t′_{ν}) = χ̌_{jk_{1}...k_{ν}}^{(ν)}(t − t′_{1}, …, t − t′_{ν}) δ(x − x′_{1}) … δ(x − x′_{ν}) .

Thus,

\begin{matrix}\frac{(2π)^{\frac{ν−1}{2}}}{ϵ_{0}} P_{j}^{(ν)}(x, t) \\ = \frac{1}{\sqrt{2π}} ∫ χ_{jk_{1}...k_{ν}}^{(ν)}(ω_{1}, …, ω_{ν}) Ẽ^{k_{1}}(x, ω_{1}) … Ẽ^{k_{ν}}(x, ω_{ν}) e^{−i(ω_{1}+...+ω_{ν})t} dω_{1} … dω_{ν} \end{matrix}

(3.4)

Draft, November 5, 2011

3See Appendix A.1.

4We use Einstein’s summation convention meaning, e.g., χ_{jk} E^{k} ≡ ∑_{k=1}^{3} χ_{jk} E^{k}.

5Recall that the reciprocal basis {b^{1}, b^{2}, b^{3}} is characterized by:

b_{j} · b^{k} = \{ \begin{matrix}1 & for j = k, \\ 0 & else . \end{matrix}

6Recall (A.1).

holds for the (generalized) susceptibilities

χ_{jk_{1}...k_{ν}}^{(ν)}(ω_{1}, …, ω_{ν}) =^{def} (2π)^{− \frac{n}{2}} ∫ χ̌_{jk_{1}...k_{ν}}^{(ν)}(t_{1}, …, t_{ν}) e^{i(ω_{1} t_{1}+...+ω_{ν} t_{ν})} dt_{1} … dt_{ν}

and (3.3) is equivalent to

χ_{jk_{1}...k_{ν}}^{(ν)}(ω_{1}, …, ω_{ν}) = χ_{jk_{π̌(1)}...k_{π̌(ν)}}^{(ν)}(ω_{π̌(1)}, …, ω_{π̌(ν)}) .  (3.5)

The Fourier transform of (3.4) is^{7}

\begin{matrix}\frac{(2π)^{\frac{ν−1}{2}}}{ϵ_{0}} P̃_{j}^{(ν)}(x, ω) \\ = ∫ χ_{jk_{1}...k_{ν}}^{(ν)}(ω_{1}, …, ω_{ν}) Ẽ^{k_{1}}(x, ω_{1}) … Ẽ^{k_{ν}}(x, ω_{ν}) δ (ω_{1} + … + ω_{ν} − ω) dω_{1} … dω_{ν} . \end{matrix}

(3.6)

An optical medium modeled by (3.6) is called a linear optical medium if all the nonlinear susceptibilities χ_{jk_{1}...k_{ν}}^{(ν)}(ω_{1}, …, ω_{ν}) , ν > 1 , vanish.^{8}

Usually,

P(x, t) = P_{el}(x, t)

is assumed for optical media and the susceptibilities should be caculated by (timedependent) quantum mechnical perturbation theory.^{9} For the tensor χ_{jk}^{(1)}(ω) of the linear susceptibility first order perturbation theory is sufficient.

3.1.2 Various Electric Polarization Effects

See (Römer, 1994, Chapter 5).

3.1.3 Perturbative Solution of Maxwell’s Equations

If we replace P by P^{(1)} in (2.1) and χ^{↔} by χ^{↔}^{(1)} in the definition of the permittivity ϵ^{↔} then the Fourier transformed Maxwell equations (1.18)–(1.21) become equivalent to:

curl H̃(x, ω) = −i ω ϵ_{0} ϵ^{↔}(ω) Ẽ(x, ω) + j̃_{ex}(x, ω) − i ω P̃^{nl}(x, ω) ,

(3.7)

curl B̃(x, ω) = i ω H̃(x, ω) ,  (3.8)
div (ϵ_{0} ϵ^{↔}(ω) Ẽ(x, ω)) = ρ̃_{ex}(x, ω) − div(P̃^{nl}(x, ω)),

(3.9)

div B̃(x, ω) = 0 ,  (3.10)

Draft, November 5, 2011

7For a listing of the dependence of the lowest nonlinear susceptibilities on the symmetry classes for crystals see (Brunner and Junge, 1982, Sect. 3.2.2.1).

8Note that nonlinearity does not necessarily mean that χ_{jk_{1}...k_{ν}}^{(ν)}(ω, …, ω) ≠ 0 for some ν > 1 . 9See (Armstrong et al., 1962a).

64 CHAPTER 3. NONLINEAR OPTICAL MEDIA where

P̃^{nl}(x, ω) =^{def} P̃(x, ω) − P̃_{j}^{(1)}(x, ω) b^{j} .

Now we have to replace (2.12) by the nonlinear equation^{10}

ĤẼ(x, ω) = −i ω μ_{0} j̃_{cr}(x, ω) − ω^{2}μ_{0} P̃^{nl}(x, ω; Ẽ(x, .)) (3.11)

in which we have written P̃^{nl}(x, ω; Ẽ(x, .)) instead of P̃^{nl}(x, ω) in order to indicate the functional (nonlinear) dependence on Ẽ(x, .) , determined by the nonlinear susceptibilities, and where

Ĥ Ẽ(x, ω) =^{def} (Δ_{x} + (\frac{ω}{c} )^{2} ϵ^{↔}_{c}(ω)) Ẽ(x, ω) − grad (div Ẽ(x, ω)) .

(3.12)

Motivated by the results of 2.1.3, let us assume that there is a unique (matrix-valued, restarted) response function ř^{↔}(x, t) of the corresponding linear medium such that

R(x, t; ȷ) =^{def} \frac{1}{\sqrt{2π}} ∫ ř^{↔}(x − x′, t − t′) ȷ (x′, t′) dV_{x′} dt′  (3.13)

fulfills the conditions

ȷ (x, t) = 0 for (|x| , t) ∉ [0, +R] × [0, +∞)
⟹ R(x, t; ȷ) = 0 for t ∉[0, \frac{|x|−R}{c}]

and

Ĥ R̃(x, ω; j̃) = −i ω μ_{0} j̃(x, ω)  (3.14)

for all sufficiently well-behaved ȷ, where

R̃(x, ω; j̃) =^{def} \frac{1}{\sqrt{2π}} ∫ R(x, t; ȷ) e^{+i ω t} dt
= ∫ r^{↔}(x − x′, ω) j̃(x′, ω) dV_{x′} .  (3.15)

Let us only consider the case

j̃_{cr} = 0 .

Then Ẽ(x, ω) is a solution of (3.11) iff the linear equation

Ĥ Ẽ_{0}(x, ω) = 0 .  (3.16)

Draft, November 5, 2011

10Again, (3.9) serves as a definition of ρ̃_{ex}(x, ω) for ω ≠ 0 .

3.2. CLASSICAL NONLINEAR OPTICAL EFFECTS 65 is fulfilled for

Ẽ_{0}(x, ω) = Ẽ(x, ω) − R̃(x, ω; j̃Ẽ) ,

where

j̃Ẽ(x, ω) =^{def} −iω P̃^{nl}(x, ω; Ẽ(x, .)) .  (3.17)

This suggests the following perturbative solution of (3.11):

Take a solution Ẽ_{0}(x, ω) of (3.16) and define recursively

Ẽ_{ν}(x, ω) =^{def} Ẽ_{0}(x, ω) + R̃(x, ω; j̃_{Ẽ_{ν−1}}) for ν = 1, 2, 3, …

(compare Section 5.1.3 von (Lücke, ein)). Then

Ẽ(x, ω) = lim_{ν→∞} Ẽ_{ν}(x, ω) ,

if the limit exists (in a suitable topology), is a solution of (3.11). Certainly, one will try to get along with the first order approximation

Ẽ(x, ω) ≈ Ẽ_{0}(x, ω) − iω ∫ r^{↔}(x − x′, ω) P̃^{nl}(x′, ω; Ẽ_{0}(x′, .)) dV_{x′} (3.18)

(for properly chosen Ẽ_{0}(x, ω)) as far as possible. Note that, by (3.11)–(3.13), we have

Ĥ Ẽ(x, ω) = −ω^{2}μ_{0} P̃^{nl}(x, ω; Ẽ_{0}(x, .))  (3.19)

in this approximation.

3.2 Classical Nonlinear Optical Effects^{11}

3.2.1 Phase Matching

Let us discuss the approximations (3.18) and

P̃_{j}^{nl}(x, ω) ≈ P̃_{j}^{(2)}(x, ω) =_{(3.6)} ∫ χ_{jkl}^{(2)}(ω′, ω − ω′) Ẽ^{k}(x, ω′) Ẽ^{l}(x, ω − ω′) dω′ . (3.20)

By (3.19), then, we have to expect that

\begin{matrix}ω = ω_{1} + ω_{2} , \\ χ_{jkl}^{(2)}(ω_{1}, ω_{2}) Ẽ_{0}^{k}(x, ω_{1}) Ẽ_{0}^{l}(x, ω_{1}) ≠ 0 \end{matrix} \} for suitable ω_{1}, ω_{1} ∈ ℝ

⟹ Ẽ(x, ω) ≠ 0

Draft, November 5, 2011

11A freeware program for calculating nonlinear frequency conversion processes is offered at:

www.sandia.gov/imrl/XWEB1128/xxtal.htm

and hence

(Ẽ_{0}(x, ω_{1}) ≠ 0 ≠ Ẽ_{0}(x, ω_{2}) ⟹ Ẽ(x, ω_{1} + ω_{2}) ≠ 0) ∀ω_{1}, ω_{2} ∈ ℝ

holds for sufficiently nontrivial^{12} χ^{(2)} . Especially for ω_{1} = ω_{2} this means second harmonic generation. Of course, the crucial question is how strong these effects are.

Actually, as can be explained by simple models, Miller’s rule^{13}

Δ_{jkl}(ω_{1}, ω_{2}) =^{def} \frac{χ_{jkl}^{(2)}(ω_{1}, ω_{2})}{χ_{jj}^{(1)}(ω_{1} + ω_{2}) χ_{kk}^{(1)}(ω_{1}) χ_{ll}^{(1)}(ω_{2})} ≈ independent of ω_{1}, ω_{2} , (3.21)

w.r.t. the principal directions in the medium, is valid for a large variety of crystals. Therefore the dispersion relations^{14} for the linear susceptibilities χ_{jk}^{(1)} , qualitatively described by Exercise 1, are of great importance also for the second order susceptibilities.

Let us analyze (3.18)/(3.19) for the special case that the solution Ẽ_{0}(x, ω) of (3.16) is of the form^{15}

\begin{aligned}Ẽ_{0}(x, ω) & = & Ẽ_{0}(x, ω) (δ(ω_{1} − ω) + δ(ω_{2} − ω)) \\ & & +\overline{Ẽ_{0}(x, −ω)}(δ(ω_{1} + ω) + δ(ω_{2} + ω)) . \end{aligned}  (3.22)

Then we have

Ẽ_{0}^{k}(x, ω′) Ẽ_{0}^{l}(x, ω − ω′)

= Ẽ_{0}^{k}(x, ω′) Ẽ_{0}^{l}(x, ω − ω′) (δ(ω_{1} − ω′) + δ(ω_{2} − ω′))(δ(ω_{1} − ω + ω′) + δ(ω_{2} − ω + ω′))

In the approximation (3.20), this together with

χ_{jkl}^{(2)}(−ω′_{1}, −ω′_{2}) = \overline{χ_{jkl}^{(2)}(ω′_{1}, ω′_{2})}

+ Ẽ_{0}^{k}(x, ω′) \overline{Ẽ_{0}^{l}(x, ω′ − ω)}(δ(ω_{1} − ω′) + δ(ω_{2} − ω′))(δ(ω_{1} + ω − ω′) + δ(ω_{2} + ω − ω′))

+ Ẽ_{0}^{k}(x, −ω′) Ẽ_{0}^{l}(x, ω − ω′)(δ(ω_{1} + ω′) + δ(ω_{2} + ω′))(δ(ω_{1} − ω + ω′) + δ(ω_{2} − ω + ω′))

+ \overline{Ẽ_{0}^{k}(x, −ω′)} \overline{Ẽ_{0}^{l}(x, ω′ − ω)}(δ(ω_{1} + ω′) + δ(ω_{2} + ω′))(δ(ω_{1} + ω − ω′) + δ(ω_{2} + ω − ω′)) .

Draft, November 5, 2011 12For inversion-symmetric media χ^{(2)} must vanish, because both P_{j}^{(2)} and Ẽ^{l} change sign under total spatial reflection.

13See (Miller, 1964). Obviously, this rule is consistent with the generally valid relations

χ_{jlk}^{(2)}(ω_{2}, ω_{1}) = χ_{jkl}^{(2)}(ω_{1}, ω_{2}) = \overline{χ_{jkl}^{(2)}(−ω_{1}, −ω_{2})} , χ_{jl}^{(1)}(−ω) = \overline{χ_{jl}^{(1)}(ω)} .

14Recall the remark on page 37 of Chapter 2.

15We shall see, however, that such a choice may be inappropriate for first order approximation.

3.2. CLASSICAL NONLINEAR OPTICAL EFFECTS 67 gives

P̃^{nl}(x, ω; Ẽ_{0}(x, .)) = P̃_{0}(x, ω) + \overline{P̃_{0}(x, −ω)}

where^{16}

P̃_{0}(x, ω) =^{def} 2 δ(ω_{1} + ω_{2} − ω) χ_{jkl}^{(2)}(ω_{1}, ω_{2}) Ẽ_{0}^{k}(x, ω_{1}) Ẽ_{0}^{l}(x, ω_{2})
+δ(2 ω_{1} − ω) χ_{jkl}^{(2)}(ω_{1}, ω_{1}) Ẽ_{0}^{k}(x, ω_{1}) Ẽ_{0}^{l}(x, ω_{1})
+δ(2 ω_{2} − ω) χ_{jkl}^{(2)}(ω_{2}, ω_{2}) Ẽ_{0}^{k}(x, ω_{2}) Ẽ_{0}^{l}(x, ω_{2})
+2 δ(ω_{1} − ω_{2} − ω) χ_{jkl}^{(2)}(−ω_{2}, ω_{1}) \overline{Ẽ_{0}^{k}(x, ω_{2})} Ẽ_{0}^{l}(x, ω_{1})
+δ(ω)(χ_{jkl}^{(2)}(ω_{1}, −ω_{1}) Ẽ_{0}^{k}(x, ω_{1}) \overline{Ẽ_{0}^{l}(x, ω_{1})}
+χ_{jkl}^{(2)}(ω_{2}, −ω_{2}) Ẽ_{0}^{k}(x, ω_{2}) \overline{Ẽ_{0}^{l}(x, ω_{2})}) .

Therefore, in the approximation (3.18) Ẽ(x, ω) is of the form

Ẽ(x, ω) = Ẽ_{0}(x, ω) + ∑_{ω′∈M_{ω_{1},ω_{2}}} Ẽ(x, ω) δ(ω − ω′) ,

where

M_{ω_{1},ω_{2}} =^{def} {ω_{1} + ω_{2}, 2 ω_{1}, 2 ω_{2}, ω_{1} − ω_{2}, −ω_{1} − ω_{2}, −2 ω_{1}, −2 ω_{2}, ω_{2} − ω_{1}} .

The static part of P̃^{nl}(x, ω; Ẽ_{0}(x, .)) (optical rectification^{17}) does not show up in (3.18) because of the ω-factor.^{18} Let us assume, for simplicity, that

{ω_{1}, ω_{2}} ∩ M_{ω_{1},ω_{2}} = ∅ .

Then special consequences of (3.19) are the equation

Ĥ Ẽ(x, 2 ω_{1}) = − (2 ω_{1})^{2} μ_{0} χ_{jkl}^{(2)}(ω_{1}, ω_{1}) Ẽ^{k}(x, ω_{1}) Ẽ^{l}(x, ω_{1}) b^{j}  (3.23)

for second harmonic generation, and the equation

Ĥ Ẽ(x, ω_{1} + ω_{2}) = −2 (ω_{1} + ω_{2})^{2} μ_{0} χ_{jkl}^{(2)}(ω_{1}, ω_{2}) Ẽ^{k}(x, ω_{1}) Ẽ^{l}(x, ω_{2}) b^{j}  (3.24)

for sum-frequency generation, where we have set

Ẽ(x, ω_{1}) = Ẽ_{0}(x, ω_{1}) for j = 1, 2 .  (3.25)

Draft, November 5, 2011

16Recall Footnote 13.

17See, e.g., (Shen, 1984, Sect. 5.1) for a discussion of this effect.

18But it does contribute to Ẽ_{0}(x, 0) . Note that Ĥ Ẽ(x, 0) = 0 does not imply ρ̃_{ex}(x, 0) = 0 .

More precisely, (3.24) is a consequence of

Ẽ(x, ω_{1} + ω_{2})

= −i (ω_{1} + ω_{2}) ∫ r^{↔}(x − x′, ω_{1} + ω_{2}) (2 χ_{jkl}^{(2)}(ω_{1}, ω_{2}) Ẽ^{k}(x′, ω_{1}) Ẽ^{l}(x′, ω_{2}) b^{j}) dV_{x′}

(3.26)

in very the same way as (3.19) is a consequence of (3.18).

Let us specialize further to^{19}

Ẽ_{0}(x, ω) = E_{e_{1}}(ω) exp (i \frac{ω}{c} N_{e_{1}}(ω) e_{1} · x) ∀ ω ∈ \{ω_{1}, ω_{2}\}  (3.27)

and

{b_{1}, b_{2}, b_{3}} = {e_{1}, e_{2}, e_{3}} right-handed orthonormal basis of ℝ^{3} .

Then (3.25) and (3.26) imply

Ẽ(x, ω) = Ẽ(x^{1}, ω) ∀ ω ∈ {ω_{1}, ω_{2}, ω_{1} + ω_{2}}  (3.28)

and therefore

Δ_{x} Ẽ(x, ω) − ∇_{x} (∇_{x} · Ẽ(x, ω)) = (\frac{∂}{∂x^{1}} )^{2} P̂_{⊥e_{1}} Ẽ(x, ω)  (3.29)

holds for ω ∈ {ω_{1}, ω_{2}, ω_{1} + ω_{2}}. Especially for ω = ω_{1} + ω_{2} this means, according to the definition of Ĥ (Equation (3.12)), that (3.24) is equivalent to

\begin{matrix}ϵ_{0} ( ϵ^{↔}_{c}(ω_{1} + ω_{2}) Ẽ(x, ω_{1} + ω_{2}) + (\frac{∂}{∂x^{1}} )^{2} P̂_{⊥e_{1}} Ẽ(x, ω_{1} + ω_{2})) \\ = −2 χ_{jkl}^{(2)}(ω_{1}, ω_{2}) Ẽ^{k}(x, ω_{1}) Ẽ^{l}(x, ω_{2}) e_{j} , \end{matrix}

i.e. to the equations

(ϵ_{0} ϵ^{↔}_{c}(ω_{1} + ω_{2}) Ẽ(x, ω_{1} + ω_{2})) · e_{1} = −2 χ_{1kl}^{(2)}(ω_{1}, ω_{2}) Ẽ^{k}(x, ω_{1}) Ẽ^{l}(x, ω_{2})

and

(ϵ_{0} ϵ^{↔}_{c}(ω_{1} + ω_{2}) Ẽ(x, ω_{1} + ω_{2})) · e_{j} + ϵ_{0} (\frac{∂}{∂x^{1}} )^{2} Ẽ^{j}(x, ω_{1} + ω_{2})

= −2 χ_{jkl}^{(2)}(ω_{1}, ω_{2}) Ẽ^{k}(x, ω_{1}) Ẽ^{l}(x, ω_{2}) for j = 2, 3 .

(3.30)

For simplicity, let us assume that the medium is uniaxial with axis along e_{1} . Then

N_{e_{1}}(ω) = N_{e_{1}}^{ord}(ω) =_{(2.69)} \sqrt{ϵ_{c}^{2}(ω)} ,

( ϵ^{↔}_{c}(ω) Ẽ(x, ω)) · e_{j} = ϵ_{c}^{2}(ω) Ẽ^{j}(x, ω) for j = 2, 3

(3.31)

Draft, November 5, 2011

19Compare with (2.30).

and, corresponding to (3.27), the slowly varying amplitude approximation^{20}

|(\frac{∂}{∂x^{1}} )^{2} A(x^{1}, ω)| ≪ |\frac{ω}{c} N_{e_{1}}(ω)\frac{∂}{∂x^{1}} A(x^{1}, ω)|  (3.32)

should be applicable to

A(x^{1}, ω) =^{def} exp (−i \frac{ω}{c} N_{e_{1}}(ω) e_{1} · x) Ẽ(x, ω)  (3.33)

for ω = ω_{1} + ω_{2} . (3.30)–(3.33) imply

\frac{∂}{∂x^{1}} A^{j}(x^{1}, ω_{1} + ω_{2}) = C^{j} e^{−i ΔK x^{1}} for j = 2, 3 ,  (3.34)

where

C^{j} =^{def} i c \frac{ω_{1} + ω_{2}}{N_{e_{1}}(ω_{1} + ω_{2})} μ_{0} χ_{jkl}^{(2)}(ω_{1} + ω_{2}) E_{e_{1}}^{k}(ω_{1}) E_{e_{1}}^{l}(ω_{2})

and

ΔK =^{def} \frac{ω_{1} + ω_{2}}{c} N_{e_{1}}(ω_{1} + ω_{2}) − \frac{ω_{1}}{c} N_{e_{1}}(ω_{1}) − \frac{ω_{2}}{c} N_{e_{1}}(ω_{2}) .

The general solution of (3.34) is^{21}

A^{j}(x^{1}, ω_{1} + ω_{2}) = \frac{i}{ΔK} C^{j} (e^{−i ΔK x^{1}} − 1) + Ĉ^{j}
= x^{1} C^{j} e^{− \frac{i}{2} ΔK x^{1}} \frac{sin (\frac{1}{2} ΔK x^{1})}{\frac{1}{2} ΔK x^{1}} + Ĉ^{j} .

(3.35)

with arbitrary Ĉ^{j} = Ĉ^{j}(ω_{1} + ω_{2}) . Thus we see:

1. Since lim_{x→0} \frac{sin x}{x} = 1 :

|A^{j}(x^{1}, ω_{1} + ω_{2}) − Ĉ^{j}| ≈ e^{ℑ(ΔK) x^{1}/2} |x^{1} C^{j}| for |ΔK x^{1}| ≪ 1 ; j = 2, 3 .

2. Therefore, for sufficiently good phase matching |ΔK| ≈ 0 arbitrary increase of |Ẽ(x, ω_{1} + ω_{2})| = |e^{− \frac{ω_{1}+ω_{2}}{c} ℑ(N_{e_{1}}(ω_{1}+ω_{2})) x^{1}} A^{j}(x^{1}, ω_{1} + ω_{2})| in the direction of propagation is possible in spite of absorption related to ℑ (N) .

3. This indicates that low order perturbative approximations may have to be restricted to sufficiently small regions and that Ẽ_{0}(x, ω) has to be adapted to the region under consideration.

Draft, November 5, 2011

20Compare Footnote 48 of Chapter 2.3. Now, however, we allow for ℑ (N_{e_{1}}(ω)) ≠ 0 .

21Note that (3.35) is consistent with (3.32) if |ΔK| ≪ |\frac{ω_{1}+ω_{2}}{c} N_{e_{1}}(ω_{1} + ω_{2})|.

Actually, under the conditions considered above, good phase matching is very unlikely for ω_{1}, ω_{2} > 0 . To simplify the argument, let us consider the special case ω_{1} = ω_{2} and μ = 1 . Then

\begin{aligned}ΔK = 0 & ⟺ & N_{e_{1}}(2 ω_{1}) = N_{e_{1}}(ω_{1}) \\ & =_{(3.31)} & ϵ_{c}^{2}(2 ω_{1}) = ϵ_{c}^{2}(ω_{1}) \end{aligned}

and the latter condition is usually violated in sufficiently lossless media as can be understood by inspection of dispersion relations.

Fortunately, there are uniaxial media (with frequency-independent optical axis) for which type-I phase matching

\frac{ω_{1} + ω_{2}}{c} N_{e_{1}}^{exo}(ω_{1} + ω_{2}) ≈ \frac{ω_{1}}{c} N_{e_{1}}^{ord}(ω_{1}) + \frac{ω_{2}}{c} N_{e_{1}}^{ord}(ω_{2})  (3.36)

or type-II phase matching

\frac{ω_{1} + ω_{2}}{c} N_{e_{1}}^{exo}(ω_{1} + ω_{2}) ≈ \frac{ω_{1}}{c} N_{e_{1}}^{exo}(ω_{1}) + \frac{ω_{2}}{c} N_{e_{1}}^{ord}(ω_{2})  (3.37)

is possible without substantial damping if the optical axis is suitably oriented (not along e_{1}).

Exercise 7

a) Using (2.72), show that

{|N_{s}^{exo}(ω)|^{2} : s ∈ ℝ^{3} , s · s = 1}
= [min {|ϵ_{c}^{∥}(ω)| , |ϵ_{c}^{⊥}(ω)|} , max {|ϵ_{c}^{∥}(ω)| , |ϵ_{c}^{⊥}(ω)|}] ,

where ϵ_{c}^{∥} denotes the nondegenerate and ϵ_{c}^{⊥} the degenerate eigenvalue of ϵ^{↔}^{c} .

b) For ω_{1} = ω_{2} and the standard low-loss case

|ϵ_{c}^{⊥}(ω_{1})| < |ϵ_{c}^{⊥}(2 ω_{1})|

show that

(3.36) ⟹ |ϵ_{c}^{∥}(2 ω_{1})| ≤ |ϵ_{c}^{⊥}(ω_{1})| .

3.2.2 Three-Wave Mixing

Let us consider a uniaxial medium with axis along e_{0} ∈ ℝ and positive frequencies ω_{1}, ω_{2} for which the phase matching conditions are such that a solution of (3.11) (for j̃_{cr}(x, ω) = 0) with the following properties exists:

1. For every sufficiently small region G in x-space there is a good first order perturbative approximation (3.18) with a suitably chosen zero order approximation (solution of (3.16)) consisting only of

(a) two ordinary plane waves^{22} of the form

e_{ord} A_{G}(ω) e^{i \frac{ω}{c} N_{e_{1}}^{ord}(ω) x^{1}} δ(ω_{j} − ω)
+ e_{ord} \overline{A_{G}(ω) e^{i \frac{ω}{c} N_{e_{1}}^{ord}(ω) x^{1}}} δ(ω_{j} + ω) ; j = 1, 2

where

e_{ord} =^{def} \frac{e_{1} × e_{0}}{|e_{1} × e_{0}|} ,  (3.38)

and

(b) an extraordinary plane wave^{23} of the form

e_{exo} A_{G}(ω) e^{i \frac{ω}{c} N_{e_{1}}^{exo}(ω) x^{1}} δ(ω_{1} + ω_{2} − ω)
+ e_{exo} \overline{A_{G}(ω) e^{i \frac{ω}{c} N_{e_{1}}^{exo}(ω) x^{1}}} δ(ω_{1} + ω_{2} + ω) ,

where

e_{exo} =^{def} \frac{( ϵ^{↔}^{c}(ω))^{−1}(e_{1} × (e_{1} × e_{0}))}{|( ϵ^{↔}^{c}(ω))^{−1}(e_{1} × (e_{1} × e_{0}))|} .  (3.39)

2. There is a good slowly varying amplitude approximation for Ẽ(x, ω) . More precisely: For some A_{j}(x^{1}, ω) fulfilling^{24}

|(\frac{d}{dx^{1}} )^{2} A(x^{1}, ω)| ≪ |\frac{ω}{c} N_{e_{1}}(ω)\frac{∂}{∂x^{1}} A(x^{1}, ω)| ∀ x^{1} ∈ ℝ , ω ∈ ℝ (3.40)

with

N_{e_{1}}(ω) = \{ \begin{matrix}N_{e_{1}}^{ord}(ω) & for ω = ω ∈ \{ω_{1}, ω_{2}\}, \\ N_{e_{1}}^{exo}(ω) & for ω = ω_{1} + ω_{2} . \end{matrix}

we have

Ẽ(x, ω) ≈ e_{ord} A(x^{1}, ω) e^{i \frac{ω}{c} N_{e_{1}}^{ord}(ω) x^{1}}(δ(ω_{1} − ω) + δ(ω_{2} − ω))
+e_{exo} A(x^{1}, ω) e^{i \frac{ω}{c} N_{e_{1}}^{exo}(ω) x^{1}} δ(ω_{1} + ω_{2} − ω)

in some neighborhood of {ω_{1}, ω_{2}, ω_{1} + ω_{2}}.

Draft, November 5, 2011

22Recall (2.70) and (2.71). 23Recall (2.73) and Remark 2 in Section 2.2.3. 24Compare (3.32).

Under these conditions — according to (3.19), (3.18), and (3.5) — the following equations are approximately valid:

Ĥ (A(x^{1}, ω_{1}) e^{i \frac{ω_{1}}{c} N_{e_{1}}^{ord}(ω_{1}) x^{1}} e_{ord})
= − (ω_{1})^{2} (A(x^{1}, ω) e^{i \frac{ω}{c} N_{e_{1}}^{ord}(ω) x^{1}})_{|_{ω=ω_{1}+ω_{2}}} \overline{A(x^{1}, ω_{2}) e^{i \frac{ω_{2}}{c} N_{e_{1}}^{ord}(ω_{2}) x^{1}}} a ,

(3.41)

Ĥ (A(x^{1}, ω_{2}) e^{i \frac{ω_{2}}{c} N_{e_{1}}^{ord}(ω_{1}) x^{1}} e_{ord})
= − (ω_{2})^{2} (A(x^{1}, ω) e^{i \frac{ω}{c} N_{e_{1}}^{ord}(ω) x^{1}})_{|_{ω=ω_{1}+ω_{2}}} \overline{A(x^{1}, ω_{1}) e^{i \frac{ω_{1}}{c} N_{e_{1}}^{ord}(ω_{1}) x^{1}}} b ,

(3.42)

Ĥ (A(x^{1}, ω_{1} + ω_{2}) e^{i \frac{ω_{1}+ω_{2}}{c} N_{e_{1}}^{exo}(ω_{1}+ω_{2}) x^{1}} e_{exo})
= − (ω_{1} + ω_{2})^{2} A(x^{1}, ω_{1}) e^{i \frac{ω_{1}}{c} N_{e_{1}}^{ord}(ω_{1}) x^{1}} A(x^{1}, ω_{2}) e^{i \frac{ω_{2}}{c} N_{e_{1}}^{ord}(ω_{2}) x^{1}} c ,

(3.43)

where:

a_{j} =^{def} 2 μ_{0} χ_{jkl}^{(2)}(ω_{1} + ω_{2}, −ω_{2}) e_{exo}^{k} e_{ord}^{l} ,
b_{j} =^{def} 2 μ_{0} χ_{jkl}^{(2)}(ω_{1} + ω_{2}, −ω_{1}) e_{exo}^{k} e_{ord}^{l} ,
c_{j} =^{def} 2 μ_{0} χ_{jkl}^{(2)}(ω_{1}, ω_{2}) e_{ord}^{k} e_{ord}^{l} .

(3.44)

The coupling of the components with frequencies ω_{1} , ω_{2} , and ω_{1} + ω_{2} via equations (3.41)–(3.43) is called three-wave mixing.

For simplicity, let us consider only the favorable case^{25}

e_{0} = e_{2} .  (3.45)

Then^{26}

e_{ord} = e_{3} , e_{exo} = −e_{2}  (3.46)

and, therefore, also for the extraordinary sum-frequency component the propagation direction^{27} is e_{1} . Application of P̂_{⊥e_{1}} to (3.41)–(3.43) and use of (3.40) gives the

Draft, November 5, 2011

25 Since beams (suitable superpositions of plane wave solutions) are used in real experiments the propagation directions of all components should essentially coincide for most efficient sumfrequency generation. See (Armstrong et al., 1962b, Sect. VI) for adaption to general e_{0} .

26Note that in this case

( ϵ^{↔}_{c}(ω_{1} + ω_{2}))^{−1}(e_{1} × (e_{1} × e_{0})) = −( ϵ^{↔}^{c}(ω_{1} + ω_{2}))^{−1}e_{0} = −ϵ_{∥}^{c}(ω_{1} + ω_{2}) e_{0} .

27Recall Exercise 6.

3.2. CLASSICAL NONLINEAR OPTICAL EFFECTS 73 approximate equations^{28}

\frac{∂}{∂x^{1}} A(x^{1}, ω_{1})  = \frac{i ω_{1} c}{N_{e_{1}}^{ord}(ω_{1})} e^{i x^{1}Δ_{1}K} A(x^{1}, ω_{1} + ω_{2}) \overline{A(x^{1}, ω_{2})} a_{3} ,
\frac{∂}{∂x^{1}} A(x^{1}, ω_{2})  = \frac{i ω_{2} c}{N_{e_{1}}^{ord}(ω_{2})} e^{i x^{1} Δ_{2}K} A(x^{1}, ω_{1} + ω_{2}) \overline{A(x^{1}, ω_{1})} b_{3} ,
\frac{∂}{∂x^{1}} A(x^{1}, ω_{1} + ω_{2}) = −\frac{i (ω_{1} + ω_{2}) c}{N_{e_{1}}^{ord}(ω_{1} + ω_{2})} e^{i x^{1}Δ_{3}K} A(x^{1}, ω_{1}) A(x^{1}, ω_{2}) c_{2} ,

(3.47)

where:

Δ_{1}K =^{def} \frac{ω_{1}}{c} N_{e_{1}}^{ord}(ω_{1}) − \frac{ω_{1} + ω_{2}}{c} N_{e_{1}}^{exo}(ω_{1} + ω_{2}) + \frac{ω_{2}}{c} \overline{N_{e_{1}}^{ord}(ω_{2})} ,
Δ_{2}K =^{def} \frac{ω_{2}}{c} N_{e_{1}}^{ord}(ω_{2}) − \frac{ω_{1} + ω_{2}}{c} N_{e_{1}}^{exo}(ω_{1} + ω_{2}) + \frac{ω_{1}}{c} \overline{N_{e_{1}}^{ord}(ω_{1})} ,
Δ_{3}K =^{def} \frac{ω_{1} + ω_{2}}{c} N_{e_{1}}^{exo}(ω_{1} + ω_{2}) − \frac{ω_{1}}{c} N_{e_{1}}^{ord}(ω_{1}) − \frac{ω_{2}}{c} N_{e_{1}}^{ord}(ω_{2}) .

Note that (3.44) and (3.46) imply

\begin{aligned}a_{3} & = & −2 μ_{0} χ_{323}^{(2)}(ω_{1} + ω_{2}, −ω_{2}) , \\ b_{3} & = & −2 μ_{0} χ_{323}^{(2)}(ω_{1} + ω_{2}, −ω_{1}) , \\ c_{2} & = & +2 μ_{0} χ_{233}^{(2)}(ω_{1}, ω_{2}) . \end{aligned}

We consider only the case

Δk =^{def} Δ_{1}K = Δ_{2}K = −Δ_{3}K ∈ ℝ .  (3.48)

(lossless media) and^{29}

K =^{def} a_{3} = b_{3} = −c_{2} ∈ ℝ .  (3.49)

Then the system of equations (3.47) simplifies to

\begin{aligned}A′_{1}(x) & = & \frac{i (ω_{1})^{2} K}{k_{1}} e^{+i x Δk} A_{3}(x) \overline{A_{2}(x)} , \\ A′_{2}(x) & = & \frac{i (ω_{2})^{2} K}{k_{2}} e^{+i x Δk} A_{3}(x) \overline{A_{1}(x)} , \\ A′_{3}(x) & = & \frac{i (ω_{3})^{2} K}{k_{3}} e^{−i x Δk} A_{1}(x) A_{2}(x) , \end{aligned}  (3.50)

where

A_{j}(x) =^{def} A(x, ω_{j}) for j = 1, 2, 3 ,
ω_{3} =^{def} ω_{1} + ω_{2} ,
k_{j} =^{def} \frac{ω_{j}}{c} N_{e_{1}}^{ord}(ω_{j}) for j = 1, 2 ,
k_{3} =^{def} \frac{ω_{3}}{c} N_{e_{1}}^{ord}(ω_{3}) .

Draft, November 5, 2011

28Recall (3.12) and (3.29).

29See (Miller, 1964).

If we write the A_{j} in polar form

A_{j}(x) = ρ_{j}(x) e^{iφ_{j}(x)} ∀ j ∈ {1, 1, 3}

and define

θ(x) =^{def} x Δk + φ_{3}(x) − φ_{1}(x) − φ_{2}(x)

then (3.50) becomes equivalent to

\begin{aligned}ρ′_{1}(x) + i φ′_{1}(x) ρ_{1}(x) & = & \frac{i (ω_{1})^{2} K}{k_{1}} ρ_{2}(x) ρ_{3}(x) e^{+i θ(x)} , \\ ρ′_{2}(x) + i φ′_{2}(x) ρ_{2}(x) & = & \frac{i (ω_{2})^{2} K}{k_{2}} ρ_{3}(x) ρ_{1}(x) e^{+i θ(x)} , \\ ρ′_{3}(x) + i φ′_{3}(x) ρ_{3}(x) & = & \frac{i (ω_{3})^{2} K}{k_{3}} ρ_{1}(x) ρ_{2}(x) e^{−i θ(x)} . \end{aligned}

Separated into real and imaginary parts these equations read:

ρ′_{1}(x) = −\frac{(ω_{1})^{2} K}{k_{1}} ρ_{2}(x) ρ_{3}(x) sin θ(x) , φ′_{1}(x) ρ_{1}(x) = −ρ′_{1}(x) cot θ(x) ,

ρ′_{2}(x) = −\frac{(ω_{2})^{2} K}{k_{2}} ρ_{3}(x) ρ_{1}(x) sin θ(x) , φ′_{2}(x) ρ_{2}(x) = −ρ′_{2}(x) cot θ(x) ,

ρ′_{3}(x) = +\frac{(ω_{3})^{2} K}{k_{3}} ρ_{1}(x) ρ_{2}(x) sin θ(x) , φ′_{3}(x) ρ_{3}(x) = +ρ′_{3}(x) cot θ(x) .

(3.51)

The equations on the l.h.s of (3.51) imply that^{30}

W =^{def} ∑_{j=1}^{3} \frac{k_{j}}{ω_{j}} |A_{j}(x)|^{2} is independent of x .

Therefore, with the definitions

u_{j} =^{def} \sqrt{\frac{k_{j}}{(ω_{j})^{2} W}} ρ_{j} ∀ j ∈ \{1, 2, 3\}

and

ζ =^{def} K \sqrt{W \frac{(ω_{1})^{2}(ω_{2})^{2}(ω_{3})^{2}}{k_{1} k_{2} k_{3}}} x

the system of equations (3.51) become equivalent to^{31}

u′_{1}(ζ) = −u_{2}(ζ) u_{3}(ζ) sin θ(ζ) , φ′_{1}(ζ) u_{1}(ζ) = −u′_{1}(ζ) cot θ(ζ) ,
u′_{2}(ζ) = −u_{3}(ζ) u_{1}(ζ) sin θ(ζ) , φ′_{2}(ζ) u_{2}(ζ) = −u′_{2}(ζ) cot θ(ζ) ,
u′_{3}(ζ) = +u_{1}(ζ) u_{2}(ζ) sin θ(ζ) , φ′_{3}(ζ) u_{3}(ζ) = +u′_{3}(ζ) cot θ(ζ) .

(3.52)

Draft, November 5, 2011

30By (2.37), if μ = 1 , \frac{2}{Z_{0}} W is the total power flow of the components with frequencies

ω_{1}, ω_{2}, ω_{1} + ω_{3} (mediated over time), since we assumed the medium to be lossless.

31Thanks to the different physical dimensions of variables we do not need extra symbols for functions when identifying f(ξ) with f(z) in spite of z ≠ ζ.

3.2. CLASSICAL NONLINEAR OPTICAL EFFECTS 75 The equations on the r.h.s. of (3.52) imply

θ′(ζ) = ΔS + cot θ(ζ) \frac{d}{dζ} ln(u_{1}(ζ) u_{2}(ζ) u_{3}(ζ))

resp.

cos θ(ζ) \frac{d}{dζ} (u_{1}(ζ) u_{2}(ζ) u_{3}(ζ)) − u_{1}(ζ) u_{2}(ζ) u_{3}(ζ) θ′(ζ) sin θ(ζ)

= −ΔS u_{1}(ζ) u_{2}(ζ) u_{3}(ζ) sin θ(ζ) ,

(3.53)

where

ΔS =^{def} \frac{Δk}{K} \sqrt{\frac{k_{1} k_{2} k_{3}}{W (ω_{1})^{2}(ω_{2})^{2}(ω_{3})^{2}}} .

Since the last equation on the l.h.s. of (3.52) implies

u_{1}(ζ) u_{2}(ζ) u_{3}(ζ) sin θ(ζ) = \frac{1}{2} \frac{d}{dζ} (u_{3}(ζ))^{2}

and since

cos θ(ζ) \frac{d}{dζ} (u_{1}(ζ) u_{2}(ζ) u_{3}(ζ)) − u_{1}(ζ) u_{2}(ζ) u_{3}(ζ) θ′(ζ) sin θ(ζ)

= \frac{d}{dζ} (u_{1}(ζ) u_{2}(ζ) u_{3}(ζ) cos θ(ζ)) ,

(3.53) implies

cos θ(ζ) = \frac{Γ − \frac{1}{2}(u_{3}(ζ))^{2} ΔS}{u_{1}(ζ) u_{2}(ζ) u_{3}(ζ)} , Γ = const. . (3.54)

This may be used to eliminate θ in, e.g., the last equation on the l.h.s. of (3.52):

\frac{d}{dζ} (u_{3}(ζ))^{2} = ±2 \sqrt{(u_{1}(ζ) u_{2}(ζ) u_{3}(ζ))^{2} − (Γ − \frac{1}{2}(u_{3}(ζ))^{2}ΔS)^{2}} .  (3.55)

Thanks to the equations on the l.h.s. of (3.52), we have are constant:

m_{1} =^{def} (u_{2}(ζ))^{2} + (u_{3}(ζ))^{2} = const. ,
m_{2} =^{def} (u_{3}(ζ))^{2} + (u_{1}(ζ))^{2} = const. ,
m_{3} =^{def} (u_{1}(ζ))^{2} − (u_{2}(ζ))^{2} = const. .

(3.56)

This together with (3.54) gives

ζ = ±\frac{1}{2} ∫_{(u_{3}(0))^{2}}^{(u_{3}(ζ))^{2}} \frac{dλ}{\sqrt{λ (m_{2} − λ) (m_{1} − λ) − (Γ − \frac{1}{2} λ ΔS)^{2}}} .  (3.57)

76 CHAPTER 3. NONLINEAR OPTICAL MEDIA Now the intensities (u_{j}(ζ))^{2} may be determined from (3.54)–(3.56) and the (boundary) values of the electromagnetic wave at ζ = 0 . For a detailed discussion see (Armstrong et al., 1962b).

For the simple case

u_{1} = u_{2} (second harmonic generation^{32}) ,
ΔS = 0 (perfect phase matching) ,
u_{3}(0) = 0 (second harmonic vanishing at the boundary) ,
Γ = 0 (no singularity in (3.54)) .

(3.56) and (3.57) imply

\begin{aligned}ζ & = & ±\frac{1}{2} ∫_{(u_{3}(0))^{2}}^{(u_{3}(ζ))^{2}} \frac{dλ}{\sqrt{λ} (m_{1} − λ)} \\ & = & ±\begin{matrix}tanh & −1 & (u_{3}(ζ)/\sqrt{m_{1}}) \\ √ & \overline{m_{1}} \end{matrix} \end{aligned}

for ζ > 0 , hence

u_{3}(ζ) = ^{+} \sqrt{m_{1}} tanh ( ^{+} \sqrt{m_{1}} |ζ|)

inside the medium.

3.2.3 Four-Wave Mixing

See (Shen, 1984, Chapter 14).

Draft, November 5, 2011

32We suppress an additional scaling of amplitudes that is necessary in this case.

3.3 Quantum Aspects of Three-Wave Mixing

3.3.1 Quantized Radiation Inside Linear Media

Let us consider a uniaxial linear crystal with (essentially) real permittivity

ϵ^{↔}(ω) = ˆ1 + χ^{↔}^{(1)}(ω) = ( \begin{matrix}ϵ_{1}(ω) & 0 & 0 \\ 0 & ϵ_{2}(ω) & 0 \\ 0 & 0 & ϵ_{2}(ω) \end{matrix} ) w.r.t. the ONB (e_{1}, e_{2}, e_{3}) .

Then, as shown in 2.2.3, for every (real) direction s there are monochromatic plane waves of the forms

E_{s,ω}^{o}(x, t) ∝ (s ⊗ e_{1}) ℜ (e^{−i(ω t− \frac{ω}{c} n^{o}(ω) s·x)}) (ordinary wave)

and

E_{s,ω}^{e}(x, t) ∝ (s × (s ⊗ e_{1})) ℜ (e^{−i(ω t− \frac{ω}{c} n_{s}^{e}(ω) s·x)}) (extraordinary wave) ,

where

n^{o}(ω) =^{def} \sqrt{ϵ_{2}(ω)} (independent of s)

and^{33}

n_{s}^{e}(ω) =^{def} 1/\sqrt{(\frac{s^{1}}{n^{o}(ω)})^{2} + \frac{(s^{2})^{2} + (s^{3})^{2}}{ϵ_{1}(ω)}} .

Correspondingly, the quantized electromagnetic field inside the χ^{↔}^{(1)}-crystal is of the form

Êχ^{↔}_{(1)}(x, t) = (Ê_{o}^{(+)}(x, t) + Ê_{e}^{(+)}(x, t)) + h.c. ,

where

Ê_{o}^{(+)}(x, t) =^{def} (2π)^{− \frac{3}{2}} ∫ f_{o}(k) â_{1}(k) \underbrace{\underbrace{ϵ_{1}(k)}_{=^{def} \frac{k}{|k|} ×e_{1}} e^{−i(ω_{o}(k) t−k·x)}}_{ordinary classical wave} \frac{dV_{k}}{\sqrt{2 |k|}}

and Ê_{e}^{(+)}(x, t) =^{def} (2π)^{− \frac{3}{2}} ∫ f_{e}(k) â_{2}(k) \underbrace{\underbrace{ϵ_{2}(k)}_{=^{def} \frac{k}{|k|} ×(\frac{k}{|k|} ×e_{1})} e^{−i(ω_{e}(k) t−k·x)}}_{extraordinary classical wave} \frac{dV_{k}}{\sqrt{2 |k|}} .

Draft, November 5, 2011

33Recall that the optical axis is along e_{1} : n_{e_{1}}^{e}(ω) = n^{o}(ω) .

Of course, the Hamiltonian Ĥχ^{↔}_{(1)} has to fulfill

\frac{i}{ℏ} [Ĥχ^{↔}_{(1)} , Êχ^{↔}_{(1)}(x, t)]_{−} = \frac{∂}{∂t} Êχ^{↔}_{(1)}(x, t) .

Therefore (up to an additive constant) we have

Ĥχ^{↔}_{(1)} = ∫ (ℏ ω_{o}(k) (â_{1}(k))^{†}â_{1}(k) + ℏ ω_{e}(k) (â_{2}(k))^{†}â_{2}(k)) dV_{k} .

Correspondingly, the generator of translations, often misinterpreted^{34} as momentum observable, is

P̂χ^{↔}_{(1)} = ∫ ℏ k ((â_{1}(k))^{†}â_{1}(k) + (â_{2}(k))^{†}â_{2}(k))dV_{k} .

The amplitudes f_{o}(k) , f_{e}(k) are fixed essentially by identification of the Hamiltonian

with the quantum version of ∫ (H_{0}(x, t) + U(x, t))dV_{x} , i.e. by postulating^{35} that

∫ (ℏ ω_{o}(k) (â_{1}(k))^{†}â_{1}(k) + ℏ ω_{e}(k) (â_{2}(k))^{†}â_{2}(k)) dV_{k} = ∫ Ĥ(x, t) dV_{x} (3.58)

holds for some quantum field Ĥ(x, t) fulfilling the conditions

lim_{t→−∞} ⟨Φ | Ĥ(x, t) Φ⟩ = 0 for sufficiently well-behaved Φ ∈ H_{field}  (3.59)

and

\frac{∂}{∂t} Ĥ(x, t) = :ϵ_{0} Ê(x, t) · \frac{∂}{∂t}Ê(x, t) + \frac{1}{μ_{0}} B̂(x, t) · \frac{∂}{∂t}B̂(x, t) + Ê(x, t) · P̂(x, t): ,

(3.60)

where^{36}

P̂(x, t) =^{def} P̂^{(+)}(x, t) + (P̂^{(+)}(x, t))^{†} ,

P̂^{(+)}(x, t) =^{def} (2π)^{− \frac{3}{2}} ∫ f_{o}(k) â_{1}(k) ϵ_{0} ϵ^{↔}(ω_{o}(k))ϵ_{1}(k) e^{−i(ω_{o}(k) t−k·x)} \frac{dV_{k}}{\sqrt{2 |k|}}

+ (2π)^{− \frac{3}{2}} ∫ f_{e}(k) â_{2}(k) ϵ_{0} ϵ^{↔}(ω_{e}(k))ϵ_{2}(k) e^{−i(ω_{o}(k) t−k·x)} \frac{dV_{k}}{\sqrt{2 |k|}} .

Draft, November 5, 2011

34See (Garrison and Chiao, 2004), in this connection.

35The requirements (3.58)–(3.60), adopted to (1.15)–(1.17), guarantee that the energy expection values in coherent states coincide with the corresponding classical energies.

36Recall (2.1) and (3.6).

Straightforward calculation yields

∫ Ĥ(x, t) dV_{x} = ∫ \frac{(n^{o}(k))^{2}}{c^{2} |k| μ_{0}} |f_{o}(k)|^{2} (â_{1}(k))^{†}â_{1}(k) dV_{k}

+ ∫ \frac{(n^{e}(k))^{2}}{c^{2} |k| μ_{0}} |f_{e}(k)|^{2} (â_{2}(k))^{†}â_{2}(k) dV_{k}

(3.61)

and thus

f_{j}(k) = i c |k| \sqrt{μ_{0} ℏ c} \sqrt{\frac{ω_{j}(k)}{c |k| (n^{j}(k))^{2}}} , for j ∈ \{o , e\} ,

by (3.58), up to irrelevant phase factors.^{37}

Proof of (3.61): ....................

Final remark: One should not be surprised that the interaction with the infinitely extended medium cannot be described in the naive interaction picture:

For fixed t, unless χ^{↔} = ˆ1 , the quantum field Êχ^{↔}_{(1)}(x, t) is

not unitarily equivalent to the free field Ê(x, t) .

3.3.2 Nonlinear Perturbation

Let us now consider a finitely extended χ^{(1)}-χ^{(2)} crystal embedded^{38} in the χ^{(1)} cystal, i.e. that

χ̌^{↔}^{(1)}(x, t; x′_{1}, t′_{1}) = δ(x − x′_{1}) (2π)^{− \frac{1}{2}} ∫ ( \begin{matrix}ϵ_{1}(ω) & 0 & 0 \\ 0 & ϵ_{2}(ω) & 0 \\ 0 & 0 & ϵ_{2}(ω) \end{matrix} ) e^{−i ω(t−t′_{1})} dω_{1}

and

χ̌_{jk_{1}k_{2}}^{(2)}(x, t; x′_{1}, t′_{1}; x′_{2}, t′_{2})

= χ_{G}(x) δ(x − x′_{1}) δ(x − x′_{2}) \frac{1}{2π} ∫ χ_{jk_{1}k_{2}}^{(2)}(ω_{1}, ω_{2}) e^{−i (ω_{1}(t−t′_{1})+ω_{2}(t−t′_{2}))} dω_{1} dω_{2}

Draft, November 5, 2011

37Obviously, such phase factors could be removed by corrsponding redefinition of the annihilation operators a_{1}(k), a_{2}(k) not changing their (canonical) commutation relations.

38This way we (essentially) prohibit diffraction at the boundaries of the χ^{(1)}-χ^{(2)} crystal.

80 CHAPTER 3. NONLINEAR OPTICAL MEDIA for some bounded region G whereas

χ̌^{(ν)} = 0 ∀ ν > 2 .

The Hamiltonian of the quantized theory now is^{39}

Ĥ = Ĥχ^{↔}_{(1)} + ∫_{G} Ĥ_{int}(x, 0) dV_{x}  (3.62)

with Ĥ_{int}(x, 0) being the (normally ordered) quantum version of H_{int}(x, t) , the latter being characterized by

lim_{t→−∞} H_{int}(x, t) = 0 for sufficiently well-behaved E(x, t)  (3.63)

and

\frac{∂}{∂t} H_{int}(x, t) = E^{j}(x, t) · \frac{∂}{∂t} P_{j}^{(2)}(x, t) ,  (3.64)

where

P_{j}^{(2)}(x, ω) =_{(3.6)} \frac{ϵ_{0}}{\sqrt{2π}} ∫ χ_{jk_{1}k_{2}}^{(2)}(ω_{1}, ω_{2}) Ẽ^{k_{1}}(x, ω_{1}) Ẽ^{k_{2}}(x, ω_{2}) δ(ω_{1} + ω_{2} − ω) dω_{1} dω_{2} .

Inserting the latter into (3.64) we get

\frac{∂}{∂t} H_{int}(x, t) = −i \frac{ϵ_{0}}{(2π)^{\frac{3}{2}}} ∫ Ẽ^{j}(x, ω′) (ω − ω′) χ_{jk_{1}k_{2}}^{(2)}(ω_{1}, ω_{2}) Ẽ^{k_{1}}(x, ω_{1})

Ẽ^{k_{2}}(x, ω_{2}) δ(ω_{1} + ω_{2} − (ω − ω′)) dω_{1} dω_{2} dω′ e^{−i ω t} dω

= −i \frac{ϵ_{0}}{(2π)^{\frac{3}{2}}} ∫ (ω_{1} + ω_{2}) e^{−i (ω′+ω_{1}+ω_{2}) t} χ_{jk_{1}k_{2}}^{(2)}(ω_{1}, ω_{2})
Ẽ^{j}(x, ω′) Ẽ^{k_{1}}(x, ω_{1}) Ẽ^{k_{2}}(x, ω_{2}) dω_{1} dω_{2} dω′

and thus, thanks to (3.5),

\frac{1}{2} \frac{∂}{∂t} H_{int}(x, t) = −i \frac{ϵ_{0}}{(2π)^{\frac{3}{2}}} ∫ ω_{1} e^{−i (ω_{1}+ω_{2}+ω_{3}) t} χ_{k_{3}k_{1}k_{2}}^{(2)}(ω_{1}, ω_{2})
Ẽ^{k_{1}}(x, ω_{1}) Ẽ^{k_{2}}(x, ω_{2}) Ẽ^{k_{3}}(x, ω_{3}) dω_{1} dω_{2} dω_{3}
= −i \frac{ϵ_{0}}{(2π)^{\frac{3}{2}}} ∫ ω_{2} e^{−i (ω_{1}+ω_{2}+ω_{3}) t} χ_{k_{3}k_{1}k_{2}}^{(2)}(ω_{1}, ω_{2})
Ẽ^{k_{1}}(x, ω_{1}) Ẽ^{k_{2}}(x, ω_{2}) Ẽ^{k_{3}}(x, ω_{3}) dω_{1} dω_{2} dω_{3} .

Draft, November 5, 2011

39Recall (1.15)–(1.17).

3.3. QUANTUM ASPECTS OF THREE-WAVE MIXING 81 For situations in which also ∫ ω_{1} e^{−i (ω_{1}+ω_{2}+ω_{3}) t} χ_{k_{3}k_{1}k_{2}}^{(2)}(ω_{1}, ω_{2}) Ẽ^{k_{1}}(x, ω_{1}) Ẽ^{k_{2}}(x, ω_{2}) Ẽ^{k_{3}}(x, ω_{3}) dω_{1} dω_{2} dω_{3} = ∫ ω_{3} e^{−i (ω_{1}+ω_{2}+ω_{3}) t} χ_{k_{3}k_{1}k_{2}}^{(2)}(ω_{1}, ω_{2}) Ẽ^{k_{1}}(x, ω_{1}) Ẽ^{k_{2}}(x, ω_{2}) Ẽ^{k_{3}}(x, ω_{3}) dω_{1} dω_{2} dω_{3}

(3.65)

holds and χ_{k_{3}k_{1}k_{2}}^{(2)}(ω_{1}, ω_{2}) can be replaced by the constant tensor χ_{k_{3}k_{1}k_{2}}^{(2)} this implies

\frac{3}{2} H_{int}(x, t)

= ϵ_{0} χ_{k_{3}k_{1}k_{2}}^{(2)} ∫ e^{−i (ω_{1}+ω_{2}+ω_{3}) t} Ẽ^{k_{1}}(x, ω_{1}) Ẽ^{k_{2}}(x, ω_{2}) Ẽ^{k_{3}}(x, ω_{3}) dω_{1} dω_{2} dω_{3}
= ϵ_{0} χ_{k_{3}k_{1}k_{2}}^{(2)} E^{k_{1}}(x, t) E^{k_{2}}(x, t) E^{k_{3}}(x, t) .

Hence, if P̂_{II} is the projector onto a corresponding subspace of H_{field} , we have

P̂_{II} Ĥ_{int}(x, 0)P̂_{II} = \frac{2}{3} ϵ_{0} P̂_{II} :Êχ^{↔}^{(1)}^{k_{1}}(x, 0) Êχ^{↔}^{(1)}^{k_{2}}(x, 0) Êχ^{↔}^{(1)}^{k_{3}}(x, 0): P̂_{II}

...

3.3.3 Type-II Spontaneous Down Conversion

...

82 CHAPTER 3. NONLINEAR OPTICAL MEDIA

Photodetection

4.1 Simple Detector Models

4.1.1 Single Localized Detectors

The counting rate at time t for a photodetector localized at x in an almost monochromatic classical radiation field with complex vector potential is (the local space-time average of)

∑_{j_{1},j_{2}=1}^{3} A^{(−)}^{j_{1}}(x, t) A^{(+)}^{j_{2}}(x, t) η^{j_{1}j_{2}} ,

where the η^{j_{1}j_{2}} (depending on the mean frequency of the radiation) characterize the efficiency of the detector.^{1}

Remark: For almost monochromatic (not quantized) light with mean angular frequency ω_{0}

E(x, t) = −\frac{∂}{∂t} A(x, t)

implies:

E^{(−)}^{j_{1}}(x, t) E^{(+)}^{j_{2}}(x, t) ≈ (ω_{0})^{2} A^{(−)}^{j_{1}}(x, t) A^{(+)}^{j_{2}}(x, t) .

For quantized almost monochromatic radiation in the (pure) state ϕ one can show^{2} that the corresponding detection rate is given in first order perturbation theory the formally similar expression^{3}

P = ∑_{j_{1},j_{2}=1}^{3} ⟨ϕ | Â^{(−)}^{j_{1}}(x, t) Â^{(+)}^{j_{2}}(x, t) ϕ⟩ η^{j_{1}j_{2}} .

Draft, November 5, 2011

1For achievable efficiencies see (Jackson and Hockney, 2004).

2See Section 7.2.2.

3Hence, in this approximation, the counting rate in the vacuum state is zero.

Special cases:

1. For ϕ = χâ(α) we have

Â^{(+)}(x, t) ϕ = α A^{(+)}(x, t) ϕ

with

A^{(+)}(x, t) =^{def} ⟨Ω | Â^{(+)}(x, t) â^{†} Ω⟩

and hence

P = |α|^{2} ∑_{j_{1},j_{2}=1}^{3} A^{(−)}^{j_{1}}(x, t) A^{(+)}^{j_{2}}(x, t) η^{j_{1}j_{2}} .

2. For the (normalized) n-photon state

ϕ = \frac{1}{\sqrt{n!}} (â^{†})^{n} Ω

we have

⟨ϕ | Â^{(−)}^{j_{1}}(x, t) \underbrace{Â^{(+)}^{j_{2}}(x, t) ϕ}_{= \frac{n}{\sqrt{n!}} [A_{0}^{(+)}(x,t)j_{2}(x,t),â^{†}]_{−}(â^{†})^{n−1} Ω}⟩ = n A^{(−)}^{j_{1}}(x, t) A^{(+)}^{j_{2}}(x, t) ,

in spite of

⟨ϕ | Ê(x, t) ϕ⟩ = 0 .

Remarks:

1. The phases of the 1-photon state vectors do not play any role for photodetection!

2. Therefore^{4} the counting rates in the mixed state^{5}

\begin{aligned}ρ̂â(α) & =^{def} & \frac{1}{2π} ∫_{0}^{2π} |χâ(|α| e^{i φ})⟩⟨χâ(|α| e^{i φ})| \\ & = & ∑_{ν=0}^{∞} |\frac{α^{ν}}{ν!} |^{2} |(â^{†})^{n} Ω⟩⟨(â^{†})^{n} Ω| \end{aligned}

are the same as in the (pure) coherent state χâ(α) .

3. Usually, experimental checks of predictions for n-photon states are performed via post selection.^{6}

Draft, November 5, 2011

4See (Sanders et al., 2003) and (Nemoto and Braunstein, 2003) in this connection. 5This state corresponds fairly well to the field of a single-mode laser inside the resonant cavity

in case of high pump intensity; see (Mandel and Wolf, 1995, Sect. 18.5.2) and (Wiseman, 2004). 6Since coherent superposition with the vacuum state, at least, are unavoidable.

4.1.2 Independent Detectors

For an almost monochromatic classical radiation field with complex vector potential A^{(+)}(x, t)

\underbrace{P_{A^{(+)}}(x_{1}, t_{1}; x_{2}, t_{2}; …; x_{n}, t_{n})}_{=^{def} ∏_{ν=1}^{n} ( ∑_{j_{ν},j′_{ν}=1}^{3} Â^{(−)}^{j_{ν}}(x_{ν}, t_{ν}) Â^{(+)}^{j′_{ν}}(x_{ν}, t_{ν}) η_{ν}^{j_{ν}j′_{ν}} )} dt_{1} ⋯ dt_{n}

is the probability that for all ν ∈ {1, …, n} the detector localized at x_{ν} fires during the time interval [t_{ν}, t_{ν} + dt_{ν}] . Here η_{ν}^{j_{ν}j′_{ν}} characterizes the efficiency of the ν-th detector and we assume that the detectors have no (essential) effect on each other.

Correspondingly, for a coherent state χ this probability can be shown to be

P_{χ}(x_{1}, t_{1}; …; x_{n}, t_{n}) dt_{1} ⋯ dt_{n}

in first order perturbation theory,^{7} where

P_{χ}(x_{1}, t_{1}; …; x_{n}, t_{n})

=^{def} ∑_{j_{1},...,j′_{n}=1}^{3}⟨ϕ | Â^{(−)}^{j_{1}}(x_{1}, t_{1}) ⋯ Â^{(−)}^{j_{n}}(x_{n}, t_{n}) Â^{(+)}^{j′_{n}}(x_{n}, t_{n}) ⋯

⋯ Â^{(+)}^{j′_{1}}(x_{1}, t_{1}) ϕ⟩ η_{1}^{j_{1}j′_{1}} ⋯ η_{n}^{j_{n}j′_{n}} ∀ ϕ ∈ H_{field} .

This together with the general rule (6.11) and the optical equivalence theorem shows:

For every physically relevant state ˆ= ρ̂ of the radiation field — to first order of quantum mechanical perturbation theory — the probability that for all ν ∈ {1, …, n} the detector localized at x_{ν} fires during the time interval [t_{ν}, t_{ν} + dt_{ν}] is

Pρ̂(x_{1}, t_{1}; x_{2}, t_{2}; …; x_{n}, t_{n}) dt_{1} ⋯ dt_{n} ,

where:

Pρ̂(x_{1}, t_{1}; x_{2}, t_{2}; …; x_{n}, t_{n})

=^{def} ∑_{j_{1},...,j′_{n}=1}^{3} Tr((ρ̂Â^{(−)}^{j_{1}}(x_{1}, t_{1}) ⋯ Â^{(−)}^{j_{n}}(x_{n}, t_{n}) Â^{(+)}^{j_{n}}(x_{n}, t_{n}) ⋯
⋯ Â^{(+)}^{j_{1}}(x_{1}, t_{1})) η_{1}^{j_{1}j′_{1}} ⋯ η_{n}^{j_{n}j′_{n}} .

(4.1)

We may also consider idealized ‘distributed’ detectors^{8} which sample the field at various points of space-time and are sensitive to the resulting effective field. A

Draft, November 5, 2011

7See Section 7.2.2.

8See (Klauder and Sudarshan, 1968, p. 150).

typical example is Young’s double-slit experiment, considered in 4.3.1, where the actual detector is sensitive to the field sampled from both slits.^{9} The probability density for arrays of such detectors may be given by

Pρ̂(x_{1}, t_{1}; …; x_{n}, t_{n})

≈ ∑_{j_{1},...,j_{n},j′_{1},...,j′_{n}=1}^{3} Tr (ρ̂Â_{eff, 1}^{(−)}^{j_{1}}(x_{1}, t_{1}) ⋯ Â_{eff, n}^{(−)}^{j_{n}}(x_{n}, t_{n})Â_{eff, n}^{(+)}^{j′_{n}}(x_{n}, t_{n}) ⋯
⋯ Â_{eff, 1}^{(+)}^{j′_{1}}(x_{1}, t_{1})) η_{1}^{j_{1}j′_{1}} ⋯ η_{n}^{j_{n}j′_{n}} ,

(Â_{eff, ν}^{(−)}^{j}(x, t))^{∗} =^{def} Â_{eff, ν}^{(+)}^{j}(x, t) =^{def} ∫ Â_{0}^{(+)}^{k}(x − x′, t − t′) S_{ν}^{jk}(x′, t′) dV_{x′} dt′ ,

instead of (4.1), with suitable distributions S_{ν}^{jk}(x′, t′) .

All this suggests that, in principle, all the correlation functions

G(n,m) ; ρ̂(x_{1}, t_{1}, j_{1}; …; x_{n+m}, t_{n+m}, j_{n+m})

=^{def} Tr (ρ̂(∏_{ν=1}^{n} Â_{0}^{(−)}^{j_{ν}}(x_{ν}, t_{ν})) ∏_{μ=n+1}^{n+m} Â_{0}^{(+)}^{j_{μ}}(x_{μ}, t_{μ}))

(4.2)

with \underline{\underline{n = m}} ∈ ℕ are determined by the rate information from all possible counting arrangements.

Remark: If G(n,m) ; ρ̂ exist for all n, m ∈ ℕ — as is usually the case for physically realizable ρ̂ — then the totality of all these correlation functions characterizes ρ̂ uniquely, as may be shown by methods of axiomatic field theory (see, e.g., Corollary 2.2.16 of (Lücke, qft)). Actually, according to (1.31), every G(n,m) ; ρ̂ is already fixed by its values at t_{1} = … = t_{n+m} = 0 .

4.2 Quantum Theory of Coherence

4.2.1 Correlation Functions in General

As explained in 4.1.2, the counting rates of corresponding detector arrays for the quantum state ρ̂ of the radiation field are given (to first approximation) by the η_{ν}^{j_{ν}j′_{ν}} and the correlation functions

G(n,m) ; ρ̂(x_{1}, t_{1}, j_{1}; …; x_{n}, t_{n}, j_{n}) =^{def} Tr(ρ̂(∏_{ν=1}^{n} Â^{(−)}^{j_{ν}}(x_{ν}, t_{ν})) ∏_{μ=n+1}^{n+m} Â^{(+)}^{j_{μ}}(x_{μ}, t_{μ})) .

Draft, November 5, 2011

9See, e.g. (Ficek and Swain, 2001) and (Brukner and Zeilinger, 2002).

In the following we make use of the identification

x_{ν} ≡ (x_{ν}, t_{ν}, j_{ν}) ∈ M =^{def} ℝ^{3} × ℝ × {1, 2, 3} .

In this notation we have, e.g.,^{10}

G_{ρ̂^{I}}^{(n,m)}(x_{1}; …; x_{n+m}) = (G_{ρ̂^{I}}^{(n,m)}(x_{n+m}; …; x_{1}))^{∗} ∀ n, m ∈ ℕ , x_{1}, …, x_{n+m} ∈ M . (4.3)

Lemma 4.2.1 The density operator ρ̂^{I} on H_{field} is a coherent state,^{11} i.e. there is a mode â and a complex number α with

ρ̂^{I} = |D̂â(α) Ω⟩⟨D̂â(α) Ω| ,  (4.4)

iff there is a function V (x) on M with

G_{ρ̂^{I}}^{(n,m)}(x_{1}; …; x_{n+m}) = ∏_{ν=1}^{n}(V (x_{ν}))^{∗} ∏_{μ=n+1}^{n+m} V (x_{μ}) ∀ n, m ∈ ℕ , x_{1}, …, x_{n+m} ∈ M .

(4.5)

More precisely, (4.4) implies (4.5) for^{12}

V (x, t, j) = ⟨D̂â(α) Ω | \underbrace{Â_{0}^{(+)}^{j}(x, t) D̂â(α) Ω}_{∼_{(1.84)} D̂â(α) Ω}⟩

=_{(1.84)} α ⟨Ω | Â^{(+)}(x, t) â^{†} Ω⟩ .

(4.6)

Outline of proof: The density operator ρ̂^{I} is uniquely fixed by the correlation functions G_{ρ̂^{I}}^{(n,m)} (recall the remark at the end of Sectionnlqo-S-indDet). Since, obviously, (4.4) and (4.6) imply (4.5) this proves the statement.

The state ρ̂^{I} of the radiation field is said to have Mth-order coherence^{13} if there is a function V (x) on M with

G_{ρ̂^{I}}^{(n,n)}(x_{1}; …; x_{2n}) = ∏_{ν=1}^{n}(V (x_{ν}))^{∗} ∏_{μ=n+1}^{2n} V (x_{μ})  (4.7)

for all n ∈ {1, …, M}. It is said to have infinite-order coherence if it has Mth-order coherence for all M ∈ ℕ.

Draft, November 5, 2011

10For additional properties see (Walls and Milburn, 1995, Section 3.2).

11Recall Sections 1.2.4 and 1.2.4.

12Comparison of (4.6) with (4.9) confirms: \left⟨{ \atop n̂}\right⟩ = |α|^{2} .

13Here we use the terminology of, e.g., (Klauder and Sudarshan, 1968; Walls and Milburn, 1995; Ficek and Swain, 2001) rather than that of, e.g., (Mandel and Wolf, 1995; Hariharan and Sanders,

Remarks:

1. In a more restricted sense only the

G(n,n) ; ρ̂(\underline{x}_{1}; …; \underline{x}_{n}; \underline{x}_{n}; …; \underline{x}_{1})

are relevant. But for ‘distributed’ detectors also the G^{(n,n)} with different arguments are relevant.

2. According to (Mandel and Wolf, 1995, Section 12.5) also the G_{ρ̂^{I}}^{(n,m)} with n ≠ m are experimentally accessible, in principle, if nonlinear dielectric media are exploited in the counting arrangement.

3. Both the pure (coherent) states

e^{− \frac{1}{2} |α|^{2}} ∑_{n=0}^{∞} \frac{1}{n!} (α e^{i φ_{n}} â^{†})^{n} Ω (= χâ(α) für φ_{n} = 0)

and the mixed state

ρ̂â(|α|) = e^{−|α|^{2}} ∑_{n=0}^{∞} \frac{|α|^{2n}}{n!} |\frac{1}{\sqrt{n!}} (â^{†})^{n} Ω⟩⟨\frac{1}{\sqrt{n!}} (â^{†})^{n} Ω|

have infinite order coherence.

4. For these states the probability p(n) for exactly n photons ‘being present’ is given by a Poisson distribution:

p(n) = \frac{⟨n⟩}{n!} e^{−⟨n⟩}
⟨n⟩ = |α|^{2}  (expectation value for the photon number) .

4.2.2 Correlation Functions of Single-Mode States

Lemma 4.2.2 For every single-mode state, i.e. every ρ̂^{I} fulfilling^{14}

ρ̂^{I} = P̂_{Hâ} ρ̂^{I} P̂_{Hâ}

for some mode â (as considered in 1.2.4),

G_{ρ̂^{I}}^{(n,m)}(x_{1}; …; x_{n+m}) = g_{nm} ∏_{ν=1}^{n}(V (x_{ν}))^{∗} ∏_{μ=n+1}^{n+m} V (x_{μ}) ∀ n, m ∈ ℕ , x_{1}, …, x_{n+m} ∈ M

(4.8)

Draft, November 5, 2011

14As usual, we denote by P̂_{Hâ} the orthogonal projection onto the subspace Hâ. The subspace Hâ was defined below Equation (1.82).

4.2. QUANTUM THEORY OF COHERENCE 89

holds with

V (x, t, j) = \sqrt{\left⟨{ \atop n̂}\right⟩} [Â_{0}^{(+)}^{j}(x, t) , â^{†}]_{−} , \left⟨{ \atop n̂}\right⟩ =^{def} Tr (ρ̂^{I} â^{†}â) .  (4.9)

and

g_{nm} =^{def} \begin{matrix}Tr & (ρ̂^{I} (â^{†})^{n} â^{m}) \\ \left⟨{ \atop n̂}\right⟩ & (n+m)/2 \end{matrix} ∀ n, m ∈ ℕ .  (4.10)

Outline of proof:

Tr (ρ̂^{I} ∏_{ν=1}^{n} Â_{0}^{(−)}^{j_{ν}} (x_{ν}, t_{ν}) ∏_{μ=n+1}^{n+m} Â_{0}^{(+)}^{j_{μ}}(x_{μ}, t_{μ}))

= ∑_{γ,ρ=0}^{∞} \frac{1}{γ! ρ!} ⟨(â^{†})^{γ} Ω|ρ̂^{I} (â^{†})^{ρ} Ω⟩⟨(â^{†})^{ρ} Ω| ∏_{ν=1}^{n} Â_{0}^{(−)}^{j_{ν}} (x_{ν}, t_{ν})·

· ∏_{μ=n+2}^{n+m} Â_{0}^{(+)}^{j_{μ}}(x_{μ}, t_{μ}) \underbrace{Â_{0}^{(+)}^{j_{n+1}}(x_{n+1}, t_{n+1}) (â^{†})^{γ} Ω}_{=_{(1.82),(1.37)}\left⟨{ \atop n̂}\right⟩^{− \frac{1}{2}} V (x_{n+1}) γ (â^{†})^{γ−1}Ω}⟩

= \frac{1}{\left⟨{ \atop n̂}\right⟩^{\frac{n+m}{2}}} ∑_{ρ=n}^{∞} ∑_{γ=m}^{∞} \frac{1}{(ρ − n)! (γ − m)!} ∏_{ν=1}^{n}(V (x_{ν}))^{∗} ∏_{μ=n+1}^{n+m} V (x_{μ}) ⟨(â^{†})^{γ} Ω|ρ̂^{I} (â^{†})^{ρ} Ω⟩

\underbrace{⟨(â^{†})^{ρ−n} Ω | (â^{†})^{γ−m} Ω⟩}_{=(ρ−n)! δ_{ρ,γ+n−m}}

= \frac{1}{\left⟨{ \atop n̂}\right⟩^{\frac{n+m}{2}}} ∏_{ν=1}^{n}(V (x_{ν}))^{∗} ∏_{μ=n+1}^{n+m} V (x_{μ}) \underbrace{∑_{γ=0}^{∞} \frac{1}{γ!} ⟨(â^{†})^{γ+m} Ω|ρ̂^{I} (â^{†})^{γ+n} Ω⟩}_{=Tr(â^{m} ρ̂^{I} (â^{†})^{n})} .

Theorem 4.2.3 For every state ρ̂^{I} of the radiation field the following three statements are equivalent:^{15}

1. |G_{ρ̂^{I}}^{(1,1)}(x_{1}; x_{2})|^{2} = G_{ρ̂^{I}}^{(1,1)}(x_{1}; x_{1}) G_{ρ̂^{I}}^{(1,1)}(x_{2}; x_{2}) ∀ x_{1}, x_{2} ∈ M.

2. ρ̂^{I} has 1st-order coherence.

3. ρ̂^{I} is a single-mode state.

Outline of proof:^{16} Assume the first statement to be valid and G_{ρ̂^{I}}^{(1,1)}(x, y) sufficiently well-behaved.^{17} Then, using the shorthand notation

G(x, y) =^{def} G_{ρ̂^{I}}^{(1,1)}(x, y) ,

Draft, November 5, 2011

15In this context it is essential that, for n = 1 , (4.7) is an exact equality. Unfortunately this can never be checked experimentally.

16We essentially follow (Klauder and Sudarshan, 1968, Section 8–2 B). 17Otherwise we could consider suitably ‘smeared’ versions of G_{ρ̂^{I}}^{(1,1)}(x, y) ; see (4.17) and (4.18),

below.

90 CHAPTER 4. PHOTODETECTION

we have

|G(x, y)| = \sqrt{G(x, x)} \sqrt{G(y, y)}

and hence

G(x, y) = \underbrace{r(x)}_{≥0} r(y) e^{i φ(x,y)} ∀ x, y ∈ ℝ^{3} × ℝ × \{1, 2, 3\}  (4.11)

for some real-valued function φ(x, y) , where

r(x) =^{def} \sqrt{G(x, x)} .

Since

G(x, y) = (G(y, x))^{∗} ,

φ has to fulfill

φ(x, y) = −φ(y, x) ∈ ℝ ∀ x, y ∈ ℝ^{3} × ℝ × {1, 2, 3} . (4.12)

Consider arbitrary x_{1}, x_{2}, x_{3} ∈ ℝ^{3} × ℝ × {1, 2, 3} fulfilling

r(x_{j}) ≠ 0 ∀ j ∈ {1, 2, 3}  (4.13)

and define

\begin{matrix}M_{k}^{j} & =^{def} & e^{i φ(x_{j},x_{k})} ∀ j, k ∈ \{1, 2, 3\} , \\ 𝕄 & =^{def} & ( M_{k}^{j}) . \end{matrix}  (4.14)

Then (4.12) implies

𝕄 = 𝕄^{∗}  (4.15)

and hence

𝕄 = ( \begin{matrix}1 & M_{2}^{1} & M_{3}^{1} \\ (M_{2}^{1})^{∗} & 1 & M_{3}^{2} \\ (M_{3}^{1})^{∗} & (M_{3}^{2})^{∗} & 1 \end{matrix} ) .

Therefore we have

\frac{1}{2} det(𝕄) = ℜ(M_{2}^{1}M_{3}^{2}M_{1}^{3}) − 1

and, consequently,

det(𝕄) ≥ 0 ⟹ M_{2}^{1}M_{3}^{2}M_{1}^{3} = 1 .  (4.16)

Another consequence of (4.14) is

\begin{aligned}∑_{j,k=1}^{3}((r(x_{j}) ζ_{j}))^{∗} (r(x_{k}) ζ_{k})M_{k}^{j} & = & ∑_{j,k=1}^{3} (ζ_{j})^{∗} ζ_{k} G(x_{j}, x_{k}) \\ & ≥ & 0 ∀ ζ_{1}, ζ_{,}ζ_{3} ∈ ℂ , \end{aligned}

which, together with (4.13) and(4.15), implies

det(𝕄) ≥ 0 .

The latter, together with (4.16) implies

M_{2}^{1}M_{3}^{2}M_{1}^{3} = 1

and hence

φ(x_{1}, x_{2}) + φ(x_{2}, x_{3}) + φ(x_{3}, x_{1}) = 0 ,

4.2. QUANTUM THEORY OF COHERENCE 91

by (4.14). This, together with (4.12) and (4.14), shows that

𝕄^{2} = 3 𝕄 .

Therefore, by (4.15), 𝕄/3 is a projection operator, i.e.

M_{k}^{j} = A^{j} (A^{k})^{∗} ∀ j, k ∈ {1, 2, 3}

holds for suitable A^{1}, A^{2}, A^{3} ∈ ℂ with |A^{1}|^{2} + |A^{2}|^{2} + |A^{3}|^{2} = 3 . Since

M_{j}^{j} = 1 ∀ j ∈ {1, 2, 3} ,

we have

M_{k}^{j} = e^{i φ_{j}} e^{−i φ_{k}}

for suitable φ_{1}, φ_{2}, φ_{3} ∈ ℝ. Together with (4.14) this implies

φ(x_{j}, x_{k}) = φ_{j} − φ_{k} ∀ j, k ∈ {1, 2, 3}

and hence

r(x) r(y) ≠ 0 ⟹ φ(x, y) = φ(x) − φ(y) ∀ x, y ∈ ℝ^{3} × ℝ × {1, 2, 3} for some real-valued φ(x) . This, together with (4.11), finally implies

G(x, y) = (V (x))^{∗} V (y) ∀ x, y ∈ ℝ^{3} × ℝ × {1, 2, 3}

with

V (x) =^{def} r(x) e^{i φ(x)} ∀ x ∈ ℝ^{3} × ℝ × {1, 2, 3} .

This proves, that the first statement implies the second one.

Now assume that the second statement is valid. Then, since^{18}

â_{g} =^{def} ∫ (g(x))^{∗} · Â_{0}^{(+)}(x, 0) dV_{x} (4.17)
=_{(1.30)} \sqrt{μ_{0} ℏ c} ∑_{j=1}^{2}(g(k))^{∗} · ϵ_{j}(k) â_{j}(k) \frac{dV_{k}}{\sqrt{|k|}}

shows that all modes (1.58) are of the form â_{g} , there is a sequence of complex numbers z_{1}, z_{2}, … with

Tr(ρ̂^{I} â_{ν}^{†} â_{μ}) = (z_{ν})^{∗} z_{μ} ∀ ν, μ ∈ ℤ .  (4.18)

We need only consider the physically relevant case

0 < ∑_{ν=1}^{∞} |z_{ν}|^{2} < ∞ .

Then the definition

b̂ =^{def} \begin{matrix}∑ & {∞ \atop μ=1} & (z_{ν}) & ∗ & â_{ν} \\ √ & \overline{∑_{ν=1}^{∞} |z_{ν}|^{2}} \end{matrix}  (4.19)

Draft, November 5, 2011

18We denote by g(k) the Fourier transform of g(x) : g(k) =^{def} (2π)^{−3/2} ∫ g(x) e^{−i x·k} dV_{x} ∀ k ∈ ℝ^{3} .

92 CHAPTER 4. PHOTODETECTION

is allowed and specifies a mode b̂ (|b̂^{†} Ω| = 1). We may choose a complete system of modes {b̂_{ν} : ν ∈ ℕ} (of type (1.58)) with

b̂_{1} = b̂ .  (4.20)

Then

b̂_{μ} = ∑_{ν=1}^{∞} u_{μν} â_{ν} ∀ μ ∈ ℕ  (4.21)

holds for suitable complex u_{μ}ν fulfilling

∑_{λ=1}^{∞}(u_{μλ})^{∗} u_{νλ} = δ_{μν} ∀ μ, ν ∈ ℕ .  (4.22)

(4.19)–(4.21) imply

u_{1ν} = \begin{matrix}(z_{ν}) & ∗ \\ √ & \overline{∑_{ν=1}^{∞} |z_{ν}|^{2}} \end{matrix} ∀ ν ∈ ℕ .  (4.23)

We conclude that

\begin{aligned}Tr(ρ̂^{I} b̂_{λ}^{†} b̂_{λ}) & =_{(4.21),(4.18)} & ∑_{ν,μ=1}^{∞}(u_{λμ} z_{μ})^{∗} (u_{λν} z_{ν}) \\ & =_{(4.23)} & \underbrace{∑_{ν,μ=1}^{∞} (u_{λμ})^{∗} u_{1μ} u_{λ_{ν}} (u_{1ν})^{∗}}_{=_{(4.23)} δ_{1λ}} ∑_{ν=1}^{∞} |z_{ν}|^{2} \end{aligned}

and hence

Tr(ρ̂^{I} b̂_{λ}^{†} b̂_{λ}) = 0 ∀ λ > 1 .

This shows that ρ̂^{I} characterizes a single mode state with mode b̂_{1} .

That the third statement implies the first one is a direct consequence of 4.2.2.

A direct consequence of Lemma 4.2.2 and (1.63) is:

g_{νν} = \underbrace{\frac{n (n − 1) ⋯ (n − (ν − 1))}{n^{ν}}}_{=0 for ν>n} for single-mode n-photon states with n ≠ 0 .

This especially shows:

Single-mode n-photon states with n ≠ 0 have coherence of only first order.^{19}

Draft, November 5, 2011

19Since for these states we have g_{11} = 1 ≠ \frac{n − 1}{n} = g_{22} .

4.2. QUANTUM THEORY OF COHERENCE 93 Lemma 4.2.2 also shows that^{20}

g_{nn} = n! for ρ̂ ∝ e^{−β â^{†} â} , β > 0 .

Thus:

Also the single-mode thermal states have coherence of only first order. Corollary 4.2.4 The state ρ̂^{I} of the radiation field has infinite-order coherence iff

|G_{ρ̂^{I}}^{(n,n)}(x_{1}; …; x_{2n})|^{2} = ∏_{μ=1}^{2n} G_{ρ̂^{I}}^{(1,1)}(x_{μ}; x_{μ}) ∀ n ∈ ℕ , x_{1}, …, x_{n+m} ∈ M .

Corollary 4.2.5 If ρ̂^{I} is a state describing photons of the single mode â then ρ̂^{I} has infinite-order coherence iff^{21}

p_{}_{ρ̂^{I}}(m) = \frac{\left⟨{ \atop n̂}\right⟩^{m}}{m!} e^{−\left⟨{ \atop n̂}\right⟩} ∀ m ∈ ℤ_{+} ,  (4.24)

where

p_{}_{ρ̂^{I}}(m) =^{def} Tr (ρ̂^{I} \frac{1}{m!} |(â^{†})^{m} Ω⟩⟨(â^{†})^{m} Ω|)  (4.25)

Draft, November 5, 2011

20Since Tr(e^{−β â^{†} â}(â^{†})^{n}â^{n})

\begin{matrix}=_{(1.63)} & Tr((e^{−β â^{†} â} (â^{†} â)(â^{†} â − 1) … (â^{†} â − (n − 1))) \\ =_{(1.64)} & ∑_{μ=0}^{∞} \frac{1}{μ!} ⟨(â^{†})^{μ} Ω | e^{−β â^{†} â} (â^{†} â)(â^{†} â − 1) … (â^{†} â − (n − 1) (â^{†})^{μ} Ω⟩ \\ = & ∑_{μ=0}^{∞} λ^{μ} μ (μ − 1) … (μ − (n − 1)) \\ = & λ^{n} (\frac{d}{dλ})^{n} \underbrace{∑_{μ=0}^{∞} λ^{μ}}_{=1/(1−λ)} \\ = & λ^{n} \frac{n!}{(1 − λ)^{n+1}} \\ = & \frac{n! e^{−n β}}{(1 − e^{−β})^{n+1}} , \end{matrix}

and, therefore, especially

Tr (e^{−β â^{†} â}) = \frac{1}{1 − e^{−β}} , Tr (e^{−β â^{†} â} N̂) = \frac{e^{−β}}{(1 − e^{−β})^{2}} .

21In other words: m ↦ p_{}_{ρ̂^{I}}(m) is a Poisson distribution.

94 CHAPTER 4. PHOTODETECTION denotes the probability for exactly m photons ‘being present’ in this state and

\left⟨{ \atop n̂}\right⟩ =^{def} Tr (ρ̂^{I} â^{†}â) .

Outline of proof: From the general relations

∑_{ν=0}^{∞} (1 + λ)^{ν} p_{}_{ρ̂^{I}}(ν)  =  ∑_{ν=0}^{∞} (1 + λ)^{ν} \frac{1}{ν!} ⟨(â^{†})^{ν} Ω | ρ̂^{I} (â^{†})^{ν} Ω⟩
=  ∑_{ν=0}^{∞} \frac{1}{ν!} ⟨(â^{†})^{ν} Ω | ρ̂^{I} e^{ν ln(1+λ)} (â^{†})^{ν} Ω⟩
=  ∑_{ν=0}^{∞} \frac{1}{ν!} ⟨(â^{†})^{ν} Ω | ρ̂^{I} e^{ln(1+λ) â^{†}â} (â^{†})^{ν} Ω⟩
=  Tr (ρ̂^{I} e^{ln(1+λ) â^{†}â})
=_{Lemma ??} Tr (ρ̂^{I} :e^{λ â^{†}â}:)  (4.26)

and

p_{}_{ρ̂^{I}}(m) = \frac{1}{m!} ((\frac{d}{dλ})^{m} ∑_{ν=0}^{∞} (1 + λ)^{ν} p_{}_{ρ̂^{I}}(ν))_{|_{λ=−1}}

we conclude that

p_{}_{ρ̂^{I}}(m) = \frac{1}{m!} (\frac{d}{dλ})^{m} ∑_{ν=0}^{∞} \frac{λ^{ν}}{ν!} Tr (ρ̂^{I} (â^{†})^{ν} â^{ν})_{|_{λ=−1}} ∀ m ∈ ℤ_{+} .  (4.27)

If ρ̂^{I} has infinite-order coherence, then (4.27) and Lemma 4.2.2 imply (4.24). Conversely, if (4.24) holds then

Tr (ρ̂^{I} (â^{†})^{m} â^{m}) = ((\frac{d}{dλ})^{m} Tr (ρ̂^{I} :e^{λ â^{†}â}:))_{|_{λ=0}}
=_{(4.26)} ((\frac{d}{dλ})^{m} ∑_{ν=0}^{∞} (1 + λ)^{ν} p_{}_{ρ̂^{I}}(ν))_{|_{λ=0}}
=_{(4.24)} ((\frac{d}{dλ})^{m} ∑_{ν=0}^{∞} (1 + λ)^{ν} \frac{\left⟨{ \atop n̂}\right⟩^{ν}}{ν!} )_{|_{λ=0}} e^{−\left⟨{ \atop n̂}\right⟩}
= ((\frac{d}{dλ})^{m} e^{λ\left⟨{ \atop n̂}\right⟩})_{|_{λ=0}}
= \left⟨{ \atop n̂}\right⟩^{m}

together with Lemma 4.2.2 implies that ρ̂^{I} has infinite-order coherence.

4.3 Simple Interference Experiments

4.3.1 Young’s Double-Slit Experiment

For the ‘distributed’ detector of Young’s double-slit experiment

[figure: .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; x_{1}; x_{2}; x; P_{0}; R_{1}; R_{2}; double slit; screen]

the first order correlation function G(1,1) ; ρ̂_{in}(\underline{x}_{1}; \underline{x}_{2}) of the quantized radiation field before the apertures determines the counting rates.

One can show^{22} that the first order degree of coherence

g(1) ; ρ̂_{in}(\underline{x}_{1}; \underline{x}_{2}) =^{def} \frac{G(1,1) ; ρ̂_{in}(\underline{x}_{1}; \underline{x}_{2})}{\sqrt[+]{G(1,1) ; ρ̂_{in}(\underline{x}_{1}; \underline{x}_{1}) G(1,1) ; ρ̂_{in}(\underline{x}_{2}; \underline{x}_{2})}}

determines the visibility

V =^{def} \frac{⟨I⟩_{max} − ⟨I⟩_{min}}{⟨I⟩_{max} + ⟨I⟩_{min}} .

For instance, we have

V(x, t) ≈ |g(1) ; ρ̂_{in}(x_{1}, t − \frac{R_{1}}{c} , 3 ; x_{2}, t − \frac{R_{2}}{c} , 3)| for vertical polarization .

4.3.2 Hanbury-Brown-Twiss Effect

For the so-called Hanbury-Brown-Twiss effect (interference of photocurrents) the second order correlation function

G(2,2) ; ρ̂_{in}(\underline{x}_{1}; …; \underline{x}_{4}) mit (\underline{x}_{3}; \underline{x}_{4}) = (\underline{x}_{2}; \underline{x}_{1})

is the relevant entity. Note that, for independent simple detectors,

Pρ̂_{in}(x_{1}, t_{1}; x_{2}, t_{2}) ∝ \{ \begin{matrix}product of the photocurrents \\ of both detectors \end{matrix}

Draft, November 5, 2011

22See, e.g. (Mandel and Wolf, 1995, Sect. 4.3.1).

96 CHAPTER 4. PHOTODETECTION if

η_{ν}^{j_{ν}j′_{ν}} ∝ δ_{j_{ν}j′_{ν}} ∀ ν ∈ {1, 2} , j_{ν}, j′_{ν} ∈ {1, 2, 3} .

Especially for coherent states of the form

χ = e^{− \frac{|α|^{2}+|β|^{2}}{2}} e^{α â^{†}+β b̂^{†}}Ω
= (e^{− \frac{|α|^{2}}{2}} e^{α â^{†}}Ω) ⊗ (e^{− \frac{|β|^{2}}{2}} e^{β b̂^{†}}Ω)
ˆ= double light source

with â ⊥ b̂ we have

\begin{aligned}Â^{(+)}^{j}(x, t) χ & = & ⟨Ω | Â^{(+)}^{j}(x, t) (α â^{†} + β b̂^{†}) Ω⟩ χ \\ & = & (A_{a}^{(+)}^{j}(x, t) + A_{b}^{(+)}^{j}(x, t)) χ \end{aligned}

where

A_{a}^{(+)}^{j}(x, t) =^{def} ⟨Ω | Â^{(+)}^{j}(x, t) α â^{†} Ω⟩ ,
A_{b}^{(+)}^{j}(x, t) =^{def} ⟨Ω | Â^{(+)}^{j}(x, t) β b̂^{†} Ω⟩ .

If

\begin{aligned}A_{a}^{(+)}^{j}(x, t) & ≈ & \underbrace{A_{a}^{j}}_{>0} e^{−i(ω t−k_{a}·x+φ_{a})} , \\ A_{b}^{(+)}^{j}(x, t) & ≈ & \underbrace{A_{b}^{j}}_{>0} e^{−i(ω t−k_{b}·x+φ_{b})} \end{aligned}

in the considered space-time region then this implies

G(2,2) ; ρ̂_{in}(\underline{x}_{1}; \underline{x}_{2}; \underline{x}_{2}; \underline{x}_{1}) = |A_{a}^{j_{1}} e^{−i(ω t_{1}−k_{a}·x_{1}+φ_{a})} + A_{b}^{j_{1}} e^{−i(ω t_{1}−k_{b}·x_{1}+φ_{b})}| ×
× |A_{a}^{j_{2}} e^{−i(ω t_{2}−k_{a}·x_{2}+φ_{a})} + A_{b}^{j_{2}} e^{−i(ω t_{2}−k_{b}·x_{2}+φ_{b})}|
= (|A_{a}^{j_{1}} |^{2} + |A_{b}^{j_{1}} |^{2} + A_{a}^{j_{1}} A_{b}^{j_{1}} (e^{−i(φ_{a}−φ_{b})} e^{(k_{a}−k_{b})·x_{1}} + c.c.))×
× (|A_{a}^{j_{2}} |^{2} + |A_{b}^{j_{2}} |^{2} + A_{a}^{j_{2}} A_{b}^{j_{2}} (e^{−i(φ_{a}−φ_{b})} e^{(k_{a}−k_{b})·x_{2}} + c.c.))

and hence

⟨G(2,2) ; ρ̂_{in}(\underline{x}_{1}; \underline{x}_{2}; \underline{x}_{2}; \underline{x}_{1})⟩ = (|A_{a}^{j_{1}}|^{2} + |A_{b}^{j_{1}} |^{2}) (|A_{a}^{j_{2}}|^{2} + |A_{b}^{j_{2}} |^{2})

+2 A_{a}^{j_{1}} A_{a}^{j_{2}} A_{b}^{j_{1}} A_{b}^{j_{2}} \underbrace{cos((k_{a} − k_{b}) · (x_{1} − x_{2}))}_{interference term!} ,

where by ⟨ ⟩ we indicate averaging over one (or both) of the angles φ_{a}, φ_{b}. Hence for

ρ̂ = \frac{e^{−|α|^{2}−|β|^{2}}}{(2π)^{2}} ∫_{φ_{a}=0}^{2π} ∫_{φ_{b}=0}^{2π} |e^{α e^{i φa} â^{†}+β e^{i φ_{b}} b̂^{†}}Ω⟩⟨e^{α e^{i φa} â^{†}+β e^{i φ_{b}} b̂^{†}}Ω| dφ_{a} dφ_{b}

4.3. SIMPLE INTERFERENCE EXPERIMENTS 97

we have Pρ̂(x_{1}, t_{1}; x_{2}, t_{2}) ≈ C_{1} + C_{2} cos((k_{a} − k_{b}) · (x_{1} − x_{2}))

with suitable constants C_{1}, C_{2} . The interference term may be used to determine the angular distance of distant pairs of corresponding light sources:

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . . . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . .

φ φ

x_{1} x_{2} k_{a} k_{b} k_{a} k_{b}

P_{0}

|k_{a}| = |k_{b}| = \frac{ω}{c} ,
∠ (k_{a} − k_{b}, x_{1} − x_{2}) ≪ 1 ,
\underline{\underline{(k_{a} − k_{b}) · (x_{1} − x_{2})}} ≈ |k_{a} − k_{b}| |x_{1} − x_{2}|
≈ \underline{\underline{φ c ω |x_{1} − x_{2}|}} .

‘Corresponding light sources’ are, e.g.:

• Almost monochromatic lasers with stochastic phase (ˆ= ρ̂â(|α|), ρ̂b̂(|β|)).

• Almost monochromatic thermal (single-mode) light sources (tar light).

Note that for ‘corresponding light sources’

G(1,1) ; ρ̂(\underline{x}; \underline{x}) = konstant

holds (within the considered space-time region) and hence the (spatial) interference is determined by the second order degree of coherence^{23}

g(2) ; ρ̂(\underline{x}_{1}; \underline{x}_{2}) =^{def} \frac{G(2,2) ; ρ̂(\underline{x}_{1}; \underline{x}_{2}; \underline{x}_{2}; \underline{x}_{1})}{G(1,1) ; ρ̂(\underline{x}_{1}; \underline{x}_{1}) G(1,1) ; ρ̂(\underline{x}_{2}; \underline{x}_{2})} .

Draft, November 5, 2011

23For the relation to g(1) ; ρ̂ for stationary polarized light see (Mandel and Wolf, 1995, p. 708).

98 CHAPTER 4. PHOTODETECTION

Applications of Non-Classical Light

5.1 Criteria for Non-Classical Light

According to Glauber a state ρ̂ of the radiation field is called classical if in Theorem 1.2.2 the ρ_{N} can be chosen nonnegative.^{1}

The optical equivalence theorem explains why interference experiments with typical interference pattern — as in Young’s double-slit experiment or the Hanbury-Brown-Twiss effect — do not show any striking difference between classical and non-classical light.

Inequalities, however, which are valid for classical light may be violated by nonclassical light, since inequalities are not invariant under averaging with non-positive weights.

For states ρ̂_{Fock} with strictly bounded photon number this is quite obvious, since they fulfill the condition

G(n,n) ; ρ̂_{Fock} ≡ 0 for sufficiently large n ,  (5.1)

whereas for classical light we always have

∑_{j_{1},...,j_{n}=1}^{3} ∫ G(n,n) ; ρ̂(\underline{x}_{1}; …; \underline{x}_{n}) dx_{1} ⋯ dx_{n} > 0 .

Unfortunately, preparation of states with strictly bounded photon number is impossible and, moreover, (5.1) cannot be rigorously checked.

Better suited for experimental checks is the inequality

|G(2,2) ; ρ̂(\underline{x}_{1}; \underline{x}_{2}; \underline{x}_{2}; \underline{x}_{1})|^{2} ≤ G(2,2) ; ρ̂(\underline{x}_{1}; \underline{x}_{1}; \underline{x}_{1}; \underline{x}_{1}) G(2,2) ; ρ̂(\underline{x}_{2}; \underline{x}_{2}; \underline{x}_{2}; \underline{x}_{2})
for classical states ρ̂ ∈ B(H_{field}) .

(5.2)

Draft, November 5, 2011

1See (Johansen, 2004) for critical remarks on Glauber’s classicality criterion. See also (Miranowicz et al., 2010).

100 CHAPTER 5. APPLICATIONS OF NON-CLASSICAL LIGHT

Outline of proof for (5.2): Let {â_{1}, â_{2}, …} be a maximal family of pairwise orthogonal modes and define

\begin{matrix}G_{α_{1},...,α_{N}}^{(n,m)} (\underline{x}_{1}; …; \underline{x}_{n+m}) \\ =^{def} Tr (|χ_{α_{1},...,α_{N}} ⟩⟨χ_{α_{1},...,α_{N}} | (∏_{ν=1}^{n} Â^{(−)}^{j_{ν}} (x_{ν}, t_{ν})) ∏_{μ=n−1}^{n+m} Â^{(+)}^{j_{μ}}(x_{μ}, t_{μ})) . \end{matrix}

Then

G_{α_{1},...,α_{N}}^{(2,2)} (\underline{x}_{1}; \underline{x}_{2}; \underline{x}_{2}; \underline{x}_{1}) = G_{α_{1},...,α_{N}}^{(1,1)} (\underline{x}_{1}; \underline{x}_{1}) \underbrace{G_{α_{1},...,α_{N}}^{(1,1)} (\underline{x}_{2}; \underline{x}_{2})}_{≥0} .

For classical states this implies

|G(2,2) ; ρ̂(\underline{x}_{1}; \underline{x}_{2}; \underline{x}_{2}; \underline{x}_{1})|^{2}
=  lim_{N→∞}|∫ \underbrace{ρ_{N}(ξ_{1}, …, η_{N})}_{≥0} G_{ξ_{1}+i η_{1},...,ξ_{N} +i η_{N}}^{(2,2)} (\underline{x}_{1}; \underline{x}_{2}; \underline{x}_{2}; \underline{x}_{1}) \frac{dξ_{1} dη_{1}}{π} ⋯ \frac{dξ_{N} dη_{N}}{π} |^{2}
=  lim_{N→∞}|∫ ( ^{+} \sqrt{ρ_{N}(ξ_{1}, …, η_{N})} G_{ξ_{1}+i η_{1},...,ξ_{N} +i η_{N}}^{(1,1)} (\underline{x}_{1}; \underline{x}_{1}))^{∗} ×

× ^{+} \sqrt{ρ_{N}(ξ_{1}, …, η_{N})} G_{ξ_{1}+i η_{1},...,ξ_{N} +i η_{N}}^{(1,1)} (\underline{x}_{2}; \underline{x}_{2}) \frac{dξ_{1} dη_{1}}{π} ⋯ \frac{dξ_{N} dη_{N}}{π} |^{2}

≤_{Schwarz} lim_{N→∞} ∫ ρ_{N}(ξ_{1}, …, η_{N}) \underbrace{|G_{ξ_{1}+i η_{1},...,ξ_{N} +i η_{N}}^{(1,1)} (\underline{x}_{1}; \underline{x}_{1})|^{2}}_{=G_{ξ_{1}+i η_{1},...,ξ_{N} +i η_{N}}^{(2,2)} (\underline{x}_{1};\underline{x}_{1};\underline{x}_{1};\underline{x}_{1})} \frac{dξ_{1} dη_{1}}{π} ⋯ \frac{dξ_{N} dη_{N}}{π} ×

× ∫ ρ_{N}(ξ_{1}, …, η_{N}) \underbrace{|G_{ξ_{1}+i η_{1},...,ξ_{N} +i η_{N}}^{(1,1)} (\underline{x}_{2}; \underline{x}_{2})|^{2}}_{=G_{ξ_{1}+i η_{1},...,ξ_{N} +i η_{N}}^{(2,2)} (\underline{x}_{2};\underline{x}_{2};\underline{x}_{2};\underline{x}_{2})} \frac{dξ_{1} dη_{1}}{π} ⋯ \frac{dξ_{N} dη_{N}}{π}

=  G(2,2) ; ρ̂(\underline{x}_{1}; \underline{x}_{1}; \underline{x}_{1}; \underline{x}_{1}) G(2,2) ; ρ̂(\underline{x}_{2}; \underline{x}_{2}; \underline{x}_{2}; \underline{x}_{2}) .

Definition 5.1.1 Let G = O × (τ_{1}, τ_{2}) be an open space-time region and let ρ̂^{I} be a density operator on H_{field} . Then ρ̂^{I} is called stationary in G iff^{2}

G_{ρ̂^{I}}^{(n,m)}(x_{1}; …; x_{n+m}) = G_{ρ̂^{I}}^{(n,m)}(x_{1} + τ; …; x_{n+m} + τ) ,

where

x_{ν} + τ =^{def} (x_{ν}, j_{ν}, t_{ν} + τ) ,

holds for all n, m ∈ ℕ and for all x_{1}, …, x_{n+m} ∈ M and τ ∈ ℝ with

x_{1}, …, x_{n+m}, x_{1} + τ, …, x_{n+m} + τ ∈ G .

Remark: of course, also for stationary states there are stochastic fluctuations.

Draft, November 5, 2011

2It is stationary in all of space-time iff e^{− \frac{i}{ℏ} Ĥ_{field} τ} ρ̂^{I} e^{+ \frac{i}{ℏ} Ĥ_{field} τ} = ρ̂^{I} ∀ τ ∈ ℝ.

Corollary 5.1.2 Let G be as in Definition 5.1.1 and let ρ̂ be a classical state of the radiation field that is stationary in G. Then

G(2,2) ; ρ̂(x; x + τ; x + τ; x) ≤ G(2,2) ; ρ̂(x; x; x; x)  (5.3)

holds for all x ∈ M and all τ ∈ ℝ with x, x + τ ∈ G .

Outline of proof:

\begin{aligned}(\underbrace{G(2,2) ; ρ̂(\underline{x}; \underline{x}; \underline{x}; \underline{x})}_{=(\underbrace{G(1,1) ; ρ̂(\underline{x}; \underline{x})}_{≥0})^{2}})^{2} & =_{stat.} & G(2,2) ; ρ̂(\underline{x}; \underline{x}; \underline{x}; \underline{x}) G(2,2) ; ρ̂(\underline{x} + τ; \underline{x} + τ; \underline{x} + τ; \underline{x} + τ) \\ & ≥_{(5.2)} & |\underbrace{G(2,2) ; ρ̂(\underline{x}; \underline{x} + τ; \underline{x} + τ; \underline{x})}_{≥0}|^{2} . \end{aligned}

While bunching

G(2,2) ; ρ̂(\underline{x}; \underline{x} + τ; \underline{x} + τ; \underline{x}) < G(2,2) ; ρ̂(\underline{x}; \underline{x}; \underline{x}; \underline{x}) ,

is possible for stationary classical light Corollary 5.1.2 shows that anti-bunching

G(2,2) ; ρ̂(\underline{x}; \underline{x} + τ; \underline{x} + τ; \underline{x}) > G(2,2) ; ρ̂(\underline{x}; \underline{x}; \underline{x}; \underline{x}) ,

is not possible for stationary classical light!

Bunching is typical for thermal light:

If a detector fires then with high probability the intensity of the radiation field is considerable which — for coherent states — implies high probability for detection of additional photons at that instant. Therefore, strong stochastic fluctuations of the momentary coherent state of classical radiation support bunching, i.e.:

The probability for detecting a second photon almost immediately after a first one decreases with increasing time delay.

Anti-bunching is to be expected if one can arrange that single photons arrive in (essentially) fixed time intervals. This could be observed for resonance fluorescence.^{3}

Draft, November 5, 2011

3See (Mandel and Wolf, 1995, Sect. 15.6.5) and (Kurtsiefer et al., 2000).

A typical experiment of this kind is described, e.g., in (Carmichael, 2001):

[figure: .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; atoms; laser beam; ρ̂_{in}; ρ̂_{out}; non-classical; cavity]

Tuning of laser light and cavity to a certain atomic transition makes the states of the single atoms evolve in periodic cycles causing periodic stochastic fluctuations of the (stationary) radiation leaking out of the cavity. Naively formulated.

An atom that has just emitted a photon has to be reexcited before it can emit another photon.

In this case, by the way, also the inequality g^{(2)}(0) ≥ 1 is violated which has to be fulfilled for classical light because of

⟨(A − ⟨A⟩)^{2}⟩ = ⟨A^{2}⟩ − ⟨A⟩^{2} .

5.2 The Action of Beam Splitters on Photons

5.2.1 General Action of Linear Optical Devices^{4}

From experience we know that the following is a fairly accurate description of the influence of linear optical media on the quantized electromagnetic field:

The expectation values of the quantized field are solutions of the classical macroscopic Maxwell equations for the linear medium.

In this approximation the action of passive linear optical devices like beam splitters, phase shifters etc. is as follows:^{5}

Draft, November 5, 2011

4See also (Leonhardt, 2003).

5Note that the time evolution of the quantized electromagnetic field in the Heisenberg picture fixes the Hamiltonian of the field up to an additive constant.

If \left\{{ \atop â_{ν}}\right\}_{ν∈ℕ} is a maximal family of pairwise orthogonal modes and if

⟨Ω | Â^{(+)}(x, t) â_{ν}^{†} Ω⟩ ⟼ ⟨Ω | Â^{(+)}(x, t) b̂_{ν}^{†} Ω⟩

agrees with the transformation of the complex vector potential in the corresponding classical optics (for every ν ∈ ℕ) then the state transformation for the quantized electromagnetic field in the interaction picture is

f(â_{1}^{†}, …, â_{2}^{†}, …) Ω ⟼ f(b̂^{†}, …, b̂_{2}^{†}, …) Ω

for sufficiently well-behaved f .

For coherent states this is a direct consequence of (1.84). Therefore, according to the optical equivalence theorem, this holds for all states of the quantized radiation field.

5.2.2 Classical Description of Lossless Beam Splitters

Consider the action of a beam splitter, as sketched in Figure 5.1, on an incoming monochromatic classical light beam corresponding to the complex vector potential

A_{in}^{(+)}(x, t) = (2π)^{−3/2}\sqrt{μ_{0}ℏc} ∫ a_{in}(k)e^{−i(ω t−k·x)} \frac{dV_{k}}{\sqrt{2 |k|}} ,

where^{6}

a_{in}(k) = z_{in}^{1} a_{1}^{V}(k) + z_{in}^{2} a_{2}^{V}(k) ,

a_{j}^{V}(k) =^{def} (\frac{ω}{c} )^{\frac{3}{2}} δ(k − k_{j}) e_{2} ∀ j ∈ \{1, 2\}

(5.4)

and^{7}

k_{1} =^{def} \frac{ω}{c} \frac{e_{1} − e_{3}}{\sqrt{2}} , k_{2} =^{def} \frac{ω}{c} \frac{e_{1} + e_{3}}{\sqrt{2}} .

Linearity of the (plane) beam splitter implies that there are r_{1}, r_{2}, t_{1}, t_{2} ∈ ℂ such that the outcoming light beam corresponds to a complex vector potential

A_{out}^{(+)}(x, t) = (2π)^{−3/2}\sqrt{μ_{0}ℏc} ∫ a_{out}(k)e^{−i(ω t−k·x)} \frac{dV_{k}}{\sqrt{2 |k|}}

with

a_{out}(k) = z_{out}^{1} a_{1}^{V}(k) + z_{out}^{2} a_{2}^{V}(k)

Draft, November 5, 2011

6The z^{j} are just complex numbers without any physical dimension.

7Actually, it would be sufficient that k_{1} · e_{3} = −k_{2} · e_{3} . Then, however the action of the beam splitter would depend on k_{1} · e_{3} , in general.

.^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}z_{out}^{2} a_{2}^{V} z_{in}^{1} a_{1}^{V} z{2 \atop in} aV2  z{1 \atop out} aV1

Figure 5.1: Classical Action of a Beam Splitter

and^{8}

( \begin{matrix}z_{out}^{1} \\ z_{out}^{2} \end{matrix} ) = 𝕊 ( \begin{matrix}z_{in}^{1} \\ z_{in}^{2} \end{matrix} ) , 𝕊 =^{def} ( \begin{matrix}t_{1} & r_{2} \\ r_{1} & t_{2} \end{matrix} ) .

For the fields

E_{1,in}(x, t) =^{def} (2π)^{−3/2} \sqrt{\frac{μ_{0} ℏ ω^{2}}{2c}} (i ω z_{in}^{1} e^{−i(ω t−k_{1}·x)} − i ω \overline{z_{in}^{1}} e^{+i(ω t−k_{1}·x)}) e_{2}
B_{1,in}(x, t) =^{def} (2π)^{−3/2} \sqrt{\frac{μ_{0} ℏ ω^{2}}{2c}} (i z_{in}^{1} e^{−i(ω t−k_{1}·x)} − i \overline{z_{in}^{1}} e^{+i(ω t−k_{1}·x)}) k_{1} × e_{2}

corresponding to the complex vector

A_{1,in}^{(+)}(x, t) =^{def} (2π)^{−3/2}\sqrt{μ_{0}ℏc} ∫ z_{in}^{1}a_{1}^{V}(k)e^{−i(ω t−k·x)} \frac{dV_{k}}{\sqrt{2 |k|}}

with a_{1}^{V}(k) given by (5.4) we have

E_{1,in}(x, t) × \frac{1}{μ_{0}} B_{1,in}(x, t) = \frac{1}{μ_{0} c} E_{1,in}(x, t)(x, t) × (\frac{k_{1}}{|k_{1}|} × E_{1,in}(x, t))
= \frac{1}{μ_{0} c} |E_{1,in}(x, t)|^{2} \frac{k_{1}}{|k_{1}|} − \underbrace{E_{1,in}(x, t) · \frac{k_{1}}{|k_{1}|}}_{=0} E_{1,in}(x, t) .

Since^{9}

∫0^{\frac{2π}{ω}} (i z_{in}^{1} e^{−i(ω t−k_{1}·x)} − i \overline{z_{in}^{1}} e^{+i(ω t−k_{1}·x)})^{2} dt = \frac{2π}{ω} (|z_{in}^{1}|^{2} + |z_{in}^{1}|^{2})

this gives for the intensity, i.e. the modulus of the time-averaged energy current density, of this stationary field

|\frac{ω}{2π} ∫0^{\frac{2π}{ω}} E_{0}(x, t) × \frac{1}{μ_{0}} B_{0}(x, t) dt| = (2π)^{−3} \frac{ℏ ω^{4}}{2 c^{2}} (|z_{in}^{1}|^{2} + |z_{in}^{1}|^{2}) .

Draft, November 5, 2011

8Of course, 𝕊 depend on the frequency and also on the polarization of the incoming radiation. 9Note that

∫0^{\frac{2π}{ω}} e^{±i 2ω t} dt = 0 .

Since similar results hold for the fields in the other arms of the interferometer, for a lossless beam splitter the condition

|z_{in}^{1}|^{2} + |z_{in}^{2}| = |z_{out}^{1}|^{2} + |z_{out}^{2}|

has to be fulfilled. This means that 𝕊 has to be unitary, hence of the form

𝕊 = ( \begin{matrix}ζ^{1} & −\overline{ζ^{2}} e^{iψ} \\ ζ^{2} & \overline{ζ^{1}} e^{iψ} \end{matrix} ) , ψ ∈ ℝ , ζ^{1}, ζ^{2} ∈ ℂ , |ζ^{1}|^{2} + |ζ^{2}|^{2} = 1 .

The latter may also be written in the form

𝕊 = e^{i φ} ( \begin{matrix}a & −\overline{b} \\ b & \overline{a} \end{matrix} ) ϕ ∈ ℝ , a, b ∈ ℂ , |a|^{2} + |b|^{2} = 1  (5.5)

(with φ = ψ/2 , a = ζ^{1} e^{−iψ/2} , and b = ζ^{2} e^{−iψ/2}) or

𝕊 = e^{i φ} ( \begin{matrix}a & \overline{b} \\ b & −\overline{a} \end{matrix} ) ϕ ∈ ℝ , a, b ∈ ℂ , |a|^{2} + |b|^{2} = 1

(with φ = (ψ + π) /2 + π, a = ζ^{1} e^{−i(ψ+π)/2} , and b = ζ^{2} e^{−i(ψ+π)/2}). Anyway, the transmittivity T resp. the reflectivity R is the same for both incoming rays:

R = |r_{1}|^{2} = |r_{2}|^{2} , T = |t_{1}|^{2} = |t_{2}|^{2} , R + T = 1 .

Beam splitters with R = T are called 50/50-beam splitters. Beam splitters with r_{1} = r_{2} and t_{1} = t_{2} are called symmetric. Thus:

𝕊 = \frac{e^{iφ}}{\sqrt{2}} ( \begin{matrix}1 & ∓i \\ ∓i & 1 \end{matrix} ) for symmetric 50/50 beam splitters  (5.6)

(see Exercise E34c) of (Lücke, eine)).

5.2.3 Transformation of Photon Modes

Let us describe the effect of beam splitters on photons as a scattering process in the interaction picture.^{10}

The photon mode corresponding to the classical mode a_{1}^{V}(k) resp. a_{2}^{V}(k) is (\frac{ω}{c})^{\frac{3}{2}} â^{V}(k_{1}) resp. (\frac{ω}{c})^{\frac{3}{2}} â^{V}(k_{2}) , where

â^{V}(k) = â_{2}(k) ∀ k ⊥ e_{2} ,

for the choice

ϵ_{2}(k) =^{def} e_{2} ∀ k ⊥ e_{2} ,

Draft, November 5, 2011

10See also (Yurke et al., 1986).

and the beam splitter acts according to^{11}

\overline{z_{in}^{1}} â^{V}(k_{1}) + \overline{z_{in}^{2}} â^{V}(k_{2}) ⟼ \overline{z_{out}^{1}} â^{V}(k_{1}) + \overline{z_{out}^{2}} â^{V}(k_{2})
= (t_{1} \overline{z_{in}^{1}} + r_{2} \overline{z_{in}^{2}}) â^{V}(k_{1}) + (r_{1} \overline{z_{in}^{1}} + t_{2} \overline{z_{in}^{2}}) â^{V}(k_{2})

for arbitrary z_{in}^{1}, z_{in}^{2} ∈ ℂ. This is equivalent to linearity plus

( \begin{matrix}â_{−}^{V}(k_{1}) \\ â_{−}^{V}(k_{2}) \end{matrix} ) =^{def} ( \begin{matrix}â^{V}(k_{1}) \\ â^{V}(k_{2}) \end{matrix} ) ⟼ ( \begin{matrix}â_{+}^{V}(k_{1}) \\ â_{+}^{V}(k_{2}) \end{matrix} ) =^{def} \underbrace{( \begin{matrix}\overline{t_{1}} & \overline{r_{1}} \\ \overline{r_{2}} & \overline{t_{2}} \end{matrix} )}_{=𝕊^{∗}=𝕊^{−1}} ( \begin{matrix}â^{V}(k_{1}) \\ â^{V}(k_{2}) \end{matrix} ) . (5.7)

Remark: The scattering matrix Ŝ_{0} in the interaction picture fulfills

Ŝ_{0} Ω = Ω

and

â_{+}^{V}(k_{j}) = Ŝ_{0} â_{−}^{V}(k_{j}) Ŝ_{0}^{−1} ∀ j ∈ {1, 2} .

The scattering matrix Ŝ in the Heisenberg picture — used, e.g., in (Mandel and Wolf, 1995, Section 12.12) and (Bowmeester et al., 2002, Chapter 4) — fulfills

Ω_{in} = Ŝ Ω_{out}

and

â_{in}^{V}(k_{j}) = Ŝ â_{out}^{V}(k_{j}) Ŝ^{−1} ∀ j ∈ {1, 2} ,

where^{12}

( \begin{matrix}â_{in}^{V}(k_{1}) \\ â_{in}^{V}(k_{2}) \end{matrix} ) = 𝕊^{−1} ( \begin{matrix}â_{out}^{V}(k_{1}) \\ â_{out}^{V}(k_{2}) \end{matrix} ) .

See Section 2.3.1 of (Lücke, qft) for a detailed discussion of the two pictures of scattering theory.

For (improper) incoming n-photon states (5.7) means that the beam splitter acts as

∑_{j_{1},...,j_{n}∈{1,2}} c_{j_{1},...,j_{n}} (â^{V}(k_{j_{1}}) ⋯ â^{V}(k_{j_{n}}))^{†} Ω

⟼ ∑_{j_{1},...,j_{n}∈{1,2}} c_{j_{1},...,j_{n}} (∑_{j′_{1}=1}^{2} S_{j_{1}}^{j′_{1}}â^{V}(k_{j′_{1}}))^{†} ⋯ (∑_{j′_{n}=1}^{2} S_{j_{n}}^{j′_{n}}â^{V}(k_{j′_{n}}))^{†} Ω ,

(5.8)

Draft, November 5, 2011

11Recall (1.58) and (1.60). 12Some authors mystify the fact that, e.g.,

â_{out}^{V}(k_{1}) = t_{1} â_{in}^{V}(k_{1}) + r_{2} â_{in}^{V}(k_{2})

holds even if the input situation at the lower arm of the beam splitter corresponds to the vacuum.

where

( \begin{matrix}S_{1}^{1} & S_{2}^{1} \\ S_{1}^{2} & S_{2}^{2} \end{matrix} ) =^{def} ( \begin{matrix}t_{1} & r_{2} \\ r_{1} & t_{2} \end{matrix} ) .

Obviously, the total number of photons is not changed by the beam splitter.

Let us consider, for example, the special action^{13}

â^{V}(k_{1})^{†} â^{V}(k_{2})^{†} Ω ⟼ \frac{1}{2} (â^{V}(k_{1}) + i â^{V}(k_{2}))^{†} (i â^{V}(k_{1}) + â^{V}(k_{2}))^{†}
= \frac{i}{2} (â^{V}(k_{1})^{†} â^{V}(k_{1})^{†} + â^{V}(k_{2})^{†} â^{V}(k_{2})^{†}) Ω

of a (symmetric, 50/50-) beam splitter with

𝕊 = \frac{1}{\sqrt{2}} ( \begin{matrix}1 & −i \\ −i & 1 \end{matrix} ) .  (5.9)

Here the incoming state is a 2-photon state with one photon definitely entering the beam splitter from above and the other photon definitely entering the beam splitter from below. As if agreed upon before, the photons leave the beam splitter either both on the upper side (with probability \frac{1}{2}) or both on the lower side. This so-called Hong-Ou-Mandel effect effect is often coined 2-photon interference,^{14} even though the relative phase of the two photon modes is irrelevant, in this case. Needless to say, there is no classical analogue for this coalescence effect.

If the action of a beam splitter with S-matrix (5.9) does not depend on the polarization then it transforms the (improper) Bell states

Φ_{=}^{±}(k_{1}, k_{2}) =^{def} \frac{1}{\sqrt{2}} (â^{V}(k_{1}) â^{V}(k_{2}) ± â^{H}(k_{1}) â^{H}(k_{2}))^{†} Ω
Φ_{≠}^{±}(k_{1}, k_{2}) =^{def} \frac{1}{\sqrt{2}} (â^{V}(k_{1}) â^{H}(k_{2}) ± â^{H}(k_{1}) â^{V}(k_{2}))^{†} Ω ,

(5.10)

where

â^{H}(k) = â_{1}(k) , ϵ_{2}(k) = e_{2} × \frac{k}{|k|} ∀ \underbrace{k}_{≠0} ⊥ e_{2} ,

Draft, November 5, 2011

13Recall Footnote 7. For more general input k-configurations see (Scully and Zubairy, 1999, Sect. 4.4.3), where the Heisenberg picture is used.

14See (Di Giuseppe et al., 2003) for experimental verification and (McDonald and Wang, 2003; Lim and Beige, 2005) for generalization. This effect may be exploited for measurement of the spectral density matrix of 1-photon states (Wasilewski et al., 2007). See also (Kim and Grice, 2003) and (Zavatta et al., 2004) for additional aspects of two-photon interference. For an extensive treatment of two-photon interference in beam splitters see (Wang, 2003) and references given there.

according to

Φ_{=}^{±}(k_{1}, k_{2}) ⟼ \frac{1}{2\sqrt{2}} ∑_{j=1}^{2} (â^{V}(k_{j}) â^{V}(k_{j}) + â^{H}(k_{j}) â^{H}(k_{j}))^{†} Ω ,

Φ_{≠}^{+}(k_{1}, k_{2}) ⟼ \frac{i}{\sqrt{2}} ∑_{j=1}^{2} (â^{V}(k_{j}) â^{H}(k_{j}))^{†} Ω ,
Φ_{≠}^{−}(k_{1}, k_{2}) ⟼ \frac{1}{\sqrt{2}} (â^{V}(k_{1}) â^{H}(k_{2}) − â^{H}(k_{1}) â^{V}(k_{2}))^{†} Ω .

Note that Φ_{≠}^{−} is the only Bell state that is transformed into a state with exactly one photon leaving the beam splitter on each side.^{15} This effect was exploited in the first experiment on (partial) quantum teleportation (Bowmeester et al., 1997); see also (Gisin et al., 2003).

Comment: By straightforward calculation (see 5.3.3) one can show that χ(k_{1}) ⊗ Φ_{≠}^{−}(k_{2}, k_{3})

= \frac{1}{2} (Φ_{=}^{+}(k_{1}, k_{2}) ⊗ Û_{1} χ(k_{3}) − Φ_{≠}^{+}(k_{1}, k_{2}) ⊗ Û_{2} χ(k_{3})
+ Φ_{=}^{−}(k_{1}, k_{2}) ⊗ Û_{3} χ(k_{3}) − Φ_{≠}^{−}(k_{1}, k_{2}) ⊗ χ(k_{3}))

holds for

χ(k) =^{def} (λ_{V} â^{V}(k) + λ_{H} â^{H}(k))^{†}Ω

and

Û_{j} χ(k) =^{def} \{ \begin{matrix}(−λ_{H} â^{V}(k) + λ_{V} â^{H}(k))^{†}Ω & for j = 1 , \\ (λ_{V} â^{V}(k) − λ_{H} â^{H}(k))^{†}Ω & for j = 2 , \\ (λ_{H} â^{V}(k) + λ_{V} â^{H}(k))^{†}Ω & for j = 3 . \end{matrix}

This allows for quantum teleportation: Alice checks in which of the four Bell states photon 1 and 2 (corresponding to k_{1} and k_{2}) are and communicates (by classical means) the result to Bob. If, e.g., Alice found Φ_{=}^{+}(k_{1}, k_{2}) then Bob just has to apply Û_{1}^{−1} to photon 3 (corresponding to k_{3}) in order to get its polarization state equal to the unknown^{16} original polarization state of photon 1.

Of great practical use are also polarizing beam splitters, transmitting the horizontal modes and reflecting the vertical modes. Appropriately oriented, these devices act according to

( \begin{matrix}â^{H}(k_{1}) \\ â^{H}(k_{2}) \\ â^{V}(k_{1}) \\ â^{V}(k_{2}) \end{matrix} ) ⟼ ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{matrix} ) ( \begin{matrix}â^{H}(k_{1}) \\ â^{H}(k_{2}) \\ â^{V}(k_{1}) \\ â^{V}(k_{2}) \end{matrix} ) (5.11)

Draft, November 5, 2011

15See also (Pittman and Franson, 2003) in this connection. 16See (Caves et al., 2002) concerning the notion of unknown state in quantum mechanics.

.^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} â^{V}(k_{1})  ∼ â^{V}(k_{1}) BS1  BS2

Figure 5.2: Balanced Mach-Zehnder interferometer

5.2.4 Mach-Zehnder Interferometer

The quantum effect of beam splitters on one-photon states is already very puzzling. Consider, for example a Mach-Zehnder interferometer as sketched in Fig. 5.2. Its action on an incoming single photon in the (improper) mode â^{V}(k_{1}) corresponds (up to irrelevant phase factors) to the following sequence of transformations:

â^{V}(k_{1})^{†} Ω ⟼_{BS1} \frac{1}{\sqrt{2}} (â^{V}(k_{1}) + i â^{V}(k_{2}))^{†} Ω

⟼_{mirrors} \frac{1}{\sqrt{2}} (â^{V}(k_{2}) + i â^{V}(k_{1}))^{†} Ω

⟼_{BS2} \frac{1}{2}((i â^{V}(k_{1}) + â^{V}(k_{2})) + i (â^{V}(k_{1}) + i â^{V}(k_{2})))^{†}Ω
= i â^{V}(k_{1})^{†} Ω .

For classical light there is no surprise:

The interferometer is balanced in the sense that there is complete destructive interference for the waves transmitted through BS2 from below and reflected at BS2 from above. If part of the waves is screened within the interferometer then this interference is disturbed and light is emitted also from the upper side of BS2.

For the quantized electromagnetic field in a one-photon state, however, there is an obvious problem:

If, e.g., the lower path within the interferometer is blocked then, with equal probability, the photon may leave BS2 either on the upper or the lower side. Now the photon, considered as a particle, can only leave BS2 if it takes the upper path within the interferometer. Then, however, the photon should behave as if the lower path were not blocked at all, and there is no explanation for the photon’s ability to leave BS2 on the upper side.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

. .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . . . . . . . . . . . . . . .

. . . . .

. . . . . . . . . . . . . . . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . . . . .

. . . . . . . . . . . . . . . .

.

.

.

.

.

.

.

.

.

.

.

.

.

. .

.

.

.

.

. .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . . . . . . . . . . . . . .

. . . . . .

. . . . . . . . . . . . . . . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . . . . . . . . . . . . . .

.

.

.

.

.

. .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . . . . . . . . . . . . . .

. . . . . .

. . . . . . . . . . . . . . . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

PBS1 PBS2 M1

M2

switching out and polarization measurement after N cycles polarization rotation by φ = \frac{π}{2N}

Figure 5.3: Principle of ‘interaction free’ measurements.

If the photon leaves the interferometer on the upper side of BS2 it ‘knows’ that the lower path is blocked although it never entered that region. Obviously, the photon is more than just a ’particle’. The described effect inspired Elitzur and Vaidman to suggest what they coined interaction free measurements (Elitzur and Vaidman, 1993). The principle of a refined version, devised by Kwiat et al. (Kwiat et al., 1995; Kwiat et al., 1999), is sketched in Fig. 5.3:

The interferometer is built with polarizing beam splitters rather than with polarization independent ones. A horizontally polarized photon will be inserted such that it enters the polarization rotator first. If there is no object inside the interferometer — built by PBS1, M1, PBS2, and M2 — then a photon entering the interferometer will pass it without any change of polarization. If, however, the path via M2 is blocked, then the photon will either be absorbed — with probability sin^{2} \frac{π}{2N} — or pass with horizontal polarization. Thus, when the photon is switched out after N cycles (if not already absorbed) and checked for horizontal or vertical polarization, there are three possible outcomes:

1. The photon will not be detected at all. This may be due to failure of the detector or absorption by some object inside the interferometer. If an object is inserted as described, the probability for absorption of the photon is N sin^{2} \frac{π}{2N} — which may be made arbitrarily small.

2. The photon will be detected with horizontal polarization. This can only happen if the absorber is inserted into the interferometer.

3. The photon will detected with vertical polarization. This can only happen if the absorber is not inserted into the interferometer.

This way, with probability arbitrarily close to 1, an object may be detected without interacting with it in any way.

5.3. APPLICATIONS TO QUANTUM INFORMATION PROCESSING 111

5.3 Applications to Quantum Information Processing

5.3.1 Single-Photon States and Quantum Cryptography

In general, non-classical effect require sufficient — preferably individual — control of photons.

A standard technique for preparing single-photon states is sketched in Figure 5.4.

.

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

single photons

D_{2} D_{1}

J

J_{⊥} BSP

1 2 BBO

laser puls

“EPR pair” creation

Figure 5.4: Possible preparation of single-photon states.

A laser pulse incident on a BBO (barium borate) crystal produces a pair of photon with polarization state

Ψ_{−} =^{def} \frac{1}{\sqrt{2}}( \underbrace{V}_{=( \begin{matrix}1 \\ 0 \end{matrix} )} ⊗ \underbrace{H}_{=( \begin{matrix}0 \\ 1 \end{matrix} )} −H ⊗ V)

via spontaneous down conversion^{17} and this internal state fulfills the condition

\begin{matrix}J ∈ ℂ^{3} \\ ∥J∥ = 1 \end{matrix} \} ⟹ Ψ_{−} = \frac{1}{\sqrt{2}} (J ⊗ J_{⊥} − J_{⊥} ⊗ J) ,

where:

( \begin{matrix}a \\ b \end{matrix} )_{⊥} =^{def} ( \begin{matrix}+b^{∗} \\ −a^{∗} \end{matrix} ) ∀ a, b ∈ ℂ .

From this we conclude:

• The internal quantum state of the ensemble of those photons running to the right and having a partner making detector D_{2} fire after passing a J-J_{⊥} beam splitter has to be described (according to quantum mechanics) by the density operator ρ̂ = |J⟩⟨J|.

Draft, November 5, 2011

17See, e.g., (Gatti et al., 2003) for details.

• The internal quantum state of the ensemble of all photons running to the right — irrespective of the behavior of their partners — corresponds to

ρ̂ = \frac{1}{2} ˆ1 = \frac{1}{2} |J⟩⟨J| + \frac{1}{2} |J_{⊥}⟩⟨J_{⊥}| .

The result of a J-J_{⊥} test (“which detector fires?”) on photon 1 (left running) predetermines the result of an eventually following J-J_{⊥} test on photon 2 (running to the right):

J_{⊥} for photon 1 ⟹ J for photon 1
J for photon 1 ⟹ J_{⊥} for photon 1

(strict correlations).

If it were possible to produce exact copies of the unknown state of photon 2, owned by Bob (far away from Alice ), then Bob could produce an ensemble of photons with internal state J resp. J_{⊥} if photon 1, owned by Alice, just passed a J-J_{⊥} beam splitter and made detector 2 resp. 1 fire. Since the states of (sufficiently large) ensembles can be determined (quantum state tomography) Bob could immediately check Alice’s choice of J-J_{⊥} alternative. This could be exploited by Alice for superluminal communication of information to Bob encoded in different choices of J-J_{⊥} alternatives.^{18}

Conclusion:

The principles of special relativity exclude the possibility of producing exact copies of unknown quantum states (no cloning theorem).

The reason:

Every attempt to gain information not available a priori about a quantum system usually results in uncontrollable changes of the system’s state.

The no cloning theorem opens up the possibility of single-photon quantum cryptography:

If Alice and Bob both are provided with some random N-bit sequence

k = (k_{1}, …, k_{N})

known only to them they can use this for encryption^{19}

b = (b_{1}, …, b_{N}) ⟼ b ⊕ k =^{def} (b_{1} ⊕ k_{1}, …, b_{N} ⊕ k_{N})

Draft, November 5, 2011

18See(Herbert, 1982) in this connection.

19As usual, ⊕ means modulo-2 addition:

b ⊕ k =^{def} (b + k) mod 2 ∀ b, k ∈ {0, 1} .

of any N-bit information b . Alice can send the N-bit sequence b + k through some public channel to Bob without being afraid that any eavesdropper could make use of this. Bob however can decrypt the message according to

b ⊕ k ⟼ (b ⊕ k) ⊕ k = b

(thanks to k ⊕ k = (0, …, 0)). Thus:

The essential point for secure secret information exchange is the ability of Alice and Bob to create a common random key k known only to them.

A well established^{20} technology for secret key distribution is based on the socalled BB84 protocol suggested by (Bennet and Brassard, 1984):

Alice and Bob agree upon two alternatives J-J_{⊥} and J′-J′_{⊥} with

|⟨J | J′⟩|^{2} = \frac{1}{2} .

Moreover they agree on certain instants of time at which Alice will send a photon to Bob with internal state being randomly chosen from the set {J, J_{⊥}, J′, J′_{⊥}}. Bob randomly chooses a test of either the J-J_{⊥} or the J′-J′_{⊥} alternative for the received photon. After all photons are sent and tested Alice and Bub determine, via public communication, those instances for which by chance the had chosen the same alternative. For these instances their results are strictly correlated (in the ideal case)

— unless the sending war perturbed (by an eavesdropper, for example). If these correlations are confirmed for certain randomly chosen instance then the other N results, kept secret, provide the common key.

The crucial point, however, is to make sure that (essentially) only single photons are sent^{21} in order to prohibit — thanks to the no cloning theorem — eavesdropping.

5.3.2 Entangled Pairs of Photons

Figure 5.4 also serves as a good illustration of the following well-known consideration by Einstein, Rosen, and Podolsky (EPR):

1. According to EPR the result of any test of a J, J_{⊥} alternative for photon 2 should be predetermined (determinism) since: Without influencing photon 2 in any way (locality) one \underline{could} test the J, J_{⊥} alternative on photon 1 and use the result to predict — thanks to the strict correlation of the photons — the outcome of a corresponding subsequent test on photon 2 with certainty.

Draft, November 5, 2011

20See http://www.idquantique.com. For more recent implementations of quantum key distribution see (Sasaki et al., 2011) and references given there.

21See (Alleaume et al., 2003) and references given there for realizations of single-photon sources.

2. From this point of view quantum mechanics appears to be incomplete (existence of hidden variables) since there is no quantum mechanical state predicting for all J-J_{⊥} alternatives the outcome of a corresponding test with certainty.

This should be confronted with another well-know consideration by Bell which — in a slightly modified form — is as follows:

Let Φ be a (large but finite) ensemble of photon pairs in the state Ψ_{−}. Then, according to EPR the definition

Φ_{J,J′} =^{def} \{pairs from Φ with: \{ \begin{matrix}J predeterminded for photon 1 and \\ J′ predetermined for photon 2 \end{matrix} \}

should be allowed (\underline{determinism}) for every ensemble of EPR pairs (photons 1 and 2) and

Φ_{J} =^{def} Φ_{J,J′} ∪ Φ_{J,J′_{⊥}}

should be independent of J′ (\underline{locality}). The \underline{correlations} of the EPR pairs then imply

Φ_{J,J′} = Φ_{J} ∩ Φ_{J′} ∀ J, J′

(5.12)

and, consequently, for all Jones vectors J_{2}, J_{2}, J_{3} the inequality^{22}

|Φ_{J_{1},J_{2}}| ≤ |Φ_{J_{1},J_{3}}| + |Φ_{J_{2},J_{3}_{⊥}}|  (5.13)

has to be fulfilled.

Outline of proof: The inequality follows from

χ_{1} χ_{2} ≤ χ_{1} χ_{3} + χ_{2} (1 − χ_{3}) ∀ χ_{1}, χ_{2}, χ_{3} ∈ {0, 1} ,

since^{23}

|Φ_{J_{ν}} | = ∑_{x∈Φ_{Jν}} χ_{Φ_{Jν}} (x) , |Φ_{J_{ν}_{⊥}}| = ∑_{x∈Φ_{Jν}} (1 − χ_{Φ_{Jν}} (x))

and

χ_{Φ_{Jν} ∩Φ_{Jμ}} (x) = χ_{Φ_{Jν}} (x) χ_{Φ_{Jμ}} (x) .

Therefore (5.12) contradicts the quantum mechanical predictions

\begin{matrix}|Φ_{J}| / |Φ| = \frac{1}{2} , \\ |Φ_{J′,J_{⊥}}| / |Φ| = \frac{1}{2} |⟨J | J′⟩|^{2} \end{matrix} \} ∀ J, J′ . (5.14)

Draft, November 5, 2011 22By |Φ_{ν}| we denote the number of elements of the set Φ_{ν} .

23As usual we denote by χ_{Φ′} the characteristic function of the set Φ′ :

χ_{Φ′}(x) =^{def} \{ \begin{matrix}1 & for x ∈ Φ′, \\ 0 & sonst . \end{matrix}

5.3. APPLICATIONS TO QUANTUM INFORMATION PROCESSING 115

This is because for, e.g.,

J_{ν} =^{def} ( \begin{matrix}cos & (\frac{ν π}{8}) \\ sin & (\frac{ν π}{6}) \end{matrix} ) ∀ ν ∈ \{1, 2, 3\}

(5.14) implies

\frac{|Φ_{J_{1},J_{2}}|}{|Φ|} = \frac{1}{2} cos^{2} (π/6)
= 3/8
≥ 2/8
= \frac{1}{2} cos^{2}(π/3) + \frac{1}{2} (1 − cos^{2}(π/6))
= \frac{|Φ_{J_{1},J_{3}}| + |Φ_{J_{2},J_{3}_{⊥}}|}{|Φ|}

rather than (5.13). \underline{Naive} proposal:

Check (5.14) experimentally.

The vast majority of physicists considers (5.14) to be experimentally confirmed^{24} and conclude:

Local determinism as assumed by EPR is inconsistent with physical reality.

Concerning the actually remaining loopholes see, e.g., (Gisin and Gisin, 1999), (Semenov and Vogel, 2010), (Morgan, 2008), (Stobinśka et al., 2010), (Gerhardt et al., 2011), and (Wittmann et al., 2011).

Generally speaking, we could take the attitude that every “no-go theorem” comes with some small-print, that is, certain conditions and assumptions that are considered totally natural and reasonable by the authors, but which may be violated in the real world.

(’t Hooft, 2002, p. 309)

The seeming conflict of quantum mechanics with local realism appears even more dramatic if state vectors of the form

\frac{1}{\sqrt{2}} (H ⊗ H ⊗ H + V ⊗ V ⊗ V)

(Greenberger-Horne-Zeilinger states) are considered (Pan et al., 2000).

Draft, November 5, 2011

24See (Aspect, 2004),(Kaltenbaek et al., 2003), (Simon and Irvine, 2003), and (Rosenberg et al., 2005) for experimental checks. For classical light, of course, the corresponding Bell inequality is fulfilled.

5.3.3 Quantum Teleportation

As discussed in 5.3.2, the internal quantum state ˆ= Ψ_{−} of the EPR pairs exhibits correlations which are incompatible with local realism.

Even without deeper insight into the origin of these correlations they may be exploited for new technologies:

Their use for preparation of single-photon states with prescribed polarization J was already mentioned:^{25}

[figure: .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; D_{2}; D_{1}; J; J_{⊥}; BSP; 1; 2; BBO; laser pulse; destructive measurement; info; selection; J]

An especially valuable application is quantum teleportation, basing on the identity

2 J ⊗ Ψ_{−} = Φ_{+} ⊗ Û_{1} J + Ψ_{+} ⊗ Û_{2} J + Φ_{−} ⊗ Û_{3} J + Ψ_{−} ⊗ Û_{4} J  (5.15)

where:

Ψ_{±} =^{def} \frac{1}{\sqrt{2}}(( \begin{matrix}1 \\ 0 \end{matrix} ) ⊗ ( \begin{matrix}0 \\ 1 \end{matrix} ) ± ( \begin{matrix}0 \\ 1 \end{matrix} ) ⊗ ( \begin{matrix}1 \\ 0 \end{matrix} )),
Φ_{±} =^{def} \frac{1}{\sqrt{2}}(( \begin{matrix}1 \\ 0 \end{matrix} ) ⊗ ( \begin{matrix}1 \\ 0 \end{matrix} ) ± ( \begin{matrix}0 \\ 1 \end{matrix} ) ⊗ ( \begin{matrix}0 \\ 1 \end{matrix} )),

Û_{j} ( \begin{matrix}α \\ β \end{matrix} ) =^{def} \{ \begin{matrix}( \begin{matrix}−β \\ α \end{matrix} ) & for j = 1 , \\ ( \begin{matrix}−α \\ β \end{matrix} ) & for j = 2 , \\ ( \begin{matrix}β \\ α \end{matrix} ) & for j = 3 , \\ − ( \begin{matrix}α \\ β \end{matrix} ) & for j = 4 . \end{matrix}

Draft, November 5, 2011

25See (Migdall et al., 2002) for possibility to increase the efficiency of such sources.

Outline of proof: With the identification

(Φ_{1}, Φ_{2}, Φ_{3}, Φ_{4}) = (Φ_{+}, Ψ_{+}, Φ_{−}, Ψ_{−})

we have

Φ_{j} = J ⊗ f_{j}(J) + J_{⊥} ⊗ f_{j}(J_{⊥}) ∀ J ∈ ℂ^{2} , j ∈ {1, …, 4} ,

for the anti-linear mappings

f_{j}(J) =^{def} ∑_{ν=1}^{2} ⟨J ⊗ e_{ν} | Φ_{j}⟩ e_{ν} ∀ J ∈ ℂ^{2} ,

not depending on the choice of orthonormal basis {e_{1}, e_{2}} of ℂ^{2} . This implies

(P̂_{Φ_{j}} ⊗ 1) (J ⊗ Φ_{4})
= (P̂_{Φ_{j}} ⊗ 1) (J ⊗ f_{j}(J) ⊗ f_{4}(f_{j}(J)) + J ⊗ f_{j}(J)_{⊥} ⊗ f_{4}(f_{j}(J)_{⊥}))
= (P̂_{Φ_{j}} ⊗ 1) (J ⊗ f_{j}(J) ⊗ (f_{4} ∘ f_{j})(J))
= ⟨Φ_{j} | J ⊗ f_{j}(J)⟩ Φ_{j} ⊗ (f_{4} ∘ f_{j})(J) ∀ J ∈ ℂ^{2} , j ∈ {1, …, 4} .

Since {Φ_{1}, …, Φ_{4}} is an orthonormal basis of ℂ^{2} ⊗ ℂ^{2} this together with

2 (f_{4} ∘ f_{j})(J) = Û_{j} J ∀ j ∈ {1, …, 4}

proves the statement.

Remark: Quantum teleportation correspond to the following virtual flow of information:

Since the initial (internal) state of photon 1 was (the unknown) J , photon 2 must have ‘been’ in the initial state f_{j}(J) if the result of the Bell measurement on photons 1 and 2 was Φ_{j} . Then, because of the correlation of photons 2 and 3 the resulting state of photon 3 must be 2 f_{4} ∘ f_{j}(J) . Therefore application of Û_{j}^{−1} turns the state of photon 3 into J .

This kind of reasoning has been elaborated into a remarkable theory.^{26}

Since the four Bell states Ψ_{±}, Φ_{±} form an orthonormal basis of ℂ^{2} ⊗ ℂ^{2} we just have to test in which of these 2-photon states states photons 1 and 2 ‘are’ (Bell test) in order to know the transformation U_{j} turning the state of photon 3 into the unknown J :

.^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} Bell state  J Û_{j}^{−1} Ψ^{−} Bell test Alice  Bob Info J  2  3 1

Draft, November 5, 2011 26See (Coecke, 2004, Theorem 3.3), (Abramsky and Coecke, 2004), and references given there.

Once Alice and Bob share an EPR pair (photons 2 and 3) they do not need a quantum channel any more in order to teleport the unknown initial state J of photon 1 onto photon 3. However, the necessary classical communication (from Alice to Bob) of the result of the Bell test can only be performed with subluminal speed. Note also that the Bell test destroys the original state of photon 1.

Conclusion:

If Alice and Bob have the possibility to order, store, and release partners of shared EPR pairs then they can exchange unknown quantum information (in principle) reliably without any chance for potential eavesdroppers.

One possibility to implement a complete Bell test results form the following action of (nonlinear) BBO crystals^{27}

Φ_{±} ⟼_{Typ-I} \frac{1}{\sqrt{2}} (Ĥ′ ± V′) ,
Ψ_{±} ⟼_{Typ-II} \frac{1}{\sqrt{2}} (Ĥ′ ± V′) .

According to (Kim et al., 2001) this may be exploited as sketched in Figure 5.5. If detector the D_{j} fires thenAlice instructs Bob to apply Û_{j}.

A severe problem of long distance quantum teleportation is the exponential increase of photoabsorption (especially in optical fibers). In principle this problem can be mastered exploiting teleportation of entanglement,^{28} sketched in figure 5.6.

Teleporting the correlations of photons 0 and 1, described by Ψ_{−}, onto the pair of photons 0 and 3 relies on the identity

2 \sqrt{2} Ψ_{−} ⊗ Ψ_{−} = ∑_{j=1}^{4} ( \begin{matrix}1 \\ 0 \end{matrix} ) ⊗ Φ_{+} ⊗ Û_{j} ( \begin{matrix}0 \\ 1 \end{matrix} ) − ( \begin{matrix}0 \\ 1 \end{matrix} ) ⊗ Φ_{+} ⊗ Û_{j} ( \begin{matrix}1 \\ 0 \end{matrix} )

following from (5.15), which we may also write as

|Ψ_{−}⟩_{0,1} ⊗ |Ψ_{−}⟩_{2,3} = ∑_{j=1}^{4} |Φ_{j}⟩_{1,2} ⊗ |(ˆ1 ⊗ Û_{j}) Ψ_{−}⟩_{0,3} .

However, useful application of such methods requires the possibility to check success of the necessary operations. This could be provided by implementations of

Draft, November 5, 2011

27We are writing H′, V′ instead of H, V in order to indicate doubling of frequency via conversion of the pairs into single photons.

28See (Duan et al., 2001). Another theoretical but barely practicable possibility to fight absorption is described in (Gingrich et al., 2003).

.^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} 1  2 BBO′-I BBO-I BBO′-II BBO-II +45^{∘}  -45^{∘} +45^{∘} -45^{∘} D_{3} D_{1} D_{2}  D_{4}

Figure 5.5: Possibility of a complete Bell test.

.^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  Bell state  Û_{j}^{−1} Ψ^{−} Bell test Alice  Bob info 2  3 1 0  Ψ^{−}  Ψ_{−}

Figure 5.6: Teleportation of Entanglement quantum nondemolition (QND) measurements of the photon number which are of great interest by themselves.

One suggestion to implement such measurements for small photon numbers n is described in (Munro et al., 2005):

.^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} Ψ = ∑_{ν=0}^{n} α_{ν} (â^{†})^{ν} Ω^{\underbrace{χ = e^{− \frac{|α|^{2}}{2}} e^{α ĉ^{†}}Ω}_{weak}} Kerr medium  \} ∝ ∑_{ν=0}^{n} α_{ν} (â^{†})^{ν} Ω ⊗ \underbrace{e^{α e^{i ν δ ĉ^{†}}}Ω}_{almost orthogonal}

Such implementations require extremely strong cross-Kerr nonlinearities which can be provided^{29} by electromagnetically induced transparency (EIT), treated in 8.3.2.

Such nonlinearities can also be used to implement CNOT gates, i.e. 2-qubit gates acting according to

[figure: H; H; V(H); H(V) ,; V; V; V(H); V(H) .]

One just has to construct a nonlinear phase gate (NS) acting according

(α ˆ1 + β â_{H}^{†} + γ (â_{H}^{†})^{2}) Ω ⟼_{NS} (α ˆ1 + β â_{H}^{†} − γ (â_{H}^{†})^{2}) Ω .

Then the arrangement sketched in Figure 5.7 represents a CNOT gate. Here 2photon interference is exploited at the Hadamard beam splitters.

Remark: For the detailed action of the (non-symmetric) Hadamard beam splitters ...................^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} see Figure 3.2 of (Lücke, qip).

The CNOT gate is also called measurement gate since it may be used for QND measurement of the polarization of the target photon:^{30}

[figure: αV + β H; V] } αV ⊗ V + β H ⊗ H .

Destructive ‘measurement’ of the (lower) ancillary photon (with initial state V) yields an ideal test of the V-H alternative^{31} for the (upper) target photon.

Draft, November 5, 2011

29See (Ottaviani et al., 2006).

30Note that is acts like a copying machine if only the states V and H are allowed for the target photon.

31Tests for general J-J_{⊥} alternatives may be easily implemented by additional unitary singlephoton transformations.

[figure: .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; NS; NS]

Figure 5.7: Implementation of a CNOT gate.

Note that CNOT gates may be used to produce EPR pairs:

[figure: H; H; H] } Ψ_{−} .

Already for this reason also indeterministic CNOT gates are of considerable interest — if appropriate quantum memory is available.^{32} here indeterministic means:

The gate acts as desired only with a certain probability. However, proper action is highlighted by some ancillary system (similarly to QND measurements).

One possibility for implementing indeterministic gates exploits the following effect:^{33}

Draft, November 5, 2011

32See (Pittman and Franson, 2002) and (Liu et al., 2001; Juzeliunas et al., 2003; Fleischhauer and Lukin, 2002; van der Wal et al., 2003) for possibilities of storing quantum information.

33See (Sanaka et al., 2003).

A beam splitter acting according to

.^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} \sqrt{R_{j}} â_{j}(k_{2}) â_{j}(k_{1})  \sqrt{1 − R_{j}} â_{j}(k^{1}) â_{j}(k^{2})  \sqrt{1 − R_{j}} â_{j}(k_{2}) \sqrt{R_{j}} â_{j}(k^{1})^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}

also acts according to

(â_{1}^{†}(k_{1}))^{m} (â_{2}^{†}(k_{1}))^{n}â_{2}^{†}(k_{2}) Ω

⟼ ∑_{μ=0}^{m} ∑_{ν=0}^{n+1} c_{μν}(â_{1}^{†}(k_{2}))^{μ} (â_{2}^{†}(k_{2}))^{ν} (â_{1}^{†}(k_{1}))^{m−μ} (â_{2}^{†}(k_{1}))^{n+1−μ} Ω ,

(5.16)

where we especially have

c_{mn} = ( ^{+} \sqrt{R_{1}})^{m} ( ^{+} \sqrt{R_{2}})^{n−1} (R_{2} − n (1 − R_{2}))  (5.17)

corresponding to the two possibilities, sketched in Figure 5.8, for exactly one photon leaving the lower right arm of the beam splitter and having polarization ˆ= H . For m = 0 , n = 1 and R_{2} = 1/2 equation (5.17) reflects the familiar 2-photon interference:

c_{01} = 0 for R_{2} = \frac{1}{2} .

As indicated in Figure 5.9 the upper arms of the beam splitter can be considered as input and output ports of an indeterministic NS′ gate in the sense that

success ˆ= \{ \begin{matrix}â_{2}(k_{1}) ˆ= state of the \\ lower output of the beam splitter , \end{matrix}

if an ancillary photon in the state â_{2}(k_{2}) is used as lower input for the beam splitter.

For suitable choice of R_{1}, R_{2} two such NS′ gates can be combined with two Hadamard beam splitters ^{...................^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}} and two Hadamard gates H to yield an indeterministic CNOT gate as sketched in Figure 5.10.

[figure: .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; â_{1}(k_{2}); â_{1}(k_{1}); â_{1}(k_{2}); â_{1}(k_{1}); â_{2}(k_{1}); â_{2}(k_{2}); â_{1}(k_{2}); â_{1}(k_{1}); â_{1}(k_{2}); â_{1}(k_{1}); â_{2}(k_{1}); â_{2}(k_{2}); â_{2}(k_{1}); â_{2}(k_{2}); â_{2}(k_{2}); â_{2}(k_{1}); â_{2}(k_{1}); â_{2}(k_{2}); â_{2}(k_{1}); â_{2}(k_{2}); â_{2}(k_{1}); â_{2}(k_{2}); m; n-1; m; n]

Figure 5.8: Selected Transitions at the Beam Splitter.

[figure: .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; â_{2}(k_{2})^{†} Ω; ?^{â_{2}(k_{1})^{†} Ω}; NS′; Ψ = (â_{1}^{†}(k′_{1}))^{m}(â_{2}^{†}(k′_{1}))^{n} Ω; c_{mn} Ψ]

Figure 5.9: Indeterministic NS Gate.

[figure: .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; NS′; NS′; H; H]

Figure 5.10: Indeterministic CNOT Gate.

The Hadamard beam splitters act according to

Ψ_{−} =^{def} \frac{1}{\sqrt{2}}(â_{1}^{†}(k_{1}) â_{2}^{†}(k_{2}) − â_{2}^{†}(k_{1}) â_{1}^{†}(k_{2})) Ω

⟼ \frac{1}{\sqrt{2}}((â_{1}^{†}(k_{1}) + â_{1}^{†}(k_{2}))(â_{2}^{†}(k_{2}) − â_{2}^{†}(k_{1}))

− (â_{2}^{†}(k_{1}) + â_{2}^{†}(k_{2}))(â_{1}^{†}(k_{2}) − â_{1}^{†}(k_{1}))) Ω

= Ψ_{−} ,

Ψ_{+} =^{def} \frac{1}{\sqrt{2}}(â_{1}^{†}(k_{1}) â_{2}^{†}(k_{2}) + â_{2}^{†}(k_{1}) â_{1}^{†}(k_{2})) Ω

⟼ Ψ_{0} =^{def} \frac{1}{\sqrt{2}}(â_{1}^{†}(k_{2}) â_{2}^{†}(k_{2}) − â_{1}^{†}(k_{1}) â_{2}^{†}(k_{1})) Ω ,

â_{j}^{†}(k_{1}) â_{j}^{†}(k_{2}) Ω

⟼ \frac{1}{2}((â_{j}^{†}(k_{1}) + â_{j}^{†}(k_{2}))(â_{j}^{†}(k_{2}) − â_{j}^{†}(k_{1})) Ω

= Ψ_{j} =^{def} \frac{1}{2}((â_{j}^{†}(k_{2}))^{2} − (â_{j}^{†}(k_{1}))^{2}) Ω .

According to Figure 5.9 and equation (5.17) joint succesfull operation of both NS′ gates has the following effect on the relevant intermediate states:^{34}

Ψ_{−} ⟼ \underbrace{c_{10} c_{01}}_{= ^{+} \sqrt{R_{1}R_{2}}(2 R_{2}−1)} Ψ_{−} ,
Ψ_{0} ⟼  \underbrace{c_{11} c_{00}}_{=c_{10} c_{01}} Ψ_{0} ,
Ψ_{1} ⟼  \underbrace{R_{1}R_{2}}_{=c_{20} c_{00}} Ψ_{1} ,
Ψ_{2} ⟼ \underbrace{R_{2} (3 R_{2} − 2)}_{=c_{02} c_{00}} Ψ_{2} .

Since the two Hadamard gates (without the NS′ gates but combined with the necessary mirrors) form a balanced Mach-Zehnder interferometer (see Figure 3.4 of (Lücke, qip)). The whole arrangement outlined in Figure 5.10, but without the two Hadamard gates, acts in case of success according to

Ψ ⟼ \{ \begin{matrix}−R_{1}R_{2} Ψ & for Ψ ∝ H ⊗ H , \\ +R_{1}R_{2} Ψ & for Ψ ⊥ H ⊗ H , \end{matrix}

if

− R_{2} (3 R_{2} − 2) = R_{1}R_{2} = ^{+} \sqrt{R_{1}R_{2}} (2 R_{2} − 1) .  (5.18)

Draft, November 5, 2011

34Equation (5.17) implies

\begin{aligned}c_{10} & = & ^{+} \sqrt{R_{1}R_{2}} , & c_{01} & = & (2 R_{2} − 1) , & c_{11} & = & ^{+} \sqrt{R_{1}} (2 R_{2} − 1) , \\ c_{00} & = & ^{+} \sqrt{R_{2}} , & c_{20} & = & R_{1} ^{+} \sqrt{R_{2}} , & c_{02} & = & ^{+} \sqrt{R_{2}} (3 R_{2} − 2) . \end{aligned}

Addition of the two Hadamard gates, finally, converts it into an indeterministic CNOT gate. The conditions (5.18) are fulfilled for

R_{2} = \frac{3 + \sqrt{2}}{7} ≈ 0.63 , R_{1} = \frac{5 − 3 \sqrt{2}}{7} ≈ 0.1 .

The corresponding probability for successful operation^{35} is |R_{1}R_{2}|^{2} ≈ 0.0047 .

A crucial discovery by (Gottesman and Chuang, 1999) is the following (see also Sect. 3.1.3 of (Lücke, qip)):

[figure: Bell test; Bell test; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; ≡; j; Ψ_{−}; k; Bell test; Ψ_{−}; j; Ψ_{−}; k; Bell test; Ψ_{−}; Û_{j}; Û_{k}; ≡; Û′_{j}; Û′_{k}]

This suggests the following implementation of deterministic CNOT gate:

Bell test ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  jk Bell test ^{Û′_{j}}^{Û′_{k}}  ≡^{Ψ_{−}^{(4)}}

Here an ancillary 4-photon state Ψ_{−}^{(4)} is used that may be prepared by means of an indeterministic CNOT gate according to^{36}

\begin{matrix}Ψ_{−} \{ \\ Ψ_{−} \{ \end{matrix}  [figure]  \} Ψ_{−}^{(4)}

This, in principle, allows scalable quantum computing using only linear optical devices, photon number resolving detectors, and quantum memory.

Draft, November 5, 2011

35For a more efficient, slightly more complicated, implementation see (Sanaka and Resch, 2003). 36See (Knill et al., 2000; Knill et al., 2001) and (Ralph et al., 2002; Dowling et al., 2004).

126 CHAPTER 5. APPLICATIONS OF NON-CLASSICAL LIGHT

Coupling of Quantum Systems

Now it is customary to define the method of measuring physical quantities without defining these quantities themselves. In fact we have no satisfactory reason for ascribing objective existence to physical quantities as distinguished from the numbers obtained when we make the measurements which we correlate with them … It would evidently be philosophically more exact if we spoke of “making measurements” of this, that, or the other type instead of saying that we measure this, that, or the other “physical quantity” … however, we can continue to use the old language, reinterpreting the terms “physical quantity” and “dynamical variable,” and allowing them to stand for the corresponding operator which fixes the nature of the measurement under consideration.

(Kemble, 1937, Section 36b)

6.1 Closed Systems

6.1.1 States and Observables in the Heisenberg Picture

In the Heisenberg picture the physical state of a closed^{1} quantum system is characterized by a density operator ρ̂^{H} on its state space H — being a Hilbert space — and a time-dependent injection

Q ⟼ P̂_{Q}^{H}(t)

of the (naively assumed) testable ‘properties’ Q into the (orthogonal) projectors P̂_{Q}^{H}(t) on H such that:^{2}

p_{Q,t}(ρ̂^{H}) =^{def} Tr(ρ̂^{H} P̂_{Q}^{H}(t)) = {probability for: “Q at time t” to be confirmed by optimal tests .

(6.1)

Draft, November 5, 2011

1For a discussion of the applicability of the concept of a “closed system” (isolated system) see (Leggett, 2001, Lecture 3).

2If every step of a test for ‘Q at time t’ is delayed by Δt then this amount to a test for ‘Q at time t + Δt’. We do not yet require that Q refers to an instant of time.

Remarks:

1. A density operator (statistical operator) is a positive trace

class operator ρ̂ with

Tr(ρ̂) = 1

(see, e.g., Definition 8.3.19 of (Lücke, eine)).

2. The simplest density operators are those of the form

ρ̂ = |Ψ⟩⟨Ψ| , Ψ ∈ H ,

Those states are called vector states and fulfill ^{3}

Tr(ρ̂P̂) = ⟨Ψ | P̂ Ψ⟩ = ∥P̂ Ψ∥^{2}  (6.2)

for all projectors P̂ on H_{field} .

3. Every density operator ρ̂ on H_{field} may be written in the form

ρ̂ = ∑_{ν=1}^{∞} \underbrace{λ_{ν}}_{≥0} |ϕ_{ν}⟩⟨ϕ_{ν}| , ∑_{ν=1}^{∞} λ_{ν} = 1 = ∥Φ_{μ}∥ ∀ μ ∈ ℕ ,

suggesting the following naive interpretation:^{4}

The actual state is one of the vector states ϕ_{1}, ϕ_{2}, … and λ_{ν} is the probability for this being ϕ_{ν} .

4. Note that (6.1) is assumed only over a time interval during which the system can be considered as closed. Thus, especially, the preparation of the state has to be done before this time interval; and nontrivial tests (‘measurements’) — since inevitably changing the state — are allowed only after this time interval.

5. More generally effects, i.e. bounded Operators Â on H with 0 ≤ Â ≤ ˆ1 , could be used^{5} instead of the projectors P̂_{Q}^{H}(t) .

The (naive) idea behind (6.1) is the following:

• P̂_{Q}^{H}(t) projects H onto the closed subspace corresponding to those vector states for which “Q at time t” will be confirmed by every optimal test.

Draft, November 5, 2011

3Of course, Tr(ρ̂) = 1 implies ∥Ψ∥ = 1 . 4In general, however, there is no unique choice for the states ϕ_{ν} . See (Timpson and Brown,

2005) in this connection.

5See (Davies, 1976, Section 2.2), (Kraus, 1983, § 1), or (Busch et al., 1995) for the corresponding physical interpretation.

• ¬P̂_{Q}^{H}(t) =^{def} ˆ1 − P̂_{Q}^{H}(t) projects H onto the closed subspace corresponding to those vector states for which “Q at time t” will never be confirmed by any optimal test.

• Since ∥P̂_{Q}^{H}(t) Ψ∥^{2} + ∥¬P̂_{Q}^{H}(t) Ψ∥^{2} = 1 it seems natural to interpret

∥P̂_{Q}^{H}(t) Ψ∥^{2} = Tr(|Ψ⟩⟨Ψ| P̂_{Q}^{H}(t))

as the probability for “Q at time t” in a state described by ρ̂^{H} = |Ψ⟩⟨Ψ|.

• This motivates (6.1) for vector states.

• Due to linearity of the trace this implies (6.1) for states represented by density operators of the form

ρ̂^{H} = ∑_{ν=0}^{∞} \underbrace{p_{ν}}_{≥0} |Ψ⟩_{ν}⟨Ψ|_{ν} , ∑_{ν=0}^{∞} p_{ν} = 1 ,  (6.3)

if,^{6} for every ν, p_{ν} is (naively) identified with the probability for the system to be in the vector state corresponding to Ψ_{ν} .

• Since every density operator can be written in the form (6.3), thanks to the Hilbert-Schmidt theorem (see, e.g., Corollary 8.3.18 of (Lücke, eine)), this implies (6.1) for all states.

There is a one-one correspondence between self-adjoint operators Â (see, e.g., Definition 8.3.7 (Lücke, eine)) and families \left\{{ \atop ÊÂ ; λ}\right\}_{λ∈ℝ} of projectors fulfilling the following requirements (see Theorem 8.3.24 of (Lücke, eine)):

1.

(λ_{1} ≤ λ_{2} ⟹ ⟨Ψ | ÊÂ ; λ_{1}Ψ⟩ ≤ ⟨Ψ | Ê_{λ_{2}}Ψ⟩) ∀ Ψ ∈ H , λ_{1}, λ_{2} ∈ ℝ .

2.

lim_{ϵ→−∞} ⟨Ψ | ÊÂ ; λΨ⟩ = 0 ∀ Ψ ∈ H , λ ∈ ℝ .

3.

lim_{ϵ→+∞} ⟨Ψ | ÊÂ ; λΨ⟩ = ⟨Ψ | Ψ⟩ ∀ Ψ ∈ H , λ ∈ ℝ .

4.

lim_{ϵ→+0} ⟨Ψ | ÊÂ ; λ+ϵΨ⟩ = ⟨Ψ | ÊÂ ; λΨ⟩ ∀ Ψ ∈ H , λ ∈ ℝ .

Draft, November 5, 2011

6See also Section 6.2.2 and (Zeh, 2000) for mixed states arising for subsystems from vector states of a composed system.

5.

DÂ = {Ψ ∈ H : ∫ |λ|^{2} d⟨Ψ | ÊÂ ; λ Ψ⟩ < ∞} .

6.

⟨Ψ | Â Ψ⟩ = ∫ λ d⟨Ψ | ÊÂ ; λ Ψ⟩ ∀ Ψ ∈ DÂ .

Therefore self-adjoint operators are interpreted as observables of physical quantities — like ‘energy at time t’ — in the following sense:

Tr(ρ̂^{H} ÊÂ ; λ) = {probability for: “A ∈ (−∞, λ]” to be confirmed by optimal tests .

Consequently:

Tr(ρ̂^{H} Â) = expectation value of A in the state ˆ= ρ̂^{H} .

(6.4)

Projectors are just observables of physical quantities for which only 0 and 1 are ‘possible values’.

6.1.2 Time Evolution and Projective Measurements

For a closed quantum system the time evolution is assumed to be given by a selfadjoint operator Ĥ called the Hamiltonian of the system, in the sense that

P̂_{Q}^{H}(t) = e^{+ \frac{i}{ℏ} Ĥ t} P̂_{Q}^{H}(0) e^{− \frac{i}{ℏ} Ĥ t} ∀ t ∈ ℝ  (6.5)

holds for all ‘testable properties’ Q.

Remark: See, e.g., (Lücke, 1996, Section2.5) for a justification of this assumption and Definition 8.3.30 of (Lücke, eine) for the definition of functions of self-adjoint operators.

This allows switching to the Schrödinger picture

Tr(ρ̂_{t}^{S} P̂_{Q}^{S}) = {probability for “Q at time t” to be confirmed by optimal tests ,  (6.6)

where

ρ̂_{t}^{S} =^{def} e^{− \frac{i}{ℏ} Ĥ t} ρ̂^{H} e^{+ \frac{i}{ℏ} Ĥ t} ,  (6.7)
P̂_{Q}^{S} =^{def} P̂^{H}(0)
= e^{− \frac{i}{ℏ} Ĥ t} P̂_{Q}^{H}(t) e^{+ \frac{i}{ℏ} Ĥ t} .  (6.8)

Note that, formally, (6.7) is equivalent to the Liouville equation

iℏ \frac{∂}{∂t} ρ̂_{t}^{S} = [Ĥ , ρ̂_{t}^{S}]_{−}  (6.9)

— to be suitably interpreted.

If the Hamiltonian is of the form

Ĥ = Ĥ_{0} + V̂  (6.10)

with e^{− \frac{i}{ℏ} Ĥ_{0} t} known and V̂ ‘small’ in some sense then it may be convenient to switch to the interaction picture

Tr(ρ̂_{t}^{I} P̂_{Q}^{I}(t)) = {probability for “Q at time t” to be confirmed by optimal tests ,  (6.11)

where

ρ̂_{t}^{I} =^{def} e^{+ \frac{i}{ℏ} Ĥ_{0} t} e^{− \frac{i}{ℏ} Ĥ t} ρ̂^{H} e^{+ \frac{i}{ℏ} Ĥ t} e^{− \frac{i}{ℏ} Ĥ_{0} t} ,  (6.12)
P̂_{Q}^{I}(t) = e^{+ \frac{i}{ℏ} Ĥ_{0} t} P̂_{Q}^{H}(0) e^{− \frac{i}{ℏ} Ĥ_{0} t} .  (6.13)

The reason will become evident in Section 6.1.3.

For simplicity we require that for every ‘testable property’ Q there is a projective measurement, i.e. a test changing the Heisenberg state according to^{7}

ρ̂^{H} ⟼ p_{Q,t}(ρ̂^{H}) ρ̂_{Q,t}^{H} + (1 − p_{Q,t}(ρ̂^{H}))ρ̂_{¬Q,t}^{H}  (6.14)

with

ρ̂_{Q,t}^{H} =^{def} \frac{P̂_{Q}^{H}(t)ρ̂^{H}P̂_{Q}^{H}(t)}{Tr(P̂_{Q}^{H}(t)ρ̂^{H}P̂_{Q}^{H}(t))}

describing the subensemble of all those individual systems for which ‘Q at time t’ has been confirmed and

ρ̂_{¬Q,t}^{H} =^{def} \frac{P̂_{¬Q}^{H}(t)ρ̂^{H}P̂_{¬Q}^{H}(t)}{Tr(P̂_{¬Q}^{H}(t)ρ̂^{H}P̂_{¬Q}^{H}(t))} , P̂_{¬Q}^{H}(t) = ˆ1 − P̂_{Q}^{H}(t) ,

describing the subensemble of all those individual systems for which ‘Q at time t’ is denied by the test. This assumption is consistent in the sense that

Tr(ρ̂^{H}) = Tr(p_{Q,t}(ρ̂^{H}) ρ̂_{Q,t}^{H} + (1 − p_{Q,t}(ρ̂^{H}))ρ̂_{¬Q,t}^{H}) .

Draft, November 5, 2011

7Recall (6.1).

Note that

p_{Q′,t}(ρ̂^{H}) = p_{Q′,t}(p_{Q,t}(ρ̂^{H}) ρ̂_{Q,t}^{H} + (1 − p_{Q,t}(ρ̂^{H}))ρ̂_{¬Q,t}^{H}) ∀ ρ̂^{H}
⟺ [P̂_{Q}^{H}(t) , P̂_{Q′}^{H}(t)]_{−} = 0 ,

since

Tr(ρ̂Â) = Tr(Â ρ̂)  (6.15)

holds for all trace class operators ρ̂ and all bounded operators Â. Therefore ‘testable properties’ Q , Q′ with commuting P̂_{Q}^{H}(t) , P̂_{Q′}^{H}(t) are called compatible.^{8}

6.1.3 Time Dependent Perturbation Theory^{9}

In the interaction picture the time-dependent density operator (6.12) fulfills the differential equation^{10}

iℏ\frac{d}{dt}ρ̂_{t}^{I} = [V̂^{I}(t), ρ̂_{t}^{I}]_{−} ,  (6.16)

where

V̂^{I}(t) =^{def} e^{+ \frac{i}{ℏ} Ĥ_{0} t} V̂ e^{− \frac{i}{ℏ} Ĥ_{0} t} .  (6.17)

(6.16) is (formally) equivalent to the integral equation

ρ̂_{t}^{I} = ρ̂_{0}^{I} − \frac{i}{ℏ} ∫_{0}^{t} [V̂^{I}(t′), ρ̂_{t′}^{I}]_{−} dt′

which may be solved for given ρ̂_{0}^{I} by iteration

ρ̂_{t}^{(0)} =^{def} ρ̂_{0}^{I} ,
ρ̂_{t}^{(ν)} =^{def} ρ̂_{0}^{I} − \frac{i}{ℏ} ∫_{0}^{t} [V̂^{I}(t′), ρ̂_{t′}^{(ν−1)}]_{−} dt′ for ν = 1, 2, …

(6.18)

and taking the limit^{11}

ρ̂_{t}^{I} = lim_{ν→∞} ρ̂_{t}^{(ν)}

Draft, November 5, 2011

8Note that

[P̂_{Q}^{H}(0) , P̂_{Q′}^{H}(0)]_{−} = 0 ⟺_{(6.5)} ([P̂_{Q}^{H}(t) , P̂_{Q′}^{H}(t)]_{−} = 0 ∀ t ∈ ℝ) .

9In this subsection we do not care about mathematical details since even for a divergent perturbative expansion the lowest order may provide useful results.

10Compare Equation (6.9).

11For simplicity, we do not specify the type of convergence.

if V̂^{I}(t) is ‘sufficiently small’. For typical applications the approximation

ρ̂_{t}^{I} ≈ ρ̂_{t}^{(2)} = ρ̂_{0}^{I} − \frac{i}{ℏ} ∫_{0}^{t} [V̂^{I}(t′), ρ̂_{0}^{I}]_{−} dt′

−ℏ^{−2} ∫_{0}^{t} (∫_{0}^{t′} [V̂^{I}(t′), [V̂^{I}(t′′), ρ̂_{0}^{I}]_{−}]_{−} dt′′) dt′

is good enough. Note that, thanks to (6.15),

P̂ρ̂_{0}^{I} = ρ̂_{0}^{I} P̂ = 0

⟹ Tr(ρ̂_{t}^{(2)}P̂) = ℏ^{−2} ∫_{[0,t]×[0,t]} Tr(V̂^{I}(t′) ρ̂_{0}^{I} V̂^{I}(t′′) P̂) dt′′ dt′ . (6.19)

This is in agreement with Dyson’s perturbation theory for state vectors.^{12}

6.2 Bipartite Systems

6.2.1 Composition of Distinguishable Systems

Let S_{1} and S_{2} be distinguishable quantum systems not interacting with each other. In order to describe S_{1} and S_{2} together as a single quantum system S in the Heisenberg picture one takes the state space H = H_{1} ⊗ H_{2} , where H_{1} resp. H_{2} is the state space of S_{1} resp. S_{2} .

Remark: See, e.g., Section 8.4.2 of (Lücke, eine) for the definition of the tensor products ⊗.

The Hamiltonian of S is

Ĥ = Ĥ_{1} ⊗ ˆ1 + ˆ1 ⊗ Ĥ_{2} ,  (6.20)

where Ĥ_{1} resp. Ĥ_{2} is the Hamiltonian of S_{1} resp. S_{2} , and therefore

e^{+ \frac{i}{ℏ} Ĥ t} (P̂_{Q_{1}}^{H}(0) ⊗ P̂_{Q_{2}}^{H}(0)) e^{− \frac{i}{ℏ} Ĥ t} = P̂_{Q_{1}}^{H}(t) ⊗ P̂_{Q_{2}}^{H}(t) ∀ t ∈ ℝ  (6.21)

holds for all ‘testable properties’ Q_{1} and Q_{2} of S_{1} resp. S_{2} .

If the Heisenberg state of S is given by

ρ̂^{H} = ρ̂_{1}^{H} ⊗ ρ̂_{2}^{H}  (6.22)

then

Tr(ρ̂^{H} (P̂_{Q_{1}}^{H}(t) ⊗ P̂_{Q_{2}}^{H}(t))) = Tr(ρ̂_{1}^{H} P̂_{Q_{1}}^{H}(t) ) Tr(ρ̂_{2}^{H} P̂_{Q_{2}}^{H}(t) ) .

This suggests the interpretation

Tr(ρ̂^{H} (P̂_{Q_{1}}^{H}(t) ⊗ P̂_{Q_{2}}^{H}(t))) = {probability for: “Q_{1} and Q_{2} at time t” to be confirmed by optimal tests .  (6.23)

Draft, November 5, 2011

12See Appendix A.4.

As long as (6.20) holds (6.23) is equivalent to

Tr(ρ̂_{t}^{S} (P̂_{Q_{1}}^{S} ⊗ P̂_{Q_{2}}^{S})) = {probability for: “Q_{1} and Q_{2} at time t” to be confirmed by optimal tests ,

(6.24)

thanks to (6.21). However, if an interaction between S_{1} and S_{2} is introduced, corresponding to the addition of a perturbation

V̂ ≠ V̂_{1} ⊗ ˆ1 + ˆ1 ⊗ V̂_{2} ∀ V̂_{1} , V̂_{2}  (6.25)

on the r.h.s. of (6.20), then (6.21) is no longer consistent for all Q_{1}, Q_{2} . Then we try to keep (6.24) for as many Q_{1}, Q_{2} as possible.^{13}

Even for the restricted class of Q_{1}, Q_{2} do correlations arise, i.e.

Tr(ρ̂_{t}^{S} (P̂_{Q_{1}}^{S} ⊗ P̂_{Q_{2}}^{S})) ≠^{i.g.} Tr(ρ̂_{t}^{S} (P̂_{Q_{1}}^{S} ⊗ ˆ1))Tr(ρ̂_{t}^{S} (ˆ1 ⊗ P̂_{Q_{2}}^{S})) ,

since:

Even if the Schrödinger state at time 0 is factorized, i.e. if

ρ̂_{0}^{S} = ρ̂_{1} ⊗ ρ̂_{2} ,

ρ̂_{t}^{S} is typically not even separable^{14} for t > 0 .

Correlations of subsystems also arise for classical systems. But:

The subsystems S_{1} , S_{2} of S should be considered as classically correlated at time t only if ρ̂_{t}^{S} is separable (Werner, 1989).

6.2.2 Partial States

Now let us consider a bipartite system S as described in 6.2.1 and ask only for properties Q_{1} of the subsystem S, i.e. for the partial state of S_{1} given by the mapping

(Q_{1}, t) ⟼ Tr(ρ̂_{t}^{S} (P̂_{Q_{1}}^{S} ⊗ ˆ1)) = {probability for: “Q_{1} at time t” to be confirmed by optimal tests . .

Draft, November 5, 2011

13Problems arise whenever the relation between two observables depends on the dynamics; a typical example being the observables of position and velocity.

14A (momentary Schrödinger) state with density operator ρ̂ is called separable if for every ϵ > 0 there exist N ∈ ℕ, p_{1}, …, p_{N} ≥ 0 , and density operators ρ̂_{1,ν}, …, ρ̂_{2,N} with

Tr |ρ̂ − ∑_{ν=1}^{N} p_{ν} ρ̂_{1,ν} ⊗ ρ̂_{2,ν} | < ϵ .

Otherwise the state is called entangled.

One may show that (for every t) there is a density operator Tr_{2}(ρ̂_{t}^{S}) on H_{1} , called the partial trace of ρ̂_{t}^{S} w.r.t. H_{2} , for which

Tr(ρ̂_{t}^{S} (P̂_{Q_{1}}^{S} ⊗ ˆ1)) = Tr(Tr_{2}(ρ̂_{t}^{S}) P̂_{Q_{1}}^{S}) ∀ Q_{1} .  (6.26)

Obviously, Tr_{2}(ρ̂_{t}^{S}) represents the partial Schrödinger state of S_{1} at time t. But:

Even if ρ̂_{t}^{S} represents a vector state of S its partial trace w.r.t. H_{2} usually does not represent a vector state^{15} of S_{1} .

Moreover, the interaction between S_{1} and S_{2} has to be very special^{16} for all the Tr_{2}(ρ̂_{t}^{S}) with t > 0 to be determined by Tr_{2}(ρ̂_{0}^{S}), at all. Usually, therefore, the Heisenberg picture does not work for subsystems.

6.2.3 Composition of Indistinguishable Systems

Two quantum systems S_{1} , S_{2} are called indistinguishable if they may be described in the same way, especially with

H_{1} = H_{2} , Ĥ_{1} = Ĥ_{2} , {Q_{1}} = {Q_{2}} .

Then there is a natural decomposition

H_{1} ⊗ H_{2} = H^{+} ⊕ H^{−} ,

where H^{σ} denotes the Hilbert subspace of H_{1} ⊗ H_{2} generated by all vectors Ψ of the form

Ψ = ψ_{1} ⊗ ψ_{2} + σψ_{2} ⊗ ψ_{1} , ψ_{1}, ψ_{2} ∈ H_{1} ,

and the composition described in 6.2.1 has to be modified in the following way:^{17}

1. The state space of the composed system is H^{σ} with σ = +resp. σ = − if S_{1} is a Boson resp. a Fermion.

2. Only those properties of the composed system can be considered as ‘testable properties’ which do not depend on the enumeration of the subsystems. Thus, for instance, if Q_{1} ≠ Q_{2} are compatible properties of S_{1} then

(‘Q_{1} for S_{1}’ ∧ ‘Q_{2} for S_{2}’) ∨ (‘Q_{2} for S_{1}’ ∧ ‘Q_{1} for S_{2}’)

is a ‘testable property’ of the composed system represented by the projector

(P̂_{Q_{1}}^{S} ⊗ P̂_{Q_{2}}^{S} + P̂_{Q_{2}}^{S} ⊗ P̂_{Q_{1}}^{S} − (P̂_{Q_{1}}^{S} P̂_{Q_{2}}^{S}) ⊗ (P̂_{Q_{1}}^{S} P̂_{Q_{2}}^{S}))/^{\}H^{σ}

in the Schrödinger picture. But

‘Q_{1} for S_{1}’ ∧ ‘Q_{2} for S_{2}’

cannot be considered as a testable.

Draft, November 5, 2011

15The evolution of an initial partial vector state into a mixed state is called decoherence.

16See Section 6.3

17Here we assume V̂ = 0 , for simplicity.

3. The Hamiltonian has to be replaced by

(Ĥ_{1} ⊗ ˆ1 + ˆ1 ⊗ Ĥ_{2} + V̂ ) /^{\}H^{σ}

with V̂ leaving H^{σ} invariant.

Recall that particles with integer spin like photons are Bosons while particles with half-integer spin like electrons are Fermions.

For multipartite systems and the possibility of parastatistics we refer to (Ohnuki and Kamefuchi, 1982).

6.3 Open Systems

See (Davies, 1976; Kraus, 1983; Carmichael, 1993; Alicki, 2003).

138 CHAPTER 6. COUPLING OF QUANTUM SYSTEMS

Perturbation Theory of Radiative Transitions

Being able to predict things or describe them, however accurately, is not at all the same thing as understanding them.

(Deutsch, 1997, p. 2)

7.1 Single Atoms in General

7.1.1 Naive Interaction Picture

Let S_{1} be a single 1-electron atom with approximate Hamiltonian^{1}

Ĥ_{atom} =^{def} \frac{p̂_{can}^{2}}{2m} + qcA^{0}(x̂)

acting on H_{atom} = L^{2}(ℝ^{3}) and let S_{2} be the electromagnetic radiation system with state space H_{field} and Hamiltonian

Ĥ_{field} = \frac{1}{2} ∫ (ϵ_{0} : Ê_{⊥}(x, 0) · Ê_{⊥}(x, 0) : + \frac{1}{μ_{0}} : B̂(x, 0) · B̂(x, 0) :) dV_{x} ,

where Ê_{⊥}(x, 0) resp. B̂(x, 0) denotes the observable of the electric^{2} resp. magnetic field at time t = 0 .

Draft, November 5, 2011

1We assume that the atom’s center of mass fixed at x = 0 and neglect spin-orbit interaction and the like. q < 0 is the electron’s charge, m > 0 its (reduced) mass , c ≈ 3 · 10^{−10}cm the velocity of light in vacuum, and cA^{0}(x̂) the electric (binding) potential of the nucleus. p̂_{can} denotes the electron’s canonical momentum operator and x̂ its position operator:

(p̂_{can}^{j} ψ) (x) = \frac{ℏ}{i} \frac{∂}{∂x^{j}} ψ(x) , (x̂^{j} ψ) (x) = x^{j} ψ(x) .

2Note that only in the absence of charges the total electric field Ê(x, 0) coincides with its transversal part Ê_{⊥}(x, 0) (recall Footnote 33 of Chapter 1).

Agreement: From now on we identify operators B̂ on H_{atom} with the corresponding operators B̂ ⊗ˆ1_{field} on H_{atom} ⊗H_{field} . Similarly we identify operators Ĉ on H_{field} with the corresponding operators ˆ1_{atom} ⊗ Ĉ on H_{atom} ⊗ H_{field} .

Since the interaction between S_{1} and S_{2} cannot be switched off let us consider these systems as components of a closed bipartite system, as described in 6.2.1, with minimal coupling^{3}

V̂ = −\frac{q}{2m}(p̂_{can} · Â (x̂, 0) + Â (x̂, 0) · p̂_{can})

+\frac{q^{2}}{2m}:Â^{2}(x̂, 0): ,

(7.1)

where we assume that the naive interaction picture works, i.e. that the state space and the time zero fields Ê_{⊥}(x, 0) , B̂(x, 0) , Â(x, 0) can be chosen identical with those of the free theory described in 1.2.2.

Warning: Actually, as stressed in (Hoffmann, 1994, Sect. 3.5), V̂ is not well-defined. This is a simple consequence of the equations

Â_{0}^{j_{1}}(x_{1}, t_{1}) Â_{0}^{j_{2}}(x_{2}, t_{2}) = :Â_{0}^{j_{1}}(x_{1}, t_{1}) Â_{0}^{j_{2}}(x_{2}, t_{2}):
+ ⟨Ω | Â_{0}^{j_{1}}(x_{1}, t_{1}) Â_{0}^{j_{2}}(x_{2}, t_{2}) Ω⟩

(7.2)

and

⟨Ω | Â_{0}^{j_{1}}(x_{1}, t_{1}) Â_{0}^{j_{2}}(x_{2}, t_{2}) Ω⟩

= ⟨Ω | [(Â_{0}^{j_{1}})^{(+)}(x_{1}, t_{1}) , ((Â_{0}^{j_{2}})^{(+)}(x_{2}, t_{2}))^{†}]_{−} Ω⟩
=_{(1.45)} (2π)^{−3} μ_{0} ℏ c ∫ (δ_{j_{1}j_{2}} − \frac{k^{j_{1}}k^{j_{2}}}{|k|^{2}} )e^{ik(x_{1}−x_{2})} e^{−i c|k|(t_{1}−t_{2})} \frac{dV_{k}}{2 |k|}

(7.3)

for the free field operators Â_{0}(x, t) . Therefore we should replace Â(x, 0)

by ∫ Â_{0}(x−y, 0) φ(y) dy with φ ∈ S(ℝ^{3}) (introduce a ultraviolet cutoff

this way) and determine the limit dynamics for φ → δ. However, we are not going to elaborate on such mathematical details.^{4}

Draft, November 5, 2011

3Our notation should be understood in the sense that, e.g.,

⟨ψ′ ⊗ χ′ | Â (x̂, 0) (ψ ⊗ χ)⟩ =^{def} ∫ (ψ′(x))^{∗} ψ(x) ⟨χ′ | Â (x, 0) χ⟩_{field} dV_{x}

holds for sufficiently well-behaved ψ ⊗ χ, ψ′ ⊗ χ′ ∈ H. See (Simon, 1971) in this context. 4See (Fröhlich et al., 2003) for such techniques.

Remark: (7.2) is a special case of Wick’s theorem; see, e.g., Sect. 3.2.1 of (Lücke, qft).

Now the time evolution of the composed system is generated by the total Hamiltonian

Ĥ = Ĥ_{atom} + Ĥ_{field} + V̂  (7.4)
= \frac{1}{2m}:(\underbrace{p̂_{can} − q Â(x̂, 0)}_{= p̂_{kin} =^{def} m \frac{i}{ℏ}[Ĥ,x̂]_{−}})^{2} : + qcA^{0}(x̂) + Ĥ_{field} .  (7.5)

For (sufficiently well-behaved) vector states the dynamical law (6.7) suggests validity of the Schrödinger equation^{5}

iℏ \frac{d}{dt}Ψ_{t} = Ĥ Ψ_{t} ∀ t ∈ ℝ ,  (7.6)

where ρ̂_{t}^{S} = |Ψ_{t}⟩⟨Ψ_{t}|.

Remark: See Section 9.1.1 of (Lücke, eine) for arbitrary vector states.

Assuming^{6}

\begin{matrix}Ψ_{t} ≈ ψ_{t} ⊗ χ_{t} , \\ ∥χ_{t}∥ = 1 \end{matrix} \} ∀ t ∈ ℝ  (7.7)

and taking the inner product of (7.6) with ψ′ ⊗ χ_{t} gives

∫ (ψ′(x))^{∗} iℏ \frac{∂}{∂t}ψ_{t}(x) dV_{x}

=  ⟨ψ′ ⊗ χ_{t} | iℏ \frac{d}{dt} (ψ_{t} ⊗ χ_{t})⟩ − ⟨ψ′ | ψ_{t}⟩⟨χ_{t}|iℏ \frac{d}{dt} χ_{t}⟩
≈_{(7.7),(7.6)} ⟨ψ′ ⊗ χ_{t} | Ĥ (ψ_{t} ⊗ χ_{t})⟩ − ⟨ψ′ | ψ_{t}⟩⟨χ_{t}|iℏ \frac{d}{dt} χ_{t}⟩
=_{(7.4),(7.1)} ⟨ψ′ | Ĥ_{atom} ψ_{t}⟩ + ⟨ψ′ | ψ_{t}⟩⟨χ_{t}|(Ĥ_{field} − iℏ \frac{d}{dt})χ_{t}⟩

−\frac{q}{2m} ∫ (ψ′(x))^{∗} (p̂_{can} · A_{ext}(x, t) + A_{ext}(x, t) · p̂_{can}) ψ_{t}(x) dV_{x}

+\frac{q^{2}}{2m} ∫ (ψ′(x))^{∗} ⟨χ_{t} |:Â^{2}(x, 0): χ_{t}⟩ ψ_{t}(x) dV_{x} ,

where

A_{ext}(x, t) =^{def} ⟨χ_{t} | Â(x, 0) χ_{t}⟩ .

Draft, November 5, 2011

5Note, however, that the time evolution of ρ̂_{t}^{S} — contrary to that of Ψ_{t} — is not changed by addition of a constant to Ĥ.

6This possibility seems reasonable for coherent states χ_{0} ; see end of 7.2.1.

Since ψ′ is arbitrary, this establishes the exterior field formalism

iℏ \frac{∂}{∂t}ψ_{t}(x) ≈ (\frac{1}{2m} (\frac{ℏ}{i}∇ − qA_{ext}(x, t))^{2} + qcA_{ext}^{0}(x, t) + qcA^{0}(x)) ψ_{t}(x) , (7.8)

if (7.7) holds ,

with suitably defined A_{ext}^{0}(x, t) . We postpone the discussion under which conditions (7.7) is reasonable.^{7}

7.1.2 Electric Field Representation^{8}

Let us choose some φ ∈ S (ℝ^{3}) with

φ(x) = (φ(x))^{∗} ≈ δ(x)  (7.9)

and exploit the Wigner symmetry corresponding to the unitary operator

Û = exp (−\frac{i}{ℏ} q x̂ · ∫ Â(y, 0) φ(y) dy) .  (7.10)

Remark: See,e.g., Section 8.3.4 of (Lücke, eine) for the notion of Wigner symmetry.

This means that all operators B̂ have to be replaced by

AdÛ B̂ =^{def} Û B̂ Û^{−1}  (7.11)

without changing the physical interpretation. The new observable for the transversal part of the electric field, for example, in the Schrödinger picture is now

AdÛ Ê_{⊥}(x, 0) = Ê_{⊥}(x, 0) − \frac{1}{ϵ}_{0}P̂_{⊥}^{(0)}(x) ,  (7.12)

where

P̂_{⊥}^{(0)}(x) =^{def} ϵ_{0} ∑_{j=1}^{3} q x̂^{j} \frac{i}{ℏ} ∫ [Â^{j}(y, 0), Ê_{⊥}(x, 0)]_{−} φ(y) dy .  (7.13)

Proof of (7.12): As usual, we (formally) define

adÂ B̂ =^{def} [Â, B̂]_{−} .

Draft, November 5, 2011

7See (Gemmer and Mahler, 2001) for the case of bipartite systems with finite dimensional state spaces.

8Here we follow (Cohen-Tannoudji et al., 1992, Appendix 5), to some extent, rather than (Mandel and Wolf, 1995, Section 14.1.3). See also (Scully and Zubairy, 1999, App. 5.A) for the corresponding exterior field formalism.

Then (1.37) implies that ad_{ln(Û)}Ê_{⊥}(x, 0) is a function of x̂ and, consequently,

(ad_{ln(Û)})^{ν} Ê_{⊥}(x, 0) = 0 ∀ ν > 1 .

Therefore (7.12) follows from the Campbell-Hausdorff formula^{9}

Ad_{exp(Â)} B̂ = exp(adÂ) B̂  (7.14)

and

ad_{ln(Û)}Ê_{⊥}(x, 0) = −\frac{i}{ℏ} q ∑_{j=1}^{3} x̂^{j} ∫ [Â^{j}(y, 0), Ê_{⊥}(x, 0)]_{−} φ(y) dy .

Similarly we get:

AdÛ x̂ = x̂ ,

AdÛ p̂_{kin} = p̂_{kin} + q \frac{i}{ℏ} ∑_{j=1}^{3} [p̂_{can}, x̂^{j}]_{−} ∫ Â^{j}(y, 0) φ(y) dy

= p̂_{can} − q Â (x̂, 0) + q ∫ Â(y, 0) φ(y) dy  (7.15)
AdÛ B̂(x, 0) =_{(1.48)} B̂(x, 0) ,

AdÛ Ĥ = \frac{1}{2m}:(AdÛ p̂_{kin})^{2} : + qcA^{0}(x̂) + Ĥ_{field}

− ∫ P̂_{⊥}^{(0)}(x) · Ê_{⊥}(x, 0) dV_{x} + \frac{1}{2ϵ_{0}} ∫ (P̂_{⊥}^{(0)}(x))^{2} dV_{x} .(7.16)

Proof of (7.16): According to (7.5) it is sufficient to show

AdÛ Ĥ_{field} = Ĥ_{field} − ∫ P̂_{⊥}^{(0)}(x) · Ê_{⊥}(x, 0) dV_{x} + \frac{1}{2ϵ_{0}} ∫ (P̂_{⊥}^{(0)}(x))^{2} dV_{x} .

This follows from

AdÛ Ĥ_{field} = \frac{1}{2} ∫ (ϵ_{0} : (AdÛ Ê_{⊥}(x, 0))^{2} : + \frac{1}{μ_{0}} : B̂^{2}(x, 0) :) dV_{x}

and (7.12).

Draft, November 5, 2011

9Formally (7.14) is a consequence of the fact that Ad_{exp(λÂ)} B̂ and exp(ad_{λÂ}) B̂, considered as operator-valued functions of λ, fulfill the same first order differential equation and the same initial condition for λ = 0 .

Note that (1.49) and (7.13) imply^{10}

P̂_{⊥}^{(0)}(x) = q ∑_{j,j′=1}^{3} x̂^{j} e_{j′} ∫ δ_{⊥}^{jj′}(y − x) φ(y) dy .  (7.17)

Since

∫ δ_{⊥}^{jj′}(y − x) Ê_{⊥}^{j′}(x) dV_{x} = Ê_{⊥}^{j}(y) ,

(7.17) implies

∫ P̂_{⊥}^{(0)}(x) · Ê_{⊥}(x, 0) dV_{x} = q x̂ · ∫ Ê_{⊥}(y, 0) φ(y) dy .

Substituting the latter into (7.16) we get

AdÛ Ĥ = (\frac{1}{2m}:(AdÛ p̂_{kin})^{2} : + qcA^{0}(x̂) + \frac{1}{2ϵ_{0}} ∫ (P̂_{⊥}^{(0)}(x))^{2} dV_{x}) + Ĥ_{field}
−q x̂ · ∫ Ê_{⊥}(y, 0) φ(y) dy .

(7.18)

Therefore we call the new representation, resulting from the Wigner symmetry transformation AdÛ , the electric field representation.

Warning: The new observable of the (transversal part of the) electric field is AdÛ Ê_{⊥}(x, 0) rather than Ê_{⊥}(x, 0) . According to (7.12)/(7.13), both coincide only ‘outside’ the atom.

7.1.3 First Order Perturbation Theory

Let us go back to the representation described in 7.1.1 which we shall call the Coulomb representation. Then we have

Ĥ_{0} =^{def} Ĥ_{atom} + Ĥ_{field}

Draft, November 5, 2011

10The observable of the transversal polarization density is the operator-valued distribution

P̂_{⊥}(x) =^{def} q ∑_{j,j′=1}^{3} x̂^{j} e_{j′}δ_{⊥}^{jj′}(x̂ − x)
= ϵ_{0} ∑_{j=1}^{3} q x̂^{j} \frac{i}{ℏ} [Â^{j}(x̂, 0), Ê_{⊥}(x, 0)]_{−} .

7.1. SINGLE ATOMS IN GENERAL 145 as unperturbed Hamiltonian and

V̂^{I}(t) =_{(6.17)} e^{+ \frac{i}{ℏ} Ĥ_{0} t} V̂ e^{− \frac{i}{ℏ} Ĥ_{0} t}

with V̂ given by (7.1). Let us consider the special case

\begin{matrix}ρ̂_{0}^{I} & = & |ψ ⊗ χ⟩⟨ψ ⊗ χ| , \\ P̂ & = & |ψ̌ ⊗ χ̌⟩⟨ψ̌ ⊗ χ̌| , \\ P̂ ρ̂_{0}^{I} & = & ρ̂_{0}^{I} P̂ = 0 . \end{matrix} \}  (7.19)

Then^{11}

p_{t} (ψ ⊗ χ → ψ̌ ⊗ χ̌)

=^{def} Tr(ρ̂_{t}^{I} P̂)
≈_{(6.19)} ℏ^{−2} |∫_{0}^{t} ⟨ψ̌ ⊗ χ̌ | V̂^{I}(t′) (ψ ⊗ χ)⟩ dt′|^{2}
=_{(6.17)} ℏ^{−2} |∫_{0}^{t} ⟨ψ̌_{t′}^{(0)} ⊗ χ̌_{t′}^{(0)} | V̂ (ψ_{t′}^{(0)} ⊗ χ_{t′}^{(0)})⟩ dt′|^{2}
=_{(7.1)} ℏ^{−2}|∫_{0}^{t}(−\frac{q}{m} ∫ ∑_{j=1}^{3} ⟨χ̌_{t′}^{(0)} | Â^{j}(x, 0) χ_{t′}^{(0)}⟩ (ψ̌_{t′}^{(0)}(x))^{∗} \frac{ℏ}{i} \frac{∂}{∂x^{j}} ψ_{t′}^{(0)}(x) dV_{x}

+ \frac{q^{2}}{2m} ∫ ⟨χ̌_{t′}^{(0)} | :Â^{2}(x, 0): χ_{t′}^{(0)}⟩ (ψ̌_{t′}^{(0)}(x))^{∗} ψ_{t′}^{(0)}(x) dV_{x})dt′|^{2} ,

= ℏ^{−2}|∫_{0}^{t}(−\frac{q}{m} ∫ ∑_{j=1}^{3} ⟨χ̌ | Â^{j}(x, t′) χ⟩ (ψ̌_{t′}^{(0)}(x))^{∗} \frac{ℏ}{i} \frac{∂}{∂x^{j}} ψ_{t′}^{(0)}(x) dV_{x}

+ \frac{q^{2}}{2m} ∫ ⟨χ̌ | :\binom{}{Â}^{2}(x, t′): χ⟩ (ψ̌_{t′}^{(0)}(x))^{∗} ψ_{t′}^{(0)}(x) dV_{x})dt′|^{2} ,

where we used

Â(x, t) = e^{+ \frac{i}{ℏ} Ĥ_{field} t} Â(x, 0) e^{− \frac{i}{ℏ} Ĥ_{field} t}  (7.20)

and

ψ_{t}^{(0)} =^{def} e^{− \frac{i}{ℏ} Ĥ_{atom} t} ψ , ψ̌_{t}^{(0)} =^{def} e^{− \frac{i}{ℏ} Ĥ_{atom} t} ψ̌ ,

χ_{t}^{(0)} =^{def} e^{− \frac{i}{ℏ} Ĥ_{field} t} χ , χ̌_{t}^{(0)} =^{def} e^{− \frac{i}{ℏ} Ĥ_{field} t} χ̌ .

Draft, November 5, 2011

11Note that ∇ · Â(x, 0) = 0 .

Usually ⟨χ̌ | :\binom{}{Â}^{2}(x, t′): χ⟩ will be neglected:^{12}

p_{t}(ψ ⊗ χ → ψ̌ ⊗ χ̌)

≈ \frac{q^{2}}{m^{2}ℏ^{2}} |∫_{0}^{t}(∫ ∑_{j=1}^{3} ⟨χ̌ | Â^{j}(x, t′) χ⟩ (ψ̌_{t′}^{(0)}(x))^{∗} \frac{ℏ}{i} \frac{∂}{∂x^{j}} ψ_{t′}^{(0)}(x) dV_{x})dt′|^{2}

if (7.19) holds .

(7.21)

For obvious reasons^{13}

p_{t}(ψ ⊗ χ → ψ̌ ⊗ χ̌) =_{(6.7),(6.12)} Tr(ρ̂_{t}^{S} e^{− \frac{i}{ℏ} Ĥ_{0} t} P̂ˆ e^{+ \frac{i}{ℏ} Ĥ_{0} t})

is called the transition probability for the transition of the Schrödinger state ˆ= e^{− \frac{i}{ℏ} Ĥ_{0} t} (ψ ⊗ χ) into the Schrödinger state ˆ= e^{− \frac{i}{ℏ} Ĥ_{0} t} (ψ̌ ⊗ χ̌) at time t.

In typical quantum optical applications also the approximation

∫ ∑_{j=1}^{3} ⟨χ̌ | Â^{j}(x, t′) χ⟩ (ψ̌_{t′}^{(0)}(x))^{∗} \frac{ℏ}{i} \frac{∂}{∂x^{j}} ψ_{t′}^{(0)}(x) dV_{x}

≈ ∫ ∑_{j=1}^{3} ⟨χ̌ | Â^{j}(0, t′) χ⟩ (ψ̌_{t′}^{(0)}(x))^{∗} \frac{ℏ}{i} \frac{∂}{∂x^{j}} ψ_{t′}^{(0)}(x) dV_{x}
= ⟨χ̌ | Â(0, t′) χ⟩ · ⟨ψ̌_{t′}^{(0)} | p̂_{can} ψ_{t′}^{(0)}⟩
= ⟨χ̌ | Â(0, t′) χ⟩ · ⟨ψ̌_{t′}^{(0)} | m \frac{i}{ℏ} [Ĥ_{atom}, x̂]_{−} ψ_{t′}^{(0)}⟩
= \frac{m}{q} ⟨χ̌ | Â(0, t′) χ⟩ · \frac{d}{dt′} ⟨ψ̌_{t′}^{(0)} | q x̂ ψ_{t′}^{(0)}⟩

is justified. Substituting this into (7.21) we get the so-called electric dipole approximation

p_{t}(ψ ⊗ χ → ψ̌ ⊗ χ̌)

≈ ∫_{[0,t]^{2}} ∑_{j_{1},j_{2}=1}^{3} ⟨χ | Â^{j_{1}}(0, t_{1}) χ̌⟩⟨χ̌ | Â^{j_{2}}(0, t_{2}) χ⟩ f_{ψ,ψ̌}^{j_{1}j_{2}}(t_{1}, t_{2}) dt_{1} dt_{2} ,

in the Coulomb representation, where

f_{ψ,ψ̌}^{j_{1}j_{2}}(t_{1}, t_{2}) =^{def} ℏ^{−2} (\frac{d}{dt_{1}} ⟨ψ_{t_{1}}^{(0)} | q x̂^{j_{1}} ψ̌_{t_{1}}^{(0)}⟩) (\frac{d}{dt_{2}} ⟨ψ̌_{t_{2}}^{(0)} | q x̂^{j_{2}} ψ_{t_{2}}^{(0)}⟩) .

(7.22)

Obviously, the f-factor gives rise to selection rules:^{14}

⟨ψ̌_{t′}^{(0)} | q x̂ ψ_{t′}^{(0)}⟩ = 0 ∀ t′ ∈ [0, t] ⟹ p_{t}(ψ ⊗ χ → ψ̌ ⊗ χ̌) ≈ 0 .  (7.23)

Draft, November 5, 2011

12See (Mandel and Wolf, 1995, End of Section 14.1.2) for justification.

13Recall (6.6).

14The l.h.s. of (7.23) is often implied by symmetries.

Remark: Note that

⟨χ̌ | Â(x, t) Ω⟩ ≠^{i.g.} 0 (= ⟨Ω | Â(x, t) Ω⟩) .

Therefore (7.22) may be non-negligible even for χ = Ω . This explains, e.g., the occurrence of spontaneous decay^{15} of excited atomic states in the vacuum.

7.2 Photoelectric Detection of Light

7.2.1 Ionisation of One-Electron Atoms

Let us assume that the projection P̂_{ion} onto the subspace of H_{atom} corresponding to ionisation is of the form

P̂_{ion} = ∫_{Ě≥0} \underbrace{σ(Ě, Ω̌)}_{≥0} |ψ̌(Ě, Ω̌)⟩⟨ψ̌(Ě, Ω̌)| dĚ dΩ̌ ,  (7.24)

where the ψ̌(Ě, Ω̌) are (improper) eigenstates of Ĥ_{atom} :

Ĥ_{atom} ψ̌(Ě, Ω̌) = Ě ψ̌(Ě, Ω̌) .

Moreover, let \left\{{ \atop χ̌_{ν}}\right\}_{ν∈ℕ} be a maximal orthonormal system (MONS) of H_{field} . Then, if

Ĥ_{atom} ψ = E ψ , E < 0 ,

the probability p_{t}^{ion}(ψ, χ) for ionisation at time t is

p_{t}^{ion}(ψ, χ)
= Tr(ρ̂_{t}^{I} (P̂_{ion} ⊗ ˆ1))
= ∑_{ν=1}^{∞} ∫_{Ě≥0} σ(Ě, Ω̌) p_{t}(ψ ⊗ χ → ψ̌(Ě, Ω̌) ⊗ χ̌_{ν}) dĚ dΩ̌

≈_{(7.22)} ∫_{[0,t]^{2}} ∑_{j_{1},j_{2}=1}^{3} ⟨χ | Â^{j_{1}}(0, t_{1}) Â^{j_{2}}(0, t_{2}) χ⟩ I_{ψ}^{j_{1}j_{2}}(t_{1} − t_{2}) dt_{1} dt_{2} ,

where I_{ψ}^{j_{1}j_{2}}(t_{1} − t_{2})

=^{def} ∫_{Ě≥0} σ(Ě, Ω̌) (E − Ě)^{2} ⟨ψ | q x̂^{j_{1}} ψ̌(Ě, Ω̌)⟩⟨ψ̌(Ě, Ω̌) | q x̂^{j_{2}} ψ⟩ ·
· e^{+ \frac{i}{ℏ}(E−Ě)(t_{1}−t_{2})} dĚ dΩ̌
= ∫_{Ě≥0} σ(Ě, Ω̌) f_{ψ,ψ̌(Ě,Ω̌)}^{j_{1}j_{2}}(t_{1}, t_{2}) dĚ dΩ̌ .

Draft, November 5, 2011

15See, e.g., (Baym, 1969, Chapter 13).

148 CHAPTER 7. RADIATIVE TRANSITIONS This together with (7.2) gives

\begin{matrix}p_{t}^{ion}(ψ, χ) \\ ≈ ∫_{[0,t]^{2}} ∑_{j_{1},j_{2}=1}^{3} ⟨χ | :Â^{j_{1}}(0, t_{1}) Â^{j_{2}}(0, t_{2}): χ⟩ I_{ψ}^{j_{1}j_{2}}(t_{1} − t_{2}) dt_{1} dt_{2} + V_{t}(ψ) , \end{matrix} (7.25)

where

V_{t}(ψ) =^{def} ∫_{[0,t]^{2}} ∑_{j_{1},j_{2}=1}^{3} ⟨Ω | Â^{j_{1}}(0, t_{1}) Â^{j_{2}}(0, t_{2}) Ω⟩ I_{ψ}^{j_{1}j_{2}}(t_{1} − t_{2}) dt_{1} dt_{2}
=_{(7.3)} (2π)^{−3} μ_{0} ℏ c ∑_{j_{1},j_{2}=1}^{3} ∫ (δ_{j_{1}j_{2}} − \frac{k^{j_{1}}k^{j_{2}}}{|k|^{2}} ) ·
· (∫_{[0,t]^{2}} I_{ψ}^{j_{1}j_{2}}(t_{1} − t_{2}) e^{−i c|k|(t_{1}−t_{2})} dt_{1} dt_{2}) \frac{dV_{k}}{2 |k|} . (7.26)

Since

⟨Ω | :Â^{j_{1}}(0, t_{1}) Â^{j_{2}}(0, t_{2}): Ω⟩ = 0 ,

we have

V_{t}(ψ) ≈_{(7.25)} p_{t}^{ion}(ψ, Ω)

and, therefore, expect V_{t}(ψ) to be negligible.

Additional consideration: By change of variables^{16}

(t_{1}, t_{2}) ⟶ (τ, t′) =^{def} (t_{1} − t_{2}, \frac{t_{1} + t_{2}}{2} )

we see — in the typical case,^{17} that I_{ψ}^{j_{1}j_{2}}(τ) is sufficiently concentrated around τ = 0 —- that

∫_{[0,t]^{2}} I_{ψ}^{j_{1}j_{2}}(t_{1} − t_{2}) e^{−i c|k|(t_{1}−t_{2})} dt_{1} dt_{2} = ∫_{−t}^{+t} (I_{ψ}^{j_{1}j_{2}}(τ) e^{−i c|k|τ} ∫|\frac{τ}{2} |^{t−|\frac{τ}{2} |} dt′) dτ
= ∫_{−t}^{+t} (t − |τ|) I_{ψ}^{j_{1}j_{2}}(τ) e^{−i c|k|τ} dτ
≈ t ∫_{−t}^{+t} I_{ψ}^{j_{1}j_{2}}(τ) e^{−i c|k|τ} dτ
≈ t ∫_{−∞}^{+∞} I_{ψ}^{j_{1}j_{2}}(τ) e^{−i c|k|τ} dτ

Draft, November 5, 2011

16Note that

(t_{1}, t_{2}) ∈ [0, t]^{2} ⟺ 0 ≤ t′ ± \frac{τ}{2} ≤ t ⟺ |\frac{τ}{2} | ≤ t′ ≤ t − |\frac{τ}{2} | .

17See comments to (Mandel and Wolf, 1995, (14.2-13)). for relevant t. Together with (7.26) and

∫_{−∞}^{+∞} I_{ψ}^{j_{1}j_{2}}(τ) e^{−i c|k|τ} dτ
= ∫_{Ě≥0} σ(Ě, Ω̌) (E − Ě)^{2} ⟨ψ | q x̂^{j_{1}} ψ̌(Ě, Ω̌)⟩⟨ψ̌(Ě, Ω̌) | q x̂^{j_{2}} ψ⟩ ·

·(\underbrace{∫_{−∞}^{+∞} e^{− \frac{i}{ℏ}(Ě−(E−ℏc|k|))τ} dτ}_{=2π δ(Ě−^{(E − ℏc |k|)}_{<0})})dĚ dΩ̌

= 0
this implies V_{t}(ψ, χ) ≈ 0 .

Therefore, by (7.25),

p_{t}^{ion}(ψ, χ) ≈ ∫_{[0,t]^{2}} ∑_{j_{1},j_{2}=1}^{3} ⟨χ | :Â^{j_{1}}(0, t_{1}) Â^{j_{2}}(0, t_{2}): χ⟩ I_{ψ}^{j_{1}j_{2}}(t_{1} − t_{2}) dt_{1} dt_{2} (7.27)

for relevant t. Obviously,

for normalized coherent states χ :

⟨χ | :Â^{j_{1}}(x_{1}, t_{1}) Â^{j_{2}}(x_{2}, t_{2}): χ⟩ = A_{χ}^{j_{1}}(x_{1}, t_{1}) A_{χ}^{j_{2}}(x_{2}, t_{2}) ,
where^{18} A_{χ}^{j}(x, t) =^{def} ⟨χ | Â^{j}(x, t) χ⟩ .

(7.28)

Since, on the other hand,

Tr(ρ̂_{t}^{I} (P̂_{ion} ⊗ |χ⟩⟨χ|))

\begin{matrix}= & ∑_{ν=1}^{∞} ∫_{Ě≥0} σ(Ě, Ω̌) p_{t}(ψ ⊗ χ → ψ̌(Ě, Ω̌) ⊗ χ̌_{ν}) dĚ dΩ̌ \\ ≈_{(7.22)} & ∫_{[0,t]^{2}} ∑_{j_{1},j_{2}=1}^{3} ⟨χ | Â^{j_{1}}(0, t_{1}) χ⟩⟨χ | Â^{j_{2}}(0, t_{2}) χ⟩ I_{ψ}^{j_{1}j_{2}}(t_{1} − t_{2}) dt_{1} dt_{2} , \end{matrix}

we see that

p_{t}^{ion}(ψ, χ) ≈ Tr(ρ̂_{t}^{I} (P̂_{ion} ⊗ |χ⟩⟨χ|)) .

This indicates that the following holds within the limits of our approximations:

If the (partial) state of the radiation field is a coherent one at time t = 0 , then the interaction influences only the time evolution of the atom’s (partial) state.^{19}

Draft, November 5, 2011

18Usually, the positive-frequency part A_{χ}^{(+)}^{j}(x, t) of A_{χ}^{j}(x, t) is called the analytic signal associated with A_{χ}^{j}(x, t) .

19Nevertheless the total energy is conserved thanks to the interaction term (7.1).

Recalling the definitions of I_{ψ}^{j_{1}j_{2}}(t_{1}−t_{2}) and Â(x, t) and interchanging integration over t_{1}, t_{2} with integration over Ě, Ω̌ we easily see that (7.27) implies

p_{t}^{ion}(ψ, χ) ≈ ∫_{[0,t]^{2}} ∑_{j_{1},j_{2}=1}^{3} ⟨χ | Â^{(−)}^{j_{1}}(0, t_{1}) Â^{(+)}^{j_{2}}(0, t_{2}) χ⟩ I_{ψ}^{j_{1}j_{2}}(t_{1} − t_{2}) dt_{1} dt_{2}

(7.29)

for sufficiently well-behaved states χ.

7.2.2 Simple Photodetectors

A typical photodetector consists of an assembly of ‘atoms’ to be ionized by the incident radiation. The electrons thus emitted from the ‘atoms’ are counted.^{20} In the following by ‘ detector’ we always mean a photodetector of this type.

First of all, let us consider the special case that the radiation field is initially in a coherent state χ (in the interaction picture). Then, as we have seen, this state remains essentially unchanged by the ionisation processes. Therefore the above considerations of the previous section make it plausible^{21} that the counting rate at time t_{0} of such a detector, localized at x_{0} , is

\begin{matrix}P_{|χ⟩⟨χ|}(x_{0}, t_{0}) \\ ≈ (\frac{∂}{∂t} ∫_{[t_{0},t_{0}+t]^{2}} ∑_{j_{1},j_{2}=1}^{3}⟨χ| Â^{(−)}^{j_{1}}(x_{0}, t_{1}) Â^{(+)}^{j_{2}}(x_{0}, t_{2}) χ⟩ K^{j_{1}j_{2}}(t_{1} − t_{2}) dt_{1} dt_{2})_{|_{t=0}} , \end{matrix}

(7.30)

where K_{ψ}^{j_{1}j_{2}}(t_{1} − t_{2}) is a characteristic function of the detector.

Remarks:

1. Here, by ‘counting rate at time t_{0}’ we mean the production rate for photoelectrons at time t_{0} . Of course, actual registration needs some extra time.^{22}

2. Strictly speaking, in (7.30), we should not use the derivative w.r.t. t at t = 0 but rather choose some ‘relevant’ t and divide by this t. This is because p_{t}^{ion}(ψ, χ) becomes proportional to t^{2} for too small t.

3. Generalization of the latter is also the reason for the so-called quantum Zeno effect:^{23}

Draft, November 5, 2011

20Actually, what we mean by ‘atom’ here is an electron bound in some photosensitive material. Ionisation of ordinary atoms would require frequencies above the optical domain.

21See (Mandel and Wolf, 1995, Sections 414.2–14.4) for further details.

22See, e.g., (Paul, 1995, Chapter 5).

23See (Lynds, 2003) for Zeno’s paradoxes and, concerning quantum mechanics, (Misra and Sudarshan, 1977).

A quantum mechanical system does not change its state as long as it is permanently checked, by ideal tests, whether it is in this state.

4. Justification of (7.30) as a consequence of (7.29) is not as straightforward, as usually tacitly assumed: (7.29) is the probability for finding the atom ionized if measured at time t. So in order to relate (7.29) to actual ionisation we have to assume that such measurements automatically occur at random times. However, these ‘measurements’ must not appear too often in order to avoid the quantum Zeno effect!

In typical cases we may substitute η^{j_{1}j_{2}} δ(t_{1} − t_{2}) for K^{j_{1}j_{2}}(t_{1} − t_{2}) , where η^{j_{1}j_{2}} describes the efficiency of the counter (slowly varying with the frequency range of the radiation field). Then (7.30) becomes^{24}

P_{|χ⟩⟨χ|}(x_{0}, t_{0}) ≈ ∑_{j_{1},j_{2}=1}^{3}⟨χ| Â^{(−)}^{j_{1}}(x_{0}, t_{0}) Â^{(+)}^{j_{2}}(x_{0}, t_{0}) χ⟩ η^{j_{1}j_{2}} .  (7.31)

Now let us consider n detectors located at the positions x_{1}, …, x_{n} and define the corresponding probability density ^{25}

P_{|χ⟩⟨χ|}(x_{1}, t_{1}; …; x_{n}, t_{n}) =^{def} (\frac{∂}{∂t′_{1}} ⋯ \frac{∂}{∂t′_{n}} p_{χ}(x_{1}, t_{1}, t′_{1}; …; x_{n}, t_{n}, t′_{n}))_{|_{t′_{1}=...=t′_{n}=0}} ,

for the radiation field being initially in the coherent state χ, where p_{χ}(x_{1}, t_{1}, t′_{1}; …; x_{n}, t_{n}, t′_{n}) is the probability for the following effect:^{26}

For all ν ∈ {1, …, n} the detector localized at x_{ν} ‘clicks’ during the time interval (t_{ν}, t′_{ν}) .

Then, assuming that the coherent state suffers no relevant changes by the detectors, we get by (7.31)

P_{|χ⟩⟨χ|}(x_{1}, t_{1}; …; x_{n}, t_{n})
≈ ∏_{ν=1}^{n} ∑_{j_{ν},j′_{ν}=1}^{3}⟨χ| Â^{(−)}^{j_{ν}}(x_{ν}, t_{ν}) Â^{(+)}^{j′_{ν}}(x_{ν}, t_{ν}) χ⟩ η_{ν}^{j_{ν}j′_{ν}}

= ∑_{j_{1},...,j_{n},j′_{1},...,j′_{n}=1}^{3} ⟨χ| Â^{(−)}^{j_{1}}(x_{1}, t_{1}) ⋯ Â^{(−)}^{j_{n}}(x_{n}, t_{n})Â^{(+)}^{j′_{n}}(x_{n}, t_{n}) ⋯
⋯ Â^{(+)}^{j′_{1}}(x_{1}, t_{1}) χ⟩ η_{1}^{j_{1}j′_{1}} ⋯ η_{n}^{j_{n}j′_{n}} ,

(7.32)

Draft, November 5, 2011

24Recall Footnote 16.

25Here Remark 1 to (7.30) applies accordingly.

26For instance, P_{}_{ρ̂^{I}}(x_{1}, t_{1}; x_{2}, t_{2}) is measured in the Hanbury Brown-Twiss experiment; see, e.g., (Mandel and Wolf, 1995, Sections 9.9, 9.10, and 14.6.1).

where, for ν ∈ {1, …, n}, the η_{ν}^{j_{1}j_{2}} characterize the efficiency of the detector localized at x_{ν} .

Since P_{}_{ρ̂^{I}}(x_{1}, t_{1}; …; x_{n}, t_{n}) — to be defined for general ρ̂^{I} accordingly^{27} — is linear (and suitably continuous) in ρ̂^{I} Theorem 1.2.2 tells us that

P_{}_{ρ̂^{I}}(x_{1}, t_{1}; …; x_{n}, t_{n})
= lim_{N→∞} ∫ ρ_{N}(α_{1}, …, α_{N})P_{Φ_{α_{1},...,α_{N}}} (x_{1}, t_{1}; …; x_{n}, t_{n})\frac{dα_{1}}{π} ⋯ \frac{dα_{N}}{π}

holds with suitable functions ρ_{N} with^{28}

ρ̂^{I} = lim_{N→∞} ∫ ρ_{N}(α_{1}, …, α_{N}) |Φ_{α_{1},...,α_{N}} ⟩⟨Φ_{α_{1},...,α_{N}} | \frac{dα_{1}}{π} ⋯ \frac{dα_{N}}{π} .  (7.33)

This together with (7.32) and the optical equivalence theorem gives^{29}

P_{}_{ρ̂^{I}}(x_{1}, t_{1}; …; x_{n}, t_{n})

\begin{aligned}≈ & ∑_{j_{1},...,j_{n},j′_{1},...,j′_{n}=1}^{3} Tr (ρ̂^{I} Â^{(−)}^{j_{1}}(x_{1}, t_{1}) ⋯ Â^{(−)}^{j_{n}}(x_{n}, t_{n})Â^{(+)}^{j′_{n}}(x_{n}, t_{n}) ⋯ \\ & ⋯ Â^{(+)}^{j′_{1}}(x_{1}, t_{1})) η_{1}^{j_{1}j′_{1}} ⋯ η_{n}^{j_{n}j′_{n}} . \end{aligned}

(7.34)

Draft, November 5, 2011

27From now on we denote by ρ̂^{I} the density matrix of the (partial) state of the electromagnetic field — rather than of the total system — in the interaction picture.

28In spite of (7.33), ρ̂^{I} need not be an ordinary mixture of coherent states, since the ρ_{N} need not be nonnegative functions.

29Admittedly, this argument involves an interchange of limits that ought to be justified.

Semiclassical Treatment of Few Level Atomic Systems

8.1 The Exterior Electric Field Representation

Let us recall the exterior field formalism for a single atom as introduced in 7.1.1:

The atom’s state space is

H_{atom} = L^{2}(ℝ^{3}, d^{3}x)

and its free Hamiltonian is Ĥ_{atom} = \frac{1}{2m} (\frac{ℏ}{i}∇_{x})^{2} + \underbrace{q c A_{atom}^{0}(x)}_{Bindungspotential} .

Interaction with the (optical) radiation field is introduced via minimal coupling:^{1}

Ĥ_{atom} ⟼ Ĥ^{S}(t) =^{def} \frac{1}{2m} (\frac{ℏ}{i}∇_{x} − q A_{ext}(x, t))^{2} + q c A_{atom}^{0}(x) . As usual, we evaluate the Schrödinger equation

i \frac{d}{dt} ℏ ψ_{t} = Ĥ^{S}(t) ψ_{t}

only in the so-called dipole approximation

Ĥ^{S}(t)ψ_{t}(x) = (\frac{1}{2m} (\frac{ℏ}{i}∇_{x} − q A(t))^{2} + q c A_{atom}^{0}(x)) ψ_{t}(x) ,

where

A(t) =^{def} A_{ext}(0, t) ∀ t ∈ ℝ .

Draft, November 5, 2011

1Here we use the Schrödinger picture.

By change of representation

ψ_{t} ↦ ϕ_{t} =^{def} Û(t) ψ_{t} ,

where

Û(t) =^{def} e^{− \frac{i}{ℏ} q x·A(t)} ,

the Schrödinger equation (in dipole approximation) becomes

i ℏ \frac{∂}{∂t} ϕ_{t}(x) ≈ Ĥ_{cl}(t) ϕ_{t}(x) ,
where:^{2} Ĥ_{cl}(t) =^{def} −\frac{ℏ^{2}}{2m}Δ_{x} − q x · E(t) + q c A_{atom}^{0}(x) ,
E(t) =^{def} −\frac{∂}{∂t}A_{ext}(0, t) .

(8.1)

Warning: Recall that — strictly speaking — the usual interpretation of

−\frac{ℏ^{2}}{2m}Δ_{x} + q c A_{atom}^{0}(x)

as observable of the energy of the atom is not correct.^{3}

8.2 2-Level Systems

Let us assume that (within the considered time interval essentially) only a ground state |g⟩ and an excited state |e⟩ are occupied^{4} corresponding to the energy eigenvalues E_{g}, E_{e} :

Ĥ_{atom} |g⟩ = E_{g} |g⟩ , Ĥ_{atom} |e⟩ = E_{e} |e⟩ .

Then, because of

ρ̂_{atom}^{S}(t) ≈ P̂_{H_{atom}^{(2)}(t)}ρ̂_{atom}^{S}(t)P̂_{H_{atom}^{(2)}(t)} ,

restriction to the 2-dimensional subspace

H_{atom}^{(2)}(t) =^{def} L({|g⟩ , |e⟩})

Draft, November 5, 2011

2Note that:

i ℏ \frac{∂}{∂t} Û(t) ψ_{t}(x) = (q x · \frac{∂}{∂t}A(t)) Û(t) ψ_{t}(x) , Û(t) \frac{ℏ}{i} ∇_{x} Û(t)^{−1} = \frac{ℏ}{i} ∇_{x} + q A(t) .

3See also (Scully and Zubairy, 1999, 5.1.3) and (Shore, 1990a, End of § 3.11).

4For simplicity we assume that interaction with the electromagnetic field does not change the level scheme.

8.2. 2-LEVEL SYSTEMS 155 of H_{atom} is adequate:

iℏ \frac{d}{dt} ρ̂_{atom}^{(2)}(t) ≈ [Ĥ_{cl}^{(2)}(t), ρ̂_{atom}^{(2)}(t)]_{−}
where: ρ̂_{atom}^{(2)}(t) =^{def} \frac{P̂_{H_{atom}^{(2)}(t)} ρ̂_{atom}^{S}(t) /^{\} H_{atom}^{(2)}(t)}{Spur (P̂_{H_{atom}^{(2)}(t)} ρ̂_{atom}^{S}(t) P̂_{H_{atom}^{(2)}(t)})} ,
Ĥ_{cl}^{(2)}(t) =^{def} P̂_{H_{atom}^{(2)}(t)}Ĥ_{cl}(t) /^{\} H_{atom}^{(2)}(t) .

(8.2)

W.r.t. the basis

(|e⟩ = ( \begin{matrix}1 \\ 0 \end{matrix} ) , |g⟩ = ( \begin{matrix}0 \\ 1 \end{matrix} ))

we have

|g⟩⟨e| = b̂ =^{def} ( \begin{matrix}0 & 0 \\ 1 & 0 \end{matrix} ) ,
|e⟩⟨g| = b̂^{†} = ( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ) ,
|e⟩⟨e| = b̂^{†} b̂ = ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) ,
|g⟩⟨g| = b̂ b̂^{†} = ( \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} ) = τ̂^{0} − b̂^{†} b̂

and hence

Ĥ_{cl}^{(2)}(t) = E_{e} |e⟩⟨e| + E_{g} |g⟩⟨g| − qP̂_{H_{atom}^{(2)}(t)}x · E_{ext}(x, t)/^{\}H_{atom}^{(2)}(t)
= ℏ ω_{0}b̂^{†} b̂ + \underbrace{E_{g} τ̂^{0}}_{irrelevant} −qP̂_{H_{atom}^{(2)}(t)}x · E_{ext}(x, t)/^{\}H_{atom}^{(2)}(t) ,  (8.3)

where:

ω_{0} =^{def} \frac{E_{e} − E_{g}}{ℏ} .

For

E_{ext}(x, t) = 0 ,

because of

b̂^{†} b̂ = \frac{τ̂^{0} + τ̂^{3}}{2} ,

the Hamiltonian may be written as

Ĥ_{cl}^{(2)}(t) = \frac{ℏ ω_{0}}{2} τ̂^{3} + \underbrace{\frac{E_{g} + E_{e}}{2} τ̂^{0}}_{irrelevant}  (8.4)

and the Liouville equation (8.2) simplifies to

iℏ \frac{d}{dt}ρ̂_{atom}^{(2)}(t) = [\frac{ℏ ω_{0}}{2} τ̂^{3}, ρ̂_{atom}^{(2)}(t)]_{−} .  (8.5)

(8.5) can easily solved by means of the Bloch representation^{5}

ρ̂_{atom}^{(2)}(t) = \frac{1}{2} (τ̂^{0} + \underbrace{x(t)}_{∈ℝ^{3}} ·τ̂) .  (8.6)

Here

τ̂^{0} =^{def} ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) , τ̂^{1} =^{def} ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) , τ̂^{2} =^{def} ( \begin{matrix}0 & −i \\ +i & 0 \end{matrix} ) , τ̂^{3} =^{def} ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} )

(8.7)

are the well-known Pauli-matrices^{6} — forming an orthonormal basis of B (H_{m}) w.r.t. the Hilbert-Schmidt norm and obeying the relations

\begin{aligned}τ̂^{ν}τ̂^{ν} & = & ˆ1 & ∀ ν ∈ \{0, 1, 2, 3\} , \\ τ̂^{j}τ̂^{k} & = & − τ̂^{k}τ̂^{j} & ∀ j, k ∈ \{1, 2, 3\} , j ≠ k , \\ τ̂^{1}τ̂^{2} & = & i τ̂^{3} , \\ τ̂^{2}τ̂^{3} & = & i τ̂^{1} , \\ τ̂^{3}τ̂^{1} & = & i τ̂^{2} \end{aligned}  (8.8)

(see, e.g., Section 7.2.2 of (Lücke, eine)). In this connection let us note that

τ̂^{3} = b̂^{†}b̂ − b̂ b̂^{†} = 2 b̂^{†}b̂ − τ̂^{0} ,  (8.9)

where the Fermi annihilation operator

b̂ =^{def} \frac{τ̂^{1} − i τ̂^{2}}{2} = ( \begin{matrix}0 & 0 \\ 1 & 0 \end{matrix} )  (8.10)

acts as

b̂ |g⟩ = 0 , b̂ |e⟩ = |g⟩ , b̂^{†} |g⟩ = |e⟩ , b̂^{†} |e⟩ = 0 .  (8.11)

and fulfills the canonical anti-commutation relations

[b̂, b̂^{†}]_{+} = τ̂^{0} , [b̂, b̂]_{+} = 0 .  (8.12)

In the Bloch representation (8.5) is equivalent to

i \frac{ℏ}{2} ẋ(t) · τ̂ = \frac{ℏ ω_{0}}{4} x(t) · [τ̂^{3}, τ̂]_{−}
= i \frac{ℏ}{2} ω_{0} (x^{1}(t) τ̂^{2} − x^{2}(t) τ̂^{1})
= i \frac{ℏ}{2} ω_{0} e_{3} · (x(t) × τ̂) ,

hence also to

ẋ(t) = ω_{0} e_{3} × x(t) ;

i.e.:

The Bloch vector x(t) rotates^{7} with constant angular frequency ω_{0} e_{3} .

Draft, November 5, 2011

5See (Leggett, 2001, p. 8) for generalization.

6A widespread notation is σ_{ν} instead of τ̂^{ν} .

7Since spontaneous decay was not taken into account.

8.2.1 Interpretation of the Bloch Vector

Obviously, positivity of ρ̂_{atom}^{(2)}(t) implies

det (ρ̂_{atom}^{(2)}(t)) ≥ 0

and hence

|x(t)| ≤ 1 ,

since

det (ρ̂_{atom}^{(2)}(t)) =_{(8.6)} det(\frac{1}{2} (τ̂^{0} + \underbrace{x(t)}_{∈ℝ^{3}} ·τ̂))

= \frac{1 − |x(t)|^{2}}{4} .

Since, moreover,

ρ̂_{atom}^{(2)}(t) projector ⟺ (ρ̂_{atom}^{(2)}(t))^{2} = ρ̂_{atom}^{(2)}(t)
⟺ \frac{1}{4} (τ̂^{0} + 2 x(t) · τ̂ + |x(t)|^{2} τ̂^{0}) = \frac{1}{2}(τ̂^{0} + x(t) · τ̂)
⟺ |x(t)|^{2} = 1

we conclude:

ρ̂_{atom}^{(2)}(t) describes a \{ \begin{matrix}pure state & if |x(t)| = 1 , \\ mixed state & if |x(t)| < 1 . \end{matrix}

Especially we have

\frac{1}{2} (τ̂^{0} + ( \begin{matrix}sin ϑ cos φ \\ sin ϑ sin φ \\ cos ϑ \end{matrix} ) · τ̂) = ( \begin{matrix}cos^{2} \frac{ϑ}{2} & \frac{1}{2} e^{−i φ} sin ϑ \\ \frac{1}{2} e^{+i φ} sin ϑ & sin^{2} \frac{ϑ}{2} \end{matrix} )

= |( \begin{matrix}e^{−i \frac{φ}{2}} cos \frac{ϑ}{2} \\ e^{+i \frac{φ}{2}} sin \frac{ϑ}{2} \end{matrix} )⟩⟨( \begin{matrix}e^{−i \frac{φ}{2}} cos \frac{ϑ}{2} \\ e^{+i \frac{φ}{2}} sin \frac{ϑ}{2} \end{matrix} )|

and^{8}

ρ̂_{x} =^{def} \frac{1}{2} (τ̂^{0} + x · τ̂)
= \frac{1 + |x|}{2} ρ̂\frac{x}{|x|} + \frac{1 − |x|}{2} ρ̂_{− \frac{x}{|x|}} ∀ x ∈ ℝ^{3} \ \{0\} .  (8.13)

Furthermore, we have

ρ̂_{atom}^{(2)}(t) = ρ̂_{x(t)} ⟹ \left⟨{ \atop μ̂}\right⟩(t) = ℜ(q ⟨g| x̂ |e⟩) x^{1}(t) + ℑ(q ⟨g| x̂ |e⟩) x^{2}(t) , (8.14)

Draft, November 5, 2011

8The vector x characterizing ρ̂_{x} is called Bloch vector resp. Stokes vector if ρ̂_{x} describes electron spin resp. photon polarization. The set of all normalized Bloch vectors resp. Stokes vectors is called Bloch sphere resp. Poincare sphere. In both cases |x| is called the degree of polarization.

158 CHAPTER 8. FEW-LEVEL ATOMIC SYSTEMS where

μ̂ =^{def} q P̂_{H_{atom}^{(2)}(t)}x̂ P̂_{H_{atom}^{(2)}(t)}

denotes the observable of the atom’s electric dipole moment (truncateded to H_{atom}^{(2)}(t)).

Outline of proof for (8.14): The statement follows from

\left⟨{ \atop μ̂}\right⟩(t) =^{def} Tr (ρ̂_{atom}^{(2)}(t) μ̂)
= Tr (\frac{1}{2} (τ̂^{0} + x(t) · τ̂)(q ⟨g| x̂ |e⟩ b̂ + h.c.))
= \frac{1}{2} Tr(x(t) · τ̂ (q ⟨g| x̂ |e⟩ b̂ + h.c.))

and

Tr (τ̂ b̂) = \frac{1}{2} Tr (τ̂τ̂^{1}) − \frac{i}{2} Tr (τ̂τ̂^{2})
= e_{1} − i e_{2} ,
Tr (τ̂ b̂^{†}) = e_{1} + i e_{2} .

8.2.2 Rotating Wave Approximation

For E(t) ≠ 0 we have

q P̂_{H_{atom}^{(2)}(t)}x̂ · E(t) P̂_{H_{atom}^{(2)}(t)} = μ̂ · E(t) .

This, together with (8.2) and (8.3) implies

iℏ \frac{d}{dt} ρ̂_{atom}^{(2)}(t) ≈ [ℏ ω_{0} b̂^{†}b̂ − μ̂ · E_{ext}(0, t), ρ̂_{atom}^{(2)}(t)]_{−} .  (8.15)

In case^{9}

E_{ext}(0, t) ≈ E e^{−i ω t} + c.c.  (8.16)

it is easier to determine

ρ̂_{ω}(t) =^{def} e^{+i ω b̂^{†}b̂ t} ρ̂_{atom}^{(2)}(t) e^{−i ω b̂^{†}b̂ t}

for which (8.15) implies

iℏ \frac{d}{dt} ρ̂_{ω}(t) ≈ [ℏ (ω_{0} − ω) b̂^{†}b̂ − μ̂_{ω}(t) · E_{ext}(0, t), ρ̂_{ω}(t)]_{−} ,

where

μ̂_{ω}(t) =^{def} e^{+i ω b̂^{†}b̂ t} μ̂ e^{−i ω b̂^{†}b̂ t} .

Usually the conditions

⟨g| x̂ |g⟩ = ⟨e| x̂ |e⟩ = 0

Draft, November 5, 2011

9In (Mandel and Wolf, 1995, Eq. (15.3-9)) E(t) is allowed to vary slowly with time.

are fulfilled.^{10} Then

e^{+i ω b̂^{†}b̂ t} b̂ e^{−i ω b̂^{†}b̂ t} =_{(8.12)} b̂ e^{−i ω t}

implies

μ̂_{ω}(t) · E_{ext}(0, t) = (q ⟨g| x̂ |e⟩ b̂ e^{−i ω t} + h.c.) · (E e^{−i ω t} + c.c.)
= (q ⟨g| x̂ |e⟩ · E^{∗} b̂ + h.c.)
+ (q ⟨g| x̂ |e⟩ · E b̂ e^{−i 2 ω t} + h.c.) ,

and this is usually replaced by the so-called rotating wave approximation^{11}

μ̂_{ω}(t) · E_{ext}(0, t) ≈ q ⟨g| x̂ |e⟩ · E^{∗} b̂ + h.c. .

All approximations together result in the equation

iℏ \frac{d}{dt} ρ̂_{ω}(t) ≈ [ℏ (ω_{0} − ω) b̂^{†}b̂ − \frac{ℏ}{2} (Ω b̂ + h.c.) , ρ̂_{ω}(t)]_{−} ,

where^{12}

Ω =^{def} \frac{2 q}{ℏ} ⟨g| x̂ |e⟩ · E^{∗} .

For suitable choice of the origin of the time scale we have

Ω = |Ω| Rabi frequency

and, in the Bloch representation

ρ̂_{ω}(t) = \frac{1}{2} (τ̂^{0} + x_{ω}(t) · τ̂) ,

this implies

\begin{aligned}iℏ ẋ_{ω}(t) · τ̂ & = & x_{ω}(t) · [ℏ (ω_{0} − ω) b̂^{†}b̂ − \frac{ℏ}{2} Ω (b̂ + b̂^{†}) , τ̂]_{−} \\ & = & iℏ((ω_{0} − ω) e_{3} × x_{ω}(t)) · τ̂ − \frac{ℏ}{2} Ω x_{ω}(t) · [τ̂^{1}, τ̂]_{−} . \end{aligned}

Using

\frac{i}{2}[τ̂^{j}, τ̂]_{−} = e_{j} × τ̂ ∀ j ∈ \{1, 2, 3\}  (8.17)

we conclude that^{13}

ẋ_{ω}(t) = Ω_{Δ} × x_{ω}(t) ,

Draft, November 5, 2011

10Especially if |g⟩ , |e⟩ are parity eigenstates.

11For exact solutions of related problems not using the rotating wave approximation approximation see (Angelo and Wreszinski, 2005) and (Chen et al., 2010).

12Note the used convention: E_{ext}(0, t) = ℜ(2 E e^{−i ω t}).

13Thus:

ρ̂_{ω}(t) stationary ⟺ x_{ω}(t) ∝ Ω_{Δ} .

where:

Ω_{Δ} =^{def} Δ e_{3} − Ω e_{1} ,
Δ =^{def} ω_{0} − ω detuning .

Correspondingly we have

x_{ω}(t) = D̂_{−ϑ e_{2}} D̂_{|Ω_{Δ}|t e_{3}} D̂_{+ϑ e_{2}} x_{ω}(0)

with

D̂_{±ϑ e_{2}} = ( \begin{matrix}\frac{Δ}{|Ω_{Δ}|} & 0 & ±\frac{Ω}{|Ω_{Δ}|} \\ 0 & 1 & 0 \\ ∓\frac{Ω}{|Ω_{Δ}|} & 0 & \frac{Δ}{|Ω_{Δ}|} \end{matrix} )

and

D̂_{φ e_{3}} = ( \begin{matrix}cos φ & − sin φ & 0 \\ sin φ & cos φ & 0 \\ 0 & 0 & 1 \end{matrix} ) ∀ φ ∈ ℝ .

Especially for

x_{ω}(0) = −e_{3} ,

therefore,

x_{ω}(t) = ( \begin{matrix}\frac{Ω Δ}{|Ω_{Δ}|^{2}} (1 − cos (|Ω_{Δ}| t)) \\ −\frac{Ω}{|Ω_{Δ}|} sin (|Ω_{Δ}| t) \\ −\frac{Δ^{2} + Ω^{2} cos (|Ω_{Δ}| t)}{|Ω_{Δ}|^{2}} \end{matrix} ) .

Remarks:

1. We see that the population of |g⟩ and |e⟩ oscillates with angular

frequency |Ω_{Δ}| = \sqrt{(ω_{0} − ω)^{2} + Ω^{2}} .

2. For |Δ| ≫ Ω these Rabi oscillations are negligible.

3. If Ω_{Δ} is ‘slowly’ changed with time and the initial angle between ρ̂_{ω}(t) and Ω_{Δ}(t) is ‘small’ then this angle remains essentially constant.

4. This adiabatic following of x_{ω}(t) may be used for specific state

preparation:

For instance, starting with x_{ω}(t) = −e_{3} and Δ ≫ Ω decrease ω until −Δ ≫ Ω . Then x_{ω} changes essentially into +e_{3} , i.e. the atom’s state is changed from |g⟩ essentially to |e⟩.

The exact value of ω_{0} does not matter in this connection!

5. Of course:

x(t) = D̂_{ω t e_{3}} x_{ω}(t) .

6. Spontaneously emitted radiation — recognizable as fluorescence but not considered so far — could be taken into account to some extent, as far as ρ̂_{ω}(t) is concerned by inserting phenomenological damping terms into the evolution equation.^{14}

7. The Rabi oscillation are correlated with a modulation of the amplitude of the emitted light. Therefore side-bands show up at the

angular frequencies^{15} ω ± \sqrt{(ω_{0} − ω)^{2} + Ω^{2}} .

8.3 3-Level Atoms

8.3.1 Population Trapping

Now assume that the atom is coupled to the classical field

E_{ext}(0, t) ≈ (E_{p} e^{−i ω_{p}t} + E_{c} e^{−i ω_{c}t}) + c.c.

and that maximally only the three levels |g⟩ , |g′⟩ , |e⟩ are (essentially) occupied. Then, if

⟨g| x̂ |e⟩ · E_{c}^{∗} = ⟨g′| x̂ |e⟩ · E_{p}^{∗} = 0 ,
⟨e| x̂ |e⟩ = ⟨g| x̂ |g⟩ = ⟨g′| x̂ |g′⟩ = ⟨g′| x̂ |g⟩ = 0 ,  (8.18)

in dipole and rotation wave approximation the Liouville equation restricted to

H_{atom}^{(3)}(t) =^{def} L({|g⟩ , |e⟩ , |g′⟩})

is

iℏ \frac{d}{dt} ρ̂_{atom}^{(3)}(t) = [Ĥ_{cl}^{(3)}(t), ρ̂_{atom}^{(3)}(t)]_{−} ,  (8.19)

where:^{16}

Ĥ_{cl}^{(3)}(t) =^{def} Ĥ_{atom}^{(3)} − (\frac{ℏ}{2} Ω_{p} e^{+i(ω_{p} t+ϕ_{p})} |g⟩⟨e| + h.c.)

− (\frac{ℏ}{2} Ω_{c} e^{+i(ω_{c} t+ϕ_{c})} |g′⟩⟨e| + h.c.) ,

Ω_{p} =^{def} \frac{2 q}{ℏ} ⟨g| x̂ |e⟩ · E_{p}^{∗} e^{−iϕ_{p}} ≥ 0 ,  (8.20)
Ω_{c} =^{def} \frac{2 q}{ℏ} ⟨g′| x̂ |e⟩ · E_{c}^{∗} e^{−iϕ_{c}} ≥ 0 .

Draft, November 5, 2011

14See (Scully and Zubairy, 1999, 5.3) and 8.3.2.

15See (Mandel and Wolf, 1995, Eq. (15.3-22)) for the special case x(0) = −e_{3} .

16 Note that

Ĥ_{atom}^{(3)} = E_{e} |e⟩⟨e| + E_{g} |g⟩⟨g| + E_{g′} |g′⟩⟨g′| .

162 CHAPTER 8. FEW-LEVEL ATOMIC SYSTEMS With the definitions

ρ̂_{rot}(t) =^{def} Û_{rot}(t) ρ̂_{atom}^{(3)}(t) Û_{rot}(−t) ,
Û_{rot}(t) =^{def} e^{−i (ω_{p}t+ϕ_{p})|g⟩⟨g|} e^{−i (ω_{c}t+ϕ_{c})|g′⟩⟨g′|} ,
Ĥ_{rot}(t) =^{def} Û_{rot}(t) Ĥ_{atom}^{(3)} Û_{rot}(−t) + ℏ ω_{c} |g′⟩⟨g′| + ℏ ω_{p} |g⟩⟨g| − E_{e}

this gives

iℏ \frac{d}{dt} ρ̂_{rot}(t) = [Ĥ_{rot}(t), ρ̂_{rot}(t)]_{−} .  (8.21)

Using the Campbell-Hausdorff formula^{17}

eÂ B̂ e^{−Â} = e^{adÂ} B̂ , adÂ \binom{}{B̂} =^{def} [Â, B̂]_{−} ∀ Â, B̂ ,

and

[|g⟩⟨g| , |g⟩⟨e|]_{−} = |g⟩⟨e| , [|g′⟩⟨g′| , |g′⟩⟨e|]_{−} = |g′⟩⟨e|

we get

Ĥ_{rot}(t) = −ℏ Δ_{p} |g⟩⟨g| − ℏ Δ_{c} |g′⟩⟨g′| − \frac{ℏ}{2} Ω_{p}(|g⟩⟨e| + |e⟩⟨g|)

− \frac{ℏ}{2} Ω_{c}(|g′⟩⟨e| + |e⟩⟨g′|),

(8.22)

where

Δ_{p} =^{def} \frac{E_{e} − E_{g}}{ℏ} − ω_{p} , Δ_{c} =^{def} \frac{E_{e} − E_{g′}}{ℏ} − ω_{c} .

For the special case

Δ_{p} = Δ_{c} = Δ

this gives

Ĥ_{Δ} =^{def} Ĥ_{rot}(t) + Δ = −\frac{ℏ}{2} ( \begin{matrix}0 & Ω_{p} & 0 \\ Ω_{p} & −2Δ & Ω_{c} \\ 0 & Ω_{c} & 0 \end{matrix} )

w.r.t. the basis (|g⟩ , |e⟩ , |g′⟩) and hence

Ĥ_{Δ} ( \begin{matrix}Ω_{c} \\ 0 \\ −Ω_{p} \end{matrix} ) = 0 .

This implies that the dark state

ρ̂_{atom}^{(3)}(t) = |ψ_{dark}(t)⟩⟨ψ_{dark}(t)| ,

Draft, November 5, 2011

17See Footnote 35 of Chapter 1.

(not depending on Δ), where

ψ_{dark}(t) =^{def} \frac{1}{\sqrt{Ω_{c}^{2} + Ω_{p}^{2}}} (Ω_{c} e^{+i(ω_{p} t+ϕ_{p})} |g⟩ − Ω_{p}e^{+i(ω_{c} t+ϕ_{c})} |g′⟩) ,

is a solution of the Liouville equation (8.19) — which is also easy to check directly. Thanks to (8.18) the expectation value of the atom’s electric dipole moment vanishes in this state. Hence the atom has no optical influence.

Remarks:

1. By suitable adiabatic variation of Ω_{p}, Ω_{c} the population may be transferred completely from |g⟩ to |g′⟩.^{18}

2. Extremely weak pump radiation (with angular frequency ω_{p}) has to be treated quantum mechanically since in this case the spontaneously emitted portion of the total radiation with angular frequency ω_{p} can no longer be neglected.^{19}

8.3.2 Electromagnetically Induced Transparency

Of course, (8.21) can be solved explicitly for Ĥ_{rot}(t) given by (8.22). For simplicity, however, let us assume strict resonance

Δ_{p} = Δ_{c} = 0 .  (8.23)

Then we have

Ĥ_{rot}(t) |g⟩ = −\frac{ℏ}{2} Ω_{p} |e⟩ ,
Ĥ_{rot}(t) |g′⟩ = −\frac{ℏ}{2} Ω_{c} |e⟩ ,
Ĥ_{rot}(t) |e⟩ = −\frac{ℏ}{2}(Ω_{p} |g⟩ + Ω_{c} |g′⟩)

and hence

Ĥ_{rot}(t)(Ω_{c} |g⟩ − Ω_{p} |g′⟩) = 0 ,  (8.24)
Ĥ_{rot}(t) (|e⟩ ± \frac{Ω_{p}}{Ω} |g⟩ ± \frac{Ω_{c}}{Ω} |g′⟩) = −\frac{ℏ}{2}(Ω_{p} |g⟩ + Ω_{c} |g′⟩) ∓ \frac{ℏ}{2} (\frac{Ω_{p}^{2}}{Ω} + \frac{Ω_{c}^{2}}{Ω} ) |e⟩
= ∓\frac{ℏ}{2} Ω (|e⟩ ± \frac{Ω_{p}}{Ω} |g⟩ ± \frac{Ω_{c}}{Ω} |g′⟩) ,  (8.25)

Draft, November 5, 2011

18See (Kuklinski et al., 1989). For the possible implementation of a universal set of quantum gates based on adiabatic population transfer see (Moller et al., 2007).

19See (Fleischhauer and Lukin, 2000) for this case. In (Kuang et al., 2003) both the pump and the coupling radiation are treated quantum mechanically.

where

Ω =^{def} \sqrt{Ω_{p}^{2} + Ω_{c}^{2}} .

(8.24) corresponds to population trapping for the special case (8.23). (8.24) and (8.25) show that the normalized vectors

ψ_{0} =^{def} \frac{1}{Ω}(Ω_{c} |g⟩ − Ω_{p} |g′⟩)
ψ_{±} =^{def} \frac{1}{\sqrt{2}} (|e⟩ ± (\frac{Ω_{p}}{Ω} |g⟩ + \frac{Ω_{c}}{Ω} |g′⟩))

are eigenvectors of Ĥ_{rot}(t) :

Ĥ_{rot}(t) Ψ_{0} = 0 ,
Ĥ_{rot}(t) Ψ_{±} = ∓\frac{ℏ}{2} Ω ψ_{±} .

(8.26)

Note that

ψ_{+} − ψ_{−} = \frac{\sqrt{2}}{Ω}(Ω_{p} |g⟩ + Ω_{c} |g′⟩)  (8.27)

and, therefore,

ψ_{+} − ψ_{−} + \sqrt{2} \frac{Ω_{c}}{Ω_{p}} ψ_{0} = (\sqrt{2} \frac{Ω_{p}}{Ω} + \sqrt{2} \frac{Ω_{c}^{2}}{Ω Ω_{p}} ) |g⟩
= \sqrt{2} \frac{Ω}{Ω_{p}} |g⟩ .

The latter implies^{20}

e^{− \frac{i}{ℏ} Ĥ_{rot}(t) t} |g⟩

=_{(8.26)} \frac{Ω_{c}}{Ω} ψ_{0} + \frac{1}{\sqrt{2}} \frac{Ω_{p}}{Ω} (e^{+ \frac{i}{2} Ω t} ψ_{+} − e^{− \frac{i}{2} Ω t} ψ_{−})
= \frac{Ω_{c}}{Ω} ψ_{0} + i \frac{Ω_{p}}{Ω} sin (\frac{Ω}{2} t) |e⟩ + (\frac{Ω_{p}}{Ω} )^{2} sin (\frac{Ω}{2} t) |g⟩ + \frac{Ω_{p} Ω_{c}}{Ω^{2}} cos (\frac{Ω}{2} t) |g′⟩ ,

i.e. the probability for finding the atom in the excited state at time T is

|⟨e| e^{− \frac{i}{ℏ} Ĥ_{rot}(t) t} |g⟩|^{2} = |\frac{Ω_{p}}{Ω} sin (\frac{Ω}{2} t)|^{2} ≤ (\frac{Ω_{p}}{Ω} )^{2} ≪ 1 for Ω_{p} ≪ Ω

if the state at time t = 0 is given by |g⟩.

This explains the effect of electromagnetically induced transparency (EIT):

The coupling radiation ˆ= Ω_{c} applied to the atom initially in the ground state |g⟩ essentially prohibits transitions into the excited state |e⟩, if Ω_{p} ≪ Ω , thus making the atom transparent for probe radiation ˆ= Ω_{p} .

Draft, November 5, 2011

20If a damping term is added to Ĥ_{rot}(t) then |g⟩ will be driven into the corresponding dark state; see (Scully, 1994).

8.3.3 Change of Linear Susceptibility^{21}

In order to calculate the susceptibility for the pump frequency, fading out of the transient perturbation — see Exercise 38a) of (Lücke, ein) in this context — has to be incorporated. A convenient heuristic way to achieve this is modifying the evolution equation into^{22}

iℏ \frac{d}{dt} ρ̂_{atom}^{(3)}(t) = [Ĥ_{cl}^{(3)}(t), ρ̂_{atom}^{(3)}(t)]_{−} − i \frac{ℏ}{2}[Γ̂, ρ̂_{atom}^{(3)}(t)]_{+}

by means of a phenomenological relaxation matrix

Γ̂ = \underbrace{γ_{g}}_{=0} |g⟩⟨g| + \underbrace{γ_{e}}_{>0} |e⟩⟨e| + \underbrace{γ_{g′}}_{≥0} |g′⟩⟨g′| .

For the interaction picture^{23}

ρ̂^{I}(t) =^{def} e^{+ \frac{i}{ℏ} Ĥ_{atom}^{(3)} t} ρ̂_{atom}^{(3)}(t) e^{− \frac{i}{ℏ} Ĥ_{atom}^{(3)} t} ,
V̂^{I}(t) =^{def} e^{+ \frac{i}{ℏ} Ĥ_{atom}^{(3)} t} (Ĥ_{cl}^{(3)}(t) − Ĥ_{atom}^{(3)}) e^{− \frac{i}{ℏ} Ĥ_{atom}^{(3)} t}

this implies

iℏ \frac{d}{dt} ρ̂^{I}(t) = [V̂^{I}(t), ρ̂^{I}(t)]_{−} − i \frac{ℏ}{2}[Γ̂, ρ̂^{I}(t)]_{+}  (8.28)

with

V̂^{I}(t) = − (\frac{ℏ}{2} Ω_{p} e^{−i(Δ_{p}t−ϕ_{p})} |g⟩⟨e| + h.c.)

− (\frac{ℏ}{2} Ω_{c} e^{−i(Δ_{c}t−ϕ_{c})} |g′⟩⟨e| + h.c.) .

In order to determine — for given coupling radiation — the linear atomic susceptibility

χ^{↔}_{atom}^{(1)}(ω_{p}) E_{p} = lim_{T →+∞} ∫_{0}^{T} Ω_{p} \frac{∂}{∂Ω_{p}} \left⟨{ \atop μ̂}\right⟩ (t) e^{+i ω_{p} t} dt
μ̂ =^{def} q P̂_{H_{atom}^{(3)}(t)}x̂ /^{\} H_{atom}^{(3)}(t)

for the pump beam we develop the functions

ρ_{jk}(t) =^{def} ⟨j| ρ̂^{I}(t) |k⟩ ∀ j, k ∈ {g, e, g′}

into power series of Ω_{p} :

ρ_{jk}(t, Ω_{p}) = ∑_{n=0}^{∞} ρ_{jk}^{[n]}(t) (Ω_{p})^{n} .

Draft, November 5, 2011

21We essentially follow the representation in (Scully and Zubairy, 1999, Sect. 7.3).

22Actually, one should also add i ℏ\sqrt{Γ̂} ρ̂_{atom}^{(3)}(t) \sqrt{Γ̂} on the r.h.s. in order to get a Lindblad equation preserving the trace and positivity of ρ̂_{atom}^{(3)}(t); more generally, see (Ottaviani et al., 2006). However, this would not effect the result for the linear susceptibility.

23See 6.1.2.

166 CHAPTER 8. FEW-LEVEL ATOMIC SYSTEMS Then we have

ϵ_{0} χ^{↔}_{atom}^{(1)}(ω_{p}) E_{p}

= lim_{T →+∞} ∫_{0}^{T} Ω_{p} Tr(e^{− \frac{i}{ℏ} Ĥ_{atom}^{(3)} t} \underbrace{∑_{j,k∈\{g,e,g′\}} ρ_{jk}^{[1]}(t) |j⟩⟨k|}_{=( \frac{∂}{∂Ωp} ρ̂^{I}(t))_{|_{Ωp=0}}} e^{+ \frac{i}{ℏ} Ĥ_{atom}^{(3)} t} μ̂) e^{+i ω_{p} t} dt
= lim_{T →+∞} ∫_{0}^{T} ∑_{j,k∈\{g,e,g′\}} ρ_{jk}^{[1]}(t) e^{− \frac{i}{ℏ} (E_{j}−E_{k})t} ⟨k| q x̂ |j⟩ Ω_{p} e^{+i ω_{p} t} dt
=_{(8.18)} lim_{T →+∞} ∫_{0}^{T} (ρ_{eg}^{[1]}(t) e^{− \frac{i}{ℏ} (E_{e}−E_{g})t} ⟨g| q x̂ |e⟩ Ω_{p} + c.c.) e^{+i ω_{p} t} dt

+ lim_{T →+∞} ∫_{0}^{T} (ρ_{eg′}^{[1]}(t) e^{− \frac{i}{ℏ}(E_{e}−E_{g′})t} ⟨g′| q x̂ |e⟩ Ω_{p} + c.c.) e^{+i ω_{p} t} dt .

(8.29)

With

[V̂^{I}(t), |g⟩⟨g|]_{−} = \frac{ℏ}{2} Ω_{p} e^{−i(Δ_{p}t−ϕ_{p})} |g⟩⟨e| − h.c. ,
[V̂^{I}(t), |g′⟩⟨g′|]_{−} = \frac{ℏ}{2} Ω_{c} e^{−i(Δ_{c}t−ϕ_{c})} |g′⟩⟨e| − h.c. ,
[V̂^{I}(t), |e⟩⟨e|]_{−} = − (\frac{ℏ}{2} Ω_{p} e^{−i(Δ_{p}t−ϕ_{p})} |g⟩⟨e| − h.c.)

− (\frac{ℏ}{2} Ω_{c} e^{−i(Δ_{c}t−ϕ_{c})} |g′⟩⟨e| − h.c.) ,

[V̂^{I}(t), |e⟩⟨g|]_{−} = \frac{ℏ}{2} Ω_{p} e^{−i(Δ_{p}t−ϕ_{p})}(|e⟩⟨e| − |g⟩⟨g|)

− \frac{ℏ}{2} Ω_{c} e^{−i(Δ_{c}t−ϕ_{c})} |g′⟩⟨g| ,

[V̂^{I}(t), |e⟩⟨g′|]_{−} = \frac{ℏ}{2} Ω_{c} e^{−i(Δ_{c}t−ϕ_{c})}(|e⟩⟨e| − |g′⟩⟨g′|)

− \frac{ℏ}{2} Ω_{p} e^{−i(Δ_{p}t−ϕ_{p})} |g⟩⟨g′| ,

[V̂^{I}(t), |g⟩⟨g′|]_{−} = −\frac{ℏ}{2} Ω_{p} e^{+i(Δ_{p}t−ϕ_{p})} |e⟩⟨g′| + \frac{ℏ}{2} Ω_{c} e^{−i(Δ_{c}t−ϕ_{c})} |g⟩⟨e| .

and

[Γ̂, |j⟩⟨k|]_{+} = (γ_{j} + γ_{k}) |j⟩⟨k| ∀ j, k ∈ {g, e, g′}

8.3. 3-LEVEL ATOMS 167 together with (8.28) we also conclude that

ρ̇_{gg}^{[0]}(t) = 0 ,
ρ̇_{ee}^{[0]}(t) = −γ_{e} ρ_{ee}^{[0]}(t) ,
ρ̇_{g′g′}^{[0]}(t) = −γ_{g′} ρ_{g′g′}^{[0]}(t) ,
ρ̇_{g′e}^{[0]}(t) = −\frac{γ_{g′} + γ_{e}}{2} ρ_{g′e}^{[0]}(t)

(8.30)

and

ρ̇_{eg}^{[1]}(t) = \frac{i}{2} e^{+i(Δ_{p}t−ϕ_{p})} (ρ_{gg}^{[0]}(t) − ρ_{ee}^{[0]}(t))

+\frac{i}{2} Ω_{c} e^{+i(Δ_{c}t−ϕ_{c})}ρ_{g′g}^{[1]}(t) − \frac{γ_{e}}{2} ρ_{eg}^{[1]}(t) ,

ρ̇_{g′g}^{[1]}(t) = \frac{i}{2} Ω_{c} e^{−i(Δ_{c}t−ϕ_{c})}ρ_{eg}^{[1]}(t)

−\frac{i}{2} e^{+i(Δ_{c}t−ϕ_{c})} ρ_{g′e}^{[0]}(t) − \frac{γ_{g′}}{2} ρ_{g′g}^{[1]}(t) ,

ρ̇_{eg′}^{[1]}(t) = \frac{i}{2} Ω_{c} e^{+i(Δ_{c}t−ϕ_{c})} (ρ_{g′g′}^{[1]}(t) − ρ_{ee}^{[1]}(t))

+\frac{i}{2} e^{+i(Δ_{p}t−ϕ_{p})} ρ_{gg′}^{[0]}(t) − \frac{γ_{e} + γ_{g′}}{2} ρ_{eg′}^{[1]}(t) .

(8.31)

For the special case

ρ̂^{I}(t_{0}) = |g⟩⟨g| ∀ Ω_{p}  (8.32)

(8.30) implies

ρ_{gg}^{[0]}(t) = 1 , ρ_{ee}^{[0]}(t) = ρ_{g′g′}^{[0]}(t) = ρ_{g′e}^{[0]}(t) = 0

and hence, by (8.31),

ρ̇_{eg}^{[1]}(t) = \frac{i}{2} e^{+i(Δ_{p}t−ϕ_{p})} + \frac{i}{2} Ω_{c} e^{+i(Δ_{c}t−ϕ_{c})} ρ_{g′g}^{[1]}(t) − \frac{γ_{e}}{2} ρ_{eg}^{[1]}(t) ,
ρ̇_{g′g}^{[1]}(t) = \frac{i}{2} Ω_{c} e^{−i(Δ_{c}t−ϕ_{c})}ρ_{eg}^{[1]}(t) − \frac{γ_{g′}}{2} ρ_{g′g}^{[1]}(t) .

If, moreover, we assume

ω_{c} = ω_{eg′}  (8.33)

then me may conclude

\begin{aligned}Ṙ(t) & = & −M̂ R(t) + A , \\ R(t_{0}) & = & 0 , \end{aligned}  (8.34)

168 CHAPTER 8. FEW-LEVEL ATOMIC SYSTEMS where:

R(t) =^{def} ( \begin{matrix}e^{−i(Δ_{p}t−ϕ_{p})} ρ_{eg}^{[1]}(t) \\ e^{−i(Δ_{p}t−ϕ_{p})} ρ_{g′g}^{[1]}(t) \end{matrix} ) ,
M̂ =^{def} ( \begin{matrix}\frac{γ_{e}}{2} + iΔ_{p} & −\frac{i}{2} Ω_{c} e^{−i ϕ_{c}} \\ − \frac{i}{2} Ω_{c} e^{+i ϕ_{c}} & \frac{γ_{g′}}{2} + iΔ_{p} \end{matrix} ) ,
A =^{def} ( 0^{\frac{i}{2}} ) .

The solution of the initial value problem (8.34) is

R(t) = M̂^{−1}A − e^{−M̂(t−t_{0})} M̂^{−1} A
⟶_{t_{0}→−∞} M̂^{−1}A .  (8.35)

Sketch of proof for (8.35): Since

det(M̂ − iΔ_{p} − E) = \frac{1}{4} (γ_{e} − E)(γ_{g′} − E) + (Ω_{c}/2)^{2} ,

M̂ − iΔ_{p} has only eigenvalues E with ℜ(E) = \frac{1}{2} (γ_{e} + γ_{g′}) > 0 .

Therefore,^{24} assuming (8.32) and (8.33), asymptotically (for large t) we have

ρ_{eg}^{[1]}(t) = \frac{i}{2} \frac{e^{+i(Δ_{p}t−ϕ_{p})}(\frac{γ_{g′}}{2} + iΔ_{p})}{(\frac{γ_{e}}{2} + iΔ_{p})(\frac{γ_{g′}}{2} + iΔ_{p}) + (Ω_{c}/2)^{2}} .

For

\frac{E_{e} − E_{g}}{ℏ} > 0 < \frac{E_{e} − E_{g′}}{ℏ} ≠ ω_{p} > 0

this, together with

Ω_{p} =_{(8.20)} \frac{2 q}{ℏ} ⟨e| x̂ |g⟩ · E_{p} e^{+iϕ_{p}}

and (8.29), implies^{25}

ϵ_{0} χ^{↔}_{atom}^{(1)}(ω_{p}) E_{p} = ρ_{eg}^{[1]}(t) e^{i(ω_{p}−ω_{eg})t} ⟨g| q x̂ |e⟩ Ω_{p}
= \frac{i}{ℏ} \frac{⟨g| q x̂ |e⟩ (⟨e| q x̂ |g⟩ · E_{p})(\frac{γ_{g′}}{2} + iΔ_{p})}{(\frac{γ_{e}}{2} + iΔ_{p})(\frac{γ_{g′}}{2} + iΔ_{p}) + (Ω_{c}/2)^{2}} .

Draft, November 5, 2011

24Note that

a d − b c ≠ 0 ⟹ ( \begin{matrix}a & b \\ c & d \end{matrix} )^{−1} = ( \begin{matrix}+d & −c \\ −b & +a \end{matrix} ) /(a d − b c) ∀ a, b, c, d ∈ ℂ .

25We do not show explicitly that the second contribution on the r.h.s. of (8.29) vanishes.

8.3. 3-LEVEL ATOMS 169 Thus, especially for

E_{p} ∝ ⟨e| x̂ |g⟩

we have

χ^{↔}_{atom}^{(1)}(ω_{p}) E_{p} = χ_{atom}(ω_{p}) E_{p} ,

where

χ_{atom}(ω_{p}) =^{def} \frac{i}{ϵ_{0} ℏ} \frac{|⟨g| q x̂ |e⟩|^{2} (\frac{γ_{g′}}{2} + iΔ_{p})}{(\frac{γ_{e}}{2} + iΔ_{p})(\frac{γ_{g′}}{2} + iΔ_{p}) + (Ω_{c}/2)^{2}} .

Together with

\frac{1}{(\frac{γ_{e}}{2} + iΔ_{p})(\frac{γ_{g′}}{2} + iΔ_{p}) + (Ω_{c}/2)^{2}}

= \frac{\frac{γ_{e}}{2} \frac{γ_{g′}}{2} − (Δ_{p})^{2} + (Ω_{c}/2)^{2} − i (\frac{γ_{e}}{2} + \frac{γ_{g′}}{2} ) Δ_{p}}{|\frac{γ_{e}}{2} \frac{γ_{g′}}{2} − (Δ_{p})^{2} + (Ω_{c}/2)^{2} + i (\frac{γ_{e}}{2} + \frac{γ_{g′}}{2} ) Δ_{p}|^{2}}
= \frac{\frac{γ_{e}}{2} \frac{γ_{g′}}{2} − (Δ_{p})^{2} + (Ω_{c}/2)^{2} − i (\frac{γ_{e}}{2} + \frac{γ_{g′}}{2} ) Δ_{p}}{(\frac{γ_{e}}{2} \frac{γ_{g′}}{2} − (Δ_{p})^{2} + (Ω_{c}/2)^{2})^{2} + (\frac{γ_{e}}{2} + \frac{γ_{g′}}{2} )^{2} (Δ_{p})^{2}}

this, finally, gives:

ℜ(χ_{atom}(ω_{p})) = \frac{|⟨g| q x̂ |e⟩|^{2}}{ϵ_{0} ℏ} \frac{Δ_{p} ((\frac{γ_{g′}}{2} )^{2} + (Δ_{p})^{2} − (Ω_{c}/2)^{2})}{(\frac{γ_{e}}{2} \frac{γ_{g′}}{2} − (Δ_{p})^{2} + (Ω_{c}/2)^{2})^{2} + (\frac{γ_{e}}{2} + \frac{γ_{g′}}{2} )^{2} (Δ_{p})^{2}} ,
ℑ(χ_{atom}(ω_{p})) = \frac{|⟨g| q x̂ |e⟩|^{2}}{ϵ_{0} ℏ} \frac{(Δ_{p})^{2} \frac{γ_{e}}{2} + \frac{γ_{g′}}{2} (\frac{γ_{e}}{2} \frac{γ_{g′}}{2} + (Ω_{c}/2)^{2})}{(\frac{γ_{e}}{2} \frac{γ_{g′}}{2} − (Δ_{p})^{2} + (Ω_{c}/2)^{2})^{2} + (\frac{γ_{e}}{2} + \frac{γ_{g′}}{2} )^{2} (Δ_{p})^{2}} .

Defining

ξ(λ) =^{def} \frac{λ (λ^{2} − 1)}{(λ^{2} − 1)^{2} + (\frac{2 γ_{e}}{Ω_{c}} )^{2} λ^{2}} ,
η(λ) =^{def} \frac{λ^{2}}{(λ^{2} − 1)^{2} + (\frac{2 γ_{e}}{Ω_{c}} )^{2} λ^{2}}
= \frac{1}{(λ − 1/λ)^{2} + (\frac{2 γ_{e}}{Ω_{c}} )^{2}} für λ ≠ 0

we get

γ_{g′} = 0 < Ω_{c} ⟹ \{ \begin{matrix}ℜ(χ_{atom}(ω_{p})) = \frac{2 |⟨g| q x̂ |e⟩|^{2}}{ϵ_{0} ℏ Ω_{c}} ξ(\frac{2 Δ_{p}}{Ω_{c}} ) , \\ ℑ(χ_{atom}(ω_{p})) = \frac{2 |⟨g| q x̂ |e⟩|^{2}}{ϵ_{0} ℏ Ω_{c}} \frac{γ_{e}}{Ω_{c}} η(\frac{2 Δ_{p}}{Ω_{c}} ) , \end{matrix}

Figure 8.1: Graphs of the functions ξ(λ) (red,below) and η(λ) (green,above).

max_{0≠λ∈ℝ} η(λ) = λ(1) .

For the special case Ω_{c} = 2 γ_{e} the graphs of the functions ξ(λ) and η(λ) are sketched in Figure 8.1.

Quite generally the complex refractive index of an isotropic medium with linear susceptibility χ(ω_{p}) is characterized by

N (ω_{p}) =^{def} \sqrt{1 + χ(ω_{p})} ∈ ℝ_{+} + i ℝ .

Because of

( ^{+} \sqrt{\frac{|z| + ℜ(z)}{2}} + i ^{+} \sqrt{\frac{|z| − ℜ(z)}{2}} sign (ℑ(z)))^{2} = z ∀ z ∈ ℂ \ ℝ .

the latter is equivalent to

ℜ(N (ω_{p})) = ^{+} \sqrt{\frac{|1 + χ(ω_{p})| + ℜ(1 + χ(ω_{p}))}{2}} ,

ℑ(N (ω_{p})) = ^{+} \sqrt{\frac{|1 + χ(ω_{p})| − ℜ(1 + χ(ω_{p}))}{2}} sign(ℑ(χ(ω_{p}))) .

Cavity QED

The general idea for modeling the coupled system of an atom and radiation in a high Q cavity is the following:

• There are only a few resonant modes dominating the radiation in the cavity. Therefore, in a reasonable approximate description, all other modes of the quantized radiation field may be disregarded.

• There are only a few levels of the atom involved in the interaction with the cavity field. Therefore, in a reasonable approximate description, the internal state space of the atom subsystem may be restricted to describe just these levels.

9.1 The Jaynes-Cummings Model

In the simplest model of cavity QED the radiation field is restricted to a single, essentially monochromatic mode â and the atom is described as a two-level quantum system at some fixed position in the cavity.

9.1.1 Quantized Radiation Field for Selected Modes

Let a_{1}, a_{2} , … be a maximal family of pairwise orthogonal modes. Then, by (1.37),

f_{ν}^{j}(k) = [â_{j}(k), â_{ν}^{†}]_{−} ∀ j ∈ {1, 2} , ν ∈ ℝ , k ∈ ℝ^{3} .  (9.1)

(1.61) together with (1.33) implies

Ê_{â_{1},...,â_{n}}^{(+)}(x) =^{def} P̂_{H_{â_{1},...,ân}} Ê_{0}^{(+)}(x, 0) P̂_{H_{â_{1},...,ân}}
= ∑_{ν=1}^{n} Eâ_{ν}(x) â_{ν} ∀ n ∈ ℕ ,

(9.2) where

Eâ_{ν}(x) =^{def} −i (2π)^{− \frac{3}{2}} \sqrt{μ_{0}ℏc} ∑_{j=1}^{2} ∫ c |k| ϵ_{j}(k) [â_{j}(k), â_{ν}^{†}]_{−} e^{i k·x} \frac{dV_{k}}{\sqrt{2 |k|}} .  (9.3)

(1.61) together with (1.53) implies

\begin{aligned}Ĥ_{â_{1},...,â_{n}} & =^{def} & P̂_{H_{â_{1},...,ân}} Ĥ_{field} P̂_{H_{â_{1},...,ân}} \\ & = & ∑_{ν,μ=1}^{n} h_{νμ} â_{ν}^{†} â_{μ} ∀ n ∈ ℕ , \end{aligned}  (9.4)

where

h_{νμ} =^{def} ∑_{j=1}^{2} ∫ ℏ c |k| (f_{ν}^{j}(k))^{∗} f_{μ}^{j}(k) dV_{k}
= (h_{ν′ν})^{∗} ∀ ν, μ ∈ ℕ .

(9.5)

(9.4) simplifies to

Ĥ_{â_{1},...,â_{n}} = ∑_{ν=1}^{n} h_{νν} â_{ν}^{†} â_{ν}

if the modes â_{ν} are non-overlapping, i.e. if

ν ≠ μ ⟹ f_{ν}^{j}(k) f_{μ}^{j}(k) = 0 ∀ j ∈ {1, 2} .

Remark: Note that restriction to the non-overlapping modes â_{1}, …, â_{n} can only be consistent if, e.g., the â_{1}^{†} Ω, …, â_{n}^{†} Ω have (sufficiently) sharp energy. Actually for cavity modes this is only possible due to a change of boundary conditions resulting in a change of the dynamics. Thus restricting the Hamiltonian of the radiation field to some non-overlapping modes is much more drastic than restricting the Hamiltonian of an atom to a few levels.

9.1.2 Interaction of the Two-Level System with the Cavity Field

Let the atom be localized at x = 0 . Then the electric field representation discussed in 7.1.2 suggests using^{1}

Ĥ_{cav} =^{def} (P̂_{H_{m}} ⊗ P̂_{Hâ}) (Ĥ_{atom} + Ĥ_{field} − q x̂ · Ê_{0}(0, 0)) (P̂_{H_{m}} ⊗ P̂_{Hâ})
= Ĥ_{m} + Ĥâ − q x̂_{m} · Êâ ,  (9.6)

Draft, November 5, 2011

1Recall the agreement made in 7.1.1.

where

Êâ =^{def} Ê(+) ; â(0) + (Ê(+) ; â(0))^{†} , x̂_{m} =^{def} P̂_{H_{m}} x̂ P̂_{H_{m}} ,  (9.7)

as an approximate Hamiltonian for interacting system of the two-level atom and the quantized single-mode radiation field.

Remark: Note that in the electric field representation p̂_{can} is to be interpreted as the (approximate) observable of the mechanical — rather than canonical — momentum of the atom’s electron! Correspondingly, H_{atom} is to be interpreted as the (approximate) observable of the internal energy of the one-electron atom.

Assuming^{2}

⟨Ψ_{ν} | x̂ Ψ_{ν}⟩ = 0 for ν = 1, 2  (9.8)

and defining

μ_{01} =^{def} q ⟨Ψ_{0} | x̂ Ψ_{1}⟩  (9.9)

we get

q x̂_{m} = μ_{01} b̂ + μ_{01}^{∗} b̂^{†}  (9.10)

for the (restricted) atomic dipole operator in the matrix representation. This is a simple consequence of (9.7), (9.2), and (8.11). Moreover, defining

E =^{def} −i (2π)^{− \frac{3}{2}} \sqrt{μ_{0}ℏc} ∑_{j=1}^{2} ∫ c |k| ϵ_{j}(k) [â_{j}(k), â_{ν}^{†}]_{−} \frac{dV_{k}}{\sqrt{2 |k|}}

we get

Êâ = E â + E^{∗}â^{†}  (9.11)

for the restricted electric field operator as a direct consequence of (9.7), (9.2), and (9.3).

Let us consider the special case^{3}

μ_{01} = μ_{m} \frac{e_{1} + i e_{2}}{\sqrt{2}} , E = E_{0} \frac{e_{1} + i e_{2}}{\sqrt{2}} .  (9.12)

Then, up to an irrelevant additive constant, Ĥ_{cav} coincides with the Jaynes-Cummings Hamiltonian

Ĥ_{JC} =^{def} ℏ ω_{0} b̂^{†}b̂ + ℏ ω â^{†}â + λ â^{†}b̂ + λ^{∗} â b̂^{†} ,  (9.13)

Draft, November 5, 2011

2This assumption is usually fulfilled for the stationary states Ψ_{0}, Ψ_{1} of the atom.

3The first condition corresponds to Ψ_{1} → Ψ_{0} being a Δm = ±1-transition. If the conditions (9.12) are not fulfilled (9.13) is called rotating-wave approximation; see, e.g., (Schleich, 2001, Section 14.8.2).

174 CHAPTER 9. CAVITY QED where

ω =^{def} ℏ^{−1} ∑_{j,j′=1}^{2} ∫ c |k| [â, â_{j}^{†}(k)]_{−} [â_{j′}(k), â^{†}]_{−} dV_{k} , λ =^{def} μ_{m} E_{0} .

Proof: Thanks to

(e_{1} + i e_{2})^{2} = 0 , (e_{1} + i e_{2}) (e_{1} − i e_{2}) = 2

we have

− q x̂_{m} · Êâ =_{(9.10)–(9.12)} (μ_{m} \frac{e_{1} + i e_{2}}{\sqrt{2}} b̂ + μ_{m}^{∗} \frac{e_{1} − i e_{2}}{\sqrt{2}} b̂^{†})

· (E_{0} \frac{e_{1} + i e_{2}}{\sqrt{2}} â + E_{0}^{∗} \frac{e_{1} − i e_{2}}{\sqrt{2}} â^{†})

= λ â^{†}b̂ + λ^{∗} â b̂^{†} . (9.14)

(9.2), (9.3), and (9.1) imply

Ĥâ = ℏ ω â^{†}â .  (9.15)

(8.4) and (8.9), finally, imply

Ĥ_{m} − ℏ ω_{0} b̂^{†}b̂ ∝ ˆ1 ,  (9.16)

Now, the statement is a direct consequence of (9.14)–(9.16).

Without loss of generality, we assume that^{4}

λ = λ^{∗} .

With

H^{(0)} =^{def} {α |0, 0⟩ : α ∈ ℂ}

and

H^{(n+1)} =^{def} \{( \begin{matrix}α_{1} \\ α_{0} \end{matrix} )_{n} =^{def} α_{0} |0, n + 1⟩ + α_{1} |1, n⟩ : α_{0}, α_{1} ∈ ℂ\} ∀ n ∈ ℤ_{+} ,

where

|j, n⟩ =^{def} Ψ_{j} ⊗ \frac{1}{\sqrt{n!}} (â^{†})^{n} Ω ∀ j ∈ \{0, 1\} , n ∈ ℤ_{+} .

we have

H_{m} ⊗ Hâ = ⨁_{n=0}^{∞} H^{(n)}

and, obviously, Ĥ_{JC} leaves all the H^{(n)} invariant. For every n ∈ ℤ_{+} , since

â^{†} |j, n⟩ = \sqrt{n + 1} |j, n + 1⟩ , â |j, n + 1⟩ = \sqrt{n + 1} |j, n⟩ ,

Draft, November 5, 2011

4Otherwise we could change â by a phase factor to arrange this.

we have

Ĥ_{JC} ( \begin{matrix}α_{1} \\ α_{0} \end{matrix} )_{n} = Ĥ_{JC}^{(n+1)} ( \begin{matrix}α_{1} \\ α_{0} \end{matrix} )_{n} ∀ α_{0}, α_{1} ∈ ℂ

with

Ĥ_{JC}^{(n+1)} =^{def} ( \begin{matrix}ℏ ω_{0} + n ℏ ω & λ \sqrt{n + 1} \\ λ\sqrt{n + 1} & (n + 1) ℏ ω \end{matrix} )

= Ě_{n} τ̂^{0} + λ\sqrt{n + 1} τ̂^{1} + ℏ \frac{δ}{2} τ̂^{3}  (9.17)

and

Ě_{n} =^{def} ℏ (ω_{0} + (n + 1/2) ω) , δ =^{def} ω_{0} − ω (detuning) .  (9.18)

In order to exploit (9.17), note that^{5}

e^{i φ·τ^{}}ˆτ̂ e^{−i φ·τ^{}}ˆ = D̂_{2 φ} τ̂ ∀ φ ∈ ℝ^{3} .  (9.19)

Sketch of proof: A straightforward calculation using (8.8) shows that

f̂(φ) = f̂_{1}(φ) =^{def} e^{i φ e·τ^{}}ˆτ̂ e^{−i φ e·τ^{}}ˆ

is a solution of the initial value problem

\frac{d}{dφ} f̂(φ) = 2 e × f̂(φ) , f̂(0) = τ̂ .

Since also

f̂(φ) = f̂_{2}(φ) =^{def} D̂_{2 φ e} τ̂

is such a solution, both f̂_{1}(φ) and f̂_{2}(φ) must coincide.

Thus

e^{+i \frac{i}{ℏ} Ĥ_{JC}^{(n+1)} t} τ̂ e^{−i \frac{i}{ℏ} Ĥ_{JC}^{(n+1)} t} = D̂_{2Ω_{n}t e′_{n}} τ̂ ,  (9.20)

where

e′_{n} =^{def} ( \begin{matrix}\sqrt{n + 1} λ/ℏ \\ 0 \\ δ/2 \end{matrix} ) /Ω_{n}

and

Ω_{n} =^{def} \sqrt{(\frac{λ}{ℏ})^{2} (n + 1) + (\frac{δ}{2})^{2}} (Rabi frequency) .  (9.21)

Note that, according to (8.13), the nutation of the Bloch vector (relative to e_{3}) corresponds to transitions between the two levels of the atom with a circular frequency Ω_{n}.

Draft, November 5, 2011

5By D̂φ we denote rotation around the axis along φ by the angle |φ|.

176 CHAPTER 9. CAVITY QED

Assume that

Ψ̂^{(n+1)} = ( \begin{matrix}α_{1} \\ α_{0} \end{matrix} )_{n} ∈ H^{(n+1)} , α_{1} ≥ 0 ,

is a normalized eigenstate of Ĥ_{JC}^{(n+1)} , n ∈ ℤ_{+} , Then, by (9.20),

( \begin{matrix}α_{1} (α_{0} + α_{0}^{∗}) \\ −i α_{1} (α_{0} − α_{0}^{∗}) \\ |α_{1}|^{2} − |α_{0}|^{2} \end{matrix} ) = ⟨Ψ̂^{(n+1)} | τ̂ Ψ̂^{(n+1)}⟩ = σe′_{n}

holds for either σ = +1 or σ = −1 . This implies σ α_{0} ≥ 0 and, together with normalization:

(α_{1})^{2} + (α_{0})^{2} = 1 , (α_{1})^{2} − (α_{0})^{2} = \frac{σ δ}{2 Ω_{n}} .

Thus,

α_{1} = ^{+} \sqrt{\frac{1}{2} + \frac{σ δ}{4 Ω_{n}}} , α_{0} = σ ^{+} \sqrt{\frac{1}{2} − \frac{σ δ}{4 Ω_{n}}}

holds for either σ = +1 or σ = −1 . We conclude that

\{Ψ̂_{σ}^{(n+1)} =^{def} ^{+} \sqrt{\frac{1}{2} + \frac{σ δ}{4 Ω_{n}}} |1, n⟩ + σ ^{+} \sqrt{\frac{1}{2} − \frac{σ δ}{4 Ω_{n}}} |0, n + 1⟩ : σ ∈ \{−, +\}\}

is a maximal orthonormal set of eigenvectors of Ĥ_{JC}^{(n+1)} :

Ĥ_{JC}^{(n+1)} Ψ̂_{±}^{(n+1)} = (Ě_{n} ± ℏ Ω_{n}) Ψ̂_{±}^{(n+1)} .  (9.22)

Proof of (9.22): The statement is a direct consequence of

\begin{aligned}(Ĥ_{JC}^{(n+1)} − Ě_{n} τ̂^{0}) ( \begin{matrix}α_{1} \\ α_{0} \end{matrix} )_{n} & =_{(9.17)} & ( \begin{matrix}λ\sqrt{n + 1} α_{0} + ℏ \frac{δ}{2} α_{1} \\ λ\sqrt{n + 1} α_{1} − ℏ \frac{δ}{2} α_{0} \end{matrix} )_{n} \\ & =_{(9.17)} & ( \begin{matrix}(λ\sqrt{n + 1} \frac{α_{0}}{α_{1}} + ℏ \frac{δ}{2}) α_{1} \\ (λ\sqrt{n + 1} \frac{α_{1}}{α_{0}} − ℏ \frac{δ}{2}) α_{0} \end{matrix} )_{n} \end{aligned}

and

λ\sqrt{n + 1} σ \sqrt{\frac{2 Ω_{n} ∓ σ δ}{2 Ω_{n} ± σ δ}} ± ℏ \frac{δ}{2} = \frac{σ λ\sqrt{n + 1} (2 Ω_{n} ∓ σ δ)}{\sqrt{4 (Ω_{n})^{2} − δ^{2}}} ± ℏ \frac{δ}{2}
=_{(9.21)} \frac{σ λ\sqrt{n + 1} (2 Ω_{n} ∓ σ δ)}{2 \frac{λ}{ℏ} \sqrt{n + 1}} ± ℏ \frac{δ}{2}

= σ ℏ Ω_{n} .

9.1. THE JAYNES-CUMMINGS MODEL 177

Since

e^{i φ·τ^{}}ˆ = cos |φ| τ̂^{0} + i sin |φ| \frac{φ}{|φ|} · τ̂ ∀ φ ∈ ℝ^{3}

(see Exercise E25b) of (Lücke, eine)), (9.17) also implies

e^{− \frac{i}{ℏ} Ĥ_{JC}^{(n+1)} t} = e^{− \frac{i}{ℏ} Ê_{n} t} (cos (Ω_{n} t) τ̂^{0} − i sin (Ω_{n} t) (\frac{λ \sqrt{n + 1}}{ℏ Ω_{n}} τ̂^{1} + \frac{δ}{2Ω_{n}} τ̂^{3})) .

178 CHAPTER 9. CAVITY QED

Appendix A

A.1 Functional Taylor Expansion

Of course, the polarization of a nonlinear medium should be a nonlinear functional of electromagnetic field. The first order functional derivative characterizes this functional only approximately (fortunately with very high precision for usual fields). This is why a functional Taylor expansion is needed.

Let T be some complex vector space of functions on ℝ^{n} with norm^{1} ∥.∥ and let F be a continuous (complex) functional on T , i.e. a continuous mapping from T into ℂ. Then the functional derivative^{2} of F at g ∈ T , if it exists, is the unique bounded linear Functional L_{g} on T fulfilling

lim_{ϵ→+0} sup_{φ∈T_{∥φ∥<ϵ}} \frac{1}{ϵ} |F (g + φ) − F (g) − L_{g}(φ)| = 0

and is identified with the generalized function \frac{δ}{δg(x)} F (g) :

∫_{ℝ^{n}} (\frac{δ}{δg(x)} F (g)) φ(x) dV_{x} =^{def} L_{g}(φ) ∀ φ ∈ T .

Note that

∫_{ℝ^{n}} (\frac{δ}{δg(x)} F (g)) φ(x) dV_{x} = lim_{ϵ→+0} \frac{F (g + ϵ φ) − F (g)}{ϵ} ∀ φ ∈ T

if L_{g} exists. More generally, the n-fold functional derivative

\frac{δ}{δg(x_{1})} … \frac{δ}{δg(x_{ν})} F (g)

of F at g is iteratively defined by

∫_{ℝ^{(ν+1)n}} (\frac{δ}{δg(x_{1})} … \frac{δ}{δg(x_{ν+1})} F (g)) φ_{1}(x_{1}) … φ_{ν+1}(x_{ν+1}) dV_{x_{1}} … dV_{x_{ν+1}}

=^{def} ∫_{ℝ^{n}} (\frac{δ}{δg(x)} F_{φ_{2},...,φ_{ν+1}}(g)) φ_{1}(x) dV_{x} ∀ φ_{1}, …, φ_{ν+1} ∈ T ,

Draft, November 5, 2011

1The linear space T need not be complete w.r.t. ∥.∥.

2This is a special case of the Fréchet derivative (see, e.g., (Choquet-Bruhat et al., 1978, p. 71)).

where

F_{φ_{2},...,φ_{ν+1}}(g)

=^{def} ∫_{ℝ^{νn}} (\frac{δ}{δg(x_{2})} … \frac{δ}{δg(x_{ν+1})} F (g)) φ_{2}(x_{2}) … φ_{ν+1}(x_{ν+1}) dV_{x_{2}} … dV_{x_{ν+1}}

∀ φ_{2}, …, φ_{ν+1} ∈ T .

This amounts to

∫_{ℝ^{νn}} (\frac{δ}{δg(x_{1})} … \frac{δ}{δg(x_{ν})} F (g)) φ_{1}(x_{1}) … φ_{ν}(x_{ν}) dV_{x_{1}} … dV_{x_{ν}}

= (\frac{∂}{∂ϵ_{1}} … \frac{∂}{∂ϵ_{ν}} F (g + ϵ_{1} φ_{1} + … ϵ_{ν} φ_{ν}))_{|_{ϵ_{1}=...=ϵν =0}} ∀ φ_{1}, …, φ_{ν} ∈ T

(A.1)

whenever \frac{δ}{δg(x_{1})} … \frac{δ}{δg(x_{ν})} F (g) exists. (A.1) especially implies

((\frac{∂}{∂ϵ})^{ν} F (g + ϵ φ))_{|_{ϵ=0}}

= ∫_{ℝ^{νn}} (\frac{δ}{δg(x_{1})} … \frac{δ}{δg(x_{ν})} F (g)) φ(x_{1}) … φ(x_{ν}) dV_{x_{1}} … dV_{x_{ν}} ∀ φ ∈ T .

Therefore, for sufficiently well-behaved F , we have the functional Taylor expansion

F (φ) = (e^{∫ dV_{x} φ(x) \frac{δ}{δg(x)}} F (g))_{|_{g=0}}
=^{def} F (0) + ∑_{ν=1}^{∞} \frac{1}{ν!} ∫ F_{ν}(x_{1}, …, x_{ν}) φ(x_{1}) … φ(x_{ν}) dV_{x_{1}} … dV_{x_{ν}} ,

where

F_{ν}(x_{1}, …, x_{ν}) =^{def} (\frac{δ}{δg(x_{1})} … \frac{δ}{δg(x_{ν})} F (g))_{|_{g=0}} ∀ ν ∈ ℕ .

More precisely, we have

F (φ) = F (0) + ∑_{ν=1}^{n} \frac{1}{ν!} ∫ F_{ν}(x_{1}, …, x_{ν}) φ(x_{1}) … φ(x_{ν}) dV_{x_{1}} … dV_{x_{ν}}

+\frac{1}{n!} ∫_{0}^{1} (1 − λ)^{n} (\frac{d}{dλ})^{n+1} F (λφ) dλ ,

This formalism has a straightforward generalization to vector-valued functions g on ℝ^{n}.

A.2 Elementary Quantum Mechanics^{3}

A.2.1 Scalar Particles in ℝ^{1}

(In the Schrödinger picture) the (pure) state of a single, free, 1-dimensional, scalar, quantum mechanical ‘particle’ of mass m is always given by a wave function Ψ(x, t) fulfilling the free Schrödinger equation

i \frac{∂}{∂t} Ψ(x, t) = \overbrace{\frac{1}{2m} p̂^{2}}^{Ĥ_{kin} =^{def}} Ψ(x, t) ,  (A.2)

where

p̂ =^{def} \frac{ℏ}{i} \frac{∂}{∂x} .  (A.3)

Its basic interpretation is

∫_{G} |Ψ(x, t)|^{2} dx = probability for ‘position at time t inside G’

(A.4)

for all (sufficiently well-behaved) region G ⊂ ℝ and requires^{4}

∫ |Ψ(x, t)|^{2} dx = 1  (A.5)

for all t ∈ ℝ. Simple asymptotic considerations^{5} show that the spatial Fourier transform

Ψ̃(p, t) =^{def} \frac{1}{\sqrt{2πℏ}} ∫ Ψ(x, t) e^{\frac{i}{ℏ} p x}dx  (A.6)

(adapted to quantum mechanics) has a similar interpretation w.r.t. linear momentum:^{6}

∫G̃ |Ψ̃(p, t)|^{2} dp = probability for ‘momentum at time t inside G̃’.

(A.7)

The latter is consistent in the sense that

∫ |Ψ̃(p, t)|^{2} dp = ∫ |Ψ(x, t)|^{2} dx  (A.8)
= 1 .

Draft, November 5, 2011

3For mathematical details see, e.g., (Lücke, fuan).

4As usual, we write ∫ for ∫_{−∞}^{∞} resp. ∫_{ℝ}. One can easily show that (A.2) guarantees that (A.5) holds for all t if fulfilled for t = 0 .

5See, e.g., (Lücke, 1995, Section 5).

6Actually, (A.2) implies the l.h.s. of (A.7) — contrary to the l.h.s. of (A.4) — to be independent of t.

182 APPENDIX A. APPENDIX Defining

Ψ_{t}(x) =^{def} Ψ(x, t) ,
⟨Ψ | Φ⟩ =^{def} ∫ \overline{Ψ(x)}Φ(x) dx ,

and the position observable x̂ by

(x̂ Ψ) (x) =^{def} x Ψ(x)  (A.9)

one gets

⟨Ψ_{t} | x̂ Ψ_{t}⟩ = ∫ x |Ψ_{t}(x)|^{2} dx

and hence^{7} ⟨Ψ_{t} | x̂ Ψ_{t}⟩ = ⟨x(t)⟩ =^{def} \{ \begin{matrix}expectation value for the position variable \\ x at time t in the state given by Ψ_{t} . \end{matrix}

(A.10)

Similarly one has ⟨Ψ_{t} | p̂Ψ_{t}⟩ = ⟨p(t)⟩ =^{def} \{ \begin{matrix}expectation value for the momentum variable \\ p at time t in the state given by Ψ_{t} , \end{matrix}

(A.11)

since

Φ(x, t) = p̂Ψ(x, t) ⟹ Φ̃(p, t) = p Ψ̃(x, t)

and, therefore, the generalization

∫ \overline{Φ(x)} Ψ(x) dx = ∫ \overline{Φ̃(p)} Ψ̃(p) dp  (A.12)

of (A.8) implies

⟨Ψ_{t} | p̂Ψ_{t}⟩ = ∫ p |Ψ̃(p, t)|^{2} dp .  (A.13)

In view of (A.11), p̂ is called momentum observable. (A.10) and (A.11) are consistent in the sense that

(A.2) ⟹ \frac{d}{dt} ⟨x(t)⟩ = \frac{1}{m} ⟨p(t)⟩ = constant.  (A.14)

Similarly to (A.14) one gets

⟨Ψ_{t} | \underbrace{Ĥ_{kin}}_{= \frac{p̂^{2}}{2m} =− \frac{ℏ^{2}}{2m}(\frac{∂}{∂x})^{2}} Ψ_{t}⟩ = ∫ \frac{p^{2}}{2m} |Ψ̃(p, t)|^{2} dp

Draft, November 5, 2011

7Note, however, that ⟨x(t)⟩ may be infinite.

and, therefore,

⟨Ψ_{t} | \frac{p̂^{2}}{2m} Ψ_{t}⟩ = ⟨E_{kin}(t)⟩ =^{def} \{ \begin{matrix}expectation value for the kinetic energy \\ at time t in the state given by Ψ_{t} . \end{matrix} (A.15)

If the particle is not free but interacting with an external force field with potential^{8} V (x) then, of course, (A.2) has to be modified. The basic interpretations (A.4) and (A.7), however, are maintained. Therefore, also (A.10) resp. (A.11) has to be maintained for (A.9) resp. (A.3). But now, in addition, we have

⟨Ψ_{t} | V (x̂) Ψ_{t}⟩ = ⟨E_{kin}(t)⟩ =^{def} \{ \begin{matrix}expectation value for the potential energy \\ at time t in the state given by Ψ_{t} . \end{matrix}

(A.16)

Assuming that the expectation value of the total energy is the sum of the expectation values for kinetic and potential energy, one concludes from (A.15) and (A.16) that

⟨Ψ_{t} | Ĥ Ψ_{t}⟩ = ⟨E(t)⟩ =^{def} \{ \begin{matrix}expectation value for the total energy \\ at time t in the state given by Ψ_{t} , \end{matrix}

(A.17)

where the operator

Ĥ =^{def} \frac{p̂^{2}}{2m} + V (x̂) (Hamiltonian )

(A.18)

is the observable for the total energy. The latter suggests replacing (A.2) by the time dependent Schrödinger equation

i \frac{∂}{∂t} Ψ(x, t) = Ĥ Ψ(x, t) ,  (A.19)

coinciding with (A.2) for V = 0 . This is consistent in the sense that — generalizing (A.14) — (A.19) implies the Ehrenfest equations

\frac{d}{dt} ⟨x(t)⟩ = \frac{1}{m} ⟨p(t)⟩ , \frac{d}{dt} ⟨p(t)⟩ = ∫ (−\frac{d}{dx} V (x)) |Ψ(x, t)|^{2} dx

and ⟨E(t)⟩ =constant.

A.2.2 Energy Eigenstates of the Harmonic Oscillator

One of the main objectives of quantum mechanics is to determine the (possibly improper) energy eigenfunctions, i.e. those Ψ(x) for which there is an energy value E with

Ĥ Φ(x) = E Ψ(x) .  (A.20)

Draft, November 5, 2011

8For particles interacting with a general electromagnetic field see 7.1.1.

184 APPENDIX A. APPENDIX These eigenfunctions give rise to the special solutions

Ψ(x, t) = e^{− \frac{i}{ℏ} E t} Ψ(x)  (A.21)

of (A.19) called stationary, since

(A.21) ⟹ \frac{∂}{∂t} |Ψ(x, t)|^{2} = 0 .

Let us solve (A.20) for the (quantum mechanical) harmonic oscillator, characterized by

V (x) = \frac{m}{2} ω^{2} x^{2} .  (A.22)

This problem is easily solvable by means of the operators

â =^{def} \frac{p̂ − i m ω x̂}{\sqrt{2 m ℏ ω}} , â^{†} =^{def} \frac{p̂ + i m ω x̂}{\sqrt{2 m ℏ ω}}  (A.23)

fulfilling the commutation relations^{9}

[â, â^{†}]_{−} = 1  (A.24)

and hence the equations

\begin{aligned}(â^{†}â) â & = & â (â^{†}â − 1) , \\ (â^{†}â) â^{†} & = & â^{†}(â^{†}â + 1) . \end{aligned}

Iterating these equations gives^{10}

(â^{†}â) â^{ν} = â^{ν} (â^{†}â − ν) ∀ ν ∈ ℕ , (A.25)
(â^{†}â)(â^{†})^{ν} = (â^{†})^{ν} (â^{†}â + ν) ∀ ν ∈ ℕ .  (A.26)

Since

Ĥ = \frac{ℏ}{2} ω (â^{†} â + â â^{†})
= ℏ ω (â^{†} â + \frac{1}{2}) ,  (A.27)

we have

Ĥ Ψ = E Ψ ⟹ \{ \begin{matrix}Ĥ (â^{ν} Ψ) & =_{(A.25)} & (E − ν) â^{ν} Ψ \\ Ĥ ((â^{†})^{ν} Ψ) & =_{(A.26)} & (E + ν) (â^{†})^{ν} Ψ \end{matrix}

Draft, November 5, 2011

9We simply write z for zˆ1 (with z ∈ ℂ) in the following. Note that, in this sense, [p̂, x̂] = \frac{ℏ}{i} . 10Note that, e.g., (A.26) also follows from

â (â^{†})^{ν} =_{(1.82)} (â^{†})^{ν} â + ν (â^{†})^{ν−1} .

for all (sufficiently well-behaved) Ψ . Thus, if Ψ is an eigenfunction of Ĥ with eigenvalue E then

⟨â^{ν} Ψ | Ĥ â^{ν} Ψ⟩ = (E − ν) ⟨â^{ν} Ψ | â^{ν} Ψ⟩

could become negative if â^{ν} Ψ were nontrivial for all ν ∈ ℕ. The latter, however is impossible because of

⟨Φ | Ĥ Φ⟩ =_{(A.27)} ℏ ω ⟨â Φ | â Φ⟩ + \frac{1}{2} ℏ ω ⟨Φ | Φ⟩
> 0 ∀ Φ ≠ 0 .

Therefore:

For every eigenfunction Ψ of Ĥ there is some μ ∈ ℤ_{+} with

â (â^{μ} Ψ) = 0 .  (A.28)

Now, â^{μ} Ψ is fixed — up to some factor — by (A.28):

â (â^{μ} Ψ) = 0 ⟹_{(A.23),(A.3)} \frac{d}{dx} (â^{μ} Ψ) (x) = −\frac{m ω}{ℏ} x (â^{μ} Ψ) (x)
⟹  \frac{d}{dx} ln (â^{μ} Ψ) (x) = −\frac{m ω}{ℏ} x
⟹ (â^{μ} Ψ) (x) ∼ e^{− \frac{m ω}{2 ℏ} x^{2}} .

We conclude that

Ĥ Ψ = E Ψ  ⟹  0 ≠ â^{μ} Ψ ∼ Ω for some μ ∈ ℤ_{+} ,  (A.29)

where^{11}

Ω(x) =^{def} (\frac{m ω}{π ℏ} )^{\frac{1}{4}} e^{− \frac{m ω}{2 ℏ} x^{2}} , â Ω = 0 .  (A.30)

Iterating

\begin{aligned}(â^{†})^{ν} â^{ν} & = & (â^{†})^{ν−1} (â^{†}â) â^{ν−1} \\ & =_{(A.25)} & (â^{†})^{ν−1} â^{ν−1}(â^{†}â − (ν − 1)) \end{aligned}

we get

(â^{†})^{ν} â^{ν} = (â^{†}â)(â^{†}â − 1) ⋯ (â^{†}â − (ν − 1)) ∀ ν ∈ ℕ  (A.31)

and conclude:^{12}

Ĥ (â^{†})^{μ} Ω =_{(A.26),(A.30)} (μ + \frac{1}{2}) ℏ ω Ω ∀ μ ∈ ℤ_{+} ,

Ĥ Ψ = E Ψ ⟹_{(A.29),(A.31),(A.27)} Ψ ∼ (â^{†})^{μ} Ω for some μ ∈ ℤ_{+} ,

Draft, November 5, 2011

11Note that ∫ e^{−x^{2}} dx = \sqrt{π} (more generally, see (2.83)), hence ⟨Ω | Ω⟩ = 1 .

12Note that

â^{μ} Ψ ≠ 0 ⟹ (â^{†})^{μ} â^{μ} Ψ ≠ 0 ,

A.2.3 Coherent States

Recall^{13} that the operators

â_{osc} =^{def} \frac{\frac{ℏ}{i} \frac{d}{dx} − i m ω x}{\sqrt{2 m ℏ ω}} , â_{osc}^{†} = \frac{\frac{ℏ}{i} \frac{d}{dx} + i m ω x}{\sqrt{2 m ℏ ω}}

of the 1-dimensional harmonic oscillator with Hamiltonian

Ĥ_{osc} =^{def} − \frac{ℏ^{2}}{2 m} (\frac{d}{dx})^{2} + \frac{1}{2} m ω^{2} x^{2}

also obey the commutation relation

[â_{osc}, â_{osc}^{†}]_{−} = 1

and that

Ĥ_{osc} = \frac{ℏ}{2} ω (â_{osc}^{†} â_{osc} + â_{osc} â_{osc}^{†}) ,

hence

Ĥ_{osc} = ℏ ω (â_{osc}^{†} â_{osc} + \frac{1}{2}) .

Here the ground state

Ω_{osc}(x) =^{def} (\frac{m ω}{π ℏ} )^{\frac{1}{4}} e^{− \frac{m ω}{2 ℏ} x^{2}}

corresponds to the vacuum:

â_{osc} Ω_{osc} = 0 .

Therefore, since

α â_{osc}^{†} − \overline{α} â_{osc} = ℑ(α) \sqrt{\frac{2 ℏ}{m ω}} \frac{d}{dx} + i ℜ(α) \sqrt{\frac{2 m ω}{ℏ}} x ,

the corresponding coherent states, according to (1.79) and (1.74), are

Ψ_{â_{osc},α}(x) = (\frac{m ω}{π ℏ} )^{\frac{1}{4}} e^{i ℜ(α)ℑ(α)} e^{i ℜ(α)\sqrt{\frac{2 m ω}{ℏ}} x} e^{− \frac{m ω}{2 ℏ} (x+ℑ(α)\sqrt{\frac{2 ℏ}{m ω}} )^{2}} .

This shows that the coherent states of the harmonic oscillator are just displacements in position and momentum of the ground state, hence have the same minimal uncertainty product^{14}

Δx Δp = \frac{ℏ}{2} = \frac{ℏ}{2} Δ(a + a^{†}) Δ(i a − i a^{†}) .

Draft, November 5, 2011

13See A.2.2.

14Quite generally we have

∥(Â − a)Ψ∥^{2} ∥(B̂ − b)Ψ∥^{2} ≥ |⟨(Â − a) Ψ | (B̂ − b) Ψ⟩|^{2}
= \frac{1}{4} ∥\underbrace{⟨Ψ | [Â, B̂]_{−} Ψ⟩}_{real}∥^{2} + \frac{1}{4} ∥\underbrace{⟨Ψ | [Â − a, B̂ − b]_{+} Ψ⟩}_{imaginary}∥^{2} .

for all selfadjoint operators Â, B̂, all real numbers a, b, and every vector Ψ ∈ D_{B̂Â} ∩ D_{ÂB̂} .

In this sense, the coherent states of the quantized electromagnetic field correspond to coherent classical fields as closely as possible.^{15}

A.3 Construction of Field Operators

Denote by 𝔉 the complex vector space of all truncated sequences

𝔣 = {f_{0}, f_{1}, f_{2}, …}

of (sufficiently well-behaved) functions^{16} f_{ν}(k_{1}, j_{1}; …; k_{ν}, j_{ν}) over (ℝ^{3} × {1, 2, 3})^{ν} fulfilling^{17}

f_{ν}(k_{π1}, j_{π1}; …; k_{πν}, j_{πν}) = f_{ν}(k_{1}, j_{1}; …; k_{ν}, j_{ν}) ∀ π ∈ S_{ν}

and

∑_{j_{1},...,j_{ν}=1}^{3} ∫ |f_{ν}(k_{1}, j_{1}; …; k_{ν}, j_{ν})|^{2} dV_{k_{1}} … dV_{k_{ν}} < ∞ .

Of course, the linear structure is meant to be the natural one:

\begin{matrix}z 𝔣 & = & \{z f_{0}, z f_{1}, z f_{2}, …\} ∀ z ∈ ℂ , 𝔣 ∈ 𝔉 , \\ 𝔣 + 𝔣′ & =^{def} & \{f_{0} + f′_{0}, f_{1} + f′_{1}, f_{2} + f′_{2}, …\} ∀ 𝔣, 𝔣′ ∈ 𝔉 . \end{matrix}

With the inner product

⟨𝔣 | 𝔣′⟩ =^{def} \overline{f_{0}} f′_{0} + ∑_{j_{1},...,j_{ν}=1}^{3} ∫ \overline{f_{ν}(k_{1}, j_{1}; …; k_{ν}, j_{ν})} f′_{ν}(k_{1}, j_{1}; …; k_{ν}, j_{ν})dV_{k_{1}} … dV_{k_{ν}}

F becomes a euklidean vector space (pre-Hilbert space). Now let us introduce operator-valued functions b̂_{j}(k) by^{18}

𝔣^{−} = b̂_{j}(k) 𝔣

⟺^{def} f_{ν}^{−}(k_{1}, j_{1}; …; k_{ν}, j_{ν}) = ℓ^{− \frac{3}{2}} \sqrt{ν} f_{ν+1}(k_{1}, j_{1}; …; k_{ν}, j_{ν}; k, j) ∀ ν ∈ ℤ_{+} ,

where ℓ is some fixed length necessary for adjusting the physical dimensions. Then

for the adjoint operators (b̂_{j}(k))^{†}, characterized by

⟨𝔣 | b̂_{j}(k) 𝔣′⟩ = ⟨(b̂_{j}(k))^{†} 𝔣 | 𝔣′⟩ ∀ 𝔣, 𝔣′ ∈ 𝔉 ,

Draft, November 5, 2011

15States with Δ(a + a^{†}) < 1 or Δ(i a − i a^{†}) < 1 are called squeezed states (see (Scully and Zubairy, 1999, Sections 2.5–2.8) for a discussion of these states).

16The physical dimension of the values of these functions should be such that the integrals ∫ |f_{ν}(k_{1}, j_{1}; …; k_{ν}, j_{ν})|^{2} dV_{k_{1}} … dV_{k_{ν}} are just real numbers without any physical dimension.

17As usual, we denote by S_{ν} the group of all permutations of (1, …, ν) .

18Of course ‘…; k_{ν}, j_{ν}’ should be dropped for ν ≤ 1 and also ‘k_{1}, j_{1}’ if ν = 0 .

we have 𝔣^{+} = (b̂_{j}(k))^{†} 𝔣 iff f_{0}^{+} = 0 and:

\begin{matrix}ℓ^{\frac{3}{2}} f_{ν+1}^{+}(k_{1}, j_{1}; …; k_{ν+1}, j_{ν+1}) \\ = \frac{1}{\sqrt{ν + 1}} ∑_{μ=1}^{ν} δ_{j_{n}uj} δ (k_{μ} − k) f_{ν+1}(k_{1}, j_{1}; …; k_{μ} \, j_{μ} \; …; k_{ν+1}, j_{ν+1}) ∀ ν ∈ ℤ_{+} . \end{matrix}

Consequently, we have the commutation relations

[b̂_{j}(k), b̂_{j′}(k′)]_{−} = 0 , [b̂_{j}(k), (b̂_{j′}(k′))^{†}]_{−} = ℓ^{−3} δ_{jj′} δ(k − k′)

and the definition

â_{j}(k) =^{def} ℓ^{\frac{3}{2}} ϵ_{j}(k) · b̂(k) ∀ j ∈ \{1, 2\} , k ∈ ℝ^{3} ,

where

b̂(k) =^{def} ∑_{j=1}^{3} b̂_{j}(k) e_{j} ,

yields a realization of the canonical commutation relations (1.37) fulfilling (1.38) for

Ω =^{def} (1, 0, 0, …) .

Note that

∑_{j=1}^{2} ϵ_{j}(k) â_{j}(k) = b̂(k) − \frac{k}{|k|} · b̂(k) ∀ k ≠ 0 .

This shows that Â^{(+)}(x, t) does not depend on the choice of polarization vectors ϵ_{j}(k) in (1.31).

A.4 Dyson Series

In the Schrödinger picture the state vectors (wave functions) Ψ_{t} = Ψ_{t}^{S} describe the momentary situation of the quantum system at time t. Thus, if Â = Â^{S} is the observable of a physical quantity A then ⟨Ψ_{t}^{S} | Â^{S} Ψ_{t}^{S}⟩ is the corresponding expectation value for A at time t. The time evolution is described by the Schrödinger equation

iℏ\frac{∂}{∂t}Ψ_{t}^{S} = Ĥ^{S}(t) Ψ_{t}^{S} .  (A.32)

Now assume that the Hamiltonian in the Schrödinger picture is of the form

Ĥ^{S}(t) = Ĥ_{0}^{S} + V̂^{S}(t) ,  (A.33)

Where Ĥ_{0}^{S}(t) corresponds to some known ‘free evolution’ and V̂^{S}(t) is some perturbation of the ‘free’ Hamiltonian Ĥ_{0}^{S}(t) . Then we may get a perturbative solution of (A.32) by switching to the interaction picture:^{19}

Ψ_{t}^{S} ⟼ Ψ_{t}^{I} =^{def} e^{\frac{i}{ℏ} Ĥ_{0}^{S} t} Ψ_{t}^{S} ,  (A.34)
Â^{S}(t) ⟼ Â^{I}(t) =^{def} e^{+ \frac{i}{ℏ} Ĥ_{0}^{S} t}Â^{S}(t) e^{− \frac{i}{ℏ} Ĥ_{0}^{S} t} .  (A.35)

This way (A.32) becomes equivalent to

iℏ\frac{∂}{∂t}Ψ_{t}^{I} = V̂^{I}(t) Ψ_{t}^{I} .  (A.36)

Sure, (A.36) has the same formal structure as (A.32). Now, however, for the sufficiently small T the mapping^{20}

I : Φ(t) ⟼ −\frac{i}{ℏ} ∫_{0}^{t} V̂^{I}(t) Φ(t′) dt′  (A.37)

may be contracting with respect to the norm

∥Φ∥_{T} =^{def} ∫_{0}^{T} ∥Φ(t)∥ dt;

i.e. the may be some λ < 1 with

∥Φ∥_{T} ≤ λ ∥Φ∥_{T} ∀ Φ .

If this is the case we have

Ψ_{t}^{I} = ∑_{ν=0}^{∞}(I^{ν}Ψ_{0}^{I})(t) ∀ t ∈ [0, T]  (A.38)

(compare Section 5.1.3 of (Lücke, ein)), since (A.36) is equivalent to the integral equation

Ψ_{t}^{I} = Ψ_{0}^{I} − \frac{i}{ℏ} ∫_{0}^{t} V̂^{I}(t′) Ψ_{t′}^{I} dt′ .

Using time ordering^{21}

T (V̂^{I}(t_{π1}) ⋯ V̂^{I}(t_{πν})) =^{def} V̂^{I}(t_{1}) ⋯ V̂^{I}(t_{ν}) ∀ t_{1} ≤ … ≤ t_{ν} , π ∈ S_{ν}

Draft, November 5, 2011

19See (Lücke, qft, Sect. 3.1.3) for a qualitative discussion.

20Here we assume that Φ(t) is a (sufficiently well behaved) function of t with values in the state space H and denote by ∥.∥ the norm corresponding to the inner product of H.

21As usual, we denote by S_{ν} the set of all permutations of 1, …, ν.

we may rewrite (A.38) as Dyson series:^{22}

Ψ_{t}^{I} = T (e^{− \frac{i}{ℏ} ∫_{0}^{t} V̂^{I}(t′) dt′}) Ψ_{0}^{I}
=^{def} Ψ_{0}^{I}(0) + ∑_{ν=1}^{∞} \frac{1}{ν!} (−\frac{i}{ℏ})^{ν} ∫_{[0,t]^{ν}} T (V̂^{I}(t_{1}) ⋯ V̂^{I}(t_{ν})) Ψ_{0}^{I} dt_{1} ⋯ dt_{ν} .

(A.39)

Usually only the first order approximation

Ψ_{t}^{I} ≈ Ψ_{0}^{I} − \frac{i}{ℏ} ∫_{0}^{t} V̂^{I}(t′) Ψ_{0}^{I} dt′ ∀ t ∈ [0, T]  (A.40)

is used — as justified for sufficiently small T . This implies for the transition probability

|⟨Φ_{0}^{0} | Ψ_{t}^{I}⟩| ≈ ℏ^{−2} |∫_{0}^{t} ⟨Φ_{0}^{0} | V̂^{I}(t′) Ψ_{0}^{I}⟩ dt′|^{2} if ⟨Φ_{0}^{0} | Ψ_{t}^{I}⟩ = 0 .  (A.41)

Remark: Since

|∫_{0}^{t} ⟨Φ_{0}^{0} | V̂^{I}(t′) Ψ_{0}^{I}⟩ dt′|^{2} = ∫_{[0,t]×[0,t]} ⟨Φ_{0}^{0} | V̂^{I}(t_{1}) Ψ_{0}^{I}⟩ ⟨Ψ_{0}^{I} | V̂^{I}(t_{2}) Φ_{0}^{0}⟩ dt_{1} dt_{2} ,

the corresponding formula for mixed states with density matrix ρ̂_{t}^{I} is

trace(ρ̂_{t}^{I} |Φ_{0}^{0}⟩⟨Φ_{0}^{0}|) ≈ ℏ^{−2} ∫_{[0,t]×[0,t]} ⟨Φ_{0}^{0} | V̂^{I}(t_{1}) ρ̂_{0}^{I} V̂^{I}(t_{2}) Φ_{0}^{0}⟩ dt_{1} dt_{2} ,

in agreement with (6.19).

Draft, November 5, 2011

22Of course, we tacitly assume also t ↦ V̂^{I}(t) to be sufficiently well behaved.

Bibliography

Abramsky, S. and Coecke, B. (2004). A categorical semantics of quantum protocols. quant-ph/0402130, pages 1–16. 117

Agranovich, V. M. and Ginzburg, V. L. (1984). Crystal Optics with Spatial Dispersion and Excitons, volume 42 of Solid-State Sci. Springer-Verlag, 2. edition. 36

Agrawal, G. P. (1999). Nonlinear fiber optics. Optics and Photonics. Academic Press. 61

Alicki, R. (2003). Controlled quantum open systems. Lecture Notes in Physics, 622:121–139. quant-ph/0302132. 137

Alleaume, R., Treussart, F., Courty, J.-M., and Roch, J.-F. (2003). Photon statistics characterization of a single photon source. quant-ph/0312084, pages 1–22. 113

Angelo, R. M. and Wreszinski, W. F. (2005). Quantum NOT operation and integrability in two-level systems. quant-ph/0502102, pages 1–17. 159

Armstrong, J., Bloembergen, N., Ducuing, J., and Pershan, P. S. (1962a). Interactions between light waves in a nonlinear dielectric. Phys. Rev., 127:1918–1939. 63

Armstrong, J. A., Bloembergen, N., Ducuing, J., and Pershan, P. S. (1962b). Interactions between light waves in a nonlinear dielectric. Phys. Rev., 127:1918–1939. 72, 76

Aspect, A. (2004). Bell’s theorem : The naive view of an experimentalist. quantph/0402001, pages 1–34. 115

Azzam, R. M. A. and Bashara, N. M. (1987). Ellipsometry and Polarized Light. Elsevier Science B.V. 44

Bachor, H.-A. (1998). A Guide to Experiments in Quantum Optics. Wiley-VCH. 3

Baym, G. (1969). Lectures on Quantum Mechanics. Benjamin, New York. 147 Bennet, C. H. and Brassard, G. (1984). Quantum cryptography, public key distribu-

tion and coin tossing. In Proceedings of International Conference on Computer Systems and Signal Processing, pages 175–179, New York. IEEE. 113

Bloembergen, N. (1996a). Encounters in Nonlinear Optics, volume 16 of World Scientific Series in 20th Century Physics. World Scientific.

Bloembergen, N. (1996b). Nonlinear Optics. World Scientific. 10, 43

Borchers, H.-J. (1990). Eine Bemerkung zur Ausbreitung von Licht in dispergierenden Medien. In Nachrichten der Akademie der Wissenschaften in Göttingen aus dem Jahre 1990, pages 1–8. Vandenhoeck & Ruprecht, Göttingen. Mathematisch-Physikalische Klasse. 41

Bowmeester, D., Howell, J. C., and Lamas-Linares, A. (2002). Quantum information science using photons. Lecture Notes in Physics, 587:149–197. 106

Bowmeester, D., Pan, J., Mattle, K., Eibl, M., Weinfurter, H., and Zeilinger, A. (1997). Experimental quantum teleportation. Nature, 390:575–579. 108

Brukner, C. and Zeilinger, A. (2002). Young’s experiment and the finiteness of information. Phil. Trans. R. Soc. Lond. A, 360:1061. quant-ph/0201026. 86

Brunner, W. and Junge, K. (1982). Lasertechnik. Dr. Alfred Hüthig Verlag, Heidelberg. 63

Busch, P., Grabowski, M., and Lahti, P. J. (1995). Operational Quantum Physics, volume m 31 of Lecture Notes in Physics. Springer-Verlag. 128

Carmichael, H. (1993). An Open Systems Approach to Quantum Optics, volume m 18 of Lecture Notes in Physics. Springer-Verlag. 137

Carmichael, H. J. (2001). Quantum fluctuations of light: A modern perspective on wave/particle duality. quant-ph/0104073. 102

Caves, C. M., Fuchs, C. A., and Schack, R. (2002). Unknown quantum states: The quantum de Finetti representation. J. Math. Phys., 43:4537–4559. quantph/0104088. 108

Chen, Q.-H., Liu, T., Zhang, Y.-Y., and Wang, K.-L. (2010). Solutions to the Jaynes-Cummings model without the rotating-wave approximation. arXiv:1011.3280 [quant-ph], pages 1–6. 159

Choquet-Bruhat, Y., Dewitt-Morette, C., and Dillard-Bleick, M. (1978). Analysis, Manifolds and Physics. North-Holland Publishing Company. 179

Coecke, B. (2004). The logic of entanglement. quant-ph/0402014, pages 1–8. 117 Cohen-Tannoudji, C., Dupont-Roc, J., and Grynberg, G. (1992). Atom-Photon

Interactions. John Wiley & Sons, Inc. 142

Davies, E. B. (1976). Quantum Theory of Open Systems. Academic Press. 128, 137

Deutsch, D. (1997). The Fabric of Reality. Penguin Books. 139

Di Giuseppe, G., Atature, M., Shaw, M. D., Sergienko, A. V., Saleh, B. E. A., Teich, M. C., Miller, A. J., Nam, S. W., and Martinis, J. (2003). Observation of bosonic coalescence of photon pairs. quant-ph/0306131, pages 1–4. 107

Dowling, J. P., Franson, J. D., Lee, H., and Milburn, G. J. (2004). Towards linear optical quantum computers. quant-ph/0402090, pages 1–9. 125

Duan, L.-M., Lukin, M. D., Cirac, J. I., and Zoller, P. (2001). Long-distance quantum communication with atomic ensembles and linear optics. Nature, 414:413– 418. quant-ph/0105105. 118

Elitzur, A. and Vaidman, L. (1993). Quantum-mechanical interaction-free measurements. Found. Phys., 23:987–997. 110

Ficek, Z. and Swain, S. (2001). Quantum interference in optical fields and atomic radiation. quant-ph/0109100, pages 1–30. 86, 87

Fleischhauer, M. and Lukin, M. D. (2000). Dark-state polaritons in electromagnetically induced transparency. Phys. Rev. Lett., 84:5094. quant-ph/0001094. 163

Fleischhauer, M. and Lukin, M. D. (2002). Quantum memory for photons: Darkstate polaritons. Phys. Rev. Lett., 65:022314. 121

Fredenhagen, K. (Winter Term 2000/2001). Quantenfeldtheorie. Notes from lec-

tures held at Hamburg University, http://unith.desy.de/research/aqft/
lecture_notes. 16

Fröhlich, J. (1977). Application of commutator theorems to the integration of representations of Lie algebras and commutation relations. Commun. Math. Phys., 54:135–150. 26

Fröhlich, J., Griesemer, M., and Schlein, B. (2003). Asymptotic completeness for Compton scattering. mp arc/03-504, pages 1–63. 140

Gardiner, C. W. and Zoller, P. (2000). Quantum Noise, volume 56 of Springer Series in Synergetics. Springer, 2. edition. A Handbook of Markovian and Non-Markovian Quantum Stochastic Methods with Applications to Quantum Optics.

Garrison, J. C. and Chiao, R. Y. (2004). Canonical and kinetic forms of the electromagnetic momentum in an ad hoc quantization scheme for a dispersive dielectric. quant-ph/0406222. 78

Gatti, A., Zambrini, R., Miguel, M. S., and Lugiato, L. A. (2003). Multi-photon, multi-mode polarization entanglement in parametric down-conversion. quantph/0306133, pages 1–22. 111

Gemmer, J. and Mahler, G. (2001). Entanglement and the factorizationapproximation. Eur. Phys J. D, 17:385–393. quant-ph/0109140. 142

Gerhardt, I., Liu, Q., Lamas-Linares, A., Skaar, J., Scarani, V., Makarov, V., and Kurtsiefer, C. (2011). Experimentally faking the violation of bell’s inequalities. arXiv:1106.3224 [quant-ph], pages 1–5. 115

Gingrich, R. M., Ko, P., Lee, H., Vatan, F., and Dowling, J. P. (2003). An all linear optical quantum memory based on quantum error correction. Phys. Rev. Lett., 91:217901. quant-ph/0306098. 118

Gisin, N. and Gisin, B. (1999). A local hidden variable model of quantum correlation exploiting the detection. Phys. Lett. A, 260:323–327. quant-ph/9905018. 115

Gisin, N., Marcikic, I., de Riedmatten, H., Tittel, W., and Zbinden, H. (2003). Quantum communications with time-bin entangled photons: long distance quantum teleportation and quantum repeaters. quant-ph/0301181, pages 1–4. 108

Gjurchinovski, A. (2002). Polarization measurements - a numerical approach. Physica Macedonica, 52:69–76. physics/0401105. 44

Gomez-Reino, C., Perez, M. V., and Bao, C. (2002). Gradient-Index Optics. Springer. Fundamentals and Applications. 57

Gottesman, D. and Chuang, I. L. (1999). Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations. Nature, 402:390–393. quant-ph/9908010. 125

Green, H. S. (1953). A general method of field quantization. Phys. Rev., 90:270–273. 20

Grigorenko, A. N. (2006). Negative refractive index in artificial metamaterials. Optics Letters, 31:2483–2485. physics/0603216. 44

Hariharan, P. and Sanders, B. C. (1996). Quantum phenomena in optical interferometry. Progress in Optics, 36:49–128. 87

Hentschel, M. and Schomerus, H. (2002). Fresnel laws at curved dielectric interfaces of microresonators. Phys. Rev. E, 65:045603(R). physics/0111160. 44 Hentschel, M. and Schomerus, H. (2006). Correcting ray optics at curved dielectric

microresonator interfaces: Phase-space unification of Fresnel filtering and the Goos-Haenchen shift. Phys. Rev. Lett., 96:243903. physics/0606024. 44

Herbert, N. (1982). FLASH—A superluminal communicator based upon a new kind of quantum measurement. Foundations of Physics, 12:1171–1181. 112

Hodgson, J. N. (1970). Optical Absorption and Dispersion in Solids. Chapman and Hall Ltd.

Hoffmann, F. (1994). Zur Strahlungstheorie wasserstoffähnlicher Atome unter Berücksichtigung der Schwerpunktsbewegung. Master’s thesis, Intitut für Mathematische Physik, Technische Universität Braunschweig. 140

Jackson, D. J. and Hockney, G. M. (2004). Detector efficiency limits on quantum improvement. quant-ph/0402201, pages 1–7. 83

Jackson, J. D. (1975). Classical Electrodynamics, Second Edition. John Wiley & Sons, Inc., New York·London·Sydney. 44

Johansen, L. M. (2004). Nonclassicality of thermal radiation. J. Opt. B: Quantum Semiclass. Opt., 6:L21–L24. quant-ph/0402105. 99

Jones, R. C. (1941). New calculus for the treatment of optical systems. J. Opt. Soc. Am., 31:488. 44

Juzeliunas, G., Masalas, M., and Fleischhauer, M. (2003). Storing and releasing light in a gas of moving atoms. Phys. Rev. A, page 023809. quant-ph/0210123. 121

Kaltenbaek, R., Aspelmeyer, M., Jennewein, T., Brukner, C., Pfennigbauer, M., Leeb, W. R., and Zeilinger, A. (2003). Proof-of-concept experiments for quantum physics in space. quant-ph/0308174, pages 1–15. 115

Kemble, E. C. (1937). The Fundamental Principles of Quantum Mechanics with Elementary Applications. Dover Publications, Inc., New York. 127

Kim, Y.-H. and Grice, W. P. (2003). Quantum interference with distinguishable photons through indistinguishable pathways. quant-ph/0304086, pages 1–6. 107

Kim, Y.-H., Kulik, S. P., and Shih, Y. (2001). Quantum teleportation with a complete Bell state measurement. Phys. Rev. Lett., 86:1370. quant-ph/0010046. 118

Kinsler, P., Favaro, A., and McCall, M. W. (2009). Four poynting theorems. Eur. J. Phys., 30:983. arXiv:0908.1721 [physics.optics]. 11 Klauder, J. R. (2010). On the role of coherent states in quantum foundations.

arXiv:1008.4307 [quant-ph], pages 1–9. 24

Klauder, J. R. and Sudarshan, E. C. G. (1968). Fundamentals of Quantum Optics. W. A. Benjamin, Inc. 34, 85, 87, 89

Klein, A. (1999). Localization of light in randomized periodic media. In Fouque, J.- P., editor, Diffuse Waves in Complex Media, pages 73–92. Kluwer, The Netherlands.

Kliger, D. S., Lewis, J. W., and Randall, C. E. (1990). Polarized Light in Optics and Spectroscopy. Academic Press, Inc. 44

Klyshko, D. N. (1988). Photons and Nonlinear Optics. Gordon and Breach Science Publishers.

Knill, E., Laflamme, R., and Milburn, G. (2000). Efficient linear optics quantum computation. Nature, 409:46. quant-ph/0006088. 125

Knill, E., Laflamme, R., and Milburn, G. J. (2001). A scheme for efficient quantum computation with linear optics. Nature, 409:46–52. 125

Kraus, K. (1983). Sates, Effects, and Operations, volume 190 of Lecture Notes in Physics. Springer-Verlag. 128, 137

Kuang, L.-M., Chen, G.-H., and Wu, Y.-S. (2003). Nonlinear optical properties of an electromagnetically-induced transparency medium interacting with two quantized fields. J. Optics B, 5:341–448. quant-ph/0402041. 163

Kuklinski, J. R., Gaubatz, U., Hioe, F. T., and Bergmann, K. (1989). Adiabatic population transfer in a three-level system driven by delayed laser pulses. Phys. Rev. A, 40:6741–6744. 163

Kurtsiefer, C., Mayer, S., Zarda, P., and Weinfurther, H. (2000). A stable solid-state source of single photons. Phys. Rev. Lett., 85:290. 101

Kwiat, P. G., Weinfurther, H., Herzog, T., Zeilinger, A., and Kasevich, M. A. (1995). Interaction-free measurement. Phys. Rev. Lett., 74:4763–4766. 110

Kwiat, P. G., White, A. G., Mitchell, J. R., Nairz, O., Weihs, G., Weinfurter, H., and Zeilinger, A. (1999). High-efficiency quantum interrogation measurements via the quantum Zeno effect. Phys. Rev. Lett., pages 4725–4728. quant-ph/9909083. 110

Lamb, Jr., W. E. (1995). Anti-photon. Appl. Phys. B, pages 77–84. 13

Landau, L. and Lifschitz, E. (1967). Elektrodynamik der Kontinua, volume VIII of Lehrbuch der Theoretischen Physik. Akademie-Verlag, Berlin. 10, 12, 21 Leggett, A. J. (2001). Qubits, cbits, decoherence, quantum measurement and en-

vironment. In Fundamentals of Quantum Information, pages 3–45. Springer. 127, 156

Leonhardt, U. (2003). Quantum physics of simple optical instruments. Rept. Prog. Phys, 66:1207–1250. quant-ph/0305007. 102

Lewis, G. N. (1926). The conservation of photons. Nature, pages 874–875. 13

Lim, Y. L. and Beige, A. (2005). Generalised Hong-Ou-Mandel experiments with bosons and fermions. New J. Phys., 7:155. quant-ph/0505034. 107

Liu, C., Dutton, Z., Behroozi, C. H., and Vestergaard Hau, L. (2001). Observation of coherent optical information storage in an atomic medium using halted light pulses. Nature, 409:490–493. 121

Lodenquai, J. (1991). Electromagnetic wave propagation in media with complex refractive indices and transmission through absorbing films. Am. J. Phys., 59:248–254. 49

Loudon, R. (2000). The Quantum Theory of Light. Oxford Science Publications, third edition. 3, 24

Lücke, W. (1995a). Nonlinear Schrödinger dynamics and nonlinear observables. In Doebner, H.-D., Dobrev, V. K., and Nattermann, P., editors, Nonlinear, Deformed and Irreversible Quantum Systems, pages 140–154. World Scientific. 181

Lücke, W. (1995b). Solutions of the inhomogeneous wave equation with unusual propagation character and global solution of the Poisson equation. J. Phys. A, 28:5393–5400. 10

Lücke, W. (1996). Axiomatic quantum theory. Acta Phys. Pol., 27:2357–2385. 130

Lücke, W. (edyn). Elektrodynamik .

http://www.wolfgang-luecke.de/skripten/edyn.pdf. 9, 10, 25

Lücke, W. (ein). Mathematische Methoden der Physik.

http://www.wolfgang-luecke.de/skripten/ein.pdf. 14, 19, 36, 65, 165,

189

Lücke, W. (eine). Ergänzungen zu “Mathematische Methoden der Physik”.

http://www.wolfgang-luecke.de/skripten/eine.pdf. 28, 34, 53, 105, 128,

129, 130, 133, 141, 142, 156, 177

Lücke, W. (ftm). Mathematische Methoden der Physik: Funktionentheoretische Methoden.

http://www.wolfgang-luecke.de/skripten/ftm.pdf. 13, 37, 41, 59

Lücke, W. (fuan). Mathematische Methoden der Physik: Funktionalanalytische Methoden.

http://www.wolfgang-luecke.de/skripten/fuan.pdf. 181

Lücke, W. (qft). Particles and fields.

http://www.wolfgang-luecke.de/skripten/qft.pdf. 13, 16, 19, 86, 106,

141, 189

Lücke, W. (qip). Introduction to quantum information processing.

http://www.wolfgang-luecke.de/skripten/qip.pdf. 120, 124, 125

Lücke, W. (rel). Spezielle Relativitätstheorie .

http://www.wolfgang-luecke.de/skripten/rel.pdf. 9

Lynds, P. (2003).  Zeno’s paradoxes: A timely solution.  http://

philsci-archive.pitt.edu/archive/00001197 , pages 1–9. 150

Mandel, L. and Wolf, E. (1995). Optical coherence and quantum optics. Cambridge University Press. 3, 14, 32, 36, 84, 87, 88, 95, 97, 101, 106, 142, 146, 148, 150, 151, 158, 161

McDonald, K. T. and Wang, L. (2003). Bunching of photons when two beams pass through a beam splitter. quant-ph/0312032, pages 1–8. 107

Migdall, A. L., Branning, D., Castelletto, S., and Ware, M. (2002). Single photon source with individualized single photon certifications. Proc. of the SPIE, 4821:455–465. quant-ph/0209103. 116

Miller, R. C. (1964). Optical second harmonic generation in piezoelectric crystals. Appl. Phys. Lett., 5:17–19. 66, 73

Mills, D. L. (1998). Nonlinear Optics. Springer-Verlag. Basic Concepts. 37, 42, 61

Milonni, P. W. (1994). The Quantum Vacuum. An Introduction to Quantum Electrodynamics. Academic Press, Inc. 18

Miranowicz, A., Bartkowiak, M., Wang, X., xi Liu, Y., and Nori, F. (2010). Testing nonclassicality in multimode fields: a unified derivation of classical inequalities. arXiv:1004.3182 [quant-ph], pages 1–16. 99

Misra, B. and Sudarshan, E. C. G. (1977). The Zeno’s paradox in quantum theory. J. Mathem. Phys., page 756. 150

Mizrahi, S. S. and Dodonov, V. V. (2002). Creating quanta with ‘annihilation’ operator. quant-ph/0207035, pages 1–8. 16

Moller, D., Madsen, L. B., and Molmer, K. (2007). Geometric phase gates based on stimulated raman adiabatic passage in tripod systems. quant-ph/0701240, pages 1–6. 163 Monzón, J. J. and Sánchez-Soto, L. L. (2001). Constructing fresnel reflection coef-

ficients by ruler and compass. physics/0111129, pages 1–5. 47

Morgan, P. (2008). Violation of Bell inequalities through the coincidence-time loophole. arXiv:0801.1776 [quant-ph], pages 1–4. 115

Munro, W. J., Nemoto, K., Beausoleil, R. G., and Spiller, T. P. (2005). A highefficiency quantum non-demolition single photon number resolving detector. Phys. Rev. A, 71:033819. quant-ph/0310066. 120

Nemoto, K. and Braunstein, S. L. (2003). Quantum coherence: myth or fact? quant-ph/0312108, pages 1–3. 84

Ohnuki, Y. and Kamefuchi, S. (1982). Quantum Field Theory and Parastatistics. Springer Verlag, Berlin. 136

Ohtsu, M. and Hori, H. (1999). Near-Field Nano-Optics. From Basic Principles to Nano-Fabrication and Nano-Photonics. Kluwer Academic / Plenum Publishers. 50

Ottaviani, C., Rebic, S., Vitali, D., and Tombesi, P. (2006). Quantum phase gate operation based on nonlinear optics: Full quantum analysis. Phys. Rev. A, 73:010301(R). quant-ph/0507137. 120, 165

Paesler, M. A. and Moyer, P. J. (1996). Near-Field Optics. Theory, Instrumentation, and Applications. John Wiley& Sons, Inc. 50

Pan, J.-W., Bouwmeester, D., Daniell, M., Weinfurter, H., and Zeilinger, A. (2000). Experimental test of quantum nonlocality in three-photon Greenberger-Horne-Zeilinger entanglement. Nature, 403:515–519. 115

Paul, H. (1995). Photonen. Eine Einführung in die Quantenoptik. B.G. Teubner, Stuttgart. 150

Pittman, T. and Franson, J. D. (2003). Violation of Bell’s inequality with photons from independent sources. Phys. Rev. Lett., 90:240401. quant-ph/0301169. 108

Pittman, T. B. and Franson, J. D. (2002). Cyclical quantum memory for photonic qubits. Phys. Rev. A, 66:062302. 121

Ralph, T. C., White, A. G., Munro, W. J., and Milburn, G. J. (2002). Simple scheme for efficient linear optics quantum gates. Phys. Rev. A, 65:012314. quant-ph/0108049. 125

Reider, G. (1997). Photonik. Springer-Verlag. 44, 59, 61

Römer, H. (1994). Theoretische Optik. Konzepte der Theoretischen Physik. VCH Verlagsgesellschaft mbH., Weinheim. 37, 51, 63 Rosenberg, D., Lita, A. E., Miller, A. J., and Nam, S. W. (2005). Noise-free high-

efficiency photon-number-resolving detectors. Phys. Rev. A, 71:061803. quantph/0506175. 115

Rullière, C., editor (1998). Femtosecond Laser Pulses. Springer.

Saleh, B. E. A. and Teich, M. C. (1991). Fundamentals of Photonics. John Wiley & Sons, Inc. 3, 36

Sanaka, K., Jennewein, T., Pan, J.-W., Resch, K., and Zeilinger, A. (2003). Experimental nonlinear sign shift for linear optics quantum computation. Phys. Rev. Lett., 92:017902. quant-ph/0308134. 121

Sanaka, K. and Resch, K. (2003). A feasible gate for scalable linear optics quantum computation using polarization optics. quant-ph/0312226, pages 1–10. 125

Sanders, B. C., Bartlett, S. D., Rudolph, T., and Knight, P. L. (2003). Photonnumber superselection and the entangled coherent-state representation. Phys. Rev. A, 68:042329. quant-ph/0306076. 84

Sasaki, M., Fujiwara, M., Ishizuka, H., Klaus, W., Wakui, K., Takeoka, M., Tanaka, A., Yoshino, K., Nambu, Y., Takahashi, S., Tajima, A., Tomita, A., Domeki, T., Hasegawa, T., Sakai, Y., Kobayashi, H., Asai, T., Shimizu, K., Tokura, T., Tsurumaru, T., Matsui, M., Honjo, T., Tamaki, K., Takesue, H., Tokura, Y., Dynes, J. F., Dixon, A. R., Sharpe, A. W., Yuan, Z. L., Shields, A. J., Uchikoga, S., Legre, M., Robyr, S., Trinkler, P., Monat, L., Page, J.-B., Ribordy, G., Poppe, A., Allacher, A., Maurhart, O., Langer, T., Peev, M., and Zeilinger, A. (2011). Field test of quantum key distribution in the Tokyo QKD Network. arXiv:1103.3566 [quant-ph], pages 1–21. 113

Sauter, E. G. (1996). Nonlinear Optics. John Wiley & Sons, Inc.

Schleich, W. P. (2001). Quantum Optics in Phase Space. WILEY-VCH. 173

Scully, M. O. (1994). Resolving conundrums in lasing without inversion via exact solutions to simple models. Quantum Opt., 6:203–215. 164

Scully, M. O. and Zubairy, M. S. (1999). Quantum Optics. Cambridge University Press. 13, 30, 107, 142, 154, 161, 165, 187

Semenov, A. A. and Vogel, W. (2010). Fake violations of quantum physics. arXiv:1004.3700 [quant-ph], pages 1–4. 115

Shen, Y. R. (1984). The Principles of Nonlinear Optics. John Wiley & Sons, Inc. 3, 67, 76

Shore, B. W. (1990a). The theory of coherent atomic excitation, volume 1 Simple Atoms and Fields. John Wiley & Sons. 154 Shore, B. W. (1990b). The theory of coherent atomic excitation, volume 2 Multilevel

Atoms and Incoherence. John Wiley & Sons.

Simon, B. (1971). Quantum mechanics for Hamiltonians defined as quadratic forms. Princeton series in physics. Princeton University Press. 140

Simon, C. and Irvine, W. T. (2003). Robust long-distance entanglement and a loophole-free Bell test with ions and photons. Phys. Rev. Lett., page 110405. quant-ph/0303023. 115

Skaar, J. and Seip, K. (2006). Bounds for the refractive indices of metamaterials. J. Phys. D: Appl. Phys., 39:1226–1229. physics/0507001. 44

Smith, F. G. and King, T. A. (2000). Optics and Photonics: An Introduction. John Wiley & Sons, Ltd.

Stobinśka, M., Horodecki, P., Buraczewski, A., Chhajlany, R. W., Horodecki, R., and Leuchs, G. (2010). Macroscopic polarization entanglement and loopholefree Bell inequality test. arXiv:0909.1545 [quant-ph], pages 1–13. 115

Stratton, J. A. (1941). Electromagnetic Theory. MacGraw-Hill Book Company. 44

’t Hooft, G. (2002). How does god play dice? (Pre-)determinism at the Planck scale. In Bertlmann, R. A. and Zeilinger, A., editors, Quantum [Un]speakables, chapter 22, pages 307–316. Springer. 115

Timpson, C. G. and Brown, H. R. (2005). Proper and improper separability. International Journal of Quantum Information, 3(4):679–690. quant-ph/0402094. 128

van der Wal, C. H., Eisaman, M. D., Andr, A., Walsworth, R. L., Phillips, D. F., Zibrov, A. S., and Lukin, M. D. (2003). Atomic memory for correlated photon states. Science, pages 196–200. 121

Vogel, W., Welsch, D.-G., and Wallentowitz, S. (2001). Quantum Optics An Introduction. Wiley-VCH, 2. edition. 3

Walls, D. F. and Milburn, G. J. (1995). Quantum Optics. Springer-Verlag, 2. edition. 87

Wang, K. (2003). Quantum theory of two-photon wavepacket interference in a beam splitter. quant-ph/0311011, pages 1–33. 107

Wasilewski, W., Kolenderski, P., and Frankowski, R. (2007). Spectral density matrix of a single photon measured. quant-ph/0702200, pages 1–5. 107

Werner, R. F. (1989). Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model. Phys. Rev. A, 40:4277–4281. 134 Wiseman, H. M. (2004). Defending continuous variable teleportation: Why a laser

is a clock, not a quantum channel. Journal of Optics B, pages S849–S859. quant-ph/0403137. 84

Wittmann, B., Ramelow, S., Steinlechner, F., Langford, N. K., Brunner, N., Wiseman, H., Ursin, R., and Zeilinger, A. (2011). Loophole-free quantum steering. arXiv:1111.0760 [quant-ph], pages 1–10. 115

Wolf, E. and Nieto-Vesperinas, M. (1985). J. Opt. Soc. Am. A, 2:886. 50

Yariv, A. (1997). Optical Electronics in Modern Communications. Oxford University Press. 3

Yeh, P. and Gu, C. (1999). Optics of Liquid Crystal Displays. Wiley Series in Pure and Applied Optics. John Wiley & Sons, Inc.

Yurke, B., McCall, S. L., and Klauder, J. R. (1986). SU(2) and SU(1,1) interferometers. Phys. Rev. A, 33:4033–4054. 105

Zavatta, A., Viciani, S., and Bellini, M. (2004). Recurrent fourth-order interference dips and peaks with a comb-like two-photon entangled state. Phys. Rev. A, 70:023806. quant-ph/0312082. 107

Zeh, H. D. (2000). The meaning of decoherence. Lect. Notes Phys., 538:19–42. quant-ph/9905004. 129 Bibliography 203

Index

4-current density, 9, 10

absorption, 35 adiabatic, 160 annihilation operators, 16 anti-bunching, 101

Baker-Hausdorff formula, 26 biaxial, 56 birefringent, 55 Boson, 135 Brewster angle, 51 bunching, 101

Campbell-Hausdorff formula, 26 canonical commutation relations, 16 causality

Einstein-, 19 causality condition, 36, 37 charge

excess, 10 classically correlated, 134 coherent

state, 34 coherent state, 27 commutation relations

canonical, 16 compatible, 132 complex

conductivity, 38

dielectric constant, 38 complex conductivity, 38 complex dielectric ‘constant’, 38 conductivity

complex, 38 constitutive equations, 35 continuity equation, 9, 13 contracting, 189 correlation functions, 86 correlations, 134 Coulomb

condition, 14 creation operators, 21 crossing relations, 37 current density induced, 35

dark state, 162

density operator, 128 detuning, 160, 175 dielectric constant complex, 38

dipole

electric

density, 10 magnetic density, 10 dipole approximation, 153 direction, 42 dispersion, 35

relations, 37 down-conversion

spontaneous, 18 Dyson series, 25

effects, 128

Einstein causality

Einstein causality, 19 electric

dipole density, 10

field

observable, 18 electric field representation, 144 electromagnetically induced transparency,

164 excess charge, 10 exterior field formalism, 142

factorized, 134 Fermion, 135 field

electromagnetic, 35 classical, 9, 14

mode, 21

operators, 14, 34 quantized tensor, 15

first order, 190 first order degree of coherence, 95 force density, 9 Fourier transform, 12, 37 functional derivative, 179

gauge

radiation-, 14 generalized

polarization, 10, 35

Hamiltonian, 130, 183 Heisenberg picture, 15, 20

impermeability, 37 indeterministic, 121 index ellipsoid, 54 indistinguishable, 135 infinite-order coherence, 87 intensity, 104 interaction picture, 131

Johnson noise, 18

key, 113

linear functional, 35 media, 35

Lorentz covariance, 15

magnetic

dipole density, 10 maximal orthonormal system, 22 set of modes, 22

Maxwell equations, 9, 35, 38 measurement gate, 120 medium

nonisotropic, 35 Miller’s rule, 66 minimal coupling, 140, 153 mode, 88 modes, 21

maximal set, 22 orthogonal, 22

momentum

observable, 20 momentum observable, 182

noise Johnson, 18 quantum, 18 non-overlapping, 172 nonlinear

multi-photon process, 14 quantum optics, 23 normal ordering, 21, 34 number

of photons, 20

observable

momentum, 20 observables, 130 optical axis, 52 optical equivalence theorem, 34 optical medium

homogeneou, 37

linear, 35 optical rectification, 67 optical tunneling, 50 orthogonal

modes, 22

partial state, 134 partial trace, 135 permittivity, 37 perturbation, 134, 189 phase matching, 69 photon

number, 20 polarization, 43

degree of, 157 generalized, 10, 35

polarizing beam splitter, 108 position observable, 182 potential

vector

complex, 15 potentials

electromagnetic retarded, 10

Poynting theorem, 11 vector, 12

probability amplitudes, 22 probability density, 151 process

nonlinear, 14 projective measurement, 131

quadrature components, 29 quantum

noise, 18 quantum teleportation, 116

radiation

gauge, 14 reflectance, 49 reflectivity, 105 relaxation matrix, 165 retarded

solution, 25 rotating wave approximation, 159

second harmonic generation, 66, 67 second order degree of coherence, 97 selection rules, 146 single-mode state, 88 slowly varying amplitude approxima-

tion, 69 specific conductivity, 37 sphere

Bloch, 157

Poincare, 157 spontaneous

down-conversion, 18 transition, 18

state coherent, 27, 34 vacuum-, 16 state space, 127 stationary, 184 statistical operator, 128 sum-frequency generation, 67 susceptibility, 37

teleportation of entanglement, 118 tensor

field, 35 theorem

optical equivalence, 34 three-wave mixing, 72 time ordering, 189 total reflection, 50 transition

spontaneous, 18 transition probability, 146, 190 transmittance, 49 transmittivity, 105

uniaxial, 56

vacuum, 16

fluctuations, 18 vacuum impedance, 43 vector

Bloch, 157 Stokes, 157 vector states, 128 visibility, 95

wave function, 181