OXFORD MASTER SERIES IN PHYSICS

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OXFORD MASTER SERIES IN PHYSICS
The Oxford Master Series is designed for final year undergraduate and beginning graduate students in physics
and related disciplines. It has been driven by a perceived gap in the literature today. While basic undergraduate
physics texts often show little or no connection with the huge explosion of research over the last two decades,
more advanced and specialized texts tend to be rather daunting for students. In this series, all topics and their
consequences are treated at a simple level, while pointers to recent developments are provided at various stages.
The emphasis is on clear physical principles like symmetry, quantum mechanics, and electromagnetism which
underlie the whole of physics. At the same time, the subjects are related to real measurements and to the
experimental techniques and devices currently used by physicists in academe and industry. Books in this series
are written as course books, and include ample tutorial material, examples, illustrations, revision points, and
problem sets. They can likewise be used as preparation for students starting a doctorate in physics and related
fields, or for recent graduates starting research in one of these fields in industry.

CONDENSED MATTER PHYSICS
1.
2.
3.
4.
5.
6.

M. T. Dove: Structure and dynamics: an atomic view of materials
J. Singleton: Baud theory and electronic properties of solids
A. M. Fox: Optical properties of solids
S. J. Blundell: Magnetism in condensed matter

J. F. Annett: Superconductivity
R. A. L. Jones: Soft condensed matter

ATOMIC, OPTICAL, AND LASER PHYSICS
7.
8.
9.
15.

C. J. Foot: Atomic Physics
G. A. Brooker: Modern classical optics
S. M. Hooker, C. E. Webb: Laser physics
A. M. Fox: Quantum optics: an introduction

PARTICLE PHYSICS, ASTROPHYSICS, AND COSMOLOGY
10. D. H. Perkins: Particle astrophysics
11. Ta-Pei Cheng: Relativity, gravitation, and cosmology

STATISTICAL, COMPUTATIONAL, AND THEORETICAL PHYSICS
12. M. Maggiore: A modern introduction to quantum field theory
13. W. Krauth: Statistical mechanics: algorithms and computations
14. J. P. Sethna: Entropy, order parameters, and complexity

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Quantum Optics
An Introduction

MARK FOX

Department of Physics and Astronomy
University of Sheffield

1
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3

Great Clarendon Street, Oxford OX2 6DP
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First published 2006
All rights reserved. No part of this publication may be reproduced,

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or as expressly permitted by law, or under terms agreed with the appropriate
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Oxford University Press, at the address above
You must not circulate this book in any other binding or cover
and you must impose the same condition on any acquirer
British Library Cataloguing in Publication Data
Data available
Library of Congress Cataloging in Publication Data
Fox, Mark (Anthony Mark)
Quantum optics : an introduction/Mark Fox.
p. cm. — (Oxford master series in physics ; 6)
Includes bibliographical references and index.
ISBN-13: 978–0–19–856672–4 (hbk. : acid-free paper)
ISBN-10: 0–19–856672–7 (hbk. : acid-free paper)
ISBN-13: 978–0–19–856673–1 (pbk. : acid-free paper)
ISBN-10: 0–19–856673–5 (pbk. : acid-free paper)
1. Quantum optics. I. Title. II. Series.
QC446.2.F69 2006
535 .15—dc22
Typeset by Newgen Imaging Systems (P) Ltd., Chennai, India
Printed in Great Britain
on acid-free paper by
Antony Rowe, Chippenham

2005025707

ISBN 0–19–856672–7 978–0–19–856672–4

ISBN 0–19–856673–5 (Pbk.) 978–0–19–856673–1 (Pbk.)
10 9 8 7 6 5 4 3 2 1

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Preface
Quantum optics is a subject that has come to the fore over the last 10–20
years. Formerly, it was regarded as a highly specialized discipline, accessible only to a small number of advanced students at selected universities.
Nowadays, however, the demand for the subject is much broader, with
the interest strongly fuelled by the prospect of using quantum optics in
quantum information processing applications.
My own interest in quantum optics goes back to 1987, when I attended
the Conference on Lasers and Electro-Optics (CLEO) for the first
time. The ground-breaking experiments on squeezed light had recently
been completed, and I was able to hear invited talks from the leading researchers working in the field. At the end of the conference, I
found myself sufficiently interested in the subject that I bought a copy
of Loudon’s Quantum theory of light and started to work through it in
a fairly systematic way. Nearly 20 years on, I still consider Loudon’s
book as my favourite on the subject, although there are now many more
available to choose from. So why write another?
The answer to this question became clearer to me when I tried to
develop a course on quantum optics as a submodule of a larger unit
entitled ‘Aspects of Modern Physics’. This course is taken by undergraduate students in their final semester, and aims to introduce them to
a number of current research topics. I set about designing a course to
cover a few basic ideas about photon statistics, quantum cryptography,
and Bose–Einstein condensation, hoping that I would find a suitable text
to recommend. However, a quick inspection of the quantum optics texts
that were available led me to conclude that they were generally pitched
at a higher level than my target audience. Furthermore, the majority

were rather mathematical in their presentation. I therefore reluctantly
concluded that I would have to write the book I was seeking myself. The
end result is what you see before you. My hope is that it will serve both
as a useful basic introduction to the subject, and also as a tasty hors
d’oeuvre for the more advanced texts like Loudon’s.
In developing my course notes into a full-length book, the first problem that I encountered was the selection of topics. Traditional quantum
optics books like Loudon’s assume that the subject refers primarily to
the properties of light itself. At the same time, it is apparent that the
subject has broadened considerably in its scope, at least to many people
working in the field. I have therefore included a broad range of topics
that probably would not have found their way into a quantum optics
text 20 years ago. It is probable that someone else writing a similar text

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vi

Preface

would make a different selection of topics. My selection has been based
mainly on my perception of the key subject areas, but it also reflects my
own research interests to some extent. For this reason, there are probably more examples of quantum optical effects in solid state systems than
might normally have been expected.
Some of the subjects that I have selected for inclusion are still developing very rapidly at the time of writing. This is especially true of the topics
in quantum information technology covered in Part IV. Any attempt to
give a detailed overview of the present status of the experiments in these
fields would be relatively pointless, as it would date very quickly. I have
therefore adopted the strategy of trying to explain the basic principles
and then illustrating them with a few recent results. It is my hope that

the chapters I have written will be sufficient to allow students who are
new to the subjects to understand the fundamental concepts, thereby
allowing them to go to the research literature should they wish to pursue
any topics in more detail.
At one stage I thought about including references to a good number of
internet sites within the ‘Further Reading’ sections, but as the links to
these sites frequently change, I have actually only included a few. I am
sure that the modern computer-literate student will be able to find these
sites far more easily than I can, and I leave this part of the task to the
student’s initiative. It is a fortunate coincidence that the book is going
to press in 2005, the centenary of Einstein’s work on the photoelectric
effect, when there are many articles available to arouse the interest of
students on this subject. Furthermore, the award of the 2005 Nobel
Prize for Physics to Roy Glauber “for his contribution to the quantum
theory of optical coherence” has generated many more widely-accessible
information resources.
An issue that arose after receiving reviews of my original book plan
was the difficulty in making the subject accessible without gross oversimplification of the essential physics. As a consequence of these reviews,
I suspect that some sections of the book are pitched at a slightly higher
level than my original target of a final-year undergraduate, and would
in fact be more suitable for use in the first year of a Master’s course.
Despite this, I have still tried to keep the mathematics to a minimum as
far as possible, and concentrated on explanations based on the physical
understanding of the experiments that have been performed.
I would like to thank a number of people who have helped in the various stages of the preparation of this book. First, I would like to thank
all of the anonymous reviewers who made many helpful suggestions and
pointed out numerous errors in the early versions of the manuscript.
Second, I would like to thank several people for critical reading of
parts of the manuscript, especially Dr Brendon Lovett for Chapter 13,
and Dr Gerald Buller and Robert Collins for Chapter 12. I would like

to thank Dr Ed Daw for clarifying my understanding of gravity wave
interferometers. A special word of thanks goes to Dr Geoff Brooker for
critical reading of the whole manuscript. Third, I would like to thank
Sonke Adlung at Oxford University Press for his support and patience

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Preface

throughout the project and Anita Petrie for overseeing the production
of the book. I am also grateful to Dr Mark Hopkinson for the TEM picture in Fig. D.3, and to Dr Robert Taylor for Fig. 4.7. Finally, I would
like to thank my doctoral supervisor, Prof. John Ryan, for originally
pointing me towards quantum optics, and my numerous colleagues who
have helped me to carry out a number of quantum optics experiments
during my career.
Sheffield
June 2005

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Contents

List of symbols

xv

List of abbreviations

xviii

I Introduction and background

1

1 Introduction
1.1 What is quantum optics?
1.2 A brief history of quantum optics
1.3 How to use this book

3
3
4
6

2 Classical optics
2.1 Maxwell’s equations and electromagnetic waves
2.1.1 Electromagnetic fields
2.1.2 Maxwell’s equations
2.1.3 Electromagnetic waves
2.1.4 Polarization
2.2 Diffraction and interference
2.2.1 Diffraction

2.2.2 Interference
2.3 Coherence
2.4 Nonlinear optics
2.4.1 The nonlinear susceptibility
2.4.2 Second-order nonlinear phenomena
2.4.3 Phase matching

8
8
8
10
10
12
13
13
15
16
19
19
20
23

3 Quantum mechanics
3.1 Formalism of quantum mechanics
3.1.1 The Schră
odinger equation
3.1.2 Properties of wave functions
3.1.3 Measurements and expectation values
3.1.4 Commutators and the uncertainty principle
3.1.5 Angular momentum

3.1.6 Dirac notation
3.2 Quantized states in atoms
3.2.1 The gross structure
3.2.2 Fine and hyperfine structure
3.2.3 The Zeeman effect
3.3 The harmonic oscillator
3.4 The Stern–Gerlach experiment
3.5 The band theory of solids

26
26
26
28
30
31
32
34
35
35
39
41
41
43
45

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Contents

4 Radiative transitions in atoms
4.1 Einstein coefficients
4.2 Radiative transition rates
4.3 Selection rules
4.4 The width and shape of spectral lines
4.4.1 The spectral lineshape function
4.4.2 Lifetime broadening
4.4.3 Collisional (pressure) broadening
4.4.4 Doppler broadening
4.5 Line broadening in solids
4.6 Optical properties of semiconductors
4.7 Lasers
4.7.1 Laser oscillation
4.7.2 Laser modes
4.7.3 Laser properties

48
48
51
54
56
56
56
57
58
58
59
61

61
64
67

II Photons

73

5 Photon statistics
5.1 Introduction
5.2 Photon-counting statistics
5.3 Coherent light: Poissonian photon statistics
5.4 Classification of light by photon statistics
5.5 Super-Poissonian light
5.5.1 Thermal light
5.5.2 Chaotic (partially coherent) light
5.6 Sub-Poissonian light
5.7 Degradation of photon statistics by losses
5.8 Theory of photodetection
5.8.1 Semi-classical theory of photodetection
5.8.2 Quantum theory of photodetection
5.9 Shot noise in photodiodes
5.10 Observation of sub-Poissonian photon statistics
5.10.1 Sub-Poissonian counting statistics
5.10.2 Sub-shot-noise photocurrent

75
75
76
78

82
83
83
86
87
88
89
90
93
94
99
99
101

6 Photon antibunching
6.1 Introduction: the intensity interferometer
6.2 Hanbury Brown–Twiss experiments and
classical intensity fluctuations
6.3 The second-order correlation function g (2) (τ )
6.4 Hanbury Brown–Twiss experiments with photons
6.5 Photon bunching and antibunching
6.5.1 Coherent light
6.5.2 Bunched light
6.5.3 Antibunched light
6.6 Experimental demonstrations of photon antibunching
6.7 Single-photon sources

105
105


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108
111
113
115
116
116
117
117
120


Contents xi

7 Coherent states and squeezed light
7.1 Light waves as classical harmonic oscillators
7.2 Phasor diagrams and field quadratures
7.3 Light as a quantum harmonic oscillator
7.4 The vacuum field
7.5 Coherent states
7.6 Shot noise and number–phase uncertainty
7.7 Squeezed states
7.8 Detection of squeezed light
7.8.1 Detection of quadrature-squeezed
vacuum states
7.8.2 Detection of amplitude-squeezed light
7.9 Generation of squeezed states
7.9.1 Squeezed vacuum states
7.9.2 Amplitude-squeezed light

7.10 Quantum noise in amplifiers

126
126
129
131
132
134
135
138
139

8 Photon number states
8.1 Operator solution of the harmonic oscillator
8.2 The number state representation
8.3 Photon number states
8.4 Coherent states
8.5 Quantum theory of Hanbury Brown–Twiss
experiments

151
151
154
156
157

III Atom–photon interactions

165


9 Resonant light–atom interactions
9.1 Introduction
9.2 Preliminary concepts
9.2.1 The two-level atom approximation
9.2.2 Coherent superposition states
9.2.3 The density matrix
9.3 The time-dependent Schră
odinger equation
9.4 The weak-eld limit: Einsteins B coefficient
9.5 The strong-field limit: Rabi oscillations
9.5.1 Basic concepts
9.5.2 Damping
9.5.3 Experimental observations of
Rabi oscillations
9.6 The Bloch sphere

167
167
168
168
169
171
172
174
177
177
180

10 Atoms in cavities
10.1 Optical cavities

10.2 Atom–cavity coupling
10.3 Weak coupling
10.3.1 Preliminary considerations
10.3.2 Free-space spontaneous emission

194
194
197
200
200
201

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139
142
142
142
144
146

160

182
187


xii

Contents


10.3.3 Spontaneous emission in a single-mode
cavity: the Purcell effect
10.3.4 Experimental demonstrations of
the Purcell effect
10.4 Strong coupling
10.4.1 Cavity quantum electrodynamics
10.4.2 Experimental observations of strong coupling
10.5 Applications of cavity effects

202
204
206
206
209
211

11 Cold atoms
11.1 Introduction
11.2 Laser cooling
11.2.1 Basic principles of Doppler cooling
11.2.2 Optical molasses
11.2.3 Sub-Doppler cooling
11.2.4 Magneto-optic atom traps
11.2.5 Experimental techniques for laser cooling
11.2.6 Cooling and trapping of ions
11.3 Bose–Einstein condensation
11.3.1 Bose–Einstein condensation as a phase
transition
11.3.2 Microscopic description of Bose–Einstein

condensation
11.3.3 Experimental techniques for Bose–Einstein
condensation
11.4 Atom lasers

233
236

IV Quantum information processing

241

12 Quantum cryptography
12.1 Classical cryptography
12.2 Basic principles of quantum cryptography
12.3 Quantum key distribution according to
the BB84 protocol
12.4 System errors and identity verification
12.4.1 Error correction
12.4.2 Identity verification
12.5 Single-photon sources
12.6 Practical demonstrations of quantum cryptography
12.6.1 Free-space quantum cryptography
12.6.2 Quantum cryptography in optical fibres

243
243
245
249
253

253
254
255
256
257
258

13 Quantum computing
13.1 Introduction
13.2 Quantum bits (qubits)
13.2.1 The concept of qubits
13.2.2 Bloch vector representation of single qubits
13.2.3 Column vector representation of qubits

264
264
267
267
269
270

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216
216
218
218
221
224
226

227
229
230
230
232


Contents xiii

13.3 Quantum logic gates and circuits
13.3.1 Preliminary concepts
13.3.2 Single-qubit gates
13.3.3 Two-qubit gates
13.3.4 Practical implementations of qubit operations
13.4 Decoherence and error correction
13.5 Applications of quantum computers
13.5.1 Deutsch’s algorithm
13.5.2 Grover’s algorithm
13.5.3 Shor’s algorithm
13.5.4 Simulation of quantum systems
13.5.5 Quantum repeaters
13.6 Experimental implementations of quantum
computation
13.7 Outlook
14 Entangled states and quantum teleportation
14.1 Entangled states
14.2 Generation of entangled photon pairs
14.3 Single-photon interference experiments
14.4 Bell’s theorem
14.4.1 Introduction

14.4.2 Bell’s inequality
14.4.3 Experimental confirmation of Bell’s theorem
14.5 Principles of teleportation
14.6 Experimental demonstration of teleportation
14.7 Discussion

270
270
272
274
275
279
281
281
283
286
287
287
288
292
296
296
298
301
304
304
305
308
310
313

316

Appendices
A

Poisson statistics

321

B

Parametric amplification
B.1 Wave propagation in a nonlinear medium
B.2 Degenerate parametric amplification

324
324
326

C

The density of states

330

D

Low-dimensional semiconductor structures
D.1 Quantum confinement
D.2 Quantum wells

D.3 Quantum dots

333
333
335
337

E

Nuclear magnetic resonance
E.1 Basic principles
E.2 The rotating frame transformation
E.3 The Bloch equations

339
339
341
344

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xiv

Contents

F

Bose–Einstein condensation
F.1 Classical and quantum statistics

F.2 Statistical mechanics of Bose–Einstein condensation
F.3 Bose–Einstein condensed systems

346
346
348
350

Solutions and hints to the exercises

352

Bibliography

360

Index

369

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List of symbols
The alphabet only contains 26 letters, and the use of the same symbol to represent different quantities is
unavoidable in a book of this length. Whenever this occurs, it should be obvious from the context which
meaning is intended.
a
ˆ
a

ˆ†
a
a
a0
A
Aij
b
B
Bij
B
ci
C
CV
d
dij
D
D
Dp
E
Eg
EX
E
E0
f
f (T )
fij
F
F
FFano
FP

g
g(E)
g(k)
g(ω)
gν (ν)

annihilation operator
creation operator
length parameter
unit vector
Bohr radius
area
Einstein A coefficient
unit vector
magnetic field (flux density)
Einstein B coefficient
magnetic field gradient
amplitude coefficient
capacitance
heat capacity at constant volume
distance; slit width
nonlinear optical coefficient tensor
diameter
electric displacement
momentum diffusion coefficient
energy
band-gap energy
exciton binding energy
electric field
electric field amplitude

frequency
fraction of condensed particles
oscillator strength
force; total angular momentum
finesse
Fano factor
Purcell factor
degeneracy; nonlinear coupling
density of states at energy E
state density in k-space
density of states at angular frequency ω
spectral lineshape function

gω (ω)
gF
gJ
gN
gs
g0
g (1) (τ )
g (2) (τ )
G
h
H
ˆ
H
H
H
Hn (x)
ˆi

i
I
Irot
Is
I
I
Iz
j
ˆj
J
k
k
ˆ
k
l
lz
L
L
Lc

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spectral lineshape function
hyperfine g-factor
Land´e g-factor
nuclear g-factor
electron spin g-factor
atom–cavity coupling constant
first-order correlation function
second-order correlation function

gain; Grover operator
strain
magnetic field
Hamiltonian
Hadamard operator
perturbation
Hermite polynomial
unit vector along the x-axis
electrical current
optical intensity; nuclear spin
moment of inertia
saturation intensity
nuclear angular momentum
identity matrix
z-component of nuclear angular momentum
current density; angular momentum (single
electron)
unit vector along the y-axis
angular momentum
wave vector
modulus of wave vector; spring constant
unit vector along the z-axis
orbital angular momentum (single electron)
z-component of orbital angular momentum
(single electron)
length; mean free path
orbital angular momentum
coherence length



xvi

Lw
m
m0
m∗
m∗e
mH
M
M
Mx
My
Mz
n

List of symbols

quantum well thickness
mass
electron rest mass
effective mass
electron effective mass
mass of hydrogen atom
matrix
magnetization
x-component of the magnetization
y-component of the magnetization
z-component of the magnetization
refractive index; photon number; number
of events

nonlinear refractive index
n2
refractive index for ordinary ray
no
refractive index for extraordinary ray
ne
n
mean photon number
n(E)
thermal occupancy of level at energy E
nBE (E) Bose–Einstein distribution function
nFD (E) Fermi–Dirac distribution function
N
number of atoms, particles, photons,
counts, time intervals, data bits
stopping number of absorption–emission
Nstop
cycles
ˆ
O
operator
p
momentum; probability
p
electric dipole moment
ˆ
p
momentum operator
P
pressure; power

P
probability
Pij
probability for i → j transition
P
electric polarization
q
charge; generalized position coordinate;
qubit
Q
quality factor
r
radius; amplitude reflection coefficient
r
position vector
rˆ
position operator
R
reflectivity; net absorption rate; electrical
resistance
R
pumping rate; count rate
rotation operator about Cartesian axis i
Ri (θ)
s
squeeze parameter; saturation parameter
s
spin angular momentum (single electron)
z-component of spin angular momentum
sz

(single electron)
S
Clauser, Horne, Shimony, and Holt
parameter
S
spin angular momentum

t
te
T
Tˆ
T
Tc
Top
Tosc
Tp
T1
T2
u
u(ν)
u(ω)
U
ˆ
U
v
V
Vˆ
Vij
w
W

Wij
x
ˆ
x
x
ˆ
X
X1,2
y
ˆ
y
Yl,ml
z
ˆ
z
Z
α
β
γ
Γ
δ
δ(x)
δij
∆
ε

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time; amplitude transmission coefficient
expansion time

temperature; time interval
kinetic energy operator
time interval; transmission
critical temperature
gate operation time
oscillation period
pulse duration
longitudinal (spin–lattice) relaxation time
transverse (spin–spin) relaxation time;
dephasing time
initial velocity
spectral energy density at frequency ν
spectral energy density at angular
frequency ω
energy density
unitary operator
velocity
volume; potential energy
perturbation; potential energy operator
perturbation matrix element
Gaussian beam radius
count rate in time interval T
transition rate
position coordinate
unit vector along the x-axis
position coordinate operator
X operator
quadrature field
position coordinate
unit vector along the y-axis

spherical harmonic function
position coordinate
unit vector along the z-axis
atomic number; Z operator; partition
function; impedance
coherent state complex amplitude;
damping coefficient
spontaneous emission coupling factor
gyromagnetic ratio; damping rate; decay
rate; linewidth; gain coefficient
torque
frequency detuning
Dirac delta function
Kronecker delta function
detuning in angular frequency units
error probability


List of symbols xvii

r

θ
Θ
η
κ
λ
λdeB
µ
µ

µij
µR
ν
νL
νvib
ξ
ρ
ρij
ρ
σ
σs

relative permittivity
angle; polar angle
rotation angle; pulse area
quantum efficiency
photon decay rate
wavelength
de Broglie wavelength
reduced mass; chemical potential; mean
value
magnetic dipole moment
dipole moment for i → j transition
relative magnetic permeability
frequency
laser frequency
vibrational frequency
dipole orientation factor; optical loss;
emission probability per unit time per
unit intensity

density matrix
element of density matrix
energy density of black-body radiation
charge density
standard deviation; electrical conductivity
scattering cross-section

τ
τc
τcollision
τD
τG
τR
τNR
φ
ϕ
χ
χ(n)
(2)
χijk
χM
Φ
Ψ
ψ
ω
ωL
Ω
Ω
ΩR


lifetime
coherence time
time between collisions
detector response time
gravity wave period
radiative lifetime
non-radiative lifetime
optical phase
wave function; optical phase; azimuthal
angle
electric susceptibility; spin wave
function
nth-order nonlinear susceptibility
second-order nonlinear susceptibility
tensor
magnetic susceptibility
photon flux; wave function
wave function
wave function
angular frequency
Larmor precession angular frequency
solid angle; angular frequency
angular velocity vector
Rabi angular frequency

List of quantum numbers
In atomic physics, lower and upper case letters refer to individual electrons or whole atoms respectively.
F
I
j, J

l, L
MF
MI

total angular momentum (with nuclear
spin included)
nuclear spin
total electron angular momentum
orbital angular momentum
magnetic (z-component of total angular
momentum including hyperfine
interactions)
magnetic (z-component of nuclear spin)

mj , M J
ml , M L
ms , MS
n
s, S

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magnetic (z-component of total angular
momentum)
magnetic (z-component of orbital angular
momentum)
magnetic (z-component of spin angular
momentum)
principal
spin



List of abbreviations
AC
AOS
APD
B92
BB84
BBO
BS
BSM
CHSH
CW
DBR
DC
EPR
EPRB
FWHM
HBT
LD
LED
LHV
LIGO
LISA
LO
MBE
MOCVD
NMR
PBS
PC

PD
PMT
QED
RF
rms
SNL
SNR
SPAD
STP
TEM
VCSEL

alternating current
acousto-optic switch
avalanche photodiode
Bennett 1992
Bennett–Brassard 1984
β-barium borate
beam splitter
Bell-state measurement
Clauser–Horne–Shimony–Holt
continuous wave
distributed Bragg reflector
direct current
Einstein–Podolsky–Rosen
Einstein–Podolsky–Rosen–Bohm
full width at half maximum
Hanbury Brown–Twiss
laser diode
light-emitting diode

local hidden variables
light interferometer gravitational wave observatory
laser interferometer space antenna
local oscillator
molecular beam epitaxy
metalorganic chemical vapour epitaxy
nuclear magnetic resonance
polarizing beam splitter
Pockels cell
photodiode
photomultiplier tube
quantum electrodynamics
radio frequency
root mean square
shot-noise level
signal-to-noise ratio
single-photon avalanche photodiode
standard temperature and pressure
transmission electron microscope
vertical-cavity surface-emitting laser

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Part I
Introduction and
background

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1

Introduction
1.1

What is quantum optics ?

Quantum optics is the subject that deals with optical phenomena that
can only be explained by treating light as a stream of photons rather
than as electromagnetic waves. In principle, the subject is as old as
quantum theory itself, but in practice, it is a relatively new one, and
has really only come to the fore during the last quarter of the twentieth
century.
In the progressive development of the theory to light, three general
approaches can be clearly identified, namely the classical, semiclassical, and quantum theories, as summarized in Table 1.1. It goes
without saying that only the fully quantum optical approach is totally
consistent both with itself and with the full body of experimental data.
Nevertheless, it is also the case that semi-classical theories are quite adequate for most purposes. For example, when the theory of absorption of
light by atoms is first considered, it is usual to apply quantum mechanics
to the atoms, but treat the light as a classical electromagnetic wave.
The question that we really have to ask to define the subject of quantum optics is whether there are any effects that cannot be explained in
the semi-classical approach. It may come as a surprise to the reader that
there are relatively few such phenomena. Indeed, until about 30 years
ago, there were only a handful of effects—mainly those related to the

vacuum field such as spontaneous emission and the Lamb shift—that
really required a quantum model of light.
Let us consider just one example that seems to require a photon
picture of light, namely the photoelectric effect. This describes the
ejection of electrons from a metal under the influence of light.
The explanation of the phenomenon was first given by Einstein in 1905,
when he realized that the atoms must be absorbing energy from the light
beam in quantized packets. However, careful analysis has subsequently
shown that the results can in fact be understood by treating only the
atoms as quantized objects, and the light as a classical electromagnetic
wave. Arguments along the same line can explain how the individual
pulses emitted by ‘single-photon counting’ detectors do not necessarily
imply that light consists of photons. In most cases, the output pulses can
in fact be explained in terms of the probabilistic ejection of an individual
electron from one of the quantized states in an atom under the influence of a classical light wave. Thus although these experiments point us
towards the photon picture of light, they do not give conclusive evidence.

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1.1 What is quantum
optics ?

3

1.2 A brief history of
quantum optics
1.3 How to use this book

4
6


Table 1.1 The three different approaches used to model the interaction
between light and matter. In classical
physics, the light is conceived as electromagnetic waves, but in quantum
optics, the quantum nature of the
light is included by treating the light
as photons.

Model
Classical

Atoms

Light

Hertzian Waves
dipoles
Semi-classical Quantized Waves
Quantum
Quantized Photons


4 Introduction
Table 1.2 Subtopics of recent European Quantum Optics Conferences

Year

Topic

1998

1999
2000
2002
2003

Atom cooling and guiding, laser spectroscopy and squeezing
Quantum optics in semiconductor materials, quantum structures
Experimental technologies of quantum manipulation
Quantum atom optics: from quantum science to technology
Cavity QED and quantum fluctuations: from fundamental
concepts to nanotechnology

Source: European Science Foundation, .

It was not until the late 1970s that the subject of quantum optics as
we now know it started to develop. At that time, the first observations
of effects that give direct evidence of the photon nature of light, such as
photon antibunching, were convincingly demonstrated in the laboratory.
Since then, the scope of the subject has expanded enormously, and it now
encompasses many new topics that go far beyond the strict study of light
itself. This is apparent from Table 1.2, which lists the range of specialist
topics selected for recent European Quantum Optics Conferences. It is
in this widened sense, rather than the strict one, that the subject of
quantum optics is understood throughout this book.

1.2

A brief history of quantum optics

We can obtain insight into the way the subject of quantum optics fits into

the wider picture of quantum theory by running through a brief history
of its development. Table 1.3 summarizes some of the most important
landmarks in this development, together with a few recent highlights.
In the early development of optics, there were two rival theories,
namely the corpuscular theory proposed by Newton, and the wave theory expounded by his contemporary, Huygens. The wave theory was
convincingly vindicated by the double-slit experiment of Young in 1801
and by the wave interpretation of diffraction by Fresnel in 1815. It was
then given a firm theoretical footing with Maxwell’s derivation of the
electromagnetic wave equation in 1873. Thus by the end of the nineteenth century, the corpuscular theory was relegated to mere historical
interest.
The situation changed radically in 1901 with Planck’s hypothesis that
black-body radiation is emitted in discrete energy packets called quanta.
With this supposition, he was able to solve the ultraviolet catastrophe
problem that had been puzzling physicists for many years. Four years
later in 1905, Einstein applied Planck’s quantum theory to explain the
photoelectric effect. These pioneering ideas laid the foundations for the
quantum theories of light and atoms, but in themselves did not give
direct experimental evidence of the quantum nature of the light. As mentioned above, what they actually prove is that something is quantized,
without definitively establishing that it is the light that is quantized.

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1.2

A brief history of quantum optics 5

Table 1.3 Selected landmarks in the development of quantum optics, including a few recent highlights.
The final column points to the appropriate chapter of the book where the topic is developed


Year
1901
1905
1909
1909
1927
1956
1963
1972
1977
1981
1985
1987
1992
1995
1995
1997
1997
2002

Authors

Development

Chapter

Planck
Einstein
Taylor
Einstein

Dirac
Hanbury Brown and Twiss
Glauber
Gibbs
Kimble, Dagenais, and Mandel
Aspect, Grangier, and Roger
Slusher et al.
Hong, Ou, and Mandel
Bennett, Brassard et al.
Turchette, Kimble et al.
Anderson, Wieman, Cornell et al.
Mewes, Ketterle et al.
Bouwmeester et al., Boschi et al.
Yuan et al.

Theory of black-body radiation
Explanation of the photoelectric effect
Interference of single quanta
Radiation fluctuations
Quantum theory of radiation
Intensity interferometer
Quantum states of light
Optical Rabi oscillations
Photon antibunching
Violations of Bell’s inequality
Squeezed light
Single-photon interference experiments
Experimental quantum cryptography
Quantum phase gate
Bose–Einstein condensation of atoms

Atom laser
Quantum teleportation of photons
Single-photon light-emitting diode

5
5
14
5
8
6
8
9
6
14
7
14
12
10, 13
11
11
14
6

The first serious attempt at a real quantum optics experiment was
performed by Taylor in 1909. He set up a Young’s slit experiment, and
gradually reduced the intensity of the light beam to such an extent that
there would only be one quantum of energy in the apparatus at a given
instant. The resulting interference pattern was recorded using a photographic plate with a very long exposure time. To his disappointment, he
found no noticeable change in the pattern, even at the lowest intensities.
In the same year as Taylor’s experiment, Einstein considered the

energy fluctuations of black-body radiation. In doing so, he showed that
the discrete nature of the radiation energy gave an extra term proportional to the average number of quanta, thereby anticipating the modern
theory of photon statistics.
The formal theory of the quantization of light came in the 1920s
after the birth of quantum mechanics. The word ‘photon’ was coined
by Gilbert Lewis in 1926, and Dirac published his seminal paper on the
quantum theory of radiation a year later. In the following years, however, the main emphasis was on calculating the optical spectra of atoms,
and little effort was invested in looking for quantum effects directly
associated with the light itself.
The modern subject of quantum optics was effectively born in 1956
with the work of Hanbury Brown and Twiss. Their experiments on correlations between the starlight intensities recorded on two separated
detectors provoked a storm of controversy. It was subsequently shown
that their results could be explained by treating the light classically and
only applying quantum theory to the photodetection process. However,

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6 Introduction

their experiments are still considered a landmark in the field because
they were the first serious attempt to measure the fluctuations in the
light intensity on short time-scales. This opened the door to more sophisticated experiments on photon statistics that would eventually lead to
the observation of optical phenomena with no classical explanation.
The invention of the laser in 1960 led to new interest in the subject. It
was hoped that the properties of the laser light would be substantially
different from those of conventional sources, but these attempts again
proved negative. The first clues of where to look for unambiguous quantum optical effects were given by Glauber in 1963, when he described
new states of light which have different statistical properties to those
of classical light. The experimental confirmation of these non-classical

properties was given by Kimble, Dagenais, and Mandel in 1977 when
they demonstrated photon antibunching for the first time. Eight years
later, Slusher et al. completed the picture by successfully generating
squeezed light in the laboratory.
In recent years, the subject has expanded to include the associated disciplines of quantum information processing and controlled light–matter
interactions. The work of Aspect and co-workers starting from 1981
onwards may perhaps be conceived as a landmark in this respect. They
used the entangled photons from an atomic cascade to demonstrate violations of Bell’s inequality, thereby emphatically showing how quantum
optics can be applied to other branches of physics. Since then, there
has been a growing number of examples of the use of quantum optics in
ever widening applications. Some of the recent highlights are listed in
Table 1.3.
This brief and incomplete survey of the development of quantum
optics makes it apparent that the subject has ‘come of age’ in recent
years. It is no longer a specialized, highly academic discipline, with few
applications in the real world, but a thriving field with ever broadening
horizons.

1.3

How to use this book

The structure of the book is shown schematically in Fig. 1.1. The book
has been divided into four parts:
Part
Part
Part
Part

I Introduction and background material.

II Photons.
III Atom–photon interactions.
IV Quantum information processing.

Part I contains the introduction and the background information that
forms a starting point for the rest of the book, while Parts II–IV contain
the new material that is being developed.
The background material in Part I has been included both for revision
purposes and to fill in any small gaps in the prior knowledge that has

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