Table Of ContentOXFORD MASTER SERIES IN PHYSICS
OXFORD MASTER SERIES IN PHYSICS
TheOxfordMasterSeriesisdesignedforfinalyearundergraduateandbeginninggraduatestudentsinphysics
andrelateddisciplines.Ithasbeendrivenbyaperceivedgapintheliteraturetoday.Whilebasicundergraduate
physicstextsoftenshowlittleornoconnectionwiththehugeexplosionofresearchoverthelasttwodecades,
moreadvancedandspecializedtextstendtoberatherdauntingforstudents.Inthisseries,alltopicsandtheir
consequencesaretreatedatasimplelevel,whilepointerstorecentdevelopmentsareprovidedatvariousstages.
Theemphasisisonclearphysicalprincipleslikesymmetry,quantummechanics,andelectromagnetismwhich
underlie the whole of physics. At the same time, the subjects are related to real measurements and to the
experimentaltechniquesanddevicescurrentlyusedbyphysicistsinacademeandindustry.Booksinthisseries
arewrittenascoursebooks,andincludeampletutorialmaterial,examples,illustrations,revisionpoints,and
problemsets.Theycanlikewisebeusedaspreparationforstudentsstartingadoctorateinphysicsandrelated
fields,orforrecentgraduatesstartingresearchinoneofthesefieldsinindustry.
CONDENSED MATTER PHYSICS
1. M. T. Dove: Structure and dynamics: an atomic view of materials
2. J. Singleton: Baud theory and electronic properties of solids
3. A. M. Fox: Optical properties of solids
4. S. J. Blundell: Magnetism in condensed matter
5. J. F. Annett: Superconductivity
6. R. A. L. Jones: Soft condensed matter
ATOMIC, OPTICAL, AND LASER PHYSICS
7. C. J. Foot: Atomic Physics
8. G. A. Brooker: Modern classical optics
9. S. M. Hooker, C. E. Webb: Laser physics
15. 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
Quantum Optics
An Introduction
MARK FOX
Department of Physics and Astronomy
University of Sheffield
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Preface
Quantumopticsisasubjectthathascometotheforeoverthelast10–20
years.Formerly,itwasregardedasahighlyspecializeddiscipline,acces-
sibleonlytoasmallnumberofadvancedstudentsatselecteduniversities.
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.
Myowninterestinquantumopticsgoesbackto1987,whenIattended
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 lead-
ing 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
bookasmyfavouriteonthesubject,althoughtherearenowmanymore
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 under-
graduatestudentsintheirfinalsemester,andaimstointroducethemto
a number of current research topics. I set about designing a course to
cover a few basic ideas about photon statistics, quantum cryptography,
andBose–Einsteincondensation,hopingthatIwouldfindasuitabletext
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
concludedthatIwouldhavetowritethebookIwasseekingmyself.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 prob-
lem 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
subjecthasbroadenedconsiderablyinitsscope,atleasttomanypeople
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
text20yearsago.Itisprobablethatsomeoneelsewritingasimilartext
vi Preface
would make a different selection of topics. My selection has been based
mainlyonmyperceptionofthekeysubjectareas,butitalsoreflectsmy
own research interests to some extent. For this reason, there are proba-
blymoreexamplesofquantumopticaleffectsinsolidstatesystemsthan
might normally have been expected.
SomeofthesubjectsthatIhaveselectedforinclusionarestilldevelop-
ingveryrapidlyatthetimeofwriting.Thisisespeciallytrueofthetopics
in quantum information technology covered in Part IV. Any attempt to
giveadetailedoverviewofthepresentstatusoftheexperimentsinthese
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
allowingthemtogototheresearchliteratureshouldtheywishtopursue
any topics in more detail.
AtonestageIthoughtaboutincludingreferencestoagoodnumberof
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
surethatthemoderncomputer-literatestudentwillbeabletofindthese
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
theoryofopticalcoherence”hasgeneratedmanymorewidely-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 over-
simplificationoftheessentialphysics.Asaconsequenceofthesereviews,
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.
Despitethis,Ihavestilltriedtokeepthemathematicstoaminimumas
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 var-
ious 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
Preface vii
throughout the project and Anita Petrie for overseeing the production
of the book. I am also grateful to Dr Mark Hopkinson for the TEM pic-
ture 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 3
1.1 What is quantum optics? 3
1.2 A brief history of quantum optics 4
1.3 How to use this book 6
2 Classical optics 8
2.1 Maxwell’s equations and electromagnetic waves 8
2.1.1 Electromagnetic fields 8
2.1.2 Maxwell’s equations 10
2.1.3 Electromagnetic waves 10
2.1.4 Polarization 12
2.2 Diffraction and interference 13
2.2.1 Diffraction 13
2.2.2 Interference 15
2.3 Coherence 16
2.4 Nonlinear optics 19
2.4.1 The nonlinear susceptibility 19
2.4.2 Second-order nonlinear phenomena 20
2.4.3 Phase matching 23
3 Quantum mechanics 26
3.1 Formalism of quantum mechanics 26
3.1.1 The Schro¨dinger equation 26
3.1.2 Properties of wave functions 28
3.1.3 Measurements and expectation values 30
3.1.4 Commutators and the uncertainty principle 31
3.1.5 Angular momentum 32
3.1.6 Dirac notation 34
3.2 Quantized states in atoms 35
3.2.1 The gross structure 35
3.2.2 Fine and hyperfine structure 39
3.2.3 The Zeeman effect 41
3.3 The harmonic oscillator 41
3.4 The Stern–Gerlach experiment 43
3.5 The band theory of solids 45
