OpticsAnIntroduction1stEdition MarkFox
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Quantum Optics An Introduction Oxford Master Series in Physics Mark Fox
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OXFORDMASTERSERIESINPHYSICS OXFORDMASTERSERIESINPHYSICS TheOxfordMasterSeriesisdesignedforfinalyearundergraduateandbeginninggraduatestudentsinphysics andrelateddisciplines.Ithasbeendrivenbyaperceivedgapintheliteraturetoday.Whilebasicundergraduate physicstextsoftenshowlittleornoconnectionwiththehugeexplosionofresearchoverthelasttwodecades, moreadvancedandspecializedtextstendtoberatherdauntingforstudents.Inthisseries,alltopicsandtheir consequencesaretreatedatasimplelevel,whilepointerstorecentdevelopmentsareprovidedatvariousstages. Theemphasisisonclearphysicalprincipleslikesymmetry,quantummechanics,andelectromagnetismwhich underliethewholeofphysics.Atthesametime,thesubjectsarerelatedtorealmeasurementsandtothe experimentaltechniquesanddevicescurrentlyusedbyphysicistsinacademeandindustry.Booksinthisseries arewrittenascoursebooks,andincludeampletutorialmaterial,examples,illustrations,revisionpoints,and problemsets.Theycanlikewisebeusedaspreparationforstudentsstartingadoctorateinphysicsandrelated fields,orforrecentgraduatesstartingresearchinoneofthesefieldsinindustry.
CONDENSEDMATTERPHYSICS 1.M.T.Dove: Structureanddynamics:anatomicviewofmaterials
2.J.Singleton: Baudtheoryandelectronicpropertiesofsolids
3.A.M.Fox: Opticalpropertiesofsolids
4.S.J.Blundell: Magnetismincondensedmatter
5.J.F.Annett: Superconductivity
6.R.A.L.Jones: Softcondensedmatter
ATOMIC,OPTICAL,ANDLASERPHYSICS 7.C.J.Foot: AtomicPhysics
8.G.A.Brooker: Modernclassicaloptics
9.S.M.Hooker,C.E.Webb: Laserphysics
15.A.M.Fox: Quantumoptics:anintroduction
PARTICLEPHYSICS,ASTROPHYSICS,ANDCOSMOLOGY 10.D.H.Perkins: Particleastrophysics
11.Ta-PeiCheng: Relativity,gravitation,andcosmology
STATISTICAL,COMPUTATIONAL,ANDTHEORETICALPHYSICS 12.M.Maggiore: Amodernintroductiontoquantumfieldtheory
13.W.Krauth: Statisticalmechanics:algorithmsandcomputations
14.J.P.Sethna: Entropy,orderparameters,andcomplexity
QuantumOptics AnIntroduction MARKFOX
DepartmentofPhysicsandAstronomy
UniversityofSheffield
GreatClarendonStreet,OxfordOX26DP
OxfordUniversityPressisadepartmentoftheUniversityofOxford. ItfurtherstheUniversity’sobjectiveofexcellenceinresearch,scholarship, andeducationbypublishingworldwidein
OxfordNewYork
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PublishedintheUnitedStates byOxfordUniversityPressInc.,NewYork
c OxfordUniversityPress2006
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Firstpublished2006
Allrightsreserved.Nopartofthispublicationmaybereproduced, storedinaretrievalsystem,ortransmitted,inanyformorbyanymeans, withoutthepriorpermissioninwritingofOxfordUniversityPress, orasexpresslypermittedbylaw,orundertermsagreedwiththeappropriate reprographicsrightsorganization.Enquiriesconcerningreproduction outsidethescopeoftheaboveshouldbesenttotheRightsDepartment, OxfordUniversityPress,attheaddressabove
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LibraryofCongressCataloginginPublicationData Fox,Mark(AnthonyMark) Quantumoptics:anintroduction/MarkFox. p.cm.—(Oxfordmasterseriesinphysics;6) Includesbibliographicalreferencesandindex.
ISBN-13:978–0–19–856672–4(hbk.:acid-freepaper)
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ISBN-10:0–19–856673–5(pbk.:acid-freepaper) 1.Quantumoptics.I.Title.II.Series. QC446.2.F692006 535 .15—dc222005025707
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ISBN0–19–856673–5(Pbk.)978–0–19–856673–1(Pbk.) 10987654321
Preface Quantumopticsisasubjectthathascometotheforeoverthelast10–20 years.Formerly,itwasregardedasahighlyspecializeddiscipline,accessibleonlytoasmallnumberofadvancedstudentsatselecteduniversities. Nowadays,however,thedemandforthesubjectismuchbroader,with theintereststronglyfuelledbytheprospectofusingquantumopticsin quantuminformationprocessingapplications.
Myowninterestinquantumopticsgoesbackto1987,whenIattended theConferenceonLasersandElectro-Optics(CLEO)forthefirst time.Theground-breakingexperimentsonsqueezedlighthadrecently beencompleted,andIwasabletohearinvitedtalksfromtheleadingresearchersworkinginthefield.Attheendoftheconference,I foundmyselfsufficientlyinterestedinthesubjectthatIboughtacopy ofLoudon’s Quantumtheoryoflight andstartedtoworkthroughitin afairlysystematicway.Nearly20yearson,IstillconsiderLoudon’s bookasmyfavouriteonthesubject,althoughtherearenowmanymore availabletochoosefrom.Sowhywriteanother?
TheanswertothisquestionbecameclearertomewhenItriedto developacourseonquantumopticsasasubmoduleofalargerunit entitled‘AspectsofModernPhysics’.Thiscourseistakenbyundergraduatestudentsintheirfinalsemester,andaimstointroducethemto anumberofcurrentresearchtopics.Isetaboutdesigningacourseto coverafewbasicideasaboutphotonstatistics,quantumcryptography, andBose–Einsteincondensation,hopingthatIwouldfindasuitabletext torecommend.However,aquickinspectionofthequantumopticstexts thatwereavailableledmetoconcludethattheyweregenerallypitched atahigherlevelthanmytargetaudience.Furthermore,themajority wererathermathematicalintheirpresentation.Ithereforereluctantly concludedthatIwouldhavetowritethebookIwasseekingmyself.The endresultiswhatyouseebeforeyou.Myhopeisthatitwillserveboth asausefulbasicintroductiontothesubject,andalsoasatasty hors d’oeuvre forthemoreadvancedtextslikeLoudon’s.
Indevelopingmycoursenotesintoafull-lengthbook,thefirstproblemthatIencounteredwastheselectionoftopics.Traditionalquantum opticsbookslikeLoudon’sassumethatthesubjectrefersprimarilyto thepropertiesoflightitself.Atthesametime,itisapparentthatthe subjecthasbroadenedconsiderablyinitsscope,atleasttomanypeople workinginthefield.Ihavethereforeincludedabroadrangeoftopics thatprobablywouldnothavefoundtheirwayintoaquantumoptics text20yearsago.Itisprobablethatsomeoneelsewritingasimilartext
wouldmakeadifferentselectionoftopics.Myselectionhasbeenbased mainlyonmyperceptionofthekeysubjectareas,butitalsoreflectsmy ownresearchintereststosomeextent.Forthisreason,thereareprobablymoreexamplesofquantumopticaleffectsinsolidstatesystemsthan mightnormallyhavebeenexpected.
SomeofthesubjectsthatIhaveselectedforinclusionarestilldevelopingveryrapidlyatthetimeofwriting.Thisisespeciallytrueofthetopics inquantuminformationtechnologycoveredinPartIV.Anyattemptto giveadetailedoverviewofthepresentstatusoftheexperimentsinthese fieldswouldberelativelypointless,asitwoulddateveryquickly.Ihave thereforeadoptedthestrategyoftryingtoexplainthebasicprinciples andthenillustratingthemwithafewrecentresults.Itismyhopethat thechaptersIhavewrittenwillbesufficienttoallowstudentswhoare newtothesubjectstounderstandthefundamentalconcepts,thereby allowingthemtogototheresearchliteratureshouldtheywishtopursue anytopicsinmoredetail.
AtonestageIthoughtaboutincludingreferencestoagoodnumberof internetsiteswithinthe‘FurtherReading’sections,butasthelinksto thesesitesfrequentlychange,Ihaveactuallyonlyincludedafew.Iam surethatthemoderncomputer-literatestudentwillbeabletofindthese sitesfarmoreeasilythanIcan,andIleavethispartofthetasktothe student’sinitiative.Itisafortunatecoincidencethatthebookisgoing topressin2005,thecentenaryofEinstein’sworkonthephotoelectric effect,whentherearemanyarticlesavailabletoarousetheinterestof studentsonthissubject.Furthermore,theawardofthe2005Nobel PrizeforPhysicstoRoyGlauber“forhiscontributiontothequantum theoryofopticalcoherence”hasgeneratedmanymorewidely-accessible informationresources.
Anissuethataroseafterreceivingreviewsofmyoriginalbookplan wasthedifficultyinmakingthesubjectaccessiblewithoutgrossoversimplificationoftheessentialphysics.Asaconsequenceofthesereviews, Isuspectthatsomesectionsofthebookarepitchedataslightlyhigher levelthanmyoriginaltargetofafinal-yearundergraduate,andwould infactbemoresuitableforuseinthefirstyearofaMaster’scourse. Despitethis,Ihavestilltriedtokeepthemathematicstoaminimumas faraspossible,andconcentratedonexplanationsbasedonthephysical understandingoftheexperimentsthathavebeenperformed.
Iwouldliketothankanumberofpeoplewhohavehelpedinthevariousstagesofthepreparationofthisbook.First,Iwouldliketothank alloftheanonymousreviewerswhomademanyhelpfulsuggestionsand pointedoutnumerouserrorsintheearlyversionsofthemanuscript. Second,Iwouldliketothankseveralpeopleforcriticalreadingof partsofthemanuscript,especiallyDrBrendonLovettforChapter13, andDrGeraldBullerandRobertCollinsforChapter12.Iwouldlike tothankDrEdDawforclarifyingmyunderstandingofgravity wave interferometers.AspecialwordofthanksgoestoDrGeoffBrookerfor criticalreadingofthewholemanuscript.Third,Iwouldliketothank SonkeAdlungatOxfordUniversityPressforhissupportandpatience
throughouttheprojectandAnitaPetrieforoverseeingtheproduction ofthebook.IamalsogratefultoDrMarkHopkinsonfortheTEMpictureinFig.D.3,andtoDrRobertTaylorforFig.4.7.Finally,Iwould liketothankmydoctoralsupervisor,Prof.JohnRyan,fororiginally pointingmetowardsquantumoptics,andmynumerouscolleagueswho havehelpedmetocarryoutanumberofquantumopticsexperiments duringmycareer.
Sheffield June2005
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Contents Listofsymbols xv
Listofabbreviations xviii
IIntroductionandbackground 1
1Introduction 3
1.1Whatisquantumoptics?3
1.2Abriefhistoryofquantumoptics4
1.3Howtousethisbook6
2Classicaloptics 8
2.1Maxwell’sequationsandelectromagnetic waves8
2.1.1Electromagneticfields8
2.1.2Maxwell’sequations10
2.1.3Electromagnetic waves10
2.1.4Polarization12
2.2Diffractionandinterference13
2.2.1Diffraction13
2.2.2Interference15
2.3Coherence16
2.4Nonlinearoptics19
2.4.1Thenonlinearsusceptibility19
2.4.2Second-ordernonlinearphenomena20
2.4.3Phasematching23
3Quantummechanics 26
3.1Formalismofquantummechanics26
3.1.1TheSchr¨odingerequation26
3.1.2Propertiesof wave functions28
3.1.3Measurementsandexpectationvalues30
3.1.4Commutatorsandtheuncertaintyprinciple31
3.1.5Angularmomentum32
3.1.6Diracnotation34
3.2Quantizedstatesinatoms35
3.2.1Thegrossstructure35
3.2.2Fineandhyperfinestructure39
3.2.3TheZeemaneffect41
3.3Theharmonicoscillator41
3.4TheStern–Gerlachexperiment43
3.5Thebandtheoryofsolids45
4Radiativetransitionsinatoms 48
4.1Einsteincoefficients48
4.2Radiativetransitionrates51
4.3Selectionrules54
4.4Thewidthandshapeofspectrallines56
4.4.1Thespectrallineshapefunction56
4.4.2Lifetimebroadening56
4.4.3Collisional(pressure)broadening57
4.4.4Dopplerbroadening58
4.5Linebroadeninginsolids58
4.6Opticalpropertiesofsemiconductors59
5Photonstatistics 75
5.1Introduction75
5.2Photon-countingstatistics76
5.3Coherentlight:Poissonianphotonstatistics78
5.4Classificationoflightbyphotonstatistics82
5.5Super-Poissonianlight83
5.5.1Thermallight83
5.5.2Chaotic(partiallycoherent)light86
5.6Sub-Poissonianlight87
5.7Degradationofphotonstatisticsbylosses88
5.8Theoryofphotodetection89
5.8.1Semi-classicaltheoryofphotodetection90
5.8.2Quantumtheoryofphotodetection93
5.9Shotnoiseinphotodiodes94
5.10Observationofsub-Poissonianphotonstatistics99
5.10.1Sub-Poissoniancountingstatistics99
5.10.2Sub-shot-noisephotocurrent101
6Photonantibunching
6.1Introduction:theintensityinterferometer105
6.2HanburyBrown–Twissexperimentsand classicalintensityfluctuations108
6.3Thesecond-ordercorrelationfunction g (2) (τ )111
6.4HanburyBrown–Twissexperimentswithphotons113
6.5Photonbunchingandantibunching115
6.5.1Coherentlight116
6.5.2Bunchedlight116
6.5.3Antibunchedlight117
6.6Experimentaldemonstrationsofphotonantibunching117
6.7Single-photonsources120
7Coherentstatesandsqueezedlight
126
7.1Light wavesas classicalharmonicoscillators126
7.2Phasordiagramsandfieldquadratures129
7.3Lightasaquantumharmonicoscillator131
7.4Thevacuumfield132
7.5Coherentstates134
7.6Shotnoiseandnumber–phaseuncertainty135
7.7Squeezedstates138
7.8Detectionofsqueezedlight139
7.8.1Detectionofquadrature-squeezed vacuumstates139
7.8.2Detectionofamplitude-squeezedlight142
7.9Generationofsqueezedstates142
7.9.1Squeezedvacuumstates142
7.9.2Amplitude-squeezedlight144
7.10Quantumnoiseinamplifiers146
8Photonnumberstates
151
8.1Operatorsolutionoftheharmonicoscillator151
8.2Thenumberstaterepresentation154
8.3Photonnumberstates156
8.4Coherentstates157
8.5QuantumtheoryofHanburyBrown–Twiss experiments160
IIIAtom–photoninteractions
9Resonantlight–atominteractions 167
9.1Introduction167
9.2Preliminaryconcepts168
9.2.1Thetwo-levelatomapproximation168
9.2.2Coherentsuperpositionstates169
9.2.3Thedensitymatrix171
9.3Thetime-dependentSchr¨odingerequation172
9.4Theweak-fieldlimit:Einstein’s B coefficient174
9.5Thestrong-fieldlimit:Rabioscillations177
9.5.1Basicconcepts177
9.5.2Damping180
9.5.3Experimentalobservationsof Rabioscillations182
9.6TheBlochsphere187
10Atomsincavities 194
10.1Opticalcavities194
10.2Atom–cavitycoupling197
10.3Weakcoupling200
10.3.1Preliminaryconsiderations200
10.3.2Free-spacespontaneousemission201
10.3.3Spontaneousemissioninasingle-mode cavity:thePurcelleffect202
10.3.4Experimentaldemonstrationsof thePurcelleffect204 10.4Strongcoupling206
10.4.1Cavityquantumelectrodynamics206
10.4.2Experimentalobservationsofstrongcoupling209
10.5Applicationsofcavityeffects211
11.2Lasercooling218
11.2.1BasicprinciplesofDopplercooling218
11.2.2Opticalmolasses221
11.2.3Sub-Dopplercooling224
11.2.4Magneto-opticatomtraps226
11.2.5Experimentaltechniquesforlasercooling227
11.2.6Coolingandtrappingofions229
11.3Bose–Einsteincondensation230
11.3.1Bose–Einsteincondensationasaphase transition230
11.3.2MicroscopicdescriptionofBose–Einstein condensation232
11.3.3ExperimentaltechniquesforBose–Einstein condensation233 11.4Atomlasers236
12.1Classicalcryptography243
12.2Basicprinciplesofquantumcryptography245
12.3Quantumkeydistributionaccordingto theBB84protocol249
12.4Systemerrorsandidentityverification253
12.4.1Errorcorrection253
12.4.2Identityverification254
12.5Single-photonsources255
12.6Practicaldemonstrationsofquantumcryptography256
12.6.1Free-spacequantumcryptography257
12.6.2Quantumcryptographyinopticalfibres258
13.1Introduction264
13.2Quantumbits(qubits)267
13.2.1Theconceptofqubits267
13.2.2Blochvectorrepresentationofsinglequbits269
13.2.3Columnvectorrepresentationofqubits270
13.3Quantumlogicgatesandcircuits270
13.3.1Preliminaryconcepts270
13.3.2Single-qubitgates272
13.3.3Two-qubitgates274
13.3.4Practicalimplementationsofqubitoperations275
13.4Decoherenceanderrorcorrection279
13.5Applicationsofquantumcomputers281
13.5.1Deutsch’salgorithm281
13.5.2Grover’salgorithm283
13.5.3Shor’salgorithm286
13.5.4Simulationofquantumsystems287
13.5.5Quantumrepeaters287
13.6Experimentalimplementationsofquantum computation288
13.7Outlook292
14Entangledstatesandquantumteleportation 296
14.1Entangledstates296
14.2Generationofentangledphotonpairs298
14.3Single-photoninterferenceexperiments301
14.4Bell’stheorem304
14.4.1Introduction304
14.4.2Bell’sinequality305
14.4.3ExperimentalconfirmationofBell’stheorem308 14.5Principlesofteleportation310
14.6Experimentaldemonstrationofteleportation313 14.7Discussion316
Appendices
APoissonstatistics
BParametricamplification
B.1Wavepropagationinanonlinearmedium324
B.2Degenerateparametricamplification326
CThedensityofstates
DLow-dimensionalsemiconductorstructures
D.1Quantumconfinement333
D.2Quantumwells335
D.3Quantumdots337
ENuclearmagneticresonance
E.1Basicprinciples339
E.2Therotatingframetransformation341
E.3TheBlochequations344
Listofsymbols Thealphabetonlycontains26letters,andtheuseofthesamesymboltorepresentdifferentquantitiesis unavoidableinabookofthislength.Wheneverthisoccurs,itshouldbeobviousfromthecontextwhich meaningisintended.
ˆ a annihilationoperator
ˆ a† creationoperator
a lengthparameter
a unitvector
a0 Bohrradius
A area
Aij Einstein A coefficient
b unitvector
B magneticfield(fluxdensity)
Bij Einstein B coefficient
B magneticfieldgradient
ci amplitudecoefficient
C capacitance
CV heatcapacityatconstantvolume
d distance;slitwidth
dij nonlinearopticalcoefficienttensor
D diameter
D electricdisplacement
Dp momentumdiffusioncoefficient
E energy
Eg band-gapenergy
EX excitonbindingenergy
E electricfield
E 0 electricfieldamplitude
f frequency
f (T )fractionofcondensedparticles
fij oscillatorstrength
F force;totalangularmomentum
F finesse
FFano Fanofactor
FP Purcellfactor
g degeneracy;nonlinearcoupling
g (E )densityofstatesatenergy E
g (k )statedensityin k -space
g (ω )densityofstatesatangularfrequency ω
gν (ν )spectrallineshapefunction
gω (ω )spectrallineshapefunction
gF hyperfine g -factor
gJ Land´ e g -factor
gN nuclear g -factor
gs electronspin g -factor
g0 atom–cavitycouplingconstant
g (1) (τ )first-ordercorrelationfunction
g (2) (τ )second-ordercorrelationfunction
G gain;Groveroperator
h strain
H magneticfield
ˆ H Hamiltonian
H Hadamardoperator
H perturbation
Hn (x)Hermitepolynomial
ˆ i unitvectoralongthe x-axis
i electricalcurrent
I opticalintensity;nuclearspin
Irot momentofinertia
Is saturationintensity
I nuclearangularmomentum
I identitymatrix
Iz z -componentofnuclearangularmomentum
j currentdensity;angularmomentum(single electron)
ˆ j unitvectoralongthe y -axis
J angularmomentum
k wavevector
k modulusof wavevector;springconstant
ˆ k unitvectoralongthe z -axis
l orbitalangularmomentum(singleelectron)
lz z -componentoforbitalangularmomentum (singleelectron)
L length;meanfreepath
L orbitalangularmomentum
Lc coherencelength
Lw quantumwellthickness
m mass
m0 electronrestmass
m∗ effectivemass
m∗ e electroneffectivemass
mH massofhydrogenatom
M matrix
M magnetization
Mx x-componentofthemagnetization
My y -componentofthemagnetization
Mz z -componentofthemagnetization
n refractiveindex;photonnumber;number ofevents
n2 nonlinearrefractiveindex
no refractiveindexforordinaryray
ne refractiveindexforextraordinaryray
n meanphotonnumber
n(E )thermaloccupancyoflevelatenergy E
nBE (E )Bose–Einsteindistributionfunction
nFD (E )Fermi–Diracdistributionfunction
N numberofatoms,particles,photons, counts,timeintervals,databits
Nstop stoppingnumberofabsorption–emission cycles
ˆ O operator
p momentum;probability
p electricdipolemoment
ˆ p momentumoperator
P pressure;power
P probability
Pij probabilityfor i → j transition
P electricpolarization
q charge;generalizedpositioncoordinate; qubit
Q qualityfactor
r radius;amplitudereflectioncoefficient
r positionvector
ˆ r positionoperator
R reflectivity;netabsorptionrate;electrical resistance
R pumpingrate;countrate
Ri (θ )rotationoperatoraboutCartesianaxis i
s squeezeparameter;saturationparameter
s spinangularmomentum(singleelectron)
sz z -componentofspinangularmomentum (singleelectron)
S Clauser,Horne,Shimony,andHolt parameter
S spinangularmomentum
t time;amplitudetransmissioncoefficient
te expansiontime
T temperature;timeinterval
ˆ T kineticenergyoperator
T timeinterval;transmission
Tc criticaltemperature
Top gateoperationtime
Tosc oscillationperiod
Tp pulseduration
T1 longitudinal(spin–lattice)relaxationtime
T2 transverse(spin–spin)relaxationtime; dephasingtime
u initialvelocity
u(ν )spectralenergydensityatfrequency ν
u(ω )spectralenergydensityatangular frequency ω
U energydensity
ˆ
U unitaryoperator
v velocity
V volume;potentialenergy
ˆ
V perturbation;potentialenergyoperator
Vij perturbationmatrixelement
w Gaussianbeamradius
W countrateintimeinterval T
Wij transitionrate
x positioncoordinate
ˆ x unitvectoralongthe x-axis
ˆ x positioncoordinateoperator
X Xoperator
X1,2 quadraturefield
y positioncoordinate
ˆ y unitvectoralongthe y -axis
Yl,ml sphericalharmonicfunction
z positioncoordinate
ˆ z unitvectoralongthe z -axis
Z atomicnumber;Zoperator;partition function;impedance
α coherentstatecomplexamplitude; dampingcoefficient
β spontaneousemissioncouplingfactor
γ gyromagneticratio;dampingrate;decay rate;linewidth;gaincoefficient
Γ torque
δ frequencydetuning
δ (x)Diracdeltafunction
δij Kroneckerdeltafunction
∆detuninginangularfrequencyunits
ε errorprobability
r relativepermittivity
θ angle;polarangle
Θrotationangle;pulsearea
η quantumefficiency
κ photondecayrate
λ wavelength
λdeB deBroglie wavelength
µ reducedmass;chemicalpotential;mean value
µ magneticdipolemoment
µij dipolemomentfor i → j transition
µR relativemagneticpermeability
ν frequency
νL laserfrequency
νvib vibrationalfrequency
ξ dipoleorientationfactor;opticalloss; emissionprobabilityperunittimeper unitintensity
ρ densitymatrix
ρij elementofdensitymatrix
ρ energydensityofblack-bodyradiation chargedensity
σ standarddeviation;electricalconductivity
σs scatteringcross-section
Listofquantumnumbers τ lifetime
τc coherencetime
τcollision timebetweencollisions
τD detectorresponsetime
τG gravitywavep eriod
τR radiativelifetime
τNR non-radiativelifetime
φ opticalphase
ϕ wave function;opticalphase;azimuthal angle
χ electricsusceptibility;spin wave function
χ(n) nth-ordernonlinearsusceptibility
χ(2) ijk second-ordernonlinearsusceptibility tensor
χM magneticsusceptibility
Φphotonflux; wave function
Ψwave function
ψ wave function
ω angularfrequency
ωL Larmorprecessionangularfrequency
Ωsolidangle;angularfrequency
Ω angularvelocityvector
ΩR Rabiangularfrequency
Inatomicphysics,loweranduppercaselettersrefertoindividualelectronsorwholeatomsrespectively.
F totalangularmomentum(withnuclear spinincluded)
I nuclearspin
j , J totalelectronangularmomentum
l , L orbitalangularmomentum
MF magnetic(z -componentoftotalangular momentumincludinghyperfine interactions)
MI magnetic(z -componentofnuclearspin)
mj , MJ magnetic(z -componentoftotalangular momentum)
ml , ML magnetic(z -componentoforbitalangular momentum)
ms , MS magnetic(z -componentofspinangular momentum)
n principal
s, S spin
Listofabbreviations ACalternatingcurrent
AOSacousto-opticswitch
APDavalanchephotodiode
B92Bennett1992
BB84Bennett–Brassard1984
BBO β-bariumborate
BSbeamsplitter
BSMBell-statemeasurement
CHSHClauser–Horne–Shimony–Holt
CWcontinuous wave
DBRdistributedBraggreflector
DCdirectcurrent
EPREinstein–Podolsky–Rosen
EPRBEinstein–Podolsky–Rosen–Bohm
FWHMfullwidthathalfmaximum
HBTHanburyBrown–Twiss
LDlaserdiode
LEDlight-emittingdiode
LHVlocalhiddenvariables
LIGOlightinterferometergravitational wave observatory
LISAlaserinterferometerspaceantenna
LOlocaloscillator
MBEmolecularbeamepitaxy
MOCVDmetalorganicchemicalvapourepitaxy
NMRnuclearmagneticresonance
PBSpolarizingbeamsplitter
PCPockelscell
PDphotodiode
PMTphotomultipliertube
QEDquantumelectrodynamics
RFradiofrequency
rmsrootmeansquare
SNLshot-noiselevel
SNRsignal-to-noiseratio
SPADsingle-photonavalanchephotodiode
STPstandardtemperatureandpressure
TEMtransmissionelectronmicroscope
VCSELvertical-cavitysurface-emittinglaser
PartI Introductionand background This page intentionally left blank
Introduction 1.1Whatisquantumoptics? Quantumopticsisthesubjectthatdealswithopticalphenomenathat canonlybeexplainedbytreatinglightasastreamofphotonsrather thanaselectromagnetic waves.Inprinciple,thesubjectisasoldas quantumtheoryitself,butinpractice,itisarelativelynewone,and hasreallyonlycometotheforeduringthelastquarterofthetwentieth century.
Intheprogressivedevelopmentofthetheorytolight,threegeneral approachescanbeclearlyidentified,namelythe classical, semiclassical,and quantum theories,assummarizedinTable1.1.Itgoes withoutsayingthatonlythefullyquantumopticalapproachistotally consistentbothwithitselfandwiththefullbodyofexperimentaldata. Nevertheless,itisalsothecasethatsemi-classicaltheoriesarequiteadequateformostpurposes.Forexample,whenthetheoryofabsorptionof lightbyatomsisfirstconsidered,itisusualtoapplyquantummechanics totheatoms,buttreatthelightasaclassicalelectromagnetic wave.
Thequestionthatwereallyhavetoasktodefinethesubjectofquantumopticsiswhetherthereareanyeffectsthatcannotbeexplainedin thesemi-classicalapproach.Itmaycomeasasurprisetothereaderthat therearerelativelyfewsuchphenomena.Indeed,untilabout30years ago,therewereonlyahandfulofeffects—mainlythoserelatedtothe vacuumfieldsuchasspontaneousemissionandtheLambshift—that reallyrequiredaquantummodeloflight.
Letusconsiderjustoneexamplethatseemstorequireaphoton pictureoflight,namelythe photoelectriceffect.Thisdescribesthe ejectionofelectronsfromametalundertheinfluenceoflight. TheexplanationofthephenomenonwasfirstgivenbyEinsteinin1905, whenherealizedthattheatomsmustbeabsorbingenergyfromthelight beaminquantizedpackets.However,carefulanalysishassubsequently shownthattheresultscaninfactbeunderstoodbytreatingonlythe atomsasquantizedobjects,andthelightasaclassicalelectromagnetic wave. Argumentsalongthesamelinecanexplainhowtheindividual pulsesemittedby‘single-photoncounting’detectorsdonotnecessarily implythatlightconsistsofphotons.Inmostcases,theoutputpulsescan infactbeexplainedintermsoftheprobabilisticejectionofanindividual electronfromoneofthequantizedstatesinanatomundertheinfluenceofaclassicallight wave.Thus althoughtheseexperimentspointus towardsthephotonpictureoflight,theydonotgiveconclusiveevidence.
1.1Whatisquantum optics?3
1.2Abriefhistoryof quantumoptics4 1.3Howtousethisbook6
Table1.1 Thethreedifferentapproachesusedtomodeltheinteraction betweenlightandmatter.Inclassical physics,thelightisconceivedaselectromagneticwaves,butinquantum optics,thequantumnatureofthe lightisincludedbytreatingthelight asphotons.
ModelAtomsLight ClassicalHertzianWaves dipoles Semi-classicalQuantizedWaves QuantumQuantizedPhotons
Table1.2 SubtopicsofrecentEuropeanQuantumOpticsConferences
YearTopic 1998Atomcoolingandguiding,laserspectroscopyandsqueezing 1999Quantumopticsinsemiconductormaterials,quantumstructures 2000Experimentaltechnologiesofquantummanipulation 2002Quantumatomoptics:fromquantumsciencetotechnology 2003CavityQEDandquantumfluctuations:fromfundamental conceptstonanotechnology
Source:EuropeanScienceFoundation,http://www.esf.org.
Itwasnotuntilthelate1970sthatthesubjectofquantumopticsas wenowknowitstartedtodevelop.Atthattime,thefirstobservations ofeffectsthatgivedirectevidenceofthephotonnatureoflight,suchas photonantibunching,wereconvincinglydemonstratedinthelaboratory. Sincethen,thescopeofthesubjecthasexpandedenormously,anditnow encompassesmanynewtopicsthatgofarbeyondthestrictstudyoflight itself.ThisisapparentfromTable1.2,whichliststherangeofspecialist topicsselectedforrecentEuropeanQuantumOpticsConferences.Itis inthiswidenedsense,ratherthanthestrictone,thatthesubjectof quantumopticsisunderstoodthroughoutthisbook.
1.2Abriefhistoryofquantumoptics Wecanobtaininsightintothewaythesubjectofquantumopticsfitsinto thewiderpictureofquantumtheorybyrunningthroughabriefhistory ofitsdevelopment.Table1.3summarizessomeofthemostimportant landmarksinthisdevelopment,togetherwithafewrecenthighlights.
Intheearlydevelopmentofoptics,thereweretworivaltheories, namelythecorpusculartheoryproposedbyNewton,andthe wave theoryexpoundedbyhiscontemporary,Huygens.The wave theorywas convincinglyvindicatedbythedouble-slitexperimentofYoungin1801 andbythewaveinterpretationofdiffractionbyFresnelin1815.Itwas thengivenafirmtheoreticalfootingwithMaxwell’sderivationofthe electromagnetic wave equationin1873.Thusbytheendofthenineteenthcentury,thecorpusculartheorywasrelegatedtomerehistorical interest.
Thesituationchangedradicallyin1901withPlanck’shypothesisthat black-bodyradiationisemittedindiscreteenergypacketscalled quanta. Withthissupposition,hewasabletosolvetheultravioletcatastrophe problemthathadbeenpuzzlingphysicistsformanyyears.Fouryears laterin1905,EinsteinappliedPlanck’squantumtheorytoexplainthe photoelectriceffect.Thesepioneeringideaslaidthefoundationsforthe quantumtheoriesoflightandatoms,butinthemselvesdidnotgive directexperimentalevidenceofthequantumnatureofthelight.Asmentionedabove,whattheyactuallyproveisthat something isquantized, withoutdefinitivelyestablishingthatitisthe light thatisquantized.
Table1.3 Selectedlandmarksinthedevelopmentofquantumoptics,includingafewrecenthighlights. Thefinalcolumnpointstotheappropriatechapterofthebookwherethetopicisdeveloped
YearAuthorsDevelopmentChapter 1901PlanckTheoryofblack-bodyradiation5 1905EinsteinExplanationofthephotoelectriceffect5 1909TaylorInterferenceofsinglequanta14 1909EinsteinRadiationfluctuations5 1927DiracQuantumtheoryofradiation8 1956HanburyBrownandTwissIntensityinterferometer6 1963GlauberQuantumstatesoflight8 1972GibbsOpticalRabioscillations9 1977Kimble,Dagenais,andMandelPhotonantibunching6 1981Aspect,Grangier,andRogerViolationsofBell’sinequality14 1985Slusher etal.Squeezedlight7 1987Hong,Ou,andMandelSingle-photoninterferenceexperiments14 1992Bennett,Brassard etal.Experimentalquantumcryptography12 1995Turchette,Kimble etal.Quantumphasegate10,13 1995Anderson,Wieman,Cornell etal.Bose–Einsteincondensationofatoms11 1997Mewes,Ketterle etal.Atomlaser11 1997Bouwmeester etal.,Boschi etal.Quantumteleportationofphotons14 2002Yuan etal.Single-photonlight-emittingdiode6
Thefirstseriousattemptatarealquantumopticsexperimentwas performedbyTaylorin1909.HesetupaYoung’sslitexperiment,and graduallyreducedtheintensityofthelightbeamtosuchanextentthat therewouldonlybeonequantumofenergyintheapparatusatagiven instant.Theresultinginterferencepatternwasrecordedusingaphotographicplatewithaverylongexposuretime.Tohisdisappointment,he foundnonoticeablechangeinthepattern,evenatthelowestintensities.
InthesameyearasTaylor’sexperiment,Einsteinconsideredthe energyfluctuationsofblack-bodyradiation.Indoingso,heshowedthat thediscretenatureoftheradiationenergygaveanextratermproportionaltotheaveragenumberofquanta,therebyanticipatingthemodern theoryofphotonstatistics.
Theformaltheoryofthequantizationoflightcameinthe1920s afterthebirthofquantummechanics.Theword‘photon’wascoined byGilbertLewisin1926,andDiracpublishedhisseminalpaperonthe quantumtheoryofradiationayearlater.Inthefollowingyears,however,themainemphasiswasoncalculatingtheopticalspectraofatoms, andlittleeffortwasinvestedinlookingforquantumeffectsdirectly associatedwiththelightitself.
Themodernsubjectofquantumopticswaseffectivelybornin1956 withtheworkofHanburyBrownandTwiss.Theirexperimentsoncorrelationsbetweenthestarlightintensitiesrecordedontwoseparated detectorsprovokedastormofcontroversy.Itwassubsequentlyshown thattheirresultscouldbeexplainedbytreatingthelightclassicallyand onlyapplyingquantumtheorytothephotodetectionprocess.However,
theirexperimentsarestillconsideredalandmarkinthefieldbecause theywerethefirstseriousattempttomeasurethefluctuationsinthe lightintensityonshorttime-scales.Thisopenedthedoortomoresophisticatedexperimentsonphotonstatisticsthatwouldeventuallyleadto theobservationofopticalphenomenawithnoclassicalexplanation.
Theinventionofthelaserin1960ledtonewinterestinthesubject.It washopedthatthepropertiesofthelaserlightwouldbesubstantially differentfromthoseofconventionalsources,buttheseattemptsagain provednegative.ThefirstcluesofwheretolookforunambiguousquantumopticaleffectsweregivenbyGlauberin1963,whenhedescribed newstatesoflightwhichhavedifferentstatisticalpropertiestothose ofclassicallight.Theexperimentalconfirmationofthesenon-classical propertieswasgivenbyKimble,Dagenais,andMandelin1977when theydemonstratedphotonantibunchingforthefirsttime.Eightyears later,Slusher etal.completedthepicturebysuccessfullygenerating squeezedlightinthelaboratory.
Inrecentyears,thesubjecthasexpandedtoincludetheassociateddisciplinesofquantuminformationprocessingandcontrolledlight–matter interactions.TheworkofAspectandco-workersstartingfrom1981 onwardsmayperhapsbeconceivedasalandmarkinthisrespect.They usedtheentangledphotonsfromanatomiccascadetodemonstrateviolationsofBell’sinequality,therebyemphaticallyshowinghowquantum opticscanbeappliedtootherbranchesofphysics.Sincethen,there hasbeenagrowingnumberofexamplesoftheuseofquantumopticsin everwideningapplications.Someoftherecenthighlightsarelistedin Table1.3.
Thisbriefandincompletesurveyofthedevelopmentofquantum opticsmakesitapparentthatthesubjecthas‘comeofage’inrecent years.Itisnolongeraspecialized,highlyacademicdiscipline,withfew applicationsintherealworld,butathrivingfieldwitheverbroadening horizons.
1.3Howtousethisbook ThestructureofthebookisshownschematicallyinFig.1.1.Thebook hasbeendividedintofourparts:
PartI Introductionandbackgroundmaterial.
PartII Photons.
PartIII Atom–photoninteractions.
PartIV Quantuminformationprocessing.
PartIcontainstheintroductionandthebackgroundinformationthat formsastartingpointfortherestofthebook,whilePartsII–IVcontain thenewmaterialthatisbeingdeveloped.
ThebackgroundmaterialinPartIhasbeenincludedbothforrevision purposesandtofillinanysmallgapsinthepriorknowledgethathas
beenassumed.Afewexercisesareprovidedattheendofeachchapter tohelpwiththerevisionprocess.Thereare,however,twosectionsin Chapter2thatmightneedmorecarefulreading.Thefirstisthediscussionofthefirst-ordercorrelationfunctioninSection2.3,andthesecond istheoverviewofnonlinearopticsinSection2.4.Thesetopicsarenot routinelycoveredinintroductoryopticscourses,anditisrecommended thatreaderswhoareunfamiliarwiththemshouldstudytherelevant sectionsbeforemovingontoPartsII–IV.
Thenewmaterialdevelopedinthebookhasbeenwritteninsucha waythatPartsII–IVaremoreorlessindependentofeachother,and canbestudiedseparately.Atthesametime,thereareinevitablyafew cross-referencesbetweenthedifferentparts,andthemainoneshavebeen indicatedbythearrowsinFig.1.1.AllofthechaptersinPartsII–IV containworkedexamplesandanumberofexercises.Outlinesolutions tosomeoftheseexercisesaregivenatthebackofthebook,together withthenumericalanswersforallofthem.Thebookconcludeswithsix appendices,whichexpandonselectedtopics,andalsopresentabrief summaryofseveralrelatedsubjectsthatareconnectedtothemain themesdevelopedinPartsII–IV.
Fig.1.1 Schematicrepresentationof thedevelopmentofthethemeswithin thebook.Thefiguresinbracketsrefer tothechapternumbers.
2.1Maxwell’sequationsand electromagnetic waves8
2.2Diffractionand interference13
2.3Coherence16
2.4Nonlinearoptics19 Furtherreading24 Exercises24
Classicaloptics Itisappropriatetostartabookonquantumopticswithabriefreview oftheclassicaldescriptionoflight.Thisdescription,whichisbasedon thetheoryofelectromagnetic wavesgovernedbyMaxwell’sequations, isadequatetoexplainthemajorityofopticalphenomenaandformsa verypersuasivebodyofevidenceinitsfavour.Itisforthisreasonthat mostopticstextsaredevelopedintermsof waveandray theory,with onlyabriefmentionofquantumoptics.Thestrategyadoptedinthis bookwillthereforebethatquantumtheorywillbeinvokedonlywhen theclassicalexplanationsareinadequate.
Inthischapterwegiveanoverviewoftheresultsofelectromagnetism andclassicalopticsthatarerelevanttothelaterchaptersofthebook. Itisassumedthatthereaderisalreadyfamiliarwiththesesubjects, andthematerialisonlypresentedinsummaryform.Thechapteralso includesashortoverviewofthesubjectofclassicalnonlinearoptics. Thismaybelessfamiliartosomereaders,andisthereforedevelopedat slightlygreaterlength.AshortbibliographyisprovidedintheFurther Readingsectionforthosereaderswhoareunfamiliarwithanyofthe topicsthataredescribedhere.
2.1Maxwell’sequationsand electromagnetic waves Olderelectromagnetismtextstendto call H themagneticfieldand B either the magneticfluxdensity orthe magneticinduction.However,itis nowcommonpracticetospecifymagneticfieldsinunitsoffluxdensity, namelyTesla.Moreover,itcanbe arguedthat B isthemorefundamentalquantity,sincetheforceexperiencedbyachargewithvelocity v in amagneticfielddependson B through F = q v × B .Amoredetailedexplanationofthedifferencebetween B and H andajustificationfortheuseof B forthemagneticfieldmaybefoundin Brooker(2003, §1.2).Thedistinctionis oflittlepracticalimportanceinoptics, becausethetwoquantitiesareusually linearlyrelatedtoeachotherthrough eqn2.8.
Thetheoryoflightaselectromagnetic waveswasdevelopedbyMaxwell inthesecondhalfofthenineteenthcenturyandisconsideredasoneof thegreattriumphsofclassicalphysics.Inthissectionwegiveasummary ofMaxwell’stheoryandtheresultsthatfollowfromit.
2.1.1Electromagneticfields Maxwell’sequationsareformulatedaroundthetwofundamentalelectromagneticfields:
• the electricfield E ;
• the magneticfield B
Twoothervariablesrelatedtothesefieldsarealsodefined,namelythe electricdisplacement D ,andtheequivalentmagneticquantity H . Sincebothincludetheeffectsofthemedium,wemustbrieflyreview
howwequantifythewaythemediumrespondstothefieldsbefore formulatingtheequationsthathavetobesolved.
Thedielectricresponseofamediumisdeterminedbythe electric polarization P ,whichisdefinedastheelectricdipolemomentper unitvolume.Theelectricdisplacement D isrelatedtotheelectricfield E andtheelectricpolarization P through: D = 0 E + P
Inanisotropicmedium,themicroscopicdipolesalignalongthedirection oftheappliedelectricfield,sothatwecanwrite:
where 0 isthe electricpermittivity offreespace(8 854 × 10 12 Fm 1 inSIunits)and χ isthe electricsusceptibility ofthemedium.By combiningeqns2.1and2.2,wethenfind:
D = 0 r E , (2.3)
where r =1+ χ. (2.4)
r isthe relativepermittivity ofthemedium. Theequivalentofeqn2.1formagneticfieldsis H = 1 µ0 B M , (2.5)
where µ0 isthemagneticpermeabilityofthevacuum(4π × 10 7 Hm 1 in SIunits)and M isthe magnetization ofthemedium,whichisdefined asthemagneticmomentperunitvolume.Inanisotropicmaterial,the magneticsusceptibility χM isdefinedaccordingto:
M = χM H , (2.6) sothateqn2.5canberearrangedtogive: B = µ0 (H + M )
µ0 (1+ χM )H
µ0 µr H , (2.7)
where µr =1+ χM isthe relativemagneticpermeability ofthe medium.Infreespace,where χM =0,thisreducesto:
Inoptics,itisusuallyassumedthatthemagneticdipolesthatcontribute to χM aretooslowtorespond,sothat µr =1.Itisthereforenormalto relate B to H througheqn2.8,andtousetheminterchangeably.
Inanisotropicmaterials,thevalueof χ dependsonthedirectionofthefield relativetotheaxesofthemedium.It isthereforenecessarytouseatensor torepresenttheelectricsusceptibility. Innonlinearmaterials,thepolarization dependsonhigherpowersoftheelectric field.SeeSection2.4.
Magneticmaterialsaretooslowto respondatopticalfrequenciesbecause themagneticresponsetime T1 (see eqnE.21inAppendixE)ismuchlonger thantheperiodofanopticalwave (∼10 15 s).Bycontrast,theelectric susceptibilityisnon-zeroatopticalfrequenciesbecauseitincludesthecontributionsofthedipolesproducedby oscillatingelectrons,whichcaneasily respondonthesetime-scales. B = µ0 H . (2.8)
Theequationforthedisplacementofa wavewithvelocity v propagatinginthe x-directionis:
2.1.2Maxwell’sequations Thelawsthatdescribethecombinedelectricandmagneticresponseofa mediumaresummarizedin Maxwell’sequations ofelectromagnetism:
Equation2.16representsageneralizationofthistoawavethatpropagates inthreedimensions.
where isthefreechargedensity,and j isthefreecurrentdensity.The firstofthesefourequationsisGauss’slawofelectrostatics.Thesecond istheequivalentofGauss’slawformagnetostaticswiththeassumption thatfreemagneticmonopolesdonotexist.Thethirdequationcombines theFaradayandLenzlawsofelectromagneticinduction.Thefourthis astatementofAmpere’slaw,withthesecondtermontheright-hand sidetoaccountforthedisplacementcurrent.
2.1.3Electromagnetic waves Wave-likesolutionstoMaxwell’sequationsarepossiblewithnofree charges( =0)orcurrents(j =0).Toseethis,wesubstitutefor D and H ineqn2.12usingeqns2.3and2.8respectively,giving:
Wethentakethecurlofeqn2.11andeliminate ∇× B usingeqn2.13:
Finally,byusingthevectoridentity
andthefactthat ∇ · E =0(seeeqn2.9with =0and D givenby eqn2.3)weobtainthefinalresult:
Equation2.16describeselectromagnetic waves withaspeed v givenby
Infreespace r =1andthespeed c isgivenby:
Fig.2.1 Theelectricandmagnetic fieldsofanelectromagneticwaveform aright-handedsystem.Part(a)shows thedirectionsofthefieldsinawave polarizedalongthe x-axisandpropagatinginthe z -direction,whilepart (b)showsthespatialvariationofthe fields.
Inadielectricmedium,thespeedisgiveninsteadby:
where n isthe refractiveindex.Itisapparentfromeqn2.19that
whichallowsustorelatetheopticalpropertiesofamediumtoits dielectricproperties.
TheusualsolutionstoMaxwell’sequationsaretransverse waves with theelectricandmagneticfieldsatrightanglestoeachother.Consider awaveof angularfrequency ω propagatinginthe z -directionwiththe electricfieldalongthe x-axis,asshowninFig.2.1.With E y = E z =0 and Bx = Bz =0,theMaxwellequations2.11and2.13reduceto:
Itispossibletofindsolutionsto Maxwell’sequationsthatarenottransverseinsomespecialsituations.One oftheseisthecaseofametalwaveguide.Anotheristhatofamaterial with r =0atsomeparticularfrequency.(SeeExercise2.1.)
Thesehavesolutionsoftheform:
where E x0 isthe amplitude, φ isthe opticalphase,and k isthe wave vector givenby:
λm beingthe wavelengthinsidethemedium.Onsubstitutionofeqn2.22 intoeqn2.21,wefind:
Theelectricandmagneticfieldscan alsobedescribedbycomplexfieldswith E x (z,t)= E x0 ei(kz ωt+φ) , and By (z,t)= By 0 ei(kz
Theuseofcomplexsolutionssimplifies themathematicsandisusedextensivelythroughoutthisbook.Physically measurablequantitiesareobtainedby takingtherealpartofthecomplex wave.Theopticalphase φ isdeterminedbythestartingconditionsofthe sourcethatproducesthelight. E x (z,t)= E x0 cos(kz ωt + φ) By (z,t)= By 0 cos(kz ωt + φ) (2.22)
Theequivalentrelationshipfor Hy 0 is Hy 0 = E x0 /Z, (2.25) where Z isthe waveimp edance: Z = µ0 0 r , (2.26) whichtakesthevalueof377Ωinfreespace.
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