Department ofElectronic Systems Engineering University ofEssex UNIVERSITYPRESS UNIVERSITY PRESS Great Clarendon Street, Oxford ox2 6DP Oxford University Press is aof the It furthers the University's objectives of excellence and education by publishing worldwide in OxfordNew York AthensAucklandBangkokCalcutta Cape TownChennaiDar esIstanbul KarachiKualaMadrid NairobiParisSaoSingaporeWarsaw and associated companies in BerlinIbadan Oxford is a registered trade mark of Oxford in the UK and in certain other countries in the United States by Oxford University PressNew York Oxford University Press, 1973, 1983,2000 The moral rights of the author have been asserted Database right Oxford University Press (maker) First edition 1973 Second edition 1983 Third edition 2000 Ker)rmcted 2001 reserved. 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You must not circulate this book inother hn-.rlln,o-and you must impose this sameon any Arecord forthis book is available from the British Library of Congress Cataloging in Publication Data Data available ISBN 0 19 850177 3(Hbk) ISBN 0 19 850176 5(Pbk) 'l'rn"'\ac.e>f- by the author p..,.,n!-t=rlin Great Britain on add-free paper by Biddies Ltd, Guildford and King's purposeof thebookremainsprovisionof a theory neededanofquantum isintended to bridge thegap between standard '"' ............,_,., ...........L....theorystatistical mechanics,astaught theory needed toquantum thequantumhereisthus accountof the fromattempts mostaremadein ...,"1-'"""A.JL/1..11. ... ..., ..............developmentsintheoverpast18yearsor of parametric down-conversion asa key process ...... ...,.u...., ........ L> .:U. lightnecessitatesnotatreatmentof the "'"' ""'..............thatphotonpair L>"-1 ............., .."" .............light.Again, interference properties. on lecture courses givenauthor to final-graduateto more advanced post-Previousthehave been widely levels of course. This newedition isalso designed asa monograph,treatments of the theory ..."',.."''riderivations(see 'UQtntl'A.mOptics forthe moremathematics).Wellover throughthetexttoencouragestudentstousethe toalltheoriginal sourcesof quantumbut,instead,referencesare the author findsclear andasaids to under-verywellservedbymoreadvancedmonographs, a comprehensive listingattime of writing, Resource letter:CQ0-1: Coherence in quantum (1997)) . .................... J1. ...........is restricted in various other ways but, most impor-of theelectromagneticradiationforwhichthe Preface author owesagreatforhelpfuladvicetomanycolleagues,including Babiker,,.... ..... ,, .......... ,.....G.Barton,Blow,M.J.Collett,J.H. Eberly,vanJ.Fraile,Grangier, M.Harris, N.Imoto, E . .......""............ .Jl. .......... ,0. Jedrkiewicz, J.Jeffers, P.L. Knight, V.Lembessis, G.Leuchs, F. De Martini,S .J.D.Phoenix,E.A.Power,T .J. Shepherd,Squire,Walther,J.P.Woerdman,andR.G. Woolley.toS.M.Barnett,whoread through the entire manuscriptmade anumber of most useful suggestions for the removal of mistakesand clarification of obscurities.Thesupport and encouragement of M ~ r yLoudonproduction of the book are deeply appreciated. Permissiontoreproduceoradapttablesanddiagramswasgenerously granted bythefollowingauthorsandpublishers:1.16,M.Harrisandthe American Institute ofFig.1.18, K.A.H.van Leeuwen and the American Instituteof Physics;3.4and3.5,A.Squirethesis;Fig.5.6,M.G. Raymerof America;5. 9,Kimbleandthe Society of5.14,J.MlynekNature,copyright (1997) Macmillan Magazineswith help fromBreitenbach; Figs.5.18and 5.19, Grangier andSciences; Fig.6.2, W.Martienssen andSciences; Fig. 6.3,MandeltheInstitute of Physics; Fig.8.6, S.Ezekiel and the American Institute of Physics; Fig.8.9,H.Walther and Elsevier Science, copy-right(1997),helpLange;Fig.8.11,Blatt,unpublisheddata; Fig.8.ClausertheAmericanInstituteof Physics;Fig.8.17,M. Gavrila andtheAmericanInstituteof Physics;Fig.9.3,Simaan,unpub-lished calculation:Figs.and 9.6,theOptical Society of America;Figs.9.12 and 9.13, Institute of Physics Publishing. 1 2 3 1.1Density of field modes in a cavity 1.2Quantizationfield energy 1.3Planck's 1Fluctuations in photon number 1.5Einstein's A and B coefficients 1.6Characteristicsthe three Einstein transitions 1.7Optical excitationtwo-level atoms 1.8Theory of optical attenuation 1.9Population inversion:optical amplification 1.10The laser 1.11Radiation pressure References 1Time-dependent quantum mechanics 2.2Form of the interaction Hamiltonian 2.3ExpressionsEinstein coefficients 2.4The Dirac delta-function and Fermi's golden rule 2.5Radiativeand linear susceptibility 2.6Doppler broadening and composite lineshape 2.7The optical Bloch equations 2.8Power broadening 2.9Collision broadening 2.10Bloch equations and rate equations References 3.1Models of chaotic light sources 3.2The lossless optical beam-splitter 3.3The Mach-Zehnder interferometer 3.4Degree of first -order coherence 3.5Interferenceand frequencyspectra 3.6of chaotic light 1 3 4 7 10 13 16 19 23 27 31 35 40 44 46 46 49 52 57 60 65 68 72 76 79 81 82 83 88 91 94 100 103 vn1Contents 4 6 Degreesecond-order ,...""t"'""'r'"'"r'""' 3.8The Brown-Twiss -ant,,.r+,:.rnTY'It::.t""'r Semiclassical theory of '-J'LJ\1.." ...., ....'"~...... ~................. 'V'"'" References 5.Single-mode Number states 5.3states Chaotic light The squeezed vacuum Squeezedstates Beam-splitter Single-photon Arbitrary single-arm N onclassicallight 1 ContentsIX Quantum theorydirect detection271 6.11Homodyne278 6.The electromagnetic vacuum284 References286 7 288 7.1Single-moderate equations289 7.2Solutions foratomic populations292 7.3Single-modetheory297 7.4Fluctuationslight304 7.5Travelling-wave attenuation310 7.6Travelling-wave amplification319 7.7Dynamics ofatom-radiation systen1324 7.8The source-field expression328 7.9Emission by aatom331 References337 8339 8.1The scattering cross-section340 8.2Resonance fluorescence344 8.3Weak incident348 8.4Single-atom resonance fluorescence352 8.5Quantum jumps360 8.6Two-photon cascade emission365 8.7Kramers-Heisenberg formula371 8.8Elastic374 8.9Inelastic378 381 9383 383 389 393 398 404 9.61 absorption417 Conclusion425 426 429 The use of theword'photon'todescribe the quantum of electromagnetic radia-tioncanlead toconfusionmisunderstanding.It isoftenused inthe context of interferenceexperiments,forexampleYoung'sslits,suchphrasesas 'whichslitdoesthephotonpassthrough?'and'wheredothephotonshitthe screenwhenone of theslitsiscovered up'.impressionisgivenof a fuzzy globule of light that travels thiswayor that way through pieces of optical equip-ment or that light beamsconsistsof streamsof theglobules,like bulletsfroma machine gun.Lambhasevenargued that thereisnosuch thingasa photon[ 1] and he has proposed thatword should be used only under licence by properly qualified people! It is, however, difficult to disagree with some of his concerns. Nevertheless, the word isextremely convenient and its avoidance often leads tolengthycircumlocutions.Theadoptionof thephotonbythequantum-optics communityiswidespreadandthepresentbookfollowscurrentusage,with sometimesimprecisestatementsthat couldamount tomisuseof theword.The intentionof thisIntroductionistolimit thedamagethatmight otherwiseoccur bybriefly explaining theconcept of the photonasused in the text.It should be mentioned that theword itself wasinvented byLewis[2],witha meaning quite differentfromthatadoptedinsubsequentwork.Therelevanthistoryiswell reviewed by Lamb [ 1]. The idea of the photon ismost easily expressed foranelectromagnetic field confinedinsideaclosedopticalresonator,orperfectly-reflectingcavity.The field excitations are then limited to an infinite discrete set of spatial modes deter-minedbytheboundaryconditionsatthe~ a v i t ywalls.Theallowedstanding-wavespatial variations ofelectromagnetic fieldinthe cavityare identical in theclassicalandquantumtheoriesbut thetimedependencesof eachmodeare governed byclassicaland quantu1nharmonic-oscillator equations,respectively. Unlikeitsclassicalcounterpart,aquantum harmonicoscillatorof angularfre-quencym can only be excited byinteger multiplesoflim, the integersnbeing eigenvalues of the oscillator number operator. A single spatial mode whose asso-ciated harmonicoscillator isinitsnth excited state unambiguouslycontainsn photons.Each photon hasa more-or-lessuniform spatialdistributionwithinthe cavity,proportional to the square modulus of the complex fieldamplitude of the mode function.The single-mode photons are thus delocalized. Theseideasareoftenextended toopenopticalsystems,withapparatusof finitesizebut noidentifiable cavity.Thediscretestanding-wavemodesof the closed cavity are replaced by discrete travelling-wave modes that propagate from sources to detectors.The excitation of one photona single travelling mode is frequentlyconsidered indiscussion of interference experiments, for example Young'sslitsor theMach-Zehnder interferometer.Eachspatial 1nodein these systemsincludesinputlight waves,both pathsthroughthe interferometer,and ' I quantumof1900 when[foundthat he could account for measurementsthe spectral distributionthe electromagnetic energy radiated by a thermal source by postulating that the energy of a harmonic oscillator is quantized. That is, aoscillator of angular frequencyOJcan have only energiesthatareintegermultiplesof thefundamentalquantumlim,where li = hj2rcandhisPlanck's original constant.Einstein[2]showed how the photoelectric effect could be explained byhypothesisa corpuscularity of electromagnetic radiation.quantum of radiationwasnamed a photon much 1926. ThePlanck and Einstein stimulated much of the early ""tJ ... A ......,,, .....of quantum mechanics. Anothermainstreamtheinitialformulationof thequantumtheorywas concernedof atomicspectrallines.Theinteractionof electromagnetic radiationatoms wasdiscussedEinstein[4]in1917.His theoryof theabsorptionemissionof light byanatomdependsuponsimple phenomenological considerations but it leads to predictionsare reproduced by the more formal quantum mechanics developed later. The use of quantum theory is,course, essential for even a gross descriptionthe nature of the states of an atom andtheory isoutstandingly successful inability topredict the finest details of atomic energy-level structures. The use of the quantum theory is not, however, essentialthe description of manyofpropertiesof visiblelight.Theformalquantizationof theelectro-magnetic field was performed by Dirac [5], who showed in 1927 that the wavelike propertiesof thefieldcouldbepreservedinconjunctionwiththeconceptsof creationanddestructionof photons,but classicaltheoriescontinued toprovide adequate interpretations ofobserved properties of light beams. Thus, in1909, soon after the early contributions of Planck and Einstein, Taylor [6]failed to find any changes from the classical fringesof a Young interferometer when the light source wasso feeble thatone photon at a time waspresent in the apparatus. Higher-order experimentsontheinterference betweenopticalintensitiesrather than amplitudes, performed in1956 by Hanbury Brown and Twiss [7], could also be explained in terms of classical electromagnetic theory. The photoelectric effect itself was shown [8]to be well described by the so-called semiclassical theory, in which the atomic part of the experimental system is treated by quantum theory but classical theory is used forradiation. Thequantum theoryof light,or the fullquantum theory,inwhichquantum mechanics isapplied both to the radiation and to the atoms with which it interacts, cameintoitsownafterinventionof thelaser byMaiman[9]in1960.The 4 measurements ...... .._...,..., ............u.vu.It provides v ~ . ~...... ...,,......experiments, manyquantum-optical manifestations of The applications of ""v1n""""'1rn'"""f"coften involve ..., ....... ;""' .... . ~ ~ . j...... """" ............of chaptersaredevoted to ............... ..., ....procedure or provide c1a.sstca1 more general concernsof thepresent .........J..............., .....and the basic a semiclassical theory .....................,..........aseriesof ..... J> ..,,.., ..... opticalenergy toa set of detectors. passingand Nevertheless,results size,shape and nature Density offield modes in a cavity5 theassumed cavity,andcan be applied tomoregeneral unconfined optical systems.Thismodelisadoptedfortheearlier partof thebook,andtheoften-hypothetical cavity is not removed until Chapter 6. Planck's law expressesspectral distribution of electromagnetic radiation in the interior of a cavityequilibrium at temperatureTheradiationis called black-body radiation; its frequencydistribution is the same asthat radiated byaperfectly-black bodyattemperatureThecalculationof thefrequency distribution breaks up into two distinct parts. We first consider the spatial depen-denceof thefieldinthecavityandderiveanexpressionforthenumber of its different modes of excitation.the second part, we consider the time dependence of the fieldand calculateenergy carried by each mode at temperatureT.To make the calculations assimple aspossible,we choose a cubic cavity of sideL, withaxesdefinedasin1.1.The wallsof thecavityareassumed perfectly conducting, and the tangential components ofelectric fieldt)must accord-ingly vanish at the boundaries. The first part of the calculation is entirely classical and we show in Chapter 4 thatthespatialdependenceof theelectromagneticfieldisnotaffectedbythe quantization. The electricin empty space must satisfy the wave equation n2E() = _!_ d2E(r,t) vr,t 22 , cdt wherecis the velocitytogether withMaxwell equation V .. E(r,t) = 0. The solution that satisfiesboundary conditionscomponents Ex(r,t) =Ex(t)cos(kxx)sin(kyy)sin(kzz) Ey(r,t) = Ey(t)sin(kxx)cos(kyy)sin(kzz) E2(r,t) = E2(t)sin(kxx)sin(kyy)cos(kzz), z ----L-----y Fig.1..Geometry of the optical cavity. (1.1.1) (1.1.2) (1.1.3) 6Planck's radiation wavevectork (. 1,2, 3, ....(.1 =0.(. wavevectorska zero ..,.._,,....... IU',.., .......... . ~.... ~ . . . , . Quantization of the field energy7 andthe polarizationlldefinesaspatialmodeof theradiationfield.Any excitation fieldinthe cavity can be expressed asalinear sumthecontributionsthesefieldtnodes.needanexpressionforthe number ofthe magnitude of their wavevector between the valueskis justnumber of lattice pointsintheoctant of a spherical shell boundedradiikandk + dk.kvery much larger that the lattice constantref L, the ... number is So of (1.1.7) account oftwo polarizations. modes,defined as...... u....... U' ........of modesper unit specifiedis obtained from (1.1.7)as ( 1.1.8) naturethe cavity used isrelated toitswavevector in (JJ=(1.1.9) of (1.1.10) modes . ._,.._..,.JI..I..U.A.-...'-..O.'V.I.H.>can be

),(1.1.1 kA-=1, 2 ...,11-' ............................. ..., ........... ...,......... ....,of the electromagnetic calculation determinestheamount of energy stored The second stage each fieldmode at 8Planck's radiation law and Anequation for isobtainedsubstitution ),withthe use of(1.1 a2E(t)/ ot2 =-m2E(t). isa simple-harmonic-motion =E(O)e -imt' isa constant vector arbitrary magnitude According to classical is given byintegral - 1 f t)2+ -z cavity c =(EoJlo)-112_ t)are equation X s'-4 .................... .._, ......... ..,..... field energy given by ( 1.2.2)canhave ......... ..., .............oscillatorequation( 1 mechanically ratherclassically only the discrete values =(n +t)nm, n =1,3, .... accordingly assign these mode asrepresented =(n+t)nm. coefficients theelectric into (1.2.1) whoseisas (1.2.2) can in (1.2.3) ofspace are (1 vectors,related (1 (1.2.6) to (1 Quantization of the field energy9 n En 5 inw II.Photon creation 3i nw Photondestruction 1 2~tzw ------------- 0 Fig.1.3.The sixenergy levelsa "1..,. .........................hnro.,.......,,....,...,,,'"'oscillator. This quantization condition clearly puts restrictions on the possible magnitudes of the amplitude1.2.2).However, it isnot necessary toconsider these consequencesfieldsasclassical quantities and impose the quantization only onlevel, one must represent the electromagnetic fieldsoperators, as4.4, but for many problems the approachabove is adequate. essence oftheory of thefield isthe association of a quantum harmonic oscillator with each mode offield.energy levels of suchanoscillator areshown in Fig.1.3.When themode energyisgiven by eqn (1.2.7),thecorrespondingoscillatorisinitsnthexcitedstate.Forn = 0the oscillator is in its groundbut a finite amount of energylim/2isstill present 1nfield.Thisisenergy ofoscillatoritssignificance is discussed6.12.most experiments,however,the factorthatgovernsthe observationsisthe degreeexcitation above theground-state.Thenth excited state hasnquanta ofliminaddition to the zero-point energy.The quanta are called photons, and one speaks of nphotonsexcited in the mode of the radiationfield.photonissaidtohavecreated(destroyed)whenthe electromagnetic energy inmode is increased (decreased)a single quantum. Figure1.4 shows thelevels of the harmonic oscillators associated with the firstten field modesorder of increasing frequency.The figuredoesnot show the two independent transversely-polarized modes for each ofoscillators. Planck's radiation law and 01110110 wLfnc~ 2v2 _exp(-EnfkBT) - Lexp(-EnfkBT) n U = exp(-= unl 00un. /n=O 001 un=--1-U' n=O hence 111210201120021102 ~ 5 012 /5 (1.) .2.6)is (1 1 (1 Planck's law11 P(n) = (1- U)Un.(1.3.5) mean number(n)photons excitedthe field mode at temperatureTis therefore (n) ==(1-nn =(1-U)U_j_ Lun =-aun1-U' (1.3.6) and with the use(1 (n) = 1 exp( limfkBT)1 (1.3.7) This important result isthe Planck thermal excitation function.It is plotted in Fig. 1asa function of theor photon frequencym . Theresults(1.1.10)(1.3.7)give,respectively,thenumberof radiation field modes per unit volume whose frequencies lie in the rangem tom + dm,and their mean energy(n)limattemperatureTinexcessof thezero-pointenergy. These expressions combine togive Planck's law forthe mean energydensityof (n) 0 2 3 1iwlk8T Fig.1.5.Mean number(n)of photons of frequencym that are thermally excited at temperatureT. (Wr(m))dm = (n)limp(m)dm = -1 (1 radiative energydensity ........ ,..., ................_frequency range volumeper are-3s.It is r..J..L'V ........... '""asa function of (1.3.9) (.3.10) temperatures. >>) .(1.3.11) 1.5 Fig.1.6.Planck'sat temperature Fluctuations in photon number13 This formof radiationwasderived by Rayleigh[11]in1900, shortly before Planck's formulation of the more general law.Rayleigh's law isthe classical limit obtained when Planck's constantlitendstozero.exponentialineqn(1.3.8) becomesverylargeatlowtemperatures,whereradiativeenergydensityis given by Wien's formula (1.3.12) Both approximate formsPlanck's law break down wheniscomparable to kBT. The total energy density of the photons in the cavity isobtained by integration of eqn(1.3.8)as (1.3.13) whereuseismadeof astandard integral[12].Theproportionalityof thetotal energy tothe fourthpower of the temperature isthe Stefan-Boltzmann radiation law, formulated in1879. An impressive demonstration of Planck's radiation law is obtained from satel-lite measurements[13]ofspectrum of thermal radiation present in space, the so-called cosmic background radiation. The measurements fit the theoretical black-bodyspectrum fora temperatureT= 2. 728 0. 004with uncertainties that are much smaller than the line thickness in Fig.1.6 .. Problem 1.2Derive an expression similar to eqn (1.3.13) but for the number of photonsperunitvolumeexcitedacavityattemperatureT. Show withhelp of another standard integral[12]that the cos-mic background radiation contains about5 x 105 photons per litre. The occurrence of photon absorption and emission processes causes the numbers of photonseach modethe radiation field in the cavity to fluctuate.The fluc-tuations take place on a characteristic time scale that is discussed in detail in 3 .1. However, some average properties of the fluctuations can be deduced without any knowledgeof the time scalesinvolved.We make use of the ergodic propertyof the fluctuationsinthermallight[14],considered furtherin3.3,which ensures that time averages are equivalent toaveragesover a large number of exactly similar systems, each maintained in a fixed state.fictitious collection of iden-ticalsystems iscalled anensemble; the membersof the ensemble are distributed among their various possible states in accordancethe appropriate probability distribution for the system considered. the photons in a particular field mode of a cavity, the ensemble consists of Planck's radiation law and 0.5 0.4 0.3 0.2 (n)= 0.1 P(n) 0.0 0.2 0.1 (n)=5 0.0 n mean 11) + (n) )l+n ( Fluctuations in photon number15 ings obtained from a large number of successive measurements of the number of photonstheselectedmode., It isseenthatn = 0alwayshasthelargest probability of occurrence,P(n)fallsoff monotonically with increasingn,a consequence of the decreasing Boltzmann probability of occupation for the higher levelsof themodeharmonicoscillator.Notethattheprobabilitydistributions show no particular feature atn = (n). The Planck probability functionP(n)issometimes called the thermal distribu-tion or the geometric distribution. It isoften useful to characterize a distribution by its factorial moments. Thefactorial moment is defined as (n(n -l)(n- 2) ... (n- + 1)) =-l)(n- 2) ... (n- r + l)P(n),(1.4.3) n whereris any positiveThe first factorialmoment is the same asthe mean value of the number of excited photons evaluated in eqn (1.3.6). The same method of differentiation is used to obtain the higher factorial moments. Problemthatfactorial momentthe Planck probability distri-bution,eqn.3.5)or (1.4.2),is (n(n- - 2) ... (n- r + 1)) = r!(nt.(1.4.4) The size of the fluctuation in photon ......... J, .........,'"'...is characterized by the variance of the distribution, defined as (L1n)2 =(n- (n) )2 P(n) = (n2)- (n)2.(1.4.5) n second factorial moment obtained from(1gives (n(n -1)) = 2(n)2,( 1.4.6) and hence (1.4.7) = ((n)2 + (n) )112. (1.4.8) The magnitude ofinn,by,isapproximatelyequalto (n)for(n) >> 1.similar equalityof fluctuationtomeanisfoundthe classicaltheoryof 3.6.thefluctuationgivenby(1.4.8)always exceeds the mean value to some extent and the additional contribution is associated naturephotons.wide spreaddistributions is Planck's radiation law and the repeated measurements of nenvisagedderivations are realized IIJ ....................,...,by photocount experiments, 10.Asisdiscussed are . . - . ~ ~ ............ ...-......cavity interesting system n-.cuYI01n1risneeded ...... ..-u .... JL....._,...,...,.in more detail3.9 11-/ ..... ,.., ... v...,...,,""' .......distribution reproduces ....................... .JL.W'-''-"experiments with perfectly measurement ofof photons ashorterthe charac-measurement hasa finite resolving itssizetofluctuationtime (1.5.1) (1 Einstein's A and B coefficients17 limB21 (W(co)) Spontaneous AbsorptionStimulated emissionemission Fig.1.8.three Einstein radiative transitions. andmeasurethefractionitsinitialintensitythatsurvives.Thus,inatypical experiment, the mean energy density of radiation at frequencym consistsof the thermalcontributionbyeqn(1.3.8)plusacontributionfromexternal sourcesof electromagneticradiation,denotedbyasubscriptE,andthetotal energy density is (1.5.3) This quantity isagain the mean radiative energy per unit volume per unit angular frequency range, with unitsJm-3s. The energy density of the externally produced radiationgenerallyvarieswithpositioninthecavityanditdoesnothavethe spatiallyhomogeneousisotropicdistributionof thethermalradiation.For example,it may take the form of a parallel beam of light whose strength fallsoff with distance of propagation, but these spatial characteristics areignored forthe present.It ishoweveranessentialrequirementforthevalidityof theEinstein theorythatthetotalenergydensityisaslowly-varyingfunctionofminthe vicinity of the atomic transition frequency. The probabilities of photonabsorptionand emissionaredefinedasfollows. Consider a single atom in state 2.There is a finite probability that the atom sponta-neouslyfallsintothelowerstate1andemitsaphotonofenergylim.The probability per unit time for occurrence of this process is denoted A21 (spontaneous emission rate).(1.5.4) Now consider an atomstate1.In the absence of any radiation of frequency m there isnowayinwhich the atom can pass intostate 2,asit isimpossible to conserve energy insucha transition.However,inthepresenceof radiation,the upwardstransition1 ~2canproceedbyabsorptionofaphotonlim.The transition isassumed to occur at a rate proportional to the radiative energy density, and the probability per unitfor this process is denoted B12 (W(m))(absorption rate).( 1.5.5) These two processes are intuitively reasonable, but it isnot so obvious that the presence of the radiation should also enhance the rate of transition from the upper state tothe lower state.However,weshall show that such anenhancement must Planck's radiation law (W(m)) absorptions populations, whose rates of ............. .II.A,Fo, .... jdt=-population toradiative processes sections.specialcaseis vanish iseasily solved for Some important relations .._,...Jll.''-'-"'.11.'-'ll.J- ofsteady-state external ...""114.-.'"""'""'rilThe solution for mean numbers of atoms in law N1 _g1 exp(- E1fksT)_ N2 - g2 exp(- E2fkBT)-two substitutioneqn.5.9) gives Thisexpressionfor = ( (W(m)).(1 (1 eqn (1 1""1'""'"coefficients are no.a,...... ,"" .nl ~ " " ".... ,..,.,,.,......,.",.of thermal ( (1.5.11) energydensityfollows Characteristics of the three Einstein transitions19 Einsteincoefficientsandanapplicationofthe atomic levelHowever, the result must beconsistentlawforthesamephysicalquantity,givenbyeqn (1.3.8). The two expressions are identical attemperaturesTonly if (1.5.12) and (1.5.13) The three Einstein coefficients are thereforeinterrelated,and the transition rates between a pairlevels canbe expressedterms of a single coefficient. It is evidentfrom(1.5.1)thatconsistencytheEinsteintheoryand Planck's law could not be achieved withoutpresence of the stimulated emis-sion process.It isshown10 that thisprocessarisesnaturally inthequantum theory of photon emission. above relations between the Einstein coefficients are derived by consider-ationof acavitythermalequilibrium,wheretheradiativeenergyishomo-geneousisotropicTheyholdgenerallyforanyspatially-isotropic distribution of radiative energy density. However, the external light beams used in experiments do not usuallythis property, asexample of a parallel light beam.Nevertheless, the(1.5.12)and (1.5.13)continue toapply for the interaction of light with agasor fluidof atomsor moleculeswhose orientations are randomly distributed. The interaction of the radiation with the gas asa whole is then isotropic, even though the interaction with a single atom or molecule may be anisotropic.orientation-averagedBcoefficientsare independent of the geo-metry of the light beam used in an experiment. These remarks do not necessarily cover the interaction of light withmatter in the solid state,where the constituent atomsor moleculesmaylocked ina common orientation togiveanisotropic optical properties. We treat here only atoms or molecules ina gasor fluid where optical isotropy prevails. Finally,wenoteanotherwayof expressingthermalphotonprobability distribution that is easily obtained from eqns (1.3.2),(1.3.5)and (1.5.10)as (1.5.14) Thedistributionisthusexpressed entirelyintermsof quantitiesrelatedtothe atomic energy levels. A wide variety of radiative processes can be treated by solution of the rate equation ( 11) sum ((n)+l),(1 at z0.7( (1 ( ( Characteristics of the three Einstein transitions21 with photon energy lim::::;3 x 10-19 J(1.6.8) as typical values in therange. The absorption and stimulated emission associated withbeams from the externalsourcescan,of course,makeimportantcontributionstotheradiative processesthatoccurwithvisiblelight.It isinterestingto considertherelative sizes of the two emission rates that occur for the light sources available in practice. Wedefineasaturationradiativeenergydensity,socalled forreasonsthatare clarified in the following section, as (1.6.9) It follows from( 1 and the saturation energy density is that which equalizes the rates of stimulated and spontaneousemission.Forvisiblefrequencyineqn(1.6.7),thesaturation radiative energyunitin a frequency rangedmis (1.6.11) strengthof alight source isusuallyexpressed intermsof itsintensity, obtainedbymultiplicationtheenergydensitythevelocityof light.The corresponding saturationdensity is defined as (1.6.12) andnumericalofsaturation intensity for the frequencygiven in eqn (1.6.7)is (1.6.13) Thusforthe exampleof a conventionallight sourcewithanangular frequency bandwidth6 x 1010 s-1,beam intensity is (1.6.14) The strongest availablelight sourcesintensities that are at least an ordermagnitude 1JJUU..................than this saturation intensity.such light beams, almost allof the photonsare absorbed by the atomsarereradiated byspon-taneousemission.Lasersourcescan easily have intensitiesthat exceed the Planck's radiation ..:.... u.uu:LU.IL."''-'emission probability 1 (11 Optical excitation of two-level atoms.23 causesascatteringofandaconsequentattenuationthebeam.Such scatteringisthemicroscopicsourceof theapparentabsorptionof alight beam during its passage through an atomic gas. Atomicexcitedstatesvisiblefrequencyhavenegligiblethermal populations at room temperature. We consider atoms that have a single low-lying state, the atomic ground state,and excited states at visible or higher frequencies. thelower level inFig.1.8taken to bestate,allNatomsarein this state in thermalSelectivestate can often or atomsan. . . . , . ~...."....... ,, ..........source whose resonancecondition(1.5.1)onlyoneatomic behave essentiallyasaatom with the (1.5.2). Less optical ...........,'-' ............ -... simple, B" " ' - " ~ " ' . . _... . . _ ~ . . _ ~ . . _.... u Bnr..t:.TT11"'101n"I"C" (11) ( ( use (1.7.4) 24Planck's radiation law and the Einstein coefficients 02468 Fig.1. radiativeenergy thesteady-stateasa ( Ws+ (W) )(W) (ws+2(W) )ws (1 (1 ( Optical excitation of two-level atoms25 2.5 2 (1) '""' 1.5 i=l 0 p ..... tl.l 1 '""' 0.5 ---------------Spontaneous 0 01234 {W)/Ws Fig.1.11.Mean ratesEinstein transitions in units of theAcoefficient as functionsof the radiative energy density. Nowconsider thedependencesof themeanatomic populations,which are readily obtained .............., ...of eqn (1.7.1) withuse of eqn (1.5.2). ProblemProve thatgeneral solution of eqn (1.7.1)is (0)- NWs + 2B(W))t] W8+2 W +NWs+(W). W8 +2(W) The solution forN2(t)followsfromeqn(1.5.2). (1.7.8) conditions where allatoms are in their ground states at timet= 0,whenthe external light beam is...... 11. ..............on, the mean numberatoms in their excited states at later times is N(t) = N(W) 2 W8 +2(W) - exp( -(A+ 2B(W) )t ]} .(1.7.9) Thedependence of thestate populationonthetimeisillustrated inFig. 1.12.The population growslinearlyatshort timesbut it approachesthesteady-state valueN2 ineqn(1.7.4)asymptoticallyat long times.Energy is transferred fromlight to the atomsthe beam is firston and the atomic excited-statepopulationgrows.transferof energyceaseswhenthesteadystateis achieved,and the only effect of the interaction between light and atoms isthena redistribution of the propagation directions of the radiation. If the external light beam isnow turned off again,the excited atomsreturn to theirgroundstatesandtheirexcitationenergyisconvertedtoradiationbythe spontaneousemission process.Lett = 0be redefined asthe instant at which the 26Planck's radiation law and 1.0 ~ .........:s0.5 ';; 0 . 0 ~ - - - - - - ~ - - - - - - ~ - - - - - - - - ~ - - - - - - ~ - - - - - - ~ 02345 (A+2B(W))t Fig.1.t. incident beam isremoved. With rate equation1) reduces to dN2jdt ==-N2A, solution decaying atom emits a same exponential sion isan experimental means of ....... ....., .... ..., .............. ,!'-. rocalA , denoted ==ljA, is the fluorescent or radiative lifetime 1.5Derive expressions energy as functions (i)anatom initially energy density (ii)anatom ............ ,..... " ....... .} switched off at Verify by integration over equal but opposite (1 (11) intensity fluorescent emis-.............. ..., ....,.,..... JL Acoefficient.recip-(1.7.12) atomic excitation a total energy transfers are .-rnrooCYJVII1.6The light beamatoms is now repeatedly..................... ..... onand off,withrtheonandoff periods. Themeanexcited atomssettlesa regularIJ'""'"'""" ......... aftermanyon/off cyclesplace.Sketchtheexpected form of this regularshow thatmaximum ..........,............... ... Theory of optical attenuation27 of excited atoms is _(W)1- exp(-(A + 2B(W) )r] N2Ct\nax- N[] . Ws+ 2(W)1- exp-2(A + B(W))r (1.7.13) Derive the limiting forms of this expression forr ~ooandr ~0 and explainin physical terms. results for the atomic population derived above refer to the mean number of atoms in the excited state.Alternatively,N2j Ncan be regarded asthe proba-bility of findinga selected atom in its excited state or asthe fraction of time s p ~ n t intheexcitedstate.Of course,aparticularatommakesrepeatedupwardand downward transitions asit absorbs and emits photons, these transitions occurring at random time intervalsaccordance withthe knownabsorptionand emission probabilities. Observationssingle-atom absorption and emission are considered in8.4 and 8.5.For anN -atom gas, the numbers of atoms in the different levels fluctuate around their mean values even in the steady state but the atomic popula-tion fluctuationsare not treated here. Problem 1.7Anatomtransition frequencygiven by eqn(1.6.7)and radi-ative lifetime10-7 sisilluminated bya light beam whose energy density equals the saturation value. What fractionof the time does the atom spend on average in its excited state in steady-state condi-tions? What are the average numberseach second of (i)absorp-tions,(ii) spontaneous emissions and (iii) stimulated emissions? Theprocessesconsideredtheprecedingsectionassumearadiativeenergy density(W) that isindependent of position. The attenuation of a light beam causes its energy density and intensity to falloff with propagation distance. In the present section we derive expressions for the attenuation based on the Einstein theory of atomic transitions. Weassumption that only the atomic ground state and a single excited state are significantly populated, but the optical intensity isnow a function of the coordinatez parallel to the beam propagation direction. The treat-mentsin thisand the followingsectionassume a travelling-wave light beam,in contrast to the cavity-mode standing waves of previous sections. It is useful to precedemicroscopic derivation with a summary of the stan-dardtheoryof attenuationbasedonthemacroscopicMaxwellequations.The beam intensity is determined by the Poynting vector (1.8.1) Textbooks of electromagnetic theory or optics[15,16]show that the mean inten-sity(/)of a light beam of frequencym varieswithpropagationdistanceinan 28 attenuating dielectric material according to (J(z)) =(J(O))exp[-K(OJ)z], = 20JK(OJ)jc. quantityis...., ......... ...., ...... extinctioncoefficientK( OJ)is...... ..., ...............,..... 1]( m)via the [ 1]( (J))+ i K( OJ) ]2 = (OJ) := + dielectric functione( m)v.U. .I.U'V\..U."-'0 and double primes ..... ........ ".,...,. ...m at nonzero and attenuation (1.8.3). plethora susceptibility e(m) =1 + x(m). susceptibilityX( OJ)provides dielectric andform can be vU..I.vU.I..U.II..'-'U exampleof thecalculationisgiven dielectric functionfollows coefficients (1 (1 (1. (1. measurable refractive index and ... u.........., ...... ...., ...... ......... and imaginary parts of nowconsider thesame ..-. ... r''"'"""='0 ........... .....throughanatomic gas attenuation rate, and hence the ................................ >,....., .....,.ll .... ..., ......., ..... JL theory.gasisassumedtobe..., ............... ....,............ u..... Y itsre>TrIJ.I.V .... . l"'n:annrt:>in beam photonmust use the right-hand side of atomicabsorption dielectric properties "''"'""'1""''""rland the condition ( IJ.I.Jl ..J\I..'U'Jl .. "'arescatteredout sideexpressestherate rateatwhichare twowaysof computing steady state. ........... ...., ...................the rate of attenuation Theory of optical attenuation29 the beam, but the descriptions of the atomic transition and of the light beam must firstbe refined.For theatoms,it isassumedsothattheyallhavethesame sharply-defined transition frequencym.However,asdiscussed in2.5and 2.6, there isalways some statistical spread in the frequenciesof the photons that atoms canabsorbandemit,evenwhenthesamepair of statesisconsidered foreach atom.The distribution of frequenciesis described by a lineshape functionF( m), defined so thatF( m)dmisfractionof transitionsforwhichthephotonfre-quency lies in a small rangedmaroundm.The lineshape functionis normalized such that JdmF(m) = 1.(1.8.7) 0 Some typical functionsare illustrated in Fig.2.4.For thelight beam, the energy density(W)in the attenuating gas is now a function not only of the frequencym butalsoof thepropagationcoordinatez.It followsfromeqn(1.7.4)thatthe atomic populationsarealsofunctionsofz,althoughthethermalmotionof the atoms in a gas tends to homogenize them. With these preliminaries, the net rate of change of the beam energy density at coordinatez isobtainedthe right-hand side of eqn (1.8.6) as d ~ ~ )=Nl- N2 F(m)B(W)Iim, (1.8.8) whereVis the sample to 'V.!I.U.Jl.ILL'VThe relation is converted by the use of eqn ( 1. 7.4) a(w)w --- s()F( m)B(W)Iim, dtws+2wv ( 1.8.9) where the saturationdensity is defined by eqn (1.6.9). The attenuation coefficient of eqns(1.8.2)and (1.8.3), however,governs the spatial rateof changeofmeanintensitydensity(/),whereastheEinstein theoryprovides eqn ( 1forthe timedependence of the mean energydensity (W). The two densities are related by simple scaling with the velocity of light (/) = c(W),(1.8.10) asin the relation (1.6.12) between the saturation densities.derivatives of the densitiesarerelated by consideration of the energy loss in athinslab of the gas perpendicular to the propagation direction. The equivalence of the representations of the loss in terms of therate of change of (W)and the spatial rate of change of (/) gives the relation J(W)/ dt = J(I)/ dz. (1.8.11) 30Planck's radiationand the Einstein coefficients Conversionof eqn( 1.8.9)tothe................... ...,. ... ..energydensity after some rearrangement, to _1 (1 + 2(I)J a( I)= -K(m) (I)Is' ( 11 = NF(m)BiimjVc.(1.8.1 1.8that12) is In( (I(z)) J+ 2 = (1 (I(O)) that (I(O))- (I(z)) ~(11 foris its (I(z)) = (I(O))(1 00 J 0 v' (1 Population inversion: optical amplification31 (I(z))- (/(0)) = -!IsK(m)z = -(NF(m)Ahmj2V)z,(1.8.18) whereeqns( 1.6.12)and.8.13)have been used.beam intensitythusfalls linearlywithpropagationdistanceataratedeterminedbytheAcoefficient, instead of the more rapid exponential fall-off determined by theBcoefficient for low intensities. The decreaseattenuation rate as the atomic transition experiences saturation is caused by the approachthe spontaneous emission rate to the maximum value of A/2, shownFig.1.11.Further increasesinbeam intensitydonot pro- . duce proportionate increasesthe scattering of energy out of the beam,and the fractional change(/)traversing the gas decreases. The gas is itself said to be saturable;it becomes progressively more transparent with increasing intensity,a phenomenon known as bleaching. The macroscopic theory of attenuation based on Maxwell'sequationsdoesnotapplyinsaturationconditions.Saturationmust obviously be avoided in measurements of the attenuation coefficient if the results are used to determine thecoefficient by eqn (1.8.17). numberN2 of ...... . ~....................,.....larger thanthenumberN1 of (1.8.8)thebeam intensityatomstheirground..::"-"'"'-"..::1 growswithdistanceIJV.I.Jl\.A.VJ, .. ,.,...,issimilartoeqn 16)....... . ~ . . _....... ..._,LJ ........ .........., ..... , ........ ~ ~ . .. -v ....~ " ' " " ' ' " ' T T ' ' ~ " ' ~ " " " "isnowcondition inversion,or sometimes asa negative-tempera-...., ....." ..........u.............s law( 110)impliesT < 0. '""... k',rl>'lncannotoccuranditis 1.it is not achievedthe Einstein theory bya two-levelatom.populationinver-of energy levels inthat use additional U.J..U.Il.!J ... ....,U ..process makes usethree statesand the theory 1 ,_,_.._,,_.",_.'VIJ ...L'VIU.J!.ril!:'1n1111C 2.1, .....,.,.-t .... velements they are related we set it isrealfortransitionsn-..""i"''"'""" .....li"'b,..,,,..,.ra..,............... .,.

wavefunctions.calculations xl2 taken as real quantities. The basic equations(2.1.11)are now simplified with the use results and notation of 11"\IC)1AQil"r1t"'Q11'"'""'to o/ cos( rot)exp( -iro0t )C2 = i(2.2.1 Form of the interaction Hamiltonian51 and (2.2.12) These equations are exact within the restriction toa pair of atomic states. Problem 2.1Considertransition from thelS ground state of atomic hydro-gen tothe excited 2Pxstate. Using the standard normalized wave-functionsthese states[1,2], prove that (2.2.13) The transition frequency is m0 = 3mRf4,(2.2.14) (2.2.15) Show that a fieldstrengthE0 of about2.5 x 1011 vm-1,corres-ponding toa light-beam intensity of order1020 wm-2,isneeded to makeVequal tom0. For transitions in the visible region of the spectrum and for light beams of normal-ly available intensities q; A1nf"1"'Arlr'lrlr'IArf':li-A emission ismen-in2.8. linear susceptibilityis coefficientsinto(2.5 .5). Bcoefficient, o/orE0.Integration of side leadstoasolution for toatom, subjected tothe ...,J!.,_. the radiation.It is thereforeo.n1l"\rnnr1o.1-,:> (2.5.8),the 1 {exp[ i( w0 + w )t] C2(t)=-z1l.+-...:::.._... ____;___;::. Wo+ W-1 Ysp .... '"',..._._._ ..... JL.U.J ......J,,.., ....condition of a term of ordero/2 .to =1. Radiative broadening and linear susceptibility63 whereeqn(2.2.1 0)beenused.Thedipolemoment of thesingleatom must now be related to the polarization of the gas. The random atomic orientations are taken into account bysame replacement (2.3.19) asmade in the calculation of theBcoefficient. Thend(t)given bythe modified eqn (2.5.14) represents the averaged dipole momentatom at timetand the macroscopic polarization of the gas is P(t) = Nd(t)jV,(2.5.15) where the justification for simple multiplication byNisdiscussedin8.1.The linear susceptibility obtained by comparison with eqn (2.5.2) is (2.5.16) where eqns(2.3.21) and (2.5.11)are used. The susceptibility at frequency- m is given by a similar expression that satisfies xc -m) = x* cm).(2.5 .17) The attenuationof thegasisrelatedtothe imaginary part of the susceptibility by K( m) = mx"( m)/cTJ( m),(2.5 .18) (1.8.3)to(1.8.5).We consider a gasthat issufficientlydilute indexto be close to unity atfrequencies.It is clear that the bracket of eqn (2.5.16) greatly exceeds the second term when misclosetom0 andYspisverymuchsmallerthanm0.Wetherefore neglect thesecond term,makingoncemoretherotating-waveapproxi-'fi""n some (2.6.5)are m can be neglected .... Jl ..'V.F.""" ......."".. m =mo~mo (1 + vlz ) . 1-(v1zfc)c widthof is thus ()112 = 2m0 2ksTin2/Mc2 Itisconvenientton.o1r1""" ........,,., ..... .,_,.uas ( 2)-1/2 =2n:L1 property dmF0(m) = 1. isthesamefunctionas z side termonright at 1 1 eqn Doppler broadening and composite lineshape67 (2.4.7). The quantityL12 isvariance of the Gaussian distribution, with (2.6.13) TheGaussianlineshapeillustratedinFig.2.4togetherwiththeLorentzian lineshape. Both lines are plotted with the same FWHM, that is28 = 2 y,and they enclose equalareas.It isseen that theGaussian line isthe more sharply peaked and it fallsoff very rapidlyaway fromthe central region. In contrast, the Loren-line has tails thatsome way from the region of the Problem 2. 8Determinesame multiplication factor for the Gaussian lineshape as considered in Problem 2. 7 for the Lorentzian lineshape. The Gaussianoften occurs for inhomogeneous broadening mecha-nisms, where different atoms absorb or emit light at different frequencies because a statistical spreadsome parameter that determines the transition frequency. The atomic velocity isrelevant statistical parameter in Doppler broadening. The atoms that absorb at a particular frequency are, in principle, distinguishable from the rest by their component of velocity parallel to the absorbed light beam. Another example occurs for absorption by atoms embeddedcrystals, where variations in the local strain can produce shifts in atomic transition frequency. Whentheabsorptionfrequenciesaresignificantlyaffectedbytwoor more line-broadening processes,it isnecessarytodeterminethe resultingcomposite lineshape.Considerthecombinationof twoline-broadeningmechanismsthat individuallygenerate normalized lineshape functionsF1 (m)andF2 (ro).The composite lineshape is 00 F(m) =JdvF1(v)F2(m+ m0- v),(2.6.14) wherero0 is the common central frequency of the two distributions. In words, the integration associateseach frequency component in the shapeF1 abroadened distribution appropriate tomechanism that generates the shapeF2.The integral is invariant under interchange of F1 andF2. Any number of line-broadening mechanisms can be combined by repeated applicationsof eqn(2.6.14)andthefinalshapeisindependentof theorder in which the contributions are combined. Specific examples of combined lineshapes are easier to evaluate intemporal domain instead of the frequencydomain of eqn(2.6.14),andthetopicisreconsideredin3.5.However,wegive herethe combination of the Lorentzian lineshape of eqn (2.5 .20)with theGaussian line-shape function of eqn (2.6.11), where andwisaformof complexerrorfunction[5].Thereisnosimpleanalytic expression for the linewidth intermediate between the .L.JVJlV.llJ.'t."'-'.....,..... limitsL.\0andr.:IV.ll'.... t..ll ....,.....is approximate rotating-wave .... Pt-'A'"'''"................. ....nowlook coefficients todirectly .:>IPmPT"'"tcof the atomic off-diagonalor diagonalelements,or/./"-''"'""'"'""'""''-"' satisfy Pu + P22= 1. average .... u,"' .. ...,.., .. ..:> .......... ,, ... ...,...,.after Voigt "-""Y.:>L:>.IlU..UHnt!sn:ape;s,toit""""'"'''"''"'"' calculations in 2.3are that contains a smooth atomic transition. o/orEoare ...(2.7.1) (2.7.2) aregenerallycomplex eqn1.9) ................. 'V .............elements in eqn eqn (2.3.17). Equations of motion for--------_1 .a..u.u.'-L.ll.n..PIPn-lt=>-nt"care easily found. (2.7.4) substitution from eqns (2.2.1it followsthat dp22fdt = -dpllfdt = -i o/ - exp( -im0t )p21}(2. 7 .5) Theoptical Bloch equations69 and (2.7.6) These are exact equationsthe density matrix elements. The solution of the equations is greatly facilitated if the rotating-wave approxi-mation is made. With the same justification as explained in connection with eqns (2.3.4)and (2.3.7),the effects of the terms that oscillate at frequencym0 + m are negligible compared to the effects of the terms that oscillate at frequencym0 - m, whenmisclosetom0.Removalof therapidly-oscillatingtermsfromeqns (2.7 .5)and (2.7 .6) produces the equations of motion and dp22/dt = -dp11/dt = -i v{ exp[i( m0- m)t ]p12- exp[ -i( m0- m)t )p21} (2.7.7) (2.7.8) These areknownastheopticalBlochequations.Theyare similar toequations derived by Bloch to describe the motion of a spin in an oscillatory magnetic field. The quantum mechanics of the two-level atom considered here is formally identical tothat of aspin systemthere are many analogiesbetween the influences of oscillatory fields on the two systems [6]. The optical Bloch equations can be solved without further approximation but, before doing so, it is convenient to remove the oscillatory factors by means of the substitutions (2.7.9) Eqns(2.7.7)and (2.7.8)become dp22/dt = -dpll/dt = -i o/(p12-jJ21) (2.7.10) and (2.7.11) wherePu= Pu,the tildes on the diagonal elements being introduced for notational uniformity. We take trial solutionsthe form Pij(t) =Pij(O)exp(/tt).(2.7.12) fourequationsthat result fromsubstitutionintoeqns(2.7.10)and(2.7.11) are written in matrix form as Quantum mechanics of the 0 0 = 1 0 X + =0. == = + 1 =0=01 = N N 1.0 !Cl_0.5 = The optical Bloch equations71 o/t ._... ,,,,_ .. " excitationj522 forthe ratios _ . _ . ~ . . ~...... ,_._.......... ....,"""againstcurves. +cos(t.at)}. (2.7.20) atombyeqn of thedetuning under ....,., .. , ........ ..,............... ,u.. probability is (2.7.19).... orn,r-.oo (2.7.21) occur in oscillation (2.7 .22) (2.7.23) 72Quantum 1nechanics of the relationis satisfied bytheexplicit ...,. ....,. ..............., .... ..., remove These changes are considered (2.7 .9).It isclearly states abovesolutionsare a single frequencyof p12 can be observed issmallcomparison .......... "' .............frequenciesm0. atomic dependencesofPij ..... t- elements. necessaryto ".,..,.,..,, .........."'.,"emission . ..:>IJ'U'.I.J. ... I.IU.l'V'VU..:>OYnif-o>ns-nr.and the ratios of rrii,.,.T ... r.o.r1nTbeam, determinant are now 2 3n 4nSn Power broadening75 1.0 4 5 Fig.2. 7.Time dependences of the degree of atomic excitationj522 foraweak incidentbeamandtheof detuninglmo- mltoradiativedampingYsp indicated against the curves. and the solution for the 13...,.,...state population with the initial conditions of eqn (2.7.18)is _rv2/4 P22(t)= 2 2 {1+exp(-2Y8pt)-2cos((m0 -m)t]exp(-Yspt)}. (mo-m)+ Ysp (2.8.12) Figure 2.7shows the time dependence of the atomicin the weak-beam limit forseveralvaluesdetuning.resultgeneralizeseqn(2.3.9),to which it reducesof zeroYsp. Allof thecurvesin2.5,2.6and2.7showthesameinitialquadratic dependenceontimeforTHTHT Dopplerandpowercollisionprocess.The....,.._,,,_..., __,._._ ........... .Juuu ....,Jland collision broadening, typical features of a vAJI.C..&.VII-..1."-' theory of collision consider just enough mechanism and its incorporation collisionsbetweentheatoms theoryof gases[12], between collisions Jl. ......, ..... u."" p( r)dr = (1jr0 )exp(-rjr0)dr, the mean periodr0 lS =0.existence areconsequences equations. The atom is no 1isa two-levelatom. their application to has so far covered vV.I..li.L!JJL'Vii-'-'U.byatreatment of experimentsare 1 .,..,. ......""'is used3.1to........ "-""--&""' ... study[11] thenature equations. The occurrence .............,'-' ......process.Accordingto anatom hasa period rangertor+is 1) (2.9.2) Collision broadening77 Heredis the distance between the centres of the atoms during a collision andM is the atomic mass. The effect of a collision on the atomic energy levels and wavefunctions is quite complicated.The energylevelsareshifted duringcollisionby the forcesof interactionbetweenthecollidingatoms,andthewavefunctionbecomessome linear combination of the wavefunctions of the unperturbed atom. It is permissible to ignore any absorption or emission of light that occurs within the duration of the collision, if this is sufficiently brief. The collisions then influence optical proces-ses only via the changesatomic states from those preceding to those follow-ingthecollision.Weassumeinwhatfollowsthecollisiondurationsare indeed very short compared tothe average timer0 of theperiodsof freeflight during which the atomic states remain unchanged. Two categories of collision can be distinguished.Inelastic collisions cause a change inthestate of the atom fromone energyleveltoanother.Their effect is represented by an additional decay rate for the atomic level populations and they appear in the optical Bloch equations via an appropriate addition to the radiative decay rate. Elastic collisions leave the atom in the same energy level as before and their effect is limited to changes in the phase of the atomic wavefunction. Changes in the phases of clandc2in eqn (2.1.8)affect the off-diagonal elements of the atomic density matrix in eqn (2. 7 .2)but the diagonalelements of eqn (2. 7.1)are unaffected. The elastic phase-interruption collisions thusintroduce anadditional decayrate intotheoff-diagonalopticalBloch equation(2.8.2)but thediagonal equation (2.8.1) is unchanged. We restrict attention tothe elastic variety of colli-sion,which has the dominant line-broadening effect for a wide range of physical conditions. The elastic off-diagonal decay rate is denotedYcoll'anditisshownin3.5 that Ycoll=lf 'io (2.9.3) The factor of 2 difference between this and the analogous relation (2.5 .11) for the radiative broadening should be noted.With collisions included,the off-diagonal optical Bloch equation (2.8.2) becomes dp12/dt =dp;l/dt =1i o/(P11- P22) + [i( roo-m)- rJP12, (2.9.4) where Y = Ysp+Ycon The expressions(2.8.3) are generalized to and (2.9.5) (2.8.4)forthesteady-state density-matrix elements (2.9.6) 78Quantum mechanics is straightforward to (2.8.6)is = 2rrNc3 Ysp( mo-m+ iy) mJv(m0 - m)2 ++ ( r j2 Ysp )'112 CTP11!.::>ra ... .._LJLL.Lt-Jeqn (2.8.7). of the subsequent ........ .., .. . . , ~ . . , " '.... broadening is .... ..,r-,, ...... 2-J,,..., ... ..... = ::::3X 1 s. 1 Bloch equations and rate equations79 Problem 2.10ProveDoppler and collisional contributions to the linewidth of antransitionareequalatagasdensityforwhichthe volume per atom isclose to/ld2,whereAistheopticalwave-length oftransition. We conclude the chaptera more detailed discussion of the relation between optical Bloch equations and the simpler rate equations for the atomic popula-tions.The basicrateequation(1.5.7)describesa rangeof opticalprocesses.It involves only the atomic populationsN1 andN2, equivalent tojj11 andjj22 in the Blochequations(2.8.1)(2.9.4).Thecoherences,oroff-diagonaldensity-matrix elementsjj12 and, are not included in the rate-equation (1.5.7). The connections between the rate equations andoptical Bloch equations are most easily studied in the limit of low beam intensity,where thesolution (1.7.9) of the rate equations gives N2(t) = (N(W)/W8){1- exp(-At)}.(2.10.1) the Bloch equations, solutionof eqns(2.8.1) population as thesame initialconditions(2.7 .18)asbefore,the (2.9.4)tolowest order ino/gives the excited-state -1 nJ2{Y/Ysp[(2Ysp- r)/Ysp ]exp(-2Yspt) P22 (t) = 1I;"v 2 2 + 2 2 - 2exp(-yt) x (roo- ro)+ Y(ro0- ro)+(2Ysp- r) [( ro0- ro )2 + r(2Ysp- r)]cos[{ ro0- ro )t] + 2{ ro0 -ro )( r- Ysp )sin[( ro0- ro )t]) [( ro0- ro)2 + r ][(roo- ro)2 + (2Ysp- rn. interest for an (2.10.2) collision broadening,wherey =Ysp,thisreducestothe result off-diagonal density-matrix elementjj21 is not of ....... ..,. .......... "...,. ... ~......but it isneeded8.3tolowest order ino/ is j521 (t) = .('T/jz)(exp{-[i( ro0- ro) + r]t} -1) 1roo-ro+ Y (2.10.3) (0) exp{ -[i( ro0 - + '=111""\1""\1""A'r1rn,-:lt"t:>solutions are valid wheno/2 > ). (2.10.5) (2. Arbitrary-band light - (>>2Ysp) optical l-hn.cf",:otnrate equations. References81 (2.10.7) The excited-state populationeqn (2.1 0.5) isthus effectively determined bythe competition between anexcitation or absorption process of rateRand radiative decayattherate2 Ysp.equivalenceofopticalBloch equationand rate equation solutions to order'])2 in this second regime, shown schematically on the right of Fig.2.8, holds for any bandwidth of the incident light. Theabovediscussionisbased onthesolutionof theBloch equationsin the presence of collision broadening,but similar rate-equationlimitscanbe estab-lished more generally when Doppler broadening isalso present [13]. Thus the rate equationsare valid in generalwhen either (i)the bandwidth of the incident light exceeds the atomic transition linewidth or (ii)the combined collision and Doppler line width greatly exceeds the radiative linewidth of the transition. The rate equa-tions involve only the atomic populations,while the optical Bloch equations also contain the atomic coherences. [1]Atkins,P.W.Friedman,R.S.,MolecularQuantumMechanics,3rd edn (Oxford University Press, Oxford,1997). [2]Merzbacher, E.,Quantum Mechanics,3rd(Wiley, New York,1998). [3]Dwight,of Integrals and Other Mathematical Data, 4th edn (Macmillan, New York,1961). [4]Landau,L.D.andLifshitz,E.M.,StatisticalPhysics,part1,3rdedn (Pergamon Press,Oxford,1980)29. [5]Abramowitz, M.Stegun, I.A., Handbook of Mathematical Functions (Dover Publications, New York,1965) chap.7. [6]Allen,andEberly,J.H.,OpticalResonanceandTwo-LevelAtoms (Dover Publications, New York,1987). [7]Rabi,I.I.,Space quantization in a gyrating magnetic field,Phys.Rev.51, 652-4 (1937). [8]Ackerhalt, J.R., Resonant laser excitation in a two-level model:a nonlinear Schrodinger equation?,Opt.Lett.6,136-8 (1981). [9]Barnett,S.M.andRadmore,P.M.,MethodsinTheoreticalQuantum Optics (Clarendon Press, Oxford,1997). [10]Butcher,P.N.andCotter,D.,TheElementsofNonlinearOptics (Cambridge University Press, Cambridge,1990). [11]Corney,A.,Atomicand LaserSpectroscopy(ClarendonPress,Oxford, 1977). [12]See 39 of ref [4]. [13]Smith, R.A., Excitation of transitions between atomic or molecular energy levelsbymonochromaticlaser radiation,Proc.R.Soc.Lond.A362,1-12(1978):(1978);368,163-75(1979). ..,..,.,..,,....,..,.,,,...transition . ...., ... ,..... . ~ . " " " " '.....light generated principle be .................. '"" ....... ""''""" isspectroscopy, and thusprovides........... v . ~ .... ,u. ........... .., ..... of theemission velocities and the r... n'""'n same gtven Models of chaotic light sources83 tiesof laserlightareonlymentionedinthepresentchapter,adetailed treatment being deferred to7.3and 7.4. The nature oftemporal fluctuations of chaotic light ismost easily appreciated by considering a model source inwhich collision broadening predominates. We ignore radiative and Doppler broadening forpresent and we suppose that the collisionsareelasticphase-interruptionvarietythat doesnot changethe atomic state. Consider a particularatom radiating light of frequencym0.Atrainof electromagneticradiationemanatessteadilyfromtheatomuntilitsuffersa collision. During a collision, the energy levels of the atom are shifted by amounts that depend on theseverityof the collision,and the radiated wave train isinter-ruptedforitsduration.thewaveof frequencym0 isresumedafterthe collision, its characteristics are identical to those that it had prior to the collision, except that the phase of the wave is unrelated to the phase before the collision. the duration ofcollision is sufficiently brief,it is permissible toignore anyradiation emittedthe collision, while the frequency is shifted fromm0. The collision-broadeningcan then be adequately represented by a model in whicheachexcitedatomalwaysradiatesatfrequencyro0,butwithrandom changesinthephaseofradiatedwaveeachtimeacollisionoccurs.The apparent spreadthefrequenciesarises because the wave is chopped into finite sections whose Fourier decompositions include frequenciesother thanro0. The wavetrainradiated bya singleatomisillustrated schematically inFig. 3.1,whichshowsthevariationof theelectricfieldamplitudeE(t)atafixed E(t) Fig.3 .1.The electric-field amplitude of the wave train radiated by a single atom. The vertical lines represent collisions separated by periods of freeflight with the mean durationr0 indicated. The quantitym0 r0 is chosen unrealistically small in order to show the random phase changes caused by the collisions. Classical theory of optical fluctuationscoherence z9X.. 1) wavebyanatomonaverage oscillation between successive VVJ......... ..... LJ. wave3.1lS exp{ -im0t + i 1), + + coherence .. +exp(itpv(t) t) 1) + Intensity fluctuations of chaotic light105 to a very good approximation. Note that this expression results from the pair term ineqn(3.6.3). The variancecycle-averaged intensity is (3.6.6) The size of fluctuation,characterized by, isthus equal to the mean valuel , as isqualitativelyevident inFigs.3.4 and3.5.Asimilar resultwasfoundineqns (1.4.7)or (1.4.8) for the fluctuationsin the number of thermally excited photons in a single cavity mode for(n) >> 1. averagesof powers of l (t)higher than the second are quite complicated ingeneral,but relativelysimpleresultsareobtainedif thenumberof radiating atoms isassumed to belarge. The dominant contribution to the average of the rthpowerof theintensitycomesfromthetermsthatinvolvethesquare moduliof productsof thephasefactorsofrdistinctatoms,asinthesecond-moment calculation in eqn (3.6.3).Thus, approximately, (l(t)r) = (ie0cEJ r..(lr!exp[i( CfJ;(t) + cpj(t) + CfJk(t)+ ... )f) l> j>k> ... (3.6.7) where the summation runsover all the different sets of ratomsand the factor of r!isthenumberof occurrencesof thesumofrphaseanglesshowninthe exponent for a given selection of ratoms. Thenvassumed to be very much larger thanr, (Iut) = r! (v >>r),(3.6.8) where the average intensity istaken from eqn (3.6.1 ). Thisresultforthemomentof theintensityfluctuationdistribution, although derived forthe collision-broadened source,isin factvalid foranykind of chaoticlight.Theoccurrenceof ther!factoreqn(3.6.8)isauniversal characteristic of chaoticThe similarity totherthfactorial moment of the thermal photon distribution in eqn (1.4.4) should be noted. Providedthatthenumberof radiatingatomsisverylarge,itispossibleto calculatenotonlythemoments,asabove,butalsotheexplicitformof the probabilitydistributionthecycle-averagedintensity.Thefirststepisthe determination of the statistical distribution of the values of a(t).It isseenfrom eqn(3.1.3)andFig.3.3a(t)isthedistancefromtheorigininanArgand diagram aftervsteps of unit length in random directions specified by the angles > 1 is Classical theory = (lfrcv)exp(-nature 2n II =1. 1 00 v.11) distribution remains -nrl.o.... an.rt.o1i'"ll'lr Fig.3.11, where I II II l~l ~ E o---7>1 I2! 11.Contours of Degree of second-order coherence107 ility distribution forl (t)theuseof eqn(3 .1Thuswiththeexpression (3.6.1)forl, theprobabilitythataninstantaneousmeasurement of thecycle-averaged intensity yieldsa value betweenl(t)andl(t) + dl(t)isp[l(t)]dl(t), where p[l(t)] = (1/l)exp( -l(t)/l).(3.6.12) Thenormalizationof thedistributioniseasilyverified.Itsformshowsthat the most probable value ofis always zero, and this feature is qualitatively evident from Figs.3.4 and 3.5.the amplitude distribution of eqn (3.6.9),the inten-sity distribution of eqn (3.6.12)isvalid for all kindsof chaotic light, irrespective of its frequencyspectrum.moments of the intensity distribution, given by 00 (l(t)r) = (1/l)J dl(t)l(tt exp(-l(t)/l) = r!lr,(3.6.13) 0 agree with the result in(3.6.8)obtained previously. The random-walk probability distributionreferstoend pointsof a large numberof walksthatallbeginattheorigin,andeachsuchpointdefinesan amplitudeaand a phaseqyfor a light beam. The collection of light beams with all possible amplitudesformsa statistical ensemble of the kind described in 1Eachbeamthe ensemble hasfixedaandqJ,andtheintensity distributionof(3.6.12)isrigorouslyvalidfortheensembleoffixed-amplitude and fixed-phaseIts application to determine time averages of a long series of instantaneous intensity measurements ona single beam of chaotic light relies once more onergodic statistics of the beam. All of the above results refer to idealized experiments that measure the instantaneous intensity. The effects of a finite detector response time, particularly on the observed second moment of the intensity fluctuations,are discussed3.8. contrast toresultschaotic light isprovided bytheclassicalstable wave of Fig. 3.1 0.There isno need to employ statistics in this case, as the cycle-averaged intensity is constant. The result analogous to eqn (3.6.6) is (3.6.14) and there are nointensity fluctuations. Theintensity-fluctuationpropertiesof chaoticlightdescribedintheprevious section refer toaverages of intensity readings taken at single instants of time. We nowconsidertwo-timemeasurementsinwhichmanypairsof readingsof the cycle-averaged intensityaretakenwitha fixedtimedelay1:.Thereadingsare againtakenat a fixedspace and onlya single polarization ismeasured. Classical theory of optical fluctuations and coherence averageofproduct ................... '"'.....of the light, analogous to (3.3.6).The measurement of .IULJL ......, ............. consider the theory It isconvenientto degree ,,,,..,"" ... ""listhelong-time field factorsin aconvention.lightbeamis It is clear from eqn (3.3.12) is g(2) ( -r) = g(2) ( r), calculations need only ments satisfy the conditions have seenthe ............. ,Fo, ...U .............., in the range 0to1. order coherence is controlled by ......, .................. .. numbers.Thustwo measurements satisfy applying this inequality to the cross resultsofNmeasurementsof averages, degreeof second-order satisfies 1::;; g(2)(0). It isnot possible to establish any correlation 1) the measure-must .7.3) it isto show .7.5) zerodelayeqn(3.7.1) .7.6) range of Degree of second-order coherence109 values is (3.7.7) The derivationsofinequalities assume a stationary light beam but they alsoapplytoaseriesmeasurementsonnonstationarylight.For theextreme example of a single opticalpulse,it isclear that measurementsata fixedobser-vation point must produce quite different results that depend on the locations of theat the times of measurement. There is no equivalence between time and statistical averaging inthiscase.However,theNmeasurements of the intensity envisagedin(3.7.4)caninprinciplebemadeatthesametimebutonthe members of anensembleNrealizations of the same optical pulse.The inequa-litiesineqns(3.7.6)and(3.7.7)continue toapplywiththisinterpretation of the averagtng,theycantakenasgeneralpropertiesof allkindsof classical light. aboveproof cannot beextendedtononzerotimedelays,andtheonly restrictionondegreesecond-order coherence then results fromthe essen-tially positive nature of thewhich gives (3.7.8) (3.7.9) which is also readily established with the use of eqn (3.7 .3).The two summations ontherightareequalforsufficientlylongandnumerousseriesof measure-ments, and the square rooteqn (3.7.9) then produces the result (3.7.10) or (3.7.11) degreeof second-order coherence canthereforenever exceed itsvaluefor zero time delay.The inequality again applies both tostationary and nonstationary light beams in the classical theory. Thesegeneralpropertiesof thedegreeof second-order coherenceareillus-trated by the examplechaotic light.suppose, asin the models used in3.4 for both collision and Doppler-broadened light, that the electric field of the plane parallellight beamisupof independent contributionsfromthedifferent radiating atomsi, Classical theory v E(t) =Ei(t).1 i=l second-order1)is (t)E* (t + + = v ++r)Ei(t)) i=l +{ ( E: (t)Ej (t + (t++(t+(t +r)Ej i::f. j 1 those terms are conjugate. of the waves (t++r)E(t)) =(t++(t)) +V(V- + +r)t} + +=+(t+r)t} 1 ~ 11:>> 1 Degree of second-order coherence111 degree of second-order coherence of Gaussian-Lorentzian light is found bysubstitutionofof first-orderfrom(3.4.7)intoeqn (3.7.16)as g(2) ( r) = 1 + exp( -2 yl= 1 + exp(- 21rlfrc)(3.7.19) The corresponding expression for Gaussian-Gaussian light emitted by a Doppler-broadened source isobtained with the use of eqn (3 .4.12) as (3.7.20) These degreesof second-order coherence for chaotic light areillustrated inFig. 3.1It isevident that the generalinequalities(3. 7. 6)and(3. 7.11)aresatisfied and both kinds of chaotichave the limiting values of second-order coherence givenineqns17)(3.7.18). peak insecond-order coherence of chaotic light forr < 'l'cis a manifestationof theof intensityfluctuationshownFigs.3.4 and 3.5. For such small delay times, the two intensity measurements to be correlated in the degreeof second-order coherence often fallwithin the same fluctuationpeak to give an enhanced contribution. For longer delay-times,r > 'l'c,the two intensities tend to be uncorrelateddegree of second-order coherence is close to unity. These propertieschaotic light contrastthose ofclassical wave of stable amplitudeandwhose electric fieldvariationisrepresented byeqn (3.4.20).It iseasilyshown by substitution of the fieldintodefinition of eqn (3.7.1)that g(2) ( -r)= 1' Stable wave -2-1 (3.7.21) 2 having Gaussian l"ru"l""'r.PnrAtime.Thedashedline a classical stable wave. Classical theory not consider isto be second-order .n"'"""" ... ""...,."lr lg(1)(rt 1'1' is ..-..o.a,.-,c.r.., rle>nr-ra.aof 3.7 isindicated by .7 =1. Degree of second-order coherence113 (3.7 .25) The""'"-' ... ,....., .........a"'"''"'caseeqn(3.7.18)does not apply. The degreessecond-order coherence derived abovetomeasurements ona single beam of light.theoryisreadilygeneralized torefer tomeasure-mentson two distinctdenoted bysubscript labelsaandb.Thebeams may differ, for example,polarizations,directions of propagation, or their non-overlappingdistributions, such that they can be distinguished experimentally.Consider measurementsontwobeamsatacommon spatialposition but attimes.Then,analogousto(3.7.1),thereare now four degrees of second-order coherence (2)_( lz (tl )lm (t2))_( E; (tl ) E , ~Ct2 )Em (t2 )Ez(ti)) gzm (tl' t2) - ()()- I*)I*) 'lz(tl)lm)\El (tl)Ez(tl)\Em(t2)Em(t2) l, m =a, a or b, b:intrabeanz degrees of coherence(3. 7 .26) l, m= a, b or b, a:interbeam degrees of coherence. The valuesof thesedegreesof second-order coherence fortwo beams that have stationaryandergodicstatistical propertiesaresubject toa restrictionobtained fromaninequality similar to(3.7.9),but for products of the intensities of beams aandb, as (3.7 .27) The relation can also be derived for statistical averages on an ensemble of pairs of nonstationarylight beams, and theinequality(3. 7 .27)representsanother general classical property of light. Some examples of nonstationary light beams are treated in6.7. simple special case of these degrees of second-order coherence occurs for measurements that do not distinguish between the contributions of the two beams. Problem 3.8Consider the light beam formed bysuperposition of two indepen-dentstationarybeams,labelledaandb,withatotalcycle-averaged intensity (3.7 .28) Showthatoveralldegreeofsecond-ordercoherencefora measurement that does not distinguish the two bemns is 1Classical theory eqn1)is can classicalstablewave, second-order coherence g(r)(rt 1'1 second-order result resultsof =1r,) The Brown-Twiss interferometer115 correlationtwoelectric-field .......... J... .......JL......, .....by theof first -order coherence of the light.of two optical intensities, conveniently expressed in terms of the degree of second-order coherence, was first measured by Hanbury Brown and Twiss[7,8].Theirtypifies all subsequent measurements of degrees of second-order coherence.measurements have a particular significance for the correspondencebetweenclassicalandquantumtheoriesof light,whichis explored5.10.Weconsider the classical theoryof theexperiment asa preliminary tothe more general discussion5.8and 5.9the framework of quantum theory. The apparatus isshown schematically in Fig.3.13.Light from a mercury arc isfiltered to retain only435.8 nm emission line and the beam isdivided into twoequalportionsbyabeam splitter,asspecified byeqn(3.2.13).The intensity of each portion ismeasured by a photomultiplier detector, the principles of whichareoutlined inand11,and theoutputsof thesedetectors,D3 and,aremultipliedtogetherinthecorrelator.Theintegratedvalueof the product over a long periodobservation provides a measurement of the intensity fluctuations.We assumean idealized arrangement in which the detectors are symmetricallyplacedrespecttothebeam splitter.Theythusmeasurethe intensitiesthe beams atlinear distancesz from the light source. Accordingtoclassicaltheory,thebeam splitter dividestheincident cycle-averaged intensity11 (z,t)two identical beamsarms3 and 4 of the experi-ment,where we use the notation of Fig.3.6 forthe input and output arms of the beam splitter.an obvious notation, (3.8.1) and the long-time average intensities in the two outputs are Thus with the degreessecond-order coherence subscripted asin eqn (3. 7 .26) to . - - - - - - - - - - - - - - - - - - - - - - - ~ Outputs to correlator - - - ~ Fig.3.13.Arrangementthe maincomponentsthe Brown-Twiss intensity interference experiment. 1Classical theory }. ( (3.8.5)is ( (2))- 1ITIT gl,l(0)-1- y2 00 (3.8.7) = (r; /2T2){exp( -2Tfrc) -1 + Semiclassical theory of optical detection117 Foraveryshorttime,T(4.3.32) 19)asit is ground-state.... 'U',, .._._,, ..(4.3.32). = -t(at + ... er'l-nt-'"'"''rilinteraction .................... u....... 'V,, ............. ,11-order coherence ability of light at .........._.. ,:;;;,. ...when super-theprevious twosections, toaanalogue of eqn (4.12.4) stationary,asdefinedin3.3,degreeof first-... ,.,.-"',.....,.,..,.form (1)_( :r (t)Ef (t + r) g( r)-1 A ), ( 4.12.5)

16). The designations of the kinds of light given in quantum definition of the degree of first-order theoutput intensity fromeqn similar to eqn (3.3.9),as (4.12.6) whererisdefined witht1 = t2 aseqn(3.3.11). The quantum theory ofBrown-Twiss interferometer parallels the classical theoryof 3.8inasimilar fashion.The experiment measuresthecorrelation of light intensities at two space-time points. With the intensities determined by two photodetectors,thecorrelationisproportionaltotransitionratefora joint absorptionof photonsattwo points.Asimple extension of the photoelectric theory of 4.11showstransition amplitude is proportional to a matrix ele-mentof

),andthetransitionrateinvolvesthesquareof the matrix element, analogous to(4.11.3). These considerations lead to the defini-tion of the quantum-mechanical degree of second-order coherence as g(2)(rt 1'1 t_( E.;:(rJ.tJ)ET(r2,t2)Ef( r2,t2).i:(r1 ,t1)) 2)- (r,tJ)f(rJ,tl))(r(r2,t2)f(r2,t2))' (4.12.7) where the angle-bracket ensemble averages are again evaluated with the use of the density operator similar to(4.12.2). The physical significance of the quantum coherenceissameasof itsclassicalcounterpart,inparticular the conditionsecond-order coherence at two space-time points is the same as eqn (3.7.23). generalthree-dimensional definitionsimplifiesfora plane parallellight beam by the replacement of rbyz, to give _(ftr(zi,ti,ti)) - (.ET(zi,ti)Ef(zi ,tl) )( .ET(z2,t2)Ef(z2,t2))' _ (.ET (t)ET u++ r)Ef (t)) - (.ETc t) .Ei ct)) 2 Jackson, 1999). 1 1 11 References179 [6]Thirunamachandran,MolecularQuantumElectro-dynamics (Academic Press, London,1984). [7]Cohen-Tannoudji,Dupont-Roc,J.andGrynberg,G.,Photonsand Atoms:toQuantumElectrodynamics(Wiley-Interscience, New [8]Babiker,and Loudon,Derivationof thePower-Zienau-Woolley .................u.......electrodynamics by gauge transformation, Proc. 439-60(1983). [9]Power,IntroductoryQuantumElectrodynamics(Longmans, London,1964). [10]Bransden,and Joachain,O.J.,IntroductiontoQuantum Mechanics (Longman, Harlow,1989); Merzbacher, E.,Quantum Mechanics,3rd edn (Wiley,New York,1998). 1]Barnett,S.M.andRadmore,P.M.,MethodsinTheoreticalQuantum Optics (Clarendon Press, Oxford,1997). [12]Power,E.A.and Zienau,S.,Coulombgaugeinnon-relativisticquantum electrodynamicstheshapeofspectrallines,Trans.R.Soc. Lond.A427-54(1959). 3]Woolley, R.G.,Molecular quantum electrodynamics, Proc.R.Soc.Lond. A557-72(1). Shore,andTheJaynes-Cummingsmodel,J.Mod. Opt.1195-238 [15]Scully,and Sargent,The concept of the photon, PhysicsToday 38-47 (March1972). [16]Glauber,Thequantumtheoryofopticalcoherence,Phys.Rev. 2529-39(1963),Coherentandincoherentstatesof theradiation field,Phys.Rev.2766-88( 1963). Single-mode field operators181 The electromagnetic field is assumed to excite atravelling-wave mode with a given wavevectorkand a polarization direction specified by the labelJL.The formsof the electric and magnetic field operators for the given mode in the inter-actionor intheHeisenbergpicture,areobtainedfromeqns (4.4.13)to(4.4.18).particularattentiontotheelectric-fieldoperator because of itselectric-dipole interactionin the definitions ofdegrees of coherence. The direction of propagation isto be thez axis and,withsubscriptskandJLremoved, the scalar electric-field operator for the given direction ofpolarization is written + _ (7;. 12 v)l/2{ A -ix""'ix} - nWcoae+ae,(5.1.1) where the positive and negative frequencyparts offieldoperator correspond explicit dependence onz 12), which simplifies to respectively totwo terms onthe right-hand side. andtisburiedphase angleX defined in = mt- kz- ~ , (5.1.2) wherek = mjcis convenient towrite the electric-field operator in ayetsimplerbyof theawkwardsquare-rootfactorfromeqn (5.1.).Thuswiththeconventionthat theelectric fieldismeasuredunitsof 2( 1imj2co V)112'operator reduces to + (5.1.3) for the selected modeareasdefined ineqns finalform onistheversion of electric-field1.3)representsa u... :--. ..... ....,....fieldmeasuredwhere phaseisphaseoflocaloscillator.hasthe propertythatthe fields......... ""''"""'""" .... . . . , ~atdifferentphaseanglesdonotgenerally commute. Thus withuse of either form of field operator(5.1.3) and the commutation relationsor (4.3.39), it is easilythat ),(5.1.4) 1.4) 1 1 1 .1 1.1 Single-mode field operators183 withvaluesofndifferingbyunity.Thefielduncertainty,controlledbythe relation(5.1.5),represents noise on the optical signal.The magnitude:A[of the noise isconventionallyby and (5.1.11) signal-to-noiseis defined as _(E(X))2 - (L\E(X) )2 . (5.1.12) ....., ...................tophotocount signal-to-noise ratiohomodyne 6.11.It providesameasuretheinformation-carrying formdetection.Thesignal-to-noiseratiois excitations considered in the following sections. It signal varieswith It is empha-formsof signal arenot 1.13) (5 .. 184Single-mode quantum optics degree ..... .n...._, ................ 'V ...must satisfy (n) ~'1.1 =0.) = Number states185 Thedegreeof second-ordercoherenceforthenumberstateIn)thusfollows trivially from eqn (5.1.15) as g(2) ( 'l") = 1- _.!._forn :2::1, n (5.2.3) andithasthevalueallowed bytheinequality(5.1.17).The classical inequality of eqn (3.7.6) for'l"= 0isviolated. The degree of second-order coher-ence for the vacuum number stateI0)isnot determined by the expressions in eqn (5.1.15)asboth their numerators and denominators vanish. The energy eigenvalue relation can be written in a variety of formsfrom eqns (4.3.11),(4.3.26)and(4.3.38), .ifln) =lim( at a+ -!-)In)= lim( x2 + 92)1n) = nm( n +-!-)In).(5.2.4) The number state thus hasquadrature-operator eigenvalue property (X2 +Y2 )In)= (n +-!-)In).(5 .2.5) quadrature-operator expectation values are (niXIn) = (niYin) = 0 (5.2.6) and (5.2.7) ThenumberstatesthusidenticalpropertiesfortheX andY quadrature operators.variancesthe smallest valuesallowed by eqn (4.3 .40)only vacuum state withn = 0 . Thus the stateI0)isanexample of a quadrature state.Figure 5.1showsa qualitativerepresentation of the Fig.5.1.Representationthequadrature-operatorexpectationvaluesforthe photon number state. quadrature properties(5 .2.5)to rp-n,ri">C'Pritthe quadrature variables. (5.2.5). expectationvaluesof determined.coherent LJA1->AA-A (5.2.6), S = (niE(z)ln) = 0, single-modestateis for homodyne detection. 2( ")2 = (L1E(X))= (nlE(X)In)= phase-independent variance ..,. ...... ,, .......... ,...,.., ratio defined smallest value fromeqn(5.1.8) the ....... "' .... ,,..,. .....of the time. expression (5.2.9) shows = (n + -!-)112 in Representation ...u.........................to a number state. of + .......u...... ...,,.., ...state,twoaxes the eigenvalue informationa noise, is '-'-"U.'-.ll'UJLI.(5.1.9) variance noiseislarger thesingle-mode at some fixedasa r.rlrH'">rnfrequencym o1 Number states187 inthefigurebytheinclusionof severalwaves,allof thesameamplitudeand frequency,butwiththeirnodesprogressivelydisplacedalongtheaxis.More accurately, the horizontal positions of the waves form a continuum,and the field at any time can take a continuous range of values between- E0 andE0. Pictorialrepresentationsof quantum-mechanicalstatesmustgenerallybe treatedwithcaution.ThusFig.5.2reproducestheexpectationvaluesgivenin eqns(5.2.8)and (5.2.9)the upper and lower cut-offs of the fielddistribution are shown astoo sharp.figure provides anaccurate representation only in the limitn >> 1. Problem 5.1Evaluateexpectationvalue(nl( E(X) )41n)andshowthatit exceeds the corresponding average for the sine waves of Fig.5.2 by anamount 3/32. As this is independent of n, the importance of the discrepancy diminishes for largen . The fieldvariationofsingle-modenumber stateshowninFig.5.2isin stark contrast with that ofclassical wave of stable amplitude and phase defined byeqn(3.4.20)andillustrated inFig.3.10.Thus,whilethenumber statehasa fairly well defined amplitudeE0 given by eqn (5.2.10), it shows no vestige of the phase angleqJof the classical wave form.Other kindsof quantum fieldexcita-tion,forexample thecoherent stateof 5.3,haveamplitudesthatarelesswell defined but phase anglesare better defined than those of the number state. It is naturaltoenquire how the phase properties of excitationsare represented inthe quantum theory of light. identification of a Hermitian operator that representsphase, in confor-mity with its physical significance, is a subtle problem that requires careful discus-sion[2,3].However, it ispossible to establish the formof the probability distri-butionP( qJ)for the phase angle quite straightforwardly. The classical phaseqJin eqn(3.4.20)cantakeanyvalue,but changesinqJbyintegermultiplesof2n make no difference to the variation of the electric field.In other words, the phase angleqJis defined modulo2 nand it can thus be restricted toany convenient2 n range, for example 0 to2rc.The same featuresof the phase are expected toapply to quantum field excitations. It is therefore appropriate to take 21C f dqJP( qJ)= 1. (5.2.11) 0 asthe normalization condition for the quantum-phase probability-distribution. The stateIn)of well-defined photon number hasa completely random phase and the quantum phase isexpected tobe complementarytothephoton number, similar to the position and momentum in quantum mechanics. Conversely, a state ofwell-definedphaseshouldhaveacompletelyrandomphoton-number distribution. It isconvenient to replace the continuous range of phase angles from 0 to2n byr + 1 equally-spaced discrete valuesq-) )2 - {j>q-) )2+- :!( 10) = 216Single-mode quantum optics The photon number operators _AtA('-123 - ai ail- ,,, from eqn (5.7.2) to "'""'"'""''"' .... ,, ...............................arrangement arm 2 input )2 =11Gate 0.5 a 0 rt.a'n""'-r-.rtt:nf"'t=>on wavevector activates a measurement on experiment measures the ......... JLA ... ...., ....... ._,n3 andin the twoarms accordanceeqn (5.8.8).However, conditionswhere has 1.5 Arbitrary single-arm input221 photon that opens the gate,furthernphotons may enter the apparatus during the time forwhich the gateopen. Figure18shows the normalized correla-tionasa functionofmean number(n)of additional photons received by the gatedetector duringthedetection period,or integration time.It isclear that the output correlation tends towards zero asthe integration time is made sufficiently short that only a singleenters the apparatus. The detailed form of variation of the correlation with(n)isderived ineqn(5.9.6).The experiments essentially confirm the expected vanishing of the quantum Brown-Twiss correlation (5.8.8). Figure19showstheresultsof asecondexperimentwiththesamegated cascade-emission sourcewith the Brown-Twiss interferometer replaced by the Mach-Zehnder.The resultsarebuilt upfromseriesof single-photontneasure-ments with increasing path differencez1 - z2. The fringeshave the Mach-Zehnder formof eqn(5.8.12)asexpected,withahigh98%visibility,andthereisno detectable difference fromclassical fringe pattern of eqn (3.5.3). calculations of the preceding section are here generalized toallow for anarbi-trary input state in arm 1,arm 2 remaining in its vacuum state. The combined input state is denotedlarb)1j0)2.It isstraightforward to repeat the previous calcu-lations,and the simplifications that result from the assumption of a vacuum state inarm 2meansand the variances of the electric fieldsin the output armsof thebeamsplitteraredeterminedbyeqns(5.7.9)and(5.7.10)withthe 20.. 0 0 -. . .. .__ .,.... 2/l -r .... ...... ....2/l (a) . 4/l (b) ..... 4/l Fig.5.19.Mach-Zehnderfringesobservedwithasingle-photoninputasa functionof thedifferencez1 - z2 expressed intermsof thewavelengthIt. Theverticalaxisshowsnumbersof photodetectionsinarm4for(a)a1 s integration time(b)a compilation of 15 such scans.(After [4]) Single-modeoptics =) 1) =+ = I2\.I2ITI21(arblal a1a[-= I2\.I2ITI2 = I2\.I2ITI2 ( n1- 1) ), two. Arbitrary single:..ann input223 counterpart. In particular,quantum Brown-Twiss correlation isobtained from eqn (3.8.5) bysubstitutionthe quantum degree of second-order coherence. The measurement ofBrown-Twiss correlation described at the end of 5.8 was not strictly performedsingle-photon inputs, asthe gate detector general-lyreceived morethanone photonduringthedetectionperiodanda correspon-dinglylarger number of photonsalsoentered theinterferometer.Theadditional photons are produceduncorrelated random emissions by different atoms in the source. Problem 5.16ConsiderBrown-Twissinterferometerwithaninputphoton number1 +, wherenhasaPoissondistributionof mean(n). Show thatmeasured degree of second-order coherence is (5.9.6) Thisof variationdegree of second-order coherence with(n)provides agoodmatchtopointsshownFig.5.1single-mode theory used here is notapplicable to the1 + ninput photons,whichwere notallinthesame modeconditionsof theexperiment[4].However,the same expressioneqn (5.9.6) is found ina more realistic calculation. Mach-Zehnderof(5.8.12) is similarly converted to photon excites state atsecond (5.9.7) source, wasattenuated to It was found ............... ...., ..... '-'was exactly the same ..::>LU.LJ...::>t..J.v0of chaotic Single-mode quantum optics ) lo)-1 (at 12- (n1 !)1/21 IO)= +IO) (n1! J/2 -m -m)4' m=Om!(n1-+ Arbitrary single-arm input225 input photons. Measurements that involve only output arm 4 are controlled by the reduced distribution P4(n4) =P3,4(n3, n3=0 andP3 (n3)isgiven by a similar expression. These distributions are used to evaluate various averages for the output photon numbers. Problem 5.17Prove the averages l(n3(n3-1)) = 12(1:(nJ(nl-1)) (n4(n4-1)) = ltii(nl (nl-1)) and verify(5.9.4).Hence show that Theresults(5.9.1),(5.9.2)and(5.9.11)showthat g(2) ( 't")=g(2) ( r) =g(2) ( r), 3,34,41,1 (5.9.11) (5.9.12) (5.9.13) and thedegreeof second-order coherenceisunchanged fromitsinput value by reflection from,or transmission through,a beam splitter.In thespecial case of a 50:50 beam splitter, eqn (5.9.12) shows that the mean-square difference in output photon number equals the mean input-photon number. Consider single-mode chaotic light asa specific example of beam-splitter input state. The number of photons in input arm 1 isnow distributed inaccordance with the probabilityP1 (n1)obtained from eqn (5.4.1). The output distribution derived in accordance with eqn (5.9.9) is (5.9.14) The reduced distribution for measurements on arm 4 alone, obtained inaccordance with eqn (5.9.10)is (5.9.15) 5. (5.7.3) ) ++= .9.1 1) Nonclassicallight227 (5.9.22) the expectationfor the individual coherent states are asgiven in eqns (5.3.16)(5.3.31)respectively,withsubstitutionof thecomplexamplitudes !](a andforoutput arms. The fringefor a Mach-Zehnder interferometeraninput coherent stateisgivenby(5.9.7),withlal2 substituted forthemeaninput photon number.Theaboveidentification of theoutput fromthefirst beam splitter with independentcoherentstatessuggeststhesimilarfringesshouldoccurinthe superposition of lightfrom independent sources.Such fringesareindeed observedthe fieldby superposition of light from two independent lasers [20,21]. The interference occurs between the probability amplitudes that a detected photon wasbyone source or the other.The interpretation of the experi-ment is the same asofMach-Zehnder interferometer with coherent-state excitation, inthere is noin which a photon can simultaneously contribute tointerference effectsbe assigned to a definite laser source. The'photon' inthis case excites a superposition state similar to(5.8.2),where 3 and 4 now labelthetwolaseroutput beamsand!I(andrrareproportionaltothebeam amplitudes. output,J ..... ,...,.............. .........,. ......... ..., ........distributions forchaotic and coherent input light given byeqns(5.9.15)(5.9.20) have thesame functionalformsasthe input distributions.is,not a general property of transmission througha beam splitter and aisprovided byinput number stateln1 )1, where eqn (5.9.gives (5.9.23) This probability is generally nonzero for allvalues of n4 from0ton1 ,unlike the singlenonzeroinput distribution,theexampleofn1 = 4is showninFig.5.20.of inputanddegreesof second-order coherence, expressed by(5.9.13), remains valid. Beam splitters playimportant rolesinexperimentalstudiesof thesqueezed states described in5.55.6, but considerationthistopic isdeferred to the treatmenthomodyne detection in6.11.The technique usesstates that excite both input armsthesplitter and these are treated in 6.8 (see also[22]). The quantum degrees of second-order coherence of the various single-mode field excitations derived in thechapter are allindependent of the time delay1:, but it isalready possible to make some useful comparisons with the corresponding classicaltheory.Theclassicaldegreeof second-ordercoherencedefinedand discussed in3.7 has magnitudes that lie inthe ranges given by eqns(3.7.7)and (3.7.8)as Single-mode quantwn optics 1.0 (a) 0.5 1- _1_ < g(2)( (n)-soo ~ limit of 0 applies states of range 1--< g(2) (n)-...UAF-,11''"' (1987). Zaheer, Adv.At. vonNeumann, (Princeton University Clausen,J., quantum-state (1999). Rev. Pfleegor, beams, Louradour, encefringes (1993). Campos, lossless A137 Mandel, L.,Sub-Poissonian..,. ...... '"' ..................:H.U.l..i..::IL..l'-'0 Opt.Lett.205-7979). resonance Rev. The single-mode theoryChapter 5isinherentlylimited tothedescriptionof experiments that use time-independent light beams.The light isessentially non-ergodic,inthat itsstatistical