Mesoscopic coherence in light scattering from cold, optically dense and disordered atomic systems

2.2 MB·

by  D. V. Kupriyanov

Mesoscopic coherence in light scattering from cold, optically dense
and disordered atomic systems
D.V.Kupriyanov,I.M.Sokolov
DepartmentofTheoreticalPhysics,St.-PetersburgStatePolytechnicUniversity
195251,St.-Petersburg,Russia
M.D.Havey
DepartmentofPhysics,OldDominionUniversity,Norfolk,VA23529
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1
0
2

n
a Abstract
J
Coherenteffectsmanifestedinlightscatteringfromcold,opticallydenseanddisorderedatomicsystemsarereviewed
1
from a primarily theoretical point of view. Development of the basic theoretical tools is then elaborated through
1
several physical atomic physics based processes which have been at least partly explored experimentally. These

] includeillustrationsdrawnfromthecoherentbackscatteringeffect,randomlasinginatomicgases,quantummemories
h
andlight-atomsinterfaceassistedbythelighttrappingmechanism. Currentunderstandingandchallengesassociated
p
withthetransitiontohighatomicdensitiesandcooperativityinthescatteringprocessisalsodiscussedinsomedetail.
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t
n
Keywords: Lightscattering,coldatoms,coherenceindisorderedsystems,quantuminterface.
a
34.50.Rk,34.80.Qb,42.50.Ct,03.67.Mn
u
q
[
c 2017.ThismanuscriptversionismadeavailableundertheCC-BY-NC-ND4.0License.
1 (cid:13)
v
6 Contents
1
8 1 Introduction 2
2
0 2 Basicapproaches,assumptionsanddefinitions 3
.
1 2.1 Thelong-wavelengthdipoleapproximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
0 2.2 Thelightcorrelationfunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
7 2.3 TheGreen’sfunctionformalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1
2.4 Graphicalimagesofthebasicinteractionsandimportantdiagramblocks . . . . . . . . . . . . . . . . 7
:
v 2.4.1 Generalanalysis: Thefieldsubsystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Xi 2.4.2 Generalanalysis: Theatomicsubsystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4.3 Weakprobe,samplesusceptibilityandthescatteringtensor. . . . . . . . . . . . . . . . . . . 11
r
a 2.4.4 Diagramexpansionofthecorrelationfunction . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.5 Theretarded-typeGreen’sfunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.6 Thelighttransportequationandthediffusionapproximation . . . . . . . . . . . . . . . . . . . . . . 16
2.6.1 Thelighttransportequation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.6.2 Thediffusionapproximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3 Coherentbackscatteringoflight 20
3.1 Observationsspecifictoatomicsystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.2 Polarizationandanisotropyeffects,anti-localization . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.3 Saturationandnon-lineareffectsincoherentbackscattering . . . . . . . . . . . . . . . . . . . . . . . 27
Emailaddress:[email protected](D.V.Kupriyanov)
PreprintsubmittedtoPhysicReports January12,2017
4 DiffusetransportinthegainedandLambda-configuredsystems 32
4.1 Randomlasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.1.1 Lightdiffusioninanamplifyingmediumandtherandomlasingeffect . . . . . . . . . . . . . 32
4.1.2 Ramanamplificationofthetrappedradiation . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.1.3 Randomlasinginainhomogeneousanddisorderedsystemofcoldatoms . . . . . . . . . . . 38
4.1.4 Coherencepropertiesoftherandomlaserradiation . . . . . . . . . . . . . . . . . . . . . . . 40
4.2 Electromagneticallyinducedtransparencywithscatteredlight . . . . . . . . . . . . . . . . . . . . . 41
4.3 Quantummemoryassistedbylighttrappingmechanism. . . . . . . . . . . . . . . . . . . . . . . . . 46
4.3.1 Importanceoftheproblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.3.2 Generaldescriptionofthememoryprotocolbasedonlightdiffusion . . . . . . . . . . . . . . 47
4.3.3 QuantummemoryforthemacroscopicanalogoftheΨ( ) Bellstate . . . . . . . . . . . . . . 49
−
4.3.4 Nanofiber-assistedquantummemory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5 Lightscatteringondenseandcomplexatomicsystems 53
5.1 Overviewoftheproblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.2 Scalarandothersimplifyingmodelsoftheatoms-lightinteraction . . . . . . . . . . . . . . . . . . . 54
5.3 Theself-consistentmacroscopicapproach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.4 Themicroscopicapproach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.4.1 Thetransitionamplitudeandthescatteringcrosssection . . . . . . . . . . . . . . . . . . . . 57
5.4.2 TheresolventoperatorandN-particleGreen’sfunction . . . . . . . . . . . . . . . . . . . . . 58
5.4.3 Theself-energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.4.4 Illustrativeexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
6 Summaryandoutlooks 62
Appendix A Entanglementoftheatom-fieldvariablesinthedipolegauge 64
Appendix B ThebasicGreen’sfunctionsoftheKeldysh’sdiagramapproach 66
Appendix B.1 Fieldsubsystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
Appendix B.2 Atomicsubsystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
Appendix C Matrixelementsofatomicdipoleandmagneticmomentoperators 67
1. Introduction
Cold atomic systems are quite challenging and promising objects for research in quantum optics and atomic
physics. This circumstance grows broadly out of a wide range of applications and fundamental explorations that
can be found in such research areas as atomic and molecular spectroscopy, metrology, quantum simulations, and
quantuminformationscience. Eachofthesesubjectsinitselfisworthyofoneormorecomprehensivereviews. The
optical responses of non-linear coherent controlled media were the subject of intensive discussions a decade ago
[1]. Atpresentthelight-matterinterfaceresultinginquantummemories,quantumentanglement,andquantumnon-
demolitionmeasurementsformsthebackgroundforvariouschallengingimplicationsofthephysicalprotocolsbased
oncoherentlycontrolledandopticallydenseatomicsystems[2,3].Surprisingly,butpracticallyindependentlyofthis,
thevariouscoherentmechanismsaffectinglightdiffusion,withemphasisontheroleofdisorderinlighttransportand
incontextoflocalizationphenomenaandwavedynamicsinrandommediahavebeenintensivelystudiedduringthe
lasttwodecades[4–6].
Inthisreviewwearemotivatedtofollowhowthesefrontierresearchdirectionsstudyingthesamephysicalobjects,
butsometimewithanaimtowardsdifferentgoals,haveanaturaltendencyforintegratingthevariousphysicalideas
and approaches. Indeed, in nearly all aspects research in the above areas depends on interaction of the physical
systemswithelectromagneticradiation,eithertopreparethesystem,tointerrogateit,orinthecaseofseveralmutually
coherentappliedfieldstomanipulatethebarephysicalpropertiesforpossiblepracticalapplications.Evenintheleast
complex case of a single very weak field, or a single photon resonantly probing a gas of two state systems, there
2
aresubtletiestotheprocessesthatcanoccur,thesebeingreflectedinthecollectiveandcooperativeresponsesofthe
system. Forthisreason,wehavebeenmotivatedtoreviewtheessentialtheoreticalandexperimentaltoolsthathave
beenused,ordeveloped,overthepasttwodecadestostudycollectiveandcooperativeeffectsincoldandessentially
atomic gases. In this, although principally focus on theoretical approacheshere, we have at the same time, and in
most cases, stayed close to physical phenomena which have been observed in the laboratory. It is intriguing that
the discussed phenomena may partly have classical precursors but the intrinsic quantum nature of atomic systems
and the interaction process dramatically modify their observable behavior as well as the dependencieson external
parameters. Theseincludethecoherentbackscatteringeffect,randomlasing,manipulationoflightdiffusionthrough
electromagneticallyinducedtransparency,andquantummemoriesbasedondiffusivelighttransport.
Anotherarea ofmotivationcomesfromrecognizingthatan atomicgas(evenif, as in ourdiscussion, notquan-
tum degenerate) at low temperatures, combined with interacting coherent fields, is essential a many body physics
problem;thegeneralopticalpropertiesofaggregatedandmesoscopicallyscaledmatterdiffersignificantlyfromthose
of individualatomic scatterers. In the case of a very dilute gasof atomsthe response of the system to appliedand
weakresonant,ornonresonant,monochromaticelectromagneticradiationcanberelevantlyunderstoodviatheentire
Green’sfunctionformalism. Thiseventuallystatesaself-consistentmacroscopicapproachbasedonMaxwelltheory
andontheconceptofsubsequentmultiplescatteringfromspatialinhomogeneities.
However,uponincreasingtheatomicdensity,thequantumopticsofsuchsystemsmodifiessuchaclassicalvision
anddevelopscorrelatedcharacteristicsintheinteractionprocess. Thisisespeciallytrueincoldandnearlymotionless
atomicgases, wherecoherencesharedbetweenappliedandscattered fields, andthe atomscomprisingthe samples,
cansurviveenvironmentalorthermaldecoherenceforsubstantialperiodsoftime. Wearethenalsointerestedinthe
evolutionofthequantumopticsofacoldatomicgasasthephysicaldensityofthegasincreases,uptothepointwhere
strongatom-atominteractionscandominatethephysics. Thecrucialpeculiarityofthelight-atominteractioninthis
caseisthatboththefarfieldradiativecouplingandnearfieldstaticinteractionmediatethecooperativedynamicsof
anatomic ensemble. Remarkableprogressof the pastdecadesin developingnumericalmethodsand approachesin
quantumelectrodynamicsgivesusatoolforexploitingthemicroscopicsimulationsforlightscatteringfromatomic
samplesofmacroscopicsize. Thelastsectionofthisreviewisdedicatedinparttoaspectsofthissubject.
2. Basicapproaches,assumptionsanddefinitions
2.1. Thelong-wavelengthdipoleapproximation
Mostofthe calculationsofinteractionprocessesofthe electromagneticfieldwith atomicsystems inthe optical
domainare performedin the dipoleapproximation. The dipoleapproximationcanbe rigorouslyintroducedvia the
Power,ZienauandWoolleycanonical/unitarygaugetransformation[7,8]. Forpracticalapplicationsthegeneraland
rather cumbersome formalism can be simplified with the assumption that the atomic size is much smaller than a
typicalwavelengthofthefieldmodesactuallycontributingtotheinteractionprocess. Suchalong-wavelengthdipole
approximation, see [9] for derivation details, leads to the following interaction Hamiltonian for an atomic system
consistingofN dipole-typescatterersinteractingwiththequantizedelectromagneticfield
N
Hˆ = dˆ(a)Eˆ(r ) + Hˆ ,
int a self
−
Xa=1
Hˆ = N 2π e dˆ(a) 2. (2.1)
self s
·
Xa=1 V Xs (cid:16) (cid:17)
Thefirstandmostimportanttermischaracteristicallyinterpretedastheinteractionofana-thatomicdipoled(a) with
theelectric field Eˆ(r) atthepointof thedipolelocation. However,as strictly definedin the dipolegauge,thelatter
quantityperformsthemicroscopicdisplacementfieldanditcanbeexpressedbyastandardexpansioninthebasisof
planewaves s k,e ,whereα = 1,2enumeratestwoorthogonaltransversepolarizationvectorsforeachk,which
kα
≡
weassumehereaslinearandreal
Eˆ(r) Eˆ(+)(r) + Eˆ( )(r)
−
≡
3
2π~ω 1/2
= s! iesaseiksr−iesa†se−iksr
Xs V h i
N
4π
= Eˆ (r)+ es(es dˆ(b))eiks(r−rb). (2.2)
⊥ Xb=1 V Xs ·
Here as and a†s are the annihilation and creation operators for the s-th field mode and the quantization scheme in-
cludesperiodicboundaryconditionsonthequantizationvolume . Thethirdlineshowsthedifferencebetweenthe
actual transverse electric field denoted as Eˆ (r) and the displacVement field. The second term in Eq.(2.1) appears
⊥
as a non-convergingself energy of the dipoles. This term is often omitted in the interaction Hamiltonian since it
doesnotprincipallyaffectthedipoledynamics,particularlywhenthedifferencebetweenthetransverseelectricand
displacementfieldsissmall. Thevectorpotentialoperatorisgivenby
Aˆ(r) Aˆ(+)(r) + Aˆ( )(r)
−
≡
2π~c2 1/2
= ω ! esaseiksr + esa†se−iksr (2.3)
Xs sV h i
andthisoperatorisnotmodifiedbythetransformationfromtheCoulombtothedipolegauge,see[9].
Letusmakea fewremarksconcerningtheapplicabilityofthedipoleapproachandhowitisrelatedtothelevel
of disorder in the atomic system. The difference between the transverse electric and displacement fields becomes
important at distances comparable with the atomic size. For a low density atomic ensemble, when the atoms are
separatedbyaradiationzonesuchthatn Ż3 1,n isthedensityofatoms,Ż = λ/2πandλisthelightwavelength,
0 0
≪
the difference between the above quantities in the interaction operatorcan be safely ignored. In terms of disorder,
classificationofsuchadiluteatomicgasperformsahomogeneousandslightlydisorderedconfigurationofthedipole
scatterers, where the transport length for light diffusion l is much longer than the light bar-wavelength Ż. The
tr
description of the light transport in such a medium can be explicitly described in terms of the Green’s function
formalismandintheself-consistentmacroscopicMaxwelltheoryapproach.
Analternatesituationoccurswhenn Ż3 > 1bringsthesystemtothestronglydisorderedregimedefinedthrough
0
the Ioffe-Regel condition l Ż, which was originally pointed out in [10], and which we simplify here for the
tr
∼
particular case of cold atomic systems. For such a dense configuration the exact definitions (2.1), (2.2) become
important. One should be quite careful with substituting (2.2) to (2.1) and then to any operator equationsbecause
any atomic variable commutes with the operator of the displacement field but not always with the operator of the
transverseelectricfield. Inadensesystemtheself-energytermcannotbeignoredandinsomecalculationalschemes
can manifestitself in the dynamicsof atomic variablesin cooperationwith anothersingularterm, namely,with the
self-contactdipoleinteraction. Thelatterappearswhenr = r = r forinteractionofa specifica-thdipolein(2.1)
a b
withthelongitudinalfieldcreatedbythesamedipoleinthesecondtermofthelastlineinEq.(2.2).
Throughoutourdiscussion,orientedtowardsexistingexperimentalcapabilities,wewillmostlydealwiththedilute
atomicconfigurationsforwhichtheabovedifficultiescanbesmoothedandarenotactuallyimportant. Butwecome
backtotheprobleminthelastpartofthisreviewinSection5,whereweconsiderthesituationwithalargedegreeof
disorder,cooperativityandlocalfieldeffects. InAppendix Awehavealsodiscussedsomefacetsofthedipolegauge
withrespecttoatomicsubsystems.
2.2. Thelightcorrelationfunction
Theintrinsiccharacteristicsoflightassociatedwithitsenergytransferandcoherenceareexpressedbyitscorrela-
tionfunction
D(E)(r,t;r,t) = Eˆ( )(r,t )Eˆ(+)(r,t) . (2.4)
µν ′ ′ ν− ′ ′ µ
D E
HeretheanglebracketsdenotethestatisticalaveragingoftheHeisenbergfieldoperatorsEµ(+)(r,t)andEν(−)(r′,t′),with
frequencyandpolarizationcomponentsgivenby(2.2). Havinginmind,inthissection,mostlyadiluteatomicsystem
withlowdensity,weshallconsiderthiscorrelationfunctionsimilarlydefinedbothoutsideandinsidethemediumand
associatethefieldoperatorswiththetransverseelectricfieldcomponents.Therigorousanddetailedanalysispresented
byGlauberinhisseminalpapers[11]introducesthegeneralsetofcorrelationfunctionsofhigherorders,whichare
4
measurablebycoincidencespectroscopyandbyhomodynedetectiontechniques. Herewerestrictourconsideration
onlytothefirstordercorrelationfunctiondefinedbyEq.(2.4).
Iftheopticalfieldlocallyperformsastationaryplanewavefreelypropagatinginthedirectionnthenthespectral
energydistributedwithitcanbeexpressedbyaFouriertransformoverthetimedifferenceτ=t t
′
−
c
I(n,r,ω) = ∞dτeiωτ D(E)(r,t¯+τ/2;r,t¯ τ/2). (2.5)
2π Z µµ −
µX=x,y −∞
Heret¯isunderstoodasatimeargumentdependencewhichvanishesundersteadystateconditions. Fortimedepen-
dent processes it is more convenient and natural to introduce the Poynting vector associated with the light energy
transportedinthedirectionn. Thiscanbedoneviatracingthecorrelationfunctionoveritspolarizationcomponents
µ= x,yintheplaneorthogonaltonandconsideringitatcoincidentspatialpointsandtimes.
c
I(n,r,t) = D(E)(r,t;r,t). (2.6)
2π µµ
µX=x,y
In some situations, see Section 2.6.1 below, these two representations can be mixed with the Wigner-type density
matrix,whichconsiderstheinstantaneouspartofthefieldenergyfluxforacertainpartofitsspectrumasdistributed
nearthepointrandpropagatinginthedirectionn. TherelevantexpressionforthecombinedquantityI(n,r,ω,t)with
t¯ twouldbegivenbyEq.(2.5),preservingtimedependenceintherighthandside.
→
Even in a dilute but optically dense medium randomly scattering light over all directions, the original incident
planelightwavewill befracturedintomanypackets. Thesefragmentswillpropagateandbere-scatteredbyatoms
moreorlessindependently. Followingsuchanaturalrandomization,thelightpropagationwouldbevisualizedasa
diffusionprocessandatanyspatialpointatacertaintimetherewouldbecontributions(2.6)arrivingfromanydirec-
tions. Thisprocesswoulddevelopincoherentlyandisexpectedtobedescribedbya diffusiontypemasterequation
forthelocallightenergy. Inapplicationsandintheliteraturetherearenumerousexamplesoflighttransportmaster
equations,see[12,13],whichoperatewiththetimedependentlocalintensity.Ageneralfeatureofthediffusivetrans-
portisthattheinitialcoherentpropertiesoflightseemcompletelylostandtheemergingwaveswouldbeunpolarized
andincoherentsuchthatonlytheoutgoingenergyfluxwouldbeaphysicallyimportantcharacteristicforthediffusion
process. However, as we will see, withoutthermallosses this naive pointof view cannotbe exactly true and there
existmanifestationsofthewavenatureandcoherenceinthemultiplescatteringprocess.
2.3. TheGreen’sfunctionformalism
Convenientexpansionsforthecorrelationfunction(2.4)canbesubsequentlybuiltupviaitsperturbationtheory
analysis. Letustransformtheoriginaldefinition(2.4),performedintheHeisenbergrepresentation,totheinteraction
representation
D(E)(r,t;r,t) = T˜S Eˆ(0, )(r,t ) TSEˆ(0,+)(r,t) , (2.7)
µν ′ ′ † ν − ′ ′ µ
D h i h iE
wherethechronologicaloperatorsT andT˜ respectivelyorderandanti-orderintimethefieldandatomicoperatorsin
thebrackets.TheevolutionoperatorS isgivenby
i
S = S( , ) = Texp ∞Hˆ(0)(t)dt . (2.8)
∞ −∞ "−~Z int #
−∞
Hereandbelowwesuperscriptby”0”thetime-dependentoperatorsintheinteractionrepresentation. Theimportant
step in introducing the representation (2.7) is that initial conditions are moved to the infinite past as is typically
assumed in scattering theory. That allows us to approach and specify Eq.(2.7) by the limits t,t where it
′
→ −∞
reproducesthecorrelationfunctionoftheincidentlight.
Furtherperturbationtheoryexpansionof(2.7)canbedoneinthesecond-quantizedformalismintroducedforboth
thefieldandatomicsubsystems. ThenthefirsttermintheinteractionHamiltonian(2.1),expressedbyatomicsecond
quantizedΨ-operators,see[14],andconsideredintheinteractionrepresentationtransformstothefollowingform
Hˆ(0)(t) = d3rdµ Eˆ(0)(r,t)Ψˆ(0) (r,t)Ψˆ(0)(r,t) + H.c., (2.9)
int − Z nm µ n † m
Xn,m
5
where,forthesakeofgenerality,weusecovariantnotationforthetensorindicesinthedipolematrixelementandfor
theelectricfieldcomponents.1 TheΨ-operatorsforthesetofthegroundstatesmaregivenby
1
Ψˆ(m0)(r,t) = √ e~ip·rbpm(t)
Xp V
1
Ψˆ(m0)†(r,t) = √ e−~ip·rb†pm(t) (2.10)
Xp V
Herebpmandb†pmarerespectivelytheannihilationandcreationoperatorsofanatomwithmomentumpintheinternal
Zeemanquantumstate m. The Ψ-operatorsinthe excitedstates ncan besimilarly definedby trivialsubstitutionof
m n. Inthecaseofnearresonantinteractiononecansmooththetimederivationandaccept
→
iω
Eˆ(0,+)(r,t) + 0 Aˆ(0,+)(r,t)
→ c
iω
Eˆ(0, )(r,t) 0 Aˆ(0, )(r,t) (2.11)
− −
→ − c
suchthateachfrequencycomponentoftheelectricfieldoperatorcanbeexpressedbythesimilarcomponentofthe
vectorpotentialoperator.Thefrequencyω shouldbeassociatedherewithanatomicresonancefrequency,whichcan
0
varyforeachparticularexcitationtransition. Also applyingthe standardassumptionsofthe rotatingwave approxi-
mation(RWA)reducestheinteractionHamiltoniantothefollowingform
ω
Hˆ(0)(t) i 0 d3rdµ Aˆ(0,+)(r,t)Ψˆ(0)(r,t)Ψˆ(0)(r,t) + H.c., (2.12)
int ≈ − c Z nm µ n † m
Xn,m
whichkeepsonlycounter-rotatingHeisenbergtermsinthefieldandatomicsubsystems.
Straightforward expansion of the evolution operators creates the operators’ products, which can be always re-
alignedtotheN-orderedform,whenthecreationandannihilationoperatorsaremovedtotheleftandrightsidesof
theaveragingexpressionrespectively. ThereareseveraltypesofGreen’sfunctionscreatedinthisprocedure,which
areassociatedwith thedifferentchronologicalorderinginthetransposedoperators. We willusethefollowingcon-
vention,see[15,16]forfurtherdetails. If,asintheexampleofEq.(2.7),anoperatorcontributesfromtheleftbrackets
(generatedbyexpansionofS andT˜-ordered),weassociateitwiththe”+”signandwealternativelyassociateitwith
†
the” ”signifitcontributesfromtherightbrackets(generatedbyexpansionofS andT-ordered). Finaltransforma-
−
tionof anyoperatorproductto its N-orderedformis knownas Wick’sTheorem[16, 17] and, in an exampleof the
fieldsubsystem,oneobtainsthefollowingsetofGreen’sfunctions
iDµ(σ1µ1σ22)(r1,t1;r2,t2) = Tσ1σ2 Aˆ(µ01)(r1,t1)Aˆ(µ02)(r2,t2) − N Aˆ(µ01)(r1,t1)Aˆ(µ02)(r2,t2)
h i h i
(2.13)
⇒
where σ ,σ equals either + or are the sign indicators showing the original location of the field operators and
1 2
T isthechronologicaloperato−rassociatedwiththislocation. ThatstandsforthatT = T, T = T˜, T isthe
σ1σ2 −− ++ +−
identicaloperatorandT isthetranspositionoperator. Intherotatingwaveapproximationonlypositivefrequency
+
−
componentsoftheGreen’sfunctions(2.13)canappearforanychronologicalordering,andthesecomponentscanbe
specifiedinadiagrambythearrowoneachwavylinedirectedfromoneendassociatedwiththecreationeventtothe
otherendassociatedwiththeannihilationevent.
FortheatomicsubsystemwekeeponlytheGaussiantypecorrelationsforanyproductconsistingofanynumber
of the atomic Ψ-operators. This approximation is often fulfilled because of the macroscopic nature of the atomic
1Weomithereandfurther(whereitisnotconfusingwiththedetailednotations)thesymbolsofinvariantsumsoverthetensorindices.
6
subsystem.ThentheatomicGreen’sfunctionsareexpressedbyallpossiblechronologicalcouplingsbetweencreation
andannihilationoperators
iGm(σ11mσ22)(r1,t1;r2,t2) = Tσ1σ2Ψˆ(m01)(r1,t1)Ψˆ(m02)†(r2,t2)
D E
(2.14)
⇒
ItisimportantthatanytranspositionoftheΨ-operatorsobeystheruleofquantumstatistics,suchthatitshouldchange
thesigninthecaseofFermionicoperators.Thispermutationrulenormallymanifestsitselfforloop-typediagramsby
anextraminussignforeachloopintheFermioniccase. LetusalsopointoutthattheappliedGaussianfactorization
isarelevantapproximationonlyforanon-degeneratequantumgasanditfailsinthecaseofquantumdegeneracy,for
example,inthecaseoftheBose-Einsteincondensate.
The exact analytical expressions of the complete set of the original unperturbed Green’s functions (2.13) and
(2.14)forthefieldandatomicsubsystemsaregiveninAppendix B.
2.4. Graphicalimagesofthebasicinteractionsandimportantdiagramblocks
2.4.1. Generalanalysis: Thefieldsubsystem
Therearetwotypesofinteractiondiagramsbetweenlightandatoms.Ifpartofthelightisoriginallyinaclassical
mode,beingasuperpositionoftheeigenstatesofthefieldannihilationoperators(2.3),thenitgeneratesthefollowing
interactiondiagrams
i
dµ (+)(r,t)
± ~ nmEµ ⇔
i
dµ ( )(r,t) (2.15)
±~ mnEµ− ⇔
Theuppergraphdescribestheexcitationtransitionfromthegroundstate m to theexcitedstate n initiatedbythe
positive frequency component of the classical field (+)(r,t). The lower|griaph respectively desc|riibes the induced
µ
E
transitionfromtheexcitedtothegroundstateinitiatedbythenegativefrequencycomponent (µ−)(r,t). Inthosecases
E
whenaclassicalfieldisappliedonemptytransitions,whichcanbeinvolvedintheexcitationprocesswithotherfields,
wewillrefertoitasacontrolmode.
Similarprocessesinitiatedbythequantizedfieldareexpressedbythediagrams
ω
0 dµ
∓ ~c nm ⇔
ω
0 dµ (2.16)
±~c mn ⇔
Inthisdiagramtheshortwavylinecanbeanedgefragmentofthepropagationfunction,givenby(2.13),butcanbe
alsoassociatedwiththecorrelationfunctionoftheexternalfieldi.e.withthelightwaveincidentonthemedium. In
thelattercase wewillnormallyassumethisfieldtobeweakandinsomeexamplesevenclassical. Nevertheless,to
distinguishitfromthefield(2.15)which,inturn,willbemostlyconsideredasastrongcontrolmodeunaffectedbythe
7
medium,wewilldisplaysuchaprobeasashortwavyline. Theprobemodecanbestronglymodifiedbythemedium
andcanbegeneratedbytheprocessesofspontaneousemission.
Iftheprobefieldoriginallyperformsacoherentpulsepropagatingthroughtheatomicsample,thenthedynamics
ofitscoherentcomponentisdescribedbythefollowinggraphicalequation
Aˆ(+)(r,t) = S T SAˆ(0,+)(r,t)
h µ i † µ
D h iE
(2.17)
⇒
Here,andfurtherinthediagrams,wewillomitevidentspecificationoftheatomicstatesandfieldpolarizations. The
self-energy part of this diagram performs the retarded-type polarization operator, and the double lines express the
”dressed”atomicpropagators(Green’sfunctions),whichaccumulatedalltheinteractionprocesses. Letusalsopoint
outthatundertherotatingwaveapproximationandinalowdensitylimitnormallythereisnovertex-typecorrection
in the structure of the polarizationoperator. One exceptionto this rule, associated with internalcorrelationsin the
coherentexcitationprocess,willbediscussedinSection3.3. Thefundamentalsolutionofequation(2.17)definesthe
positivefrequencycomponentoftheretarded-typeGreen’sfunction,whichisgivenby
i (R) (r ,t ;r ,t ) = Aˆ(+)(r ,t ), Aˆ( )(r ,t ) θ(t t )
Dµ1µ2;+ 1 1 2 2 µ1 1 1 µ−2 2 2 1− 2
Dh iE
⇒
= (2.18)
whereweputanindexofthefrequencycomponentinthesubscriptline. Thisfunctionperformstheaveragedcom-
mutatorofthecompleteHeisenbergoperatorsofthequantizedfield. Itsgraphicaldefinitionshowsthatthisfunction
can be expressed by the original vacuum lines (2.13) subsequently ”dressed” by the interactions generated in the
expansionof the evolutionaryoperator. Any double wavy line can be interpretedas the relevantlyorderedproduct
of the dressed quantum field fluctuations. For example, the subtracted term in the second line of Eq.(2.18) is the
resultofhigherorderinteractionprocessesanditincludesthecontributionofincoherentlyscatteredlightaswellas
fluorescenceassociated with the processes of optical pumping,and with other non-linearprocesses initiated by the
interactingfields. ThistermcanbeexpressedanalyticallybythefollowingproductoftheHeisenbergoperators
iDµ(−1µ+2);+(r1,t1;r2,t2) = hAˆ(µ−2)(r2,t2)Aˆ(µ+1)(r1,t1)i−hAˆ(µ−2)(r2,t2)ihAˆ(µ+1)(r1,t1)i
= T˜S Aˆ(0, )(r ,t ) TSAˆ(0,+)(r ,t ) T˜ S Aˆ(0, )(r ,t ) S S TSAˆ(0,+)(r ,t )
† µ2− 2 2 µ1 1 1 − † µ2− 2 2 † µ1 1 1
D h i h iE D h i ED h iE
(2.19)
⇒
With the acceptance of the relations (2.11) this Green’s function gives us the correlation function of the radiation
modediffuselypropagatingthroughtheatomicmedium. OtherGreen’sfunctions,dressedbytheinteraction,canbe
similarlyexpressedaccordinglytotheirorderingrules. ThelastlineinEq.(2.18)indicatesausefulsymmetryrelation
fortheretarded-typephotonpropagator.
8
Iftheprobefieldincidentonthemediumexistsinacoherentstatethenequation(2.17)isapplicableinthemost
generalcaseevenif thisfieldisstrong. Thenthelasttermofthisequationlegitimizesthe situationwhentheupper
energylevelsofatomscanbesignificantlypopulated. Thus, this termbecomesresponsibleforamplificationof the
propagatingprobemode via inducedemission and such an amplificationmanifestsitself in the retardedpropagator
(2.18) as well. In an optically dense gas the probe pulse will be immediately split into many scattered fragments,
such that equation (2.17) considered only for its forward propagationis actually insufficient. Nevertheless, further
dynamicsofeachscatteredfragmentalongashortfreepropagationsegmentbetweensubsequentscatteringeventsis
alsoexpressedbythe retardedpropagator(2.18), i.e.bythefundamentalsolutionofequation(2.17). Amplification
ofthediffuselypropagatinglightisadesirablescenariofortherandomlasing,whichwewilldiscussinSection4.1
in more detail. We will further see from this and other examples that coherenceis not completely lost in the light
diffusionprocess.
2.4.2. Generalanalysis: Theatomicsubsystem
We emphasizethattheentire dynamicsofthe lightsubsystemshouldbe consideredtogetherwiththe dynamics
of the atomic subsystem and the atomic Green’s functions (2.14) are strongly modified by the interaction with the
electromagneticfield. Letusdefinethefollowingdifferentialoperatorassociatedwithfreeevolutionofanatom
∂ ~ 1
Gˆ 1 = i + E , (2.20)
−0j ∂t 2m △j− ~ j
j 0
where jdenotesr ,t aswellasspecifiestheinternalstatem ,n ...with j = 1,2...;m istheatomicmass,andE
j j j j 0 j
denotestheenergyofthespecificatomicstateinthedefinitions(2.10),(2.14). Thenwecanintroducethefollowing
graphicalprecursorofakineticequationfortheatomicdensitymatrix
Gˆ 1 Gˆ 1
−01 − −02∗
(cid:16) (cid:17)
⇒
+   (2.21)
InthisgeneraldiagrammaticformwXCeohacceptanyelectromagnetic-typeinteractionscouplingtheatomicgroundand
excited states. The first line gives the free dynamics of the atoms, which is quantum mechanical for their internal
degreesoffreedomandassumedtobeclassicalfortheirspatialmotion. Thenexttwolinesdescribetheinteraction
withquantizedmodesandcanincludeinteractionswiththescatteredortrappedlight.Thedoublywavylinesexpress
thecorrelationfunctionsofthequantizedmodesdressedbyinteractionwiththemedium. ThelastlineinEq.(2.21)
symbolicallyindicates all possible interactionswith externalcoherentmodes, which can be opticaland microwave
and in some configurations these modes can be either depleted or modified by interaction with the medium. The
inwardoroutwardarrowsonthedresseddoubledashedlines(notshown,buthavetoattributethem)dependonthe
concretevertextypespecifiedbythebasicinteractiondiagrams(2.15).
The single particle atomic densitymatrix ρ (p,r,t) is expressedbyG( +)-typeGreen’sfunctions, see (2.14),
m1m2 −
dressedbytheinteractionprocesses
i ( +) (r ,t ;r ,t ) = Ψˆ (r ,t )Ψˆ (r ,t )
Gm−1m2 1 1 2 2 ± †m2 2 2 m1 1 1
D E
= T˜ S Ψˆ(0) (r ,t ) T SΨˆ(0)(r ,t ) (2.22)
† m2† 2 2 m1 1 1
D h i h iE
9
viathefollowingWigner-typetransform
d3p i i p2 E +E
i ( +) (r ,t ;r ,t ) = exp p(r r ) + m1 m2 (t t )
Gm−1m2 1 1 2 2 ±Z (2π~)3 (~ 1− 2 − ~"2m 2 # 1− 2 )
r +r t +t
ρ p, 1 2, 1 2 , (2.23)
× m1m2(cid:18) 2 2 (cid:19)
wheret t 0and theexternalsign, either+ or , relatesto eitherBosonicor Fermionicstatistics respectively.
1 2
− ∼ −
Insemiclassicalconditions,whenthedensitymatrixdependsontheatomicmomentumpandspatialcoordinateras
classicalvariables,thedifferenceinstatisticsvanishesinthefinalkineticequation.
If the scattered light is weak and characterizedby a small degeneracyparameter (small numberof photonsper
coherencevolume)itplaysanegligibleroleintheatomicdynamics,whichthenismostlyaffectedbyinteractionwith
vacuumquantizedmodes. Insuchasituationtheabovegraphicalequation(2.21)canbestraightforwardlyconverted
to a kinetic equation responsible for the dynamics of the single particle density matrix (2.23). The density matrix
describing the dynamics of an arbitrary atom driven by external fields obeys the following Lindblad-type master
equationwrittenintheoperatorform
∂ p i i i
+ ρˆ = Hˆ ,ρˆ Vˆ(t),ρˆ Uˆ(t),ρˆ + ˆρˆ. (2.24)
"∂t m ∇# −~ 0 − ~ − ~ R
0 h i h i h i
Here ρˆ = ρˆ(p,r,t) and the left-hand side performs a full time convective derivative of the density operator where
p/m = v is the atomic velocityand is the gradientoperator. The first term in the right-handside is responsible
0
for free dynamics of the atom driven∇by its internal Hamiltonian Hˆ and the second and third terms describe the
0
interactionswithexternalfields. WedenotetheseinteractiontermsasV(t)foropticalmodesandU(t)formicrowave
modes.
Forthemonochromaticcase,andintherotatingwaveapproximation,theinteractionoperatorsaregivenby
Vˆ(t) = −dˆ(−)E0e−iωct+ikcr−dˆ(+)E∗0eiωct−ikcr,
Uˆ(t) = mˆ( )H e iΩt mˆ(+)H eiΩt, (2.25)
− − 0 − − ∗0
wheredˆ( ) andmˆ( ) are theraising/loweringcomponentsofthe electricdipoleandmagneticmomentoperators;E
∓ ∓ 0
andH arethecomplexamplitudesoftheelectricandmagneticfieldsrespectively.Inthecaseofthemicrowavefield
0
wecanignoreitsspatialprofilesinceitswavelengthistypicallymuchlargerthanthesizeoftheatomiccloud. Inthe
caseoflinearpolarizationsforbothfieldsinadirectionalongtheZ-axistheinteractionmatrixelementsaredefined
asfollows,
V = (d ) E ,
nm˜ z nm˜ 0
U = (m ) H , (2.26)
mm˜ z mm˜ 0
wherewefollowaconventionwheren,n,...specifyanyupperstatesandm,m,...anygroundstates. If,asinmatrix
′ ′
elements(2.26), we needtodistinguishthegroundstates belongingtodifferenthyperfinesublevelsweadditionally
overscribebytildethestatesbelongingtootherhyperfinesublevels.
ThespontaneousradiativedecayoftheupperstatesandoftheopticalcoherencescontributesinEq.(2.24)bythe
followingrelaxationterms
ˆρˆ = γρ (p,r,t),
nn
R nn − ′
(cid:16) (cid:17) ′ γ
ˆρˆ = ρ (p,r,t), (2.27)
nm
R nm −2
(cid:16) (cid:17)
where γ is the natural radiative decay rate. In the important example of alkali-metal atoms the optical pumping
repopulationprocess providingthe atomic polarizationtransfer from the upperto the groundstate via spontaneous
decayisdescribedbythefollowingincoming-typeterm
(cid:16)Rˆρˆ(cid:17)m′m = γ Xnn ρn′n(p,r,t)Xq CFF0′′MM′0′1qCFF0MM01q
′
×(−)F0−F0′ h(2F0′ +1)(2F0+1)i1/2(2J+1) ( FS′ 1I FJ0′ )( SF 1I FJ0 ) (2.28)
10