Coherent backscattering of light from ultracold and optically dense atomic ensembles

0.78 MB·

by  I. M. Sokolov

Coherent backscattering oflightfrom ultracold andopticallydense atomicensembles
I.M. Sokolov1,2 and D.V. Kupriyanov1
1Department of Theoretical Physics, StatePolytechnicUniversity, 195251, St.-Petersburg, Russia
2InstituteforAnalyticalInstrumentation, RussianAcademyofSciences, 198103, St.-Petersburg, Russia
C.I. Sukenik and M.D. Havey
Old Dominion University, Department of Physics, Norfolk, Virginia 23529
Wereviewexperimentalandtheoreticalstudiesofcoherentbackscatteringofnearresonantradiationfroman
7
1 ultracoldatomicgasintheweaklocalizationregime. Recentaccomplishmentsinhighresolutionspectroscopy
0 ofatomicensemblesbasedonthecoherentbackscatteringprocessarediscussed. Wealsoproposeseveralnew
2 experimentalschemesfortime-dependentspectroscopyasappliedtomultiplescatteringintheregimeofweak
localization.
n
a
J
I. INTRODUCTION
4
2
Thestudyofopticalphenomenaisanancientquantitativescientificdiscipline,withhistoricalrootsextendingmorethantwo

] millenniaintothepast. Itwasthereforeremarkablethat, asrecentlyastwentyyearsago, in1984,a completelynewclassical
h
optics effect was reported for the first time in the scientific literature. It was then that Ishimaru and Kuga [1] reported the
p
observationof coherentbackscattering(CBS) of light froma disorderedscattering sample. This reportwas quicklyfollowed
-
m by experimental and theoretical work [2, 3] that included explanation of the effect based on the classical physical optics of
electromagneticwavescatteringinadisorderedmedium. Inthecoherentbackscatteringeffect,thereisanenhancementinthe
o
t intensityoflightscatteredinthenearlybackwardsdirectionfromaliquidoropticallydisorderedsolid. Theenhancementmay
a
be aslargeasa factorof two overthe usuallyexpectedaverageincoherentscatteredlightintensity. Further,the enhancement
.
s effect is concentrated in a cone-shaped interference profile typically a few milliradians in angular width. The fundamental
c
mechanism developed was that electromagnetic wave scattering along reciprocal, or time-reversed, multiple scattering paths
i
s preserves the relative phase. The result of phase preservation, after configurationaveraging, along the reciprocal paths leads
y
directlytopredictionofthemainfeaturesofthecoherentbackscatteringconeforsemiinfinitedisorderedscatteringsamples.
h
p Thebasiccoherentbackscatteringeffectisremarkablyrobust,andcanbeobservedinwavescatteringfromawiderangeof
[ common natural and manmade materials [4–6]. In the optical regime, the quantitative features are also quite sensitive to the

polarizationoftheincidentanddetectedwaves. TheCBSeffectisnotrestrictedtoscatteringofelectromagneticwaves,buthas
1
been observed, for example, in acoustics, ultrasonics, and in propagationof waves in the solid earth. For each of these areas
v
there has been a significant range of fundamentalstudies and developmentof practicalapplications [4, 5], particularly in the
4
1 areasofimagingordetectionofembeddedobjectswithindiffusivemedia. Anotherveryimportantassociatedresearchareais
7 connectedtolasingandwaveamplificationinrandommediahavinggain[7].
6 It is our purpose in this review to consider the more specialized and more recent developments associated with coherent
0 backscatteringoflightinultracoldatomicgaseoussamples.SuchsampleshavecharacteristicallyveryhighQandstrongoptical
.
1 resonances,makingthemuniqueincomparisontoscatteringfromclassicalcondensedorliquidsamples[8]. Further,forscatter-
0 ingfromsingleatoms,thevariedinfluencesofopticalandatomicpolarization,andtheresponsesofatomicsystemstoapplied
7 staticanddynamicfieldsarewellknowninthelinearresponseregime[9].Nonlinearresponsestoappliedelectromagneticfields
1
arealsowellstudied[10,11]. Thesecharacteristicsmaketheoreticalandexperimentalstudyofmesoscopicwavescatteringin
:
v atomicmediaanattractiveandaccessibleareaofresearch.
i Mesoscaleprocessesindenseandcoldatomicvaporscanalsodisplayaconstellationoffundamentalandpotentiallyimportant
X
phenomena. One of these, strong localization of light[4], is a dynamic research area in its own right. Strong localization of
r
lightistheopticalanalogofAndersonlocalizationofelectrons[12],inwhichenergytransportthroughamediumissuppressed
a
byinterferencesmediatedbymultiplescatteringinaspatiallyrandommedium.Tworeportsofstronglocalizationincondensed
systemshavebeenmadeintheliterature,oneintheopticalregime[13],andtheotherinthemicrowaveresponseofaquasi-one
dimensional system [14]. Localization is expected to occur in the density range given by the Ioffe-Regel condition, kl < 1,
wherekisthelocalwavevectorinthemediumandlthemeanfreepathforlightscattering. Inadiluteatomicvapor,themean
freepathisl=1/ρσ,whereρistheatomdensity,andσ thelightscatteringcrosssection[4,8]. Lightlocalizationhasnotyet
beenobservedinanatomicvapor,eventhoughthedensityandtemperatureregimeswhereitisexpectedtooccuraretechnically
accessibleusingthetechniquesofultracoldatomicphysics[15].
Inadditiontotheintriguingpossibilityofstronglightlocalizationinanatomicgas,researchintootherlinearandnonlinear
effectsinamultiplescatteringenvironmentisrelativelyundeveloped[16]. Therearealsopotentialtiestootherareasofmodern
quantum optics research, including the developments of quantum memories in the form of polaritonic excitations [17, 18]
in strongly scattering media. Manipulation of propagation and scattering in ultracold atomic gases, through application of
electromagneticallyinducedtransparencyinvariousconfigurations[19–21],couldpotentiallyleadtocoherentcontrolofoptical
2
transport properties in such media. Similar techniques have been applied to nonlinear optical phenomena [22, 23] including
four-wavemixing[24],nonlinearopticsatverylowlightlevels[25],andapplicationsofnonlinearopticswithsinglephotonsto
quantuminformationprocessing[26].
Inthisreviewwefocusourdiscussiontonear-resonancemultiplelightscattering,intheweaklocalizationregime,inensem-
blescomposedofultracoldatoms. Wereviewthevarioustheoreticalandexperimentalresultsinthisfield,withsomeemphasis
onourtheoreticaldevelopment,butwithdueattentiontothemanyrecentexperimentalachievementsinthisareaofresearch.The
main physicalobservableis the coherentbackscatteringcone, which is qualitativelycharacterizedby an overallenhancement
andangularwidth.Wediscusstheinfluence,inthecoherentbackscatteringeffect,ofensemblesize,opticaldepth,hyperfineand
Zeemanstructure,andspectraldetuningfromresonanceexcitation.Thefascinatingspatialvariationsintheangulardistribution
ofbackscatteringlightproducedbyapplicationofexternalstaticmagneticfieldsofafewgaussarealsoreviewed. Adynamic
area of theoreticalresearch, for whichthere are as yetonlya few experimentalresults, is the area of nonlinearopticaleffects
incoherentbackscattering;wepresentanoverviewofthesestudiesaswell. Ofparticularinteresthereisthetimedevelopment
oftheangulardistributionandspectralprofileofmultiplyscatterednear-resonanceradiation. Forobservationsofthetimede-
velopmentof diffusely scattered light, experimentaland theoretical results show strong variations with light polarization and
detuning. Inthebackscatteringdirection,thetheoreticalanalysisofthetimedevelopmentofthescatteredfluxrevealsavariety
oftransienteffectsinthecoherentbackscatteringenhancement.Finally,weincludeinanappendixsomedetailsofthetheoretical
developmentswhichwillbeofinteresttosomereaders.
II. THEORETICALOVERVIEW
A. Microscopicdescription
Consideranatomicensembleconsistingofatomsseparatedontheaveragebyadistancelargerthanatypicalradiativewave-
lengthλ. Letthisensemblescatterlowintensitylight,suchthattheinteractionprocesscanbedescribedproperlybyaperturba-
tiontheoryapproach. ThenintheHeisenbergformalismtheoperatorforthepositivefrequencycomponentoftheelectricfield
Eˆ(+)(r,t)atthepointrandatthemomentt,modifiedbytheprocessofmultiplescattering,canbeexpressedbythefollowing
series
Eˆ(+)(r,t) = Eˆ(+)(r,t) + Eˆ(+)(r,t)
0 a
a
X
+ ′Eˆ(+)(r,t) + ′Eˆ(+)(r,t) + ...
ab abc
ab abc
X X
Thisexpansion,writtenforthepositivefrequencycomponentoftheelectricfield,showshowthedifferentscatteringorders,
startingfromsingleviadoubleandtriplescattering,uptothehigherorders,subsequentlycontributetotheoutgoingHeisenberg
operator. Theindicesa,b,c,etc. enumeratetheatomsparticipatinginthescatteringprocess. Indoublescatteringa =b,butin
6
higherscatteringordersrecurrentscattering,inwhichsomeoftheindicescoincide,canoccur.
Theexpansion(1)canbeprovedundertheassumptionthatatamicroscopicleveleachrandomlychosenandisolatedatomof
theensemblescatterslightindependentlyfromitsenvironment. Thentheseriescanbegeneratedasanexpansionofevolution
operatorsactingontheoriginal”non-dressed”operatorEˆ(+)(r,t)ignoringanyinterferenceininteractionsrelatingtodifferent
0
and well separated atoms. This makes possible independent evaluation of each scattering amplitude as well as the radiative
correction of the excited state atomic Green function. In the case of weak interactions, when the incoming field does not
noticeablymodifythedynamicsoftheatomicsubsystem,thefinaloperatorsofthepositive/negativefieldcomponentsEˆ(±)(r,t)
preserve the canonical commutation relation between the non-perturbedoperators Eˆ(±)(r,t). Thus the transformation (1) is
0
unitary and the whole series reproducescorrectly the microscopic behavior of the Heisenberg field operator. This important
propertyisbasedontheabsenceofanylossesoflightapartfromthescatteringchannel,see[27].
As a pedagogicalexample,which will be usedthroughoutourdiscussion, letus show howa doublescatteringterm can be
3
writteninthecaseofsuccessivescatteringonatom”one”firstandonatom”two”second
1
Eˆ(+)(r,t) =
12 r r r
| − 2| 12 mX1,m′1mX2,m′2 Xν Xij
2π~ω 1/2 ω2 ω2
k 2 12
c2 c2
k,µ (cid:18) V (cid:19)
X
exp( iω t+ik r r +ik r +ikr )
2 2 2 12 12 1
× − | − |
× ek′ναˆν(mi′2m2)(ω12−k12v2)δi⊥j
× αˆj(mµ′1m1)(ω−kv1)akµ (1)
It is assumed here that, before interaction, the light subsystem is specified by the set of modes described by the wave vector
k, frequency ω = ωk and the polarization vector ekµ. Then akµ is an annihilation operator of the mode in the Schro¨dinger
representationand istherespectivequantizationvolume.Forlightscatteredfromatom”two”andpropagatingintoitsradiation
zone, similar paramVeters are respectively defined as k′, ω′ and ek′ν. If atoms are moving in space with velocities v1 and
v the input-outputtransformationsof the light frequency,as a result of successive quasi-elastic scattering events, is a direct
2
consequenceofthecombinedactionofRamanprocessesandtheDopplereffect.
ω12 = ω − ωm′1m1 + (k12−k)v1
ω′ = ω2 =ω12 − ωm′2m2 + (k2−k12)v2 (2)
ThelightscatteringisaccompaniedbyZeemantransitions m m′ and m m′ inthegroundstatesofthefirstand
| 1i→| 1i | 2i→| 2i
thesecondatoms.Theintermediateandoutputwavevectorsaregivenby
ω r r ω r r
k = 12 2− 1 2− 1
12 c r r ≈ c r r
2 1 2 1
| − | | − |
ω r r ω r r
k′ = k = 2 − 2 − 2 (3)
2 c r r ≈ c r r
2 2
| − | | − |
wheretheobservationpointrtendstoinfinityandtheapproximatedexpressions,definedbythelastequationsandignoringall
inelastic corrections, should be substituted in the Dopplerterms of (1). By r = r r we denotedthe relative distance
12 2 1
| − |
betweenatom”one”andatom”two”whicharelocatedrespectivelyatspatialpointsr andr . Consideringthereciprocalscat-
1 2
teringpath,i.e.scatteringfromatom”two”firstandatom”one”seconditisonlynecessarytotransposetheindices1 2inthe
⇔
abovetransformations,andtheoutputfrequencywillobtainadifferentmagnitudeω . But,ascanbeverifiedstraightforwardly
1
forspecificscatteringchannelssuchastheforwardandbackwarddirections,thefollowingequalityissatisfied: ω′ =ω =ω .
2 1
Themostimportantcharacteristiccontributingto(1)isthescatteringtensor,whichcanbedefinedinoperatorformasfollows
αˆ(m′m)(ω) = m′ m (dj)m′n(di)nm
ji − | ih |~(ω ω ) + i~γ /2
nm n
n −
X
m′ m α(m′m)(ω) (4)
≡ | ih | ji
Hered andd arevectorcomponentsofthetransitiondipolemomentbetweenlower m , m′ andupper n states,ω isthe
i j nm
| i | i | i
transitionfrequencyandγ isthenaturalrelaxationrateoftheupperstate. Aslongasγ hasapureradiativenaturethepartial
n n
transformationofthefieldoperatorineachscatteringevent,describedbytheamplitude(4),isunitary.
Theδ⊥-symbolisdefinedas
k k
δ⊥ =δ 12i 12j (5)
ij ij − k2
12
whichguaranteesthatthelightwavepropagatingbetweenthescatterersistransverse.
Therearenoadditionalphysicalideasnecessarytorecoverthewholeseries(1). Othertermscontributingtothisexpansion,
such as Eˆ(+)(r,t) fortriple scattering andthe higherorderterms, can be written similar to (1). For eachscattering sequence
abc
a,b,c,...thecorrespondingmultipleamplitudewillbeasubsequentproductofscatteringtensorsandthephotonGreenfunc-
tionsinvacuum,thesebeingresponsibleforthelightpropagationbetweenthescatterers.Thustheentireseriescanberecovered
byfollowingthesimplecombinatorialrulesformulatedinthebeginningofthissection.
4
B. Mesoscopicaveragingandmacroscopicdescription
We restrictourdiscussionbyapplicationtotheexperimentsdirectedtowardsmeasurementofthe firstorderinterferenceor
correlationpropertiesoflight,whicharedescribedbythefollowingcorrelationfunction
D(E)(r,t;r′,t′) = Eˆ(−)(r′,t′)Eˆ(+)(r,t) (6)
µν h ν µ i
Heretheanglebracketsdenotestatisticalaveragingovertheinitialstateofatomsandlight.Inthecaseofanultracoldatomicgas
thatisnotinaquantumdegeneratephase,thelocationofatomscanbevisualizedasthelocationofclassicalobjectsdistributed
inacertainmacroscopicvolume.Thentherearethefollowingimportantprocessesgoverningtheaveragingprocedure.
First, for a light ray propagating in any direction, there is a preferable coherent enhancementfor its forward propagation.
This means that along any ray, for a short mesoscopic scale consisting of a large number of atoms, there is only a slight
attenuation of the propagating wave. Such an attenuation comes from the events of incoherent scattering, which have small
butnotnegligibleprobability. Then an importantmodificationshouldbe madefor the propagationofthe lightwave between
anypairofneighboringatoms,seeexample(1),aswellasfortheincomingandoutgoingpartsofthelightpath. TheGreen’s
functionresponsibleforthelightpropagationinavacuumshouldbereplacedbytheFourierimageoftheretarded-typeGreen’s
functionresponsibleforthedispersionandintensityattenuationoftheforwardpropagatinglightinthebulkmedium,
1 1
δ⊥ exp[ik r ] D(R)(r ,r ,ω ) (7)
ij r 12 12 → −~ ij 1 2 12
12
where ω is given by (2). The complex conjugateof Eq.(7) transformsits right-handside to the Fourier componentsof the
12
advanced-typeGreenfunction.
In a homogeneousand isotropic medium, the Green’sfunctioncan be introducedby directmodificationof the exponential
factorinthelefthandsideofEq.(7)throughabsorptionandrefractionindicesofthemedium.Butinaninhomogeneouspolarized
atomicgasitbecomesaconsiderablymorecomplexproblem. Therefore,inanappendixwebrieflyreviewthepropertiesofthe
retardedGreen’sfunctionandshowhowitcanbecalculatedinapplicationtotheproblemoflightpropagationthroughapolarized
atomicensemble.
Afteraveraging,theactuallightwaveinthesamplecanbevisualizedasasetofunknownzigzagpaths,whoseverticesconsist
ofatomsscatteringthe lightfromthedirectionof forwardpropagation. Anyrandomlychosenpathcontainsa chainofatoms
located at the vertices and their number is just associated with the scattering order. Each chain of atomic scatterers makes a
partialcontributionto theformationoftheoutgoingwavesimilartohowitisdescribedbyexpressions(1)fortheHeisenberg
operators. Theimportantdifferenceis that, as a resultofthe mesoscopicaveraging,the seriesconvergesrapidlyandonlythe
multiplescatteringoftheloworderscontributesignificantlytotheformationofthecorrelationfunction(6).
Secondly, it is remarkable that the coherence is not completely lost for scattering in the non-forward direction. For the
light emerging from the sample in the backward direction the interference of multiple amplitudes for any selected chain of
scattererssurvivesstatisticalaveraging. Thisisknownasthecoherentbackscattering(CBS)effect,whichiscloselyrelatedto
weaklocalizationoflight. Comparingtheexpression(1)withasimilaronewrittenforthereciprocalEˆ(+)(r,t)term,theCBS
21
effectas well as the criteria of its observationcan be clearly seen. If scattered lightis detected at any randomangle, such as
k′+k=0,theinterferencecontributionbecomesquitesensitivetotheatomslocations. Inasampleconsistingofmanyatoms
6
theinterference,beingaveragedoverallpossiblecombinationsofatomicpairs,willbenegligiblecomparingwithladder(non-
interference)term. Butthiswouldnotbethecaseinobservationofthescatteringinthebackwardornear-backwarddirection,
wherek′+k 0. WeakoscillationscausedbyatomicmotionorRamantypescatteringstillsurvive. Butthesealsobecome
→
negligibleinthecaseofelasticscatteringoncoldandslowlymovingatoms.
C. Observablecharacteristicsincoherentbackscattering
Normally the relevant quantity for discussing the scattering process is a differential cross-section, which is defined as the
normalizedfluxofthescatteredlightemergingthesampleintheobservationdirection. Intermsofthecorrelationfunctionthe
cross section is givenby expression(6) consideredat coincidentspatial and time arguments: r = r′, t = t′. In this section
weillustratebycertainphysicalexamplesthatthespectrally-sensitiveandtime-dependentmeasurementsofthelightcorrelation
function (6) provide us further informationthan the measurementof the cross-section only. We concentrate ourselveson the
timedependenceofthecorrelationfunction,whichcorrespondstothefollowingobservationschemes.
First, the excitationof theensemblecanbe initiatedbya coherentlightpulse. Inthiscase theoriginalcorrelationfunction
willbefactorizedintheproduct
(E)(r,t;r′,t′) = (−)(r′,t′) (+)(r,t) (8)
Dµν Eν Eµ
5
where (+)(r,t)isacoherentfieldcomponentofthelaserlightpulse.Thenthescatteredresponse(6),whoseshapeisadistorted
µ
E
copyoftheoriginalpulseprofile,canprovideuswithcomparativeinformationabouthowthisresponseissensitivetotheeffects
ofsingleandhigherordersofthemultiplescattering.Theappropriateanalysisofthescatteredpulsecanbemadebythemethods
oftime-dependentspectroscopy.
Second,atomicmotion,whichalwaysexistsinarealisticsample,leadstoarandomlow-frequencymodulationofthescatter-
ingtermsbecauseoftheDopplereffect. Asclearlyseenintheexampleofdoublescattering(1),suchamodulationisdescribed
byvelocitiesv andv consideredasstochasticparametersinthefrequencytransformation(2). Thentheprobeofthesample
1 2
withamonochromaticcoherentwaveoffrequencyωwillbemodifiedinresponseasanon-monochromaticscatteredwavewith
the output correlation function (6) decaying as a function of t t′. Taken at coincident spatial arguments r = r′ and for a
−
point-likephotodetector,thecorrelationfunctioncanbeexpressedintheform
c
D(E)(r,t;r,t′) = e−iωRτI (τ) (9)
2π µν µν
whereω denotesthecarrierfrequencyofthescatteredlight,whichingeneralcanbeshiftedfromtheinputfrequencyωbecause
R
of the inelastic Raman effect. The outgoingintensityin any selected polarizationchannelis describedby the Pointingvector
I k′/k′(forµ=ν)andisgivenbythecorrelationfunctionconsideredatcoincidenttimest=t′.Thedependenceonτ =t′ t
µν
−
intherighthandsidecomesfromthespectraldistributionofthescatteredmodes.TheFouriertransform
∞
I(ω)= dτei(ω−ωR)τI (τ) (10)
µµ
µ=1,2 Z−∞
X
givesusthespectraldistributionofthescatteredintensityinthevicinityoftheRamanfrequency.Aswesee,theknowledgeofthe
spectraldistribution(10)providesuswithquiteimportantinformationaboutthevelocitydistributionandpossiblecorrelations
existing in an atomic ensemble confined with a magneto-optic trap. The corresponding spectral selection can be done by
heterodynedetectionandthelightbeatingspectroscopymethod[28].
Weconcludethetheoreticaloverviewbythefollowingremark. Ifanatomicsamplewereexcitedwithmonochromaticmode
andtheeffectofatomicmotionwereneglected,thenallthecharacteristicsofthescatteredlightwouldbecompletelydescribed
insidethecross-sectionformalism. Thespectrallysensitiveandtime-dependentanalysisgivesusmoreaccesstotheimportant
physicalinformationconcerningthelightpropagationandinternaldynamicsoftheatomicgasinthemagnetooptictrap.
III. EXPERIMENTALOVERVIEW
A. Apparatus
Inthissectionwegiveabroadoverviewofanexperimentalapparatususedtomeasurelightscatteringinanensembleofatoms
consistingofanultracold,dilutegasofatomic85Rbconfinedinamagneto-opticaltrap(MOT).Thedescriptionhererefersin
particulartothatofRef. [27];thephysicsoftheprocessissuchthatthedescribedapproachisquitegeneral.Inthepresentcase,
the MOT operateson the 5s2S F = 3 5p2P F = 4 hyperfinetransition and producesa nearly Gaussian cloud of
1/2 0 3/2
→
approximately108atomsatatemperature 100µK.Thepeakdensityatthecenterofthetrapis 3x1010cm−3. TheGaussian
∼ ∼
radiusofthesampleisr 1mm,determinedbyfluorescenceimaging.Measurementofthespectralvariationofthetransmitted
0
∼
lightgivesa peakopticaldepth,throughthecenterofthetrap, ofb =6 -8. Fora Gaussianatomdistributionin thetrap, the
0
weak-fieldopticaldepth,onresonanceandthroughthecenterofthetrap,isgivenbyb = √2πn σ r , wheren isthepeak
0 0 0 0 0
trapdensityandσ istheresonancecrosssection,seesectionIVA.
0
Note thatfor an isolated transitionthe near resonancecrosssection for lightscatteringσ and the respectiveopticaldepthb
varywithprobefrequencysuchthat
b
b= 0 , (1)
1+(2∆/γ)2
where∆=ω ω ,andω istheprobefrequency,whileω is(inthepresentcase)theF =3 F =4resonancefrequency.
L 0 L 0 0
− →
Separate lasers are used to provide the trapping and probe light. In both cases, a continuous wave diode laser having a
bandwidth 1MHzisused.Afulldescriptionofthemaster-slavelasersystemandvacuumhardwarecanbefoundin[27]. The
∼
laserintensityforboththetrappingandprobelightismodulatedwithanacousto-opticmodulator(AOM)-usedasanoptical
switch-whichgeneratesnearlyrectangularpulsesofadjustableduration. The20dBresponseislimitedbytheAOMtoabout
60ns. Thelaserlightissubsequentlycoupledintoasinglemodefiberopticpatchcord.ThecombinationoftheAOMswitching
andfibercouplingresultsinan 65dBattenuationofthelaserlightwhenswitchedoff. Aweakprobelaseristunedinarange
∼
of severalγ aroundthe trappingtransition. The probelaser is linearly polarized in the verticaldirection. The probebeam is
directedintotheMOTasshowninFigure1.
6
Pump Laser LP BS WP MOT
Field Lens
Field Lens
Chopper
Lenses
Display
CCD Camera
PMT
Figure1: Aschematicdiagramofatypicalexperimentalarrangement. Showninthefigureisamagnetoopticaltrap(MOT),linearpolarizers
(LP)whichselectthedetectedpolarizationchannels,abeamsplitter(BS),waveplates(WP)andaphotomultipliertube(PMT)todetectthe
fluorescencesignalsfortimedependentmeasurements. Forcontinuouswavemeasurements,acharge-coupleddevice(CCD)cameraisused.
Thegenericdisplayisamultichannelscalarfortimedependentmeasurements.
Dependingontheexperimenttobeperformed,lightscatteredbytheatomsisdetectedeitherinthebackwarddirectionona
liquidnitrogencooledcharge-coupleddevice(CCD)cameraorinadirectionatsomeotheranglerelativetotheincidentprobe
beamusingaphotomultipliertube(PMT).
B. CoherentBackscatteringMeasurements
For measurements of the CBS cone, great care must be taken to suppress multiple and back reflections from optics in the
detectionopticalpath. A majorsourceof unwantedback-scatteredlightisfromthe vacuumviewportsonthe MOT chamber.
Windows are typically wedged and V-type anti-reflection (AR) coated for 780 nm on the probe laser entrance and exit ports
andontheCCDcamera. TheARcoatingcharacteristicallyresultsinlessthan0.25%reflectivityat780nm. Additionally,the
entranceportwindowcanbemountedonaultrahighvacuumbellows,allowingredirectionofunwantedreflectionsawayfrom
thedetector.
Afterexitingthefiber,theprobebeamwasexpandedandcollimatedbyabeamexpandertoa1/e2 diameterofabout8mm.
Thepolarizationoftheresultingbeamwasselectedandthenthebeampassedthroughanonpolarizingandwedgedbeamsplitter
thattransmitsapproximatelyhalfofthelaserpowertotheatomicsample. Thebackscatteredradiationisdirectedbythesame
beam splitter to a field lens of 45 cm focal length, which condenses the light on the focal plane of the CCD camera. The
diffractionlimitedspatialresolutionwasabout100µrad,whilethepolarizationanalyzingpowerisgreaterthan2000at780nm.
Anyoneofthefourpolarizationchannelsthatarecustomarilystudiedincoherentbackscatteringcanbeselectedbyinsertingor
removingthequarterwaveplate,andadjustingthelinearpolarizationanalyzerlocatedbeforethefieldlens.
C. Time-DependentScatteringMeasurements
Formeasurementsoftime-dependentlightscattering,lightsignalsaredetectedinadirectionawayfromthecoherentbeam.
In a typical geometry, as shown in Figure 1, detection could be in a direction orthogonalto the probe laser propagationand
polarizationdirections. For example, in Ref. [29], the lightwas collected in an effectivesolid angle of about0.35mrad, and
refocussed to match the numerical aperture of a 400 µm multimode fiber. A linear polarization analyzer is placed between
the MOT and the field lens to collect signals in orthogonal linear polarization channels, which we label as parallel ( ) and
k
perpendicular( ).Thedifferentialpolarizationresponseiscalibratedagainsttheknownlinearpolarizationdirectionoftheprobe
⊥
laser, and the measured20 % differencein polarizationsensitivity is used to correctthe signals taken in the two polarization
channels.Thefiberoutputiscoupledthrougha780nm(5nmspectralwidth)interferencefiltertoanear-infraredsensitiveGaAs-
cathodephotomultipliertube.ThePMToutputisamplifiedanddirectedtoadiscriminatorandmultichannelscalar,whichserves
totime sortandaccumulatethedatainto5nsbins. Aprecisionpulsegeneratorisusedto controlthetimingoftheMOTand
probelasersandfortriggeringthemultichannelscalar.
Finally, we pointoutthatin thistype ofexperimentthe quantitativeresultsobtaineddependonthe relativediameterof the
pumpingbeamandthesamplesize. Themaineffectisinthecontributionofsinglescatteringincomparisonwiththemultiple
7
1.10
y
sit
n
e
nt
d i 1.05
e
al
c
s
1.00
-3 -2 -1 0 1 2 3
angle (mrad)
Figure2: Acoherentbackscatteringconeinthehelicitynon-preservingchannelassociatedwiththeF0 =3→F =4transitioninultracold
atomic85Rb.ReprintofFig.3fromRef.[30];copyright1999bytheAmericanPhysicalSociety.
Figure3:Dependenceontheinversesamplesizeofthefullwidthathalfmaximumofthecoherentbackscatteringconeforvariouspolarization
channels.Thepeakopticaldepthisfixedatfiveforthesecalculations.CalculationsrefertoMonte-Carlosimulationsoflightscatteringonthe
F0 =3→F =4resonancetransitioninultracoldatomic85Rb.
scatteringsignals.Forapumpbeamlargeincomparisonwiththesamples,thereissignificantsinglescatteringfromtherelatively
lowdensityperiphery,whileforanarrowprobebeamalargerportionofthesignalisduetoscatteringfromthedenserregions
ofthesample.
IV. CBSOBSERVATIONINANULTRACOLDATOMICGASINTHECONTINUOUSWAVEREGIME
A. Theenhancementfactorandconeshape
FirstobservationoftheCBS effectinanultracoldatomicgaswasreportedbyLabeyrieetal. inRef. [30]. InFigure2we
reproducetheexperimentalgraphfromthatpapershowingtheconefeatureinthespatialprofileofthebackscatteredlight. The
experimentwasdonewithatomsof85Rbandthescatteredlightwasobservedasaresponsetoaprobelasertunednear-resonance
withtheclosedhyperfinetransitionF = 3 F = 4oftherubidiumD line. Therubidiumatomsformaconvenientsample
0 2
→
formeasurementsandthedependenceofFigure2showsatypicalbehavioroftheCBSconefromanensembleoftheseatoms.
ThemeasurementsshowninFigure2[30]weremadewithacircularpolarizedcwmonochromaticprobelaser,andthescattered
lightwasdetectedinthechannelwithorthogonalhelicity,whichrelatestotheRayleighprocessforthesinglescatteringevent.
The graph shown in Figure 2 gives us the following two importantparameters of the CBS process. The main informative
8
parameteristheso-calledenhancementfactor,whichisdefinedas
I
α=1+ C (1)
I +I
L S
and shows the maximumenhancementof the backscatteredintensity. As was discussed in the theoreticaloverview,the addi-
tionalintensityinthescatteredlightisaresultofconstructiveinterferencebetweendirectandreciprocalscatteringpathsandis
describedbythecrosstermsI inthenumeratorof(1). ThedenominatorconsistsofthesinglescatteringcontributionI and
C S
non-interferingladdercontributionsI ofthesecondandhigherordersofmultiplescattering. Asisclearfromthestructureof
L
Eq. (1),forclassicaltypedipolescattererswithonlyRayleighscatteringchannels,theenhancementfactorαshouldapproach
2iftheopticaldepthb tendstoinfinity. Butitisalsoclearlyseenthatthisisnotthecasefortheexperimentalgraphshownin
0
Figure2.
Aswas pointedoutlater in [31]the relativelysmallmagnitudeof theenhancementfactorisa directconsequenceofmulti-
Zeeman-levelatomic structure. The physics of the process was reiterated within an analytical microscopic theory developed
byMu¨ller,etal. [32]. Theenhancementofafactoroftwocanbeachievedonlyifdirectandreciprocalscatteringamplitudes
describethetimereversalprocesses. Otherwise,forweakfieldscattering,theinterferencealwaysleadstoanenhancementless
thanafactoroftwo.Recoveryofthefactoroftwointheenhancementfactorwasclearlydemonstratedinexperimentsonatomic
Sr by Bidel, et al. [33]; in this case scattering is on a J = 0 J = 1 transition. The differencesbetween the Sr and Rb
0
→
cases, including the role of nonzero atom velocity, was emphasized in Wilkowski, et al. [34]. Finally, we point out that the
interferencesincoherentbackscatteringcanbedestructiveinspecialscatteringchannels. Belowwediscusssuchanexampleof
destructivelyinterferingchannels,wheretheenhancementfactorislessthanunity.
Thesecondimportantcharacteristicoftheconeshapeisitsangularwidth.Forsimpleevaluationandasapedagogicalmodel,
onecanimagineasemi-infinitehomogeneousmediumandapplythediffusionapproachtoestimatethestatisticaldistributionof
spatialseparationsofthefirstandlastscatterersinascatteringchain. Itisonlythelocationofthesescatterersthatdetermines
thephaseoftheinterferenceinthecrossterm. Thisstraightforwardlyleadstothefollowingestimationoftheconeangle
1
θ′ (2)
CBS ∼ kl
0
where l = 1/n σ is the free path for a resonancephoton migratingin the sample, n is the density of scatterers and σ is
0 0 0 0 0
theresonancecrosssection. Howevertheapplicationofthisestimationtorealexperimentswithatomicscatterersconfinedina
MOTfailsandquantitativelydisagreeswithobservabledata.
As was shown by precise Monte-Carlo modelling in [27], and was also discussed in [35], the cone angular width does
not depend on the diffusion length in the case of an atomic cloud with a Gaussian-type density distribution. For the density
distribution
r2
n(r)=n exp( ) (3)
0 −2r2
0
wheren isapeakdensityinthemiddleofthecloudandr istheradiusofthecloud,therelevantestimationofconeangleis
0 0
givenby
1
θ (4)
CBS
∼ kr
0
ThisisillustratedbytherespectivedependenciesshowninFigure3, whichshowsthelineardependenceoftheconewidthon
inversesamplesize r−1 in variouspolarizationchannelsofthe F = 3 F = 4 transitionof 85Rb. Since forthe Gaussian
0 0 →
cloudtheopticaldepthisgivenbyb =√2πσ n r weseethattheestimations(2)and(4)differbyafactorofb andexpression
0 0 0 0 0
(2)givesalargerconeanglewidththanisactuallymeasured.
TheCBS imagesin the planeorthogonaltothe incidentandbackscattereddirectionsaredifferentfor differentpolarization
channels. Normallythepolarizationchannelsarediscussedforcircularandlinearinputandoutputpolarizations. Circularpo-
larizationisnormallydefinedintermsofhelicities(hel)withrespecttotheframeofwavepropagation,whilelinearpolarization
directions (lin) are defined with respect to a laboratory frame. The scattered light is detected either in the same polarization
channelastheinputlightorinanorthogonalpolarizationchannel. Thereisaspatialasymmetryinthelinearpolarizationchan-
nel. Thisincludesa largercone widthin the verticaldirectionforthe lin lin channelwith the lines ofasymmetryalongthe
k
bisectorsofthedetectedlinearpolarizationdirectionsinthelin linchannel. Thedetailsofthefeaturesintheconeshapeare
⊥
discussedinRef. [36].
B. Theinfluenceofhyperfinestructure
Ininitialstudiesofcoherentbackscatteringinatomicsamples,itwasassumedthatfornear-resonancescatteringthehyperfine
structure of the excited atomic state is unimportant (other than for the degeneracies of the transitions under consideration).
9
F=4
120.7 MHz
52P
3/2 F=3
63.4 MHz
F=2
29.3 MHz
F=1
780 nm ω
»
F =3
52S 0
1/2 3.03 GHz
F =2
0
85
Rb, I=5/2
Figure4:Hyperfineenergylevelsofrelevanttransitionsinatomic85Rb.
ParticularlyfortheF = 3 F = 4 ”closed”transitionin85Rb thenearesthyperfinesatellite F = 3 F = 3islocated
0 0
→ →
at 120 MHz, see Figure 4. This is about20 times the naturalline width γ 5.9 MHz and arguesfor the unimportanceof
− ∼
theoff-resonanttransitions. Howeveraswaspredictedin[27]thereisanasymmetryintheCBSenhancementforthespectral
scanningneartheresonance,thisbeingcausedbytheinterferenceamongallthehyperfinetransitions. Theasymmetricshape
is more clearly seen in the case of circular polarization. This indicates the non-trivial spectral behavior of the Raman-type
cross/interferencetermsandtheRaman-typeladdertermsneartheresonance.
Theexperimentalverificationofthiseffectwasmadein[37,38]andinFigure5wereproducetheillustrativegraphfrom[37]
for the helicity preservingscattering channel. In the calculationsthe influence of possible heatingeffects, where the Doppler
broadening kv (v = 2k T/m is the most probable velocity in the atomic ensemble) was varied from 0 to 0.25γ. The
0 0 B
heatingmechanismmakestheenhancementweakerbutthespectralshapebroader.Comparingtheresultsonecanseethatthere
p
is a qualitativeagreementbetween theoreticalcalculationsand experimentaldata. However they differ quantitativelyand the
enhancement,observedintheexperimentinthewings,isevenlargerthanitstheoreticalprediction. Aswasmentionedin[37]
themainpossiblereasonofsuchacontradictionbetweenexperimentandtheoryisintheopticalpumpingprocess,whichtends
to orientatomic spins along the probe beam. For 100%orientationthis effect can increase the enhancementup to maximum
valueoftwo. Theenhancementofafactoroftwowouldbeachievablebecauseforspinorientedensemblethereisnothesingle
scattering contribution. Then there would be only two constructively interfering amplitudes if the double scattering channel
could be isolated. This effects is similar to the enhancement factor behavior in a strong magnetic field, as discussed in the
followingsection.
Weconcludethissectionbyemphasizingthattherearealsosignificantvariationsintheenhancementfactorevenforresonant
scatteringondifferentF F transitions. Thisisdueprimarilytothedifferentdegeneraciesassociated withthetransitions.
0
→
Experimentalconfirmationofthese variations, and comparisonswith Monte-Carloand modelcalculationshaverecentlybeen
reportedbyWilkowski,etal. [39].
C. Influenceofanexternalmagneticfield
TheinfluenceofanexternalmagneticfieldonthemultiplescatteringinopticallydenseatomicensembleswasstudiedinRefs.
[40–42]. Theuniqueconditionsofultracoldatomicensembles,wherethereisnegligibleinhomogeneousDopplerbroadening,
makepossiblethespectroscopicmanipulationwithamagneticfieldfortheZeemansplittingoftheatomiclevelsatthelevelof
thenaturallinewidth.InRef. [40],whichhasnodirectrelationtotheCBSphenomenon,theFaradayrotationofquasi-resonant
lightinanopticallythickcloudoflasercooledrubidiumatomswasexperimentallystudied. MeasurementsyieldalargeVerdet
10
constantintherange2000000/T/mmandamaximalpolarizationrotationof1500. TheFaradayeffectwasinitiated
Figure5: Comparisonofexperimentalandtheoreticalenhancementspectrainthehelicity-preservingpolarizationchannelassociatedwithan
F0 = 3 → F = 4resonance transition. Theoretical spectrashow modificationbyDoppler broadening, whichisvariedfromkv0 = 0to
kv0 =0.25γ,inanensembleof85Rbatomshavingapeakdensityofn0 =1.6×1010cm−3andaGaussianradiusr0=1mm.
bytheZeemansplittinginthegroundandintheupperstate,whichledtodifferencesintherefractionindicesforσ andσ
+ −
polarizationoftheprobelight. ItisjusttheabsenceoftheDopplereffectandthepossibilitytoscanthesamplenearthenatural
resonancelinewhichpermitssuchahugeVerdetconstanttobeobtained.
As a nextstep it was recognizedthata weakmagneticfield, with Zeemansplitting comparablewith the naturalline width,
shouldmodifytheCBSprocessitself. Thepresenceofthemagneticfieldmanifestsitselfinthescattererdipoleresponsetothe
electric field in a manner similar to the Hanle effect in a fluorescence geometry. As was shown in experiment[41] the polar
shape of the CBS cone in the linear polarizationscattering channelfollowsthe magnitudeand directionof the magnetic field
vector. InthetheoreticaldiscussionofRef. [41]anattemptwasmadetoclassifytheinfluenceofamagneticfieldintermsof
thewellknownFaraday,Cotton-MoutonandHanleeffects.Butasisclearinmultiplescattering,andparticularlyinthemultiple
scatteringregimeofCBS,thereisonlyaconvenientanalogywiththesebasicopticalprocesses.
The remarkable manifestation of a magnetic field was recently observed in Ref. [42]. There it was observed that
the enhancement factor can be increased with magnetic field up to its maximal value of two. This unusual behavior ap-
pears due to lifting of degeneracy in the helicity scattering channel for the spectrally selected Zeeman hyperfine transition
F = 3,M = 3 F = 4,M = 4 of 85Rb, which can be done by applying a rather strong external magnetic field. For
0 0
→
this transition there is no Raman-type scattering in the single scattering response. If only double scattering dominates in the
helicitypreservingscatteringresponsethereshouldbeafactoroftwoenhancementofthescatteredintensity.IntheFigure6we
reproducedthebasicgraphof[42],showingtheexperimentalverificationofthiseffect.Letusalsopointoutthatthereisadirect
analogy of the behavior of the enhancementfactor in a magnetic field with its spectral behavior in a spin-oriented ensemble
predictedin[37].
D. TheCBSprocessinthesaturationregime
RecentexperimentsbyChaneliere,etal. [43]ontheresonancetransitioninatomicstrontiumandbyBalik,etal. [29]inru-
bidium,haveshownthatasignificantreductioninthecoherentbackscatteringenhancementcanoccurwithincreasingintensity
oftheprobelaser. Bothnon-lineareffectsandadditionalinelasticscatteringcomponentscancontributetothisreduction. The-
oreticalandmodelstudieshaveshownsimilarqualitativeeffects[44–46],althoughtheyhaveyettobequantitativelycompared
withexperiment.
Inthestrontiumexperiment,aMOTcontaining7 107atomsatatemperatureof 1mKandwithaGaussianradiusof 0.7
× ∼ ∼
mmwasilluminatedwithnearresonantlight. Theensembleofcoldatomshadapeakopticaldepthofb =3.5andkl = 104,
0 0
resulting in a regimeof weak localization. In the absence of MOT light and magneticfield gradients, a resonantprobebeam
illuminated the sample for a variable period of 5 to 70 µs. The probe pulse duration was adjusted so as to keep the total
number of scattered photonsbelow 400 for all intensities investigated, therebyminimizing mechanicaleffects of the light on
thecoldatomsample. ACBSconewasthenrecordedinthehelicitypreservingchannel(hel hel),wheresinglescatteringis
k