ManipulatingatomswithphotonsClaudeN.Cohen-TannoudjigedeFranceetLaborat - PDF document

ManipulatingatomswithphotonsClaudeN.Cohen-TannoudjigedeFranceetLaboratoireKastlerBrosseldel'EcoleNormaleSupe75231ParisCedex05,FranceFrance()Electromagneticinteractionsplayacentralroleinlowenergyphysics.Theyareresponsibleforthecohesionofatomsandmoleculesandtheyareattheoriginoftheemissionandabsorptionoflightbysuchsystems.Thislightisnotonlyasourceofinformationonthestructureofatoms.Itcanalsobeusedtoactonatoms,tomanipu- The1997NobelPrizeinPhysicswassharedbyStevenChu,ClaudeN.Cohen-Tannoudji,andWilliamD.Phillips.Thislec-tureisthetextofProfessorCohen-Tannoudji'saddressontheoccasionoftheaward.LaboratoireKastlerBrosselisalaboratoryaf®liatedwiththeCNRSandwiththeUniversitePierreetMarieCurie.ReviewsofModernPhysics,Vol.70,No.3,July19980034-6861/98/70(3)/707(13)/$17.601998TheNobelFoundation Dissipativeeffectsandreactiveeffectsalsoappearfortheatoms,asaresultoftheirinteractionwithphotons.Theycorrespondtoabroadeningandtoashiftoftheatomicenergylevels,respectively.Sucheffectsalreadyappearwhentheatominteractswiththequantizedra-diation®eldinthevacuumstate.Itiswellknownthatatomicexcitedstatesgetanaturalwidth,whichisalsotherateatwhichaphotonisspontaneouslyemittedfromsuchstates.Atomicenergylevelsarealsoshiftedasaresultofvirtualemissionsandreabsorptionsofpho-tonsbytheatom.SucharadiativecorrectionissimplytheLambshift(Heitler,1954).Similareffectsareassociatedwiththeinteractionwithanincidentlightbeam.Atomicgroundstatesgetara-diativebroadening,whichisalsotherateatwhichphotonsareabsorbedbytheatom,ormorepreciselyscatteredfromtheincidentbeam.Atomicenergylevelsarealsoshiftedasaresultofvirtualabsorptionsandreemissionsoftheincidentphotonsbytheatom.Suchenergydisplacementsarecalledlightshifts,oracStarkshifts(BarratandCohen-Tannoudji,1961;Cohen-Tannoudji,1962).Inviewoftheirimportanceforthefollowingdiscus-sions,wegivenowabriefderivationoftheexpressions,usingtheso-calleddressed-atomapproachtoatom-photoninteractions(seeforexampleCohen-Tannoudji,Dupont-Roc,andGrynberg,1992,chapterVI).Intheabsenceofcoupling,thetwodressedstates(atominthegroundstateinthepresenceofphotons)and(atomintheexcitedstatethepresenceof1photons)areseparatedbyasplit-,whereisthedetuningbetweenthelightfrequencyandtheatomicfrequency.Theatom-lightinteractionHamiltoniancouplesthesetwostatesbecausetheatomincanabsorbonephotonandjumpto.Thecorrespondingmatrixelementofcanbewrittenas/2,wheretheso-calledRabiisproportionaltothetransitiondipolemo-mentandto .Undertheeffectofsuchacoupling,thetwostatesrepeleachother,andthestateshiftedbyanamount,whichisthelightshiftofThecontaminationofbytheunstablestate(havingawidth)alsoconferstothegroundstateawidth.Inthelimitwhere,asimpleperturbativecalculationgives: ,(1) .(2)areproportionalto,i.e.,tothelightintensity.TheyvarywiththedetuningasLorentzabsorptionanddispersioncurves,respec-tively,whichjusti®esthedenominationsabsorptiveanddispersiveusedforthesetwotypesofeffects.Forlargedetunings(variesas1/andbecomesneg-ligiblecomparedtowhichvariesas1/.Ontheotherhand,forsmalldetunings,(ismuchlarger.Inthehighintensitylimit,whenislargecom-paredto,thetwodressedstatesresultingfromthecouplingarethesymmetricandantisymmetriclinearcombinationsof.Theirsplittingisandtheysharetheinstabilityinequalparts,so/2.Onecanexplaininthiswayvariousphysi-caleffectssuchastheRabi¯oppingortheAutler-Townessplittingsofthespectrallinesconnectingtoathirdlevel(AutlerandTownes,1955).B.Manipulationofinternaldegreesoffreedom1.OpticalpumpingOpticalpumpingisoneofthe®rstexamplesofma-nipulationofatomsbylight(Kastler,1950).Itusesreso-nantexcitationofatomsbycircularlypolarizedlightfortransferringtotheatomspartoftheangularmomentumcarriedbythelightbeam.ItisbasedonthefactthatdifferentZeemansublevelsintheatomicgroundstatehaveingeneraldifferentabsorptionratesforincomingpolarizedlight.Forexample,foratransition,onlyatomsinthesublevel1/2canab--polarizedlight.Theyareexcitedinthesublevel1/2offromwhichtheycanfallbackinthe1/2byspontaneousemissionofapolarizedphoton.Theythenremaintrappedinthisstatebecausenofurther-transitioncantakeplace.Itispossibleinthiswaytoobtainhighdegreesofspinorien-tationinatomicgroundstates.2.LightshiftsOpticalpumpingisadissipativeeffectbecauseitisassociatedwithresonantabsorptionofphotonsbytheatom.NonresonantopticalexcitationproduceslightshiftsofthegroundstateZeemansublevels.Becauseofthepolarizationselectionrules,lightshiftsdependonthepolarizationoftheexcitinglightandvaryingeneralfromoneZeemansubleveltoanother.Considerforex-amplea1/2transition[Fig.1(a)].A-polarizedexcitationshiftsonlytheZeemansublevel1/2,whereasa-polarizedexcitationshiftsonlythesublevel1/2[Fig.1(b)].Magneticresonancecurvesinthegroundstate,whichareverynarrowbe-causetherelaxationtimeincanbeverylong,arethuslightshiftedbyapolarizednonresonantexcitation,andthesignofthisshiftchangeswhenonechangesthepo-larizationofthelightbeamfrom.Itisinthiswaythatlightshiftswere®rstobserved(Cohen-Tannoudji,1961).Figure1(c)givesanexampleofex-perimentalresultsobtainedbyexcitingthetransition1/2ofHgatomsbythenon-resonantlightcomingfromalamp®lledwithanotherisotope(Hg).Lightshiftscanbeconsideredfromdifferentpointsofview.First,theycanbeinterpretedasaradiativecorrec-tion,duetotheinteractionoftheatomwithanincident®eldratherthanwiththevacuum®eld.ThisiswhyAl-fredKastlercalledthem``Lampshifts.''Secondly,theyintroduceperturbationstohighprecisionmeasurementsusingopticalmethods,whichmustbetakenintoaccountClaudeN.Cohen-Tannoudji:ManipulatingatomswithphotonsRev.Mod.Phys.,Vol.70,No.3,July1998 beforeextractingspectroscopicdatafromthesemea-surements.Finally,becauseoftheirvariationfromonesubleveltoanother,theeffectoflightshiftscanbede-scribedintermsof®ctitiousmagneticorelectric®elds(Cohen-TannoudjiandDupont-Roc,1972).Thisex-plainswhythelightshiftsproducedbyanonresonantlaserstandingwavearebeingmoreandmorefrequentlyusedtoproducespatialmodulationsoftheZeemansplittingsinthegroundstateonanopticalwavelengthscale.Thiswouldnotbeeasilyachievedwithspatiallyvaryingrealmagnetic®elds.WewillseeinSectionIIinterestingapplicationsofsuchasituation.C.Manipulationofexternaldegreesoffreedom1.ThetwotypesofradiativeforcesTherearetwotypesofradiativeforces,associatedre-spectivelywithdissipativeandreactiveeffects.Dissipativeforces,alsocalledradiationpressureforcesorscatteringforces,areassociatedwiththetrans-feroflinearmomentumfromtheincidentlightbeamtotheatominresonantscatteringprocesses.Theyarepro-portionaltothescatteringrate.Considerforexampleanatominalaserplanewavewithwavevector.Be-causephotonsarescatteredwithequalprobabilitiesintwooppositedirections,themeanmomentumtrans-ferredtotheatominanabsorption-spontaneousemis-sioncycleisequaltothemomentumoftheabsorbedphoton.Themeanrateofmomentumtransfer,i.e.,themeanforce,isthusequalto.Sincesaturatesto/2athighintensity(seeSectionI.A),theradiationpressureforcesaturatesto/2.Thecorrespondingac-celeration(ordeceleration),whichcanbecommuni-catedtoanatomwithmass,isequalto,whereistherecoilveloc-ityoftheatomabsorbingoremittingasinglephoton,istheradiativelifetimeoftheexcitedstate.Forsodiumatoms,m/sands,sothatcanreachvaluesaslargeas,i.e.,10istheaccelerationduetogravity.Withsuchaforce,onecanstopathermalatomicbeaminadistanceoftheorderofonemeter,providedthatonecompensatesfortheDopplershiftofthedeceleratingatom,byusingforexampleaspatiallyvaryingZeemanshift(PhillipsandMetcalf,1982;Prodan,Phillips,andMetcalf,1982)orachirpedlaserfrequency(Ertmeretal.,1985).Dispersiveforces,alsocalleddipoleforcesorgradientforces(Askarian,1962;Ashkin,1980;LetokhovandMi-nogin,1981),canbeinterpretedintermsofpositionde-pendentlightshifts)duetoaspatiallyvaryinglightintensity(Cohen-Tannoudji,1992).Considerforexamplealaserbeamwelldetunedfromresonance,sothatonecanneglect(noscatteringprocess).Theatomthusremainsinthegroundstateandthelightshift)ofthisstateplaystheroleofapotentialenergy,givingrisetoaforcewhichisequalandoppositetoits.Suchaforcecanalsobein-terpretedasresultingfromaredistributionofphotonsbetweenthevariousplanewavesformingthelaserwaveinabsorption-stimulatedemissioncycles.Ifthedetuningisnotlargeenoughtoallowtobeneglected,sponta-neoustransitionsoccurbetweendressedstateshavingoppositegradients,sothattheinstantaneousforceoscil-latesbackandforthbetweentwooppositevaluesinarandomway.Suchadressed-atompictureprovidesasimpleinterpretationofthemeanvalueandofthe¯uc-tuationsofdipoleforces(DalibardandCohen-Tannoudji,1985).2.ApplicationsofdissipativeforcesÐDopplercoolingandmagneto-opticaltrapsWehavealreadymentionedintheprevioussubsec-tionthepossibilityofdeceleratinganatomicbeambytheradiationpressureforceofalaserplanewave.Inter-estingeffectscanalsobeobtainedbycombiningtheef-fectsoftwocounterpropagatinglaserwaves.A®rstexampleisDopplercooling,®rstsuggestedforneutralatomsbyT.W.HanschandA.L.Schawlow(1975)and,independentlyfortrappedions,byD.Wine-landandH.Dehmelt(1975).ThiscoolingprocessresultsfromaDoppler-inducedimbalancebetweentwooppo-siteradiationpressureforces.Thetwocounterpropagat-inglaserwaveshavethesame(weak)intensityandthe FIG.1.Experimentalobservationoflightshifts(fromCohen-Tannoudji,1961).Fora1/2transition(a),a-polarizednonresonantexcitationshiftsonlythesublevel[rightpartof(b)]whereasa-polarizedexcitationshiftsonlythesublevel[leftpartof(b)].ThedetuningpositiveandverylargecomparedtotheZeemansplittingsin.TheZeemansplittinginthegroundstateisthusin-creasedinthe®rstcase,decreasedinthesecondone.(c)mag-neticresonancesignalvsmagnetic®eld,inunitsofthecorre-spondingLarmorfrequency.Thecentralcurve(circles)istheresonancecurveintheabsenceoflightshift.Whenthenon-resonantlightbeamisintroduced,either(crosses)or-polarized(triangles),themagneticresonancecurveislightshiftedinoppositedirections.ClaudeN.Cohen-Tannoudji:ManipulatingatomswithphotonsRev.Mod.Phys.,Vol.70,No.3,July1998 samefrequencyandtheyareslightlydetunedtotheredoftheatomicfrequency().Foranatomatrest,thetworadiationpressureforcesexactlybalanceeachotherandthenetforceisequaltozero.Foramovingatom,theapparentfrequenciesofthetwolaserwavesareDopplershifted.Thecounterpropagatingwavegetsclosertoresonanceandexertsastrongerradiationpres-sureforcethanthecopropagatingwavewhichgetsfar-therfromresonance.Thenetforceisthusoppositetotheatomicvelocityandcanbewrittenforsmallisafrictioncoef®cient.Byusingthreepairsofcounterpropagatinglaserwavesalongthreeor-thogonaldirections,onecandamptheatomicvelocityinaveryshorttime,ontheorderofafewmicroseconds,achievingwhatiscalledan``opticalmolasses''(Chuetal.,1985).TheDopplerfrictionresponsibleforthecoolingisnecessarilyaccompaniedby¯uctuationsduetothe¯uo-rescencephotonswhicharespontaneouslyemittedinrandomdirectionsandatrandomtimes.Thesephotonscommunicatetotheatomarandomrecoilmomentum,responsibleforamomentumdiffusiondescribedbyadiffusioncoef®cient(GordonandAshkin,1980;LetokhovandMinogin,1981;Cohen-Tannoudji,1992).AsinusualBrownianmotion,competitionbetweenfric-tionanddiffusionusuallyleadstoasteadystate,withanequilibriumtemperatureproportionalto.ThetheoryofDopplercooling(Letokhov,Minogin,andPavlik,1977;WinelandandItano,1979;GordonandAshkin,1980)predictsthattheequilibriumtemperatureobtainedwithsuchaschemeisalwayslargerthanacer-tainlimit,calledtheDopplerlimit,andgivenby/2whereisthenaturalwidthoftheexcitedstateandtheBoltzmannconstant.Thislimit,whichisreachedfor/2,is,foralkaliatoms,ontheorderof100K.Infact,whenthemeasurementsbecamepreciseenough,itappearedthatthetempera-tureinopticalmolasseswasmuchlowerthanexpectedetal.,1988).Thisindicatesthatotherlasercoolingmechanisms,morepowerfulthanDopplercooling,areoperating.WewillcomebacktothispointinSectionII.Theimbalancebetweentwooppositeradiationpres-sureforcescanbealsomadepositiondependentthoughaspatiallydependentZeemanshiftproducedbyamag-netic®eldgradient.Inaone-dimensionalcon®guration,®rstsuggestedbyJ.Dalibardin1986,thetwocounter-propagatingwaves,whicharedetunedtothered()andwhichhaveoppositecircularpolarizations,areinresonancewiththeatomatdifferentplaces.Thisresultsinarestoringforcetowardsthepointwherethemagnetic®eldvanishes.FurthermorethenonzerovalueofthedetuningprovidesaDopplercooling.Infact,suchaschemecanbeextendedtothreedimensionsandleadstoarobust,largeanddeeptrapcalleda``magneto-opticaltrap''or``MOT''(Raabetal.,1987).Itcombinestrappingandcooling,ithasalargevelocitycapturerangeanditcanbeusedfortrappingatomsinasmallcell®lledwithalowpressurevapour(Monroeetal.3.Applicationsofdispersiveforces:LasertrapsandatomicWhenthedetuningisnegative(0),lightshiftsarenegative.Ifthelaserbeamisfocused,thefocalzonewheretheintensityismaximumappearsasamini-mumofpotentialenergy,formingapotentialwellwheresuf®cientlycoldatomscanbetrapped.Thisisalasertrap.Lasertrapsusingasinglefocusedlaserbeam(Chuetal.,1986)ortwocrossedfocusedlaserbeams(Adamsetal.,1995;Kuhnetal.,1997)havebeenrealized.Earlyproposals(Letokhov,1968)wereconsideringtrappingatomsattheantinodesornodesofanonresonantlaserstandingwave.Channelingofatomsinalaserstandingwavehasbeenobservedexperimentally(Salomonetal.Ifthedetuningispositive,lightshiftsarepositiveandcanthusbeusedtoproducepotentialbarriers.Forex-ample,anevanescentbluedetunedwaveatthesurfaceofapieceofglasscanpreventslowatomsimpingingontheglasssurfacefromtouchingthewall,makingthembounceoffa``carpetoflight''(CookandHill,1982).Thisistheprincipleofmirrorsforatoms.Planeatomicmirrors(Balykinetal.,1988;Kasevich,Weiss,andChu,1990)havebeenrealizedaswellasconcavemirrorsetal.,1993).II.SUB-DOPPLERCOOLINGIntheprevioussection,wediscussedseparatelythemanipulationofinternalandexternaldegreesoffree-dom,andwehavedescribedphysicalmechanismsin-volvingonlyonetypeofphysicaleffect,eitherdispersiveordissipative.Infact,thereexistcoolingmechanismsresultingfromaninterplaybetweenspinandexternaldegreesoffreedom,andbetweendispersiveanddissipa-tiveeffects.Wediscussinthissectiononeofthem,theso-called``Sisyphuscooling''or``polarization-gradientcooling''mechanism(DalibardandCohen-Tannoudji,1989;Ungaretal.,1989)(seealsoDalibardandCohen-Tannoudji,1985),whichleadstotemperaturesmuchlowerthanDopplercooling.Onecanunderstandinthiswaythesub-DopplertemperaturesobservedinopticalmolassesandmentionedaboveinSectionI.C.2.A.SisyphuseffectMostatoms,inparticularalkaliatoms,haveaZeemanstructureinthegroundstate.Sincethedetuningusedinlasercoolingexperimentsisnottoolargecomparedto,bothdifferentiallightshiftsandopticalpumpingtran-sitionsexistforthevariousgroundstateZeemansublev-els.Furthermore,thelaserpolarizationvariesingeneralinspacesothatlightshiftsandopticalpumpingratesarepositiondependent.Weshownow,withasimpleone-dimensionalexample,howthecombinationofthesevariouseffectscanleadtoaveryef®cientcoolingConsiderthelasercon®gurationofFig.2(a),consist-ingoftwocounterpropagatingplanewavesalongtheaxis,withorthogonallinearpolarizationsandwiththeClaudeN.Cohen-Tannoudji:ManipulatingatomswithphotonsRev.Mod.Phys.,Vol.70,No.3,July1998 samefrequencyandthesameintensity.Becausethephaseshiftbetweenthetwowavesincreaseslinearly,thepolarizationofthetotal®eldchangesfromandviceversaevery/4.Inbetween,itisellipticalorlinear.Considernowthesimplecasewheretheatomicgroundstatehasanangularmomentum1/2.AsshowninSectionI.B,thetwoZeemansublevels1/2undergodifferentlightshifts,dependingonthelaserpolarization,sothattheZeemandegeneracyinzeromagnetic®eldisremoved.ThisgivestheenergydiagramofFig.2(b)showingspatialmodulationsoftheZeemansplittingbetweenthetwosublevelswithape-Ifthedetuningisnottoolargecomparedto,therearealsorealabsorptionsofphotonsbytheatomfol-lowedbyspontaneousemission,whichgiverisetoopti-calpumpingtransfersbetweenthetwosublevels,whosedirectiondependsonthepolarization:1/2foraforapolarization.Herealso,thespatialmodulationofthelaserpolarizationresultsinaspatialmodulationoftheopticalpumpingrateswithaperiod/2[verticalarrowsofFig.2(b)].Thetwospatialmodulationsoflightshiftsandopticalpumpingratesareofcoursecorrelatedbecausetheyareduetothesamecause,thespatialmodulationofthelightpolarization.ThesecorrelationsclearlyappearinFig.2(b).Withthepropersignofthedetuning,opticalpumpingalwaystransfersatomsfromthehigherZee-mansubleveltothelowerone.Supposenowthattheatomismovingtotheright,startingfromthebottomofavalley,forexampleinthestate1/2ataplacewherethepolarizationis.Becauseofthe®nitevalueoftheopticalpumpingtime,thereisatimelagbetweeninternalandexternalvariablesandtheatomcanclimbupthepotentialhillbeforeabsorbingaphotonandreachthetopofthehillwhereithasthemaximumprob-abilitytobeopticallypumpedintheothersublevel,i.e.,inthebottomofavalley,andsoon[doublearrowsofFig.2(b)].LikeSisyphusintheGreekmythology,whowasalwaysrollingastoneuptheslope,theatomisrun-ninguppotentialhillsmorefrequentlythandown.Whenitclimbsapotentialhill,itskineticenergyistrans-formedintopotentialenergy.Dissipationthenoccursbylight,sincethespontaneouslyemittedphotonhasanen-ergyhigherthantheabsorbedlaserphoton[anti-StokesRamanprocessesofFig.2(b)].AftereachSisyphuscycle,thetotalenergyoftheatomdecreasesbyanamountoftheorderof,whereisthedepthoftheopticalpotentialwellsofFig.2(b),untilsmallerthan,inwhichcasetheatomremainstrappedinthepotentialwells.ThepreviousdiscussionshowsthatSisyphuscoolingleadstotemperaturessuchthat.Ac-cordingtoEq.(2),thelightshiftisproportionaltowhen4.SuchadependenceofontheRabifrequencyandonthedetuninghasbeencheckedexperimentallywithcesiumatoms(Salomonetal.,1990).Figure3presentsthevariationsofthemea-suredtemperaturewiththedimensionlessparameter.Measurementsofversusintensityfordiffer-entvaluesofshowthatdependslinearly,forlowenoughintensities,onasingleparameterwhichisthelightshiftofthegroundstateZeemansublevels. FIG.2.Sisyphuscooling.Lasercon®gurationformedbytwocounterpropagatingplanewavesalongtheaxiswithorthogo-nallinearpolarizations(a).Thepolarizationoftheresultingelectric®eldisspatiallymodulatedwithaperiod/2.Every/4,itchangesfromandviceversa.ForanatomwithtwogroundstateZeemansublevels1/2,thespatialmodulationofthelaserpolarizationresultsincorrelatedspa-tialmodulationsofthelightshiftsofthesetwosublevelsandoftheopticalpumpingratesbetweenthem(b).Becauseofthesecorrelations,amovingatomrunsuppotentialhillsmorefre-quentlythandown[doublearrowsof(b)]. FIG.3.Temperaturemeasurementsincesiumopticalmolasses(fromSalomonetal.,1990).Theleftpartofthe®gureshowsthe¯uorescencelightemittedbythemolassesobservedthroughawindowofthevacuumchamber.Thehorizontalbrightlineisthe¯uorescencelightemittedbytheatomicbeamwhichfeedsthemolassesandwhichissloweddownbyafre-quencychirpedlaserbeam.Rightpartofthe®gure:tempera-tureoftheatomsmeasuredbyatime-of-¯ighttechniquevsthedimensionlessparameterproportionaltothelightshift(istheopticalRabifrequency,thedetuningandnaturalwidthoftheexcitedstate).ClaudeN.Cohen-Tannoudji:ManipulatingatomswithphotonsRev.Mod.Phys.,Vol.70,No.3,July1998 B.ThelimitsofSisyphuscoolingAtlowintensity,thelightshiftismuchsmallerthan.ThisexplainswhySisyphuscoolingleadstotemperaturesmuchlowerthanthoseachievablewithDopplercooling.Onecannot,however,decreaseinde®nitelythelaserintensity.Thepreviousdiscussionignorestherecoilduetothespontaneouslyemittedpho-tonswhichincreasesthekineticenergyoftheatombyanamountontheorderof,whereistherecoilenergyofanatomabsorbingoremittingasinglephoton.Whenbecomesontheorderoforsmallerthan,thecoolingduetoSisyphuscoolingbecomesweakerthantheheatingduetotherecoil,andSisyphuscoolingnolongerworks.Thisshowsthatthelowesttemperatureswhichcanbeachievedwithsuchaschemeareontheorderofafew.Thisresultiscon®rmedbyafullquantumtheoryofSisyphuscooling(Castin,1991;CastinandMolmer,1995)andisingoodagreementwithexperimentalresults.TheminimumtemperatureinFig.3isontheorderof10C.OpticallatticesFortheoptimalconditionsofSisyphuscooling,atomsbecomesocoldthattheygettrappedinthequantumvibrationallevelsofapotentialwell[seeFig.3(b)].Moreprecisely,onemustconsiderenergybandsinthisperodicstructure(CastinandDalibard,1991).Experi-mentalobservationofsuchaquantizationofatomicmo-tioninanopticalpotentialwas®rstachievedinonedi-mension(seeFig.4andVerkerketal.,1992;Jessenetal.,1992).Atomsaretrappedinaspatialperiodicar-rayofpotentialwells,calleda``1D-opticallattice,''withanantiferromagneticorder,sincetwoadjacentpotentialwellscorrespondtooppositespinpolarizations.2Dand3Dopticallatticeshavebeenrealizedsubsequently(seethereviewpapersbyGrynbergandTriche,1996;Hem-merich,Weidemuller,andHansch,1996;andJessenandDeutsch,1996).III.SUBRECOILLASERCOOLINGA.Thesinglephotonrecoillimit.Howtocircumventit.Inmostlasercoolingschemes,¯uorescencecyclesnevercease.Sincetherandomrecoiltotheatombythespontaneouslyemittedphotonscan-notbecontrolled,itseemsimpossibletoreducetheatomicmomentumspreadbelowavaluecorrespond-ingtothephotonmomentum.Conditionde®nesthe``singlephotonrecoillimit.''Itisusualinlasercoolingtode®neaneffectivetemperaturetermsofthehalf-width(at1/ )ofthemomentumdistributionby.Inthetemperaturescale,conditionde®nesa``recoiltemperature'' 25\2k2 .(4)Thevalueofrangesfromafewhundrednanokelvinforalkalistoafewmicrokelvinforhelium.Itisinfactpossibletocircumventthislimitandtoreachtemperatureslowerthan,aregimewhichiscalled``subrecoil''lasercooling.Thebasicideaistocre-ateasituationwherethephotonabsorptionratewhichisalsothejumprateoftheatomicrandomwalkinvelocityspace,dependsontheatomicvelocityandvanishesfor0[Fig.5(a)].Considerthenanatomwith0.Forsuchanatom,theabsorptionoflightisquenched.Consequently,thereisnospontaneousreemissionandnoassociatedrandomrecoil.Onepro- FIG.4.Probeabsorptionspectrumofa1Dopticallattice(fromVerkerketal.,1992).Theupperpartofthe®gureshowsthetwocounterpropagatinglaserbeamswithfrequencyorthogonallinearpolarizationsformingthe1Dopticallattice,andtheprobebeamwithfrequencywhoseabsorptionismeasuredbyadetector.Thelowerpartofthe®gureshowstheprobetransmissionvs.Thetwolateralresonancescor-respondingtoampli®cationorabsorptionoftheprobeareduetostimulatedRamanprocessesbetweenvibrationallevelsoftheatomstrappedinthelight®eld(seethetwoinsets).ThecentralnarrowstructureisaRayleighlineduetotheantifer-romagneticspatialorderoftheatoms. FIG.5.Subrecoillasercooling.(a)therandomwalkoftheatominvelocityspaceissupposedtobecharacterizedbyajumpratewhichvanishesfor0.(b)asaresultofthisinhomogeneousrandomwalk,atomswhichfallinasmallin-tervalaround0remaintrappedthereforalongtime,ontheorderof,andaccumulate.ClaudeN.Cohen-Tannoudji:ManipulatingatomswithphotonsRev.Mod.Phys.,Vol.70,No.3,July1998 tectsinthiswayultracoldatoms(with0)fromthe``bad''effectsofthelight.Ontheotherhand,atomswith0canabsorbandreemitlight.Insuchabsorption-spontaneousemissioncycles,theirvelocitieschangeinarandomwayandthecorrespondingrandomwalkinspacecantransferatomsfromthe0absorbingstatesintothe0darkstateswheretheyremaintrappedandaccumulate[seeFig.5(b)].Thisremindsusofwhathap-pensinaKundt'stubewheresandgrainsvibrateinanacousticstandingwaveandaccumulateatthenodesofthiswavewheretheynolongermove.Note,however,thattherandomwalktakesplaceinvelocityspaceforthesituationconsideredinFig.5(b),whereasittakesplaceinpositionspaceinaKundt'stube.Uptonow,twosubrecoilcoolingschemeshavebeenproposedanddemonstrated.Inthe®rstone,called``Ve-locitySelectiveCoherentPopulationTrapping''(VSCPT),thevanishingof)for0isachievedbyusingdestructivequantuminterferencebetweendiffer-entabsorptionamplitudes(Aspectetal.,1988).Thesec-ondone,calledRamancooling,usesappropriatese-quencesofstimulatedRamanandopticalpumpingpulsesfortailoringtheappropriateshapeof(KasevichandChu,1992).B.BriefSurveyofVSCPTWe®rstrecalltheprincipleofthequenchingofab-sorptionby``coherentpopulationtrapping,''aneffectwhichwasdiscoveredandstudiedin1976(Alzettaetal.1976;ArimondoandOrriols,1976).Considerthe3-levelsystemofFig.6,withtwogroundstatesublevelsandoneexcitedsublevel,drivenbytwolaser®eldswithfrequencies,excitingthetransitions,respectively.LetbethedetuningfromresonanceforthestimulatedRamanprocesscon-sistingoftheabsorptionofonephotonandthestimulatedemissionofonephoton,theatomgoing.Oneobservesthatthe¯uorescenceratevanishesfor0.Plottedversus,thevariationsofaresimilartothoseofFig.5(a)withreplacedbyTheinterpretationofthiseffectisthatatomsareopti-callypumpedintoalinearsuperpositionofwhichisnotcoupledtobecauseofadestructivein-terferencebetweenthetwoabsorptionamplitudesThebasicideaofVSCPTistousetheDopplereffectformakingthedetuningofthestimulatedRamanpro-cessofFig.6proportionaltotheatomicvelocity.Thequenchingofabsorptionbycoherentpopulationtrap-pingisthusmadevelocitydependentandoneachievesthesituationofFig.5(a).Thisisobtainedbytakingthetwolaserwavescounterpropagatingalongaxisandbychoosingtheirfrequenciesinsuchawaythat0foranatomatrest.Then,foranatommovingwithavelocityalongtheaxis,theoppositeDopplershiftsofthetwolaserwavesresultinaRamanproportionaltoAmorequantitativeanalysisofthecoolingprocessetal.,1989)showsthatthedarkstate,forwhich0,isalinearsuperpositionoftwostateswhichdiffernotonlybytheinternalstate()butalsobythemomentumalongthe.(5)Thisisduetothefactthatmustbeassociatedwithdifferentmomenta,,inordertobecoupledtothesameexcitedstatebyab-sorptionofphotonswithdifferentmomenta.Furthermore,when0,thestate(5)isasta-tionarystateofthetotalatomlaserphotonssystem.AsaresultofthecoolingbyVSCPT,theatomicmomen-tumdistributionthusexhibitstwosharppeaks,centered,withawidthwhichtendstozerowhentheinteractiontimetendstoin®nity.The®rstVSCPTexperiment(Aspectetal.,1988)wasperformedonthe2metastablestateofheliumat-oms.Thetwolowerstateswerethe1Zeemansublevelsofthe2wasthe0Zeemansubleveloftheexcitedstate.Thetwocounterpropagatinglaserwaveshadthesamefrequencyandoppositecircu-larpolarizations.Thetwopeaksofthemomentumdis-tributionwerecenteredat,withawidthcorre-spondingto/2.Theinteractiontimewasthenincreasedbystartingfromacloudoftrappedprecooledheliumatomsinsteadofusinganatomicbeamasinthe®rstexperiment(Bardou,Saubameaetal.,1994).Thisledtomuchlowertemperatures(seeFig.7).Veryre-cently,temperaturesaslowas/800havebeenob-served(Saubameaetal.,1997).Infact,itisnoteasytomeasuresuchlowtempera-turesbytheusualtime-of-¯ighttechniques,becausetheresolutionisthenlimitedbythespatialextentoftheinitialatomiccloud.Anewmethodhasbeendevelopedetal.,1997)whichconsistsofmeasuringdi-rectlythespatialcorrelationfunctionoftheatoms),where)isthewavefunctionoftheatomicwavepacket.Thiscorrelationfunction,whichdescribesthedegreeofspatialcoher-encebetweentwopointsseparatedbyadistance,is FIG.6.Coherentpopulationtrapping.Athree-levelatomisdrivenbytwolaser®eldswithfrequenciesexcitingthetransitions,respectively.isthedetuningfromresonanceforthestimulatedRamanprocessinducedbetweenbythetwolaser®elds.When0,atomsareopticallypumpedinalinearsuperpositionofwhichnolongerabsorbslightbe-causeofadestructiveinterferencebetweenthetwoabsorptionClaudeN.Cohen-Tannoudji:ManipulatingatomswithphotonsRev.Mod.Phys.,Vol.70,No.3,July1998 simplytheFouriertransformofthemomentumdistribu-.ThismethodisanalogoustoFourierspec-troscopyinoptics,whereanarrowspectralline)ismoreeasilyinferredfromthecorrelationfunctionoftheemittedelectric®eld),whichistheFouriertransformof).Experimentally,themeasurementof)isachievedbylettingthetwoco-herentVSCPTwavepackets¯yapartwitharelativevelocity2duringadarkperiod,duringwhichtheVSCPTbeamsareswitchedoff.Duringthisdarkperiod,thetwowavepacketsgetseparatedbya,andonethenmeasureswithaprobepulseasignalproportionaltotheiroverlap.Figure8(a)showsthevariationswithofsuchasignal(whichisinfactequalto/2).Fromsuchacurve,onededucesatemperature/625,correspondingto/25.Figure8(b)showsthevariationsofwiththeVSCPTinteractiontime.Aspredictedbytheory(seenextsection),varieslinearlywithandcanreachvaluesaslargeas800.VSCPThasbeenextendedtotwo(Lawalletal.,1994)andthree(Lawalletal.,1995)dimensions.Fora1transition,ithasbeenshown(Ol'shaniiandMinogin,1992)thatthereisadarkstatewhichisde-scribedbythesamevector®eldasthelaser®eld.Moreprecisely,ifthelaser®eldisformedbyalinearsuperpo-sitionofplanewaveswithwavevectors1,2,...)havingthesamemodulus,one®ndsthatatomsarecooledinacoherentsuperpositionofpacketswithmeanmomentaandwithamomentumwhichbecomessmallerandsmallerastheinteractiontimeincreases.Furthermore,becauseoftheisomorphismbetweenthedeBrogliedarkstateandthelaser®eld,onecanadiabaticallychangethelasercon®gurationandtransferthewholeatomicpopulationintoasinglewavepacketortwocoherentwavepacketschosenatwill(Kulinetal.,1997).Figure9showsanexampleofsuchacoherentmanipulationofatomicwavepacketsintwodimensions.InFig.9(a),oneseesthetransversevelocitydistributionassociatedwiththefourwavepacketsobtainedwithtwopairsofcounter-propagatinglaserwavesalongtheaxisinahori-zontalplane;Figure9(b)showsthesinglewavepacket FIG.7.One-dimensionalVelocitySelectiveCoherentPopula-tionTrapping(VSCPT)experiment.Theleftpartofthe®gureshowstheexperimentalscheme.ThecloudofprecooledtrappedatomsisreleasedwhilethetwocounterpropagatingVSCPTbeamswithorthogonalcircularpolarizationsareap-pliedduringatime1ms.Theatomsthenfallfreelyandtheirpositionsaredetected6.5cmbelowonamicrochannelplate.Thedoublebandpatternisasignatureofthe1Dcoolingprocesswhichaccumulatestheatomsinastatewhichisalin-earsuperpositionoftwodifferentmomenta.Therightpartofthe®guregivesthevelocitydistributionoftheatomsdetectedbythemicrochannelplate.Thewidthofthetwopeaksisclearlysmallerthantheirseparation2,where9.2cm/sistherecoilvelocity.Thisisaclearsignatureofsubrecoilcool- FIG.8.Measurementofthespatialcorrelationfunctionofat-omscooledbyVSCPT(fromSaubameaetal.,1997).Afteracoolingperiodofduration,thetwoVSCPTbeamsareswitchedoffduringadarkperiodofduration.Thetwocoherentwavepacketsintowhichatomsarepumped¯yapartwitharelativevelocity2andgetseparatedbyadistance.ReapplyingthetwoVSCPTbeamsduringashortprobepulse,onemeasuresasignalwhichcanbeshowntobeequalto/2where)isthespatialoverlapbe-tweenthetwoidenticalwavepacketsseparatedby.From),whichisthespatialcorrelationfunctionofeachwavepacket,onedeterminestheatomicmomentumdistributionwhichistheFouriertransformof).Theleftpartofthe®guregives.Therightpartofthe®guregivesthecoolingtime,whereistherecoiltemperatureandthetemperatureofthecooledatomsdeterminedfromthewidthof).Thestraightlineisalinear®tinagreementwiththetheoreticalpredictionsofLevystatistics.Thelowesttemperature,ontheorderof/800,isequalto5nK. FIG.9.Two-dimensionalVSCPTexperiment(fromKulinetal.,1997).TheexperimentalschemeisthesameasinFig.7,butoneusesnowfourVSCPTbeamsinahorizontalplaneandatomsarepumpedintoalinearsuperpositionoffourdifferentmomentumstates,givingrisetofourpeaksinthetwo-dimensionalvelocitydistribution(a).WhenthreeofthefourVSCPTlaserbeamsareadiabaticallyswitchedoff,thewholeatomicpopulationistransferredintoasinglewavepacket(b).ClaudeN.Cohen-Tannoudji:ManipulatingatomswithphotonsRev.Mod.Phys.,Vol.70,No.3,July1998 intowhichthewholeatomicpopulationistransferredbyswitchingoffadiabaticallythreeofthefourVSCPTbeams.Similarresultshavebeenobtainedinthreedi-C.SubrecoillasercoolingandLevystatisticsQuantumMonteCarlosimulationsusingthedelayfunction(Cohen-TannoudjiandDalibard,1986;Zoller,Marte,andWalls,1987)haveprovidednewphysicalin-sightintosubrecoillasercooling(Bardou,Bouchaudetal.,1994).Figure10shows,forexample,therandomevolutionofthemomentumofanatomina1D-VSCPTexperiment.Eachverticaldiscontinuitycorre-spondstoaspontaneousemissionprocessduringwhichchangesinarandomway.Betweentwosuccessiveremainsconstant.Itclearlyappearsthattherandomwalkoftheatominvelocityspaceisanomalousanddominatedbyafewrareeventswhosedurationisasigni®cantfractionofthetotalinteractiontime.Asimpleanalysisshowsthatthedistribution)ofthetrappingtimesinasmalltrappingzonenear0isabroaddistributionwhichfallsasapowerlawinthewings.Thesewingsdecreasesoslowlythattheaverage(orthevariance)candiverge.Insuchcases,thecentrallimittheorem(CLT)canobviouslynolongerbeusedforstudyingthedistributionofthetotaltrappingtimeafterentriesinthetrappingzonesepa-ratedbyItispossibletoextendtheCLTtobroaddistributionswithpower-lawwings(GnedenkoandKolmogorov,1954;BouchaudandGeorges,1990).Wehaveappliedthecorrespondingstatistics,called``Levystatistics,''tosubrecoilcoolingandshownthatonecanobtaininthiswayabetterunderstandingofthephysicalprocessesaswellasquantitativeanalyticalpredictionsfortheasymptoticpropertiesofthecooledatomsinthelimitwhentheinteractiontimetendstoin®nity(Bardou,etal.,1994;Bardou,1995).Forexample,onepredictsinthiswaythatthetemperaturedecreasesas,andthatthewingsofthemomentumdistributiondecreaseas1/,whichshowsthattheshapeofthemomentumdistributionisclosertoaLorentzianthanaGaussian.ThisisinagreementwiththeexperimentalobservationsrepresentedinFig.8.[The®tinFig.8(a)isanexponential,whichistheFou-riertransformofaLorentzian.]Oneimportantfeaturerevealedbythistheoreticalanalysisisthenon-ergodicityofthecoolingprocess.Re-gardlessoftheinteractiontime,therearealwaysatomicevolutiontimes[trappingtimesinthesmallzoneofFig.5(a)around0]whichcanbelongerthanAnotheradvantageofsuchanewapproachisthatitallowstheparametersofthecoolinglaserstobeopti-mizedforgivenexperimentalconditions.Forexample,byusingdifferentshapesforthelaserpulsesusedinone-dimensionalsubrecoilRamancooling,ithasbeenpossibletoreachforcesiumatomstemperaturesaslowas3nK(Reicheletal.,1995).IV.AFEWEXAMPLESOFAPPLICATIONSThepossibilityoftrappingatomsandcoolingthematverylowtemperatures,wheretheirvelocitycanbeaslowasafewmm/s,hasopenedthewaytoawealthofapplications.Ultracoldatomscanbeobservedduringmuchlongertimes,whichisimportantforhighresolu-tionspectroscopyandfrequencystandards.TheyalsohaveverylongdeBrogliewavelengths,whichhasgivenrisetonewresearch®elds,suchasatomoptics,atominterferometryandBose-Einsteincondensationofdilutegases.Itisimpossibletodiscusshereallthesedevelop-ments.WereferthereadertorecentreviewssuchasAdamsandRiis(1997).Wewilljustdescribeinthissec-tionafewexamplesofapplicationswhichhavebeenrecentlyinvestigatedbyourgroupinParis.A.CesiumatomicclocksCesiumatomscooledbySisyphuscoolinghaveanef-fectivetemperatureontheorderof1K,correspondingtoarmsvelocityof1cm/s.Thisallowsthemtospendalongertimeinanobservationzonewhereamicro-wave®eldinducesresonanttransitionsbetweenthetwohyper®nelevelsofthegroundstate.Increasingdecreasesthewidthofthemicrowavereso-nancelinewhosefrequencyisusedtode®netheunitoftime.Thestabilityofatomicclockscanthusbeconsid-erablyimprovedbyusingultracoldatoms(GibbleandChu,1992;Leaetal.,1994).Intheusualatomicclocks,atomsfromathermalce-siumbeamcrosstwomicrowavecavitiesfedbythesameoscillator.Theaveragevelocityoftheatomsisseveralhundredm/s;thedistancebetweenthetwocavitiesisontheorderof1m.Themicrowaveresonancebetweenismonitoredandisusedtolockthefrequencyoftheoscillatortothecenteroftheatomicline.Thenar-rowertheresonanceline,themorestabletheatomic FIG.10.MonteCarlowavefunctionsimulationofone-dimensionalVSCPT(fromBardou,Bouchaudetal.,1994).characterizingthecooledatomsvstime.Eachverticaldiscontinuitycorrespondstoaspontaneousemissionjumpduringwhichchangesinarandomway.Betweentwosuccessivejumps,remainsconstant.Theinsetshowsazoomedpartofthesequence.ClaudeN.Cohen-Tannoudji:ManipulatingatomswithphotonsRev.Mod.Phys.,Vol.70,No.3,July1998 clock.Infact,themicrowaveresonancelineexhibitsRamseyinterferencefringeswhosewidthisdeter-minedbythetimeof¯ightoftheatomsfromonecavitytoanother.Forthelongestdevices,,whichcanbeconsideredastheobservationtime,canreach10ms,leadingtovaluesofontheorderof100Hz.MuchnarrowerRamseyfringes,withsub-Hertzline-widths,canbeobtainedintheso-called``Zachariasatomicfountain''(Zacharias,1954;Ramsey,1956).At-omsarecapturedinamagneto-opticaltrapandlasercooledbeforebeinglaunchedupwardsbyalaserpulsethroughamicrowavecavity.Becauseofgravitytheyaredecelerated;theyreturnandfallback,passingasecondtimethroughthecavity.Atomsthereforeexperiencetwocoherentmicrowavepulseswhentheypassthroughthecavity,the®rsttimeontheirwayup,thesecondtimeontheirwaydown.Thetimeintervalbetweenthetwopulsescannowbeontheorderof1sec,i.e.,abouttwoordersofmagnitudelongerthanwiththeusualclocks.Atomicfountainshavebeenrealizedforsodiumetal.,1989)andcesium(Claironetal.,1991).Ashort-termrelativefrequencystabilityof1.3,whereistheintegrationtime,hasbeenrecentlymeasuredforaonemeterhighcesiumfountainetal.,1996;Ghezali,1997).Forandforhasbeenmeasured.Infactsuchastabilityismostlikelylimitedbythehydrogenmaserwhichisusedasarefer-encesource,andtherealstability,whichcouldbemorepreciselydeterminedbybeatingthesignalsoftwofoun-tainclocks,isexpectedtoreachforaonedayintegrationtime.Inadditiontothestability,anotherveryimportantpropertyofafrequencystandardisitsaccuracy.Becauseoftheverylowvelocitiesinafoun-taindevice,manysystematicshiftsarestronglyreducedandcanbeevaluatedwithgreatprecision.Withanac-curacyof2,theBNM-LPTFfountainispres-entlythemostaccurateprimarystandard(Simonetal.1997).Afactorof10improvementinthisaccuracyisexpectedinthenearfuture.Toincreasetheobservationtimebeyondonesecond,apossiblesolutionconsistsofbuildingaclockforopera-tioninareducedgravityenvironment.Suchamicro-gravityclockhasbeenrecentlytestedinajetplanemak-ingparabolic¯ights.Aresonancesignalwithawidthof7Hzhasbeenrecordedina10environment.Thiswidthistwicenarrowerthanthatproducedonearthinthesameapparatus.Thisclockprototype(seeFig.11)isacompactandtransportabledevicewhichcanalsobeusedonearthforhigh-precisionfrequencycomparison.AtomicclocksworkingwithultracoldatomscouldnotonlyprovideanimprovementofpositioningsystemssuchastheGPS.Theycouldalsobeusedforfundamen-talstudies.Forexample,onecouldbuildtwofountains FIG.11.(Color)Themicrogravityclockprototype.Theleftpartisthe60cm60cm15cmopticalbenchcontainingthediodelasersourcesandthevariousopticalcomponents.Therightpartistheclockitself(aboutonemeterlong)containingtheopticalmolasses,themicrowavecavityandthedetectionregion.ClaudeN.Cohen-Tannoudji:ManipulatingatomswithphotonsRev.Mod.Phys.,Vol.70,No.3,July1998 clocks,onewithcesiumandonewithrubidium,inordertomeasurewithahighaccuracytheratiobetweenthehyper®nefrequenciesofthesetwoatoms.Becauseofrelativisticcorrections,thehyper®nesplittingisafunc-tionofisthe®ne-structureconstantandistheatomicnumber(Prestage,Tjoelker,andMaleki,1995).Sinceisnotthesameforcesiumandrubidium,theratioofthetwohyper®nestructuresdependsonBymakingseveralmeasurementsofthisratiooverlongperiodsoftime,onecouldcheckDirac'ssuggestioncon-cerningapossiblevariationofwithtime.Thepresentupperlimitforinlaboratorytests(Prestage,Tjoelker,andMaleki,1995)couldbeimprovedbytwoordersofmagnitude.Anotherinterestingtestwouldbetomeasurewithahigheraccuracythegravitationalredshiftandthegravi-tationaldelayofanelectromagneticwavepassingnearalargemass(Shapiroeffect,Shapiro,1964).B.GravitationalcavitiesforneutralatomsWehavealreadymentionedinSectionI.Cthepossi-bilityofmakingatomicmirrorsforatomsbyusingbluedetunedevanescentwavesatthesurfaceofapieceofglass.Concavemirrors[Fig.12(a)]areparticularlyinter-estingbecausethetransverseatomicmotionisthenstableifatomsarereleasedfromapointlocatedbelowthefocusofthemirror.Ithasbeenpossibleinthiswaytoobserveseveralsuccessivebouncesoftheatoms[Fig.12(b)],andsuchasystemcanbeconsideredasa``tram-polineforatoms''(Aminoffetal.,1993).Insuchanex-periment,itisagoodapproximationtoconsideratomsasclassicalparticlesbouncingoffaconcavemirror.Inaquantummechanicaldescriptionoftheexperiment,onemustconsiderthere¯ectionoftheatomicdeBrogliewavesbythemirror.StandingdeBrogliewavescanthenbeintroducedforsucha``gravitationalcavity,''whicharequiteanalogoustothelightstandingwavesforaFabry-Perotcavity(Wallis,Dalibard,andCohen-Tannoudji,1992).Bymodulatingatfrequencyintensityoftheevanescentwavewhichformstheatomicmirror,onecanproducetheequivalentofavibratingmirrorfordeBrogliewaves.There¯ectedwavesthushaveamodulatedDopplershift.Thecorrespondingfre-quencymodulationofthesewaveshasbeenrecentlydemonstrated(Steaneetal.,1995)bymeasuringtheen-ergychangeofthebouncingatom,whichisfoundtobeequalto,where2,...[Figs.12(c)and(d)].Thediscretenatureofthisenergyspectrumisapurequantumeffect.Forclassicalparticlesbouncingoffavibratingmirror,wouldvarycontinuouslyinacer-tainrange.C.BlochoscillationsInthesubrecoilregimewherebecomessmaller,theatomiccoherencelengthlargerthantheopticalwavelengththelasersusedtocooltheatom.Considerthensuchanultracoldatomintheperiodiclightshiftpotentialpro-ducedbyanonresonantlaserstandingwave.TheatomicdeBrogliewaveisdelocalizedoverseveralperiodsoftheperiodicpotential,whichmeansthatonecanpre-pareinthiswayquasi-Blochstates.Bychirpingthefre-quencyofthetwocounterpropagatinglaserwavesform-ingthestandingwave,onecanproduceanacceleratedstandingwave.Intherestframeofthiswave,atomsthusfeelaconstantinertialforceinadditiontotheperiodicpotential.TheyareacceleratedandthedeBrogliewave-decreases.When,thedeBrogliewaveisBraggre¯ectedbytheperiodicopti-calpotential.Insteadofincreasinglinearlywithtime,themeanvelocityoftheatomsoscillatesbackandforth.SuchBlochoscillations,whichareatextbookef-fectofsolid-statephysics,aremoreeasilyobservedwithultracoldatomsthanwithelectronsincondensedmatterbecausetheBlochperiodcanbemuchshorterthanthe FIG.12.Gravitationalcavityforneutralatoms(fromAminoffetal.,1993,andSteaneetal.,1995).(a)trampolineforatoms.Atomsreleasedfromamagneto-opticaltrapbounceoffacon-cavemirrorformedbyabluedetunedevanescentwaveatthesurfaceofacurvedglassprism.(b)numberofatomsattheinitialpositionofthetrapvstimeafterthetraphasbeenswitchedoff.Tensuccessivebouncesarevisibleinthe®gure.(c)principleoftheexperimentdemonstratingthefrequencymodulationofdeBrogliewaves.Theuppertracegivestheatomictrajectories(verticalpositionvstime).Thelowertracegivesthetimedependenceoftheintensityoftheeva-nescentwave.The®rstpulseisusedformakingavelocityselection.Thesecondpulseismodulatedinintensity.Thispro-ducesavibratingmirror,givingrisetoafrequency-modulatedre¯ecteddeBrogliewavewhichconsistsinacarrierandside-bandsatthemodulationfrequency.Theenergyspectrumofthere¯ectedparticlesisthusdiscretesothatthetrajectoriesofthere¯ectedparticlesformadiscreteset.(d)thiseffectisde-tectedbylookingatthetimedependenceoftheabsorptionofaprobebeamlocatedabovetheprism.ClaudeN.Cohen-Tannoudji:ManipulatingatomswithphotonsRev.Mod.Phys.,Vol.70,No.3,July1998 relaxationtimeforthecoherenceofdeBrogliewaves(incondensedmatter,therelaxationprocessesduetocollisionsareverystrong).Figure13showsanexampleofBlochoscillations(BenDahanetal.,1996)observedoncesiumatomscooledbytheimprovedsubrecoilRa-mancoolingtechniquedescribedinReicheletal.V.CONCLUSIONWehavedescribedinthispaperafewphysicalmecha-nismsallowingonetomanipulateneutralatomswithlaserlight.Severalofthesemechanismscanbesimplyinterpretedintermsofresonantexchangesofenergy,angularandlinearmomentumbetweenatomsandpho-tons.Afewofthem,amongthemostef®cientones,re-sultfromanewwayofcombiningwell-knownphysicaleffectssuchasopticalpumping,lightshifts,andcoher-entpopulationtrapping.Wehavegiventwoexamplesofsuchcoolingmechanisms,Sisyphuscoolingandsubre-coilcooling,whichallowatomstobecooledinthemi-crokelvinandnanokelvinranges.Afewpossibleappli-cationsofultracoldatomshavebeenalsoreviewed.TheytakeadvantageofthelonginteractiontimesandlongdeBrogliewavelengthswhicharenowavailablewithlasercoolingandtrappingtechniques.Onecanreasonablyexpectthatfurtherprogressinthis®eldwillbemadeinthenearfutureandthatnewapplicationswillbefound.Concerningfundamentalproblems,twodirectionsofresearchatleastlookprom-ising.First,abettercontrolof``pure''situationsinvolv-ingasmallnumberofatomsinwell-de®nedstatesex-hibitingquantumfeaturessuchasverylongspatialcoherencelengthsorentanglement.Inthatperspective,atomic,molecularandopticalphysicswillcontinuetoplayanimportantrolebyprovidinga``testingbench''forimprovingourunderstandingofquantumphenom-ena.Asecondinterestingdirectionistheinvestigationofnewsystems,suchasBosecondensatesinvolvingamacroscopicnumberofatomsinthesamequantumstate.Onecanreasonablyhopethatnewtypesofcoher-entatomicsources(sometimescalled``atomlasers'')willberealized,openingthewaytointerestingnewpossibili-Itisclear®nallythatallthedevelopmentswhichhaveoccurredinthe®eldoflasercoolingandtrappingarestrengtheningtheconnectionswhichcanbeestablishedbetweenatomicphysicsandotherbranchesofphysics,suchascondensedmatterorstatisticalphysics.TheuseofLevystatisticsforanalyzingsubrecoilcoolingisanexampleofsuchafruitfuldialogue.TheinterdisciplinarycharacterofthepresentresearchesonthepropertiesofBosecondensatesisalsoaclearsignoftheincreaseoftheseexchanges.Adams,C.S.,H.J.Lee,N.Davidson,M.Kasevich,andS.Chu,1995,Phys.Rev.Lett.,3577.Adams,C.S.,andE.Riis,1997,Prog.QuantumElectron.Alzetta,G.A.,A.Gozzini,L.Moi,andG.Orriols,1976,NuovoCimentoB,5.Aminoff,C.G.,A.M.Steane,P.Bouyer,P.Desbiolles,J.Dalibard,andC.Cohen-Tannoudji,1993,Phys.Rev.Lett.Arimondo,E.,andG.Orriols,1976,Lett.NuovoCimentoAshkin,A.,1980,Science,1081.Askarian,G.A.,1962,Zh.Eksp.Teor.Fiz.,1567[Sov.Phys.JETP,1088(1962)].Aspect,A.,E.Arimondo,R.Kaiser,N.Vansteenkiste,andC.Cohen-Tannoudji,1988,Phys.Rev.Lett.,826.Aspect,A.,E.Arimondo,R.Kaiser,N.Vansteenkiste,andC.Cohen-Tannoudji,1989,J.Opt.Soc.Am.B,2112.Autler,S.H.,andC.H.Townes,1955,Phys.Rev.,703.Balykin,V.I.,V.S.Letokhov,Yu.B.Ovchinnikov,andA.I.Sidorov,1988,Phys.Rev.Lett.,2137.Bardou,F.,1995,Ph.D.thesis(UniversityofParisXI,Orsay).Bardou,F.,J.P.Bouchaud,O.Emile,A.Aspect,andC.Cohen-Tannoudji,1994,Phys.Rev.Lett.,203.Bardou,F.,B.Saubamea,J.Lawall,K.Shimizu,O.Emile,C.Westbrook,A.Aspect,andC.Cohen-Tannoudji,1994,C.R.Acad.Sci.,877.Barrat,J.P.,andC.Cohen-Tannoudji,1961,J.Phys.Radium,329and423.BenDahan,M.,E.Peik,J.Reichel,Y.Castin,andC.Salomon,1996,Phys.Rev.Lett.,4508.Bouchaud,J.P.,andA.Georges,1990,Phys.Rep.,127.Castin,Y.,1991,Ph.D.thesis(UniversityofParis,Paris).Castin,Y.,andJ.Dalibard,1991,Europhys.Lett.,761.Castin,Y.,andK.Molmer,1995,Phys.Rev.Lett.,3772.Chu,S.,J.E.Bjorkholm,A.Ashkin,andA.Cable,1986,Phys.Rev.Lett.,314.Chu,S.,L.Hollberg,J.E.Bjorkholm,A.Cable,andA.Ash-kin,1985,Phys.Rev.Lett.,48.Clairon,A.,C.Salomon,S.Guellati,andW.D.Phillips,1991,Europhys.Lett.,165.Cohen-Tannoudji,C.,1961,C.R.Acad.Sci.,394.Cohen-Tannoudji,C.,1962,Ann.Phys.(Paris),423and469.Cohen-Tannoudji,C.,1992,``Atomicmotioninlaserlight,''inFundamentalSystemsinQuantumOptics,LesHouchesses-sionLIII,editedbyJ.Dalibard,J.M.Raimond,andJ.Zinn-Justin(ElsevierScience,Amsterdam),p.1.Cohen-Tannoudji,C.,andJ.Dalibard,1986,Europhys.Lett. 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