(1)ultraold atoms in an optial resonator Ph.D

ultraold atoms in an optial resonator

Ph.D. Thesis

Dávid Nagy

Supervisor: Dr. Péter Domokos

Researh Institute for Solid State Physis and Optis

Quantum Optis and Quantum Informatis Department

2010

1 Introdution 4

1.1 Laser ooling and trapping . . . 5

1.2 Bose-Einsteinondensation. . . 8

1.3 Cavity quantum eletrodynamis . . . 10

1.4 Outlineof the Thesis . . . 15

2 Theoretial bakground 18 2.1 Single atom motion insidea single mode avity . . . 18

2.1.1 JaynesCummingsmodel. . . 19

2.1.2 Quantum master equation . . . 21

2.1.3 HeisenbergLangevin equationsand bosonization . . . 23

2.1.4 Linearly polarizablepartileand the dispersivelimit. . . 24

2.2 Mean-eld theory . . . 26

2.2.1 Mean-eld modelfor thermalatoms . . . 28

2.2.2 Mean-eld theory for a BEC . . . 29

3 Mean-eld model of self-organization 32 3.1 Self-organization of athermal loudin a ring avity . . . 35

3.1.1 Model . . . 35

3.1.2 Results. . . 39

3.1.3 Summary . . . 42

3.2 Self-organization of aBose-Einstein ondensatein anoptial avity . . . 43

3.2.1 Model . . . 43

3.2.2 Steady state . . . 46

3.2.3 Colletive exitations . . . 48

3.2.4 Strongolletiveoupling regime . . . 55

4 Exess noise depletion of a Bose-Einstein ondensate inan optial av-

ity 59

4.1 Flutuations aroundthe meaneld . . . 60

4.2 Condensate depletion . . . 64

4.3 Summary . . . 68

5 Nonlinear quantum dynamis of two BEC modes dispersively oupled by an optial avity 69 5.1 The BEC-avity system and itsanalogy toopto-mehanial systems . . . 70

5.2 Eliminationof the photon eld. . . 72

5.2.1 Adiabati eliminationof the eld . . . 73

5.2.2 First-orderorretion to the adiabatielimination . . . 74

5.2.3 Eetive masterequation. . . 75

5.2.4 Relationwith linearized models . . . 77

5.3 Tunnelinginthe optialbistabilityregime . . . 79

5.4 Coherent density wave osillations. . . 84

5.5 Summary . . . 86

6 Dipole-dipole instability of atom louds in a far-detuned optial dipole trap 88 6.1 Dipole-dipoleinteration . . . 89

6.1.1 Linear polarization . . . 91

6.1.2 Cirularpolarization . . . 92

6.2 Large detuning limit . . . 92

6.3 Mean-eld approximation . . . 93

6.4 Stability analysis . . . 94

6.5 Possibilityof Bose-Einstein ondensation . . . 98

6.6 Summary . . . 99

7 Summary 100

Aknowledgements 102

Bibliography 103

List of publiations 117

Introdution

Thenatureoflighthas beenof entralinterestsinetheearliestdaysofsiene. Manyof

the greatsientists devoted theirtime toimprove ourunderstanding of light.Long after

thespeulationsoftheGreekphilosophers,therstmilestoneworkemergedinterestingly

inthe Islamiworldnearly thousand years ago.The Bookof Optis waswrittenin1021

by the Arabpolymath Ibnal-Haytham.Besides explainingthe proess of vision,he laid

downthepriniplesofoptisonanexperimentalbasis,thatgainedhimthetitleoffather

of modern optis. Seven enturies had to pass in Europe until the birth of Newton's

theory of olour published in the remarkable treatise Optiks appeared in 1704. The

following important stage was Maxwell's paper from 1864, in whih he established the

basisoflassialeletrodynamis.Einsteinhimselfhadbeenaptivatedbythemysteryof

lightsine hisyouth. Hissimple explanation ofthe photoeletri eetin termsof light

quantain1905revealedthefundamentalharateroflight.In1917,itwasalsoEinstein,

whorstmadeadistintionbetweenspontaneousandinduedquantumproesses,whih

laid ground not only for lasing, but for the laser manipulation of atoms. The piture

beameompletein1927,whenDiraquantizedtheeletromagnetield,andalulated

the Einstein oeients,establishing quantum eletrodynamis.

Forenturies, optiswasdealingwith light-matterinterationsfromtheviewpointof

light.Matterwasonlyusedtotailorthepropagationorthespetrumoflight,whereas its

propertieswere simpliedand taken intoaountasarefrativeindex.Thedevelopment

ofthelaserinthe1960smadepossibletheoppositesituation.Bytherepeatedsattering

ofphotons being inthe same quantum state,oherentmanipulationofmatterwith light

beame possible. This advaned tool revolutionized atomi physis, whih was a our-

ishingeld by that time. Lasersenabled physiists topreiselymeasure and manipulate

not onlythe internalstate of atoms (with laser spetrosopy and optialpumping), but

1.1 Laser ooling and trapping

In the optial domain, photons have large enough momentum to onsiderably modify

the atomi motion. Sattering of one photon yields generally a few m/s hange in the

veloityofalkaliatoms.Inanintense lasereld,the rateofinelastiphotonsattering is

determinedby the spontaneous emission rate of the atom, that amountsto anaelera-

tion about

10 ⁵

^{timeslarger than the} gravitationalaeleration onEarth. Aordingly, a ollimatedthermal atomi beam an be `stopped' by a ounter-propagating laser beam

withina few tens of entimeters, rst demonstrated by Phillips and Metalf in1982 [1℄.

FollowingtheproposalofHänshandShawlow[2℄,Chuwasabletooolthe slowatoms

further by exploiting the Doppler eet in 1985 [3℄. In the Doppler ooling sheme the

atoms are illuminated from all diretions by lasers whose frequeny is detuned slightly

below an atomi resonane line. Then, the Doppler-shift pulls the beams opposing the

atomi motion loser to resonane and pushes the o-propagating beams farther from

resonane, hene the atom is more likely to absorb photons with momentum opposite

to itsmotional diretion. On average, the inelastiphoton sattering yields a nonlinear

frition fore, whih an be linearized for slowveloities resultingina visous damping.

The three pairs of ounter-propagating laser beams reate a photon sea that ats like

an exeptionally visous uid, literallyalled optial molasses. Treating the atoms as

pointlikepartiles, they undergolassialBrownianmotion[4℄. Besidesfrition, theyex-

perienemomentumdiusionaswellarisingfromthe randomkiksofthespontaneously

emittedphotons. Aording totheEinstein relation,the equilibriumtemperatureof the

atomi loud is determined by the ompetition of these two eets. The lowest ahiev-

able temperaturesales with the linewidth of the atomitransitionand itis realized at

a frequeny of half linewidth below resonane. For alkali atoms this so-alled Doppler

temperatureison the order of

100µK

^.

After the ooling stage, the next hallenge was to onne the roaming atoms inside

a trap. The pioneer of optial trapping was Ashkin, who rst demonstrated the ael-

erationand trapping of miron-sizedlatexspheres in water with fousedlaser beams in

1970[5℄. Hisinvention,the optialtweezer[6℄, whihis suitabletotrapand move miro-

sopi objets (even an individual virus or a living baterium), beame widely used in

biophysis. Furthermore,with Gordon he examined the possibility of trappingatoms in

aradiationtrapby desribing thelightfores, inludingtheirdependene onthe atomi

veloityand quantumutuations[7℄. They identied thetwo types offores atingina

tion followed by spontaneous emission, and the dipole foreoriginating in the oherent

redistributionof photons due tostimulatedemission.They showed that unlikethe sat-

tering fore, the dipole fore is onservative, and the orresponding optial potentialis

proportionaltothe eld'sintensity.Moreover, itssigndepends onthe detuningbetween

the laser frequeny and the atomi resonane. When the laser is tunedbelowresonane

(red detuning) the optial potentialis negative, hene the atoms are attrated towards

the high intensity regions of the eld, behaving as high eld seekers. However, for blue

detuning the potential is positive, so the atoms are repelled out of the high intensity

regionsbeominglow eld seekers. Surprisingly,the dipoleforevanishes onthe atomi

resonane 1

.

Sine Ashkin's rst proposal [8℄ in 1978, a wide variety of dipole traps have been

developed, and they beame standard tools for manipulating and trappingold neutral

atoms[9℄.Theremarkableideaofthefar-o-resonanetrapping(FORT)madepossibleto

suppressspontaneousemissiontoanegligiblelevelbyinreasingtheatomlaserdetuning

along withthe eld intensity [10℄. The reation ofhighly onservative optialpotentials

with far-o-resonant laser beams opened up the way to atom optis, the oherent

manipulationof atomi matterwaves with light[11℄.

In ontrast to the long-termsuess of the dipoletraps, inthe beginningthey failed

to ollet the majority of atoms (

10 ³

^{out of}

10 ⁶

^{) from the optial molasses beause}

of their small trapping volume. The rst robust trapping sheme was realized in 1987

[12℄, following the idea of Jean Dalibard. The magneto-optial trap (MOT) uses the

radiation pressure fore for ombining Doppler ooling with magneto-optial trapping.

Asthe foreisveloity dependentdue tothe Doppler shift,itan bealsomadeposition

dependent through the Zeeman shift produed by a magneti eld gradient. If one has

twored-detunedounter propagatinglaserbeamsofoppositeirularpolarizationsanda

magnetieldthathangessignattheenter,thendierentZeemansublevelsarepulled

loser to resonane preferring absorption from opposite beams at opposite sides. Thus,

one an set the parametersso thatthe atomsbeing righttothe enter would preferably

absorb photons from the beam oming from the right and the atoms on the left would

1

This intriguing nature of the dipole fore an be easily understood from lassial physis. The

osillatingeletrield

E

induesanatomidipolemoment

d

,thatanbeonsideredadrivendamped

harmoni osillator.In this manner, the optial potential is reated by the dipole interation energy

− dE

,that variesin spae.An osillatordrivenbelowitsresonanerespondsinphasewiththedriving eld,resultinginnegativeinterationenergy,whereasanosillatordrivenaboveitsresonaneosillates

out of phase with the driving, yielding positive interation energy. On resonane, the osillator is

90

dominantlyabsorbphotons fromtheleftbeam,thatresultsinanet forepointingtothe

enter. Inthree dimensions, withananti-Helmholtzmagneti eld superimposed onthe

optial molasses, a restoring fore an be reated, whih pushes the atoms towards the

entre of the trap, where the magneti eld is zero. Beause of its simultaneous ooling

and trappingfaility, the MOT has beome a widespread soure of laser-ooled atoms,

thatisapabletooolandhold upto

10 ¹⁰

^{atomswithinatrappingvolumeofaboutone}

ubi entimetre.

Strikingly, muh lower temperatures were deteted in the MOT than the Doppler

limit,whihinitiatedthe ndingofanew typeofooling proess,the so-alledpolariza-

tiongradientooling[13℄.Itemergesfromthedeliateombinationoftwoeets:optial

pumping and AC Stark shift. Optial pumping is suitable to reate a one-way transi-

tion between dierent groundstate Zeeman sublevelsby irularlypolarized laser light.

Whereas,ACStarkshiftisthe splittingoftheatomiZeemansublevelsintheosillating

eletri eld of a non-resonant optial exitation. Both eets depend on the polariza-

tion of light. Generally, in a MOT, laser polarization varies in spae between dierent

linear and irular polarizations, and the light shifts of the sublevels vary aordingly,

givingrisetodistintperiodioptialpotentialsfordistintsublevels.Coolingispossible

when optial pumping transitions take plae from higer to lower potential energies. In

this ase, a moving atom ontinually limbs potentialhills at the expense of its kineti

energy, beause at the top of the hills it is pumped down to a lower energy spin state.

After the famous gure of the Greek mythology, this kind of ooling proess was hris-

tened Sisyphus ooling. Sueeding the Doppler ooling, it ould slow the atoms down

tothe reoil temperature, whihorresponds to the energysale of the reoil due tothe

absorbtionorthe emissionofasinglephoton.Foralkaliatoms,the reoiltemperatureis

onthe order of

1µK

^.

One might think that this is the fundamental limit of laser ooling, sine the last

spontaneously emitted photon would result in a random veloity of this magnitude.

Nevertheless, it is still possible to irumvent it with more sophistiated methods like

veloity-seletiveoherentpopulationtrapping[14℄andRamanooling[15℄.Byapplying

these tehniques, the temperature of the atoms ould be lowered by three orders of

magnitude,downtothe nanokelvinrangeinthemid-1990s[16,17℄.Thesemethodshave

nostrit ooling limits, however, the aompanying large atom lossgives alowerbound

onthe temperatureand anupperbound on the ahievable atom density.

Forfurtherreviewsonlaserooling,onsulttheNobelleturesofC.Cohen-Tannoudji

1.2 Bose-Einstein ondensation

Bythe early 1990s,the quest forBose-Einstein ondensationina diluteatomi gas had

beome a entral goal of the atomi physis ommunity. [21, 22℄. The most important

benet of diluteness isin the redution of the interpartileinterations, whih normally

preludes a gas from reahing quantum degeneray. Cooling a dense gas would lead to

a liquid,then a solid, whereas ooling a dilute gas an lead toa metastable phase, that

is dened by quantum statistis.

2

In fat,the density tunes the ratio between two- and

three-body ollisions among the gas partiles. At suiently low densities, the three-

bodyreombinationbeing responsiblefor moleuleformationis highlysuppressed in

favour ofthe two-bodyollisionsthat areessentialforreahingthermalequilibrium.For

instane, at typial experimentaldensities being on the order of

10 ¹⁴

^m

^{− 3}

^{, the lifetime}

of the metastable quantum gas is a few tens of seonds, whih anyway, seems like an

eternity on the timesaleof the atomi proesses.

Inasimpliedpiture,thequantum natureofagas shows up whenthe wavepakets

of itspartiles overlap. The size of the wave pakets, in turn, isapproximatelygiven by

thedeBrogliewavelengthofthepartiles,whihisinverselyproportionaltotheirthermal

veloity,henetothe squarerootofthetemperature.Therefore,the ritialtemperature

of the Bose-Einsteintransitionsaleswith the pariledensity,i.e., itis lowerfor amore

dilutegas.Atsuhlowdensitiesrequiredbymetastability,itrangesfromten nanokelvin

toten mirokelvin, that usually raises need forsubreoil ooling.

Despite of the amazing developmentof laser ooling, quantum degeneray ould not

be reahed solelyby its means.However low temperatures ould be produed with sub-

reoilooling,the aompanyingatomlosspreluded toobtainhighenoughphasespae

densities. The other route to degeneray, i.e. inreasing the atom number ina MOT at

aonstant temperature, alsofailed.Above aritialdensity, instabilitiesoureddue to

the attenuation ofthe laser beams and themultiplesattering ofspontaneously emitted

photons within the sample [23℄.

Meanwhile,severalexperimentalgroupsmadeeortstoBoseondensespin-polarized

atomi hydrogen gas using traditional ryogenis ombined with magneti elds. They

made a breakthrough by applying evaporative ooling in a magneti trap, a method

proposedby Hessin1986[24℄,andby 1991they obtained phasespae densitieswithin a

2

The only exeption is He

4

whih remains a liquid even at zero temperature, and it beomes a

superuidbelow

2 . 2

Kelvin.ForhalfaenturyitwasthesolerealizationofBose-Einsteinondensation, howeveritisastronglyinteratingsystemwithaondensatefrationlessthan

10%

.

fatorof

5

^{ofondensing.Coolingby}evaporationisastraightforwardwaytodereasethe temperature of a trapped interating atomi gas. By merely letting the highest-energy

atoms esape from the trap, the mean energy of the remaining atoms derease, hene

thegas getsolder,anditsdensityinreases soitbeomesompressedbothinrealand

inmomentum spae. This an beontinued as long asthe remnantthermalizes itself at

the new temperature, therefore two-body ollisionsplay a key role inthe proess.

Ultimately,the suesiveombinationoflaseroolingandevaporativeoolingproved

to be the most eient tehnique for reating ultraold quantum gases. The large, few

hundred

µK

^{old atomi samples produed in a MOT ould be easily loaded into a}

spatially superimposed magneti trap, whih provided an exellent ontainer for spin

polarizedatoms[25℄,thusitservedasagoodplatformforfurtheroolingbyevaporation.

Using these tehniques, the rst ahievements of Bose-Einstein ondensation in dilute

alkali gases were made in 1995 by Cornell and Wieman with rubidium [26℄ and by

Ketterle with sodium atoms [27℄. Amazingly, the ondensate fration reahed almost

hundred perent inthe experiments, meaningthat a pure quantum matterwas formed.

The physis of ultraoldquantum gases has beome a proliferating eld in the last

fteen years [28℄.Asa referene, Bose-Einstenondensates(BEC) are routinelyrealized

inmorethan fty laboratoriesworldwidefor alargevariety ofisotopesinludingmainly

alkaline and alkaline earth atoms:

7

Li,

23

Na,

41

K,

84

Sr,

85

Rb,

87

Rb,

133

Cs and a few

others like

1

H,

4

He,

52

Cr and

174

Yb [29℄. Degenerate Fermi gases are alsoreated from

6

Li and

40

K opening up new researh diretions [30℄. The attrativeness of ultraold

quantum gases stems from their simpliity, ontrollability and generality. First, they

are relatively easy toprodue, maintain and manipulate invarious forms. Seond, their

parametersareexible,highlyontrollable,andthereexistpowerfulmethodstomeasure

them. For instane, both the sign and the strength of the atomatom ollisions an be

tuned by an external magneti eld via the so-alled Fesbah resonanes [31℄. Third,

theunderstanding ofthesesystems requiresinterdisiplinaryphysisinludingnonlinear

and quantum optis, atomi and solid-state physis. Bose-Einstein ondensates an be

onsideredasoherentatomi deBrogliewavesobeyinga nonlinearwaveequation,thus

theyshowinterferenepropertiesfamiliarfromnonlinearoptislikefour-wavemixingand

solitonformation [32℄. Byouplingout atomsfrom atrapped BEC, one gets aoherent

soureof matterwave,i.e. anatomlaser [33, 34℄. Furthermore,ultraoldgasesare good

quantum simulatorsfor a number of models in solid-state physis.As an example,BCS

(Bardeen-Cooper-Shrieer)pairinganbestudiedexperimentallyinanultraoldFermi

moleules an form from the fermioni atoms, whih in turn are bosons, hene an be

Bose ondensed. The rossover between this moleularBEC and the atomi BCS state

is a subjet of urrent researh [30℄. As another example, artiial perfet rystals, or

so-alledoptial latties an be made by lling the atomi gas intothe periodi optial

potentialreated by standingwavelaser elds.Thissystem realizesthelean Hubbard

modelforeither bosoniorfermioni partilesinone,two,orthree dimensions,hene it

is able toshowquantum phase transitions,e.g. the superuidMott transition[36℄.

1.3 Cavity quantum eletrodynamis

At the end of the last entury, tehnology gave another gift to physiists, namely the

high-qualityoptialresonatorwhihprovedtobeanewpreiouselementinexperimental

quantum optis. Cavity Quantum Eletrodynamis (CQED) has not only opened up

oneptually new ways in the manipulation of atomi internal and external degrees of

freedom, but also reated a new mirosopi system onsisting of a single atom and a

single photon,the so-alled atom-photonmoleule.

Historially, CQED emerged fromthe simple idea that the radiative properties of a

dipoleanbemodiedbytayloringthe modedensityof thesurroundingeletromagneti

eld. In 1946, Purell pointed out the enhanement of spontanous emission rates for

nulear magneti moment transitions in the presene of a resonant eletri iruit [37℄.

Forty years later, the Purell eet was demonstrated experimentally for exited atoms

traversing a mirowave avity. Tuning the avity, spontaneous deay from the exited

state ould beeither enhaned or inhibitedby a few perent [38, 39, 40℄.

Theseearlyexperimentswere performedinthe perturbativeregimeof CQED,where

theresonatorwasmerelyusedtomodifythemodedensityoftheeletromagnetivauum.

Thedynamisoftheavityeldwasirrelevant,sineboththeatomsandthephotonsleft

thevolumeoftheavitytoofastforfurtherinteration.Inthemid-1990s,however,theso-

alled strong oupling regime was ahieved, in whih the single-photon Rabi frequeny

desribing atom-eld oupling exeeded the rate of avity deay and that of atomi

spontaneousemissionand the inverse ofthe interation time. Thismeans thatthe atom

and the avity eld exhange an exitation many times before it is dissipated into the

environment, orbefore the atom leaves the avity.

Atthis point,ananalogy with the formationof a diatomimoleulean beinvoked,

where the exhange of a valene eletron leads to binding and non-binding moleular

reatea new objet, anatom-photonmoleule[41℄, with an energy spetrumdisplaying

alearlyresolved normal-modesplitting.Thelowerlevelsdesribeatom-photonbinding,

whereastheupperlevelsorrespondtotherepulsionoftheatomoutoftheavityvolume.

Slowexited atoms are indeed attrated [42℄ or repelled [43℄ by the vauum eld of the

avity, when either the photon energy or the atomi energy dominates the other one,

respetively. The situation is opposite in the ase of a ground state atom interating

with a avity ontaining a single photon.This frequeny dependent behaviouralls the

dipole fore into our mind, whih we know from laser ooling. Indeed, this is the same

foreoriginatingfrom the energy of the indued atomi dipoleinthe avity eld.

Strong ouplingwasrst ahieved in1996 by Harohe in the mirowave regimewith

aresonatorbuiltfromsuperonduting niobiummirrors.Theavity eldwasoupledto

exitedstatesofhighly-exited(so-alledRydberg)atomsying throughtheavitywith

thermal veloity. A series of fundamental quantum mehanial experiments were per-

formedontheatom-photonmoleule.Forinstane, theappearane ofdisretefrequeny

omponents inthe spetrum of energy exhange between anatom and aweak oherent

eld inside the avity provided a diret proof of eld quantization [44℄. In another ex-

itingexperiment,ameasurementproess wassimulatedby atomstraversingthe avity,

whih ontainedasuperposition ofoherent states.The measurement-indued quantum

deoherene of Shrödinger at states was observed quantitatively [45℄.

In the optial domain, strong oupling was made possible by the development of

dieletri multilayer mirrors. So as to obtain high enough intraavity eld, the short

wavelength has to be ompensated with small mode volume. However, as the distane

between the mirrors is redued, the number of photon reetions inreases, therefore

extremely good mirrors are needed for keeping down the avity deay. In a typial ex-

perimentalsetup, the avity lengthis below

100µ

^{m,and the reexivity of the mirrorsis}

higher than

0.99999

^{,resulting ina nesse between}

10 ⁵

10 ⁶

^. Pitorially,this means that aphoton bounes up to

10 ⁶

^{times between the mirrors beforeexiting the system.}

High-quality optial resonators signify a breakthrough in the study of light-matter

interations whih is omparable to the appearane of the laser. While in a laser eld,

strong interation is obtained by the sattering of many idential photons onthe atom,

inaavity,strongouplingisreahed bythe reylingofthe samephotonthatimpinges

on the atom several times. This piture also explains the main dierene, that is the

bakationof the atomson the light,whih isnegligiblefora lasereld, but essential in

aavity.Thehighnumberofphotonroundtripsenhanesthesensitivityoftheresonator

fallingthrough the avity [46℄. In fat,the optial resolution being halfa wavelength in

free spae is improved proportionally to the square root of the round trip ount. Thus,

the esaping photons give a unique way to monitor single atom trajetories inside the

avity.Inthe atom-avitymirosoperealizedin2000[47,48℄,afewmirometersspatial

resolution has been ahieved fromthe miroseond-resolved analysis of the transmitted

eld intensity.

The other novel aspet of atom-avity interation in optial resonators is the me-

hanial fore ating on the atomi enter-of-mass motion. As previously disussed at

laser ooling, optial photons arry large enough momentum to exert appreiable light

fores.Wehavedistinguishedtwotypesof them,theradiationpressure foreoriginating

in photon absorption, and the dipolefore arising from the oherent sattering of pho-

tons between radiationmodes viaindued emission. Both of these fores have diusion

through the utuations of the atomi dipole. In addition, spontaneous emission gives

rise to athird, so-alled reoil diusion derivingfrom the atomi reoils indued by the

randomlyemittedphotons.Inanoptialavity,theseforesare highlytunable,however,

inanon-trivialmanner, forexampleviasuppressingorenhaning spontaneousdeay.In

ontrast to a laser eld where a stati optial potential an be assigned tothe dipole

fore , the avity provides a dynami optial potential, whih depends on the atomi

motion. Furthermore, the eld of the resonator utuates due to the photon leakage

through themirrors,whihyieldsanadditionaldiusion fore.Finally,fordepiting the

strength of the interation, letusemphasize thatin the atom-avity mirosope[47, 48℄

singleavity photonswere abletobind inorbitsingleatomsforuptoafewmilliseonds.

Strongouplingintheoptialdomainwasrstattainedin1998by theKimblegroup

(Calteh)[49℄ and ayear later by the Rempegroup (MPQ Garhing) [50℄. As aninitial

step,theseCQED experimentsused laseroolingtehniquesforproduingaold atomi

sample.Typially,alargenumberofatoms(about

10 ⁷

^{)areolletedinaMOT,andsome}

ofthemaredroppedoutorkikedoutwithalasertowardstheavityvolumeataveloity

ofafewmetersperseond.Theavityiseitherpumpeddiretlywithalaserbeamviaone

of itsmirrors (avity pumping), orfed by oherent sattering onthe atoms from alaser

that is perpendiular to the avity axis (so-alled atomi or transverse pumping). This

latter sheme has the advantage that it solves three problems simultaneously. Besides

the feeding ofthe system, atransverse slowly movinglaser standingwavean alsoserve

asan optialonveyor belt[51℄ thatis suitableto optially transportsingle atoms from

the MOT intothe avity [52℄. In addition,the lasereld provides anoptialdipoletrap

the avity, three dimensional trapping of a single atom has been ahieved [53, 54℄ for

trappingtimes up to 17seonds [55℄.

Optialavities antrap atoms forlong times if they are exploitedfor atomooling.

What the MOT realizes for alarge numberof atoms,the optialresonator attains for a

single atom, viz. simultaneous ooling and trapping. Generally, ooling means the irre-

versible damping of the atomi kineti energy through oupling to the environment. In

laser ooling, the atomi enter-of-mass motion is oupled to the surrounding vauum

modes via inelastisattering of laser photons, thusdissipation manifests itself through

spontaneousemission.In the perturbativeregimeof CQED (for weakatom-photonou-

pling), the avity an be used as a spetral lter to modify the mode density of the

eletromagneti eld around the atomi resonane. In suh a oloured vauum, those

sattering proesses an be favoured in whih the photon frequeny is onverted up-

wards taking away kineti energy. Based on this onept, a number of ooling methods

havebeen developed at the dawn of optialCQED [56, 57, 58,59℄. As a matter of fat,

in these tehniques the resonator remains a passive element that is used to improve

or generalize laser ooling shemes only having the dissipative hannel of spontaneous

emission.

In the strong ouplingregime of CQED, a novel ooling mehanism takes plae [60,

61,62,63℄.The atomissostronglyoupledtoagiven modeoftheavitythattheyshare

alldissipationhannelsof thesystem. Then,theatomikineti energyan bedissipated

through the photon loss hannel of the avity mode. For adequately hosen detunings,

ooling arises from the oupled atomeld dynamis. Astonishingly, the dipole fore

whih isonservativeina lasereld an resultinaveloity dependentfrition forein

the avity. This isbeausethe motion of the atom ats bak on the eld by modulating

the avity resonane. Hene, the eld reates a dynami optial potential whose depth

depends on the atomi position. However, the eld follows the atom's motion with a

delay that orresponds tothe avity relaxation time. Therefore, the atom enounters a

retardedoptialpotential,andtheamountofretardationdependsonitsveloity.Besides

the dissipation, diusion alsoemerges in the system in agreement with the utuation-

dissipationtheorem. Pitorially,the optiallattieutuates duetothe photons leaking

outofthe avity,thatheatstheatom.Spontaneousemissionplaysnoroleinthisooling

proess, it an even be eliminated for large enough atompump detuning. In this ase,

the nal temperatureis determined solelyby the photonloss hannel of the avity, and

inanalogytofree-spaelaser oolingitisproportionaltotheavitymodelinewidth.

oolinganrealizesub-Doppler temperatures.Therst experimentaldemonstrationwas

reported in2003 [64℄, followed by a series of experimentswith dierent parameters and

geometries [65, 66, 55℄.

The most important benet of avity ooling is that it provides a general method

to ool polarizable partiles. As it uses the dipolefore, no real optial exitations need

take plae, hene in ontrast toDoppler ooling nolosed optial yle isneessary

for the repeated sattering of photons. In the limit of large atomlaser detuning, the

far-o-resonane trapping sheme an be generalized for the avity eld whih an si-

multaneouslytrap andoolarbitrarypolarizableobjets[67℄.Inthiswise,avityooling

seemstoprovideasolutionforthe optialoolingof moleules,thatisstillasubtleopen

problem[68, 69℄.

Over reent years, the study of many-body phenomena in optial resonators has

attratedonsiderableattention.Forseveralatoms,lightmatterinterationinthestrong

oupling regime is, by nature, a many-body problem. As all atoms are oupled to the

sameavity mode,one of themfeels thehangeinthe eldthat isaused by the others.

Thus, the eld of the resonator mediates a long-range atomatom interation, whih

gives rise to interesting olletiveeets.

Although olletive light fores in an optial resonator have been reported in the

Rempe group as early as 2000, atthat stage of the experiments it was diult to trap

more than one atom in the smallavity volume [70℄. So as to observe many-body phe-

nomena,the avitymirrorswere plaedfarther apart,andfurther magnetiormagneto-

optial trappingwas used tokeep a large number of atoms inside the avity. Extending

the mode volume dereases the atom-eld oupling, however, the olletive oupling,

whih sales with the atom number, an remain large. At rst, three groups started to

set up experiments for studying olletive eets in optial resonators. Vuleti¢ at the

MITsueeded intrappingafew thousandatomsinside aonfoalmultimodeavity.In

TübingenandinHamburg,Zimmermann[71℄andHemmerih[72℄ heldnearly

10 ⁶

^atoms

inside aring avity,whih has two ounter-propagating degeneratemodes.

The prospets for the olletiveooling and trapping of adilute atomi loudin the

eld of anoptial avity were investigated theoretially for the rst time by the Ritsh

groupinInnsbruk[73,74℄.In2002,thenumerialsimulationofthetransverselypumped

system showed that for properly hosen parameters the atoms arranged into a regular

pattern whih supported eetive ooling [75℄. The initially homogeneous atom loud

illuminatedfromtheside evolved by spontaneoussymmetrybreakingintooneof the

The phenomenon is in lose analogy with Dike superradiane [76℄. Two phases of the

avity eld, diering by

π

^{, orrespond to the two possible atomi patterns. Within a}

year, this so-alled self-organization phenomenon was demonstrated experimentally by

Vuleti¢[77℄.Furthertheoretialadvanerevealed thatthis wasarealphasetransitionin

the thermodynami limit,whereit ould be wellderibed by a mean-eld model[78℄.

In a ringavity,the self-organization islosely relatedto the olletiveatomireoil

lasing (CARL) [79, 80, 81℄, whih originally introdued as a gain proess analogous to

thefreeeletronlaser[82℄.IntheHamburgandTübingenexperiments[83,84℄,oneprop-

agatingmode of the ring avity isdriven, and abovea pumping threshold, a stationary

eld builds up inthe othermode along withthe formationof a regulardensity pattern.

Hene, the non-pumped mode is fed by oherent photon sattering on the atomi grat-

ing from the pumped mode. In 2007, the CARL proess was demonstrated with both

ultraoldand Bose-Einsteinondensed atoms [85℄.

More reently, a peuliar atomavity system has been realized [86℄. At the ETH

in Zurih, the Esslinger group sueeded in trapping a Bose-Einstein ondensate of

10 ⁵

Rb atoms inside an ultra-high nesse optial miroavity. Working in the far-detuned

limit,they attained strong dispersive ouplingbetween the atoms and the eld. Even a

single atom of the ondensate realizes strong oupling to the eld, hene the olletive

oupling of so many atoms is enormous. On the one hand, this means that the avity

is highly sensitive to the dynamis of the BEC, so for instane it an be used as a

strobosope to monitor matter-wave motion [87℄. On the other hand, the eld reates

both oherent and inoherent exitations in the BEC. The avity-mediated long-range

atomatom interation has signiant eets as it is in the same order of magnitude

or even larger than the ollisionalinteration between the atoms. This exoti quantum

many-body system shows a number of generi, olletive eets [88, 89, 90℄, whih are

stillbeing explored both by theory and by experiment.

1.4 Outline of the Thesis

The entral part of this Thesis is devoted to the theoretial desription of many-body

eets ourring in aloud of old orultraoldatoms whih isdispersively oupledto a

high-nesseoptialavity.Idesribetheoupledatomelddynamisondierentlevels,

starting from a lassial mean-eld model through a mean-eld Bogoliubov theory up

to a full quantum simulation of the system in terms of a quantum master equation. In

interation between the atoms.Generallythishas anegligibleeet,however, forstrong

laser elds itan eventuate the ollapse of adense atomi loud.

Being more spei, in Chapter 2 of the Thesis, I review the basi theoretial tools

for desribing the atomeld interation in optial avities [63℄, thus providing a solid

basis for the forthominghapters.

Chapter 3dealswiththe avity-induedself-organizationintwodierentsystems.In

therst one,athermalatomiloudisoupledtothe eldof aring resonator[91℄,while

in the seond one, a BEC interats with a single-mode avity [92℄. Although a thermal

loudofatomsseems toberatherdierentfromaBEC,thephase transitiontakesplae

similarly,sine itarises fromthe same type of atomeld oupling. For both systems, I

disussthephenomenonintermsofamean-eldapproah,andIdrawaphasediagramas

funtionsoftheontrolparameters.InaseofaBEC,theBogoliubovexitationspetrum

oftheompoundatomavitysystemisalsoalulated,whihprovidesadditionalinsight

intothe phase transition.

In Chapter 4of this Thesis, I ompute the exess noise depletion of a Bose-Einstein

ondensatearisingfromtheinterationwiththeavityeld[93℄.Evenazero-temperature

BEC annot fully oupy its ground state, sine the atomatom interations kik out

atomstotheexitationmodes.Inanoptialavity,besidesthes-waveollisionstheeld-

mediated atomatom interation together with the photonloss noise is a new soure of

depletion.IuseLangevinequationslinearizedaroundthemean-eldsolutiontoalulate

the steady-state atom numberin the exited states of the ondensate. The study of the

depletion ompletes the mean-eld modelof Chapter 3, as it desribes the error of the

mean-eld approximation.

In Chapter 5, I provide a full quantum simulation for a single-mode exitation of a

BEC by a high-nesse optial avity mode [94℄. This system is formallyanalogous to a

broad lass of optomehanial systems, where miromehanial osillators are oupled

to resonator modes via the radiation pressure fore. For weak elds, the avity mode

dominantlyouples a homogeneousondensate toa single exitationmode (playingthe

role ofthe osillator),ontowhih thedynamis an berestrited. Byadiabatiallyelim-

inating the photon eld, I derive a quantum master equation for this BEC exitation

mode, whih aounts for both the oherent and the dissipative parts of the dynamis

due to the oupling of a driven, lossy mode of aresonator. Numerialsimulation of our

modelallows for exploringthe quantum limitof optomehanial systems ina lassially

bistableregime,andleadstobetterunderstandingthequantumbak-ationoftheavity

In Chapter 6 of this Thesis, I alulate the eet of the indued dipole-dipoleinter-

ation on the far-o-resonane trapping of old atoms [96℄. The laser eld indues an

atomi polarizationwhih gives rise to a radiative atomatom interation that is disre-

gardedinmostases. Nevertheless, athigh densitiesandstrongelds itan provokethe

ollapse of the loud. I apply a mean-eld approah to alulate the boundary of the

stable equilibriumregion, where the thermal motion of atoms stabilizesthe gas against

self-ontration. I draw a phase diagram, and disuss the limitations imposed by the

dipole-dipoleinstability on the parameters needed to reah Bose-Einstein ondensation

inan optialdipoletrap.

Finally,I summarize my resultsahieved during my Ph.D. inChapter 7.

Theoretial bakground

Theaimofthishapteristoprovideageneralframeworkforthedesriptionofdispersive

interation between anatomiensembleandthe eld ofa high-nesseoptialresonator,

that onstitutes a ommon basis for the olletive phenomena disussed in this Thesis.

Startingwith the understanding of the atomavity interation for a single atom and a

single avity mode, we introdue the standard approximations whih lead to dispersive

oupling. Then,weoutline theonstrutionof the mean-eldmodelsboth for athermal

atomiloud and for a Bose-Einsteinondensate.

2.1 Single atom motion inside a single mode avity

Atomsareharaterisedbyexternal(motional,i.e.positionandmomentum)andinternal

(eletroni) degrees of freedom. They interat with the eletromagneti eld via their

transition dipole moment, meaning that light ouples dierent eletroni states of the

atoms. However, in the optial domain the atomi motional states are also oupled by

the photon sattering proess. In the beginning, we are going to deal with the internal

dynamis of the atom in whih its external position appears as a parameter. Then, we

release the atom and derive the light fores dening its motion. The dissipative atom

avity dynamis an be desribed by two equivalent approahes, either by a master

equationfor theredued density operatorof thesystem, orby the HeisenbergLangevin

equationswhihinlude noiseterms toaount for dissipation.Weuse both approahes

in this Thesis, always the one wih ts the given problem best. Finally, we eliminate

the exited atomi level and explain why spontaneous emission an be negleted in the

2.1.1 JaynesCummings model

We begin withthe desription ofthe dipoleouplingbetween asingle atomand asingle

quantized mode of the eletromagneti eld. Let us onsider two energy levels of the

atom, separated by

~ ω _A

^{. The} orresponding atomi states onstitute an orthonormal basis of a losed two-dimensional Hilbert spae. The ground state

| g i

^{is assumed to be}

stable,while theexited state

| e i

^isallowedtospontaneouslydeay tothe groundstate.

It isonvenient todesribe this so-alledtwo-levelatom by assoiatinga spin-halfto it.

Thus, we an use the Paulispin operators,

σ = | g i h e | ,

^(2.1a)

σ ^† = | e i h g | ,

^(2.1b)

σ z = 1

2 ( | e i h e | − | g i h g | ),

^(2.1)

whih fullthe ommutationrelationsof thespin-half operatoralgebra.The rsttwoof

them are the atomi lowering and raising operators, whilst the third one is the atomi

inversion operator.The Hamilton operatorof the atomthen beomes

H A = ~ ω A | e i h e | = ~ ω A σ ^† σ = ~ ω A

σ z + 1

2

.

^(2.2)

Sinethe eletronwave funtionsinthe stationarystatesareentrosymmetri,the

2 × 2

matrix of the dipolemomentoperator

ˆ d = e ˆ r

^{, has zero diagonalelements,moreover its}

o-diagonalelementsbeome the same real vetor

d

^{for aproperlyhosen global phase.}

Consequently,theatomhasatransitionaldipolemomentwhihorrespondstoeletroni

transitions, and an be expressed with the spin operators as

ˆ d = d (σ + σ ^† ),

^(2.3)

with the sole o-diagonalmatrix element

d = d _eg = d _ge = h g | d ˆ | e i

^.

The eletrield operator of a quantized eletromagneti eld mode of frequeny

ω

is given in the Shrödinger pitureby

E(r) = ˆ i r ~ ω

2ǫ ₀ V e(f (r)a − f ^∗ (r)a ^† ),

^(2.4)

where

f(r)

^{isthe modefuntion,}

e

^isthe polarizationvetor,and

V

^{isthemode volume.}

The operators

a ^†

^and

a

^{are the photon reation and} annihilation operators. By on- sidering, for instane, the single TEM00 mode of a high-Q optial resonator, its mode

funtionanbesimplytaken

f (x) = cos kx

^{alongthe diretionofthe avityaxis}

x ˆ

^,with

the wavenumber

k = 2π/λ

^{. In the transverse diretion it has a Gaussian shape deter-}

minedbytheavitywaist.TheexpressionEq.(2.4)revealsthesalingoftheeletrield

with the mode volume. While in free spae

V

^{is a titious} quantization volume, in a avity,itisgivenbyawell-denedGaussianmodebetweenthemirrors;

V = R

| f (r) | ² d ³ r

^,

where

sup {| f (r) |} = 1

^{. Hene, the eld of a single photon beomes larger as the avity}

mode volumeis dereased.The Hamilton operatorof aquantized avitymode issimply

H C = ~ ω C a ^† a

^,with

ω C

^{being the frequeny of the avity resonane.}

The atomeld interation an be treated within the dipole approximation. As the

atomi radiusis muh smaller than the optial wavelengths, the spatial variation of the

eletrieldis negletedonthe atomilengthsale. Theatom isregarded asapointlike

dipole whih interats with the eletri eld at its atual position

r

^{. In this spirit, the}

interation Hamiltonianis writtenas

H AC = − dˆ ˆ E(r) = − i r ~ ω

2ǫ 0 V de (σ + σ ^† )(f(r)a − f ^∗ (r)a ^† ).

^(2.5)

Whentheenergysaleoftheinterationisdwarfedbytheatomiandphotoniexitation

energies, only those terms play important role in the dynamis, whih onserve the

exitationnumber. Therefore, in the rotating wave approximation, we negletthe terms

ontainingtwo reationortwoannihilationoperators, whih are ounter-rotating inthe

Heisenberg piture, arriving to the Jaynes-Cummings model[97℄,

H _JC ^′ = ~ ω C a ^† a + ~ ω A σ z − i ~ g(r)(σ ^† a − a ^† σ).

^(2.6)

Here, we assumed a real mode funtion and introdued the single photon Rabi fre-

queny desribing atom-photon oupling,

g (r) = g 0 f(r)

^{, with the maximum value}

g 0 = q ω C

2 ~ ǫ ⁰ V de

^{.Finally,wetransformtheaboveHamiltonoperatorintoaframerotatingwith}

thedrivingfrequeny

ω

^{thatorrespondstothelaserexitation.Asaresult,tworelevant}

frequeny parameters appear in our model, the atomi detuning

∆ A = ω − ω A

^{and the}

avity detuning

∆ _C = ω − ω _C

^{. The} transformation of the operators is straightforward, henewekeep theiroriginalnotation,however, thedetuningsrefertothe rotatingframe.

The transformed Hamilton operatorthen reads

H _JC = − ~ ∆ _C a ^† a − ~ ∆ _A σ _z − i ~ g(r)(σ ^† a − a ^† σ).

^(2.7)

Beause of the rotating wave approximation, the Jaynes-Cummings Hamilton oper-

ator preserves the number of exitation quanta inthe system. Consequently, the states

with a xed (

n + 1

^{) exitation number form an invariant subspae spanned by the ba-}

sis vetors

| e, n i

^and

| g, n + 1 i

^{. The} eigenstates of the system are usually referred to as

dressed states, and they are easily obtained by the rotation of these two basis vetors.

Note that the atom-eld oupling

g (r)

^{depends on the atomi position}

r

^{. Thus, if the}

atom being initiallyinits groundstate enters intointerationwith the eld, the system

remainsinthesame eigenstate,whihadiabatiallyfollows theslowatomimotion.The

energy ofthe orrespondingdressed state signiesareal optialpotentialfor the atomi

entralmass motion.Depending onthe sign of the detunings, a groundstate atom an

feel potential hillsorvalleysatthe intensity maxima of the eld.

2.1.2 Quantum master equation

Hithertowehavedisussed theoherentdynamisofanatomandaavity mode.Never-

theless,both of theminterat withthe environmentwhih isonstitutedbythe vauum

eldmodes.Asaresultexitationsofthesystem deayviatwopossibledissipationhan-

nels, namely, by spontaneous emission from the exited atomi level or by leaking out

of avity photons through the mirrors.These proesses are haraterized by the atomi

spontaneousemissionrate

2γ

^{andthe photon lossrateof the avity denoted by}

2κ

^.The

ontinuum of vauum modes forms a broadband reservoir, whose orrelation funtions

deay on a muh shorter time sale than that of the relevant dynamis of the system.

ThisallowsonetoinvoketheMarkov approximation,whihassumesthattheutuations

inthe reservoir are

δ

-orrelated,hene their bak-ationon the system is awhite noise.

Aordingly, the environment has no memory in the sense that information entering it

does not ome bak. In other words, the Markov approximation separates the system

(atom plus avity) from the environment (vauum modes) by slaving the environment

variables tothe system variables.

The standard formalism whih desribes the dissipative dynamis of open quantum

systems reliesonanequationof motionforthe redued density operator

ρ

^{, thatisgiven}

by traing out the environmental degrees of freedom from the density operator of the

losed grand system ontaining the environment [98℄. This quantum master equation

takes the followinggeneral form

˙ ρ = 1

i ~ [H, ρ] + L ρ,

^(2.8)

where the Hamilton operator

H

^{and the} Liouvillean superoperator

L

^{desribe the on-}

servative and the dissipative parts of the dynamis, respetively. In fat, the Neumann

equation for the density operator is amended by terms that take into aount environ-

Figure2.1: Sheme of a linear resonator showing the pumpingand lossproesses.

For simpliity, we present the master equation for a single atom interating with a

single avity mode, that an begeneralized for more atomsand modes by astraightfor-

ward summation. We allowfor both pumpingmethods applied in the experiments (see

Fig. 2.1); the diret feeding of the avity mode viaone of the mirrors (desribed by the

parameter

η

^{) and the transverse pumping sheme, where the atom is} illuminated from the side by alaser standing wave (whosestrength ismeasured by the Rabifrequeny

Ω

of the atom). We assume that the two pumps have the same frequeny and the same

phase.

Our model Hamiltonianfor the atomavitysystem then reads

H = p ²

2m − ~ ∆ A σ z − ~ ∆ c a ^† a − i ~ g(r) σ ^† a − a ^† σ

− i ~ η a − a ^†

− i ~ Ω(r) σ ^† − σ

.

^(2.9)

The rst line is the Jaynes-Cummings Hamiltonian, Eq. (2.7) omplemented by the

kineti energy of the atom, while the seond line ontains the avity and the atomi

pumpingterms. The atomeld interation term and the atomi pump term depend on

thepositionoftheatomthroughtheavitymodefuntionas

g (r) = g 0 f(r)

^,andthrough

thespatialvariationofthetransversepumpingeldas

Ω(r) = Ω h(r)

^,respetively.These terms ouplethe atomi motionaldegrees of freedom.

The Liouvilleansuperoperator isgiven by

L ρ = κ 2aρa ^† − { a ^† a, ρ } + γ

2

Z

d ² u N(u)σe ^{− ik A ur} ρe ^{ik A ur} σ ^† − { σ ^† σ, ρ }

.

^(2.10)

Here the rst term desribes the avity deay, and the seond term stands for the

atomi spontaneous emission. The notation

{ , }

^{is used for the} antiommutator. The

atomofmass

m

^{has position}

r

^andmomentum

p

^fullling

[r α , p β ] = i ~ δ αβ

^.Intheseond

term, thereis anaveraging overthe randomdiretion(denoted by the unit vetor

u

^{) of}

the spontaneouslyemittedphotonwiththe diretiondistribution

N ( u )

harateristi to the given atomi transition. The wavenumber

k A

^{orresponds to the atomi transition}

frequeny

ω A

^.Notethatthewavenumbersinoursystemarepratiallyallequal,sinethe detuningsaremuhsmallerthantheoptialfrequeniesthemselves:

∆ A , ∆ C ≪ ω, ω A , ω C

^,

therefore

k A ∼ = k C ∼ = k

^.

2.1.3 HeisenbergLangevin equations and bosonization

Theinternaldynamisof theatom dened bythe Hamiltonian(2.9) andthe Liouvillean

(2.10) is the soure of a wealth of interesting phenomena that are widely studied in

quantumoptis,suhasRabiosillationsandatomeldentanglement[99℄.Nevertheless,

wefousontheexternalatomidynamis,thereforeweshallsimplifythepresentedmodel

by eliminatingthe atomi internal degrees of freedom. To do this, we use a desription

thatisequivalenttothe masterequation,namely theHeisenberg-Langevin equationsfor

the system operators.

Asastartingpoint,weadiabatiallyseparatethetimesalesoftheatomiinternaland

external motions. We assume the that the atom is moving slowly in the eld suh that

the the orresponding frequeny is muh smaller than the deay rates of the internal

dynamis, expressed for the veloity,

kv ≪ κ, γ

^{. This means, on the one hand, that}

the atomi polarization relaxes to its steady state value dened by the eld at the

urrent atomi position. On the other hand, the external degrees of freedom beome

justparameters for the internaldynamis.

In theHeisenberg piture,the equationsof motionofthe atomiand the eldopera-

tors are given by their ommutators with the Hamiltonoperator.In ase of dissipation,

however, orrelated noise operators together with deay terms appear in the equations

of motion, in agreement with the utuation-dissipation theorem. They arise from the

interation ofthe system withthe vauumeld.The expetationvalues ofthe Langevin

noise operators are zero, whereas their orrelations desribe diusion. The Heisenberg-

Langevin equations are derived via the elimination of the vauum modes by formally

integrating their equations of motion in the Heisenberg piture, and using the Markov

approximation(seepage394ofRef.[100℄).Suhadesriptionofthesystemisequivalent

to the one provided by the master equation for the redued density operator Eq. (2.8).

of the atomi polarization.

The Heisenberg-Langevin equations for the eld and the atomi internal variables then

read

˙

a = (i∆ C − κ)a + Ng ^∗ (r)σ + η + ξ,

^(2.11a)

˙

σ = (i∆ A − γ )σ + 2g (r)σ z a + 2Ω(r)σ z + ζ,

^(2.11b)

˙

σ _z = − g(r)(σ ^† a + a ^† σ) − Ω(r)(σ ^† + σ) − γ(σ _z + 1/2) + ζ _z .

^(2.11)

The non-zero two-timeorrelation funtions of the Langevin noise operatorsare

h ξ(t 1 )ξ ^† (t 2 ) i = 2κδ(t 1 − t 2 ),

^(2.12a)

h ζ(t 1 )ζ ^† (t 2 ) i = 2γδ(t 1 − t 2 ),

^(2.12b)

h ζ _z (t ₁ )ζ ^† (t ₂ ) i = 2γ h σ ^† i δ(t ₁ − t ₂ ),

^(2.12)

h ζ(t 1 )ζ z (t 2 ) i = 2γ h σ i δ kl δ(t 1 − t 2 ),

^(2.12d)

h ζ z (t 1 )ζ z (t 2 ) i = 2γ( h σ z i + 1/2)δ(t 1 − t 2 ).

^(2.12e)

In equations(2.11), the eletromagneti eld is also a variable in ontrast to free-

spae laser ooling that leads to a oupled nonlinear atomeld dynamis. For large

atompump detuning, when the atomi saturation remains small, one an replae the

inversion operator

σ z

^{with its expetation value in the ground state, thus}

2 h σ z i ≈ − 1

^.

This approximation is alled bosonization of the atomi dipole, sine the atomi opera-

torsthensatisfythe bosoniommutationrelation

[σ, σ ^† ] = − 2σ z ≈ 1

^{.Aswenegletthe}

dynamis of the

σ z

^{operator, we also set}

ζ z

^{to zero. The} bosonization, hene, linearizes the remainingHeisenberg-Langevin equations(2.11a,b), by breakingthe atomeldor-

relation

h σ ^z a i = − 1/2 h a i

^. Interestingly, this relationis exatly valid when there is only one exitationquantum inthe system [101℄.

2.1.4 Linearly polarizable partile and the dispersive limit

Inthelarge atomidetuning limit(when

| ∆ A |

^{farexeedsthe otherparameters}

g 0 h a i , Ω

^,

kv

^{), the internal dynamis follows the external atomi motion in the radiation eld,}

hene the atomi operators

σ

^,

σ ^†

^{an be} adiabatiallyeliminated from the model. This simpliestheatomtoalinearlypolarizablepartile.Thisouldseemasevererestrition,

however, it alsoonstitutes a generalizationof our modelinthe sense that it willapply

Tehnially,oneaveragesoutthefastosillationoftheatomioperator

σ

^with

∆ A

^on

anintermediatetime sale

δt

^{suh that}

δt ≫ ∆ ⁻ _A ¹

^,but

δt

^{remainsstillsmallonthe time}

sale of the relevant dynamis. Formally, this is equivalent to the

σ

^{operator expressed}

from Eq. (2.11b) by setting the left hand side

σ ˙ = 0

^{. Introduing the operator of the}

dimensionlesseletri eld

E(r) = ˆ f (r)a + Ω(r)/g 0 ,

^(2.13)

the atomi polarizationoperatorthen beomes

σ ≈ g

i∆ A − γ E(ˆ ˆ r ) = − 1

g (iU 0 + Γ 0 ) ˆ E (ˆ r ),

^(2.14)

where

U 0 = g ₀ ² ∆ A

∆ ² _A + γ ²

^{, and}

Γ 0 = g ² ₀ γ

∆ ² _A + γ ² .

^(2.15)

Physially, the parameters

U 0

^and

Γ 0

^{orrespond to the real and imaginary parts of the}

omplexsuseptibility

χ

^{of theatom, sinethey desribethe linear}relationshipbetween the atomi polarizationand the eletrield aordingto

P = ǫ 0 χE

^{. Withthe usage of}

Eq.(2.3)and Eq.(2.14),

U ₀

^and

Γ ₀

^{areexpressed with the}suseptibility

χ = χ ^′ − iχ ^′′

^by

U ₀ = − ω C

V χ ^′ Γ ₀ = − ω C

V χ ^′′ ,

^(2.16)

where

ω _C

^{is the mode frequeny and}

V

^{isthe mode volume.}

InsertingtheaboveapproximationEq.(2.14)oftheatomioperatorsintotheoriginal

Hamiltonoperator (2.9), we get the followingeetive Hamiltonian

H eff = p ²

2m − ~ ∆ C a ^† a − i ~ η(a − a ^† ) + ~ U 0 E ˆ ^† (r) ˆ E(r).

^(2.17)

Here the lastterm an be expanded as

~ U 0 f ² (r)a ^† a + ~ η t f (r)h(r)(a + a ^† ) + ~ U 0

Ω ²

g ₀ ² h ² (r).

^(2.18)

The eetive Liouvilleanarising fromthe adiabati eliminationis

L eff ρ = κ 2aρa ^† − { a ^† a, ρ } +

− Γ ₀ { E ^† ( ˆ r)E(ˆ r), ρ } +

+ 2Γ 0

Z

d ² u N(u) ˆ E (ˆ r)e ^{− ik A uˆ r} ρe ^{ik A uˆ r} E ^† (ˆ r).

^(2.19)

These eetive operators desribe the oupled dynamis of the external motion of the

linerlypolarizablepartileandtheavityeldaordingtothequantummasterequation

(2.8). The eld reatesanoptial potentialfor the atom that isproportionalto

U ₀

^{, and}

by

U 0

^{and auses an eetive avity deay with rate}

Γ 0

^{. On a mirosopi level, the}

formerand thelatter eetsan beonnetedwith theatomiindued andspontaneous

emissionproesses. Furthermore,the atomipump

Ω

^{gives rise toan eetive pumping}

strength for the avity mode that is

η t = Ωg 0 ∆ A /(∆ ² _A + γ ² )

^{. It arises fromthe oherent}

sattering ofphotons fromthe transverse pumpintothe avity,hene itdepends onthe

atomipositionalong the avity axis

x ˆ

^{via the avity mode funtion as}

η t (x) = η t f(x)

^.

Now, weare prepared toonsider the dispersive limit ofthe atomavityinteration.

This is the same limit that is taken in the far-o-resonane trapping sheme [10℄, and

it alsoapplies to CQED [67℄. It is seen fromEq. (2.15) that

U 0

^and

Γ 0

^{sale dierently}

with the atomi detuning

∆ A

^{. Namely, for large}

∆ A

^,

U 0 ≈ g ² ₀ /∆ A

^{, while}

Γ 0 ≈ g ₀ ² γ/∆ ² _A

^.

Thus, their ratio

Γ 0 /U 0 ≈ γ/∆ A

^{tends tozero as}

∆ A

^{inreases. The depthof the optial}

potential,however,isgivenby

U 0 h a ^† a i

^{.Therefore,oneansuppress}spontaneousemission whilekeepingthe optialfores onaonstantlevelbyinresingthe atomidetuning

∆ A

together with either of the pumping strengths

η

^or

η t

^{. In the} experiments,

| ∆ A |

^an

easilybetunedup to

100 . . . 1000γ

^{atreasonablelaser powers. Inthedispersivelimit,we}

neglet the eets stemmingfrom atomi spontaneous emission,and set

γ = 0

^,

Γ 0 = 0

^.

With this, we erase the last two terms of the eetive Liouvillean operator Eq. (2.19),

saying that dissipation of the system is onlypossiblevia the avity eld mode.

2.2 Mean-eld theory

Up tothis point, we have disussed the oupled dynamisof a single atom and asingle

avitymode.Themotionofasingleatominsideaavityisanappealingsystemproduing

eets suh as dynamial avity ooling [60℄. The orrelated motion of a few atoms

interatingwiththeavityeldisalsoaninterestingproblemresultinginavitymediated

rossfrition[102℄ andmotionalentanglement[103,104℄.InthisThesis, however, weare

interested in olletive eets produed by a large number of atoms. So as to desribe

suh a system, we need to make further approximations regarding the atomi external

degrees of freedom.

One possibility is the semilassial approximation, whih assumes that the atomi

wave pakets are well loalized in both position and momentum spaes [4℄. Hene, one

an desribe the external atomi motion with lassial variables, obeying the lassial

Langevin equations derived from the operator equations of motion [63℄. Using this ap-

proximation,oneansimulatethedynamisofaoldatomiensembleoupledtoasingle

The otherpossibilityisthe mean-eldapproah,whihredues the many-bodyprob-

lem to an eetive one-body problem by assuming that all atoms move in the same

mean-eldpotentialreated by the eldof the avity.This alsoinludestheassumption

that the eld, reahing its steady-state on a faster time sale, adiabatially follows the

atomi motion. Thus, we exlude eets like avity ooling whih are based on the de-

layed dynamisof theeld.In themean-eldmodel,the avityeld isdeterminedsolely

by atomiensembleaverages,thus the bak-ationof eah individualatom isnegleted.

An important advantage of this approah is that it orresponds to the thermodynami

limitof the system, where the atom number

N → ∞

^{, the single atom oupling}

g 0 → 0

^,

while the olletive oupling desribed by

Ng ₀ ²

^or

NU 0

^{is kept onstant. In a physial}

realization of the limit, the avity volume would be inreased (by raising the avity

lenth

l cav

^{) at a onstant atomi density, whih would derease the oupling aording}

to

g ∝ 1/ √

V

^{. Sine the photon round trip time inreases with the avity length, the}

reexity of the mirrorshas tosale as

∝ 1/l _cav

^tokeep

κ

^onstant.

The mean-eldmodelan be thoughtofas aself-onsistenteld theory. The density

distributionofthe atomssimultaneouslyxesthe steadystateoftheavitymode(whih

is a oherent state), and the orresponding optial potential ats bak on the atomi

distribution. Hene, the mean-eld solution has to be determined self-onsistently. In

pratie, we alulate it by the numerial iteration of the atomi density with the self-

onsistentpotential.

In Chapter 3 the Reader will see, that the mean-eld desription gives qualitatively

similar results both for a thermal loud and a Bose-Einstein ondensate of the atoms.

Thisisso,beausetheavityelddependsexlusivelyontheatomidensitydistribution,

hene it provides the same optial potential for a given distribution irrespetive of the

motionalquantum states of the atoms.

Flutuations are absent in the mean-eld model, however, they may be taken into

aount by expanding the original equations of motion of the system around the mean

eld.Inthisontext,themean-eldtheoryanberegardedasthezeroth-orderexpansion

of the problemin the utuations.In this Thesis we are dealing with the self-onsistent

mean-eld solution, and the rst-order utuations around it. In the studied systems,

utuations play an important role beause of two reasons. First, they indue a phase

transition of the atomi loud plaed inside the resonator from a homogeneous to a

periodiallymodulatedphase.This so-alledself-organizationisinvestigatedindetail in

Chapter 3. Seond, in the ase of a BEC, ertain matter wave utuations are driven