Strong olletive oupling regime

harmoni trap, onlythe seondexitation isoupled tothe eld [108℄.

The frequeny and the deay rate of the avity eld are weakly perturbed by the

ondensate.The frequeny of the avity mode isexpeted tobe

ν _f = ∆ _C − NU ₀ B

^{, that}

depends on the olletive oupling

NU 0

^{and, through the bunhing parameter, on the}

loalizationof the groundstate

ψ 0

^{. Dereaseinthe eldmode frequeny isaompanied}

by aninrease of the stationary photonnumber

| α 0 | ²

^{in the avity.}

√ N η t [in units of ω R ] cr it ic al N U 0 [i n u n it s of ω R ]

300 250 200 150 100 50

0 -220 -240 -260 -280 -300 -320

Figure 3.8: Full phase diagram of the system for ollision parameters

Ng c = 0

^and

10 λω R

^{. For}

Ng c = 0

^{the dashed blue urve represents the phase boundary between}

homogeneous distribution and self-organized lattie. In the region between the dashed

blue and the solid red line (these urves oalese asymptotially for large

NU ₀

^{) there}

is a self-organized lattie without defets. The solid red urve gives the ritial

NU 0

belowwhihthe seondary minimaour.Right shifted, the same phaseboundary lines

are drawn by dotted green and dashed-dotted brown for

Ng c = 10λω R

^{. Parameters are}

κ = 200ω R

^,

∆ C = − 2κ

^.

exitations that are omposed of the ombination of exitations loalized in the two

dierent wells.

For simpliity, we alulate the spetrum in the ollisionless ase for

g c = 0

^{. In}

Fig. 3.9, we plot the olletive exitation frequenies (a) and the deay rates (b) as a

funtionof the pumping strength

η _t

^for

κ = 200ω _R

^and

NU ₀ = − 1000ω _R

^{. Startingfrom}

the Bogoliubov spetrum of the homogeneous ondensate, there appear two types of

exitation modes in the self-organized phase. The exitations that are loalized in the

deeper wellsare the sameasthe ones presented in Fig.3.7. Notie the lakof the dip at

the ritial point, whih is beause of the hoie

g c = 0

^{. The other type,}represented by the twoexitation modes bendingup in Fig. 3.9a ontain the exitations of the `defet'

well.Forperfet loalization,boththelowerand thehigherpotentialminima,henealso

their dierene, are proportional to

N | η t | ²

^{. Exiting the ondensate intothe defet well}

osts the energy dierene, this is the reason of the quadrati behavior asa funtion of

the pumpingstrength

√ N η t

^.

a)

√ N η t [in units of ω R ] ex ci ta ti on fr eq u en ci es [ ω R ]

120 100 80 60 40 20 0 100

80 60 40 20

b)

√ N η t [in units of ω R ] d ec ay ra te s [ ω R ]

120 100 80 60 40 20 0 0.01 0.008

0.006 0.004 0.002 0

Figure 3.9: Frequenies (a) and deay rates (b) of the six lowest olletive ondensate

exitationsintheregime,wheretheseondarypotentialminimaarepresentasafuntion

ofthetransversepumpstrength.Deayrateoftherstexitation

γ 1

^{(solidred)isdivided}

by

40

^{.Parameters are}

g c = 0

^,

NU 0 = − 1000 ω R

^,

κ = 200 ω R

^and

∆ C = − 1200 ω R

^.

weplotthewavefuntionsoftwoondensateexitationsatinreasingvaluesofthepump

strength

√ N η t

^{in Fig. 3.10. Namely, we hose the exitations with the lowest and the}

highestfrequeniesinthe rightof Fig.3.9a. Bothof themare deoupled fromtheavity

eldsothatthewavefuntionisrealandanbeinterpretedasaondensatewavefuntion

in position spae. The lowest one (a) is proportional to

sin kx

^{in the uniform phase at}

√ Nη t = 15

^{, having two nodes at}

x = 0

^and

λ/2

^{. On inreasing the pump strength}

√ Nη t

^{,theondensatewave funtiongetsmoreand moreloalized,henethis exitation}

ontrats into the primary potential well,whih is now entered at

x = λ/2

^{in the}

self-organized phase. At

√ Nη t = 120

^{, the strongly loalized BEC feels just the harmoni}

term of the optialpotential. Therefore, at this end, we get for the wavefuntion of the

exitation something lose to the rst exitation of a harmoni osillator. The other

seleted exitation is the highest one, bending up in Fig. 3.9a. It has six nodes in the

homogeneousphase,thatorrespondstothethirdexitationoftheBogoliubovspetrum.

It runs side by side with itsorthogonal pair up to

√ N η t ≈ 60

^{,where they split up. The}

lower branh tends to the fourth harmoni osillator exitation in the primary well for

√ Nη t → ∞

^{,howevertheupperbranhplottedinFig.3.10bbeomestherst exitation}

inthe seondary well entered at

x = 0

^.

δΨ +

√ N η t /ω R

x/λ a) δΨ +

3 2 1 0 -1 -2 -3

60 120 15 30

0.25 0.5 0

δΨ +

√ N η t /ω R

x/λ b) δΨ +

3 2 1 0 -1 -2 -3

60 120 15 30

0.5 0 0.25

Figure3.10:The

δψ + (x)

^{omponentsofthe}eigenvetorsoftheolletiveexitationswith the lowest (a) and the highest frequenies (b) inFig. 3.9a are plotted for hosen values

of

η t

^{. In these speial ases the other omponentsare zero.}

3.2.5 Summary

In this setion, I desribed the quantum version of the lassial self-organization phase

transition and gave a detailed aount of the steady-state and the dynamis when the

atoms formaBose-Einstein ondensatein the avity. Within the framework of a

Gross-Pitaevskiilikemean-eldmodel,Ishowed thatthe steady-stateofthe drivenondensate

is either the homogeneous distribution or a

λ

^{-periodi ordered pattern, and the two}

regimesarewellseparatedby aritialpoint.Thatis,the quantumanalogueof the

las-sialphase transitionexists forBose-ondensed atoms.Theritialpoint,orresponding

to athreshold pump power has been determined analytiallyfrom the Gross-Pitaevskii

equation, and also analytiallyfrom the spetrum of the exited states. The mean-eld

results for the self-organizationin the lassialand inthe quantum systems show

qual-itative agreement. From the expressions of the ritial points one an onlude that in

a thermal gas it is the temperature whih stabilizesthe homogeneousphase, while in a

BECthe same role isplayed mainlyby theatom-atom ollisions,andthe kineti energy

has only a minor eet. Finally, I studied the spetrum of the olletive exitations of

theompoundBECavity systemaroundthemean-eldsolution.Ifoundpolariton-like

exitation modes of the oupled light and matter wave elds. I showed that the

spe-trumoftheolletiveexitationsbeomesmoreintriateinthestrongolletiveoupling

regimeof avity QED,where the ordered phase isdetermined by anasymmetri double

wellpotential withdefet sites.

Exess noise depletion of a

Bose-Einstein ondensate in an optial

avity

In this hapter, I shall deal with the utuations of a Bose-Einstein ondensate plaed

into a single-mode high-nesse optial avity. In ontrast to the transverse pump

ge-ometry of the previous hapter, here I onsider a BECavity system that is pumped

diretly through one of the avity mirrors. Many of the properties of the BEC an be

satisfatorily aounted for by assuming a ondensate wavefuntion whih obeys a

non-linearShrödingerequation. Nevertheless, atomatomollisionsonstantly kik out

atomsfromtheone-partilegroundstate,even atzerotemperature.The atomi

popula-tioninthe exitedstatesofthe BECisalledondensatedepletion. Inadilutequantum

gas, the atomatom interations are typially weak, therefore in a BEC the fration of

nonondensedatomsduetos-wavesatteringisusuallynegligible(afewperent).Large

depletion of a BEC an indiatesome inherent many-body interation eet. In ase of

ollisions,suhobservationrequireslargeloaldensities.Anexampleisthe strong

deple-tion demonstrated as a preursor of the phase transition from superuid to Mott state

inanoptiallattie[115,116℄.Inthis hapter,Iwillshowthat aondensatedispersively

oupledtotheradiationeldinaavityanalsobesubjetedtostrongdepletion.Inthis

ase the many-bodyeet originatesfrom the long-rangeatom-atom ouplingmediated

bytheavitymode,thereforeasigniantamountofdepletionmayouratlowdensity.

ApumpedlossyavityeldinteratingwithaBECinsideisanopensystem,inwhih

thequantumutuationsoftheradiationeldisapossiblesoureofdepletion.However,

in the dispersive interation limit. As it was alulated for a probe eld propagating

freelythrough the ondensate [117℄,the depletion sales asthe absorptionwhihwillbe

suppressed by hoosing verylarge detuning. Therefore the avity is essentialin reating

aspeiouplingofthe many-atomsystem totheradiation.Beauseof thefast

round-trips of photons, the avity eld experienes the olletive behavior of atoms. This kind

of oupling gives rise to the noise ampliation mehanism analogous to the so alled

exess noise in laser physis [118,119, 120, 121℄.

In document (1)ultraold atoms in an optial resonator Ph.D (Pldal 55-60)