Model

WedesribetheoupledBECavity systemintermsof themean-eldmodelderived in

Setion 2.2.2. For ompleteness, however, we briey summarize the modelassumptions

and desribe the origin of the eetive parameters of the system.

We onsider a pure Bose-Einsteinondensate (BEC) interating with a single-mode

of a high-Q optial avity. The ondensate atoms are oherently driven from the side

by alaser eld with frequeny

ω

^{, direted} perpendiularlyto the avity axis. The laser is detuned far from the atomi transition

ω A

^{, that is,}

| ∆ A | ≫ γ

^{, where}

2γ

^{is the full}

atomi linewidth athalf maximum and the atom-pumpdetuning is

∆ _A = ω − ω _A

^{. This}

onditionensuresthattheeletroniexitationisextremelylowintheondensateatoms,

henethespontaneous photonemissionissuppressed.Atthe sametime,thelasereld is

nearly resonantwith the avity mode frequeny

ω C

^,i.e.

| ∆ C | ≈ κ

^{, where}

κ

^{isthe avity}

mode linewidth and the avity-pump detuning is

∆ C = ω − ω C

^{. The sattering of laser}

photons intothe avity is thusaquasi-resonantproess andis signiantlyenhaned by

the strong dipoleouplingbetween the atoms and the mode due to the smallvolume of

g 0

^{, whih is in the range of}

κ

^{. Therefore, although the ondensate is hardly exited, it}

an eiently satter photons into the avity.

For the sake of simpliity, we desribe the dynamis in one dimension

x

^{along the}

avity axis. The avity mode funtion is then simply

cos kx

^{. This model an apply e.g.}

toaigarshaped BECtightlyonnedinthe transverse diretionsby astrongdipoleor

magneti trap, sothat the transverse size of the ondensate

w

^{is smaller thanthe waist}

of the avity eld. The pump laser is assumed to be homogeneousalong the avity axis

therefore itis desribed by aonstant Rabifrequeny

Ω

^.

The ondensate ontains a number of

N

^{atoms assumed to have the same wave}

funtion

| ψ(t) i

^{. The avity eld is assumed to be in a oherent state desribed by the}

omplexamplitude

α

^.Theseapproximationsimplythatthequantumstateofthesystem is fatorized: entanglement between the ondensate and the avity eld [111, 112℄ is

negleted, whih an be done for large enoughavity photonnumber

| α | ²

^.

The avity eld is subjet to the strong refrativeindex eet of the optially dense

ondensate.Atthesametime,theevolutionofthe ondensatewavefuntionisdesribed

byaGross-Pitaevskii-typeequation,inludingthemehanialeetoftheradiationeld

intheavity.Thesystemofoupledmean-eldequationsisreadilyderivedinEqs.(2.26),

whih read

i ∂

∂t α = [ − ∆ C + N h U (x) i − iκ] α + N h η t (x) i ,

^(3.18a)

i ∂

∂t ψ(x, t) = p ²

2 ~ m + | α(t) | ² U(x) + 2Re { α(t) } η t (x) + Ng c | ψ(x, t) | ²

ψ(x, t).

^(3.18b)

Let us disuss the physial meaning of the terms oupling the photon and the

mat-ter elds. Eah atom shifts the avity resonane frequeny in a spatially dependent

manner by

U (x) = U ₀ cos ² kx

^{. The maximum shift is}

U ₀ = g ² /∆ _A

^{, obtained at the}

antinodes of the mode funtion. In the mean-eld approximation, the shift has to be

spatially averaged over the single-atom wave funtion, giving the frequeny shift per

atom

h U(x) i = U 0 h ψ | cos ² kx | ψ i

^{. It is worth noting that, even if the one-atom light}

shift

U 0

^{issmallompared to the avity deay}

κ

^{, one an ahieve the} interesting strong olletive oupling regime of avity QED,

N | U ₀ | > κ

^{, with a trapped BEC. Feeding}

the avity by laser sattering on the atoms appears as an eetive pump with strength

η t = Ωg/∆ A

^{. This proess has a spatial dependene inherited from the mode funtion,}

η t (x) = η t cos kx

^{, and this term alsohas tobe averaged over the ondensate wave}

fun-tion.

The well known Gross-Pitaevskii equation (GPE) in one dimension in Eq. (3.18b)

eld.Thebakationofthelightshiftisthetermproportionalto

U(x) = U 0 cos ² kx

^and

theavityphotonnumber

| α | ²

^{.Thistermisperiodiwithhalfofthewavelength}

λ

^and,in

freespae,isreferredtoasanoptiallattie.Thebakationoftheoherentsattering

ofphotonsbetween thetransversepumpandtheavitymodeisthetermproportionalto

η t (x) = η t cos kx

^{and has a periodiity of}

λ

^{. The lastterm of the GPE aounts for the}

s-waveollisionof the atoms, itsstrength is relatedtothe s-wave satteringlength

a

^by

g c = 4π ~ a/(mw ² )

^{, and depends on the transverse size}

w

^{of the ondensate. We assume}

repulsive atomatom ollisions in the system (

a > 0

^{) whih are neessary to maintain}

the stability of the ondensate in the thermodynami limit [106℄. In the following we

will onsider the ase

U 0 < 0

^{, i.e. large red detuning, where the atoms behave as high}

eldseekers.Consequently,fornonzero avityeldtheondensateatomstendtoloalize

aroundthe eld antinodes, thereby maximizingtheir ouplingto the lighteld.

It an be heked that Eqs. (3.18) with the eld amplitude replaed by

α/ √ N

^are

invariant under the saling of the parameters suh that

NU ₀

^,

Ng _c

^{, and}

√

Nη _t

^{is kept}

onstant. That is, in the mean eld model, the atom number an be inorporated in

the system parameters and the eld amplitude variable (with the proposed resaling,

the absolute square of this latter gives the photon number per atom). Therefore, in the

following the system parameters willour onlyin the form of the above ombinations.

However, we refrain fromintroduinganew notation forthe saledparameters inorder

tosignify their relationto experimental parameters.

Further simpliation for the numerial method an be obtained by making use of

the periodiity of the optial potential. We an onsider periodi wave funtions

ψ(x)

^,

and solve Eqs. (3.19) in the interval

[0, λ]

^{using periodi boundary ondition. In this}

way, we disard the dynamis within a Bloh band. We are interested in eets due

tothe ondensate-avity eld interation whih doesnot ouple states of dierent quasi

momenta. Thene it is independent of the length of the interval and we hoose the

shortest one to redue the omputational eort. By ontrast, the ollisional interation

depends on the atual atom density, hene, for a xed atom number

N

^{, the density is}

artiiallyenhanedby foldingthespae intoasingle

λ

^{interval.Therefore, the ollision}

parameterhas tobemodiedsuh that

Ng c /λ

^{orrespond tothe ollisionsattheatual}

(experimental)atom density inthe avity.

In document (1)ultraold atoms in an optial resonator Ph.D (Pldal 43-46)