Condensate depletion

parts reets the semilassialresult that the avity ooling mehanismismost eient

athalf alinewidth belowresonane [67℄.

Contrary to the ase of a onservative system, where quantum depletion is the sum of

theindividualontributionsofthe nonondensedone-partilestateoupation numbers,

here the depletion isthe double sum of ontributions frompairs of quasi-normalmodes

(

k

^and

l

^).

The denominator in Eq. (4.10) may beome small for a pair of eigenvalues (

ω _k

^and

ω l

⁾ ^if

Re { ω k } = − Re { ω l }

^,^i.e. ^when ^the quasi-normalmodes

k

^and

l

^are ^onneted ^with

the transformation

Γ

^. ^Moreover, ^the ^smaller ^the ^imaginary ^parts ^(their ^dampings), ^the

smallerthe denominator. However, aneigenvalue with asmall imaginarypart is

aom-paniedby aneigenvetorwith smallphoton omponents (diminishing

l 1

^and

l 2

^terms).

Therefore not all of the eigenvalues with small imaginary parts play a signiiant role

andthe evaluationof thetermsremainsanumerialtask.Nevertheless, theontribution

tothe depletion fromsuh pairs of quasi-normalmodes dominates the double sum.

The total steady-state depletion in the

t → ∞

^limit ^(the exponential in Eq. (4.10) is dropped) is shown in Figure 4.2 (semilog sale). The thik solid line orresponds to

the parameter set of the spetrum in Fig. 4.1. The depletion is losely onstant for a

wide rangeof

U 0

^up ^to^the ^resonane ^regime^dened ^by

| ∆ C − N h U i| . κ

^. ^At^resonane

thedepletiondiverges,inaordanewiththe fatthatthedynamialequilibriumofthe

systembeomesunstable.Itisinterestingthatthereisadipslightlybelowtheresonane.

We attribute it to the signiant hange in the shape of the ondensate wavefuntion,

from homogeneous to a strongly loalized one, whih yields a variation of the overlap

fators in the matrix

M

^. ^It ^is ^important ^to^note ^that ^the ^nite, ^large ^value ^of ^depletion

in the limit of

U 0 → 0

^was ^alulated ^by ^taking ^rst ^the

t → ∞

^limit ^of ^the ^time

dependentresultEq.(4.10).However, duringanitemeasurementtime thesystem does

notneessarilyreahthesteady statesine therelaxationtimedivergesas

U ₀ ⁻ ²

^for^small

U 0

It is instrutive to approximate the steady-state depletion analytially,sine an

an-alytial expression reveals its saling with the parameters of the system. The studied

parameter set is partiularlyappealing, sine for

0 < | U ₀ | . 0.8 ω _R

^the ^ondensate ^wave

funtion

ψ(x)

^is ^pratially^onstant,^that ^is^onrmed^by ^the^spetrumⁱⁿ^Fig.^4.1. ^The

dispersive interation with the avity eld ouples a onstant ondensatewave funtion

solely to the

cos 2kx

^F^ourier ^exitation ^mode, ^therefore ^one ^an ^restrit ^the ^dynamis

into the subspae

R ˆ ≡ [δˆ a, δˆ a ^† , δˆ c 2 , δˆ c ^† ₂ ]

^, ^where

δ Ψ(x) = ˆ √

2 cos 2kx δˆ c 2

^. ^This ^proedure

resembles the one appliedin Se. 3.2.3, however, inthe transverse pumping ase the

in-teration ouplesthe onstantondensate wave funtion tothe

cos kx

^exitation^mode,

| U 0 | [in units of ω R ]

δN

1 0.8 0.6

0.4 0.2

0 10 ⁴

10 ³

10 ²

10 ¹

10 ⁰

Figure 4.2: The depletion

δN

^of ^the ^ondensate ^as ^a ^funtion ^of ^the ^oupling ^onstant

U 0

^for ^various ^detunings. ^The ^parameters ^are

κ = 100 ω R

η = − ∆ C

N = 1000

^, ^and

∆ C = − 10 ⁴

− 10 ³

− 10 ² ω R

^for ^the ^dashed, ^solid ^and ^dotted ^lines, respetively. In the plotted range of

U 0

^, ^for ^smaller ^detunings ^(solid ^and ^dotted ^lines) ^the ^BEC-avity

systemanberesonantwiththepump,whihleadstodivergenes.Wellbelowresonane,

N | U ₀ | ≪ ∆ _C

^, ^the ^depletion ^depends ^weakly ^on

U ₀

^(see ^the ^plateau^of ^the ^solid ^and ^the

dashedlines) and on

N

^. ^In^this ^regime^the ^depletion ^sales ^with

∆ C /ω R

in ase of avity pumping, the ondensate beomes modulated for any non-zero pump

strength,thereforethewave funtionisonlyapproximatelyonstant,thankstothesmall

interation strength

U ₀

^and ^the ^low ^photon ^number ^inside ^the ^avity. ^The ^matrix

M

dened by Eq. (4.5),then takesthe following formin the restrited subspae,

M =







A 0 αX αX

0 − A ^∗ − α ^∗ X − α ^∗ X α ^∗ Y αY E 2 0

− α ^∗ Y − αY 0 − E 2







,

^(4.11)

where the onstants appearing in the matrix are

X = √

2U 0 /4

Y = √

N U 0 /2

^and

E 2 = 4ω R + | α | ² U 0 /2

^. ^In ^this ^simplied ^model ^the ^depletion ^is

δN = h c ^† ₂ c 2 i

^, ^whih ^is

alulatedbyndingtheeigenvaluesandthe orrespondingleftandrighteigenvetorsof

the matrix

M

^. ^F^or^the ^depletion, ^one ^gets ^a ^similar ^formula ^to^(4.10), ^however ^without

the spatialintegralover

x

^.^Skipping ^the ^details ^of ^the ^lengthy ^alulation,^the ^result^for

the steady-state depletion is

δN ∼ = (∆ _C − NU ₀ /2) ² + κ ²

8ω R ( − ∆ C + NU 0 /2) .

^(4.12)

ThisexpressiongivesaperfetagreementwiththenumerialresultspresentedinFig.4.2

for weak interation when

| U 0 | . 0.8 ω R

^. ^It ^explains ^the ^slight^derease ^of ^the ^plateaux,

whih is due tothe tuningof the dressed avity frequeny with the olletivelight shift

NU 0 /2

^of ^the homogeneous ondensate. Interestingly, the formula aurately desribes

the resonane of the depletion for

∆ C = − 100ω R

^(dotted ^line), ^giving^hyperboli

diver-gene when the denominator tends to zero. Nevertheless, the restrited model annot

aount for the dip and the resonane for

∆ C = − 1000ω R

^(solid ^line), ^sine ^they ^are

loated outsidethe rangewhere the ondensate wave funtion an be taken onstant.

The analytial result alsoyields the order of the steady-state value of the depletion.

One an onsider two regimes depending on the dressed avity frequeny

δ C = ∆ C − NU 0 /2

^. ^Typially, ^the ^driving ^eld ^is ^nearly ^resonan^t ^to ^the ^avity, ^hene

δ C

^is ⁱⁿ ^the

order of

κ

^. ^F^or ^instane, ^setting

δ C = − κ

^, ^the ^magnitude ^of ^the ^depletion ^beomes

δN = κ/(4ω R )

^.^Thus,^the^number^of^atoms^outside^the^ondensate^is^simplyproportional to the photon loss rate of the avity. However, in far detuned limitwhere

δ C ≫ κ

^, ^the

depletion is determined by the ratio of the avity mode frequeny

∆ _C

^and ^the ^reoil

frequeny

ω R

^, ^i.e., ^the ^relation ^of ^the ^energy ^sales ^desribing ^optial ^exitations ^and

atomimotion,respetively.ThisisillustratedbytheothertwourvesinFig.4.2.Dashed

lineorrespondstoadetuning

∆ C

^inreased^by^an^order^of^magnitude,^then^the^depletion

isalso saledup by a fator of 10.Obviously, the resonane regime ispushed out of the

plottedrange and onlythe initialplateau an be seen. Oppositely, when dereasing

∆ C

by a fator of10, the initialdepletion dereases (dotted line).However, forsuh asmall

detuning

∆ C ∼ κ

^the ^system ^is ⁱⁿ ^the ^resonane ^regime ^already ^for ^small

U 0

^values,

the wide plateauis missing and the divergene of the depletion is exhibited.As long as

the atom-avity system isfar fromresonane (on the wide plateau), the depletion

δN

^is

independentofthe atomnumber

N

^.^Similarly,^the^photon^number

| α | ²

^is^quite^irrelevant

tothe amountofdepletion,itinuenes onlytheshapeofthegroundstateinthe optial

potentialand thereby some overlap fators inthe matrix

M

In document (1)ultraold atoms in an optial resonator Ph.D (Pldal 64-68)

k

l

ω k

ω l

Re { ω k } = − Re { ω l }

k

l

Γ

l 1

l 2

t → ∞

U 0

| ∆ C − N h U i| . κ

M

U 0 → 0

t → ∞

U 0 − 2

U 0

0 < | U 0 | . 0.8 ω R

ψ(x)

cos 2kx

R ˆ ≡ [δˆ a, δˆ a † , δˆ c 2 , δˆ c † 2 ]

δ Ψ(x) = ˆ √

2 cos 2kx δˆ c 2

cos kx

| U 0 | [in units of ω R ]

δN

1 0.8 0.6

0.4 0.2

0 10 4

10 3

10 2

10 1

10 0

δN

U 0

κ = 100 ω R

η = − ∆ C

N = 1000

∆ C = − 10 4

− 10 3

− 10 2 ω R

U 0

N | U 0 | ≪ ∆ C

U 0

N

∆ C /ω R

U 0

M

M =













A 0 αX αX

0 − A ∗ − α ∗ X − α ∗ X α ∗ Y αY E 2 0

− α ∗ Y − αY 0 − E 2













,

X = √

2U 0 /4

Y = √

N U 0 /2

E 2 = 4ω R + | α | 2 U 0 /2

δN = h c † 2 c 2 i

M

x

δN ∼ = (∆ C − NU 0 /2) 2 + κ 2

8ω R ( − ∆ C + NU 0 /2) .

| U 0 | . 0.8 ω R

NU 0 /2

∆ C = − 100ω R

ω _k

U ₀ ⁻ ²

0 < | U ₀ | . 0.8 ω _R

R ˆ ≡ [δˆ a, δˆ a ^† , δˆ c 2 , δˆ c ^† ₂ ]

0 10 ⁴

10 ³

10 ²

10 ¹

10 ⁰

∆ C = − 10 ⁴

− 10 ³

− 10 ² ω R

N | U ₀ | ≪ ∆ _C

U ₀

U ₀

0 − A ^∗ − α ^∗ X − α ^∗ X α ^∗ Y αY E 2 0

− α ^∗ Y − αY 0 − E 2

E 2 = 4ω R + | α | ² U 0 /2

δN = h c ^† ₂ c 2 i

δN ∼ = (∆ _C − NU ₀ /2) ² + κ ²

∆ _C

| α | ²