Colletive exitations

Letusnowalulatethe exitationspetrumofthe oupledondensate-avitysystem as

thelinearresponseoftheself-onsistentsteady-state.Twolimitingasesanberelatively

easilyunderstood:(i)for

η _t = 0

^{thereisnoeld inthe avity andone getsbakthe}

exi-tationspetrumofahomogeneousBose-gas; (ii)for

η t → ∞

^{,deeply inthe}self-organized phase, where the optial potential an be approximated as a parabola, one expets to

obtain the exitations of a BEC in harmoni trap potential. This simpliation,

how-ever, does not perfetly apply sine the exitations are not only olletive osillations

of the atom loud, but they are polaritons involving the utuation of the eld

ampli-tude aroundits steady state. Inthe following, wewillonsider the fulltransitionrange,

inludingthe ritial point,between these limitingases.

Weneed to onsider the deviations fromthe stationary state (

ψ 0

^and

α 0

^):

α(t) = α 0 + δα(t) ,

^(3.22a)

ψ(x, t) = e ^{− iµt} [ψ 0 (x) + δψ(x, t)] .

^(3.22b)

Inserting the ansatzintoEqs. (3.18) and linearizingin

δψ

^and

δα

^{, one gets}

iδ α ˙ = Aδα+Nα 0 [ h ψ 0 | U (x) | δψ i + h δψ | U (x) | ψ 0 i ]+N [ h ψ 0 | η t (x) | δψ i + h δψ | η t (x) | ψ 0 i ] ,

(3.23a)

iδ ψ ˙ =

H 0 + Ng c | ψ 0 (x) | ² δψ + Ng c ψ ₀ ² (x)δψ ^∗

+ ψ 0 (x)U(x)(α 0 δα ^∗ + α ^∗ ₀ δα) + ψ 0 (x)η t (x)(δα + δα ^∗ ) ,

^(3.23b)

where

A = − ∆ C + N h ψ 0 | U(x) | ψ 0 i − iκ ,

^(3.24a)

and

H ₀ = p ²

2 ~ m + Ng _c | ψ ₀ (x) | ² − µ + | α ₀ | ² U (x) + 2Re { α ₀ } η _t (x) .

^(3.24b)

Beause the linearized time evolution ouples

δψ

^and

δα

^{to their omplex onjugates,}

we searh the solution inthe form

δα(t) = e ^{− iωt} δα + + e ^{iω ∗ t} δα ^∗ ₋ ,

^(3.25a)

δψ(x, t) = e ^{− iωt} δψ + (x) + e ^{iω ∗ t} δψ ₋ ^∗ (x),

^(3.25b)

where

ω = ν − iγ

^{isaomplex parameterof the osillationstandingfor frequeny}

ν

^and

damping rate

γ

^{. Equations (3.23) have to be obeyed separately for the}

e ^{− iωt}

^and

e ^{iω ∗ t}

terms, whih leads tothe lineareigenvalue equation:

ω











 δα +

δα ₋ δψ ₊ (x) δψ ₋ (x)













= M











 δα +

δα ₋ δψ ₊ (x) δψ ₋ (x)













,

^(3.26)

where

M

^isa non-Hermitianmatrix being determinedby Eqs. (3.23).Colletive exita-tionsofthe system arethe solutionsof thiseigenvalue problem.Itsimpliesifwe hoose

ψ 0 (x)

^{real,and write}

M

^{inthe basis ofthe symmetriand} antisymmetriombinations,

δα a = δα + − δα ₋ , δα _s = δα ₊ + δα ₋ , δf (x) = δψ + (x) + δψ ₋ (x) , δg(x) = δψ ₋ (x) − δψ + (x) .

(3.27)

Transformedintothis basis, the matrix

M

^is













− iκ ReA 2N (Reα 0 X + Y) 0

ReA − iκ 2iN Imα 0 X 0

0 0 0 − H 0

2iψ 0 UImα 0 − 2ψ 0 (UReα 0 + η t ) − H 0 − 2Ng c ψ ₀ ² 0













.

^(3.28)

The argument

x

^{has been omitted from the funtions}

ψ 0

^,

U

^and

η t

^{for brevity. The}

integraloperators

X

^and

Y

^{,ouplingtheondensate exitationstothe avity eld,read}

Xξ(x) = Z

dx ψ ₀ (x)U (x)ξ(x),

^(3.29a)

Yξ(x) =

Z

dx ψ ₀ (x)η _t (x)ξ(x).

^(3.29b)

Beinganon-Hermitianmatrix,

M

^{has dierentrightandleft}eigenvetorsorresponding tothe sameeigenvalue

ω

^.

The matrix has aspeial symmetry: if

ω

^{isan eigenvalue with the righteigenvetor}

(δα a , δα s , δf, δg) ,

then

− ω ^∗

^{is alsoan eigenvalue with the righteigenvetor}

( − δα ^∗ _a , δα ^∗ _s , δf ^∗ , − δg ^∗ ) .

Thusthe eigenvalues ome in pairs, havingthe same imaginary parts but the real parts

are of the opposite sign in a pair. This grouping of the eigenvetors makes sense for

eigenvalues with non-vanishingreal part.

The imaginarypart desribes dampingwhih arises from the nonadiabatiity of the

avity eld dynamis. The linear perturbation alulus of the exitations whih we

adopted here takes into aount that the eld follows the hanges of the BEC wave

funtion with a delay in the order of

1/κ

^{. Depending on the spei hoie of the}

pa-rameters, this an yield dampingor heating, whih is known as avity ooling and has

been extensively studiedinthepast deade. DampingofBEC exitationsinoptial

av-ities,whihhas beenrst studiedinRefs.[108, 113℄,isaninterestingopportunitywhih

motivatesthe realizationand the study of the oupled avity-ondensate system.

Spetrum analytially below threshold

Belowthreshold, the ondensatewave funtionis onstantand linearizationaroundthis

simple solutionleads toanalytial results. The stability of the uniform distribution an

be evaluated from the spetrum, thus the ritial point an be determined in this way,

independently of the previous alulation in Se. 3.2.2. Moreover, we obtain a detailed

desription of arestrited part of the spetrumwhih ontains polaritonexitations.

Assuming homogeneous atomi distribution and, orrespondingly, vanishing avity

eld (

ψ 0 ≡ 1

^,

α 0 = 0

^{), the only} non-trivial oupling term in the matrix

M

^{, see Eq.}

(3.28), is the

Y

^{operator dened in Eq. (3.29b). Sine}

η t (x) ∝ cos kx

^{, only the Fourier}

omponent

cos kx

^{ouples to the avity amplitude. In return, the fourth line of the}

matrix

M

^{shows that the eld utuations exite just this}

η t (x) ∝ cos kx

^ondensate

perturbation. Thus, this subspae is losed below threshold. All the other ondensate

exitations deouple from the eld and remain simply the higher Fourier omponents

with frequenies

Ω n = p

n ² ω R (n ² ω R + 2Ng c ) , (n > 1) ,

^(3.30)

suient todiagonalize the matrix

M

^{in the restrited subspae,}













− iκ δ C Nη t 0

δ C − iκ 0 0

0 0 0 − ω R

0 − 2η _t − ω _R − 2Ng _c 0













.

^(3.31)

It has the fourth order harateristiequation

(λ ² − Ω ² ₁ )

(iκ + λ) ² − δ ² _C

− 2Nη _t ² ω R δ C = 0,

^(3.32)

where

δ C = − ∆ C + ¹ ₂ NU 0

^{, and}

Ω 1

^{is the rst exitation energy in the Bogoliubov}

spetrum(3.30)ofaBECinabox.Oneanhekthat,for

η t = 0

^{,thelasttermvanishes}

and

λ 1,2 = ± Ω 1

^{for the ondensate exitation,}

λ 3,4 = ± δ C − iκ

^{forthe avity mode. For}

non-zero pumping,

η _t 6 = 0

^{, the} ondensate-like and the eld-like exitations mix. When

Ω ² ₁ ≪ κ ² + δ ² _C

^{, the frequenies} orresponding to the exitations of the free eld and to the free ondensate are well separated. Therefore, the mixing ratio is small so that the

polaritonmodes an be attributed todominantly ondensateor eld exitations.

At the onset of self-organizationthe uniform ground state

ψ ₀

^{hanges, whih,in the}

present approah, orresponds to the appearane of a zero eigenvalue in the spetrum.

In Fig. 3.6, we plot the numerial solutions of Eq. (3.32) for the lowest, dominantly

ondensate-type exitation. The real parts (solid red) tend to zero as inreasing the

pumpstrength. Oppositely, the absolutevalue of the imaginary partsof the eigenvalues

(dashed green)inreases withthe pumpstrength. The behaviourofthe eigenvaluesnear

the ritial point

√ N η _c = 65.612 ω _R

^{is magnied in the inset of Fig. 3.6. When the}

real parts reah zero at

√ N η _∗

^{, the initially idential imaginary parts split up and the}

upper branh rosses zero. This rossing is the ritial point, here the

ψ 0

steady-state beomes dynamially unstable. By expressing

η t

^{from Eq. (3.32) for}

λ = 0

^{, the same}

ritialtransverse pump amplitude

η c

^{is obtained asthat in Eq. (3.21).}

The emergene of a positive and a negative imaginary part just at the vanishing of

the real parts of an exitation is a typial feature of a ondensate's instability. In our

ase, however, oupling to the avity eld yields a negative imaginary part where the

real part beomeszero. Thus, there isanarrowrange,as shown by the inset ofFig. 3.6,

wheretherealparts remainzero,but bothimaginarypartsarenegative.This unfamiliar

ourseof the dynamialinstability isasignature of avity ooling, thusitis sofornite

avity deay rate

κ

^.

65.615 65.61

65.605 0.06

0

− 0.06

√ N η t [in units of ω R ] ei ge n va lu es [i n u n it s of ω R ]

60 50 40 30 20 10 0 4 2 0 -2 -4

Figure3.6:Eigenvalues ofthe lowest ondensate exitationofahomogeneousBEC from

Eq. (3.32) as a funtion of the pump strength

√ N η t

^{. As shown in the inset, the real}

parts (solid red) vanishes slightly below the ritial point whih is reahed when the

upperbranh of the imaginaryparts (dashed green) rosses zero. Inthe main gure the

imaginaryparts are magnied by afator of

50

^{.Parameters are the same asinFig. 3.5.}

In the limit

Ω ² ₁ ≪ κ ² + δ _C ²

^{, the rst exitation frequeny an be}approximatedby

Reλ 1 = Ω 1

s 1 − η ² _t

η ² _c ,

^(3.33a)

and the imaginary part isquadrati inthe pump strength,

Imλ 1 = − κ Ω ² ₁ δ ² _C + κ ²

η _t ²

η _c ² .

^(3.33b)

TheseapproximateexpressionstwelltheurveshowninFig.3.6.Thes-waveinteration

between atomsinreases thedeayrate ofthis partiularexitation.Theimaginarypart

vanishes, of ourse,for

κ → 0

^.

The width of the narrow range, where the

ψ 0 = 1

^{uniform steady state is stabilized}

by that the avity ooling damps out zero-energy exitations, is obtained:

η ² _c − η _∗ ² = η _c ²

κ Ω 1

δ _C ² + κ ² 2

.

^(3.34)

Similarlytothe deay rate

Imλ 1

^{,the rangeexpands oninreasingthe ollisional}

param-eter

Ng _c

^{. Finally,note thatEq. (3.34)isonsidered asmall parameterinthe estimation}

of

λ 1

^{inEqs. (3.33).}

a)

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

145 125 105 85 65 45 25 5 60 50 40 30 20 10 0

b)

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

145 125 105 85 65 45 25 5 0.08

0.06

0.04

0.02

0

Figure 3.7: Frequenies (a) and deay rates (b) of the six lowest olletive ondensate

exitationsandthe ones orrespondingtothe avityeld asafuntion ofthe transverse

pump strength. Deay rates of the lower branhes of eah pair are

0

^{. For the}

avity-dominatedmodethe frequenyandthe deay rateisdividedby5and4000,respetively,

and

γ 1

^{is divided by 2 in(b). Parameters are the same as inFig. 3.5.}

Full spetrum

Abovethreshold,theolletiveondensate-avityexitationsbelongtotheself-onsistent

ground state given by Eqs. (3.19) whih an be alulated only numerially. Therefore,

the solutionof the eigenvalue problemin Eq. (3.26)is performed alsonumeriallyusing

LAPACK. The wave funtionis dened ona spatialgrid of 200 points inthe intervalof

one wavelength.

InFig.3.7(a),weshowtherstthreeexitationfrequeniesanddampingrates(

γ 1 /2

^is

plotted)of theondensatearoundtherealgroundstate,andthe frequeny anddamping

rateofahigherexitedstateorrespondingtotheavitymode(doubledashedlines,

ν f /5

and

γ _f /4000

^{are plotted). We depit only the positive} frequenies, taking into aount the symmetry of the eigenvalues of the matrix (3.28).

Belowthreshold the wave funtion

ψ 0 (x)

^{is onstant,only the rst exitation}

cos kx

ouples tothe avity eld, and tends tozero oninreasing the pumping, as it has been

disussedbefore.Alltheotherexitationsareindependentoftheeld,andhaveonstant

energies giving bak the Bogoliubov exitation spetrum of a ondensate in a box, see

Eq.(3.30).Twoorthogonal exitationshave axed numberof nodes andare degenerate

inthis regime. One of the modes in eahpair is orthogonal tothe avity mode funtion

cos kx

^{, thusdeouples fromthe avity eld.}

al point the exitation energies drop slightly below the Bogoliubov energies given in

Eq.(3.30).Thisdipisrelatedtothe ollisionsanddisappearsfor

g _c → 0

^{.Letusmention}

that the numerialalulationof the stationary solutionofthe Gross-Pitaeskiiequation

beomes inaurateat the ritialpoint. Theonvergene of the iterativesolution slows

down, whih is aninherent onsequene of the ritiality obtained at the degeneray of

the groundand rst exited state energies. Nevertheless weheked insome pointsthat

the dip is stillthere if very high aurayis demanded inthe iterationproess.

Atthreshold,thedegenerayintheexitationpairsislifted.Thelowerbranhremains

orthogonal to the avity mode and deouples from the eld utuations. The upper

branhes orrespond topolaritonexitationsmixingondensate and eld utuations.

Far above threshold,

√ Nη t → ∞

^{, the exitation frequenies inrease linearly with}

√ Nη t

^{and are uniformly spaed. This an be understood if we} approximate the deep optial trap by a harmoni potential in the viinity of the antinodes. The adiabati

potentialin Eq. (3.1), transformed by using

cos kx ≈ 1 − (kx) ² /2

^{, gets a} harateristi harmoni frequeny proportional to the square root of the intraavity intensity. Thus,

from Eq. (3.5), the trap frequeny is linearly proportional to

√ N η t

^{. The spetrum is}

omposed of the integer multiples of this frequeny. This is obvious in the ollisionless

ase,

g c = 0

^{, where the exitation frequenies are the same as for a single partile in}

a harmoni potential.The other extreme ase, the Thomas-Fermi regime whih we are

losertowiththeparametersofFig.3.7,analsobetreated.Here,forathree-dimensional

harmoni trap the exitation frequenies are given by a linear ombinationwith integer

oeients of the three vibrational frequenies [114℄. This an be ontrated to one

dimensionalmotionbyassumingverylargefrequenies inthe transverse diretions,then

the low exitationsare obtained asinteger multiples of the longitudinaltrap frequeny.

In Fig.3.7(b), the deay rate of the exitationsorrespondingto theupperbranhes

in Fig. 3.7(a) are shown. The lowerbranh of eah pair has zero damping beause it is

orthogonal to the mode funtion of the avity eld. The lowest exitation deays with

a rate

γ ₁

^{whih exhibits a dip at threshold. In priniple, it should drop down to zero,}

as shown in Fig. 3.6representing the exat resultbelow the ritial point, however, the

aurayofthenumerialalulationbreaksdown intheviinityofthe thresholddue to

ritialslowingdown.Somewhatabovethreshold,

γ 1

^{risesbak tothe valueithadbelow}

thresholdand theninreases further withinreasing

√ Nη t

^{. Nottoofarabovethreshold,}

the weakly loalizedground state enables the oupling of higher ondensate exitations

to the avity eld. These modes beome damped, but their damping rate vanishes in

the

√ N η t → ∞

^{limit. There, as the ground state tends to a tightly loalized one in a}

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.}

In document (1)ultraold atoms in an optial resonator Ph.D (Pldal 48-55)