Multimode mean-field model for the quantum phase transition of a Bose-Einstein condensate in an optical resonator

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by  G. Konya

EPJ manuscript No.
(will be inserted by the editor)
Multimode mean-field model for the quantum phase transition
of a Bose-Einstein condensate in an optical resonator
G. K´onya1, G. Szirmai1,2, P. Domokos1a
1 Research Institute for Solid State Physics and Optics, Budapest, P.O Box 49, H-1525, Hungary
2 ICFO-Institut de Ci`encies Foto`niques, Mediterranean Techonoly Park, 08860 Castelldefels (Barcelona), Spain
the date of receipt and acceptance should be inserted later
1
1
0 Abstract. We develop a mean-field model describing the Hamiltonian interaction of ultracold atoms and
2 the optical field in a cavity. The Bose-Einstein condensate is properly defined by means of a grand-
canonical approach. The model is efficient because only the relevant excitation modes are taken into
n
account.However,themodelgoesbeyondthetwo-modesubspacenecessarytodescribetheself-organization
a
quantum phase transition observed recently. We calculate all the second-order correlations of the coupled
J
atom field and radiation field hybrid bosonic system, including the entanglement between the two types
9
of fields.
1

]
h 1 Introduction sharp transition threshold is thus expected at zero tem-
p perature,too.Thishasbeenexperimentallyevidencedre-
- A thermal cloud of cold atoms interacting with a single cently[13].Moreover,inRef.[13]aswellasinRef.[14],an
t
n mode of a high-finesse optical cavity undergoes a phase analogy of the self-organization to the celebrated Dicke-
a transition when tuning the power of a laser field which model phase transition [15,16,17] has been pointed out.
u
illuminates the atoms from a direction perpendicular to The dynamically coupled BEC wavefunction and single-
q
the cavity axis. Below a threshold pump power, the cloud mode cavity field realizes the Dicke-model with tunable
[
is homogeneous which is stabilized by thermal fluctua- parameters in the kHz range, and this is the first system

1 tions. In this phase, the optical mean field in the cav- inwhichthecriticalpointcanbereachedandinvestigated.
v
ity is zero, because the laser pump field is not scattered
0
into the cavity mode from the homogeneous distribution
6
of atoms. Above a threshold pump power, however, this
7 ηt <η crit ηt >η crit
3 solution becomes unstable. Then the atoms self-organize
1. into a wavelength-periodic crystalline order which gives modefunction
rise to Bragg-scattering from the transverse pump laser cos(kx)
0
into the cavity. The resulting cavity field traps the atoms
1
in the optical lattice distribution (see Figure 1). The self-
1
: organizationeffecthasbeenfirstpredictedinRef.[1],and
v BEC
soon experimentally observed [2]. It is closely related to
i
X the collective atomic recoil lasing transition [3] which is
r a Kuramoto-model-like synchronization phenomenon [4]. pumplaser
a A classical mean-field description of self-organization has
been presented in [5,6], which relies on a self-consistent
Fig. 1. Self-organization of a Bose-Einstein condensate
canonical distribution of the atomic ensemble at a finite
in a cavity. Below a certain threshold pump power (left),
temperature T. The mean-field theory in [7] introduces a
the ultracold atoms have a quasi-homogeneous distribu-
Vlasov-type equation in phase space which can account
tion, and the cavity field with mode function cos(kx) is
for arbitrary velocity distributions.
empty. Above threshold (right), the atoms self-organize
Thesameself-organizationeffectcanoccurinthecase
into a λ-periodic ordered lattice in which they scatter the
of Bose-Einstein condensed ultra-cold atoms (BEC) at
pump light constructively into the resonator mode. There
zero temperature [8,9,10,11,12]. For low pump power, in-
is another, λ-shifted lattice possible to be formed.
stead of the thermal fluctuations, the homogeneous phase 2
is stabilized by atom-atom collisions, or, in the lack of
collisions, ultimately by the zero-point kinetic energy. A
The Dicke-type phase transition is considered usually
a Corresponding author: [email protected] in systems with a fixed number of atoms and where the
2 G. Ko´nya, G. Szirmai, P. Domokos: Multimode mean-field model for a BEC in a cavity
individual atomic degrees of freedom span only a 2-mode ∆ κ, where κ is the cavity mode linewidth and the
C
| | ∼
Hilbert space. These assumptions are necessary to intro- cavity-pump detuning is ∆ = ω ω (all these param-
C C
−
duce the spin representation of the two-mode boson field. eters are summarized in Fig. 2). The scattering of laser
The drawback of this approach is that the role of higher
excitedmotionalmodescannotbeincludedinthedescrip-
tion.
In this paper we resort to a different approach which
allows for the generalization to a multimode treatment
m
of the matter wave field. Instead of fixing the number of o
atomsintheatomicmodes,weinvokethegrandcanonical at %
ensembledescription,onassumingthattheconstantNc is #A = ! $ !A g
the mean number of atoms in the condensate. In contrast
toRef.[8]wherethecondensatemeanfieldhasbeenwrit- or
at
ten in position space, here both the condensate and the n
o
quantumfluctuationswillbetreatedinmomentumspace. es & #C = ! $ !C
This is the most economic approach in terms of compu- r laser
tational needs, since the calculation converges very fast
!, "
R
to the exact result as one includes higher excited kinetic
energy eigenstates. We will show that the position of the
criticalpointisnotaffectedbythehighermodes.Further-
more we will confirm that the two mode model correctly Fig. 2. Summary of the system parameters used in the
describes the system and the phase transition below, and microscopic model.
in the vicinity of the critical point. Far above threshold,
however, the effect of the higher modes will become sig-
photons into the cavity is thus a quasi-resonant process.
nificant and the new multimode treatment is required.
Moreover,itissignificantlyenhancedbythestrongdipole
The paper is organized as follows. In section 2, we de-
couplingbetweentheatomsandthemodeduetothesmall
scribe the microscopic Hamiltonian model of our system.
volume of the cavity. This coupling strength is character-
Insection3,theone-particlewavefunctionsandthemode
ized by the single-photon Rabi frequency g, which is in
expansion are introduced. The backbone of our paper is
the range of κ.
section 4, where we introduce the grand canonical Hamil-
For the sake of simplicity, we describe the dynamics
tonian, which makes possible to systematically define, in
in one dimension x along the cavity axis, in which direc-
subsection 4.1, the mean-field approximation in a multi-
tionthemodefunctioniscos(kx),andthecavitylengthis
mode model. Then, in 4.2, we determine the independent
L. The atom field and the resonator mode are described
quasiparticles by means of a Bogoliubov transformation.
by the pair of bosonic annihilation and creation field op-
Thesubsection4.3isdevotedtostudyingthefluctuations
erators, Ψˆ(x), Ψˆ†(x) and aˆ, aˆ†, respectively. In the large
in the normal phase of the system (below threshold). In
detuning limit, i.e., ∆ is the far largest frequency in the
section5,thegroundstateofthefluctuationHamiltonian A
system, the excited state can be eliminated [20], and the
is analysed and the incoherent populations in the excited
many-particleHamiltonianinthereferenceframerotating
modes above the condensate are calculated numerically.
at the pump frequency ω is ((cid:126)=1) :
We show that the ground state is an entangled one of the
bipartite system of the cavity mode and the atomic mo-
tional degrees of freedom, and the entanglement is quan- Hˆ = ∆ aˆ†aˆ + +L2 Ψˆ†(x) 1 d2
tified in section 6. − C (cid:90)−L2 (cid:16)−2mdx2
+U aˆ†aˆ cos2(kx)+η cos(kx) aˆ†+aˆ Ψˆ(x)dx.
0 t
2 Microscopic model of the system
(cid:0) (cid:1)(cid:17) (2.1)
We consider an ensemble of ultracold atoms at T =0 in- The first term gives the optical field energy in the res-
teracting with a single-mode of a high-Q optical cavity onator.Thesecondtermisthekineticenergyoftheatoms.
[18,19]. The atoms are coherently driven from the side by The third term describes the dispersive interaction be-
alaserfieldwithfrequency ω,directedperpendicularly to tween the atoms and the cavity with a coupling strength
the cavity axis (see Fig. 1). The driving strength is de- U = g2/∆ . The underlying physical process is the ab-
0 A
scribed by the Rabi-frequency Ω . The laser is detuned sorptionandstimulatedemissionofacavityphoton.This
R
far below the atomic transition ω , that is, ∆ γ, scattering process, from the point of view of the atoms,
A A
where 2γ is the full atomic linewidth at half|ma|xi(cid:29)mum means an effective potential of the shape cos2(kx) with
and the (red) atom-pump detuning is ∆ = ω ω < 0. depthdependingonthephotonnumberaˆ†aˆ.Accordingto
A A
−
Thisconditionensuresthattheelectronicexcitationisex- the red atomic detuning (∆ <0) the coefficient U <0,
A 0
tremely low in the atoms, hence the spontaneous photon which implies trapping positions at the λ/2 separated
emission is suppressed. At the same time, the laser field antinodesofthecavitymodefunction.Ontheotherhand,
isnearlyresonantwiththecavitymodefrequencyω ,i.e. from the viewpoint of the cavity field, this term preserves
C
G. Ko´nya, G. Szirmai, P. Domokos: Multimode mean-field model for a BEC in a cavity 3
the photon number but gives rise to a frequency shift de- Onsubstitutingtheexpansion(3.1)into(2.1),theHamil-
pendingonthespatialdistributionoftheatomfield.This tonian can be constructed in terms of quadratic forms as
term is responsible for the optomechanical-type coupling
[21,22,23]investigatedintheexperiments[24,25].Thelast Hˆ = ∆ aˆ†aˆ+ω cˆ†M(0)cˆ
C R
term describes the effect of the pump field and results −
from the scattering between the pump laser and the cav- √2 (cid:16) (cid:17)
+ η aˆ†+aˆ cˆ†M(1)cˆ (3.3)
ity field. The back action of this scattering on the pump 2 t
lfiaesledrdisrinveinggle,caˆte†d+. Iaˆt,twhuitshatmheoucnotnssteaffnetcttirvaenlysvteorsaecpauvmitpy + 1U0aˆ(cid:0)†aˆ cˆ†(cid:1)(cid:16)M(2)+2I(cid:17) cˆ ,
4
amplitude ηt = ΩRg/∆A, and depending again on the (cid:16) (cid:16) (cid:17) (cid:17)
local matter wave field density. whereω = k2 istherecoilfrequency,Iistheunitmatrix,
R 2m
In Eq. (2.1) we consider only one type of atom-atom and the M(j) matrices are all real and symmetric,
interaction,namelytheonewhichismediatedbythecav-
ity field. This interaction is long-range [26] and has strict 02
periodicity due to the fixed momentum of the exchanged 12
photons. In contrast, the effect of s-wave scattering can  22 
take place at arbitrary momentum values and causes a M(0) = 32 (3.4a)
 
broadening of the atomic momentum distribution around  
the peaks at integer times the momentum of the cavity  · 
photon [27]. We disregard this broadening effect by as-  ·
 
suming that the cavity mediated long range interaction 0 1
dominates over s-wave scattering. Such an assumption is 1 0 √1
physically sound for a wide range of experimental para-  2 
√1 0 √1
maters [21] and helps the distillation of the effect caused M(1) = 2 2 (3.4b)
by the cavity photons.  √12 0 √12 
Here we are interested in the ground state proper-  √1 0 
 2 ·
ties and the excitation spectrum of this Hamiltonian, and  
therefore disregard the effects of the photon leakage out  · ·
of the cavity. 0 0√2
0 1 0 1
M(2) =√20 0 01  (3.4c)
1 0 00
3 Mode expansion  1 0 ·
 
 · ·
 · · ·
For the decomposition of Ψˆ(x), we can use the complete  
Note that the M(j) matrix is diagonal for j = 0, tridiag-
orthonormal set of mode functions,
onal for j =1, and pentadiagonal for j =2. Since the ki-
∞ ∞ netic energy difference between adjacent modes increases
1 2 2 with the mode index, there appears a natural cutoff ex-
, cos(nkx) , sin(nkx) .
L L L cludingthehighenergymodesfromthedynamicsandthe
(cid:114) (cid:40)(cid:114) (cid:41) (cid:40)(cid:114) (cid:41)
n=1 n=1 system effectively has only a finite number of atom field
modes.Inthisway,thismodedecompositionwillresultin
However, the sin(nkx) modes are not populated by the
a significantly reduced numerical effort compared to the
parity-conserving Hamiltonian (2.1) when the system
real-space mean-field approaches in Refs. [8,28].
starts from a homogeneous BEC (being the ground state
for U =η =0).Thus, Ψˆ(x)can beexpandedin termsof
0 t
the even modes as
4 Mean-field approximation
∞
1 2
Ψˆ(x)= cˆ0+ cos(nkx)cˆn , (3.1) We assume that the atoms form a condensate which is a
(cid:114)L n=1 (cid:114)L macroscopic mean-field ψˆ(r,t) rotating at a unique fre-
(cid:88) (cid:104) (cid:105)
quency corresponding to the chemical potential µ. There-
where the cˆn operators satisfy bosonic commutation fore, we transform into the picture given by the grand
relations. We will use the compact notations cˆ = canonical Hamiltonian
(cˆ , cˆ , ...)T,beingacolumnvector,andcˆ† = cˆ†, cˆ†, ... ,
0 1 0 1 Kˆ =Hˆ µNˆ , (4.1)
being a row vector. Then the operator of the (cid:16)total atom (cid:17) −
number is:
whichdefinesadynamicssuchthatthecondensatemean-
field is static. The chemical potential can be determined
+L ∞
Nˆ = 2 Ψˆ†(x)Ψˆ(x)dx= cˆ† cˆ =cˆ†cˆ (3.2) self-consistently, by assuming a fixed density of the con-
n n
(cid:90)−L2 n(cid:88)=0 densate atoms Nc/L.
4 G. Ko´nya, G. Szirmai, P. Domokos: Multimode mean-field model for a BEC in a cavity
Letusseparatethemeanvaluesoftheoperatorsaˆand and
cˆ from the quantum fluctuations. The mean-field part is
denoted by aˆ = √N α and cˆ = √N γ, where the Kˆ(2) =Ω(γ)aˆ†aˆ+cˆ† (M(α) µI) cˆ
γT γ =1nor(cid:104)m(cid:105)alizationcconditio(cid:104)nfi(cid:105)xesthencumberofcon- 1 − (4.8)
+ aˆ†+aˆ cˆ†M(cid:48)(α)γ+γT M(cid:48)(α)cˆ .
densed atoms to be N . Later we will justify that α and
c 2
the elements of γ can be chosen real. The √Nc multipli- (cid:0) (cid:1)(cid:16) (cid:17)
ers are included in order to make α and γ constant in the The effective cavity resonance frequency is
thermodynamic limit (N , L ). Let us displace
c
the operators, →∞ →∞ Ω(γ)= δC +uγT M(2)γ, (4.9a)
−
aˆ Ncα+aˆ, (4.2a) andthecross-couplingofthemotionalmodesviathecav-
→
ity mean field is expressed by the matrix
cˆ (cid:112)Ncγ+cˆ, (4.2b)
→
which is a canonical tran(cid:112)sformation. After the displace- M(α)=ω M(0)+yαM(1)+uα2 M(2)+2I , (4.9b)
R
ment aˆ = 0 and cˆ = 0, so the new operators cor-
(cid:104) (cid:105) (cid:104) (cid:105) (cid:16) (cid:17)
respond to the quantum-fluctuations. Note that the dis-
which is a real, symmetric matrix valued polynomial of
placement breaks the U(1) symmetry of the microscopic α. M(cid:48)(α) denotes the derivative of this polynomial with
model, which is the invariance of the Hamiltonian with
respect to α.
respect to the transformation Ψˆ(x) → Ψˆ(x)e−iϕ with ar- It can be seen that Kˆ(j) is proportional to Nc1−j/2,
bitrary phase ϕ. The expectation values of the total pho-
which implies that the third and fourth order terms dis-
ton and atom number operators can be expressed in the
appearinthethermodynamiclimit,sowedisregardthem.
displaced frame as
The mean-fields α and γ are determined by the condition
aˆ†aˆ N α2+ aˆ†aˆ (4.3a) that,inthegrandcanonicalHamiltonian,thetermslinear
c
(cid:104) (cid:105)→ (cid:104) (cid:105) inthefluctuationsaˆandcˆcollectedinKˆ(1)havetovanish.
cˆ†cˆ N + cˆ†cˆ (4.3b)
(cid:104) (cid:105)→ c (cid:104) (cid:105) Thefluctuationsaroundthemeanvaluesaredescribedby
That is, on top of the number of N condensate atoms, the bilinear Hamiltonian Kˆ(2).
c
there is an incoherent population of atoms outside the
condensate which appears due to the atom-photon inter-
action. 4.1 The mean field solution
The terms in Kˆ after the displacement should be
grouped according to the powers of aˆ and cˆ, The condition Kˆ(1) =0 leads to the system of equations:
Kˆ =Kˆ(0)+Kˆ(1)+Kˆ(2)+Kˆ(3)+Kˆ(4) (4.4) 1
Ω(γ)α + y γT M(1)γ =0. (4.10a)
2
Let us introduce new parameters which have constant
value in the thermodynamic limit (N , L ): M(α)γ =µγ . (4.10b)
c
→ ∞ → ∞
Theseequationsdefineaquasieigenvalueproblem:γisthe
1
δ =∆ N U (4.5a) eigenvector of the matrix M(α), and the smallest eigen-
C C c 0
− 2 valueisthechemicalpotential.Becausethematrixissym-
1
metric, µ will have a real value. But the value of α, and
u= N U (4.5b)
c 0
4 so the matrix itself depends on γ through the first equa-
y = 2N η (4.5c) tion,whichrenderstheproblemtobenonlinear.Itcanbe
c t
solved by iteration which, as a main virtue of the present
The zeroth order term of Kˆ(cid:112)is a c-number, approach, is a stable and fast numerical method.
Let us make an iteration step starting from the initial
valueα=0.Then,M(α=0)=ω M(0),whichisadiag-
K(0) =N δ α2+ω γT M(0)γ R
c(cid:32)− C R onal matrix and its smallest eigenvalue is µ=0. The cor-
(cid:16) (cid:17) responding normalized eigenvector is γ = (1, 0, 0, ...)T
+yα γT M(1)γ +uα2 γT M(2)γ µ , (4.6) for which Ω(γ) = δC = 0 and γT M(1)γ = 0. This
− (cid:33) yields α = 0, which−is th(cid:54)en a trivial solution describing
(cid:16) (cid:17) (cid:16) (cid:17)
the normal phase of our system: the resonator contains
soitisirrelevanttothedynamics,butgivesthemean-field
no photons and the whole condensate is in the homoge-
energy of the system. The first and second order terms
neous mode. This solution always exists but it becomes
read
unstable above acertain threshold y =√ δ ω . This
crit C R
−
1 critical point can be seen in Fig. 3 which plots α, the
Kˆ(1) = N aˆ†+aˆ Ω(γ)α+ yγT M(1)γ
c 2 mean amplitude of the cavity mode divided by √Nc, as a
(cid:18) (cid:19)
(cid:112) (cid:0) (cid:1) function of the transverse pump amplitude y. This curve
+ Nc cˆ† (M(α)−µI) γ+γT (M(α)−µI) cˆ , is calculated for various ncutoff cutoff mode indexes, from
(cid:112) (cid:16) (4(cid:17).7) ncutoff =2,correspondingexactlytotheDicke-model[14],
G. Ko´nya, G. Szirmai, P. Domokos: Multimode mean-field model for a BEC in a cavity 5
0.8 n =10 stillfarbelowtheoneneededinthereal-spacedescription
cutoff
[8].
n =5
cutoff
0.6 The role of higher-order modes is illustrated also in
n =3
cutoff Fig. 5. There is another criticality in the system, of dif-
ferent nature, which occurs when the effective mode fre-
α 0.4 ncutoff=2 quency Ω, depending on the condensate distribution γ
as shown in Eq. (4.9a), becomes negative. In this regime
0.2 there is no stable solution for the coupled atom field and
cavity mode system. The effective mode frequency can be
tuned by varying u, which leads to a divergence in α.
0 This critical point|de|pends on the cutoff mode index be-
0 1 2 3 4 5 6 7
low n . In the two mode case the divergence occurs
y/y cutoff
crit
when u = δ , however, the exact result, approached well
C
withthecutoffn =10,isatmuchsmaller u because
Fig. 3. The coherent field amplitude α as a function of cutoff | |
the atoms in higher order modes are allowed to localize
the pumping strength y for various cutoff mode numbers
much better at the antinodes of the cavity mode function
(n = 2...10). The critical point y = 10 does not
cutoff crit
and yield a larger resonance shift.
depend on n . The parameters: ω = 1, δ = 100,
cutoff R C
−
u= 20.
−
n =2 n =5
cutoff cutoff
2
1
0.8 1.5 n =3
cutoff
γ2
0
0.6
n =10
cutoff
2γj α 1
γ2
0.4 1
0.5
0.2 γ2
2
γ2
3 γ2
4 0
0
-1 -0.8 -0.6 -0.4 -0.2 0
0 1 2 3 4 5 6 7 8 9
y/y u/ δ
crit C
| |
Fig. 4. The distribution γ2 of condensate atoms in the Fig. 5. The effect of the phase shift term on α. The nu-
j
modes as a function of the pump strength. The phase merical simulation confirms that the effective cavity fre-
transition occurs in the subspace spanned by the modes quency Ω(γ) tends to zero at the divergence point. The
cˆ0 and cˆ1, but well above the threshold other modes also parameters: ωR = 1, δC = 100. The value of y is fixed
−
get involved in the dynamics. The parameters: ωR = 1, just above the critical point, y =11, ycrit =10.
δ = 100, u= 20, n =10.
C cutoff
− −
to n =10. The case n =10 is close to being ex-
cutoff cutoff
act, since the higher excited modes have γ 0 with four
n
≈ 4.2 The analysis of the fluctuations
digitprecisionforn 10.Thedistributionoftheconden-
≥
sate atoms in the modes is shown in Fig. 4. Below thresh-
old, only the homogeneous mode n = 0 is populated. At
thecriticalpoint,thepopulationγ2inthemodecoskxbe- Thevaluesofαandγ areknownfromthenumericalsolu-
1 tionof (4.10)whichalsoprovidesfortheeigenvectorsv(j)
gins to grow abruptly from zero with a finite slope, while
associated with the eigenvalues λ of M(α). The matrix
the higher mode populations start slowly with vanishing j
issymmetricandreal,sotheeigenvaluesandeigenvectors
derivative. Therefore, the two-mode approximation holds
are real. The eigenvectors form a complete orthonormal
in the vicinity of the critical point. Well above threshold
the other modes get populated. In Fig. 3, above thresh- basis: v(i)T v(j) = δij. We can arrange the eigenvalues
·
old,thesignificantdependenceoftheslopeonthenumber in increasing order: the smallest one is λ0 = µ, and the
ofexcitedmodestakenintoaccountunderlinestheimpor- corresponding eigenvector is v(0) =γ.
tanceofthemultimodeapproachascontrastedtothetwo- Let us first decouple the atomic modes interacting via
mode model. However, the mode number n = 10 is the second term of (4.8). The eigenvectors put into the
cutoff
6 G. Ko´nya, G. Szirmai, P. Domokos: Multimode mean-field model for a BEC in a cavity
columns of a matrix, thenafreeexpansionwithcharacteristictimescaleabout
√
π Nc, which is the far longest time scale. Note also that,
g0
below threshold g = 0 so the phase fluctuations are not
O= v(0) v(1) v(2) ... , (4.11) 0
  growing at all.
From now on, we neglect these fluctuations and com-
  pletely eliminate the dynamics of the mode ˆb . This ap-
definetheorthogonaltransformation,OT O=O OT =I, 0
proximationrendersthecondensatetobeaclassicalback-
· ·
which leads to the bosonic modes
groundfieldsimilartotheexternallaserpumpfieldwhich
was described by the real parameter η in the model.
bˆ =OT cˆ. (4.12) t
· This step is equivalent to projecting the atomic exci-
tation space to the one orthogonal to the condensate,
Inversely,
cˆ cˆ =cˆ γ(γTcˆ), as described in [30].
⊥
→ −
cˆ=O bˆ =v(0) ˆb +v(1) ˆb +v(2) ˆb +... , (4.13) Inthefollowing,weperformaBogoliubov-transforma-
· · 0 · 1 · 2 tionon aˆ, aˆ†, bˆ, bˆ† inordertodefinetheindependent
Sincev(0) =γ theˆb0 modedescribesthefluctuationspar- quasipar(cid:16)ticle modes (cid:17)dˆ, dˆ† , which combine excitations
allel to the condensate. Subsequently theˆb ,ˆb , ... modes
1 2
describe orthogonal excitations. of the atom field an(cid:16)d the e(cid:17)lectromagnetic field. Let us
introduce the quadrature amplitudes
The grand canonical Hamiltonian simplifies to
1
∞ xˆ = aˆ†+aˆ (4.18a)
0
Kˆ(2) = Ω(γ)aˆ†aˆ+ (λ µ)ˆb†ˆb √2Ω
j − j j (cid:0) (cid:1)
j=0 Ω
(cid:88) pˆ =i aˆ† aˆ (4.18b)
∞ 0
1 2 −
+ g aˆ†+aˆ ˆb†+ˆb , (4.14) (cid:114)
2(cid:88)j=0 j(cid:0) (cid:1)(cid:16) j j(cid:17) xˆj = 2(λ1 µ(cid:0)) ˆb†j +(cid:1)ˆbj (4.18c)
j
− (cid:16) (cid:17)
wzehreor,esignc=eλO0T=·Mµ.(cid:48)(Tαh)isγ.zeTroh-emfroedqeuiesntchyeoGfothldesˆtbo0nmeomdoedies pˆj =i(cid:112) λj2−µ ˆb†j −ˆbj , (4.18d)
(cid:114)
resulting from the U(1) symmetry breaking imposed by (cid:16) (cid:17)
the choice of real mean field γ. In the two-dimensional wherej 1,2,3,... .Notethatthe(xˆ , pˆ )quadratures
0 0
∈{ }
phase space of the Goldstone mode, the quadrature rˆ arerelatedtoaˆ,nottoˆb .Thequadraturesobeytheusual
21(ˆb†0+ˆb0)isparallelwiththecondensateandisaconstan≡t canonical commutation0relations:
of motion (commutes with the above Kˆ(2)). The orthogo-
nalquadrature√N φˆ i(ˆb† ˆb )/2correspondstophase [xˆk, pˆl]=iδkl , (4.19)
c ≡ 0− 0
fluctuations of the condensate [29]. It obeys the equation
andallothercommutatorsvanish.Usingvectornotations,
of motion
dφˆ= g0 aˆ†+aˆ , (4.15) xˆ = (xˆ0, xˆ1, xˆ2, ...)T and pˆ = (pˆ0, pˆ1, pˆ2, ...)T, the
grandcanonicalHamiltoniancanbeexpressedintermsof
dt −2√N
c
the quadratures as
(cid:0) (cid:1)
d2 g Ω(γ)
dt2 φˆ(t)=−i 20√Nc aˆ†−aˆ . (4.16) Kˆ(2) = 1 pˆT pˆ + 1 xˆT Sxˆ +const, (4.20)
(cid:0) (cid:1) 2 2
Let∆tbeasmalltimeintervalonthetimescaleofthevari-
ationofthecondensatephase.φˆ(∆t)canthenbewellap- wherec-numberswereomittedandtheSkernelmatrixis:
proximated by the second-order Taylor-expansion which
Ω2 g˜ g˜
includes the above two time derivatives. The growth of 1 2
2 ··
φtˆh(e0)pihsansoetflcuocrtruealattioednsniesitchhearrwacittehriaˆz(e0d)bnyor(cid:104)φwˆ2it(h∆aˆt)†(cid:105)(.0)S,inthcee S= gg˜˜12 (λ1−µ) (λ2 µ)2  , (4.21)
−
 
only non-vanishing term up to second order is  · · 
 
 · ·
φˆ2(∆t) = φˆ2(0) + g02 aˆ†(0)+aˆ(0) 2 ∆t2 with the off-diagonal elements g˜ = g  Ω(λ µ). It
(cid:104) (cid:105) (cid:104) (cid:105) 4Nc(cid:104) (cid:105)· follows that the Bogoliubov tranjsformja·tion amjo−unts to
(cid:0) +O(∆(cid:1) t3), (4.17) the diagonalization of the real symmetric(cid:112)matrix S. Let
U be the orthogonal matrix comprising the eigenvectors
where aˆ†(0)+aˆ(0) 2 will be given later by (4.18) and of S as its colums (UT U=U UT =I). The canonical
(cid:104) (cid:105) · ·
(5.2). However, this expectation value is close to 1 except transformation,
(cid:0) (cid:1)
for a small vicinity of the critical point where it diverges.
InthethermodynamiclimitN thephaseundergoes xˆ =U Xˆ , pˆ =U Pˆ , (4.22)
c
→∞ · ·
G. Ko´nya, G. Szirmai, P. Domokos: Multimode mean-field model for a BEC in a cavity 7
leads to respectively. This matrix is already diagonal: we can read
out that µ = 0 and λ = n2 ω . The orthogonal trans-
Kˆ(2) = 1 ∞ Pˆ2 + ω2Xˆ2 +const, (4.23) formation in (4.12) isnthe triv·iaRl one: O = I and bˆ = cˆ.
2 j j j The vector of the coupling constants between aˆ and cˆ is:
(cid:88)j=0 (cid:16) (cid:17) g = OT M(cid:48)(α)γ = y (0, 1, 0, 0, ...)T. This is an im-
where ω2 are the real eigenvalues of the matrix S. This portant r·esult: in the n·ormal phase aˆ is coupled only to
j
Hamiltonian describes independent harmonic oscillators cˆ1 and the coupling constant is g1 = y. This means that
associated with bosonic quasiparticles with ω eigenfre- thetwo-modemodel[14]isexactinthenormalphase,and
j
quencies. The annihilation and creation operators of the the Hamiltonian (4.14) is simply
quasiparticles are
∞
dˆj = ωj Xˆj + i Pˆj , (4.24) Kˆ(2) = −δC aˆ†aˆ + ωR n2cˆ†ncˆn
2 2ω n=1
(cid:114) j (cid:88)
1
dˆ† = ωj Xˆ (cid:112)i Pˆ , (4.25) + 2y aˆ†+aˆ cˆ†1+cˆ1 . (4.27)
j (cid:114) 2 j − 2ωj j (cid:0) (cid:1)(cid:16) (cid:17)
The S matrix is
where j 0,1,2,... , then the H(cid:112)amiltonian is
∈{ } δ2 y y
∞ y yC ·ω2crit
Kˆ(2) = ωj dˆj†dˆj +const. (4.26) S= · crit R 4ωR2  , (4.28)
j=0 9ω2
(cid:88)  R 
 
The spectrum, the set of the eigenvalues ω , is plotted in  ·
j  
Fig. 6 as a function of the pump strength parameter y.
with the modified coupling constant g˜ = y y , where
1 crit
The calculation relies on a number of modes ncutoff = 10 y = √ δ ω is the critical point of the p·hase transi-
crit C R
−
tioninthetwomodemodelasshownin[14].Diagonaliza-
60 tion of the first block leads to the non-trivial eigenvalues
50 ω6 δ2 +ω2 δ2 ω2 2 y2
ω2 = C R C − R +δ2 ω2 , (4.29)
± 2 ±(cid:115) 2 C R y2
40 ω (cid:18) (cid:19) crit
5
/ωR 30 ω ω Tthheepfhreaqseuetnracynsωit−iongopeosintot.zIetrofoalltowys=thyactritt,hwehpihchasiesttrhaenn-
ωj 4 1 sition occurs at the point y =y even in the multimode
crit
20 ω3 model.
ω
ω 0
10 2
5 Analysis of the ground state fluctuations in
0
arbitrary phase
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
y/y
crit
OurnextgoalistoexpressthegroundstateoftheHamil-
Fig. 6. The ω eigenfrequencies of the system. For y =0 tonian (4.8) in terms of the Fock space of the operators
j
the well-known ω = j2 ω spectrum of an ideal gas in aˆ and cˆ which have clear physical meaning. The ground
j R
a box is rendered. On in·creasing y, the lowest eigenvalue stateoftheHamiltonian(4.26)issimplythevacuumstate
tends to zero at the phase transition point. The parame- of the dˆ operators. However, since the Bogoliubov trans-
ters are the same as in Fig. 4. formation in Eq. (4.22) mixes creation and annihilation
operators, the ground state contains photonic and mo-
tionalexcitations,moreover,itwillbeanentangledstate.
which reproduces the exact result [8]. AsfollowsfromtheHamiltonianin(4.27),belowthreshold
the ground state is the two-mode squeezed vacuum.
We can use the Wigner-function to fully describe the
4.3 Fluctuations in the normal phase
ground state [31]. We make use of the fact that the
Wigner-function associated with the ground state of a
We apply the general results to the normal phase of the
bilinear Hamiltonian is always a multivariate Gaussian
system described by α = 0 and γ = (1, 0, 0, ...)T. distribution which, when centered at the origin, is fully
So, there is no coherent mean optical field in the res- determined by its covariance matrix [32]. To obtain the
onator, and the condensate is homogeneous. Then (4.9a) Wigner-function, we have to calculate then the symmet-
and (4.9b) gives Ω(γ) = δ and M(α = 0) = ω M(0), rically ordered covariance matrix.
C R
−
8 G. Ko´nya, G. Szirmai, P. Domokos: Multimode mean-field model for a BEC in a cavity
Wecanstartfromthecorrelationsbetweenthe(Xˆ,Pˆ) where k 1,2,3,... . By summing these terms, we can
∈ { }
quadraturespertainingtotheindependentquasi-particles, get the total number of atoms outside the condensate:
which are
∞
(cid:104)XˆkXˆl(cid:105)= 21ω δkl (5.1a) (cid:104)cˆ†cˆ(cid:105)=(cid:104)bˆ†bˆ(cid:105)= (cid:104)ˆb†kˆbk(cid:105)
k k=1
ω (cid:88)
(cid:104)PˆkPˆl(cid:105)= 2k δkl (5.1b) = 1 ∞ ∞ U2 ωj + λk−µ 2 , (5.5)
4 kj λ µ ω −
(cid:104){XˆkPˆl}s(cid:105)=0, (5.1c) k(cid:88)=1 (cid:88)j=0 (cid:18) k− j (cid:19)
where ... denotes symmetric ordering. which is plotted in Fig. 7 as a function of the coupling
s
Now{ w}e apply the (4.22) transformation to determine parameter y. Note that the index associated with cˆk runs
thecovariancematrixofthe(xˆ,pˆ)quadraturesassociated from 0, while the index associated with ˆbk runs from 1.
with the photonic and the atomic motional excitations: Finally, the number of atoms in the cˆk modes, which are
associatedwiththespatialharmonicfunctions,canbeob-
∞
1 1 tainedbyuseofthetransformationrule(4.13),theinverse
xˆ xˆ = U U (5.2a)
(cid:104) k l(cid:105) 2 kj lj · ω of the formulae (4.18) and the covariance matrix (5.2):
j
j=0
(cid:88)
1 ∞ ∞
(cid:104)pˆkpˆl(cid:105)= 2 UkjUlj ·ωj (5.2b) (cid:104)cˆ†ncˆn(cid:105)= OnkOnl(cid:104)ˆb†kˆbl(cid:105)
(cid:88)j=0 k(cid:88),l=1
xˆ pˆ =0 (5.2c) ∞ ∞
k l s 1
(cid:104){ } (cid:105)
= O O U U
nk nl kj lj
4 ·
k,l=1 j=0
(cid:88) (cid:88)
0.03 2 (λ µ)(λ µ) ω
k l j
− − +
·(cid:32)(cid:112) ωj (λk µ)(λl µ)
− −
1.5
0.02 (λ µ(cid:112)) (λ µ)
k l
aaˆˆ† (cid:11) 1 ˆˆcc† E −(cid:115)(λl−−µ) −(cid:115)(λk−−µ)(cid:33) (5.6)
(cid:10) D
0.01 Starting from this formula, and using the orthogonality
0.5
of the matrix O, the total number of atoms outside the
condensate in (5.5) can be verified.
0 0
0.2 0.6 1 1.4 1.8
y/ycrit 6 Entanglement in the ground state
Fig.7.Thenumberoftheincoherentphotons aˆ†aˆ (solid Let us partition the system to the cavity mode and an-
(cid:104) (cid:105)
line) and the expectation value of the total number of other part including all the motional modes of the atom
atomsoutsidethecondensate cˆ†cˆ (dashedline)nearthe field. The ground state of the system ψ , which is the
(cid:104) (cid:105) | g(cid:105)
phase transition point. The parameters: ωR = 1, δC = vacuum of the quasiparticles defined by the operators dˆ
k
100, u= 20, y =10, n =3.
crit cutoff (k = 0,1,...), exhibits bipartite entanglement between
− −
the radiation and the matter wave fields. This is sim-
ilar to the entanglement occurring in the Dicke-model
Astheapplicationoftheaboveresults,letuscalculate
[33,34,35]. The entanglement can be simply measured by
the number of incoherent photons in the ground state.
the Neumann-entropy of the cavity subsystem ,
From (4.18) and (5.2) it follows that
C
aˆ†aˆ = 1 ∞ U2 ωj + Ω 2 . (5.3) SC =−TrC(ρˆC ·lnρˆC) , (6.1)
(cid:104) (cid:105) 4 j=0 0j (cid:18)Ω ωj − (cid:19) where the reduced density matrix of the cavity mode is
(cid:88) obtained by tracing the total density matrix ρˆ= ψ ψ
g g
| (cid:105)(cid:104) |
This expression is numerically evaluated and plotted as over the atomic subsystem , i.e., ρˆC =TrA(ρˆ). Another
A
a function of the pump strength parameter y in Fig. 7. entanglement measure, which can be used, is the linear
With a similar calculation, just performing the replace- entropy,
mateonmtss 0in→eakchaonfdtΩhe→ˆb λmko−deµs, we can get the number of SC,lin =1−TrC ρˆ2C . (6.2)
k Both entropies have zero value pre(cid:0)cise(cid:1)ly if ρˆC is pure. For
∞ mixed states, they are positive, however, SC can have an
1 ω λ µ
ˆb†ˆb = U2 j + k− 2 , (5.4) arbitrarilylargevalue,whilethelinearentropyisbounded,
(cid:104) k k(cid:105) 4 (cid:88)j=0 kj (cid:18)λk−µ ωj − (cid:19) SC,lin ≤1.
G. Ko´nya, G. Szirmai, P. Domokos: Multimode mean-field model for a BEC in a cavity 9
0.1 leads to mean-field equations which have to be solved nu-
merically.Wecalculatedthenumberofincoherentphotons
0.08 and populations in the higher excited motional modes, as
well as the amount of entanglement between the matter
y 0.06 and radiation fields. This approach is suitable and will
op be used in the future to deal with other type of multi-
r
ent 0.04 modesystems,forexample,thematterwavefieldcoupled
to the radiation field in a degenerate confocal resonator
S which shows a rich phase diagram [38,39].
0.02 C
S
,lin
C This work was supported by the National Office for Research
0
andTechnologyunderthecontractERC HU 09OPTOMECH,
0.2 0.6 1 1.4 1.8
y/y and the European Science Foundation’s EuroQUAM project
crit
Cavity-Mediated Molecular Cooling. G.Sz. also acknowledges
funding from the Spanish MEC projects TOQATA (FIS2008-
Fig. 8. The bipartite entanglement measures, the
00784), QOIT (Consolider Ingenio 2010), ERC Advanced
Neumann-entropy S and the linear entropy S as a
C C,lin Grant QUAGATUA and EU STREP NAMEQUAM.
functionofthecouplingparametery.Parameters:ω =1,
R
δ = 100, u= 20, y =10, n =3.
C crit max
− −
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