The Dicke model phase transition in the quantum motion of a Bose-Einstein condensate in an optical cavity
TThe Dicke model phase transition in the quantum motion of a Bose-Einsteincondensate in an optical cavity
D. Nagy , G. K´onya , G. Szirmai , , and P. Domokos Research Institute for Solid State Physics and Optics, H-1525 Budapest P.O. Box 49, Hungary and ICFO-Institut de Ci`encies Fot`oniques, 08860 Castelldefels (Barcelona), Spain
We show that the motion of a laser-driven Bose-Einstein condensate in a high-finesse opticalcavity realizes the spin-boson Dicke-model. The quantum phase transition of the Dicke-modelfrom the normal to the superradiant phase corresponds to the self-organization of atoms fromthe homogeneous into a periodically patterned distribution above a critical driving strength. Thefragility of the ground state due to photon measurement induced back action is calculated.
PACS numbers: 05.30.Rt,37.30.+i,42.50.Nn
A thermal cloud of cold atoms interacting with a sin-gle mode of a high-finesse optical cavity can undergo aphase transition when tuning the power of a laser fieldwhich illuminates the atoms from a direction perpendic-ular to the cavity axis [1, 2, 3, 4]. Below a thresholdpower, the thermal fluctuations stabilize the homoge-neous distribution of the cloud, and photons scatteredby the atoms into the cavity destructively interfere, ren-dering the mean optical field to be zero. Above threshold,the atoms self-organize into a wavelength-periodic crys-talline order bound by the radiation field which, in thiscase, is composed of the constructive interference of pho-tons scattered off the atoms from the laser into the cavity.The same phase transition can happen for Bose-Einsteincondensed ultra-cold atoms, that is exempt from thermalfluctuations. For low pump power at zero temperature,the homogeneous phase is stabilized by the kinetic energyand the atom-atom collisions, a sharp transition thresh-old is thus expected [5, 6]. In both examples the self-organization is a non-equilibrium phase transition withthe distinct phases being stationary states of the driven-damped dynamics.In this paper we show that the Hamiltonian under-lying the spatial self-organization is analogous to theDicke-type Hamiltonian [7] and the transition to the self-organized phase can thus be identified with the superra-diant quantum phase transition [8]. Hence, the quantummotion of ultracold atoms in a cavity effectively realizesthe Dicke model and may lead to the first experimentalstudies on this paradigmatic system. The accessibility ofsuch a Hamiltonian dynamics is limited by the couplingto the environment. We explore how quantum noise infil-trates and depletes the ground state [9], imposing therebya condition on the time duration allowed for the adiabaticvariation of the macroscopically populated ground stateby means of tuning an external parameter.We consider a zero-temperature Bose-Einstein conden-sate of a number of N atoms of mass m which is in-side a high- Q optical cavity with a single quasi-resonantmode of frequency ω C . Such a system has been re-alized and manipulated in several recent experiments [10, 11, 12, 13, 14, 15]. The atoms are coherently drivenfrom the side by a pump laser field. The pump laserfrequency ω is detuned far below the atomic resonancefrequency ω A , so that the atom-pump (red) detuning∆ A = ω − ω A far exceeds the rate of spontaneous emis-sion. One can then adiabatically eliminate the excitedatomic level and the atom acts merely as a phase-shifteron the field. The dispersive atom-field interaction hasa strength U = g / ∆ A , where g is the single-photonRabi frequency at the antinode of the cavity mode. Wedescribe the condensate dynamics in one dimension alongthe cavity axis x , where the cavity mode function iscos kx . The motion perpendicular to the cavity axis re-quires a trivial generalization of the theory, and with astanding-wave side pump the self-organization effect oc-curs quite similarly in two-, and three dimensions [2].The many-particle Hamilton operator in a frame ro-tating at the pump frequency ω and with (cid:126) = 1 reads H = − ∆ C a † a + (cid:90) L Ψ † ( x ) (cid:20) − (cid:126) m d dx + U a † a cos ( kx ) + iη t cos kx ( a † − a ) (cid:21) Ψ( x ) dx, (1)where Ψ( x ) and a are the annihilation operators of theatom field and the cavity mode, respectively. The cavitylength is L , the detuning ∆ C = ω − ω C is the effectivephoton energy in the cavity. Atom-atom s-wave scatter-ing is neglected. Besides the dispersive interaction term U cos kx , there is another sinusoidal atom-photon cou-pling term describing an effective cavity-pump with theamplitude η t = Ω g / ∆ A , where Ω is the Rabi frequencyof the coupling to the transverse driving field.Self-organization is a transition from the homogeneousto a λ -periodic distribution. The minimum Hilbert-spacefor the atom field required to describe this transition isspanned by two Fourier-modes,Ψ( x ) = 1 √ L c + (cid:114) L c cos kx , (2)where c and c are bosonic annihilation operators. Inthe low excitation regime these two modes can be as- a r X i v : . [ qu a n t - ph ] D ec sumed to form a closed subspace, so c † c + c † c = N is a constant of motion giving the number of parti-cles. On invoking the Schwinger-representation in termsof the spin ˆ S with components ˆ S x = ( c † c + c † c ),ˆ S y = i ( c † c − c † c ) and the population difference ˆ S z = ( c † c − c † c ), the Hamiltonian Eq. (1) confined into thetwo-mode subspace reads H = − δ C a † a + ω R ˆ S z + iy ( a † − a ) ˆ S x / √ N + ua † a (cid:16) + ˆ S z /N (cid:17) , (3)where δ C = ∆ C − u , ω R = (cid:126) k / m , u = N U /
4, and y = √ N η t . In the first line one can recognize the fa-mous Dicke-model Hamiltonian with a coupling constant y tunable via the transverse driving amplitude η t . Thelast term is inherent to the BEC-cavity system, how-ever, it does not essentially change the conclusions tobe drawn here as long as | u | < | δ c | . This condition hasto be anyway fulfilled so that the neglect of the next ex-cited Fourier-mode c cos 2 kx be justified in Eq. (2). Inthe following, we will restrict the discussion to the pa-rameter regime which is needed for the self-organization[5]. That is, δ C < N → ∞ and V → ∞ , while the atom density ρ ∝ N/V is kept con-stant. The coupling constants u and y have been in-troduced such that they are proportional to the atomdensity, u ∝ N/V and y ∝ (cid:112) N/V (there is a fillingfactor coefficient), and thus remain constant in the ther-modynamic limit. The ground state can be determinedas in Ref. [8]. Let’s use the Holstein-Primakoff repre-sentation in which the spin- N/ b such that ˆ S − = √ N − b † b b , ˆ S + = b † √ N − b † b , and ˆ S z = b † b − N/
2. TheHamiltonian transforms into H = − δ C a † a + ω R b † b + ua † ab † b/N + i y ( a † − a ) (cid:32) b † (cid:114) − b † bN + (cid:114) − b † bN b (cid:33) . (4)Next, let’s employ the similarity transformationˆ D − ( β ) ˆ D − ( α ) H ˆ D ( α ) ˆ D ( β ), with the displacement op-erators, ˆ D ( α ) = exp { αa † − α ∗ a } and ˆ D ( β ) = exp { βb † − β ∗ b } , which does not change the spectrum of the Hamil-tonian. Formally, the transformation amounts to replac-ing b → b + β , a → a + α , and analogously for the hermi-tian adjoint operators in (4). The resulting Hamiltonianis then expanded up to second-order in the boson oper-ators. Note that the expansion of the nonlinear squareroot term can be performed only approximately, becausethe physically sensible Hilbert space for the operator b is truncated as b † b < N . Therefore the forthcoming resultsare exact up to 1 /N .There is a pair of real β and α such that the linearterms in the Hamiltonian in the displaced phase spacevanish for α = i √ N α and β = √ N β . They obey (cid:0) δ C − uβ (cid:1) α = y β (cid:113) − β , (5a) (cid:0) ω R + uα (cid:1) β = − y α − β (cid:112) − β . (5b)The trivial solution α = β = 0 always satisfies theseequations, which corresponds to the physical state of ahomogeneous condensate and no photon in the cavity.When α (cid:54) = 0 and β (cid:54) = 0, the product of the two equa-tions leads to a second-order algebraic equation, uδ C β − β + δ C ω R + y uω R + y = 0 , (6)where we used that δ C ( uω R + y ) (cid:54) = 0. There is a phys-ically sensible solution in the range 0 < β ≤ y > y crit ≡ √− δ C ω R . Then, β = δ C u (cid:32) − (cid:115) − uδ C y − y y − uδ C y (cid:33) . (7)For u = 0, which amounts to the normal Dicke model,the solution is β = y − y y with the same critical value y crit of pump amplitude. The light shift term does not in-fluence the threshold, because the zero mean fields makethis term vanish below threshold. Note also that this re-sult for y crit corresponds to the one calculated from theinstability of the Gross-Pitaevski equation (GPE) [5], ifthis latter is taken in the g c → κ → H = E + M a † a + M x + M y b † b + M x − M y (cid:16) b † + b (cid:17) + i M c ( a † − a )( b † + b ) , (8)where M = − δ C + uβ , (9a) M x = ω R + u α − yα β − β (1 − β ) / , (9b) M y = ω R + u α − yα β − β ) / , (9c) M c = 2 uα β + y − β (1 − β ) / . (9d) y/y c h a † a ih b † b i FIG. 1: (Color online) Photon (dashed red lines) and motion-ally excited atom (solid blue lines) numbers in the groundstate. Thick lines represent the contributions form the meanfields ( α and β , these are the photon and atom excitationnumbers divided by N , respectively), which can be the or-der parameters of the phase transition. Thin lines representthe incoherent excitations due to the squeezing, given by thequantum averages (cid:104) a † a (cid:105) and (cid:104) b † b (cid:105) taken in the ground state.Parameters: δ C = − ω R , u = − . ω R . The ground state is the vacuum state of the normal modeoscillators which have the eigenfrequencies ω ± = M + M x M y ± (cid:113) ( M − M x M y ) + M M y M c . (10)Below threshold, the energy gap to the first excited statevanishes as (cid:112) − y/y crit on approaching the critical point(the exponent is thus 1/2). The ground state contains ex-cited Fock states of the uncoupled photon a and atomic b modes because of the squeezing (with the coefficient M x − M y ) and two-mode squeezing (with the coefficient M c ) terms. Below threshold M x − M y = 0, thus theground state is simply the two-mode squeezed vacuum ofthe a and b modes, which is an entangled state [17, 18].The quantum average of the photon and atom motion ex-citations in the ground state can be seen in Fig. 1 togetherwith the mean field populations. The squeezing leads toa singular state at the critical point which amounts to adivergence of the incoherent excitations (cid:104) a † a (cid:105) and (cid:104) b † b (cid:105) .Therefore, the detection of the continuous transition ofthe mean field amplitudes α and β requires, e.g., ho-modyne light measurement or the observation of the in-terference between the homogeneous and the sinusoidalcomponents of the atom field.The quantum phase transition associated with theground state of the Dicke-type Hamiltonian must be in-fluenced by the cavity loss. The coupling to the en-vironment amounts to a quantum measurement of thecoupled BEC-cavity system [9, 19], and has a back ac-tion on its state. Therefore, even at zero temperature,the ground state is being depleted, which process canbe modeled as a diffusion. We calculate the rate of dif-fusion out of the ground state in the following. For a compact notation the variables are arranged in a vectorˆ R ≡ [ˆ a, ˆ a † , ˆ b, ˆ b † ]. The Heisenberg equations of motionoriginating from the quadratic Hamiltonian (8) are lin-ear and are driven by quantum noise terms associatedwith the photon field decay, ∂∂t ˆ R = M ˆ R + ˆ ξ , where thematrix M contains the coupling between the bosonic cre-ation and annihilation operators, and the noise source isˆ ξ = [ ˆ ξ, ˆ ξ † , , (cid:104) ξ ( t ) ξ † ( t (cid:48) ) (cid:105) = 2 κδ ( t − t (cid:48) ), where 2 κ is the pho-ton loss rate. We neglect the dissipative − κa and − κa † terms, because we are interested in the transient dynam-ics and not in the stationary regime of the system. Ini-tially, the dominant effect in irreversibly escaping fromthe ground state can be attributed to the infiltration ofquantum noise (a diffusion process).The left and right eigenvectors l ( k ) and r ( k ) , respec-tively, of M can be used to expand the fluctuation vectorˆ R in terms of normal modes: ˆ R ≡ (cid:80) k ˆ ρ k r ( k ) . By useof the orthogonality of the left and right eigenvectors,( l ( k ) , r ( l ) ) = δ k,l , where ( a, b ) is the scalar product, thenormal mode amplitudes are obtained as ˆ ρ k = ( l ( k ) , ˆ R ).They evolve independently asˆ ρ k ( t ) = e − iω k t ˆ ρ k (0) + (cid:90) t e − iω k ( t − t (cid:48) ) ˆ Q k ( t (cid:48) )d t (cid:48) , (11)where the projected noise is ˆ Q k ≡ ( l ( k ) , ˆ ξ ). In the presentHamiltonian problem, the normal modes form hermi-tian adjoint pairs ρ + , ρ † + with eigenfrequencies ± ω + , and ρ − , ρ †− with frequencies ± ω − from Eq. (10), respectively,where each pair corresponds to one of the normal modeoscillator of the quadratic Hamiltonian in (8). Secondorder correlations evolve as (cid:104) ˆ ρ k ( t )ˆ ρ l ( t ) (cid:105) = (cid:104) ˆ ρ k (0)ˆ ρ l (0) (cid:105) e − i ( ω k + ω l ) t + 2 κ − e − i ( ω k + ω l ) t i ( ω k + ω l ) l ( k )1 ∗ l ( l )2 ∗ . (12)The first term represents the initial condition. The dif-fusion is due to the second term in which the linear timedependence can be written as being proportional to thesin( x ) /x function, where x = ( ω k + ω l ) δt/ (cid:104) ρ † + ρ + + ρ †− ρ − (cid:105) . Using (12) forits time evolution, the first term vanishes in the groundstate, and, in the second term, ω k + ω l = 0 for both ρ † + ρ + and ρ †− ρ − terms. Thus the time evolution leadsexactly to a linear increase of the excited population,the corresponding diffusion rate is plotted in Fig. 2 withdashed line. The singularity at the critical point reflectsthat the excitation energy of one of the normal modestends to zero.Let us calculate the diffusion in terms of measurablequantities, such as the number of incoherent photons andmotionally excited atoms, δN = (cid:10) a † a + b † b (cid:11) . The inco- y/y c D i ff u s i o n [ un i t s o f κ ] FIG. 2: Diffusion out from the ground state. The rate ofincrease of normal mode excitations (dashed line) and that ofphotons and motionally excited atoms (solid line) with coarsegraining | δ C | − (cid:28) δt (cid:28) ω − R . The almost overlapping dashed-dotted line is derived from an adiabatic elimination methodand is given by the analytic result of Eq. (15). Parameters: δ C = − ω R , u = − . ω R . herent population evolves as δN ( t ) = (cid:88) k,l (cid:104) ˆ ρ k ( t )ˆ ρ l ( t ) (cid:105) (cid:16) r ( k )2 r ( l )1 + r ( k )4 r ( l )3 (cid:17) . (13)By using Eq. (12) and by approximating the sin( x ) /x function, the diffusion rate becomes δN ( t ) δt ≈ κ (cid:88) k,l l ( k )1 ∗ l ( l )2 ∗ (cid:16) r ( k )2 r ( l )1 + r ( k )4 r ( l )3 (cid:17) Θ (cid:0) δt − − | ω k + ω l | (cid:1) , (14)where Θ is the Heavyside-function. If the “time step” δt is shorter than any of the time periods ω − ± , none ofthe pairs ( k, l ) is cut off by the Heavyside-function in thesum (14). Then, it follows from the completeness relation (cid:80) k r ( k ) i l ( k ) j ∗ = δ ij that the depletion rate is zero. Onsuch a short time the quantum noise is associated withthe photon field amplitudes a and a † , and normal-orderproducts vanish at zero temperature. Diffusion in thepopulations is obtained when a coarse graining of thedynamics over a longer δt is performed. We consideronly the special case | δ C | (cid:29) ω R , when there is a largedifference between the eigenfrequencies ω ± . The timestep can be set such that | δ C | − (cid:28) δt (cid:28) ω − R , and the twopairs ( k, l ) with ω k = ω l = ± ω − ∼ ± ω R also contributeto the double sum in Eq. (14), in addition to the ( k, l )pairs with ω l = − ω k . Then, the departure from theground state appears as a regular diffusion process in themotional excitation Fock space with a finite rate even atthe critical point, which is plotted in Fig. 2 with solidline.As δ C is the far highest frequency in this example,there is an alternative avenue to the depletion rate which relies on the adiabatic elimination of the photon fieldvariables, a and a † , following the method of Ref. [19]. Itleads to an analytical approximation, which is plotted indashed-dotted line in Fig. 2, δN ( t ) δt = κ M c δ C + κ . (15)Below threshold, the diffusion rate is about ω R ( κ/ | δ C | ) ( y/y crit ) . 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