Quantum switching at a mean-field instability of a Bose-Einstein condensate in an optical lattice

a r X i v :0810.0307v 2 [c o n d -m a t .o t h e r ] 10 J a n 2009

Quantum switching at a mean-?eld instability of a Bose-Einstein condensate in an

optical lattice

V.S.Shchesnovich 1and V.V.Konotop 2

1

Centro de Ci?e ncias Naturais e Humanas,Universidade Federal do ABC,Santo Andr′e ,SP,09210-170Brazil,

2

Centro de F′?sica Te′o rica e Computacional,Universidade de Lisboa,Complexo Interdisciplinar,Avenida Professor Gama Pinto 2,

Lisboa 1649-003,Portugal;Departamento de F′?sica,Faculdade de Ci?e ncias,Universidade de Lisboa,Campo Grande,Ed.C8,Piso 6,Lisboa 1749-016,Portugal

It is shown that bifurcations of the mean-?eld dynamics of a Bose-Einstein condensate can be related with the quantum phase transitions of the original many-body system.As an example we explore the intra-band tunneling in the two-dimensional optical lattice.Such a system allows for easy control by the lattice depth as well as for macroscopic visualization of the phase transition.The system manifests switching between two selftrapping states or from a selftrapping state to a superposition of the macroscopically populated selftrapping states with the step-like variation of the control parameter about the bifurcation point.We have also observed the magni?cation of the microscopic di?erence between the even and odd number of atoms to a macroscopically distinguishable dynamics of the system.

PACS numbers:03.75.Lm;03.75.Nt

Introduction.-Since the very beginning of the quan-tum mechanics its relation to the classical dynamics con-stitutes one of the central questions of the theory.De-pendence of the energy levels distribution on the type of dynamics of the corresponding classical system [1],in general,and the quantum system response to variation of the bifurcation parameters controlling the qualitative changes of the classical behavior [2]are among the ma-jor issues [3].One of the main tools in studies of the quantum-classical correspondence is the WKB approxi-mation,where,loosely speaking,the Planck constant is regarded as a small parameter.

On the other hand,for a N -boson system the limit N →∞at a constant density,leading to the mean-?eld approximation,can also be understood as a semiclassi-cal limit.This latter approach has received a great deal of attention during the last decade [4],due its high rele-vance to the theory of Bose-Einstein condensates (BECs),many properties of whose dynamics are remarkably well described within the framework of the mean-?eld mod-els [5].More recently,it was shown [6,7]that the mean-?eld description of a few-mode N -boson system can be recast in a form similar to the WKB approximation for a discrete Schr¨o dinger equation [8],emergent for the coef-?cients of the wavefunction expansion in the associated Fock space,where 1/N plays the role similar to that of the Planck constant in the conventional WKB approxi-mation.

The mean-?eld equations of a system of interacting bosons are nonlinear,hence,they naturally manifest many common features of the nonlinear dynamics,in-cluding bifurcations of the stationary solutions caused by variation of the system parameters.One of the well stud-ied examples is a boson-Josephson junction [9],which can show either equally populated (symmetric)or strongly

asymmetric states,characterized by population of only one of the sites (the well known phenomenon of selftrap-ping [10]).Now,exploring parallels between the semi-classical approach and the mean-?eld approximation one can pose the natural question:what changes occur in a manybody system when a control parameter crosses an instability (e.g.bifurcation)point of the limiting mean-?eld system?

In the present Letter we give a partial answer showing that one of the possible scenarios is the quantum phase transition of the second type,associated with the switch-ing of the wave-function in the Fock space between the “coherent”and “Bogoliubov”states possessing distinct features.Considering a ?exible (time-dependent)control parameter,we have also found a strong sensitivity of the system to the parity of the total number of atoms N ,showing parity-dependent structure of the energy levels and the macroscopically di?erent dynamics for di?erent parity of N .Observation of the discussed phenomena is feasible in the experimental setting available nowadays.Quantum and mean-?eld models.-We consider the nonlinearity-induced intra-band tunneling of BEC be-tween the two high-symmetry X -points of a two-dimensional square optical lattice (OL).The process is described by the two mode boson Hamiltonian (see [6]for the details)?H

=1

2 and the atomic densityρ.The link with the semiclas-

sical limit is evident for the Hamiltonian in the form

(1):the Schr¨o dinger equation written in the Fock ba-

sis,|k,N?k =(b?1)k(b?2)N?k

k!(N?k)!

|0 ,depends only on the

relative populations k/N and(N?k)/N,while h serves as an e?ective“Planck constant”.

Hamiltonian(1)represents a nonlinear version of the well-known boson-Josephson model(see,e.g.[9,11]),

where unlike in the previously studied models the states are coupled by the exchange of pairs of atoms.This is a fairly common situation for systems with four-wave-

mixing,provided by the two-body interactions involving four bosons.The exchange of the bosons by pairs results in the coupling of the states with the same parity of the

population and is re?ected in the double degeneracy of all (N+1)/2energy levels for odd N,due to the symmetry

relation 2k,N?2k|Ψ1 = N?2k,2k|Ψ2 .For even N the energy levels show quasi degeneracy(see below). The mean-?eld limit of the system(1)can be formally

obtained by replacing the boson operators b j in(1)by the c-numbers b1→√N(1?x)e?iφ/4, what gives the classical Hamiltonian[6]

H=x(1?x)[2Λ?1+Λcosφ]+1

2,φ=0)and

minimum P2=(x=1

3.ForΛ<Λc it looses its

stability,and another set of stationary points x=1(S1) and x=0(S2)appears,which is a fairly general situa-tion in nonlinear boson models.The appearing solutions describe the symmetry breaking leading to selftrapping. Energy levels near the critical point.-To describe the spectrum of the Hamiltonian(1)in the vicinity of the critical valueΛc we rewrite?H in terms of the operators a1,2=(b1?ib2)/√

Λc ?V+E(Λ),E(Λ)=Λ+12N(3)

where?H0=2Λ

4N2 a?1a2+a?2a1 2. At the critical point the energy spectrum is determined by?H0:E m=2Λc

2+Λc

2.

The conditions for?V to be treated as a perturbation de-

pend on m as is seen from the diagonal matrix elements:

E m,j|?V|E m,j =12N 1?m

Spectrum in the limit N→∞.Coherent states and

selftrapping states.-ForΛ?1c?Λ?1?N?2the quantum

states corresponding to P1can be obtained by quantizing

the local classical Hamiltonian(2),i.e.by expanding it

with respect to x?1/2andφand settingφ=?ih?

1/N in E max ).The “wave function”ψ(x )=√

N k,N ?k |ψ satis?es Λh 2?x 2+(3Λ?1)

1

2 2 ψ=Eψ.(6)Eq.(6)is the negative mass quantum oscillator prob-lem with the frequency ω2

=8 3?

14

Λ

4

n +

1

2h x ?

1

2

+

(2Λ?1)

2N

[(b ?1)2+b 21].

(7)

Hamiltonian (7)can be diagonalized by the Bogoliubov

transformation c =cosh(θ)b 1?sinh(θ)b 1?,where θ=θ(Λ)>0is determined from tanh(2θ)=Λ/(1?2Λ).We get

?H

S 2=?Λ

2N

+1

(2k )!/(2k k !)C 0and C (vac )

2k ?1=0,(C 0is a nor-malization constant).

The validity condition of the approximation (7),given by ?n ,?n ?N ,can be rewritten in the form

tanh ?2(2θ)?1+N ?2,what is the same as Λ?1?Λ?1c ?N ?2.In this case,the eigenstates of (7)are well-localized in the atom-number Fock space,i.e.the coe?cients C 2k decay fast enough.The condition for this excludes the same small interval as in the perturbation theory,hence the transition between the coherent states and the self-trapping (Bogoliubov)states occurs on the interval of Λof order of N ?2.The convergence of the eigenstates

of ?H

S 2to that of the full Hamiltonian (1)turns out to be remarkably fast as it is shown in Fig.2(a).In

2S 2the associated Hamiltonian is de?ned by replacing the boson operators in Eq.(7)by the c-numbers b 1=

√Nβ.Using |α|2+|β|2=1and ?xing the

irrelevant common phase by setting βreal we get the dy-namical variables αand α?and the classical Hamiltonian in the form H S 2=12(1?|α|2){2(2Λ?1)|α|2+Λ[α2+(α?)2]},from which the stability of the point S 2(α=0)for Λ<Λc follows.

Thus,the passage through the bifurcation point Λc of the mean-?eld model,corresponds to the phase transition in the quantum many-body system on an interval of the control parameter scaling as N ?2and re?ected in the de-formation of the spectrum and dramatic change of the system wave-function in the Fock space.The described change of the system is related to the change of the sym-metry of the atomic distribution,and thus it is the second order phase transition .

In our case this scenario corresponds to loss of stability of the selftrapping solutions S 1and S 2and appearance of the stable stationary point P 1.In the quantum descrip-tion this happens by a set of avoided crossings of the top energy levels (and splitting of the quasi-degenerate en-ergy levels for even N )as the parameter Λsweeps the small interval on the order of N ?2about the critical value Λc (see Fig.1).For lower energy levels the avoided crossings appear along the two straight lines approximat-ing the classical energies of the two involved stationary

points:H (P 1)=14

(for Λ<Λc )and E max =1

4

Dynamics of the phase transition.-Let us see how the quantum phase transition shows up in the system dynam-ics whenΛis time-dependent.The selftrapping states S1 and S2,eigenstates of the Hamiltonian(1),correspond to occupation of just one of the X-points.Such an ini-tial condition can be experimentally created by switching on a moving lattice withΛ<Λc(see e.g.[7]).As the lattice parameterΛ(τ)passes the critical value from be-low,the selftrapping states are replaced by the coherent states with comparable average occupations of the two X-points.

A more intriguing dynamics is observed whenΛ(τ)is a smooth step-like function betweenΛ1andΛ2such that Λ1<Λc<Λ2.In this case,the system dynamics and the emerging states dramatically depend also on parity of the number of atoms.For?xedΛ1,2the system be-havior crucially depends on the time thatΛ(τ)spends aboveΛc.More speci?cally,one can identify two dis-tinct scenarios,which can be described as a switching dynamics between the selftrapping states at the two X-points,Fig.3(a),(b)or dynamic creation of the super-

To estimate the physical time scale,t≡t phτ= md2?⊥