Nonclassical dynamics of Bose condensates in an optical lattice in the superfluid regime
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J un Nonclassical dynamics of Bose condensates in an optical lattice in the superfluidregime
Roberto Franzosi ∗ Dipartimento di fisica Universit`a di Firenze and CNR-INFM, Via Sansone 1, I-50019 Sesto Fiorentino, Italy. (Dated: November 3, 2018)A condensate in an optical lattice, prepared in the ground state of the superfluid regime, isstimulated first by suddenly increasing the optical lattice amplitude and then, after a waiting time,by abruptly decreasing this amplitude to its initial value. Thus the system is first taken to theMott regime and then back to the initial superfluid regime. We show that, as a consequence of thisnonadiabatic process, the system falls into a configuration far from equilibrium whose superfluidorder parameter is described in terms of a particular superposition of Glauber coherent states thatwe derive. We also show that the classical equations of motion describing the time evolution of thissystem are inequivalent to the standard discrete nonlinear Schr¨odinger equations. By numericallyintegrating such equations with several initial conditions, we show that the system loses coherence,becoming insulating.
PACS numbers: 03.75.Kk, 05.30.Jp, 03.75.-b, 03.65.SqKeywords: Superfluidity,Bose-Einstein condensation, Mott-insulator, wave matter.
1. INTRODUCTION
Nowadays Bose-Einstein condensates (BECs) repre-sent one of more powerful and versatile testing groundsfor low-energy modern physics in which experimentaltests on quantum computation [1], many-body physics[2], superfluidity [3], the Josephson junction effect [6],atom optics [7], and quantum phase transitions [8] can beperformed. Condensates can be put into interaction witheach other or manipulated by means of optical lattices(OLs), which are periodic trapping potentials generatedby standing laser waves. By raising the amplitude of theOL, a condensate loaded therein is fragmented into anarray of interacting condensates. By adjusting the laseramplitudes, the system is taken into different regimes.The superfluid regime is obtained with weak optical po-tentials (OPs), where the kinetic energy dominates overthe interacting one, and the atoms hop from one well toanother. The opposite -quantum- regime, is generatedby strong OPs that suppress the tunneling of the atomsbetween the wells.In this paper, we consider a one-dimensional gas. Thisis prepared by use of a transverse harmonic confiningpotential that tightly confines the atoms so that theirmotion, in the transverse direction, is limited to the zeropoint. Along the longitudinal direction, a further har-monic potential weakly confines the atoms, and an OP isswitched on. For large enough laser amplitudes, the con-densate splits into components tightly confined at theminima of the effective potential. In what follows, weimagine abruptly adjusting the amplitude of the longi-tudinal laser at two instants, in order to first take thesystem from the superfluid regime to the quantum oneand then to take it back to the superfluid regime. In this ∗ Electronic address: franzosi@fi.infn.it way the system is taken to a nonequilibrium state (NES)that is described in terms of a particular superpositionof Glauber coherent states (CSs). The latter combina-tion is derived in the following. Although the system istaken back to a weak OP regime, as in the superfluidcase, this NES follows a nonclassical dynamics. In fact,we show that the equations of motion for the system’sorder parameter are inequivalent to the discrete nonlin-ear Schr¨odinger equation. The present study is in somesense complementary to former work [9, 10], it continuesthe work in [11], and it is motived by recent experimentsas in Refs. [8, 12, 13], where BECs are manipulated inan OL. Furthemore, the nonadiabatic procedure we de-scribe, can be straightforwardly realized in a real exper-iment, and then the dynamics of the NES, a superposi-tion of the canonical CSs can be directly observed, e.g.,by displacing the condensates with respect to the har-monic trap center, as in the experiment of Ref. [6], andby observing the oscillations of the center of the atomicdensity distribution.
2. THE MODEL
The quantum dynamics of an array of condensatesin a deep enough OL, can be described by the Bose-Hubbard (BH) model [14]. Let V H ( r ) = Σ j =1 m Ω j r j / V L ( r ) = ~ ω sin ( kr ) / (4 E r ) be the one-dimensional OP, where k is the laser wavelength, E r = ~ k / (2 m ) is the recoilenergy, and ω is the angular frequency of the parabolicapproximation of V L at each minimum. Then, the BHHamiltonian, written in terms of the boson operators a j and a + j that, respectively, annihilate and create atoms atthe j site of the lattice, reads H = X i [ U n i ( n i − λ i n i ] − T X h ij i (cid:0) a + i a j + h.c. (cid:1) , (1)where the operators n i = a + i a i count the number ofbosons at the i site, and the boson operators satisfy thestandard commutation relations [ a i , a + j ] = δ i,j . Theindices i, j ∈ Z label the local minima r i , r j , where r ℓ = πℓ/k , of V ( r ) = V L ( r ) + Σ ℓ =2 m Ω ℓ r ℓ / N = Σ j n j is a conserved quan-tity. Within the Gaussian approximation, the Hamilto-nian parameters have the following expressions in termsof the trapping potentials and of the optical one (see [11]). U := a s Ω p m ~ Ω / (2 π ) is the strength of the on-site re-pulsion, in which we have set Ω = √ ω Ω Ω . The latterapproximation is suitable in the limit of tight confinementof BECs in every well. In fact, in this case the spatialwidth of each trapped condensate does not depend, inthe first approximation, on the number of atoms in thewell, and the condensate wave function in every potentialminimum is well approximated by a Gaussian [15]. Thesite external potential is λ j := ǫ ( ω ) + j π ~ Ω / (4 E r ),where ǫ ( ω ) = ~ ( ω + Ω + Ω ) /
2, and T ( ω ) = ~ ω E r (cid:20) π − E r ~ ω − e − Er ~ ω (cid:21) e − π ~ ω Er (2)is the tunneling amplitude between neighboring sites. Itis worth stressing that the Gaussian approximation is notessential for the use of the BH model, but it is useful inorder to derive an analytic estimation of the Hamiltonianparameters.When the OL amplitude is weak enough so that T /U ≫
1, the ground-state configuration of Hamilto-nian (1) admits a factorization into a product of sitestates that catches the superfluid nature of the sys-tem. The system’s order-parameter dynamics near theground state can be studied by a time-dependent vari-ational principle (TDVP) [17]. Following the TDVPmethod, we describe the system in terms of the trial state | Ψ i = exp( iS/ ~ )Π i | z i i , which contains a product of siteGlauber CSs. In fact, in Ref. [16] it is shown that, inthe limit U = 0, N, M → ∞ at fixed density
N/M , theground state of Hamiltonian (1) with M wells, is indis-tinguishable by a product of local coherent states. Thus,in the strong hopping regime, such a state should still bea good approximation. Here | z i i := e − | z i | ∞ X n =0 z ni n ! ( a † i ) n | i (3)have the defining equation a i | z i i = z i | z i i , and the z i are complex numbers. The equations of motion forthe z j dynamical variables are derived by a variationwith respect to z j and z ∗ ℓ of the effective action S = R dt ( i Σ j ˙ z j z ∗ j − H ) that is associated with the classicalHamiltonian H ( Z, Z ∗ ) := h Z | H | Z i , where | Z i = Q j | z j i .Hence, the z j = h Ψ | a j | Ψ i represent the classical canon-ical variables of the effective Hamiltonian H and satisfythe Poisson brackets { z ∗ j , z ℓ } = iδ jℓ / ~ . The classical Hamiltonian is H = X j h U | z j | + λ j | z j | − T (cid:0) z ∗ j z j +1 + c . c . (cid:1)i , (4)where j and j +1 run on the chain sites, and the followingequations of motion result: i ~ ˙ z j = (2 U | z j | + λ j ) z j − T ( z j − + z j +1 ) , (5)together with the complex conjugate equations. The con-straint on the total number of bosons is now satisfied onaverage, the quantity N = P j | z j | being conserved.Equations (5) are the discrete version of the Gross-Pitaevskii equation [15], and the corresponding super-fluid ground state is approximately given by the discreteThomas-Fermi solution z j = r NM ′ − ( λ j − ¯ λ )2 U e iφ , (6)where M ′ = min ( M, q ) and q in turn is the maxi-mum integer such that 2 U N + q (¯ λ − λ q ) ≥
0, and¯ λ = Σ | j |∈ IM ′ λ j /M ′ .
3. DYNAMICS
After having prepared the system in the superfluidground-state configuration (6), it is taken to the Mottregime by abruptly increasing the OP depth and, afteran adjustable time τ , it is carried back to the super-fluid regime by suddenly decreasing the OP depth tothe original value. Our goal is to derive the equationsof motion that describe the dynamics of the system af-ter the latter decreasing of the OP amplitude. We willshow that these equations of motion are more compli-cated than the standard ones recalled in Eq. (5), andinequivalent to these. Furthermore, as we said above,the superfluid ground state is well approximated by aproduct Q i | z i i of CSs. Hence, the condensate in eachsite i is described by a CS | z i i , that is, a semiclassicalstate. On the contrary, we shall show that, after thisstimulation, the system will be described by a productof integrals of site CSs. This means that the semiclassi-cal nature of the site states is partially destroyed duringthe intermediate quantum regime, in spite of the systembeing in a superfluid regime.Following the procedure above, at the time t = 0 theamplitude of the OP is suddenly increased by varying ω from its initial value ω to ω = ω ≫ ω . Since thetunneling amplitude T ( ω ) in Eq. (2) is dominated bythe exponential term exp {− π ~ ω/ E r } , it will result in T ( ω ) /T ( ω ) → ω → ω . Meanwhile, also U and λ j are modified when changing the OP amplitude, but theirdependence on ω is much less dramatic. In fact, we have U ( ω ) = U p ω /ω and λ j ( ω ) = λ j + ~ ( ω − ω ) /
2. Inorder to apply the sudden approximation when the po-tential amplitude is varied, that is, for 0 < t < τ b with τ b ≪ τ , the jump in the potential depth must be fast com-pared with the tunneling time between neighboring wells,but slow enough so that no excitations are induced ineach well, that is 2 π/ω, π/ Ω , π/ Ω ≪ τ b ≪ ~ /T ( ω ).For t > τ b (we will assume τ b = 0 from now on),the system enters into the Mott regime and the classi-cal description of the system dynamics is no longer al-lowed; thus we resort to the quantum one. The appro-priate dynamics is described by the Schr¨odinger equationwith Hamiltonian (1) in which we have to set T = 0, U = U ( ω ) =: ˜ U and λ j = λ j ( ω ) =: ˜ λ j . The quantumtime evolution of the initial state (3) is | ( t ) i = Y i ∈ I M e − | zi | + ∞ X n i =0 [ z i ν i ( t )] n i √ n i ! e − in i u ( t ) | n i i , (7)where ν i ( t ) := exp[ i/ ~ ( ˜ U + ˜ λ i ) t ], u ( t ) = ˜ U t/ ~ , and the z j are those defined in (6). We want to stress that, al-though the factorization of the state vector still holds, theterm exp[ − in i u ( t )] in Eq. (7) breaks the CS structureof the initial state (3), and a j | ( t ) i 6 = α ( t ) | ( t ) i . By directcalculation, one can easily verify the following relations: h ( t ) | a + j a j | ( t ) i = | z j | ,z j ( t ) := h ( t ) | a j | ( t ) i = z j e i ~ ˜ λ j t − i | z j | sin[2 u ( t )] e − | z j | sin [ u ( t )] , h ( t ) | a + j +1 a j | ( t ) i = z ∗ j +1 z j e − | z j | + | z j +1 | ) sin [ u ( t )] e i ~ (˜ λ j − ˜ λ j +1 ) t − i ( | z j | −| z j +1 | ) sin[2 u ( t )] . (8)The system shows the characteristic scenario of phase col-lapse and revivals, observed in many BEC systems [4, 5].For 0 < t < τ , the wells’ populations h ( t ) | a + j a j | ( t ) i donot change, whereas site wave functions z j ( t ) are dy-namically active. The modulus of z j ( t ) is a periodicfunction of t , whose revival time is T m = π ~ / ˜ U . Thephase of the wave function ϕ j := arg[ z j ( t ) / | z j ( t ) | ] =˜ λ j t/ ~ − | z j | sin[2 u ( t )] of the site j , is driven by thethree time scales 2 T m , T ˜ λ j = 2 π ~ / ˜ λ j , and, for the siteswhere | z j | > π , T z j , the solution of the equation | z j | sin [ u ( t + T z j )] = | z j | sin [ u ( t )] + 2 π . Moreover, thesite-dependent external potentials ˜ λ j induce a dephasingbetween the wave functions of near sites: ϕ j − ϕ j +1 = − ( π ~ Ω / (2 j + 1) /E r − ( | z j | − | z j +1 | ) sin [ u ( t )]. Suchdephasing leads the system out of the ground-state con-figuration.After a time τ , the system is taken back to the super-fluid regime, T /U ≫
1, by abruptly decreasing the OPdepth (in a time of order τ b ≈
0) to its initial value. The t = τ initial state | ( τ ) i = Y j ∈ I M E j + ∞ X n j =0 [ z j ] n j p n j ! e − in j u | n j i (9)given by Eq. (7), where E j = exp {−| z j | / } , z j = z j ν j ( τ ), and u = u ( τ ), by the identitylim ǫ → + Z ∞−∞ dx exp[ − ( p + ǫ ) x − inx ] = exp[ − n / (4 p )] p π/p , with p = − i/ (4 u ), can be rewritten as the superpositionof product states of CSs at each site, | ( τ ) i = Y j ∈ I M Z ∞−∞ dx j √ πu e − iπ/ e ix j / (4 u ) | z j e − ix j i , (10)where the states labeled by z ′′ j = z j e − ix j are the normal-ized CSs of Eq. (3) with z j = z ′′ j .The time evolution for t > τ of the mean-field state(10) can be derived within the TDVP picture, in a waysimilar to that previously described, by resorting to a suitable superpositions of Glauber CSs . Thus we intro-duce the trial state | Ψ i = exp( iS/ ~ ) | Z i v , where | Z i v :=Π i | z i , u i v is written in terms of the states | z i , u i v thatin turn are superposition of the standard Glauber onesas | z i , u i v := Z ∞−∞ dxf ( x, u ) | z i e − ix i . (11)Here | z i e − ix i are the standard CSs given in (3), and f ( x, u ) = 12 √ πu e − iπ/ e ix / (4 u ) . (12)A remark about the trial state that we have chosen is inorder. The TDVP method provides the best approxima-tion to the true state within the restricted set of statescaught by the trial one; thus, in general, we do not knowwhat superposition of the canonical CSs gives a class ofstates broad enough to obtain a good approximation ofthe true state. However, in this case, the form of thesuperposition (11) is suggested by the fact that it in-cludes the initial condition (10). Furthermore, in thelimit τ →
0, that is, u → | z i , u i v → | z i i and, inthis way, the canonical CSs (3) and the standard dy-namics (5) are recovered. From now on we drop theexplicit dependence on u in | z i , u i v . The scalar prod-uct between the | z ℓ i v states is defined as v h z j | z ℓ i v := R dxdyf ∗ ( x, u ) f ( y, u ) h z j e − ix | z ℓ e − iy i (here the integra-tions over x and y run from −∞ to ∞ ); thus, from thedefinition (11) and by the normalization of the GlauberCSs h α | α ′ i = exp[ α ∗ α ′ − / | α | + | α ′ | )], the followingidentities can be checked by direct calculation: v h z j | z j i v =1 , v h z j | i ~ ∂ t | z j i v = i ~ z ∗ j ˙ z j − ˙ z ∗ j z j ) , v h z j | a † j a † j a j a j | z j i v = | z j | , v h z j | a † j a j | z j i v = | z j | , v h z j | a † j a j +1 | z j +1 i v = z ∗ j z j +1 e − | z j | + | z j +1 | ) sin [ u ] e i ( | z j | −| z j +1 | ) sin[2 u ] . (13)Following the TDVP procedure, we require the trial stateto be a solution of the weaker form of the Schr¨odingerequation, h Ψ | i ~ ∂ t − H | Ψ i = 0 , where ∂ t is the time derivative and H is the BH Hamil-tonian (1). From the latter equation, we obtain˙ S = i ~ v h Z | ∂ t | Z i v − v h Z | H | Z i v , and, from this and the relations in (13), we get S = Z dt [ i ~ X j
12 ( z ∗ j ˙ z j − ˙ z ∗ j z j ) − H ( Z, Z ∗ )] . (14)The effective classical Hamiltonian H ( Z, Z ∗ ) := v h Z | H | Z i v , can be derived by exploiting the identitiesin (13) and the result is H ( Z, Z ∗ ) = X j [ U | z j | + λ j | z j | ]+ − T X j [ z ∗ j z j +1 e i ( | z j | −| z j +1 | ) sin[2 u ] + c.c. ] e − | z j | + | z j +1 | ) sin [ u ] . (15)The variation of the action (14) with respect to z j and z ∗ j brings us to the classical equations of motion i ~ ˙ z j = (2 U | z j | + λ j ) z j − T × (cid:26)(cid:20) z j +1 + z ∗ j z j +1 − | z j | z j +1 (cid:21) e Φ j,j +1 + (cid:20) z j − + z ∗ j z j − − | z j | z j − (cid:21) e Φ j,j − + − z ∗ j +1 z j e Φ j +1 ,j − z ∗ j − z j e Φ j − ,j (cid:27) (16)for j ∈ I M , where Φ j,k = z ∗ j z k − ( | z j | + | z k | ) /
2, and withthe complex conjugate equations. It is worth noting that,in the present case, the dynamical variables z j , z ∗ ℓ arerelated to the expectation values of the boson operatorsin a more complicated form than in the Glauber CS case.In fact we have v h z j | a j | z j i v , = z j e − iu exp[ | z j | ( e − i u − | z j | still count the number ofatoms in each site j , see the fourth of Eqs. (13).
4. NUMERICAL SIMULATIONS
We have numerically integrated Eqs. (5) and (16) ona lattice of 256 sites, with the initial conditions (13) withthe z j given by (6), and for the values of u correspond-ing to the waiting times τ = π/ , π/ , π/
2. For a BECin an OL with
T / (2 U N ) ≈ .
01, in a harmonic trapwith λ j / (2 U ) = α ( j − and α = 0 .
01, the stan-dard equations of motion (5) imply a superfluid dynam-ics [20]. This is shown in Fig. 1 (a), where we plot the s x c m (a) s x c m (b) FIG. 1: Center of the atomic density distribution x cm = P j | z j | j , as a function of the rescaled time s = t ~ /U (bothdimensionless). (a) shows the regular oscillations of the centerof the atomic density distribution of a BEC in a harmonic trapobtained by numeric integration of Eqs. (5). (b) shows thesame quantity obtained by numerical integration of Eqs. (16)with τ = π/
10 (continuous), π/ π/ regular oscillations of the center of the atomic density dis-tribution along the chain. These oscillations have beentriggered by displacing the condensates respect to theharmonic trap center, as in the experiment of Ref. [6].On the contrary, once τ = 0, the nonstandard equationsof motion (16), with the same initial conditions, entailinsulator (dissipative) dynamics for the system. Thisis clearly shown in Fig. 1 (b), where we plot the mo-tion of the system’s center of atomic density distributionfor τ = π/
10 (continuous line), π/ π/ z j = √ k exp {− ( j − x ) /σ + ip ( j − x ) } , where x = 128 isthe initial center of the Gaussian, p = 3 π/ σ = 10 is the width of the Gaus-sian profile, and k = [ P j exp ( − ( j − x ) / (2 σ ))] − . Thedynamics of these profiles have been numerically and an-alytically studied in Ref. [18], where a dynamical stabil-ity phase diagram for these states was derived. Therein,and also here, the theoretical configuration where theharmonic trapping is off was considered, and the dynam-ics takes place on a finite lattice endowed with reflectingboundary conditions. Thus, we have performed simula-tions in the same conditions and we have chosen the com-bination of the dynamical parameters T / (2 U N ) = 4 . FIG. 2: Breather excitation obtained by numeric integrationof Eqs. (5) with a Gaussian initial profile. The breathertravels along the chain and bounces at the lattice ends. Afterany bounce, the breather is reconstructed.FIG. 3: Excitation obtained by numeric integration of Eqs.(16) with the same initial condition as in Fig. 2. The valueof the waiting time is τ = π/
2. This excitation is completelydestroyed after a few bounces at the lattice ends. Thus, in thefar-from-equilibrium situation, this excitation loses stabilityand it seems to behave like the states of the diffusion regimethat has been identified in [18]. which corresponds to the region of the phase diagram ofRef. [18] where (stable) breather excitations were identi-fied. In Fig. 2 we report the two-dimensional contour plotobtained by numeric integration of (5) with this Gaus-sian initial condition. In Fig. 2 is clear that the breatherstructure is maintained when the traveling excitation isreflected at the lattice boundaries. Figure 3 shows thesame quantity obtained by integration of Eqs. (16) withthe same initial condition as in Fig. 2, and with τ = 0 .
5. FINAL REMARKS
In the present paper we have studied how the dynam-ics of a superfluid is affected by briefly bringing the sys-tem into the insulating regime. We have shown that thesystem is taken to an excited state, described by a super-position of product states of Glauber coherent states ateach site, which we have derived. The classical equationsof motion ruling its dynamics have been derived. Fur-thermore, we have shown that these classical equationsof motion are inequivalent to the standard discrete non-linear Schr¨odinger equations that describe the dynamicsof an array of BECs [6] in the superfluid regime. By nu-merically integrating such nonstandard equations withseveral initial conditions, we have shown that the systemloses coherence, becoming insulating.The simulations we have performed show that the in-terplay between classical and quantum dynamics leadsto loss of the coherence properties of the system. Infact, the brief period in the insulating regimes changesthe superfluid wave function, which becomes a superpos-tion of product states of the site’s coherent states, thatis, a product of mean-field states. Each mean-field statehas a complicated distribution of phases at each site thatresults from the intermediate quantum dynamics. Thisdistribution of phases leads to a unique tunneling dy-namics described by a complicated hopping term. By aglance at Eqs. (16) one can guess that, as a consequenceof this unusual term, the “effective” tunneling rate be-tween close sites in the case of Eqs. (16) becomes site(population) and time dependent. For this reason thesystem loses coherence.It is also worth emphasizing that the nonadiabaticprocedure we have described in the present paper canstraightforwardly be realized in a real experiment similarto those of Refs. [12, 19]. Therefore, by displacing thecontensates with respect to the harmonic trap, as donein the experiment of Ref. [6], and by observing the oscil-lations of the center of the atomic density distribution,the effects of the nonstandard dynamics can be directlyobserved.
Acknowledgments
I thank the ESF Exchange Grant for support withinthe activity “Quantum Degenerate Dilute Systems.” Ialso thank G.-L. Oppo and V. Penna for useful discus-sions. [1] O. Mandel, et al., Nature,
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