Correlation dynamics of strongly-correlated bosons in time-dependent optical lattices
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] S e p Correlation dynamics of strongly-correlated bosons in time-dependent optical lattices
Karen Rodr´ıguez, Arturo Arg¨uelles and Luis Santos
Institut f¨ur Theoretische Physik, Leibniz Universit¨at Hannover, Appelstr. 2, D-30167, Hannover, Germany (Dated: November 15, 2018)We analyze by means of Matrix-Product-State simulations the correlation dynamics of strongly-correlated superfluid Bose gases in one-dimensional time-dependent optical lattices. We show that,as for the case of abrupt quenches, a quasi-adiabatic modulation of the lattice is characterized bya relatively long transient regime for which quasi-local single-particle correlation functions havealready converged to a new equilibrium, whereas long-range correlations and particularly the quasi-condensate fraction may still present a very significant dynamics well after the end of the latticemodification. We also address the issue of adiabaticity by considering the fidelity between thetime-evolved state and the ground-state of the final lattice.
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I. INTRODUCTION
Strongly-correlated atomic gases in optical latticesconstitute one of the most active fields in the physics ofcold gases. Nowadays spectacular progresses allow for anunprecedented degree of control of these systems, whichpermit the detailed analysis of many-body phenomena ase.g. the realization of the superfluid (SF) to Mott insu-lator (MI) transition in bosonic lattice gases [1], the 3Dfermionic Mott insulator [2, 3] or the Tonks gas [4].The high tunability and long characteristic time scalesof these systems offer an ideal scenario to investigate non-equilibrium dynamics in a way not available in traditionalcondensed matter systems. In particular, lattice hoppingrates may be easily tuned by modulating the intensity ofthe lasers creating the optical lattice, and the interactionsmay be also modified in real time by means of Feshbachresonances and time-dependent magnetic fields. Thesechanges may be produced fast enough to be consideredas a sudden quench. These quenches have attracted agrowing attention in recent years, in particular in whatconcerns the evolution of correlations and possible equili-bration after a quench [5, 6, 7, 8, 9, 10, 11, 12]. Thermal-ization (or actually the absence of it) was recently studiedin a milestone experiment performed in nearly integrableone-dimensional Bose gases by Kinoshita et al [13].On the contrary, if the modification of the system isvery slow, much slowlier than the tunneling rate, one canin principle assume the evolution as adiabatic [14, 15, 16].The issue of adiabaticity is however far from trivial, espe-cially in low dimensional gapless systems, as recently dis-cussed by Polkovnikov and Gritsev [17]. Interestingly, inthe so-called non-adiabatic scenarios, the adiabatic limitcannot be reached no matter how slow the change is in-troduced. Although this result is strictly speaking onlyapplicable in integrable harmonic systems, it was shownin Ref. [17] that this non-adiabatic scenario may be ob-tained by considering an initial non-interacting 1D or 2DBose gas in a lattice under a slow increase of the inter-action strength. The harmonic approximation remainsaccurate as long as U /Jn ≪
1, where (see below) U characterizes the on-site interactions, J is the hopping rate, and n is the filling factor. In this regime the trun-cated Wigner approximation (TWA), plus an additionalfirst-order quantum correction, allows for an accurate de-scription of the correlation dynamics [17].In this paper we study the correlation dynamics of asuperfluid Bose gas in a 1D lattice during and after amodification of the lattice depth. Contrary to the caseof a quench [5, 6, 7, 8, 9, 10, 11, 12], the modification isnot considered as instantaneous, but rather a finite linearramp. In addition, contrary to the (quasi-)harmonic sce-nario discussed by Polkovnikov and Gritsev [17] we arehere particularly interested in the correlation dynamics inthe deeply quantum regime U ≫ J at low filling n < II. MODEL AND METHODOLOGY
In the following we consider bosons in a deep latticeconstrained to the lowest energy band. In this regime,the free energy of the system is described by the BHHˆ H = X i (cid:20) − J ( t ) (cid:16) ˆ a † i ˆ a i +1 + h.c (cid:17) + U n i (ˆ n i − − µ ˆ n i (cid:21) (1)where ˆ a i (ˆ a † i ) is the annihilation (creation) operator ofa boson at the i -th site, ˆ n i is the corresponding num-ber operator, µ is the chemical potential, J ( t ) is thetime-dependent tunneling amplitude between neighbour-ing sites, and U > t = 0 the system isin the ground-state for the initial J i = J (0).As mentioned above, the system may be driven out ofequilibrium either by modifying the hopping rate (as weconsider here) or by modifying the interaction rate (e.g.by means of Feshbach resonances). In the following weconsider a time-dependent lattice depth, which leads toa linear-ramp J ( t ) = J i + ( J f − J i ) tt r for an initial timeinterval 0 < t < t r , where J f is the final hopping. For t > t r the system evolves at a constant J = J f . Weconsider sufficiently large post-ramp times such that thequantities of interest enter into a new equilibrium.We consider in our calculations a lattice with L = 60sites with open boundary conditions, which is sufficientlylarge to minimize finite-size effects at the lattice center,where we evaluate the correlations discussed below. Inour time-evolution simulations, we work in the canonicalensemble with two different total number of particles, N = 20 which leads to an average lattice-site filling ¯ n ∼ . N = 50 which leads to¯ n ∼ .
8, i.e. above HF. In all simulations discussed belowwe considered J i = 0 . U and J f = 0 . U . Thesevalues were chosen relatively close to each other to allowfor the convergence to a new equilibrium discussed belowwithin a numerically available evolution time. In spite ofthat these values are sufficiently different to allow for thestudy of the ramping adiabaticity. Note that the hoppingrates J i and J f are rather low and comparable to thecritical tunneling for on-set of MI (tip of the lowest MIlobe), which for 1D is found at J ≃ . U . In that regimequantum fluctuations are highly relevant, but due to thelow filling ¯ n considered, the system remains within the(highly-correlated) SF regime.In our calculations we first obtained the ground-stateat J = J i by means of an MPS-algorithm using a similarapproach as that of Refs. [18, 19]. In this DMRG-liketechnique, the many-body state is approximated by aMPS ansatz of the form | Ψ MP S i = d X s ,...,s N =1 T r ( A s . . . A s N N ) | s , . . . , s N i , (2)where the A ’s are matrices of a given dimension D and d is associated to the on-site dimension, which in our G ∆ ( t ) / G ∆ g s t/J (a) ∆ =0 ∆ =1 ∆ =4 ∆ =8 0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 0 5 10 15 20 25 30 35 G ∆ ( t ) / G ∆ g s t/J (b) ∆ =0 ∆ =1 ∆ =4 ∆ =8 FIG. 1: Time-evolution of several correlation functions con-sidering a ramp-time of (a) t r = 0 . J f and (b) t r = 10 . J f .The used density population is ¯ n = 0 . case represents the maximal number of atoms per siteconsidered, (for our parameters it can be safely assumedas d = 2). The MPS-algorithm performs in an efficientway a variational method using the matrix elements A s i i as variational parameters, and basically reduces to sub-sequent DMRG sweeps over the lattice until achievingenergy convergence [20].Once the ground state is obtained with J = J i weevolve the problem in real time for t > D in each step to keep thestate as faithful to the real one as possible. III. CORRELATION DYNAMICS ANDQUASI-CONDENSATE FRACTION
In the following we are particularly interested onthe dynamics of the single particle correlation functions G (∆; t ) ≡ h ˆ a † j ˆ a j +∆ i . In our numerical calculations weevaluate G (∆ , t ) at the lattice center j = 0. In equilib-rium, the 1D SF phase is characterized by a polynomialdecrease G (∆) ∼ ∆ − η with the relative distance ∆ [23](contrary to the gapped MI phase where an exponential G ( ∆ ) / G ( ) ∆ PSfrag replacements | ϕ igs i| ϕ fgs i| ϕ evol i FIG. 2: Spatial correlation for the initial state | ϕ igs i , theground-state | ϕ fgs i of the final configuration and the evolvedstate | ϕ evol i after t/J = 35 for a ramp time of t r = 10 . J f . decay is expected).We have analyzed the evolution of G (∆ , t ) /G (∆ , t = 0)for different values of ∆ and different ramping times t r .Fig. 1(a) exemplifies the case of an abrupt ramp (basi-cally an instantaneous quench) with t r = 0 . J f , whereasFig. 1(b) depicts typical results observed for a mild ramp,in this case t r = 10 . J f . Both cases are calculated at afilling factor ¯ n = 0 .
3. Note that G (∆ , t ) shows in bothcases a significant dynamics following the linear ramp(due to number conservation the density, i.e. ∆ = 0, isunaffected by the lattice modulation).Both abrupt and slow ramps lead to an evolution of thecorrelations characterized by an initial short-time scale,followed by an eventual convergence into a new equilib-rium at longer times (observe, however, that the cor-relation dynamics following the abrupt quench presentsa short-time modulation which persists well within thequasi-equilibrium region [10]). The short-time evolutionof the correlations continues well after the end ( t = t r )of the ramp (even for such a mild ramp). Notice alsothat short-distance correlations, in particular ∆ = 1 con-verge significantly quicker than correlations at larger dis-tances. As a consequence the lattice bosons experiencea transient regime characterized by a quasi-equilibriumof local or quasi-local observables coexisting with out-of-equilibrium global properties. After this transient regimethe system reaches a final equilibrium, characterized byan equilibrium correlation G (∆) (Fig. 2).The transient regime becomes particularly clear froman analysis of the quasi-condensate fraction. This frac-tion may be defined as the largest eigenvalue λ of thedensity matrix h ˆ a † i ˆ a j i of the system [24], and hence maybe considered a global property of the system, influencedby correlations at any available ∆. Although strict con-densation is prevented in 1D, quasi-condensation, charac-terized by a distinct finite λ , is possible in finite systems.The time evolution of λ ( t ) /λ (0) is depicted in Fig. 3.Note that the quasi-condensate fraction evolves at amuch longer time scale (much larger than the ramp timeeven in the mild-ramping case) than quasi-local correla-tions (it has not yet fully converged in our typical cal- λ ( t ) / N λ s t/J PSfrag replacements ¯ n = 0 . n = 0 . t t λ qe = 1 . λ qe = 1 . FIG. 3: The evolution of the quasi-condensate fraction fortwo different particle densities ¯ n = 0 . n = 0 . t r = 10 . J f ). The horizontal lines denote where thequasi-equilibrium ( λ qe ) values tend in each case. The dashedlines fit the initial curve stretch and the intersection points aredefined as a qualitative estimation of the λ variation time, t (¯ n = 0 .
3) = 17 . t (¯ n = 0 .
8) = 12 .
9. The plot is inunits of λ gs where λ gs is the quasi-condensate fraction ofthe initial ground-state. culations). The filling factor ¯ n is of course importantto determine the time scale of variation of the differ-ent correlation functions. The figure shows for a ramp t r = 10 J f the evolution of the quasi-condensate fractionfor the case of an average filling factor ¯ n = 0 . n = 0 .
3. At higher densities thequasi-equilibrium is reached faster for the same ramping(a similar behaviour is observed for G (∆)). As in Fig. 3we may define a typical time scale for the variation of λ ,which is t (¯ n = 0 .
3) = 17 . t (¯ n = 0 .
8) = 12 . IV. FIDELITY AND FINAL ENERGY
A good tool for the analysis of the adiabaticity of theramping is provided by the fidelity F = |h ϕ fgs | ϕ ( t ) i| ,i.e. the Hilbert-space distance between the time-evolvedstate | ϕ ( t ) i and the expected ground-state | ϕ fgs i calcu-lated with the final J = J f . The fidelity F is an inter-esting figure of merit for the adiabaticity of the ramping,since contrary to other figures (as the correlation func-tions discussed above) it just evolves while the rampingis on, since after the ramping the eigenstates of the finalHamiltonian are of course stationary, and consequently F remains constant. Hence, although other quantitiesrequire a rather long waiting time for comparing the fi-nal state and the time-evolved one, F provides an answerat the relatively short-time scale t r (see Fig. 4). As ex-pected the abrupt ramping t r /J f = 0 . | ϕ fis i into | ϕ fgs i (note that the overlapping is alreadyrather large, 88%, due to the relative close values of J i and J f ). As expected the milder ramping approachesfurther to | ϕ fgs i , however, the fidelity is still 5% off from | ϕ igs i . Interestingly, this indicates that even very largeramping times significantly larger than the hopping time,and for a relatively small variation of ∆ J = 0 . U , do PSfrag replacements F i d e li t y ( F ) t r /J f = 0 . t r /J f = 10 .t r /J f = 20 .t r /J f = 30 . FIG. 4: Fidelity of the time-evolved state to the ground stateat the end of the lattice modulation for t r /J f = 0 . t r /J f =10 . t r /J f = 20 . t r /J f = 30 . not guarantee a perfect transfer into the ground state ofthe final configuration. The analysis of F for even milderramps shows (see Fig. 4) that milder ramps lead indeedto more adiabatic transfers (contrary to what may beexpected in the harmonic regime [17]).Hence, although, as mentioned above, the correlations G (∆) approach at longer times to a new equilibrium,this new equilibrium is not that given by the expectedground state, but by a new distribution with a higher en-ergy. The analysis of the final energy after the rampingdoes not provide however an equally strict adiabaticityanalysis as that of the fidelity, especially for situationsas those discussed here, in which J i and J f possess rel-atively close values. In our case the difference between the energy of | ϕ igs i (or equivalently that of the systemafter the abrupt ramping) and that of | ϕ fgs i is less than1%, whereas the time-evolution with the mild rampingprovides a final energy less than 0 .
1% off that of | ϕ fgs i . V. CONCLUSIONS
We have analyzed by means of MPS techniques thedynamics of correlation functions and quasi-condensatefraction of ultra cold lattice bosons in the deeply corre-lated superfluid regime during and after a finite linearramp modulation of the hopping rate. We have shownthat the evolution is characterized by a transient non-equilibrium state in which quasi-local correlation func-tions have already converged into a new equilibriumwhereas long-range correlations and the quasi-condensatefraction present still a significant time dependence. Ad-ditionally, we have analyzed the formation (at a longertime scale) of a new equilibrium from an initial gas at zerotemperature. By considering the fidelity with respect tothe ground state of the final configuration we have shownthat even rather mild ramps do not fully guarantee a per-fect loading of the new ground state. We have howevershown that contrary to the harmonic regime [17] progres-sively milder ramps lead to a more adiabatic transfers.
Acknowledgments
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