Thermalisation in a Bose-Hubbard dimer with modulated tunneling
TThermalisation in a Bose-Hubbard dimer with modulated tunneling
R. A. Kidd, ∗ A. Safavi-Naini,
2, 1 and J. F. Corney School of Mathematics and Physics, University of Queensland, Brisbane, Queensland 4072, Australia. ARC Centre of Excellence for Engineered Quantum Systems,School of Mathematics and Physics, University of Queensland, Brisbane, QLD 4072, Australia (Dated: June 11, 2020)The periodically modulated Bose-Hubbard dimer model offers an experimentally realizable andhighly tunable platform for observing the scrambling of quantum information and the apparentthermalisation of isolated, interacting quantum many-body systems. In this work we apply thefidelity out-of-time-order correlators in Ref. [1] to establish connections between thermalisation inFloquet system, the exponential growth of FOTOCs as quantified by a non-zero quantum Lyapunovexponent, and the underlying classical transition from regular to chaotic dynamics in the dimer.Moreover, we demonstrate that a non-zero quantum Lyapunov exponent can also be inferred frommeasures quantifying the delocalisation of the Floquet modes of the system such as the Shannonentropy, which approaches unity if the system thermalises to the periodic Gibbs ensemble prediction.
I. INTRODUCTION
Periodically driven quantum systems can be engi-neered to display out-of-equilibrium exotic many-bodyphenomena, such as dynamical localisation and transi-tion from integrable to chaotic dynamics as a result ofvarying the driving parameters [2]. Hence these systemscan facilitate one of the current research fronts in quan-tum many-body dynamics: thermalisation and its con-nection to the dynamics of quantum information, corre-lations, and quantum entanglement.The connection between thermalisation and the dy-namics of quantum information is manifest in quan-tum scrambling , in which two initially commuting opera-tors become rapidly delocalised and non-commutative, asquantified by out-of-time-order correlators (OTOCs) [3–8]. However, OTOCs are not easily accessible in mostquantum simulation platforms.Time-reversal protocols similar to Loschmidt echoeshave been used to measure OTOCs in an Ising spin sys-tem [9], a nuclear magnetic resonance quantum simu-lator [10] and a nuclear spin system in a natural crys-tal [11]. For some systems, time reversal protocolscan be difficult to implement and alternative OTOCmeasurement schemes have been developed, utilising in-terferometric protocols [12], auxiliary degrees of free-dom [13], statistical correlations [14, 15], operator vari-ance measurements [1, 16] and operator eigenbasis mea-surements [17, 18].Here we consider OTOC protocols for periodicallydriven systems, which can be described with a Flo-quet formalism [19]. An isolated, nonintegrable Flo-quet system thermalises to the equivalent of an infinite-temperature state, in the sense that its few-body ob-servables approach the values predicted by the maximal-entropy ‘diagonal ensemble’ [2, 20, 21]. Such thermali-sation in isolated quantum systems is predicted by the ∗ [email protected] eigenstate thermalisation hypothesis (ETH), which per-tains to the statistical behaviour of the eigenvalues, or inthe case of a Floquet system, the quasienergies [20].In this paper we demonstrate the use of the fidelity out-of-time-order correlator (FOTOC) [1] for studying themanifestation of chaos in a system of ultra-cold bosons ina double-well potential with periodically modulated tun-nelling [22]. Our proposal can be immediately realisedexperimentally and highlights the utility of FOTOC dy-namics for characterizing scrambling and thermalisationin Floquet systems. This paper is organized as follows:we introduce the physical system of periodically modu-lated ultracold lattice-bound bosons followed by a briefintroduction of Floquet theory. We then introduce FO-TOCs in the context of quantum scrambling, showingdistinct behaviours that can be linked with the semi-classical regular and chaotic regimes. Finally we com-pare the behaviour of FOTOCs to statistical indicatorsof thermalisation, such as level-spacing parameters andspectral delocalisation of Floquet modes. II. THE BOSE-HUBBARD DIMER
We consider a system of ultra-cold bosons in a double-well potential within the tight-binding approximation.At sufficiently low temperatures, the system is describedby a two-mode model, which upon introducing the lad-der operators ˆ a j , ˆ a † j associated with the occupation inthe j th well, takes the form of a two-site Bose Hubbardmodel.The Hamiltonian governing the system dynamics canbe written ˆ H = 2 U ˆ S z − J ˆ S x , (1)where we have introduced the pseudo-angular- a r X i v : . [ qu a n t - ph ] J un momentum operators,ˆ S x = ˆ a † ˆ a + ˆ a † ˆ a , ˆ S y = ˆ a † ˆ a − ˆ a † ˆ a i , ˆ S z = ˆ a † ˆ a − ˆ a † ˆ a , (2)with ˆ S i satisfying the commutation relations [ ˆ S α , ˆ S β ] = i(cid:15) αβγ ˆ S γ . Here J is the tunneling strength and U is theon-site interaction energy.In the noninteracting limit, this quantum dimer ex-hibits Rabi oscillations akin to a pseudospin-1/2 particle.With increasing interaction strength U/J , the unmodu-lated dimer undergoes a pitchfork bifurcation at the crit-ical interaction strength U c /J = 1 /N [23]. This bifurca-tion corresponds to the onset of self-trapping, which hasbeen experimentally observed [24].In addition to the above phase transition, periodicmodulation of the coupling rate J ( t ) = J + µ cos ( ωt )introduces chaotic behaviour to the dimer model [25].The chaotic behaviour is manifest in the semiclassi-cal dynamics of the system and can be studied at themean-field level using the expectation values ( x, y, z ) ≡ ( (cid:104) ˆ S x (cid:105) , (cid:104) ˆ S y (cid:105) , (cid:104) ˆ S z (cid:105) ). Under particle conservation, the re-sulting phase space is spanned by two parameters (cid:126)r =( z, φ ), where z = (cid:104) ˆ S z (cid:105) and φ = − arg (cid:16) (cid:104) ˆ S x (cid:105) + i (cid:104) ˆ S y (cid:105) (cid:17) .In Fig. 1 we use stroboscopic Poincar´e sections, withtrajectories plotted at intervals of the modulation period,to illustrate the regular-to-chaotic transition as a func-tion of the modulation frequency. We tune between fullyregular (no driving, or effectively ω → ∞ ) to fully chaotic( ω = 0 . λ = lim t →∞ lim δ(cid:126)r → t ln (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) δ(cid:126)r ( t ) δ(cid:126)r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (3)Thus, the regular and chaotic regions in the semiclassi-cal phase space correspond to zero and non-zero values of λ , respectively. The chaotic dynamics of the modulateddimer was explored in Ref. [22], where the Lyapunov ex-ponent was correlated with various probes of chaotic be-haviour in the corresponding quantum dynamics.In the quantum regime, the dimer model is a numeri-cally tractable model which allows us to explore connec-tions between classical chaos in periodically modulatedsystems, the dynamics of quantum information as char-acterized by the growth of OTOCs, and thermalisation.In order to establish these connections we utilise Flo-quet theory to study the dynamics generated by Eq. (1)under periodic modulation of J ( t ). This periodic drivingscheme can be readily implemented in bosonic lattice sys-tems by modulating optical lattice depth [26, 27], and in- FIG. 1. Poincar´e sections of the semiclassical dynamics (blackpoints), overlaid on colour maps of coherent-state Shannonentropy in the basis of Floquet modes, for a range of driv-ing parameters, as indicated, from (a) fully chaotic to (d)fully regular. Blue markers indicate initial conditions forFig. 2. Parameters are NU = − J ( t ) = 1 + 1 . ωt ).The colour map corresponds to the Shannon entropy of the N = 1000 coherent state centred at that point in phase space,scaled by S max = log ( N + 1) to give the range [0 , terparticle interaction strength can be controlled via Fes-bach resonance [28]. In these systems, the configurabletrapping potential and interparticle interactions allowsfor arbitrary time-periodic Floquet driving schemes tobe applied to ultracold atom systems [27].The functional form of the modulation creates a time-periodic Hamiltonian ˆ H ( t ) = ˆ H ( t + T ) with driving pe-riod T = 2 π/ω . Floquet systems are characterised by thecomplex eigenvalues of the time-evolution operator overone period, which is given byˆ U ( T, | φ α (cid:105) = e − i(cid:15) α T | φ α (cid:105) , (4)where we have set (cid:126) = 1. Here | φ α (cid:105) is a Floquet modeand (cid:15) α is the associated quasienergy.In general, the Floquet Hamiltonian, ˆ H F , is defined byˆ U ( T,
0) = e − i ˆ H F T . (5)For sufficiently large driving frequency (larger than thebandwidth of the time-dependent Hamiltonian [2]), wecan find a perturbative approximation to ˆ H F using theFloquet-Magnus expansion [29],ˆ H eff = ∞ (cid:88) j =0 T j ˆΩ j = ˆΩ + ˆΩ T + O ( T ) , (6)withˆΩ = 2 U ˆ S z − J ˆ S x , (7)ˆΩ = 2 µU π (cid:16) ˆ S x + 4 ˆ S z ˆ S x ˆ S z (cid:17) + µUπ ( µ − J ) ( ˆ S y − ˆ S z ) . For large driving frequencies the contribution from theˆΩ term is negligible, in which case ˆ H eff reduces to thetime-averaged, or unmodulated, Hamiltonian (1). Wenote that the effective Hamiltonian is valid only whenthe quasienergies (cid:15) α do not exhibit level-repulsion consis-tent with random matrix theory [2]. In other words, thebreakdown of the Magnus expansion is an indication ofthe chaotic behaviour in the quantum dynamics. In thispaper, we generally determine the Floquet quasienergiesand modes numerically, using QuTiP [30].Further insight into the breakdown of the Magnus ex-pansion, and hence the failure of the effective Hamilto-nian approximation [31], can be obtained analysis of thedelocalisation of the Floquet modes in the basis of effec-tive Hamiltonian eigenstates [2]. The Shannon entropy, S , quantifies this delocalisation [31] and is defined as S n = (cid:88) m | c mn | ln (cid:16) | c mn | (cid:17) , (8)where | c mn | = (cid:104) ψ n | φ m (cid:105) is the overlap between the Flo-quet mode | φ m (cid:105) and the effective Hamiltonian eigenstate | ψ n (cid:105) , satisfying ˆ H eff | ψ n (cid:105) = E n | ψ n (cid:105) .A related delocalisation measure can be obtained bycalculating the Shannon entropy of coherent states in thebasis of Floquet modes, which yields a distribution overphase space. Fig. 1 compares the delocalisation of co-herent states across phase space in the basis of Floquetmodes to the corresponding semiclassical phase space fea-tures. Chaotic phase space regions are accompanied byhigh delocalisation, while regular regions are accompa-nied by low delocalisation. Coherent states centred onphase space fixed points exhibit near-minimal delocalisa-tion.States that are sufficiently delocalised in the Flo-quet modes, as indicated by Shannon entropy close tothe circular orthogonal ensemble prediction, S COE ≈ ln [0 . N + 1)] [2], are expected to synchronise to themodulation by asymptotically approaching a periodicGibbs ensemble, a limit cycle with a period given by themodulation period [32]. III. SCRAMBLING
The quantum Lyapunov exponent λ Q characterises therate at which quantum information is scrambled, that is, where initially local quantum information spreads overthe degrees of freedom of the system. Previous stud-ies have established that the presence of classical chaos,characterized by λ >
0, indicates that in the correspond-ing quantum system λ Q > λ Q can be extracted from the measurement of anOTOC, C ( t ) = (cid:104)| [ ˆ W ( t ) , ˆ V ] | (cid:105) , ˆ W ( t ) = ˆ U † ( t ) ˆ W ˆ U ( t ) , (9)where ˆ W and ˆ V are two operators and C ( t ) characterizesthe non-commutativity of ˆ V and and ˆ W at a later time t .For quantum many-body systems with a chaotic classicallimit, the OTOC grows exponentially as C ( t ) ∼ e λ Q t [3–5, 8, 34, 35].In the following we focus on the behaviour of a specificfamily of OTOC, the fidelity OTOC (FOTOC), whichhas been used to probe the connections between chaos,scrambling, and thermalisation in the Dicke model [1],and has been experimentally implemented in a quantumsimulator of the all-to-all Ising model composed of a crys-tal of hundreds of ions [9]. To generate the FOTOC, wechoose ˆ W = e iδ ˆ w , where ˆ w is the generator of an arbi-trary rotation, and ˆ V = | ψ (cid:105) (cid:104) ψ | , with | ψ (cid:105) the initialstate of the system.For small perturbations, δ (cid:28)
1, the FOTOC reduces tothe variance of ˆ w since C ( t ) ≈ δ var [ ˆ w ( t )] + O ( δ ) [36].This allows for a simple experimental implementation ofthe FOTOC since the variance of the particle-number dif-ference, ˆ S z , is directly measurable experimentally, evento single-particle precision [37]. Indeed, the variancealong any axis can be experimentally measured, throughappropriate rotations on the Bloch sphere around ˆ S x (interwell-tunnelling) [38] and ˆ S y (relative energy differ-ence). For the rest of this work we restrict ourselves tothis regime.To provide a local probe in a mixed phase space featur-ing both regular and chaotic dynamics, we calculate FO-TOCs for arbitrary initial coherent states on the Blochsphere. Furthermore, to ensure the FOTOC is initiallyzero, we choose the operator ˆ w such that the initial coher-ent state is an eigenstate [1, 18]. Thus for a spin coherentstate centred on azimuthal angle, φ , and polar angle, θ ,we chooseˆ w = cos( φ ) sin( θ ) ˆ S x + sin( φ ) sin( θ ) ˆ S y + cos( θ ) ˆ S z . (10)In Fig. 2 we plot the FOTOC dynamics for parame-ters in the chaotic (panels a and d) and regular (panelsb and c) phase spaces for N = 1000. We find that inthe regime where the system is classically chaotic, afteran initial delay time, the FOTOCs grow exponentiallybefore plateauing at a finite value predicted by the di-agonal ensemble for an infinite-temperature state, in ac-cordance with the Floquet eigenstate thermalisation hy-pothesis prediction. As expected [1], both the saturationtime (Ehrenfest time) and the saturation value grow withincreasing N . In Fig. 2b, the FOTOC for a stable fixedpoint in phase space remains small-valued and appears J t J t FIG. 2. Fidelity out-of-time-order correlator (FOTOC) for N = 1000 initial coherent state in: a) chaotic phase space ( ω = 0 . ω = 7) on a stable fixed point; and c) regular phase space ( ω = 7) away from a stable fixed point. Bluemarkers correspond to FOTOC initial conditions shown in Fig. 1. d) FOTOCs for various particle numbers with parametersas in (a) and δ = 10 − . Here the dotted green, thin orange and thick purple lines represent N = 100 , , λ Q = 3 . J − . Horizontal dashed lines are diagonal ensemble predictions. Parameters for allplots are NU = − J ( t ) = J + 1 . ωt ). quasiperiodic, but does not synchronise with the modu-lation period. In Fig. 2c, the FOTOC in regular phasespace away from a fixed point grows slowly and reaches alarge-valued quasiperiodic limit after about 10 Ehrenfesttimes. Unlike the chaotic FOTOC, the regular FOTOClimit has large-amplitude oscillations and does not ap-proach the infinite-temperature diagonal-ensemble pre-diction. The average value of the quasiperiodic limit isgreater than the infinite-temperature diagonal-ensembleprediction due to the state’s ring-like Q -distribution.It is worth noting that the behaviour of the FOTOCcan be a mixture of the behaviours described above if thestate has overlap with both regular and chaotic regions ofthe phase space. We indicate two such states using bluemarkers in Fig. 1b and plot their respective FOTOCsin Fig. 3. Hence, to observe pure exponential growthof a FOTOC it is important to tune the Hamiltonianparameters such that the semiclassical phase diagram haswell-defined chaotic and regular regimes that are wideenough with respect to the quantum noise of the initialstate. l n [ C ( t ) / C d i ag ] FIG. 3. Dynamics of FOTOC logarithms scaled by diagonalensemble prediction, C diag , for reference states indicated byblue markers in Fig. 1. Parameters are N = 1000, NU = − J ( t ) = J + 1 . t ). IV. THERMALISATION
The exponential growth of the FOTOC discussedabove closely resembles the behaviour of the FOTOC inthe Dicke model [1]. Thus we expect the previously es-tablished relationship between thermalisation and chaosto hold. In the case of the modulated Bose-Hubbarddimer, however, the eigenstate thermalisation hypothesis(ETH) predicts that the system will approach an infinite-temperature state [2], of the form ˆ ρ = I / ( N + 1). Thevariance of an arbitrary local operator (10) in this limitis var ( ˆ w ) = Tr (ˆ ρ ˆ w ) − Tr (ˆ ρ ˆ w ) = Tr ( ˆ w ) / ( N + 1) − Tr ( ˆ w ) / ( N + 1) = N ( N + 2) / , (11)where we have used Tr ( ˆ S α ) = 0 and Tr ( ˆ S α ˆ S β ) = δ αβ N ( N + 1)( N + 2) /
12. Therefore, C diag = δ N ( N +2) / (cid:104) r (cid:105) .Level-spacing statistics more generally have long beenused as the defining characteristic of chaos in quantumsystems [39]. In Floquet systems, the spacings betweenadjacent quasienergies within the same symmetry class ofthe Hamiltonian play the role of the eigenvalue spacingfor time-independent systems. Hence we expect circularensemble statistics when our system equilibrates to infi-nite temperature, and Poissonian statistics otherwise [2].The modulated dimer Hamiltonian (1) displays twosymmetry classes, defined by the two possible eigenval- a) b)c) d)e) f) FIG. 4. a) Quasienergy level spacing distributions for widelyintegrable ( ω = 7, grey bars) and chaotic ( ω = 0 .
5, red line)phase spaces, compared to Poisonnian and circular orthogonalensemble (COE) predictions for fully integrable and noninte-grable Hamiltonians, respectively. b) Average quasienergylevel spacing parameter, (cid:104) r (cid:105) , as a function of driving fre-quency, ω , showing the transition from COE statistics toPoissonian statistics. c) Distributions of Shannon entropy, S , of the effective Hamiltonian eigenstates in the Floquet ba-sis. The Poissonian prediction is P ( S ) = δ ( S ). d) AverageShannon entropy (cid:104)S(cid:105) as a function of modulation frequency.e) Distribution of FOTOCs sampled across phase space af-ter J t = 30, slightly after the Ehrenfest time. f) Averageof FOTOCs at J t = 30 as a function of modulation fre-quency. Parameters for all plots are N = 1000, NU = − J ( t ) = J + 1 . ωt ). ues, ±
1, of the parity operator, ˆ P = ( − i ) N e − iπ ˆ S x [40].Floquet modes must be sorted into their respective sym-metry classes before quasienergy level spacing statisticscan be calculated. As shown in Fig. 4a, the quasienergylevel-spacing distribution for low driving frequency ( ω =0 .
5) matches the circular orthogonal ensemble predic-tions, indicating nonintegrability [2]. For high driving frequency ( ω = 7), the quasienergy level spacing distri-bution matches the Poissonian distribution, indicatingintegrability [2].As the driving frequency is increased, we observe atransition from COE to Poissonian statistics. To pin-point the critical driving frequency ω c at which this oc-curs, we calculate the average level-spacing spacing pa-rameter, defined as [2], r n = min ( δ n , δ n +1 )max ( δ n , δ n +1 ) ∈ [0 , , δ n = (cid:15) n +1 − (cid:15) n . (12)The average level spacing is then given by (cid:104) r (cid:105) = N (cid:80) n r n . A high (cid:104) r (cid:105) indicates level repulsion and non-integrability, while a low value indicates level clusteringand integrability [2]. This is confirmed in Fig. 4b, wherethe (cid:104) r (cid:105) agrees with the COE prediction for low drivingfrequency, and rapidly falls away as as the driving fre-quency is increased into the regime where we expect theMagnus expansion to hold.The signatures of this transition are also evident in theShannon entropy of the system. The Shannon entropyindicates the average delocalisation of Floquet modes inthe basis of effective Hamiltonian eigenstates. To pro-duce comparable results to Figs. 4a-b, we plot distribu-tions of Shannon entropy for regular and chaotic phasespace parameters in Fig. 4c and the average Shannonentropy as function of modulation frequency in Fig. 4d.In Fig. 4d we show that the Shannon entropy rises asthe modulation frequency is decreased. Unlike the levelspacing parameter, however, the Shannon entropy neverfully reaches reach the COE prediction over the frequencyvalues plotted. This behaviour is due to the greater sen-sitivity of the Shannon entropy to finite-size effects ascompared to the level-spacing parameter. Thus, we ex-pect that far from the thermodynamic limit, it is only forvery low modulation frequency that the Shannon entropywill reach the COE prediction [2, 31, 41].For intermediate driving frequency, the phase spacedisplays both integrable and chaotic regions and the levelspacing distribution, a global indicator of chaos, lies be-tween the Poissonian and COE predictions. To differen-tiate between chaotic and integrable phase space in thisregime, we use a local indicator of chaos, the FOTOC.To generate Figs. 4e and f we calculate the long-time value of FOTOCs of coherent-state centred oper-ators (10) distributed across the phase space in a 21 by20 grid. For small modulation frequency, all phase spaceFOTOCs saturate to the infinite-temperature diagonal-ensemble prediction, as indicated by the small variancein the distribution of FOTOC values for ω = 0 . ω = 7 (greysolid distribution in Fig. 4e) most FOTOCs retain a lowvalue at t = 100 J , but some grow larger than the diag-onal ensemble prediction. The latter behaviour is due tothe large, regular shearing of the phase-space distributionof some states over time. Nevertheless, the average FO-TOC value as a function of modulation frequency showsa sharp transition between regular and chaotic regimes,as indicated in Fig. 4f.As the long-time FOTOC can attain large values forboth chaotic and regular phase space, a large-valued FO-TOC is not a sufficient indicator of quantum chaos orthermalisation. Quantum chaos can be identified fromFOTOC dynamics by the presence of exponential growthup until the Ehrenfest time and saturation to the di-agonal ensemble prediction, as indicated in Fig. 2. Inthe case of initial states with support on both regu-lar and chaotic phase space regions, the short-time FO-TOC dynamics may feature irregular growth, as shownin Fig. 3. Therefore, choosing appropriate initial coher-ent state centre and width, which can be decreased byincreasing particle number, N , is important when usingFOTOCs to diagnose chaos and thermalisation. V. CONCLUSIONS
In this work we explored the connections between clas-sical chaos, the scrambling of quantum information, andthe predictions of the eigenstate thermalisation hypoth-esis in Floquet systems. In particular, we considered a driven Bose-Hubbard dimer model, which features aregular-to-chaotic transition in its semiclassical dynam-ics. We used Floquet analysis to study the various signa-tures of this transition in the quantum dynamics of thesystem, such as the exponential growth of the fidelityout-of-time-order correlator, from which a quantum Lya-punov exponent can be extracted.Moreover, we were able to establish a link betweenthe exponential growth of the FOTOC and thermalisa-tion using several measures. First, we compared the sat-uration value of the FOTOC to the predictions of theperiodic Gibbs ensemble, as required by the eigenstatethermalisation hypothesis. Next, we showed that thisapparent thermalisation is also evident in the level spac-ing statistics. As the system transitions from integrableto chaotic, the level spacing statistics are no longer Pois-sonian but instead correspond to the circular orthogonalensemble of random matrices. Finally, we showed thatthe divergence of the Floquet-Magnus perturbative ex-pansion is another signature of the apparent thermalisa-tion of Floquet system. We used the Shannon entropy toquantify the onset of this failure via the delocalisation ofthe Floquet in the effective Hamiltonian eigenstate basis.Our results build on, and are complementary, to previ-ous studies in Refs. [1, 2, 22]. Our protocol is realisableexperimentally and provides an alternative platform toexplore ideas relevant to quantum thermodynamics andthe dynamics of quantum information. [1] R. J. Lewis-Swan, A. Safavi-Naini, J. J. Bollinger, andA. M. Rey, Nat. Commun. , 1581 (2019).[2] L. D’Alessio and M. Rigol, Phys. Rev. X , 041048(2014).[3] P. Hayden and J. Preskill, J. High Energy Phys. (09), 120.[4] Y. Sekino and L. Susskind, J. 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