Chaoticity without thermalisation in disordered lattices

Abstract

We study chaoticity and thermalization in Bose-Einstein condensates in disordered lattices, described by the discrete nonlinear Schrödinger equation (DNLS). A symplectic integration method allows us to accurately obtain both the full phase space trajectories and their maximum Lyapunov exponents (mLEs), which characterize their chaoticity. We find that disorder destroys ergodicity by breaking up phase space into subsystems that are effectively disjoint on experimentally relevant timescales, even though energetically, classical localisation cannot occur. This leads us to conclude that the mLE is a very poor ergodicity indicator, since it is not sensitive to the trajectory being confined to a subregion of phase space. The eventual thermalization of a BEC in a disordered lattice cannot be predicted based only on the chaoticity of its phase space trajectory.Introduction In this Letter, we bring together the topics of disorder, nonlinearity, chaos, ergodicity, and Bose-Einstein condensation (BEC) in optical lattices. Inspired by the realisation of disordered optical potentials in experiments with ultracold atomic gases [1, 2], we explore the question of thermalisation in such systems. Previous theoretical works have, amongst others, addressed the topics of Bose and Anderson glasses [3], Anderson localisation [4,5], and Lifshits glasses [6]. In [6], various regimes of interaction strengths were investigated, and it was found that for sufficiently strong interactions, a disordered BEC is expected. We will focus on this regime, and study the Bose-Hubbard model [7,8], which describes a bosonic gas in a lattice, in the mean-field approximation. The resulting model can be obtained by discretising the Gross-Pitaevskii equation, and is also known as the discrete nonlinear Schrödinger equation (DNLS) [9]. In this Letter, we pose and answer the following question: what is the effect of disorder on thermalisation and ergodicity in the mean-field Bose-Hubbard model / DNLS?The connection between chaoticity and thermalisation was recently discussed in the disorder-free case [10]. Intuitively, one would expect chaotic trajectories to thermalise, since unlike regular ones, they are not confined to the neighborhood of stable periodic orbits. Not being confined, the expectation is that they are able to cover the available phase space, and that the system is therefore well-described by the microcanonical ensemble, since only the energy and particle number are conserved. We show that this expectation is not correct, by explicitly demonstrating the absence of equiprobability of states on the microcanonical shell. Since the energies involved are higher than the disorder potential, classical localisation cannot be responsible for this effect.The most commonly employed method of chaos detection, which quantifies the sensitive dependence on initial conditions, is the evaluation of the maximum Lyapunov exponent (mLE) [11][12][13][14]. In [10], a positive mLE, which

Bose-Einstein condensation and many-body localization of rotational excitations of polar molecules following a microwave pulse

“…Note that the mean-field theory we study is a classical nonlinear dynamical system. Similar non-ergodic behaviour, and lack of thermalization, has also been found in recent work by Tieleman et al [26], who have shown that in the closely related mean-field theory of the Bose-Hubbard model, disorder breaks the connection between chaoticity and ergodicity even when classical localization cannot occur. We interpret the non-ergodic nature of the mean-field spin system as indicative of the many-body localization in the underlying quantum spin system.…”

“…In the presence of disorder such systems are natural candidates to show many-body localization and nonergodic behaviour [25,26] as originally envisaged by Anderson [27] in the context of disordered spin systems in solids.…”

Characteristics of chaos evolution in one-dimensional disordered nonlinear lattices

“…Chaoticity by itself is not enough to guarantee thermalization of disordered systems [40] and support subdiffusion theories. The needed, additional ingredient is the spatiotemporal fluctuations of the chaotic seeds inside the excited part of the lattice, something which was shown in [38] through the time evolution of the deviation vector (i.e.…”

Thermalization in the one-dimensional Salerno model lattice

“…It is therefore natural to expect that the application of statistical and thermodynamical approaches and related mixing, ergodicity, and energy equipartition concepts to such problems is a topic of substantial ongoing interest. Among the numerous prototypical nonlinear physical examples are: the statistics of the Fermi-Pasta-Ulam-Tsingou chains [11], chains of Josephson-junctions [12], Gross-Pitaevskii/discrete nonlinear Schrödinger lattices with different types of nonlinearities [13][14][15][16][17][18][19], Toda and Morse lattices [20], etc. The localized wave patterns that naturally emerge in such systems as a result of the interplay between lattice dispersion and nonlinearity play a crucial role in the thermalization and lattice dynamics [21].…”