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arXiv:1102.2469 · Physical Review A

Strong thermalization of the two-component Bose-Hubbard model at finite temperatures

J. M. Zhang, Chao Shen, W. M. Liu

Published Jan 27, 2012• 11 citations
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Abstract

We study thermalization of a two-component Bose-Hubbard model by exact diagonalization. Initially, the two components do not interact and are both at equilibrium but with different temperatures. As the onsite intercomponent interaction is turned on, perfect thermalization occurs. Remarkably, not merely those simple ``realistic'' physical observables thermalize but even the density matrix of the whole system---the time-averaged density matrix of the system can be well approximated by that of a canonical ensemble. We also discuss how symmetry of the quenched Hamiltonian affects the level spacing distribution and thermalization of the system.

DOI
10.1103/physreva.85.013637

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We study thermalization of a two-component Bose-Hubbard model by exact diagonalization. Initially, the two components do not interact and are both at equilibrium but with different temperatures. As the onsite intercomponent interaction is turned on, perfect thermalization occurs. Remarkably, not merely those simple ``realistic'' physical observables thermalize but even the density matrix of the whole system---the time-averaged density matrix of the system can be well approximated by that of a canonical ensemble. We also discuss how symmetry of the quenched Hamiltonian affects the level spacing distribution and thermalization of the system.
We study quench dynamics of the Bose-Hubbard model by exact diagonalization. Initially, the system is at thermal equilibrium and of a finite temperature. The system is then quenched by changing the on-site interaction strength UUU suddenly. Both the single-quench and double-quench scenarios are considered. In the former case, the time-averaged density matrix and the real-time evolution are investigated. It is found that though the system thermalizes only in a very narrow range of the quenched value of UUU, it does equilibrate or relax well into a much larger range. Most importantly, it is proven that this is guaranteed for some typical observables in the thermodynamic limit. In order to test whether it is possible to distinguish the unitarily evolving density matrix from the time-averaged (thus time-independent), fully decohered density matrix, a second quench is considered. It turns out that the answer is affirmative or negative depending on whether the intermediate value of UUU is zero or not.
Motivated by recent experiments, we study the relaxation dynamics and thermalization in the one-dimensional Bose-Hubbard model induced by a global interaction quench. Specifically, we start from an initial state that has exactly one boson per site and is the ground state of a system with infinitely strong repulsive interactions at unit filling. Using exact diagonalization and the density-matrix renormalization-group method, we compute the time dependence of such observables as the multiple occupancy and the momentum distribution function. Typically, the relaxation to stationary values occurs over just a few tunneling times. The stationary values are identical to the so-called diagonal ensemble on the system sizes accessible to our numerical methods and we further observe that the microcanonical ensemble describes the time averages of many observables reasonably well for small and intermediate interaction strength. The expectation values of observables in the canonical ensemble agree quantitatively with the time averages obtained from the quench at small interaction strengths, and qualitatively provide a good description even in parameter regimes where the microcanonical ensemble is not applicable due to finite-size effects. We discuss our numerical results in the framework of the eigenstate thermalization hypothesis. Moreover, we also observe that the diagonal and the canonical ensembles are practically identical for our initial conditions already on the level of their respective energy distributions for small interaction strengths. Finally, we discuss implications of our results for the interpretation of a recent sudden expansion experiment [Phys. Rev. Lett. 110, 205301 (2013)], in which the same interaction quench was realized.
We describe the decoherence instability of Schrödinger Cat states in the two-site Bose-Hubbard model with an attractive on-site interaction between particles. For N particles with onsite attractive energy U and hopping amplitude between sites t, Cat states exist for [Formula: see text] at zero temperature. However, they are increasingly unstable to small thermal fluctuations as the Cat itself is increasingly well-defined and its components become well-separated. For any given [Formula: see text], the decoherence temperature becomes smaller for large N. The loss of off-diagonal coherence peaks in the equilibrium density matrix is dominated by the thermal admixture of the first excited state of the many-body system with its ground state. Particle number fluctuations, described in the grand canonical ensemble also reduce coherence, but to a lesser degree than thermal fluctuations. The full density matrix of the Schrödinger Cat is obtained by exact numerical diagonalization of the many-body Hamiltonian and a narrow regime in the parameter space of the particle number, temperature, and U/t is identified where small Cat states may survive decoherence in a physical environment.
Motivated by the nonequilibrium dynamics experiment, we study the quantum quench dynamics of the JCH model with weak nearest-neighbor hopping strength by exact diagonalization. One of the crucial insights is to focus on the comparison between the time-averaged density matrix and the canonical ensemble in a wide range of values for the coupling strength between the cavity and the atom, and significant differences are found. Thus, this phenomenon can be used to infer that the system with weak nearest-neighbor hopping strength may be integrable. In order to verify our prediction, we perform the level spacing distribution of the system and find that they conform with the characteristics of the integrable system. Further, a nearly integrable system is presented by removing the degenerate energy level and fitting Brody distribution. Besides, the evolution of the momentum distribution functions of the photon and the equilibrium value predicted by the canonical ensemble are followed, which demonstrates that the system cannot be thermalized. Finally, the way how the system avoids thermalization is also discussed by the finite-size scaling of the fluctuations of the diagonal and the off-diagonal matrix elements.
We study the dynamics and thermalization of strongly correlated fermions in finite one-dimensional lattices after a quantum quench. Our calculations are performed using exact diagonalization. We focus on one- and two-body observables such as the momentum distribution function [n(k)][n(k)][n(k)] and the density-density structure factor [N(k)][N(k)][N(k)], respectively, and study the effects of approaching an integrable point. We show that while the relaxation dynamics and thermalization of N(k)N(k)N(k) for fermions is very similar to the one of hardcore bosons, the behavior of n(k)n(k)n(k) is distinctively different. The latter observable exhibits a slower relaxation dynamics in fermionic systems. We identify the origin of this behavior, which is related to the off-diagonal matrix elements of n(k)n(k)n(k) in the basis of the eigenstates of the Hamiltonian. More generally, we find that thermalization occurs far away from integrability and that it breaks down as one approaches the integrable point.