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Quantum phase-space analysis of equilibration in multi-well Bose-Hubbard systems

M. K. Olsen, C. V. Chianca

Published Aug 1, 2011• 0 citations
Publisher

Abstract

We examine the quantum dynamics and thermalisation of two, three and four-well Bose-Hubbard models using stochastic integration in the truncated Wigner phase-space representation. For the two-well model, we compare our results with those obtained by exact methods. We find that all three systems will thermalise, with the details depending on both the classical initial conditions and the initial quantum statistics. We construct an effective single-particle reduced density matrix for each of the systems, using the expectation values of operator moments, and use this to calculate an effective entropy. Knowing the expected maximum values of this entropy for each system, we are able to quantify the different approaches to equilibrium.

DOI
10.1109/iqec-cleo.2011.6193703

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We examine the quantum dynamics and thermalisation of two, three and four-well Bose-Hubbard models using stochastic integration in the truncated Wigner phase-space representation. For the two-well model, we compare our results with those obtained by exact methods. We find that all three systems will thermalise, with the details depending on both the classical initial conditions and the initial quantum statistics. We construct an effective single-particle reduced density matrix for each of the systems, using the expectation values of operator moments, and use this to calculate an effective entropy. Knowing the expected maximum values of this entropy for each system, we are able to quantify the different approaches to equilibrium.
We examine the medium time quantum dynamics and population equilibration of two-, three-, and four-well Bose-Hubbard models using stochastic integration in the truncated Wigner phase-space representation. We find that all three systems will enter at least a temporary state of equilibrium, with the details depending on both the classical initial conditions and the initial quantum statistics. We find that classical integrability is not necessarily a good guide as to whether equilibration will occur. We construct an effective single-particle reduced density matrix for each of the systems, using the expectation values of operator moments, and use this to calculate an effective entropy. Knowing the expected maximum values of this entropy for each system, we are able to quantify the different approaches to equilibrium.
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We discuss stochastic phase-space methods within the truncated Wigner approximation and show explicitly that they can be used to solve non-equilibrium dynamics of bosonic atoms in one-dimensional traps. We consider systems both with and without an optical lattice, and address different approximations in the stochastic synthesization of quantum statistical correlations of the initial atomic field. We also present a numerically efficient projection method for analyzing correlation functions of the simulation results, and demonstrate physical examples of non-equilibrium quantum dynamics of solitons and atom number squeezing in optical lattices.
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.