Exact number conserving phase-space dynamics of the M-site Bose-Hubbard model

TL;DR

This paper develops an exact phase-space description for M-site Bose-Hubbard models, deriving second-order differential equations for quantum distributions that interpolate between quantum and classical dynamics as particle number increases.

Contribution

It introduces a novel phase-space framework for Bose-Hubbard models, providing explicit evolution equations that bridge quantum and classical regimes based on particle number.

Findings

Evolution equations are second order and scale as 1/N

For large N, dynamics approximate classical Liouvillian behavior

Phase space approach aids in understanding quantum-to-classical crossover

Abstract

The dynamics of M-site, N-particle Bose-Hubbard systems is described in quantum phase space constructed in terms of generalized SU(M) coherent states. These states have a special significance for these systems as they describe fully condensed states. Based on the differential algebra developed by Gilmore, we derive an explicit evolution equation for the (generalized) Husimi-(Q)- and Glauber-Sudarshan-(P)-distributions. Most remarkably, these evolution equations turn out to be second order differential equations where the second order terms scale as 1/N with the particle number. For large N the evolution reduces to a (classical) Liouvillian dynamics. The phase space approach thus provides a distinguished instrument to explore the mean-field many-particle crossover. In addition, the thermodynamic Bloch equation is analyzed using similar techniques.