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

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.