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 $\text{SU}\left(M\right)$ 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 $\left(Q\right)$ and Glauber-Sudarshan $\left(P\right)$ 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.