| The dynamics of Bose-Einstein condensates in optical lattices can be modeled by an M-site, N-particle Bose-Hubbard Hamiltonian. The quantum phase space dynamics of such Bose-Hubbard systems can be conveniently described using SU(M) coherent states conserving the particle number. Time evolution equations for generalized Husimi distributions can be formulated as second-order partial 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 celebrated mean-field dynamics as an effective single particle approximation. This phase-space description enables us to go beyond the mean-field limit partially resolving the breakdown of the mean-field approximation. Moreover, one can extend the classical mean-field approximation in a semiclassical manner, by allowing interferences. In this way one can reconstruct properties of the many particle system from the 'classical' mean-field dynamics. |
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