We investigate the effects of phase noise and particle loss on the dynamics of a Bose-Einstein condensate in an optical lattice. Starting from the many-body master equation, we discuss the applicability of generalized mean-field approximations in the presence of dissipation as well as methods to simulate quantum effects beyond mean-field by including higher-order correlation functions. It is shown that localized particle dissipation leads to surprising dynamics, as it can suppress decay and restore the coherence of a Bose-Einstein condensate. These effects can be applied to engineer coherent structures such as stable discrete breathers and dark solitons.Comment: 7 pages, 6 figur

We discuss the dynamics of a Bose-Einstein condensate in a double-well trap subject to phase noise and particle loss. The phase coherence of a weakly-interacting condensate as well as the response to an external driving show a pronounced stochastic resonance effect: Both quantities become maximal for a finite value of the dissipation rate matching the intrinsic time scales of the system. Even stronger effects are observed when dissipation acts in concurrence with strong interparticle interactions, restoring the purity of the condensate almost completely and increasing the phase coherence significantly.

The number-conserving quantum phase space description of the Bose-Hubbard model is discussed for the illustrative case of two and three modes, as well as the generalization of the two-mode case to an open quantum system. The phase-space description based on generalized SU͑M͒ coherent states yields a Liouvillian flow in the macroscopic limit, which can be efficiently simulated using Monte Carlo methods even for large systems. We show that this description clearly goes beyond the common mean-field limit. In particular it resolves well-known problems where the common mean-field approach fails, such as the description of dynamical instabilities and chaotic dynamics. Moreover, it provides a valuable tool for a semiclassical approximation of many interesting quantities, which depend on higher moments of the quantum state and are therefore not accessible within the common approach. As a prominent example, we analyze the depletion and heating of the condensate. A comparison to methods ignoring the fixed particle number shows that in this case artificial number fluctuations lead to ambiguities and large deviations even for quite simple examples.

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.