Simulating the complete quantum mechanics of very large driven-dissipative Bose-Hubbard systems*

Phase-space descriptions of open quantum systems allow for linear or log-linear scaling of computational difficulty with system size, and adaptability to non-uniform and time-dependent systems. They also allow for straightforward calculation of many of the multi-time correlations of interest by replacing Heisenberg operators with he bare time-dependent stochastic variables.

Here, we demonstrate that the positive-P method captures the complete many-body quantum dynamics of the driven-dissipative Bose-Hubbard model across a wide range of parameters. Notably, these parameters include intermediate regimes where interactions and dissipation are comparable, and especially cases with low occupations for which common semiclassical approximations can break down. The presence of dissipation can alleviate instabilities in the method that are known to occur for closed systems, allowing the simulation of dynamics up to and including the steady state.  

We demonstrate its use in several examples with non-trivial quantum correlations, including a nonuniform 2d system with tens of thousands of sites, larger than typical experimental setups.

It is also found that the positive-P's region of applicability is complementary to that of the truncated Wigner method, and together both methods allow one to cover the majority of parameter space.