A phase-space method for the Bose–Hubbard model: application to mean-field models

We present a phase-space method for the Bose–Hubbard model based on the Q-function representation. In particular, we consider two model Hamiltonians in the mean-field approximation; the first is the standard ‘one-site’ model where quantum tunnelling is approximated entirely using mean-field terms; the second ‘two-site’ model explicitly includes tunnelling between two adjacent sites while treating tunnelling with other neighbouring sites using the mean-field approximation. The ground state is determined by minimizing the classical energy functional subject to quantum mechanical constraints, which take the form of uncertainty relations. For each model Hamiltonian, we compare the ground state results from the Q-function method with the exact numerical solution. The results from the Q-function method, which are easy to compute, give a good qualitative description of the main features of the Bose–Hubbard model including the superfluid to Mott insulator. We find the quantum mechanical constraints dominate the problem and show there are some limitations of the method particularly in the weak lattice regime.