Action on the Sphere: An Interfering Mean-Field Propagator for the Bose-Hubbard Dimer

The Bose-Hubbard system has been studied extensively both theoretically and experimentally, in particular in the context of ultracold atomic gases in optical lattices.

Even in the two-mode case the many-particle dynamics display complex interference effects resulting in revival and breakdown phenomena as well as tunnelling.

The most basic theoretical description is the mean-field approximation, which can be derived from a time-dependent variational principle assuming the many-particle wave function is an SU(2) coherent state.

Here we build on this to construct a simple initial-value coherent state propagator, summing over mean-field trajectories and keeping track of their phases, given by the corresponding mean-field actions.

This yields an approximation to the full time-dependent many-particle state, and is able to reproduce breakdown and revival dynamics.

Applying a time-slicing procedure on top of this, we are able to accurately capture many-particle tunnelling effects.

While in this paper we focus our analysis on the Bose-Hubbard dimer, the methods developed can be applied to more general SU(2) Hamiltonians, and can be extended to SU(M) systems.