Beyond Mean-Field Dynamics of Two-Mode Bose-Hubbard Model with Linear Coupling Ramping

Abstract

The mean- field Hamiltonian of two-mode Bose-Hubbard model with real and imaginary coupling constants demonstrates pitchfork bifurcation in its phase-space structure within certain interval of real coupling constant. Its mean- field dynamics have been previously studied by Zhang et al. It was shown therein that when the real coupling constant is ramped adiabatically towards the pitchfork bifurcation critical point, the classical intrinsic dynamical fluctuations assist in the selection between the two stable stationary points. Based on this finding, we set out to study the corresponding quantum Hamiltonian with the real coupling constant ramped linearly. At very slow ramping, the quantum system is able to resolve the energy difference of the two nearly degenerate lowest energy states. Therefore, it no longer demonstrates self-trapping as what is predicted by the mean-field dynamics. Such breakdown of mean- field within dynamical instability is an example of incommutability between semiclassical and adiabatic limit. To extend beyond the mean- field level, we employ the Bogoliubov backreaction method and the semiclassical phase space method to understand how the second and higher order quantum fluctuations alter the system dynamics. It turns out that both approaches yield good prediction on the dynamics of population imbalance between the two modes and the fraction of non-condensed atoms at fast and very slow ramping.