Mean-Field Dynamics of a Non-Hermitian Bose-Hubbard Dimer

We investigate an $N$-particle Bose-Hubbard dimer with an additional effective decay term in one of the sites. A mean-field approximation for this non-Hermitian many-particle system is derived, based on a coherent state approximation. The resulting nonlinear, non-Hermitian two-level dynamics, in particular, the fixed point structures showing characteristic modifications of the self-trapping transition, are analyzed. The mean-field dynamics is found to be in reasonable agreement with the full many-particle evolution.

Figure 1

Mean-field dynamics on the Bloch sphere for the Hermitian $\gamma =0$ (top) and the non-Hermitian case $\gamma =0.75$ (bottom) for $g=0$ (left) and $g=2$ (right) and $\epsilon =0$ and $v=1$.

Figure 2

Decay of the survival probability (full black curve) and the populations of site 1 (dashed red curve) and 2 (dotted blue curve) for an initial coherent state located at the south pole, for $g=0.1$, $\gamma =0.01$, $v=1$ and $N=20$ (left) and the relative deviations between many-particle and mean-field results (right).

Figure 3

Mean-field evolution of the population imbalance $s_z(t)$ (dashed blue curve) in comparison with the full many-particle system for $N=20$ particles (black curve) and an initial coherent state located at the north pole [$g=0.5$, $\gamma =0.1$ (top) and $g=2$, $\gamma =0.5$ (bottom) and $v=1$].