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Entanglement and entropy production in coupled single-mode Bose-Einstein condensates

We investigate the time evolution of the entanglement entropy of coupled single-mode Bose-Einstein condensates in a double well potential at $T=0$ temperature, by combining numerical results with analytical approximations. We find that the coherent oscillations of the condensates result in entropy oscillations on the top of a linear entropy generation at short time scales. Due to dephasing, the entropy eventually saturates to a stationary value, in spite of the lack of equilibration. We show that this long time limit of the entropy reflects the semiclassical dynamics of the system, revealing the self-trapping phase transition of the condensates at large interaction strength by a sudden entropy jump. We compare the stationary limit of the entropy to the prediction of a classical microcanonical ensemble, and find surprisingly good agreement in spite of the non-equilibrium state of the system. Our predictions should be experimentally observable on a Bose-Einstein condensate in a double well potential or on a two-component condensate with inter-state coupling.