The BBGKY hierarchy for ultracold bosonic systems: II. Applications

Abstract

In this work, which is based on our previously derived theoretical framework [1], we apply the truncated Born-Bogoliubov-Green-Kirkwood-Yvon (BBGKY) hierarchy for ultracold bosonic systems with a fixed number of particles to two out-of-equilibrium scenarios, namely tunneling in a double-well potential and an interaction quench in a harmonic trap. The efficient formulation of the theory provided in [1] allows for going to large truncation orders such that the impact of the truncation order on the accuracy of the results can be systematically explored. While the short-time dynamics is found to be excellently described with controllable accuracy, significant deviations occur on a longer time-scale for a sufficiently strong interaction quench or the tunneling scenario. Theses deviations are accompanied by exponential-like instabilities leading to unphysical results. The phenomenology of these instabilities is investigated in detail and we show that the minimal-invasive correction algorithm of the equation of motion as proposed in [1] can indeed stabilize the BBGKY hierarchy truncated at the second order.

Two-Time Correlation Functions in Dissipative and Interacting Bose-Hubbard Chains

“…Based on our method, we can treat particle interactions on a mean-field level (see Section IV) and also one-order beyond (see Section V). Higher-order expansions are possible, at the cost of very lengthy formulae, which at some point will become difficult to deal with even numerically (see e.g., [28,29] for just time-equal correlators).…”