Revealing quantum effects in bosonic Josephson junctions: a multi-configuration atomic coherent state approach

Abstract

The mean-field approach to two-site Bose–Hubbard systems is well-established and leads to non-linear classical equations of motion for population imbalance and phase difference. It can, for example, be based on the representation of the solution of the time-dependent Schrödinger equation either by a single Glauber state or by a single atomic (SU(2)) coherent state [S. Wimberger et al., Phys. Rev. A 103, 023326 (2021)]. We demonstrate that quantum effects beyond the mean-field approximation are easily uncovered if, instead, a multiconfiguration ansatz with a few time-dependent SU(2) basis functions is used in the variational principle. For the case of plasma oscillations, the use of just two basis states, whose time-dependent parameters are determined variationally, already gives a good qualitative agreement of the phase space dynamics with numerically exact quantum solutions. In order to correctly account for more non-trivial effects, like macroscopic quantum self-trapping, moderately more basis states are needed. For the onset of spontaneous symmetry breaking, however, a multiplicity of 2 gives a significant improvement already. In any case, the number of variational trajectories needed for good agreement with the full quantum results is orders of magnitude smaller than that in the semi-classical case, which is based on multiple mean-field trajectories.

Quench dynamics of interacting bosons: generalized coherent states versus multi-mode Glauber states

“…The use of GCS to treat the quantum dynamics of finite bosonic lattice models of Bose-Hubbard type has also been put forth more recently [14][15][16]. In [14], a mean-field variational approach, based on an ansatz for the solution of the time-dependent Schrödinger equation in terms of a single GCS (termed atomic coherent state) was shown to correctly predict some finite size effects in the bosonic Josephson junction (double well Bose-Hubbard system), with unsatisfactory results for more subtle quantum effects like spontaneous symmetry breaking and macroscopic self trapping, however.…”

“…In [14], a mean-field variational approach, based on an ansatz for the solution of the time-dependent Schrödinger equation in terms of a single GCS (termed atomic coherent state) was shown to correctly predict some finite size effects in the bosonic Josephson junction (double well Bose-Hubbard system), with unsatisfactory results for more subtle quantum effects like spontaneous symmetry breaking and macroscopic self trapping, however. In [16], a multi-configuration variational approach based on GCS was taken to overcome those deficiencies. In contrast to the well-established multi-configuration time-dependent Hartree method (MCTDH) [17], the basis functions used in [16] are non-orthogonal, such that special care has to be taken in terms of regularization of the equations of motion [18].…”

“…In [16], a multi-configuration variational approach based on GCS was taken to overcome those deficiencies. In contrast to the well-established multi-configuration time-dependent Hartree method (MCTDH) [17], the basis functions used in [16] are non-orthogonal, such that special care has to be taken in terms of regularization of the equations of motion [18]. This approach then led to an almost quantitative prediction also of the more elaborate quantum effects in the bosonic Josephson junction, using only a handful of basis functions (sometimes even as little as two).…”