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Bulletin of the American Physical Society

Nonequilibrium dynamics of the Bose-Hubbard model and the discrete nonlinear Schrödinger equation in one dimension

Christian B. Mendl, Johannes M. Oberreuter, Michael Knap, Herbert E. Spohn

Published Mar 15, 2017• 0 citations
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Abstract We prove that the flow of the discrete nonlinear Schrödinger equation (DNLS) is the mean field limit of the quantum dynamics of the Bose–Hubbard model for N interacting particles. In particular, we show that the Wick symbol of the annihilation operators evolved in the Heisenberg picture converges, as N becomes large, to the solution of the DNLS. A quantitative LpL^pLp L p -estimate, for any p≥1p \ge 1p≥1 p ≥ 1 , is obtained with a linear dependence on time due to a Gaussian measure on initial data coherent states.
In the limit of large particle numbers and low densities systems of cold atoms can be effectively described as macroscopic single-particle systems in a mean-field approximation. In the case of a Bose–Hubbard system, modelling bosons on a discrete lattice with on-site interactions, this yields a discrete nonlinear Schrödinger equation of Gross–Pitaevskii type. It has been recently shown that the correspondence between the Gross–Pitaevskii equation and the Bose–Hubbard system breaks down for complex extensions. In particular, for a Bose–Hubbard dimer with complex on-site energy the mean-field approximation yields a generalized complex nonlinear Schrödinger equation. Conversely, a Gross–Pitaevskii equation with complex on-site energies arises as the mean-field approximation of many-particle Lindblad dynamics rather than a complex extension of the Bose–Hubbard system. Here we address the question of how the mean-field description is modified in the presence of a complex-valued particle interaction term for a Bose–Hubbard dimer. We derive the mean-field equations of motion leading to nonlinear dissipative Bloch dynamics, related to a nontrivial complex generalization of the nonlinear Schrödinger equation. The resulting dynamics are analysed in detail. It is shown that depending on the parameter values there can be up to six stationary states, and for small values of the interaction strength there are limit cycles. Furthermore, we show how a Gross–Pitaevskii equation with a complex interaction term can be derived as the mean-field approximation of a Bose–Hubbard dimer with an additional Lindblad term modelling two-particle losses.
Abstract The dynamics of repulsive bosons condensed in an optical lattice is effectively described by the Bose-Hubbard model. The classical limit of this model, reproduces the dynamics of Bose-Einstein condensates, in a periodic potential, and in the superfluid regime. Such dynamics is governed by a discrete nonlinear Schrödinger equation. Several papers, addressing the study of the discrete nonlinear Schrödinger dynamics, have predicted the spontaneous generation of (classical) breathers in coupled condensates. In the present contribute, we shall focus on localized solutions (quantum breathers) of the full Bose-Hubbard model. We will show that solutions exponentially localized in space and periodic in time exist also in absence of randomness. Thus, this kind of states, reproduce a novel quantum localization phenomenon due to the interplay between bounded energy spectrum and non-linearity.