Mean-field dynamics of rotating bosons in strongly anisotropic traps

Abstract

The effective equations describing the mean-field dynamics of bosons in a rotating anisotropic trap are derived rigorously. It is shown that in the mean-field/strong anisotropy limit, the dynamics of an initial state that is close to a ground state of the dominant one dimensional harmonic potential is effectively decoupled, such that in the orthogonal complement of the direction of the dominant potential, it is described by a two dimensional magnetic nonlinear Schrödinger equation. Depending on the scaling of the two-body interaction, the nonlinear dynamics is given by a Hartree or the Gross–Pitaevskii equation in a rotating reference frame. For the sake of clarity of presentation, the Hartree case is discussed first before extending the analysis to the Gross–Pitaevskii one. The rigorous analysis yields explicit error bounds and new precise estimates of finite size effects.