Quantum-classical correspondence in a nonlinear Gross–Pitaevski system

Abstract

We investigate the quantum-classical correspondence of the Bose–Einstein condensate (BEC) which is described by the Gross–Pitaevski equation with the periodic, both temporally and spatially, modulation of nonlinear interaction. It is found that this system is equivalent to a generalized kicked rotor (GKR) model. Interestingly, the classical dynamics of the GKR system exhibits the regular-to-chaotic transition as time evolves, and the corresponding energy diffusion is exponentially fast. For the BEC system, the energy diffusion can be predicted by the classical mapping equation of the GKR model. However, the ratio of the energy between the BEC system and the classical GKR model is not unity, even if the effective Planck constant is very small, which is contradictory to our conventional understanding of quantum-classical correspondence. We find that the BEC system is not exactly a quantum counterpart of the classical GKR system. We construct a better quantum counterpart and derive an analytical expression for the exponentially-fast diffusion of energy, which is in perfect agreement with numerical results.