Quantum corrections to the dynamics of interacting bosons: beyond the truncated Wigner approximation

TL;DR

This paper introduces a perturbation theory for quantum fluctuations around classical dynamics of interacting bosons, extending beyond the truncated Wigner approximation by including quantum scattering corrections.

Contribution

It develops a systematic perturbation framework that incorporates quantum corrections to classical bosonic dynamics, surpassing the truncated Wigner approximation.

Findings

Recovered classical Gross-Pitaevskii equations at zero order

Derived the truncated Wigner approximation at first order

Presented numerical examples demonstrating the formalism's effectiveness

Abstract

We develop a consistent perturbation theory in quantum fluctuations around the classical evolution of a system of interacting bosons. The zero order approximation gives the classical Gross-Pitaevskii equations. In the next order we recover the truncated Wigner approximation, where the evolution is still classical but the initial conditions are distributed according to the Wigner transform of the initial density matrix. Further corrections can be characterized as quantum scattering events, which appear in the form of a nonlinear response of the observable to an infinitesimal displacement of the field along its classical evolution. At the end of the paper we give a few numerical examples to test the formalism.