Phase space representation of quantum dynamics

TL;DR

This paper develops a phase space framework for quantum dynamics in many-body systems, using perturbative expansions around classical limits, and explores quantum corrections, jumps, and applications to cold atom models.

Contribution

It introduces a unified phase space approach for quantum dynamics with perturbative expansions around classical limits, incorporating quantum jumps and corrections.

Findings

Derivation of a phase space representation based on classical limits.

Analysis of quantum corrections via stochastic jumps and nonlinear responses.

Application to cold atom models demonstrating the method's versatility.

Abstract

We discuss a phase space representation of quantum dynamics of systems with many degrees of freedom. This representation is based on a perturbative expansion in quantum fluctuations around one of the classical limits. We explicitly analyze expansions around three such limits: (i) corpuscular or Newtonian limit in the coordinate-momentum representation, (ii) wave or Gross-Pitaevskii limit for interacting bosons in the coherent state representation, and (iii) Bloch limit for the spin systems. We discuss both the semiclassical (truncated Wigner) approximation and further quantum corrections appearing in the form of either stochastic quantum jumps along the classical trajectories or the nonlinear response to such jumps. We also discuss how quantum jumps naturally emerge in the analysis of non-equal time correlation functions. This representation of quantum dynamics is closely related to the phase space methods based on the Wigner-Weyl quantization and to the Keldysh technique. We show how such concepts as the Wigner function, Weyl symbol, Moyal product, Bopp operators, and others automatically emerge from the Feynmann's path integral representation of the evolution in the Heisenberg representation. We illustrate the applicability of this expansion with various examples mostly in the context of cold atom systems including sine-Gordon model, one- and two-dimensional Bose Hubbard model, Dicke model and others.