Semiclassical evolution of correlations between observables

TL;DR

This paper develops a semiclassical framework for evolving correlations between quantum observables using phase space integrals, linking quantum dynamics to classical trajectories and identifying conditions for classical versus nonclassical evolution.

Contribution

It introduces a semiclassical approximation for quantum correlations in phase space, including an initial value scheme and analysis of classical and nonclassical phases.

Findings

Correlation evolution reduces to a phase space integral for smooth observables.

Classical evolution occurs when observables undergo independent Heisenberg evolution.

Nonclassical phase factors appear in correlated evolutions, such as quantum Loschmidt echoes.

Abstract

The trace of an arbitrary product of quantum operators with the density operator is rendered as a multiple phase space integral of the product of their Weyl symbols with the Wigner function. Interspersing the factors with various evolution operators, one obtains an evolving correlation. The kernel for the matching multiple integral that evolves within the Weyl representation is identified with the trace of a single compound unitary operator. Its evaluation within a semiclassical approximation then becomes a sum over the periodic trajectories of the corresponding classical compound canonical transformation. The search for periodic trajectories can be bypassed by an exactly equivalent initial value scheme, which involves a change of integration variable and a reduced compound unitary operator. Restriction of all the operators to observables with smooth non-oscillatory Weyl symbols reduces the evolving correlation to a single phase space integral. If each observable undergoes independent Heisenberg evolution, the overall correlation evolves classically. Otherwise, the kernel acquires a nonclassical phase factor, though it still depends on a purely classical compound trajectory: e.g. the fase for a double return of the quantum Loschmidt echo does not coincide with twice the phase for a single echo.