Semiclassical Coherent States propagator

TL;DR

This paper develops a semiclassical approximation for quantum propagators in coherent states that uses only real trajectories, simplifying calculations and avoiding root searches, with exact results for quadratic Hamiltonian systems.

Contribution

It introduces a novel semiclassical coherent states propagator that avoids complex trajectories and root searches, applicable to chaotic systems, and exact for quadratic Hamiltonians.

Findings

The method avoids complex trajectories and root searches.

It provides an explicit time-dependent propagator for chaotic systems.

The approach is exact for quadratic Hamiltonian systems.

Abstract

In this work, we derived a semiclassical approximation for the matrix elements of a quantum propagator in coherent states (CS) basis that avoids complex trajectories, it only involves real ones. For that propose, we used the, symplectically invariant, semiclassical Weyl propagator obtained by performing a stationary phase approximation (SPA) for the path integral in the Weyl representation. After what, for the transformation to CS representation SPA is avoided, instead a quadratic expansion of the complex exponent is used. This procedure also allows to express the semiclassical CS propagator uniquely in terms of the classical evolution of the initial point, without the need of any root search typical of Van Vleck Gutzwiller based propagators. For the case of chaotic Hamiltonian systems, the explicit time dependence of the CS propagator has been obtained. The comparison with a \textquotedbl{}realistic\textquotedbl{} chaotic system that derives from a quadratic Hamiltonian, the cat map, reveals that the expression here derived is exact up to quadratic Hamiltonian systems.