Semiclassics for particles with spin via a Wigner-Weyl-type calculus

TL;DR

This paper develops a semiclassical framework relating quantum spin-1/2 particle dynamics to classical Hamiltonian flow on an extended phase space, with applications to models like Rabi- and Jaynes-Cummings.

Contribution

It introduces an Egorov-type theorem combining Wigner-Weyl and Stratonovich-Weyl calculus for spin systems, extending validity to longer times for certain Hamiltonians.

Findings

Establishes a second-order accurate semiclassical approximation.

Proves Egorov theorem for specific models like Rabi- and Jaynes-Cummings.

Demonstrates the approach with a Stern-Gerlach experiment model.

Abstract

We show how to relate the full quantum dynamics of a spin-1/2 particle on R^d to a classical Hamiltonian dynamics on the enlarged phase space R^d x S^2 up to errors of second order in the semiclassical parameter. This is done via an Egorov-type theorem for normal Wigner-Weyl calculus for R^d [Lei10,Fol89] combined with the Stratonovich-Weyl calculus for SU(2) [VGB89]. For a specific class of Hamiltonians, including the Rabi- and Jaynes-Cummings model, we prove an Egorov theorem for times much longer than the semiclassical time scale. We illustrate the approach for a simple model of the Stern-Gerlach experiment.