Semiclassical Lindblad master equation for spin dynamics

TL;DR

This paper derives a semi-classical phase space master equation for spin systems using the Wigner-Moyal formalism, extending the classical Fokker-Planck limit to non-canonical Poisson brackets relevant in atomic physics.

Contribution

It generalizes the semi-classical Lindblad master equation to non-canonical Poisson brackets, including gyro-Poisson brackets for spin ensembles.

Findings

Derivation of semi-classical equations of motion for spin variables.

Connection of Bloch equations with microscopic operators.

Extension of Fokker-Planck limit to non-canonical brackets.

Abstract

We derive the semi-classical Lindblad master equation in phase space for both canonical and non-canonical Poisson brackets using the Wigner-Moyal formalism and the Moyal star-product. The semi-classical limit for canonical dynamical variables, i.e., canonical Poisson brackets, is the Fokker-Planck equation, as derived before. We generalize this limit and show that it holds also for non-canonical Poisson brackets. Examples are gyro-Poisson brackets, which occur in spin ensembles, systems of recent interest in atomic physics and quantum optics. We show that the equations of motion for the collective spin variables are given by the Bloch equations of nuclear magnetization with relaxation. The Bloch and relaxation vectors are expressed in terms of the microscopic operators: The Hamiltonian and the Lindblad functions in the Wigner-Moyal formalism.