Dynamics of mixed quantum-classical spin systems

TL;DR

This paper introduces a fully Hamiltonian framework for mixed quantum-classical spin systems that maintains consistency, positivity, and Heisenberg's uncertainty principle, advancing the theoretical foundation of quantum-classical dynamics.

Contribution

It presents the first Hamiltonian theory ensuring key consistency properties for quantum-classical spin systems, including positivity and adherence to Heisenberg's principle.

Findings

Ensures positivity of quantum and classical densities at all times.

Connects the theory to energy-balance in measurement processes.

Extends the model to multiple spins and orbital degrees of freedom.

Abstract

Mixed quantum-classical spin systems have been proposed in spin chain theory and, more recently, in magnon spintronics. However, current models of quantum-classical dynamics beyond mean-field approximations typically suffer from long-standing consistency issues, and, in some cases, invalidate Heisenberg's uncertainty principle. Here, we present a fully Hamiltonian theory of quantum-classical spin dynamics that appears to be the first to ensure an entire series of consistency properties, including positivity of both the classical and the quantum density, so that Heisenberg's principle is satisfied at all times. We show how this theory may connect to recent energy-balance considerations in measurement theory and we present its Poisson bracket structure explicitly. After focusing on the simpler case of a classical Bloch vector interacting with a quantum spin observable, we illustrate the extension of the model to systems with several spins, and restore the presence of orbital degrees of freedom.