Koopman wavefunctions and classical-quantum correlation dynamics

TL;DR

This paper develops a Hamiltonian-based framework for classical-quantum coupling that accurately models mutual influence and backreaction, ensuring positive-definite quantum states and preserving classical density signs in certain systems.

Contribution

It introduces a new Hamiltonian wave equation for classical-quantum dynamics that overcomes limitations of existing models, including positivity and sign preservation properties.

Findings

Quantum density matrix remains positive-definite.

Classical density sign is preserved in specific models.

Model is validated with an exactly solvable two-level system.

Abstract

Upon revisiting the Hamiltonian structure of classical wavefunctions in Koopman-von Neumann theory, this paper addresses the long-standing problem of formulating a dynamical theory of classical-quantum coupling. The proposed model not only describes the influence of a classical system onto a quantum one, but also the reverse effect -- the quantum backreaction. These interactions are described by a new Hamiltonian wave equation overcoming shortcomings of currently employed models. For example, the density matrix of the quantum subsystem is always positive-definite. While the Liouville density of the classical subsystem is generally allowed to be unsigned, its sign is shown to be preserved in time for a specific infinite family of hybrid classical-quantum systems. The proposed description is illustrated and compared with previous theories using the exactly solvable model of a degenerate two-level quantum system coupled to a classical harmonic oscillator.