An illustration of canonical quantum-classical dynamics

TL;DR

This paper uses the harmonic oscillator to demonstrate how hybrid quantum-classical brackets can model their interaction, revealing how classical backreaction affects quantum variables and the importance of a consistent hybrid bracket.

Contribution

It illustrates the use of hybrid dynamical brackets for quantum-classical interaction and emphasizes the necessity of a fully specified hybrid bracket for consistent dynamics.

Findings

Backreaction manifests as a nonzero commutator between classical variables.

Larger classical systems are less affected by backreaction.

Hybrid variables obey hybrid canonical relations despite pure variables violating pure canonical relations.

Abstract

Using the example of the harmonic oscillator, we illustrate the use of hybrid dynamical brackets in analyzing quantum-classical interaction. We only assume that a hybrid dynamical bracket exists, is bilinear, and reduces to the pure quantum/classical bracket when acting on pure quantum/classical variables. Any hybrid bracket obeying these natural requirements will produce the same dynamics for pure classical or quantum variables, given a hybrid Hamiltonian. Backreaction is manifested in the evolution of a nonvanishing commutator between classical variables. The more massive the classical system is, the less it is affected by backreaction. Interestingly, we show that while pure variables evolve to violate the pure canonical relations, they always obey the hybrid canonical relations. The dynamics of hybrid variables, on the other hand, is shown to require a fully specified and consistent hybrid bracket, otherwise evolution cannot be defined.