General Approach to Quantum-Classical Hybrid Systems and Geometric Forces

We present a general theoretical framework for a hybrid system that is composed of a quantum subsystem and a classical subsystem. We approach such a system with a simple canonical transformation which is particularly effective when the quantum subsystem is dynamically much faster than the classical counterpart, which is commonly the case in hybrid systems. Moreover, this canonical transformation generates a vector potential which, on one hand, gives rise to the familiar Berry phase in the fast quantum dynamics and, on the other hand, yields a Lorentz-like geometric force in the slow classical dynamics.

Figure 1

A schematic setup of a magnetic particle interacting with a spin. The particle moves freely in the $xy$ plane with a magnetic moment of ${\mathbf{m}}_F$ always pointing in the negative $z$ direction. A single spin with magnetic moment $\mathbf{\mu }$ is placed beneath the plane with a distance of $d$.