Solving quantum dynamics with a Lie algebra decoupling method

TL;DR

This paper introduces a Lie algebra decoupling method for solving quantum system dynamics, providing a pedagogical overview, mathematical proof, and applications to common Hamiltonians in quantum optics.

Contribution

It presents a new Lie algebra decoupling theorem for quantum dynamics, with applications to linear and quadratic Hamiltonians, including open-system cases.

Findings

Derives differential equations for Gaussian dynamics in quadratic Hamiltonians

Applies the theorem to examples in quantum optics

Discusses extensions beyond quadratic and open-system dynamics

Abstract

At the heart of quantum technology development is the control of quantum systems at the level of individual quanta. Mathematically, this is realised through the study of Hamiltonians and the use of methods to solve the dynamics of quantum systems in various regimes. Here, we present a pedagogical introduction to solving the dynamics of quantum systems by the use of a Lie algebra decoupling theorem. As background, we include an overview of Lie groups and Lie algebras aimed at a general physicist audience. We then prove the theorem and apply it to three well-known examples of linear and quadratic Hamiltonian that frequently appear in quantum optics and related fields. The result is a set of differential equations that describe the most Gaussian dynamics for all linear and quadratic single-mode Hamiltonian with generic time-dependent interaction terms. We also discuss the use of the decoupling theorem beyond quadratic Hamiltonians and for solving open-system dynamics.