Solving the quantum master equation of coupled harmonic oscillators with Lie algebra methods

TL;DR

This paper introduces Lie algebra-based methods to solve the quantum master equation for coupled harmonic oscillators with Markovian loss, providing new analytical tools for understanding open quantum system dynamics.

Contribution

It presents two novel Lie algebra-based approaches to analytically solve the quantum master equation in Liouville space for lossy coupled harmonic oscillators, including a Wei-Norman expansion.

Findings

Eigenstates of the Liouvillian constructed via Lie algebra facilitate density matrix decomposition.

The methods offer insights into transport properties of lossy quantum systems.

Structure analysis enables effective non-Hermitian Hamiltonian construction.

Abstract

Based on a Liouville-space formulation of open systems, we present two methods to solve the quantum dynamics of coupled harmonic oscillators experiencing Markovian loss. Starting point is the quantum master equation in Liouville space which is generated by a Liouvillian that induces a Lie algebra. We show how this Lie algebra allows to define ladder operators that construct Fock-like eigenstates of the Liouvillian. These eigenstates are used to decompose the time-evolved density matrix and, together with the accompanying eigenvalues, provide insight into the transport properties of the lossy system. Additionally, a Wei-Norman expansion of the generated time evolution can be found by a structure analysis of the algebra. This structure analysis yields a construction principle to implement effective non-Hermitian Hamiltonians in lossy systems.