Algebraic solution of the Lindblad equation for a collection of multilevel systems coupled to independent environments

TL;DR

This paper develops an algebraic method to solve the Lindblad equation for multiple multilevel quantum systems, leveraging symmetry and Lie algebra techniques to simplify calculations and find analytic solutions.

Contribution

It introduces a basis and superoperators in the permutation-symmetric subspace, enabling analytic solutions and efficient simulations of multilevel open quantum systems.

Findings

Analytic solution for three-level atoms coupled to radiation baths.

Reduced computational complexity for simulating large collections.

Basis and superoperators applicable to various multilevel systems.

Abstract

We consider the Lindblad equation for a collection of multilevel systems coupled to independent environments. The equation is symmetric under the exchange of the labels associated with each system and thus the open-system dynamics takes place in the permutation-symmetric subspace of the operator space. The dimension of this space grows polynomially with the number of systems. We construct a basis of this space and a set of superoperators whose action on this basis is easily specified. For a given number of levels, $M$, these superoperators are written in terms of a bosonic realization of the generators of the Lie algebra $\sln{M^2}$. In some cases, these results enable finding an analytic solution of the master equation using known Lie-algebraic methods. To demonstrate this, we obtain an analytic expression for the state operator of a collection of three-level atoms coupled to independent radiation baths. When analytic solutions are difficult to find, the basis and the superoperators can be used to considerably reduce the computational resources required for simulations.