MultiAtomLiouvilleEquationGenerator: A Mathematica package for Liouville superoperators and master equations of multilevel atomic systems

TL;DR

MulAtoLEG is an open-source Mathematica package that automates the generation of Liouville superoperators and master equations for multilevel atomic systems, aiding complex quantum system analysis.

Contribution

It extends existing methods to multilevel atomic systems and provides a computational tool for generating exact Liouville equations efficiently.

Findings

Enables generation of equations for complex atomic transition configurations

Supports construction of differential equations in dressed-state basis

Leverages Mathematica's capabilities for computational efficiency

Abstract

MulAtoLEG (Multi-Atom Liouville Equation Generator) is an open-source Mathematica package for generating Liouville superoperators and Liouville equations, specialized for multilevel atomic systems comprising an arbitrary number of atoms. This scheme is based on an extension to multilevel atomic systems, originally developed by Lehmberg [R. H. Lehmberg, Phys. Rev. A 2, 883 (1970)] as an adjoint master equation for ensembles of two-level emitters and later reformulated by Genes [M. Reitz, C. Sommer and C. Genes, PRX Quantum 3, 010201 (2022)] as a master equation. The package facilitates the generation of equations for complex transition configurations in alkali atoms. Although primarily designed for atomic systems, it can also generate the master and adjoint master equations for general Hamiltonians and Lindbladians. In addition, it includes functionalities to construct the differential equations in the dressed-state basis, where, in many cases, the non-unitary evolution operator can be determined explicitly. To maximize computational efficiency, the package leverages Mathematica's vectorization and sparse linear algebra capabilities. Since MulAtoLEG produces exact equations without approximations, the feasible system size is naturally limited by the available computational resources.