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

TL;DR

MulAtoLEG addresses the need for an open, symbolic tool to generate exact Liouville and adjoint Liouville equations for open quantum systems consisting of many atoms with multiple levels. It builds on the adjoint master equation of Lehmberg and the density-matrix master equation reformulation by Genes to cover arbitrary numbers of atoms and levels, with support for the dressed-state basis. The package provides a modular workflow to construct the Hamiltonian and Lindbladian from transition lists, perform rotating-frame simplifications, solve the resulting linear Liouville equations, and obtain observables, including far-field emission. This enables efficient, exact treatment of cooperative open-system dynamics in alkali-atom ensembles, constrained only by computational resources.

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.

Figures (4)

• Figure 1: Schematics of the MulAtoLEG package.
• Figure 2: Scheme of a single two-level atom. This system is characterized by the ground state $\left\vert 0 \right\rangle$ and the excited state $\left\vert 1 \right\rangle$. These two states are coupled to a coherent source characterized by the Rabi parameter $R_1$ and a decay channel characterized by the decay rate $\gamma_1$.
• Figure 3: Scheme of two two-level atoms. It is characterized by ground states $\lvert 0 \rangle_1$ and $\lvert 0 \rangle_2$, and excited states $\lvert 1 \rangle_1$ and $\lvert 1 \rangle_2$, corresponding to atoms $1$ and $2$. The quantum levels of each atom are coupled to a coherent source through the Rabi parameter $R_1$, which is common to both atoms. These two levels are also connected to decay channels characterized by the decay rate $\gamma_1$. In addition, the quantum levels of one atom are coupled to those of the other via dipole–dipole interactions, mainly determined by $\Omega_1$. Furthermore, the quantum levels are coupled through a collective decay channel with decay rate $\gamma_{1,2}$.
• Figure 4: Scheme of five two-level atoms. It is characterized by ground states $\lvert 0 \rangle_\alpha$ for $\alpha = 1,2,3,4,5$, and excited states $\lvert 1 \rangle_\alpha$ for $\alpha = 1,2,3,4,5$, corresponding to atoms $1$ through $5$. The quantum levels of each atom are coupled to a coherent source through the Rabi parameter $R_1$, which is common to all five atoms. These levels are also connected to decay channels characterized by the decay rate $\gamma_1$. In addition, the quantum levels of one atom are coupled to those of its neighbors via dipole–dipole interactions, mainly determined by $\Omega_1$. Moreover, neighboring atoms are coupled through a collective decay channel with decay rate $\gamma_{1,2}$.