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SciPost Physics Codebases

Codebase release 1.1 for MultiAtomLiouvilleEquationGenerator

P. Yanes-Thomas, Rocío Jáuregui, Santiago F. Caballero-Benítez, Daniel Sahagún, A. Kunold

Published Jun 8, 2026• 1 citation
Publisher

Abstract

MulAtoLEG (Multi-Atom Liouville Equation Generator) is a source-available 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.

DOI
10.21468/scipostphyscodeb.73-r1.1

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MulAtoLEG (Multi-Atom Liouville Equation Generator) is a source-available 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.
MulAtoLEG (Multi-Atom Liouville Equation Generator) is a source-available 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.
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.
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 M . ( ) slIn 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.
We investigate the algebraic structure underlying the Lindblad equation for finite-dimensional open quantum systems. By introducing a suitable operator representation of the Liouville superoperator, we show that the dynamics can be formulated in terms of a closed algebra of Hermitian operators that is independent of the particular physical model. This formulation reveals that dissipative dynamics requires a substantially richer algebraic structure than purely unitary evolution, thereby providing a clear characterization of the additional complexity introduced by the Lindbladian. The resulting framework naturally leads to parametrizations of the dynamical map and to differential equations governing its evolution. We further derive recursion relations that enable the efficient construction of the algebra for systems of increasing dimension. Because the algebraic basis is universal, while all model-dependent information enters through a single set of coefficients, the proposed approach significantly reduces the computational cost of constructing the Liouville superoperator compared with direct methods. To facilitate the implementation of the method, we provide a Mathematica notebook containing a one-qubit example that can be systematically extended to an arbitrary number of qubits. The proposed framework therefore provides both a general mathematical description of finite-dimensional Lindblad dynamics and a practical foundation for efficient analytical and numerical implementations.
The evolution of mixed states of a closed quantum system is described by a group of evolution superoperators whose infinitesimal generator (the quantum Liouville superoperator, or Liouvillian) determines the mixed-state counterpart of the Schrödinger equation: the Liouville–von Neumann equation. When the state space of the system is infinite-dimensional, the Liouville superoperator is unbounded whenever the corresponding Hamiltonian is. In this paper, we provide a rigorous, pedagogically-oriented, and self-contained introduction to the quantum Liouville formalism in the presence of unbounded operators. We present and discuss a characterization of the domain of the Liouville superoperator originally due to M. Courbage; starting from that, we develop some simpler characterizations of the domain of the Liouvillian and its square. We also provide, with explicit proofs, some domains of essential selfadjointness (cores) of the Liouvillian.
The Liouville equation governing the evolution of the density matrix for an atomic/molecular system is expressed in terms of a commutator between the density matrix and the Hamiltonian, along with terms that account for decay and redistribution. To find solutions of this equation, it is convenient first to reformulate the Liouville equation by defining a vector corresponding to the elements of the density operator, and determining the corresponding time-evolution matrix. For a system of N energy levels, the size of the evolution matrix is N2 × N2. When N is very large, evaluating the elements of these matrices becomes very cumbersome. We describe a novel algorithm that can produce the evolution matrix in an automated fashion for an arbitrary value of N. As a non-trivial example, we apply this algorithm to a 15-level atomic system used for producing optically controlled polarization rotation. We also point out how such a code can be extended for use in an atomic system with arbitrary number of energy levels.