QuantumCumulants.jl: A Julia framework for generalized mean-field equations in open quantum systems

David Plankensteiner, Christoph Hotter, and Helmut Ritsch

Institut für Theoretische Physik, Universität Innsbruck, Technikerstr. 21a, A-6020 Innsbruck, Austria

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Abstract

A full quantum mechanical treatment of open quantum systems via a Master equation is often limited by the size of the underlying Hilbert space. As an alternative, the dynamics can also be formulated in terms of systems of coupled differential equations for operators in the Heisenberg picture. This typically leads to an infinite hierarchy of equations for products of operators. A well-established approach to truncate this infinite set at the level of expectation values is to neglect quantum correlations of high order. This is systematically realized with a so-called cumulant expansion, which decomposes expectation values of operator products into products of a given lower order, leading to a closed set of equations. Here we present an open-source framework that fully automizes this approach: first, the equations of motion of operators up to a desired order are derived symbolically using predefined canonical commutation relations. Next, the resulting equations for the expectation values are expanded employing the cumulant expansion approach, where moments up to a chosen order specified by the user are included. Finally, a numerical solution can be directly obtained from the symbolic equations. After reviewing the theory we present the framework and showcase its usefulness in a few example problems.

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[4] Yuan Zhang, Qilong Wu, Hao Wu, Xun Yang, Shi-Lei Su, Chongxin Shan, and Klaus Mølmer, "Cavity Quantum Electrodynamics Effects of Optically Cooled Nitrogen-Vacancy Centers Coupled to a High Frequency Microwave Resonator", arXiv:2203.04102.

[5] G. Ferioli, A. Glicenstein, F. Robicheaux, R. T. Sutherland, A. Browaeys, and I. Ferrier-Barbut, "Laser-Driven Superradiant Ensembles of Two-Level Atoms near Dicke Regime", Physical Review Letters 127 24, 243602 (2021).

[6] Alessandro Cheli, "Automated Code Optimization with E-Graphs", arXiv:2112.14714.

[7] F. Robicheaux and Deepak A. Suresh, "Beyond lowest order mean-field theory for light interacting with atom arrays", Physical Review A 104 2, 023702 (2021).

[8] Anna Bychek, Christoph Hotter, David Plankensteiner, and Helmut Ritsch, "Superradiant lasing in inhomogeneously broadened ensembles with spatially varying coupling", arXiv:2105.11023.

The above citations are from Crossref's cited-by service (last updated successfully 2022-05-20 07:10:22) and SAO/NASA ADS (last updated successfully 2022-05-20 07:10:23). The list may be incomplete as not all publishers provide suitable and complete citation data.