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.
 Bela Bauer, LD Carr, Hans Gerd Evertz, Adrian Feiguin, J Freire, S Fuchs, Lukas Gamper, Jan Gukelberger, E Gull, Siegfried Guertler, et al. Journal of Statistical Mechanics: Theory and Experiment 2011, P05001 (2011).
 D. Meiser, Y. Jun, D. R. Carlson, and M. J. Holland. Phys. Rev. Lett. 102, 163601 (2009).
 Stefan A Schäffer, Mikkel Tang, Martin R Henriksen, Asbjørn A Jørgensen, Bjarke TR Christensen, and Jan W Thomsen. Phys. Rev. A 101, 013819 (2020).
 Peter Kirton and Jonathan Keeling. Phys. Rev. Lett. 118, 123602 (2017).
 Florentin Reiter, Thanh Long Nguyen, Jonathan P. Home, and Susanne F. Yelin. Phys. Rev. Lett. 125, 233602 (2020).
 Mikkel Tang, Stefan A Schäffer, Asbjørn A Jørgensen, Martin R Henriksen, Bjarke TR Christensen, Jörg H Müller, and Jan W Thomsen. Physical Review Research 3, 033258 (2021).
 Crispin Gardiner, Peter Zoller, and Peter Zoller. ``Quantum noise: a handbook of markovian and non-markovian quantum stochastic methods with applications to quantum optics''. Springer Science & Business Media.
 ``Documentation of QuantumCumulants.jl''. https://qojulia.github.io/QuantumCumulants.jl/stable/ (Accessed: December 20, 2021).
 ``GitHub repository of QuantumCumulants.jl''. https://github.com/qojulia/QuantumCumulants.jl (Accessed: December 20, 2021).
 M. Orszag. ``Quantum optics: including noise reduction, trapped ions, quantum trajectories, and decoherence''. Springer.
 Christoph Hotter, David Plankensteiner, Georgy Kazakov, and Helmut Ritsch, "Continuous multi-step pumping of the optical clock transition in alkaline-earth atoms with minimal perturbation", Optics Express 30 4, 5553 (2022).
 Nicholas Werren, Erik M Gauger, and Peter Kirton, "A quantum model of lasing without inversion", New Journal of Physics 24 9, 093027 (2022).
 Gage W. Harmon, Jarrod T. Reilly, Murray J. Holland, and Simon B. Jäger, "Mean-field Floquet theory for a three-level cold-atom laser", Physical Review A 106 1, 013706 (2022).
 Yuan Zhang, Qilong Wu, Shi-Lei Su, Qing Lou, Chongxin Shan, and Klaus Mølmer, "Cavity Quantum Electrodynamics Effects with Nitrogen Vacancy Center Spins Coupled to Room Temperature Microwave Resonators", Physical Review Letters 128 25, 253601 (2022).
 Qilong Wu, Yuan Zhang, Xigui Yang, Shi-Lei Su, Chongxin Shan, and Klaus Mølmer, "A superradiant maser with nitrogen-vacancy center spins", Science China Physics, Mechanics, and Astronomy 65 1, 217311 (2022).
 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.
 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).
 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).
 Kasper Kusmierek, Sahand Mahmoodian, Martin Cordier, Jakob Hinney, Arno Rauschenbeutel, Max Schemmer, Philipp Schneeweiss, Jürgen Volz, and Klemens Hammerer, "Higher-order mean-field theory of chiral waveguide QED", arXiv:2207.10439.
 Alessandro Cheli, "Automated Code Optimization with E-Graphs", arXiv:2112.14714.
 Anna Bychek, Christoph Hotter, David Plankensteiner, and Helmut Ritsch, "Superradiant lasing in inhomogeneously broadened ensembles with spatially varying coupling", arXiv:2105.11023.
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