Christoph Hotter

QuantumCumulants.jl is a package for the symbolic derivation of generalized mean-field equations for quantum mechanical operators in open quantum systems. The equations are derived using fundamental commutation relations of operators. When averaging these equations they can be automatically expanded in terms of cumulants to an arbitrary order. This results in a closed set of symbolic differential equations, which can also be solved numerically.

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.

Christoph Hotter

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QuantumCumulants.jl

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