Abstract
In a recent publication we showed that permutation symmetry reduces the numerical complexity of Lindblad quantum master equations for identical multi-level systems from exponential to polynomial scaling. This is important for open system dynamics including realistic system bath interactions and dephasing in, for instance, the Dicke model, multi-Λ system setups etc. Here we present an object-oriented C++ library that allows to setup and solve arbitrary quantum optical Lindblad master equations, especially those that are permutationally symmetric in the multi-level systems. PsiQuaSP (Permutation symmetry for identical Quantum Systems Package) uses the PETSc package for sparse linear algebra methods and differential equations as basis. The aim of PsiQuaSP is to provide flexible, storage efficient and scalable code while being as user friendly as possible. It is easily applied to many quantum optical or quantum information systems with more than one multi-level system. We first review the basics of the permutation symmetry for multi-level systems in quantum master equations. The application of PsiQuaSP to quantum dynamical problems is illustrated with several typical, simple examples of open quantum optical systems.
Highlights
In a recent publication we showed that permutation symmetry reduces the numerical complexity of Lindblad quantum master equations for identical multi-level systems from exponential to polynomial scaling
We have introduced a library that enables the setup of master equations for identical multi-level systems
The library provides ready-made setup functions for density matrices as well as Liouville operators. The design of these functions is centered around the sketch representation of the Liouville operators or master equation introduced in ref.[1]
Summary
As stated in the previous section we target Lindblad master equations of collections of identical, indistinguishable multi-level systems. One example for a master equation with permutational symmetry is the open Dicke model, i.e. a set of identical two-level systems coupled to a bosonic mode. Exploiting the permutational symmetry of Lindblad equations results in a polynomial complexity in the number of multi-level systems instead of an exponential complexity This is equivalent to projecting the master equation onto a subspace of special symmetrized Liouville space states. This approach is only valid if the master equation obeys the permutation symmetry These symmetrized basis states have been introduced and discussed for two-level systems by various authors[1,3,10,11,13,14,15,16,17,19,22], notably Hartmann called them generalized Dicke states[16]. Omitting the normalization makes the method numerically more stable, see ref.[1]
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