Algebraic solution of the Lindblad equation for a collection of multilevel systems coupled to independent environments
Marduk Bolaños, Pablo Barberis-Blostein · Journal of Physics A Mathematical and Theoretical · 2015
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.