An Algebraic Approach to Solving $n$-Level Open Quantum System Problems with Dynamical Invariants

A quantum system, which interacts with its environment, is called open. Its dynamics are obtained by solving a master equation. This work explores the derivation of a master equation with expanded validity regime using a dynamical (or Lewis-Riesenfeld) invariant. In general, there may exist many dynamical invariants, but finding useful nontrivial dynamical invariants gets significantly more difficult as the size of the Hilbert space increases. A powerful approach, which is explored in our work, applies algebraic considerations to the Hamiltonian to classify its symmetry algebra $\mathfrak{h}\hookrightarrow\mathfrak{su}(n)$, the generators of which can be utilized to construct an invariant. We demonstrate our progress on applying this approach to a plethora of $n$-level systems, where $n\geq 3$, to construct their dynamical invariants. We also discuss how the invariants and their eigenstates facilitate the derivation of a master equation.