Select Topics in Open Quantum Systems

The design of realistic quantum technologies relies intricately on the understanding of open quantum systems. Such open systems are often studied as an input–output theory, where time is often not explicitly parameterized. Such maps have a rich structure that has been elucidated by several authors over the years. In contrast to this, the master equation approach is a dynamical description where the infinitesimal evolution of the quantum system is studied. These two descriptions are related to each other when the underlying maps are parametrized in time or equivalently when the system dynamics is integrated for finite time. In this overview, we will briefly discuss some established results in this field alongside commenting on some recent results relating to deriving the transient and steady state dynamics of open quantum systems. We discuss structure of CP maps, review canonical forms of \(\mathcal {A}\) and \(\mathcal {B}\) maps and highlight their relationship to Lindblad equations. We review properties of Liouville superoperators and highlight some stable numerical methods to find steady states. We conclude the overview with a summary of results relating Zeno dynamics and the open systems approach to driven system engineering.