Morse theory classifies open quantum system observables

In analogy to solid state phases, different steady states of multi-photon quantum optical systems are distinguished by the behavior of observables such as populations and extinction coefficients. We describe a global analytic approach for determining the topology of the configuration space of the associated class of open quantum systems and use Morse theory to predict the changes in the observables when moving through different steady states. We present details of the transitions between, for example, Electromagnetically Induced Transparency (EIT) and Absorption (EIA), saturation, optical pumping, etc. and the computation of their exponents.