Controlling a d-level atom in a cavity

In this talk we discuss quantum control theory for a d-level atom in a cavity. The atom is described by a Graph Γ with energy levels as vertices and edges e as allowed transitions. For each such e the atom interacts (via a Jaynes-Cummings like interaction term) with a different mode of the cavity. We consider controllability of the overall system (i.e. atom and cavity) under the assumption that all atom-cavity interactions can be switched on and off individually and that the atom itself is fully controllable. Our main tools are symmetry based arguments recently introduced for the discussion of the two-level case. The basic idea is to divide the control Hamiltonians into two sets. One which is invariant under the action of an Abelian symmetry group G and a second set which breaks this symmetry. We will discuss how the group G and its action are related to the graph Γ and its fundamental groupoid, and how these structure can be used to prove full controllability - at least if Γ is acyclic. For Graphs containing cycles the situation is more difficult and the universal covering graph has to be used. We demonstrate this, using the fully connected graph on three vertices as an example.