  In this paper we study controllability of a $d$-level atom interacting with
the electromagnetic field in a cavity. The system is modelled by an ordered
graph $\Gamma$. The vertices of $\Gamma$ describe the energy levels and the
edges allowed transitions. To each edge of $\Gamma$ we associate a harmonic
oscillator representing one mode of the electromagnetic field. The dynamics of
the system (drift) is given by a natural generalization of the Jaynes-Cummings
Hamiltonian. If we add in addition sufficient control over the atom, the
overall system (atom and em-field) becomes strongly controllable, i.e. each
unitary on the system Hilbert space can be approximated with arbitrary
precision in the strong topology by control unitaries. A key role in the proof
is played by a topological *-algebra which is (roughly speaking) a
representation of the path algebra of $\Gamma$. It contains crucial structural
information about the control problem, and is therefore an important tool for
the implementation of control tasks like preparing a particular state from the
ground state. This is demonstrated by a detailed discussion of different
versions of three-level systems.
