Controlling a d-level atom in a cavity

Abstract

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.