On the controllability of some quantum electro-dynamical systems

A result (see, e.g., [3][complement BIII, page 217]) of quantum electrodynamics (based on Glauber theorem) says that classical currents and sources only generate classical light (quasi-classical states of the field). This paper provides a control theoretic interpretation of this result when the classical currents and sources are considered as control inputs: the dynamics of the quantified electrodynamics field is not controllable; the controllable part is contained into the classical dynamics. This result can be seen as the infinite dimensional analogue of the following fact (see [8]): the controllable part of the quantum harmonic oscillator corresponds to the classical dynamics of the average <q̈>=-<q>+u (u is the control input). Thus controllability can only be achieved when the field dynamics is coupled to some localized quantum dynamics. We describe here two typical models of such controlled systems: the Jaynes-Cummings model [6] (one field mode coupled to a two-level atom) and trapped ions vibrational model. Both systems correspond to experiments conducted by physicists. We recall their main approximations and validity domains. We also provide their associated formulation in terms of PDE with control.