Dynamics of a two-level trapped ion in a cavity

In the present paper we consider the case of a two-level ion in a cavity in the presence of a single mode field linearly polarized. We suppose that the ion is free to move along the polarization direction and trapped by a harmonic potential along the other two directions. By multiple path integration we derive the density matrix of the system and we study its dynamics. We assume an initial electromagnetic vacuum. This initial condition for the present system, compared with any other initial photonic state, gives new and higher order leading terms with respect to an expansion in powers of the inverse of the volume. Further after such an expansion there appears a first order term that originates from the combined interaction of the two-level system (qubit) with the quantum motion of the ion and the electromagnetic field in the cavity. We notice that the dynamics of the present system is very rich and can be studied exhaustively in the present framework.