Vacuum induced Stark shifts for quantum logic using a collective system   in a high quality dispersive cavity
  A collective system of atoms in a high quality cavity can be described by a nonlinear interaction which arises due to the Lamb shift of the energy levels due to the cavity vacuum [Agarwal et al., Phys. Rev. A 56, 2249 (1997)]. We show how this collective interaction can be used to perform quantum logic. In particular we produce schemes to realize CNOT gates not only for two-qubit but also for three-qubit systems. We also discuss realizations of Toffoli gates. Our effective Hamiltonian is also realized in other systems such as trapped ions or magnetic molecules. 
A. Gabris and G. S. Agarwal