Deterministic quantum controlled-PHASE gates based on non-Markovian environments

We study the realization of the quantum controlled-PHASE gate in an atom-cavity system beyond the Markovian approximation. The general description of the dynamics for the atom-cavity system without any approximation is presented. When the spectral density of the reservoir has the Lorentz form, by making use of the memory backflow from the reservoir, we can always construct the deterministic quantum controlled-PHASE gate between a photon and an atom, no matter the atomcavity coupling strength is weak or strong. While, the phase shift in the output pulse hinders the implementation of quantum controlled-PHASE gates in the sub-Ohmic, Ohmic or super-Ohmic reservoirs.