  We theoretically investigate the implementation of a quantum phase gate in a
system constituted by a single atom inside an optical cavity, based on the
electromagnetically induced transparency effect. Firstly we show that a probe
pulse can experience a $\pi$ phase shift due to the presence or absence of a
classical control field. Considering the interplay of the cavity-EIT effect and
the quantum memory process, we demonstrated a controlled phase gate between two
single photons. To this end, firstly one needs to store a (control) photon in
the ground atomic states. In the following, a second (target) photon must
impinge on the atom-cavity system. Depending on the atomic state, this second
photon will be either transmitted or reflected, acquiring different phase
shifts. This protocol can then be easily extended to multiphoton systems, i.e.,
keeping the control photon stored, it may induce phase shifts in several single
photons, thus enabling the generation of multipartite entangled states. We
explore the relevant parameter space in the atom-cavity system that allows the
implementation of quantum phase gates using the recent technologies. In
particular we have found a lower bound for the cooperativity of the atom-cavity
system which enables the implementation of phase shift on single photons. The
induced shift on the phase of a photonic qubit and the controlled phase gate
between single photons, combined with optical devices, enable to perform
universal quantum computation.
