  We present a full quantum treatment of a five-level atomic system coupled to
two quantum and two classical light fields. The two quantum fields undergo a
cross-phase modulation induced by electro-magnetically induced transparency.
The performance of this configuration as a two-qubit quantum phase gate for
travelling single-photons is examined. A trade-off between the size of the
conditional phase shift and the fidelity of the gate is found. Nonetheless, a
satisfactory gate performance is still found to be possible in the transient
regime, corresponding to a fast gate operation.
