Quantum Phase Gate Operation Based on Nonlinear Optics: Full Quantum   Analysis
  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. 
C. Ottaviani, S. Rebic, D. Vitali and P. Tombesi