Pulse-shape effects on photon-photon interactions in nonlinear optical quantum gates

Ideally, strong nonlinearities could be used to implement quantum gates for photonic qubits by well-controlled two-photon interactions. However, the dependence of the nonlinear interaction on frequency and time makes it difficult to preserve a coherent pulse shape that could justify a single-mode model for the time-frequency degree of freedom of the photons. In this paper, we analyze the problem of temporal multimode effects by considering the pulse shape of the average output field obtained from a coherent input pulse. It is shown that a significant part of the two-photon state transformation can be derived from this semiclassical description of the optical nonlinearity. The effect of a nonlinear system on a two-photon state can then be determined from the density-matrix dynamics of the coherently driven system using input-output theory. As an example, the resonant nonlinearity of a single two-level atom is characterized. The results indicate that the most efficient nonlinear effect may not be the widely studied single-mode phase shift, but the transfer of one of the photons to an orthogonal mode distinguished by its temporal and spectral properties.