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Electromagnetically-induced transparency (EIT) has been proposed as a way togreatly enhance cross-phase modulation, with the possibility of leading tofew-photon-level optical nonlinearities. This enhancement grows as thetransparency window width, \Delta_EIT, is narrowed. Decreasing \Delta_EIT,however, increases the response time of the effect, suggesting that for pulsesof a given duration, there could be a fundamental limit to the strength of thenonlinearity. We show that in the regimes of most practical interest - narrowEIT windows perturbed by short signal pulses- the enhancement offered by EIT isnot only in the magnitude of the nonlinear phase shift but in fact also in itsincreased duration. That is, for the case of signal pulses much shorter(temporally) than the inverse EIT bandwidth, the narrow window serves toprolong the effect of the passing signal pulse, leading to an integrated phaseshift that grows linearly with \Delta_EIT even though the peak phase shift maysaturate; the continued growth of the integrated phase shift improves thedetectability of the phase shift, in principle without bound. For manypurposes, it is this detectability which is of interest, more than the absolutemagnitude of the peak phase shift. We present analytical expressions based on alinear time-invariant model that accounts for the temporal behavior of thecross-phase modulation for several parameter ranges of interest. We concludethat in order to optimize the detectability of the EIT-based cross-phase shift,one should use the narrowest possible EIT window, and a signal pulse that is asbroadband as the excited state linewidth and detuned by half a linewidth.

Short-pulse cross-phase modulation in an electromagnetically-induced-transparency medium