Cross-Kerr nonlinearity between continuous-mode coherent states and single photons

Weak cross-Kerr nonlinearities between single photons and coherent states are the basis for many applications in quantum-information processing. These nonlinearities have so far mainly been discussed in terms of highly idealized single-mode models. We develop a general theory of the interaction between continuous-mode photonic pulses and apply it to the case of a single photon interacting with a coherent state. We quantitatively study the validity of the usual single-mode approximation using the concepts of fidelity and conditional phase. We show that high fidelities, nonzero conditional phases, and high photon numbers are compatible, under conditions where the pulses fully pass through each other and where unwanted transverse-mode effects are suppressed.

Figure 1

Fidelity function for the XPM between two pulses moving with a relative velocity $v$. Here the pulses are assumed to have identical Gaussian profiles of width $\sigma$. The initial position of the single photon is $5\sigma$ away from the pulse in coherent state. The parameters for the example are $\mathrm{n̄}={10}^3$, $\chi /v=0.01$, and $\mathit{vt}=10\sigma$. The fidelity function assumes the maximum value of 1 at $\theta =\chi /v$.

Figure 2

Fidelity function for the XPM between two co-propagating pulses in the identical Gaussian profile. The parameters for the example are $\mathrm{n̄}={10}^3$, $\chi t=0.01$. The $\delta$ function $\delta (z-z^{\prime })$ in Eq. (40) is simulated with a sharp Gaussian function $\frac{1}{2\sqrt{\pi \epsilon }}\exp (-\frac{(z-z^{\prime }){}^2}{4\epsilon })$, where $\epsilon ={10}^{-20}$. The fidelity function assumes the maximum at $\theta =0$.