Effects of noninstantaneous nonlinear processes on photon-pair generation by spontaneous four-wave mixing

We present a general model, based on a Hamiltonian approach, for the joint quantum state of photon pairs generated through pulsed spontaneous four-wave mixing, including nonlinear phase modulation and a finite material response time. For the case of a silica fiber, it is found that the pair-production rate depends weakly on the waveguide temperature, due to higher-order Raman scattering events, and more strongly on pump-pair frequency detuning. From the analytical model, a numerical scheme is derived, based on the well-known split-step method. This scheme allows computation of joint states where nontrivial effects are included, such as group-velocity dispersion and Raman scattering. In this work, the numerical model is used to study the impact of the noninstantaneous response on the prefiltering purity of heralded single photons. We find that for pump pulses shorter than 1 ps, a significant detuning-dependent change in quantum-mechanical purity may be observed in silica. This shows that Raman scattering not only introduces noise, but can also drastically change the spectral correlations in photon pairs when pumped with short pulses.

Figure 1

Feynman diagrams where wavy lines represent photons and solid lines represent phonon excitations. (a) Stokes scattering where a photon is scattered into a lower-energy photon while creating a phonon. (b) Anti-Stokes scattering where a photon is scattered into a higher-energy photon while absorbing a phonon. (c) Higher-order diagram from the combination of (a) and (b) where a phonon mediates the scattering of two incoming photons into two outgoing photons with one gaining and one losing energy corresponding to the phonon mode.

Figure 2

Generation probability of photon pairs, normalized to the case of $f_{\mathrm{R}}=0$, as a function of linear frequency detuning. The wavelength axis is shown for a 1550-nm pump. The dashed line marks $R_{\mathrm{pair}}=R_0$.

Figure 3

Value of $\mathcal{C}(\omega ,T)$ as a function of linear frequency detuning for three different temperatures with a pump positioned at $\omega =0$ or $\lambda =1550$ nm. The dashed line represents the Raman susceptibility.

Figure 4

Coincidence-to-accidental ratio of a pair-source based on degenerate SFWM as a function of linear frequency detuning. The wavelength axis assumes a $1550-\mathrm{nm}$ pump wavelength.

Figure 5

Application of steps in the developed $O\left(h^3\right)$ split-step scheme with color indicating the type of effect. The dashed arrows indicate the order in which the steps should be applied. The numbers indicate $①$ the initialization part, $②$ the repeating part, and $③$ the final part.

Figure 6

Absolute value of the joint temporal amplitude with (a) a pure electronic response and (b) a pure phononic response and the absolute value of the joint spectral amplitude with (c) a pure electronic response and (d) a pure phononic response.

Figure 7

Purity of the heralded single photon as a function of the linear frequency detuning for different pump durations. The wavelength axis assumes a pump wavelength of $1550\mathrm{nm}$. The inset shows the real part of the Raman susceptibility for reference.