Heisenberg treatment of pair generation in lossy coupled-cavity systems

We employ a quasimode representation of the adjoint master equation to study nonlinear pair-generation dynamics via spontaneous four-wave mixing in coupled-cavity systems in the presence of scattering loss. Perturbative analytic solutions for signal and idler photon-number operators are derived, which enable us to examine the effect of scattering loss on pair-generation dynamics. In particular, for a simple three-mode system, we show that any loss mismatch for the signal and idler photons increases the probability of finding unpaired photons. While decreasing the loss rates for signal and idler photons increases the pair signal brightness, it also results in higher generation rates for multiple pairs in the system, which can result in lower operational signal-to-noise ratios.

Figure 1

Schematic of four-wave mixing process. Two pump photons convert into signal and idler photons. Top: Spatial extent of the three pump, signal, and idler modes is shown. They all need to overlap over the nonlinear region of the system. Bottom: Energy conservation, ${\omega }_s+{\omega }_i=2{\omega }_p$, selects the generated pair frequencies for a given pump frequency. The diagram shows excitation over a range of frequencies which turns into pair generation over neighboring ranges of frequencies.

Figure 2

Dimensionless two-photon expectation in three cases: dashed (green) curve, when $\overline{\gamma }={\gamma }_p/2$; dotted (red) curve, when $\overline{\gamma }={\gamma }_p$; and solid (blue) curve, when $\overline{\gamma }=2{\gamma }_p$.

Figure 3

Comparison between the two-photon expectation and the only-one-photon expectation when $\overline{\gamma }={\gamma }_p$. The two-photon expectation [dashed (red) curve] is compared to the only-one-photon expectation [solid (blue) curve]. Note that the plotted quantities are both dimensionless.

Figure 4

Photon dynamics when the pump mode is lossless but ${\gamma }_s={\gamma }_i$. The two-photon expectation [dashed (red) curve] is compared to the one-photon-only expectation [solid (blue) curve]. Note that the plotted quantities are both dimensionless.

Figure 5

Dimensionless $N_1\left(t\right)$ as a function of time in three cases: solid (blue) curve, when ${\gamma }_s-{\gamma }_i=0$; dotted (red) curve, when ${\gamma }_s-{\gamma }_i=\overline{\gamma }/2$; and dashed (green) curve, when ${\gamma }_s-{\gamma }_i=\overline{\gamma }$.

Figure 6

(a) Plot of the dimensionless noise figure ${R\left(t\right)\left|G\right|}^2/{\gamma }_p^2$ in three cases: dashed (green) curve, for $\overline{\gamma }=2{\gamma }_p$; dotted (red) curve, for $\overline{\gamma }={\gamma }_p$; and solid (blue) curve, for $\overline{\gamma }={\gamma }_p/2$. (b, c) $p_{11}\mathrm{and} p_{22}$ as a function of time for the same conditions.

Figure 7

(a) Plot of the dimensionless ${R\left(t\right)\left|G\right|}^2/{\overline{\gamma }}^2$ in two cases: dashed (red) curve, when ${\gamma }_p=0$; and solid (blue) curve, when $\overline{\gamma }={\gamma }_p$. (b, c) $p_{11}\mathrm{and} p_{22}$ as a function of time for the same conditions.