Effect of loss on photon-pair generation in nonlinear waveguide arrays

We describe theoretically the process of spontaneous parametric down-conversion in quadratic nonlinear waveguide arrays in the presence of linear loss. We derive a set of discrete Schrödinger-type equations for the biphoton wave function and the wave function of one photon when the other photon in a pair is lost. We demonstrate effects arising from loss-affected interference between the generated photon pairs and show that nonlinear waveguide arrays can serve as a robust loss-tolerant integrated platform for the generation of entangled photon states with nonclassical spatial correlations.

Figure 1

Scheme of the Hamiltonian describing the photon-pair propagation involving SPDC and losses in a single waveguide. Losses are represented by beam splitters [37, 39].

Figure 2

Normalized number of photon pairs $I_s^{\left(0\right)}$ generated through SPDC in a single waveguide vs the phase mismatch $\Delta {\beta }^{\left(0\right)}$ for $z=5,A=1$, and different losses: (a) ${\gamma }_p={\gamma }_s={\gamma }_i=0$, (b) ${\gamma }_p=0,{\gamma }_s={\gamma }_i=0.5$, and (c) ${\gamma }_s={\gamma }_i=0.5,{\gamma }_p={\gamma }_s+{\gamma }_i=1$.

Figure 3

(left) Total signal intensity $I_s\left(z\right)$ and (right) the ratio of the intensity contribution when neither photon is absorbed and the full intensity $I_s^{\left(0\right)}\left(z\right)/I_s\left(z\right)$ vs the signal and idler losses in a single waveguide for different values of phase mismatch: (a), (b) $\Delta {\beta }^{\left(0\right)}=0$, (c), (d) $\Delta {\beta }^{\left(0\right)}=3$, and (e), (f) $\Delta {\beta }^{\left(0\right)}=6$. Parameters are ${\gamma }_p=0,z=5,A=1$.

Figure 4

Ratio of the intensity contribution when neither photons is absorbed and the full intensity $I_s^{\left(0\right)}\left(z\right)/I_s\left(z\right)$ vs the signal and idler loss $\gamma$ $\left({\gamma }_s={\gamma }_i=\gamma ,{\gamma }_p=0\right)$ in a single waveguide. Parameters are $A=1,z=5$, and $\Delta {\beta }^{\left(0\right)}=\{0,3,6\}$, as indicated by the legend.

Figure 5

Total signal mode intensity $I_s$ vs the propagation distance in a single waveguide for different signal and idler losses ${\gamma }_s={\gamma }_i=\gamma =\{0,0.3,0.6\}$. Parameters are $A=1,\Delta {\beta }^{\left(0\right)}=0,{\gamma }_p=0$.

Figure 6

Scheme of the experimental setup designed to measure spectral and spatial distributions of the nonlinear WGA output photon-pair intensity. The pump beam generates photon pairs that suffer losses and couple to the neighboring waveguides. (a) The output intensity distribution can be characterized using a prism and a camera. Spectral filtering can be used to choose only a signal channel to measure the signal intensity $I_s$. (b) The photon-pair correlations can be characterized by measuring the coincidences from the two single-photon detectors.

Figure 7

(left) Photon-pair correlations in $k$ space, (middle) real-space correlations, and (right) Schmidt coefficients for different signal and idler losses, ${\gamma }_s={\gamma }_i=\gamma /2$: (a)–(c) $\gamma =0$, (d)–(f) $\gamma =0.2$, (g)–(i) $\gamma =0.6$. The pump is coupled to the central waveguide, $A\left(0\right)=0$. Parameters are $z=5,C_s=C_i=1,{\gamma }_p=0,\Delta {\beta }^{\left(0\right)}=0$.

Figure 8

(left) Photon-pair correlations in $k$ space, (middle) real-space correlations, and (right) Schmidt coefficients [see Eq. (41)] for different signal and idler losses, ${\gamma }_s={\gamma }_i=\gamma /2$: (a)–(c) $\gamma =0$, (d)–(f) $\gamma =0.2$, and (g)–(i) $\gamma =0.6$. The pump is coupled in phase to two neighboring waveguides in the center, $A\left(0\right)=A\left(1\right)=1$. Parameters are $z=5,C_s=C_i=1,{\gamma }_p=0,\Delta {\beta }^{\left(0\right)}=0$.

Figure 9

(left) Schmidt number, (middle) the signal mode full intensity in real space, and (right) the ratio of signal photons coupled with idler photons to all signal photons in real space vs the signal and idler loss $\gamma$ $\left({\gamma }_i={\gamma }_s=\gamma /2\right)$ for different pump profiles: (a)–(c) pump coupled to the central waveguide, $A\left(0\right)=1$, and (d)–(f) pump coupled in phase to two neighboring waveguides, $A\left(0\right)=A\left(1\right)=1$. Parameters are $z=5,C_s=C_i=1,{\gamma }_p=0,\Delta {\beta }^{\left(0\right)}=0$.

Figure 10

(left) Schmidt number, (middle) the signal mode full intensity in real space, and (right) the ratio of signal photons coupled with idler photons to all signal photons in real space vs the idler loss ${\gamma }_i$ for different pump profiles: (a)–(c) pump coupled to the central waveguide, $A\left(0\right)=1$, and (d)–(f) pump coupled in phase to two neighboring waveguides, $A\left(0\right)=A\left(1\right)=1$. Parameters are $z=5,C_s=C_i=1,{\gamma }_p={\gamma }_s=0,\Delta {\beta }^{\left(0\right)}=0$.