Effect of second-order coupling on photon-pair statistics in waveguide structures

We investigate a quasi-one-dimensional periodic array of coupled waveguides, with one extended and one bound dimension, incorporating both first- and second-order coupling. We study the evolution of optical fields in this system, and measure quantum correlations when path-entangled photon pairs are launched into them. We observe a surprisingly large and nontrivial effect of second-order coupling on these correlations—while quantum correlations are symmetric when only first-order coupling is present, the introduction of next-nearest-neighbor coupling often breaks the symmetry to reflections.

Figure 1

Schematic of the double-row waveguide array. The fabricated array extends over 25 waveguides horizontally, only part of which contain signal and are measured. (a) The site numbering for the measured waveguides begins in the top row, and continues on the bottom one. $C_H$, $C_V$, and $C_D$ are the horizontal, vertical, and diagonal coupling constants, appropriately. (b) Input sites for the measurements with classical light. Two laser beams are coupled to vertically neighboring waveguides, with a controllable phase between them: $E_{10}=\frac{1}{\sqrt{2}},E_{29}=e^{i\varphi }\frac{1}{\sqrt{2}}$ (for these experiments, measurements were taken of $M=19$ waveguides in each row). (c) Input sites for the measurements with quantum light. Two photons are inserted in a superposition of being either in one waveguide on the top row, or its horizontal neighbor, with a controllable phase between the possibilities: $\frac{1}{2}\left(a_9^{{†}^2}+e^{i\varphi }{a}_{10}^{{†}^2}\right)|0\rangle$ (for these experiments, measurements were taken of $M=18$ in each row). (d) Image of the glass slide facet, showing a cross section of a ladder array. This image has been processed to improve contrast, reduce the effect of noise, and correct for uneven lighting.

Figure 2

Simulations of intensity propagation of classical light in a double-row lattice. Following the numbering convention, the lower site numbers represent the upper row, and the higher ones represent the lower row, separated by the dashed, orange line. Note the color scales, which are saturated at higher values to allow details to be visible at later propagation distances. (a) Intensity along propagation when a single site on the upper row is initially excited, in a lattice with diagonal coupling turned off. The dashed white line is located such that the total power in the upper and lower rows are equal ($Z=\frac{n\pi }{4C_V}$, with $n$ an odd integer. In this case, $n=5$, as $C_V$ in these simulations is lower than the one in the experiment in order to maintain clarity in the graph). All measurements were taken at such a location. (b) Same as (a), with diagonal coupling turned on. Changes are evident, but are relatively small. (c),(d) Intensity along propagation with diagonal coupling turned on, and the $y$-symmetric ($E_{10}=\frac{1}{\sqrt{2}},E_{29}=\frac{1}{\sqrt{2}}$) (c) and $y$-antisymmetric ($E_{10}=\frac{1}{\sqrt{2}},E_{29}=\frac{-1}{\sqrt{2}}$) (d) modes excited. In either case, both rows experience the same discrete diffraction pattern, with a different effective coupling constant for either excitation mode.

Figure 3

Output intensity distribution at an equal-power location (the white dashed line in Fig. 2). (a) Simulation of the intensity distribution at the output when site 10 is excited, at an equal-power location. When diagonal coupling is turned on (red, dashed line) and off (blue, solid line) power distribution changes, but not significantly. (b) Measured intensity distribution at the output, summed over both rows, when the $y$-symmetric (solid, orange line) and $y$-antisymmetric (dashed, purple line) modes are excited. (c)–(f) CCD Images [(c) and (d)] and simulated images [(e) and (f)] of the power distribution at the output of the array, when the $y$-symmetric [(c) and (e)] and $y$-antisymmetric [(d) and (f)] modes are excited. The circles mark the input waveguides.

Figure 4

Experimental setup. LD - laser diode, HWP - half-wave plate, PBS - polarizing beam splitter, QWP - quarter-wave plate, M - mirror, RR - retroreflector, PZT - piezoelectric transducer, PPKTP - nonlinear crystal, DP - dove prism, DM - dichroic mirror, CCD - camera, BS - beam splitter, IF - interference filters, D1/2 - Fibre input facets, CCU - coincidence counting unit. The elements in green frames (or marked by green arrows) are those inserted (or replaced) for the experiment using classical light.

Figure 5

Correlation matrices ${\mathrm{\Gamma }}_{q,r}$ for the double row lattice, for an input state of $\frac{1}{2}\left(a_9^{{†}^2}+e^{i\varphi }{a}_{10}^{{†}^2}\right)|0\rangle$. The orange lines separate between the correlations of different rows, i.e., in each graph, the bottom-left quadrant represents the correlations inside the upper row, the upper-left/bottom-right represents the correlations between the different rows, and the top-right corner represents the correlations inside the bottom row. (a)–(d) Simulated correlation matrices when diagonal coupling is turned off, for different phases $\varphi$. In this case, all quadrants display the same correlations, which are identical to those observed in a 1D lattice with similar input and only first-order coupling [19, 20]. (e)–(h) Same as (a)–(d), with diagonal coupling is turned on. While the correlations for the cases of $\varphi =0,\pi$ are relatively similar to those without diagonal coupling (same bunching and antibunching behavior), those for the cases of $\varphi =\frac{\pi }{2},\frac{3\pi }{2}$ show stark differences—the location of each photon is highly dependent on the location of the other, in a nontrivial manner. These dependencies are determined not only by their location within each row, but which row the are both in. (i)–(l) Experimentally measured, background-reduced correlation matrices, for a lattice with diagonal coupling as in (e)–(h). Although the results are quite noisy, the salient features are easily discerned. Contrast in these measurements may be reduced due to potential undesired overlap between the input beams and waveguides in the lower row.

Figure 6

Simulated correlation matrices ${\mathrm{\Gamma }}_{q,r}$ for a 1D lattice, for an input state of $\frac{1}{2}\left(a_9^{{†}^2}+e^{i\varphi }{a}_{10}^{{†}^2}\right)|0\rangle$ (identical to that used in the experiment). [(a) and (b)] A 1D lattice with only nearest-neighbor coupling [19, 20]. For both phases $\varphi =\frac{\pi }{2},\frac{3\pi }{2}$, the correlations are identical and symmetric about the center of the array. [(c) and (d)] Same as (a)–(d), with second-order coupling present. Here, the correlations are no longer identical nor symmetric.