Quantum Spatial Search in Two-Dimensional Waveguide Arrays

Continuous-time quantum walks (CTQWs) have shown the capability to perform efficiently the spatial search of a marked site on many kinds of graphs. However, most of such graphs are hard to realize in an experimental setting. Here we study CTQW spatial search on a planar triangular lattice by means of both numerical simulations and experiments. The experiments are performed using two-dimensional waveguide arrays fabricated by femtosecond laser pulses, illuminated by coherent light. We show that the retrieval of the marked site by the quantum walker is accomplished with higher probability than the classical counterpart, in a convenient time window placed early in the evolution.

Figure 1

The graph considered in this work is the finite portion of the triangular lattice shown on the left, having $|V|=n=31$ vertices. The target, if present, may be placed on the central site ($C$) or on a shifted position ($S$). On the right we report the microscope picture of the output facet of one waveguide array, implementing such a graph in our experiments. The waveguide arrangement follows the Bravais parameters $\overrightarrow{a}$ and $\overrightarrow{b}$ with $|\overrightarrow{a}|=|\overrightarrow{b}|$ and $\vartheta ={120}^{\circ }$. For the array in this picture $|\overrightarrow{a}|=23.4\mu \mathrm{m}$ and the target is placed on the $C$ site. Note that the target is implemented by a slight variation in the inscription parameters of the correspondent waveguide, which do not result in discernible changes in the appearance of the waveguide cross section.

Figure 2

Time evolution of the probabilities $p_w(\gamma t)$ for the quantum (blue solid line) and the classical (dashed orange line) search, for the central (a) and shifted (b) target. In the lower panels we display the quantity $R(\gamma t)$ for the central (c) and shifted (d) target. As a guide for the eye, we color in orange the region where the quantum walk outperforms the classical walk, while the yellow shadow indicates the opposite regime. The detuning $\beta /\gamma$ is set to 4.16.

Figure 3

Target probability $p_w^Q(\gamma t)$ as a function of the target onsite energy $\beta /\gamma$ and time $\gamma t$, for central ($C$) and shifted ($S$) target.

Figure 4

(a) Experimental apparatus used for the characterization of the waveguide arrays. A $\mathrm{He}$:$\mathrm{Ne}$ laser beam (633-nm wavelength) is first passed through a half-wave plate (HWP) and a polarizing beam-splitter cube, to arbitrarily attenuate the beam intensity and to select horizontal polarization state. The beam is focused by a convex lens (with focal length $f=40\mathrm{cm}$) on the input facet of the glass sample containing the waveguide arrays (WGs). The glass sample is mounted onto micrometric manipulation stages allowing three orthogonal translations and two tilts. The output facet is imaged onto a CCD camera using a $10\times$ objective. (b) On the right, the experimentally measured output distribution for the fabricated waveguide arrays, for the different values of nominal propagation coordinate $\gamma t$ and for the different target position (or absence of target). On the left, numerical simulation of the same distributions, assuming the nominal detuning $\beta /\gamma =4.1$ on the target waveguide. The intensity scale of each picture is normalized to the maximum value of that picture.

Figure 5

Numerical simulation of the propagation in waveguide arrays with the geometry as in Fig. 1, with an interwaveguide spacing $a=25\mu \mathrm{m}$ and no target site implemented (all waveguides are equal), excited at the input with a plane wave with different tilt angles (wavelength 633 nm). The intensity scale of each picture is normalized to the maximum value of that picture. One notes that even a small tilt angle can alter significantly the light distribution.

Figure 6

Numerical simulation of a classical random walk on a 31-site graph, as described in Sec. 2. We consider the case of the absence of target, target on the central site $C$ and target on a site shifted from center $S$ as in Fig. 1. At the initial time $t=0$ the walker populates with identical probabilities all sites. We choose to represent the probability distribution of the walker at each $\gamma t$ in a graphical way that matches the optical intensity distributions in Fig. 4, to facilitate comparison. The intensity scale of each picture is normalized to the maximum value of that picture.

Figure 7

Experimental values of $p_j^Q(\gamma t)$ at each waveguide output measured in the arrays with the target on the central site (top left graph) and in shifted position (top right graph). Blue circles represent the measured values for the target waveguide while crosses represent nontarget sites. Blue thin lines represent the classical probability $p_w^C(\gamma t)$ to find the target in a classical random-walk setting. In the top-right graph (target $S$), red crosses correspond to the values of the central waveguide. Experimental values of $R(\gamma t)$ are plotted (bottom left and bottom right graphs), as calculated from the experimental values of $p_w^Q(\gamma t)$ and the calculated values of $p_w^C(\gamma t)$, as plotted in the above graph.

Figure 8

Triangular graph with $N=37$ (left) and $N=61$ (right) nodes. The central ($C$), nearest-neighbor ($1N$), and second-neighbor ($2N$) targets are colored in red.

Figure 9

(a) Scaling behavior of the maximum $R(t_{\mathrm{opt}})$ as a function of the size of the graph, for three different target positions: “$C$” central, “$1N$” nearest neighbor, “$2N$” second-nearest neighbor. (b) Optimal time $t_{\mathrm{opt}}$ as a function of $N$. (c) Value of the target probabilities at the optimal time, for the quantum and classical search. The target detuning is set to $\beta /\gamma =4$.

Figure 10

$R(t_{\mathrm{opt}})$ as a function of the size $N$ of the graph and the parameter $\beta /\gamma$, for three different target positions: “$C$” central, “$1N$” nearest neighbor, “$2N$” second-nearest neighbor.