Synthetic Gauge Field for Two-Dimensional Time-Multiplexed Quantum Random Walks

Temporal multiplexing provides an efficient and scalable approach to realize a quantum random walk with photons that can exhibit topological properties. But two-dimensional time-multiplexed topological quantum walks studied so far have relied on generalizations of the Su-Shreiffer-Heeger model with no synthetic gauge field. In this work, we demonstrate a two-dimensional topological quantum random walk where the nontrivial topology is due to the presence of a synthetic gauge field. We show that the synthetic gauge field leads to the appearance of multiple band gaps and, consequently, a spatial confinement of the quantum walk distribution. Moreover, we demonstrate topological edge states at an interface between domains with opposite synthetic fields. Our results expand the range of Hamiltonians that can be simulated using photonic quantum walks.

Figure 1

(a) Schematic explaining the possible movements of a walker at spatial position ($x$, $y$) along with applied phase shifts during each step of the quantum walk. (b) Schematic of the two-dimensional quantum walk setup describing the details of the experimental setup. PD is for photodetector, BPF for band pass filter, SOA for semiconductor optical amplifier, EOPM for electro-optic phase modulator, and PC is for polarization controller.

Figure 2

(a) Experimental observations and (b) theoretical predictions of the evolution of the quantum walk distribution under no phase modulation. (c) Experimental observations and (d) theoretical predictions of the evolution of the quantum walk distribution under linearly dependent phase modulation ${\varphi }_y=y\varphi$ for the case of $\varphi =\pi /2$. The left, middle, and right columns show the distributions at time steps of 1, 5, and 9, respectively. In these plots, all the distributions are normalized to their maximum.

Figure 3

(a) Comparison of the theoretical and experimental results for the variation of the quadratic mean of $x$ under no gauge field as well as under the gauge field with $\varphi =\pi /2$. (b) Comparison of the theoretical and experimental results for the variation of the quadratic mean of $y$ under no gauge field as well as under the gauge field with $\varphi =\pi /2$. The error bars in the measurements are smaller than the size of the plotted data points. (c,d) Band structure of the system under no phase modulation ($\varphi =0$) and a linear phase modulation ($\varphi =\pi /2$).

Figure 4

(a) The schematic describing the phase modulation pattern of ${\varphi }_y=|y|\varphi$ for $\varphi =\pi /2$ in the synthetic space. (b) Band diagram of the corresponding system, which clearly shows the presence of edge states in the band gap. (c) Experimental observations and (d) theoretical predictions of the evolution of the quantum walk distribution moving along the boundary under the phase modulation of ${\varphi }_y=|y|\varphi$ for $\varphi =\pi /2$. The left, middle, and right columns show the distributions at time steps of 1, 5, and 9, respectively. In these plots, all the distributions are normalized to their maximum.