Observation of Topologically Protected Edge States in a Photonic Two-Dimensional Quantum Walk

Periodically driven systems have displayed a variety of fascinating phenomena without analogies in static systems, which enrich the classification of quantum phases of matter and stimulate a wide range of research interests. Here, we employ discrete-time quantum walks to investigate a nontrivial topological effect unique to a two-dimensional periodically driven system: chiral edge states can exist at the interface of Floquet insulators whose Chern numbers vanish. Thanks to a resource-saving and flexible fiber-loop architecture, we realize inhomogeneous two-dimensional quantum walks up to 25 steps, over an effective $51\times 51$ lattice with tunable local parameters. Spin-polarized chiral edge states are observed at the boundary of two distinct quantum walk domains. Our results contribute to establishing a well-controlled platform for exploring nontrivial topological phases.

Figure 1

Phase diagram, inhomogeneous 2DQW, and quasienergy bands. (a) Phase diagram of the 2DQW. Topological phases are distinguished by Rudner winding numbers $W=\pm 1$ (bold), whereas the Chern number is always 0 (normal typeface). Black bold lines indicate the gapless phases, separating the parameter space into distinct phases (magenta and blue). ①–③ represent different parameter choices for 2DQWs. (b) Schematic figure of spatial inhomogeneous 2DQW with effective Hamiltonians $H({\theta }_{1-},{\theta }_2)$ for $x<0$ and $H({\theta }_{1+},{\theta }_2)$ for $x\ge 0$, where ${\theta }_{1-}\ne {\theta }_{1+}$ and ${\theta }_2=\pi /2$. A finite geometry and periodic boundary conditions in the $x$ direction are assumed (dashed black line). In the $y$ direction, we employ a geometry of infinite length with translational invariance. When the left ($x<0$) and right ($x\ge 0$) Hamiltonians are described by different winding numbers, chiral edge modes appear, as marked by the green (yellow) arrowed lines at the middle (outer) edge. (c)–(d), Quasienergy bands of the inhomogeneous 2DQWs with parameter choices shown above. Chiral edge states are colored green (yellow) with positive (negative) dispersion corresponding to upward (downward) edge modes in (b). In our experiment, we simulate inhomogeneous 2DQWs with open boundary conditions, so only edge states localized at the middle edge ($x=0$) are depicted in the band structures.

Figure 2

Experimental implementation of 2DQWs with time-bin encoding. (a) Experimental setup. The initial pulse is generated from an 895 nm pulsed laser with a pulse duration of 15 ps and a repetition frequency of 30 kHz, and is coupled into the circuit by an acousto-optic modulator (AOM). The rotation operations $R({\theta }_1)$ and $R({\theta }_2)$ are, respectively, implemented by a fast-switching electro-optic modulator (EOM) and a half wave plate (HWP). The two-PBS (polarizing beam splitter) loops are used to realize polarization-dependent optical delays, where the $V$ component is delayed relative to the $H$ component by a 6 m-long single mode fiber (SMF) and 30 cm free-space path difference, representing the translation in the $x$ and $y$ directions, respectively. After every single step, around 3% of photons are reflected by a beam splitter for detection and the transmitted photons continue to propagate through the circuit. (b) Reconstruction of 2D probability distributions. The initial pulse is localized at origin $|0,0⟩$. After one 2DQW step, a single pulse is split into four discrete time bins, equivalent to a walker coherently dispersed on a $3\times 3$ lattice with the probabilities of the even sites null. Similarly for the second step and so on.

Figure 3

Observation of spin-polarized chiral edge states in inhomogeneous 2DQWs. (a)–(b) Experimentally detected normalized probability distribution $P(x,y;n)$ after $n$ steps with photons initialized at the origin (white circle), and parameters and initial polarization of: (a) ${\theta }_{1-}=\pi /4$, ${\theta }_{1+}=3\pi /4$, $|H⟩$; (b), ${\theta }_{1-}=5\pi /4$, ${\theta }_{1+}=7\pi /4$, $|V⟩$. The red line at $x=0$ indicates the boundary of inhomogeneous 2DQWs. (c) Evolution of the remained probability $P(B;n)$ within the boundary region as step numbers $n$ increase. To calculate $P(B;n)$, probabilities in the boundary region $B$ with a width of four lattice units [green unfilled rectangle in (a)] are summed. Quantum walks in (a) (blue circle) and (b) (orange square), as well as an inhomogeneous 2DQW without a topological edge (red diamond and green triangle), are shown. The parameters ${\theta }_{1\pm }$ and initial polarizations $s$ (pseudospin) are annotated in the legend. Theoretical (experimental) values are depicted as solid lines (dots). (d) Plots of the average polarization in ${\sigma }_z$ basis of photons remaining in edge-state wave packet after 25 steps as a function of the distance $|d|(|d^{\prime }|)$ of ${\theta }_{1\pm }$ away from $\pi /2(3\pi /2)$. The values of ${\theta }_{1\pm }$ are symmetrically assigned around ${\theta }_1=\pi /2(3\pi /2)$. The parameters we choose to obtain upward edge modes (yellow star), in turn, are ${\theta }_{1-}=\pi /3$ and ${\theta }_{1+}=2\pi /3$, ${\theta }_{1-}=\pi /4$ and ${\theta }_{1+}=3\pi /4$, ${\theta }_{1-}=\pi /5$ and ${\theta }_{1+}=4\pi /5$, and ${\theta }_{1-}=\pi /6$ and ${\theta }_{1+}=5\pi /6$. In terms of downward edge modes, all pairs of ${\theta }_{1\pm }$ are right shifted by $\pi$ shown as the green polygons in the inset phase diagram. Statistical errors are smaller than the dot sizes.

Figure 4

Robustness of topologically protected edge states. (a) The measured probability distributions $P(x,y;n)$ of the inhomogeneous 2DQW with a boundary including two right angles, where the first right angle is at the position (0,7) and the second is at (7,7). The red line is to guide the eye. ${\theta }_1$ for the left and right sides of the boundary are $\pi /6$ and $5\pi /6$, respectively. (b)–(d) The simulated [(b)] and experimentally measured [(c)–(d)] probability distributions $P(x,y;25)$ of the inhomogeneous 2DQW (${\theta }_{1-}=\pi /4$, ${\theta }_{1+}=3\pi /4$) with a straight-line edge at $x=0$. Only odd sites are shown as the probabilities for even sites are zero. The schematic electro-optic modulator (EOM) signals of one step period are shown below the probability distributions correspondingly. In the theoretical simulation [(b)], a sharp boundary without impurities at the boundary and no disorder in the bulk are assumed. In (c), a $2\times 5$-sized impurity is introduced along the boundary (marked as a yellow nub), where the on-site ${\theta }_1$ is $\pi /2$. In (d), a noisy EOM signal is applied, corresponding to a spatially disordered rotation operation $R({\theta }_1)$ where the value of ${\theta }_1$ for each position $(x,y)$ of $x<0$ and $x\ge 0$ is randomly generated from the range of (${\theta }_{1-}-\delta \theta ,{\theta }_{1-}+\delta \theta$) and (${\theta }_{1+}-\delta \theta ,{\theta }_{1+}+\delta \theta$), respectively, where $\delta \theta =0.46\mathrm{rad}$.