Scattering theory of topological phases in discrete-time quantum walks

One-dimensional discrete-time quantum walks show a rich spectrum of topological phases that have so far been exclusively analyzed using the Floquet operator in momentum space. In this work, we introduce an alternative approach to topology which is based on the scattering matrix of a quantum walk, adapting concepts from time-independent systems. For quantum walks with gaps in the quasienergy spectrum at 0 and $\pi$, we find three different types of topological invariants, which apply dependent on the symmetries of the system. These determine the number of protected boundary states at an interface between two quantum-walk regions. Quantum walks with an unequal number of leftward and rightward shifts per cycle are characterized by the number of perfectly transmitting unidirectional modes they support, which is equal to their nontrivial quasienergy winding. Our classification provides a unified framework that includes all known types of topology in one-dimensional discrete-time quantum walks and is very well suited for the analysis of finite-size and disorder effects. We provide a simple scheme to directly measure the topological invariants in an optical quantum-walk experiment.

Figure 1

Left: Propagation of a particle in the simple quantum walk [Eq. (2.8)] initialized in spin-up state on a single site. Right: Band structure of the simple quantum walk for different values of the rotation angle $\theta$. Generically, the spectrum is gapped around quasienergies $\epsilon =0,\pi ,$ except for the special cases $\theta =0,\pi$.

Figure 2

Scattering setting for the simple quantum walk [Eq. (2.8)]. The lattice is divided in three regions: a left lead ($x<0$), a right lead ($x\ge L$), and a scattering region in-between. Each site supports two internal spin states. The shift operators of the protocol act throughout the whole system (solid black arrows), shifting a walker with state $\uparrow$ to the right and state $\downarrow$ to the left. Rotations (dotted arrows) only change the internal state of the walker in the scattering region. (a) A walker with spin up in the left lead is incident on the scattering region. (b) Once it reaches $x=0$, it is subject to rotations and acquires a spin-down component, which is shifted in the opposite direction. The purple arrows illustrate a possible reflection process. (c) A walker with spin down is propagated away from the scattering region. While (a)–(c) depict the scattering process in time, the scattering states we consider are the corresponding quasienergy eigenstates.

Figure 3

Topological phase diagrams for three quantum-walk examples, obtained from the scattering matrix approach. All phases are labeled by their topological invariant ${\mathcal{Q}}_X$ and are furthermore encoded in brightness (${\mathcal{Q}}_{X,0}$) and hue (${\mathcal{Q}}_{X,\pi }$). (a) Topological invariant ${\mathcal{Q}}_{\text{BDI}}$ of the split-step quantum walk (7.2). (b) Topological invariant ${\mathcal{Q}}_{\text{BDI}}$ of the four-step quantum walk (7.3), where ${\theta }_1={\theta }_3$ and ${\chi }_{1,2,3}=0$, so that falls into class BDI. (c) Topological invariant ${\mathcal{Q}}_{\text{DIII}}$ of the quantum walk (7.5), with ${\theta }_2=0$. For all three examples, the length of the scattering region is $L=50$. Close to phase boundaries, where the gap closes, $r$ becomes subunitary due to finite-size effects and the invariants are not quantized. Otherwise, the quantization of the invariants is evident.

Figure 4

Disorder-averaged invariant $\langle Q_{\mathrm{ch},0}\rangle$ at $\varepsilon =0$ for the simple quantum walk as a function of mean rotation angle and disorder strength. The transition region around $\langle \theta \rangle =0$ is broadened with increasing disorder until the topological phases are not properly defined anymore (green region). System size is $L=50$, the average is taken over $n=100$ different disorder realizations.

Figure 5

Schematic layout for the experimental measurement of the reflection amplitudes of a split-step quantum walk. An incident coherent light pulse at $\tau =0$, $x=-1$ enters an array of beam splitters of two types (dark blue, light orange), where it is split and recombined repeatedly, thereby performing the quantum walk. A row of detectors at $x=-1$ measure the wave amplitudes $\langle -1,\downarrow \left|{\mathcal{F}}^{\tau }\right|-1,\uparrow \rangle$ leaving the quantum-walk region. The reflection amplitudes $r\left(0\right)$ and $r\left(\pi \right)$ are given by the sum and the alternating sum of the measured reflected amplitudes (8.1).