Rabi transport and the other finite-size effects in one-dimensional discrete-time topological quantum walk

TL;DR

This work addresses how topology and finite size shape quantum transport in one-dimensional discrete-time quantum walks (DTQWs). It combines analytical and numerical analysis to identify edge states at interfaces between topologically distinct DTQWs, derive their explicit forms, and characterize finite-size induced splittings and the resulting dynamics. A central finding is that finite length lifts the degeneracy of edge modes, enabling robust Rabi-like oscillations between opposite edges with a period that scales exponentially with system size, while ballistic transport can persist when boundary coins share the bulk phase. The results illuminate the interplay between topology and size in DTQWs and offer a framework for robust, topologically protected quantum transport with potential applications in quantum information processing.

Abstract

This paper investigates Rabi transport and finite-size effects in one-dimensional discrete-time topological quantum walks. We demonstrate the emergence of localized states at boundaries between topologically distinct phases and analyze how finite system sizes influence quantum walk dynamics. For finite lattices, we show that topology induces localized and bilocalized states, leading to Rabi-like transport as a result of degeneracy breaking due to finite-size effects. The study bridges the gap between topological protection and size-dependent dynamics, revealing transitions from ballistic motion to localized or oscillatory behavior based on the system's topological properties. Analytical and numerical methods are employed to explore the spectra and dynamics of quantum walks, highlighting the robustness of Rabi transport against disorder. The findings provide insights into controlled quantum transport and potential applications in quantum information processing.

Figures (10)

• Figure 1: Sketches of the models. Colors of the vertices indicate differences in the coin operators. (a) Homogeneous infinite chain. (b) Two homogeneous chains glued together. (c) Infinite chain with a single site defect. (d) A finite chain with reflective boundary coins. (e) A finite ring consisting of two homogeneous chains glued together. (f) Finite ring with a single site defect.
• Figure 2: Evolution of the inhomogeneous quantum walk $(\theta_\pm=\pm \pi/4, \ \delta=0, \ \zeta=0, \ \sigma=\pi/6)$ starting at the boundary ($x=0$). The initial state is $|{\Psi(t=0)}\rangle =\frac{1}{\sqrt{2}} \ |{0}\rangle\otimes1i$.
• Figure 3: Probability distribution for a homogeneous walk, with $\theta_{\mp} = \pi/4$, ($\delta=0, \ \zeta=0, \ \sigma=\pi/6)$ (a) and an inhomogeneous ones, with $\theta_{\mp} = \mp \pi/4$ (b), $\theta_{-} = - \pi/40$, $\theta_{+} = \pi/80$ (c) and $\theta_{\mp} = \mp \pi/40$ (d). For the chosen time $t=150$, odd positions are not populated at all, so the probabilities of the only even positions are presented in the figure. Red lines (c,d) represent the approximation given by Eq. \ref{['approx']}. The initial state is $|{\Psi(t=0)}\rangle =\frac{1}{\sqrt{2}} \ |{0}\rangle\otimes1i$.
• Figure 4: Energies $\omega$ versus parameter $\theta_A$ for a walk on D-cycle ($D=42$) with two (topologically distinct) segments of different sizes $d$. (See Fig. \ref{['fig1']} (e,f)). Parameter $\theta_B$ is equal to $\pi/4$ whereas $\delta=-\pi/2,\zeta=-\pi/2,\sigma=0$.
• Figure 5: Probability distribution of the quantum walk depicted in Fig. \ref{['fig1']}c with parameters $\theta_B=\pi/4$, with different $\theta_A$ (at position $x=0$). For the chosen time $t=150$, odd positions are not populated at all, so the probabilities of the only even positions are presented in the figure. The walk starts from the state $|{\Psi(t=0)}\rangle = \frac{1}{\sqrt{2}}|{0}\rangle\otimes1i.$