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arXiv:2506.19523 · New Journal of Physics

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

Andrzej Grudka, Marcin Karczewski, Paweł Kurzyński, T. P. Polak, Jan Wójcik, Antoni Wójcik

Published Feb 1, 2026• 0 citations
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Abstract

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.

DOI
10.1088/1367-2630/ae45c6

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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.
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 $\ensuremath{\pi}$, 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.
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Topological insulators host protected boundary states that are robust to disorder, making them attractive for applications and motivating their realization in engineered systems. Discrete-time quantum walks, which have been implemented in a variety of experimental platforms, can exhibit topologically nontrivial phases and their associated boundary states. Here, we introduce a class of topological quantum walks with variable step lengths that offer theoretical insights into the topological classification of such walks. For example, these walks can access higher winding numbers and support multiple edge states. They also provide context for the reduction of topological protection from a Z\mathbb{Z}Z- to Z2\mathbb{Z}_2Z2​-valued invariant when time-reversal symmetry is broken. The topologically protected edge states are investigated using a transfer-matrix approach that describes their spatial profiles and spin structures. These predictions are in excellent agreement with numerical results and, where applicable, with the Jackiw--Rebbi zero-mode solution of the Dirac equation. Taken together, our analysis provides a roadmap for designing quantum walks with control over the number, spatial profile, and spin structure of their topological edge states.
We study the dynamical localization of discrete time evolution of topological split-step quantum random walk (QRW) on a single-site defect starting from a uniform distribution. Using analytical and numerical calculations, we determine the high localization probability regions in the parameter space of the quantum walker. These regions contain two or more pairs of trapped states, forming near a lattice defect. By investigating the spectral properties of the discrete-time evolution operators, we show that these trapped states have large overlap with the initial uniformly distributed state, thus offering a simple interpretation of the localization effect. As this localization scheme could be interpreted as a variation of spatial quantum search algorithm, we compare the localization probability and time with other types of two-dimensional quantum walks that do not have topological phases and realize localization time scaling similar to Grover's algorithm. Finally we show that mechanism of localization we identified is robust against the influence of disorder.
Discrete-time quantum walks are among the branches of quantum information and computation. They are platforms for developing quantum algorithms for quantum computers. In addition, due to their universal primitive nature, discrete-time quantum walks have been used to simulate other quantum systems and phenomena that are observed in physics and chemistry. To fully utilize the potentials that the discrete-time quantum walks hold in their applications, control over the discrete-time quantum walks and their properties becomes essential. In this dissertation, we propose two models for attaining a high level of control over the discrete-time quantum walks. In the first one, we incorporate a dynamical nature for the unitary operator performing the quantum walks. This enables us to readily control the properties of the walker and produce diverse behaviors for it. We show that with our proposal, the important properties of the discrete-time quantum walks such as variance would indeed improve. To explore the potential of this proposal, we apply it in the simulations of topological phases in condensed matter physics. With our proposal, we can control the simulations and determine the type of topological phenomena that should be simulated. In addition, we confirm simulations of topological phases and boundary states that can be observed in one-, two- and three-dimensional systems. Finally, we report the emergence of exotic phase structures in form of cell-like structures that contain all types of topological phases and boundary states of certain classes. In our second proposal, we take advantage of resources available in quantum mechanics, namely quantum entanglement and entangled qubits. In this proposal, we use entangled qubits in the structure of a quantum walk and show that by tuning the initial entanglement between these qubits and how these qubits are modified through the walk, one is able to produce diverse behaviors for the quantum walk and control its behavior.
We investigate one-dimensional (1D) discrete-time quantum walks (QWs) with spatially or temporally random defects as a consequence of interactions with random environments. We focus on the QWs with chiral symmetry in a topological phase, and reveal that chiral symmetry together with the bipartite nature of the QWs brings about intriguing behaviors such as coexistence of topologically protected edge states at zero energy and Anderson transitions in the 1D chiral class at nonzero energy in their dynamics. In contrast to results of previous studies, therefore, the spatially disordered QWs can avoid complete localization due to the Anderson transition. It is further confirmed that the edge states are robust to spatial disorder but not to temporal disorder.