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arXiv:1506.08100

Zak Phase in Discrete-Time Quantum Walks

Graciana Puentes, Osvaldo P. Santillán

Published Jun 25, 2015• 1 citation
arXivHTMLPDF

Abstract

We report on a simple scheme that may present a non-trivial geometric Zak phase ($Φ_{Zak}$) structure, which is based on a discrete-time quantum walk architecture. By detecting the Zak phase difference between two trajectories connecting adjacent Dirac points where the quasi-energy gap closes for opposite values of quasi-momentum ($k$), it is possible to identify geometric invariants. These geometric invariants correspond to $|Φ_{Zak}^{+(-)}-Φ_{Zak}^{-(+)}|=π$ and $|Φ_{Zak}^{+(-)}-Φ_{Zak}^{+(-)}|=0$, we argue that this effect can be directly measured.

DOI
10.48550/arxiv.1506.08100

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We report on a simple scheme that may present a non-trivial geometric Zak phase (ΦZakΦ_{Zak}ΦZak​) structure, which is based on a discrete-time quantum walk architecture. By detecting the Zak phase difference between two trajectories connecting adjacent Dirac points where the quasi-energy gap closes for opposite values of quasi-momentum (kkk), it is possible to identify geometric invariants. These geometric invariants correspond to ∣ΦZak+(−)−ΦZak−(+)∣=π|Φ_{Zak}^{+(-)}-Φ_{Zak}^{-(+)}|=π∣ΦZak+(−)​−ΦZak−(+)​∣=π and ∣ΦZak+(−)−ΦZak+(−)∣=0|Φ_{Zak}^{+(-)}-Φ_{Zak}^{+(-)}|=0∣ΦZak+(−)​−ΦZak+(−)​∣=0, we argue that this effect can be directly measured.
We present a research article which formulates the milestones for the understanding and characterization of holonomy and topology of a discrete-time quantum walk architecture, consisting of a unitary step given by a sequence of two non-commuting rotations in parameter space. Unlike other similar systems recently studied in detail in the literature, this system does not present continous 1D topological boundaries, it only presents a discrete number of Dirac points where the quasi-energy gap closes. At these discrete points, the topological winding number is not defined. Therefore, such discrete points represent topological boundaries of dimension zero, and they endow the system with a non-trivial topology. We illustrate the non-trivial character of the system by calculating the Zak phase. We discuss the prospects of this system, we propose a suitable experimental scheme to implement these ideas, and we present preliminary experimental data.
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.
We present a study of the 2D Zak phase landscape in photonic discrete-time quantum walk (DTQW) protocols. In particular, we report numerical results for three different DTQW scenarios which preserve spatial inversion symmetry (SIS) and time-reversal symmetry (TRS), while presenting a non-trivial Zak phase structure, as a consequence of a non-vanishing Berry connection. Additionally, we propose a novel approach to break TRS in photonic systems, while preserving a vanishing Berry curvature. Our results bear a close analogy to the Aharonov-Bohm effect, stating that in a field-free multiply connected region of space the evolution of the system depends on vector potentials, due to the fact that the underlying canonical formalism cannot be expressed in terms of fields alone.
Abstract We present an analytical framework to investigate phase dynamics in discrete-time quantum walks, with a focus on geometric phases in both single- and two-particle systems. Leveraging the Cayley-Hamilton theorem, we construct the evolution operator in momentum space and derive exact closed-form solutions for the full time evolution. This allows us to explicitly extract total, dynamic, and most notably, non-adiabatic geometric phases throughout the walk. In the two-particle case, we further examine how initial entanglement influences the accumulation and interplay of these phases. Our results offer a unified and exact approach to understanding the geometric structure underlying quantum walk dynamics, with particular emphasis on non-adiabatic effects.
The topology of a single-particle band structure plays a fundamental role in understanding a multitude of physical phenomena. Motivated by the connection between quantum walks and such topological band structures, we demonstrate that a simple time-dependent, Bloch-oscillating quantum walk enables the direct measurement of topological invariants. We consider two classes of one-dimensional quantum walks and connect the global phase imprinted on the walker with its refocusing behavior. By disentangling the dynamical and geometric contributions to this phase, we describe a general strategy to measure the topological invariant in these quantum walks. As an example, we propose an experimental protocol in a circuit QED architecture where a superconducting transmon qubit plays the role of the coin, while the quantum walk takes place in the phase space of a cavity.
Discrete-time quantum walks (DTQWs) provide a convenient platform for a realization of many topological phases in noninteracting systems. They often offer more possibilities than systems with a static Hamiltonian. Nevertheless, researchers are still looking for DTQW symmetries protecting topological phases and for definitions of appropriate topological invariants. Although the majority of DTQW studies on this topic focus on the so-called split-step quantum walk, two distinct topological phases can be observed in more basic models. Here we infer topological properties of the basic DTQWs directly from the mapping of the Brillouin zone to the Bloch Hamiltonian. We show that for translation-symmetric systems they can be characterized by a homotopy relative to special points. We also propose a topological invariant corresponding to this concept. This invariant indicates the number of edge states at the interface between two distinct phases.