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arXiv:2306.12540

2D Zak Phase Landscape in Photonic Discrete-Time Quantum Walks

Graciana Puentes

Published Jun 21, 2023• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.2306.12540

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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.
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.
Quantum walks are a useful platform for studying high-efficiency energy transport. In this paper, we study the probability transfer in the discrete-time quantum walk (DTQW) in detail. With intricate design of the initial state, the probability transfer in the DTQW system can exhibit similar behaviors to those in continuous-time quantum walks (CTQWs). Furthermore, without the assumption of a biased initial state and coupling to the environment in the DTQW system, we explore how to achieve directed probability transfer in the DTQW and find that it is closely connected to the time-reversal symmetry of the effective Hamiltonian of the walk. We find that regardless of whether the number of vertices in the DTQW is even or odd, we can control the probability transfer in the DTQW by modulating the time-reversal symmetry of the system. Such control of directed probability transfer can only work well in a CTQW with odd vertices. Even when temporal or spatial disorder emerges in the DTQW, the connection between the probability transfer and the symmetry of the walk still exists. Experimental proposals for observing the probability distributions in the DTQW are discussed.
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.
We present a review of photonic implementations of discrete-time quantum walks (DTQW) in the spatial and temporal domains, based on spatial- and time-multiplexing techniques, respectively. Additionally, we propose a detailed novel scheme for photonic DTQW, using transverse spatial modes of single photons and programmable spatial light modulators (SLM) to manipulate them. Unlike all previous mode-multiplexed implementations, this scheme enables simulation of an arbitrary step of the walker, only limited, in principle, by the SLM resolution. We discuss current applications of such photonic DTQW architectures in quantum simulation of topological effects and the use of non-local coin operations based on two-photon hybrid entanglement.
A particular family of Discrete Time Quantum Walks (DTQWs) simulating fermion propagation in 2D curved space-time is revisited. Usual continuous covariant derivatives and spin-connections are generalized into discrete covariant derivatives along the lattice coordinates and discrete connections. The concepts of metrics and 2-beins are also extended to the discrete realm. Two slightly different Riemann curvatures are then defined on the space-time lattice as the curvatures of the discrete spin connection. These two curvatures are closely related and one of them tends at the continuous limit towards the usual, continuous Riemann curvature. A simple example is also worked out in full.
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.