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arXiv:1910.02949 · Physical Review A

Simulation of the multiphase configuration and phase transitions with quantum walks utilizing a step-dependent coin

S. Panahiyan, S. Fritzsche

Published Dec 12, 2019• 12 citations
arXivHTMLPDF

Abstract

Quantum walks are versatile simulators of topological phases and phase transitions as observed in condensed-matter physics. Here, we utilize a step-dependent coin in quantum walks and investigate what topological phases we can simulate with it, their topological invariants, bound states, and possibility of phase transitions. These quantum walks simulate nontrivial phases characterized by topological invariants (winding number) $\ifmmode\pm\else\textpm\fi{}1$, which are similar to the ones observed in topological insulators and polyacetylene. We confirm that the number of phases and their corresponding bound states increase step dependently. In contrast, the size of topological phase and distance between two bound states are decreasing functions of steps resulting into formation of multiple phases as quantum walks proceed (multiphase configuration). We show that, in the bound states, the winding number and group velocity are ill defined and the second moment of the probability density distribution in position space undergoes an abrupt change. Therefore, there are phase transitions taking place over the bound states and between two topological phases with different winding numbers.

DOI
10.1103/physreva.100.062115

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Quantum walks are versatile simulators of topological phases and phase transitions as observed in condensed-matter physics. Here, we utilize a step-dependent coin in quantum walks and investigate what topological phases we can simulate with it, their topological invariants, bound states, and possibility of phase transitions. These quantum walks simulate nontrivial phases characterized by topological invariants (winding number) \ifmmode±\else\textpm\fi1\ifmmode\pm\else\textpm\fi{}1\ifmmode±\else\textpm\fi1, which are similar to the ones observed in topological insulators and polyacetylene. We confirm that the number of phases and their corresponding bound states increase step dependently. In contrast, the size of topological phase and distance between two bound states are decreasing functions of steps resulting into formation of multiple phases as quantum walks proceed (multiphase configuration). We show that, in the bound states, the winding number and group velocity are ill defined and the second moment of the probability density distribution in position space undergoes an abrupt change. Therefore, there are phase transitions taking place over the bound states and between two topological phases with different winding numbers.
We study the simulation of the topological phases in three subsequent dimensions with quantum walks. We focus mainly on the completion of a table for the protocols of the quantum walk that could simulate different families of the topological phases in one, two, and three dimensions. We also highlight the possible boundary states that can be observed for each protocol in different dimensions and extract the conditions for their emergences. To further enrich the simulation of the topological phenomena, we include step-dependent coins in the evolution operators of the quantum walks. This leads to step dependence of the simulated topological phenomena and their properties which introduces dynamicity as a feature of simulated topological phases and boundary states. This dynamicity provides the step number of the quantum walk as a means to control and engineer the numbers of topological phases and boundary states, their numbers, types, and even occurrences.
Quantum walks constitute a versatile platform for simulating transport phenomena on discrete graphs including topological material properties while providing a high control over the relevant parameters at the same time. To experimentally access and directly measure the topological invariants of quantum walks, we implement the scattering scheme proposed by Tarasinski et al. [Phys. Rev. A 89, 042327 (2014)] in a photonic time multiplexed quantum walk experiment. The tunable coin operation provides opportunity to reach distinct topological phases, and accordingly to observe the corresponding topological phase transitions. The ability to read-out the position and the coin state distribution, complemented by explicit interferometric sign measurements, allowed the reconstruction of the scattered reflection amplitudes and thus the computation of the associated bulk topological invariants. As predicted, we also find localized states at the edges between two bulks belonging to different topological phases. In order to analyze the impact of disorder, we have measured invariants of two different types of disordered samples in large ensemble measurements, demonstrating their constancy in one disorder regime and a continuous transition with increasing disorder strength for the second disorder sample.
Quantum walks constitute a versatile platform for simulating transport phenomena on discrete graphs including topological material properties while providing a high control over the relevant parameters at the same time. To experimentally access and directly measure the topological invariants of quantum walks, we implement the scattering scheme proposed by Tarasinski et al. [Phys. Rev. A 89, 042327 (2014)] in a photonic time multiplexed quantum walk experiment. The tunable coin operation provides opportunity to reach distinct topological phases, and accordingly to observe the corresponding topological phase transitions. The ability to read-out the position and the coin state distribution, complemented by explicit interferometric sign measurements, allowed the reconstruction of the scattered reflection amplitudes and thus the computation of the associated bulk topological invariants. As predicted, we also find localized states at the edges between two bulks belonging to different topological phases. In order to analyze the impact of disorder, we have measured invariants of two different types of disordered samples in large ensemble measurements, demonstrating their constancy in one disorder regime and a continuous transition with increasing disorder strength for the second disorder sample.
Topological gapless phases of matter have been a recent interest among theoretical and experimental condensed matter physicists. Fermionic chains with extended nearest neighbor couplings have been observed to show unique topological transition at the multicritical points between distinct gapless phases. In this work, we show that such topological gapless phases and the transition between them can be simulated in a quantum walk. We consider a three-step discrete-time quantum walk and identify various critical or gapless phases and multicriticalities from the topological phase diagram along with their distinguished energy dispersions. We reconstruct the scaling theory based on the curvature function to study transition between gapless phases in the quantum walk. We show the interesting features observed in fermionic chains, such as diverging, sign flipping and swapping properties of curvature function, can be simulated in the quantum walk. Moreover, the renormalization group flow and Wannier state correlation functions also identify transition at the multicritical points between gapless phases. We observe the scaling law and overlapping of critical and fixed point properties at the multicritical points of the fermionic chains can also be observed in the quantum walk. Furthermore, we categorize the topological transitions at various multicritical points using the group velocity of the energy eigenstates. Finally, the topological characters of various gapless phases are captured using winding number which allows one to distinguish various gapless phases and also show the transitions at the multicritical points.
We report on the possibility of controlling quantum random walks (QWs) with a step-dependent coin (SDC). The coin is characterized by a (single) rotation angle. Considering different rotation angles, one can find diverse probability distributions for this walk including: complete localization, Gaussian and asymmetric likes. In addition, we explore the entropy of walk in two contexts; for probability density distributions over position space and walker's internal degrees of freedom space (coin space). We show that entropy of position space can decrease for a SDC with the step-number, quite in contrast to a walk with step-independent coin (SIC). For entropy of coin space, a damped oscillation is found for walk with SIC while for a SDC case, the behavior of entropy depends on rotation angle. In general, we demonstrate that quantum walks with simple initiatives may exhibit a quite complex and varying behavior if SDCs are applied. This provides the possibility of controlling QW with a SDC.
Emerged as the quantum counterpart of classical random walks, quantum walks are established precious resources in a variety of quantum sciences. Recent studies have shown that quantum walks may be characterized by topological invariants, in close analogy to condensed matter systems exhibiting topological order. Exploiting these features, quantum walks are currently used to simulate topological systems and to probe their exotic features. Here we present the implementation of a one-dimensional quantum walk protocol based on the orbital angular momentum of light, manifesting the topological phases that characterize time-periodic systems (Floquet topological insulators) showing chiral symmetry. By considering the orbital angular momentum spectrum of a light beam undergoing this quantum evolution, we show that the associated statistical moments have marked differences in distinct phases and contain information on the system topology. While varying a control parameter determining the value of the invariants, these moments in the large step-number limit exhibit a sharp variation at the phase changes. We show that these phenomena arise from the singular behavior of the dispersion relation at the transition points. The extension of our results to systems featuring different symmetries, or characterized by higher spatial dimensions, may unveil novel intriguing features associated with these complex systems.