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Detection of Zak phases and topological invariants in a chiral photonic quantum walk

Filippo Cardano, Alessio D’Errico, Alexandre Dauphin, Maria Maffei, Bruno Piccirillo, Corrado de Lisio, G. De Filippis, V. Cataudella, Enrico Santamato, Lorenzo Marrucci, Maciej Lewenstein, Pietro Massignan

Published Oct 20, 2016• 4 citations
OpenAlex

Abstract

Topological insulators are fascinating states of matter exhibiting protected edge states and robust quantized features in their bulk. Here, we propose and validate experimentally a method to detect topological properties in the bulk of one-dimensional chiral systems. We first introduce the mean chiral displacement, and we show that it rapidly approaches a multiple of the Zak phase in the long time limit. Then we measure the Zak phase in a photonic quantum walk, by direct observation of the mean chiral displacement in its bulk. Next, we measure the Zak phase in an alternative, inequivalent timeframe, and combine the two windings to characterize the full phase diagram of this Floquet system. Finally, we prove the robustness of the measure by introducing dynamical disorder in the system. This detection method is extremely general, as it can be applied to all one-dimensional platforms simulating static or Floquet chiral systems.

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Topological insulators are fascinating states of matter exhibiting protected edge states and robust quantized features in their bulk. Here, we propose and validate experimentally a method to detect topological properties in the bulk of one-dimensional chiral systems. We first introduce the mean chiral displacement, and we show that it rapidly approaches a multiple of the Zak phase in the long time limit. Then we measure the Zak phase in a photonic quantum walk, by direct observation of the mean chiral displacement in its bulk. Next, we measure the Zak phase in an alternative, inequivalent timeframe, and combine the two windings to characterize the full phase diagram of this Floquet system. Finally, we prove the robustness of the measure by introducing dynamical disorder in the system. This detection method is extremely general, as it can be applied to all one-dimensional platforms simulating static or Floquet chiral systems.
We experimentally investigate topological phenomena in one-dimensional discrete-time photonic quantum walks using a combination of methods. We first detect winding numbers of the quantum walk by directly measuring the average chiral displacement, which oscillates around quantized winding numbers for finite-step quantum walks. Topological phase transitions can be identified as changes in the center of oscillation of the measured chiral displacement. The position of topological phase transition is then confirmed by measuring the moments of the walker probability distribution. Finally, we observe localized edge states at the boundary of regions with different winding numbers. We also confirm the robustness of edge states against chiral-symmetry-preserving disorder.
Winding number and Zak phase, as topological invariants, play crucial roles in characterizing the topological phases of one-dimensional Floquet topological insulators. It is difficult to directly detect Floquet topological invariants with a unified scheme, especially for low driving frequencies. Here, by defining the dynamic winding number based on the time-averaged spin textures, we propose to use it to characterize one-dimensional Floquet topological phases with arbitrarily varying frequencies. In the high-frequency regime, both the conventional winding number and Zak phase can be fully determined by measuring the dynamic winding number of time-averaged spin textures, regardless of initial states and initial phases. In the low-frequency regime, only conventional winding number can accurately describe the system's topological phase due to the presence of large topological numbers, the Zak phase only captures the parity of the winding number and fails to characterize the number of topological edge states. However, one can still correctly characterize the Floquet topological phases through our dynamic winding number approach, in which stroboscopic measurement of spin textures is needed and the initial phases should guarantee chiral symmetry of the time-evolution operator. Our paper not only clarifies the necessity of stroboscopic measurements, but also provides a general dynamic approach for the detection of Floquet topological insulators.
Differently from the majority of the other phases of matter, which are characterized by local order parameters, the topological phases are characterized by integer or semi-integer numbers, the topological invariants, which are depending on global properties and robust against impurities or deformations. In the last decade, the study of the topological phases of matter has been developing parallel to the field of quantum simulation. Quantum simulators are fully controllable experimental platforms simulating the dynamics of systems of interest by the use of the mapping between the two Hamiltonians. These simulators represent a key resource in the study of topological phases of matter because their observation in natural systems is usually highly problematic and sometimes impossible. Quantum simulators are commonly realized with cold atoms in optical lattices or with photonic systems. The unitary and time-periodic protocols, known as quantum walks, are a versatile class of photonic quantum simulators. The purpose of this PhD thesis is to design feasible protocols to simulate and characterize topological non-interacting crystalline Hamiltonians in 1 and 2 dimensions. Moreover, this thesis contains the description of the experiments that have been completed using the theoretical proposals. In details: i) We demonstrate that the topological invariant associated to chiral symmetric 1D Hamiltonians becomes apparent through the long time limit of a bulk observable, the mean chiral displacement (MCD). This detection method converges rapidly and requires no additional elements (i.e. external fields) or filled bands. The MCD has been used to characterize the topology of a chiral-symmetric 1D photonic quantum walk and to detect a signature of the so-called topological Anderson insulating phase in a disordered chiral symmetric wire simulated with ultracold atoms. ii) We designed the protocol to measure the topological invariant that characterizes a 2D photonic quantum walk simulating a Chern insulator. A diferencia de la mayoría de las otras fases de la materia, caracterizadas por un parámetro de orden local, las fases topológicas de la materia se definen por su invariante topológico que depende de las propiedades globales del sistema y es robusto frente a la presencia de impurezas y/o deformaciones. En la última década, el estudio de las fases topológicas de la materia se ha desarrollado en paralelo con el campo de la simulación cuántica. Un simulador cuántico es unas plataformas experimental altamente controlable cuyo objetivo es simular la dinámica de un sistema de interés, mediante la correspondencia entre los dos Hamiltonianos. Estos simuladores representan un recurso clave en el estudio de las fases topológicas dado que su observación en sistemas reales es en general muy problemática y en determinadas ocasiones hasta imposible. Normalmente, los simuladores cuánticos se crean mediante átomos fríos en redes ópticas o con sistemas fotónicos. Los paseos cuánticos (quantum walks), un proceso unitario y temporalmente periódico, representan una de las clases mas versátiles de simuladores cuánticos. El propósito de esta tesis de doctorado es el diseño de protocolos para la simulación y la caracterización de Hamiltonianos topológicos no interactivos de estructuras cristalinas, tanto en una como en dos dimensiones. Además, en esta tesis se expone la descripción de experimentos llevados a cabo a partir del modelo teórico propuesto. En detalle: Demostramos que el invariante topologico asociado a la simetría quiral en una dimensión se hace aparente a partir del limite a tiempos largos de un observable del volumen (bulk), el desplazamiento quiral medio (MCD, por sus siglas en inglés). Este método de detección converge de manera rápida y no necesita de elementos adicionales (es decir, de campos externos) o bandas pobladas. El MCD ha sido utilizado para caracterizar la topología de un paseo cuántico en una dimensión con simetria quiral y para detectar la fase topológica aislante de Anderson en hilos quirales con desorden, simulados con átomos ultra fríos. Hemos diseñado un protocolo para medir el invariante topológico que caracteriza un paseo cuántico en dos dimensiones simulando un aislante de Chern.
We report the experimental detection of bulk topological invariants in nonunitary discrete-time quantum walks with single photons. The nonunitarity of the quantum dynamics is enforced by periodically performing partial measurements on the polarization of the walker photon, which effectively introduces loss to the dynamics. The topological invariant of the nonunitary quantum walk is manifested in the quantized average displacement of the walker, which is probed by monitoring the photon loss. We confirm the topological properties of the system by observing localized edge states at the boundary of regions with different topological invariants. We further demonstrate the robustness of both the topological properties and the measurement scheme of the topological invariants against disorder.
Periodically driven systems have displayed a variety of fascinating phenomena without analogies in static systems, which enrich the classification of quantum phases of matter and stimulate a wide range of research interests. Here, we employ discrete-time quantum walks to investigate a nontrivial topological effect unique to a two-dimensional periodically driven system: chiral edge states can exist at the interface of Floquet insulators whose Chern numbers vanish. Thanks to a resource-saving and flexible fiber-loop architecture, we realize inhomogeneous two-dimensional quantum walks up to 25 steps, over an effective 51×51 lattice with tunable local parameters. Spin-polarized chiral edge states are observed at the boundary of two distinct quantum walk domains. Our results contribute to establishing a well-controlled platform for exploring nontrivial topological phases.
The detection of topological phases of matter has become a central issue in recent years. Conventionally, the realization of a specific topological phase in condensed matter physics relies on probing the underlying surface band dispersion or quantum transport signature of a real material, which may be imperfect or even absent. On the other hand, quantum simulation offers an alternative approach to directly measure the topological invariant on a universal quantum computer. However, experimentally demonstrating high-dimensional topological phases remains a challenge due to the technical limitations of current experimental platforms. Here, we investigate the three-dimensional topological insulators in the AIII (chiral unitary) symmetry class, which yet lack experimental realization. Using the nuclear magnetic resonance system, we experimentally demonstrate their topological properties, where a dynamical quenching approach is adopted and the dynamical bulk-boundary correspondence in the momentum space is observed. As a result, the topological invariants are measured with high precision on the band-inversion surface, exhibiting robustness to the decoherence effect. Our Letter paves the way toward the quantum simulation of topological phases of matter in higher dimensions and more complex systems through controllable quantum phases transitions.