Supersymmetric Polarization Anomaly in Photonic Discrete-Time Quantum Walks

Abstract

Quantum anomalies lead to finite expectation values that defy the apparent symmetries of a system. These anomalies are at the heart of topological effects in electronic, photonic and atomic systems, where they result in a unique response to external fields but generally escape a more direct observation. Here, we implement an optical-network realization of a discrete-time quantum walk, where such an anomaly can be observed directly in the unique circular polarization of a topological midgap state. We base the system on a single-step protocol overcoming the experimental infeasibility of earlier multi-step protocols. The evolution combines a chiral symmetry with a previously unexplored unitary version of supersymmetry. Having experimental access to the position and the coin state of the walker, we perform a full polarization tomography and provide evidence for the predicted anomaly of the midgap states. This approach opens the prospect to dynamically distil topological states for quantum information applications. PACS numbers: 03.67.Ac, 42.50.-p, 03.65.VfIntroduction.-Quantum anomalies take a privileged position amongst fundamental physics as they equip quantum systems with robust topological effects. The historic backdrop for quantum anomalies is provided by the Atiyah-Singer index theorem for the Dirac operator [1], which states that the difference of zero modes with positive and negative chirality is a topological invariant. These zero modes are of fundamental significance not only because of their robustness against smooth deformations, but also since their definite chirality defies an apparent symmetry of the system, which results in an anomalous response to symmetry-breaking external fields. An early practical realization is the Su-Schrieffer-Heeger model for polyacetylene [2], where the anomalous properties of a midgap state result in charge fractionalization and spin-charge separation [3]. Interest in this phenomenon therefore quickly transcended the original setting of continuum and lattice field theories [4], and presently provides a major motivation for research particularly in electronic [5][6][7][8], superconducting [8-11], photonic [12-25] and ultracold atomic [26][27][28][29][30] systems. In all these settings, zero-modes represent symmetryprotected midgap states with unique finite expectation values of a relevant symmetry operator, resulting in a distinct response when probed by suitable external fields. This includes the formation of anomalous currents, as recently observed in Dirac and Weyl semimetals [31,32]. An equally early development was the relation of such anomalous behaviour to supersymmetry. In this case systems appear with partners that differ in the number of zero modes, with the prime example being a Dirac particle exposed to a magnetic field [33,34]. This feature is central to field-theoretic descriptions, but has been much less inquired in practical systems.

Enhanced localization and protection of topological edge states due to geometric frustration

“…to fall into a new category of topological insulators labeled Square-Root Topological Insulators ( √ T Is) [8,9]. The nontrivial topological features of these √ T Is are linked to their squared-Hamiltonian (H 2 ) [34,35]. Since √ T Is are bipartite, their H 2 can be put into a block diagonal form, that is, the squared model is a system of two-independent chains, one of which being topologically non-trivial.…”

Quantum Topological Photonics

“…There are also several other works that focused on simulation of different topological phenomena using the discrete‐time quantum walk of a single photon. [ 25,151 , 153,160,161 ] Figure 13c shows the conceptual diagram for a quantum walk using topological SSH waveguides. From the works discussed above, we can conclude that not only can the topological protection work for quantum walks, but also that quantum walks represent a powerful way to explore new physical phenomena, e.g., topological photonics.…”

Bulk-edge correspondence in nonunitary Floquet systems with chiral symmetry

“…Since quantum walks are Floquet systems in which time evolves in a discrete manner, topological phases can be different from those which are described by time-independent Hamiltonians. Floquet topological phases of quantum walks have been intensively studied for the last decade [51][52][53][54][55][56][57][58][59][60][61][62][63][64][65][66][67][68][69], and topological edge states have been observed in experiments of both closed [70][71][72][73] and open [36,47,74,75] systems. Specifically, much attention has been paid to Floquet systems with chiral symmetry.…”