Supersymmetric Polarization Anomaly in Photonic Discrete-Time Quantum Walks

Sonja Barkhofen, Lennart Lorz, Thomas Nitsche, Christine Silberhorn, and Henning Schomerus

Phys. Rev. Lett. 121, 260501 – Published 28 December 2018

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 multistep 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 distill topological states for quantum information applications.

Received 24 July 2018

DOI:https://doi.org/10.1103/PhysRevLett.121.260501

Physics Subject Headings (PhySH)

Quantum optics Quantum simulation Quantum walks Topological defects

Topological insulators Topological materials

Quantum Information, Science & TechnologyAtomic, Molecular & Optical

Authors & Affiliations

Sonja Barkhofen^1,*, Lennart Lorz¹, Thomas Nitsche¹, Christine Silberhorn¹, and Henning Schomerus²

¹Applied Physics, University of Paderborn, Warburger Strasse 100, 33098 Paderborn, Germany
²Department of Physics, Lancaster University, Lancaster LA1 4YB, United Kingdom

^*Corresponding author. sonja.barkhofen@uni-paderborn.de

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Vol. 121, Iss. 26 — 28 December 2018

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Images

Figure 1
Supersymmetric single-step quantum walk realizing an interface between two topologically distinct phases. (a) Coin structure in the interface configuration, where each disk represents the action of a coin that rotates the polarization by the denoted angles $φ_{1}$ or $φ_{2}$ . Across the interface the positions of these coins in the unit cells (red and blue boxes) are interchanged. (b) Alternating circular polarization of the spatially localized midgap states trapped by the interface. The fading of the color strength away from the interface indicates the intensity decay of the localized midgap state. All extended states display a linear polarization (not shown). (c) Winding of states from the bands around the three-dimensional torus ( $α$ , $β$ , $γ$ ), revealing the topological structure of the supersymmetric quantum walk on both sides of the interface. (d) Quasienergy band structure $λ (k) = \exp [- i ε (k)]$ comprising four symmetric bands (colored arcs, here shown for $φ_{1} = 1$ , $φ_{2} = 0.2$ ). We realize the midgap states pinned to $λ = \pm i$ (red dots). (e) Experimental setup using a time-multiplexing optical fiber loop; see text for details.
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Figure 2
Light trapping for the interface configuration (a),(b) compared to the interface-free bulk system (c),(d). The exemplary input polarizations are $| H ⟩$ in (a) and (c), $C_{QWP} (137 °) | H ⟩$ in (b), and $C_{HWP} (50 °) | H ⟩$ in (d), as defined in Eqs. (S17) and (S18) of Ref. [49]. The dependence of the trapped light intensity on the initial polarization is further characterized in (e) for interface (orange lines, black dots) and bulk (green lines and symbols) configuration. It shows the total intensity after step 13 at position 0 as a function of the initial polarization set by the angle $α$ of the QWP in front of the in-coupler (vertical ticks indicating error bars, experimental data; continuous curves, numerical prediction for 13 steps; dashed curve, numerical prediction for 100 steps).
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Figure 3
Anomalous polarization of the trapped midgap state from tomography of the polarization state in the interface configuration after step 17 at $x = 0$ . Note that due to the strong spatial localization the other positions are hardly occupied. The reconstructed complex density matrix from the experiment (a),(b) is compared with the numerical prediction in the $H^{'} / V^{'}$ time frame (c),(d). The input polarization is $| H ⟩$ . We observe an almost equal amplitude for the $H^{'}$ and the $V^{'}$ component on the diagonal elements of the real part, while the off-diagonal elements of the imaginary part clearly display a $π / 2$ phase shift, corresponding to right-handed circular polarization. From the experimental data we find the polarization state $(0.70 \pm 0.03) | H^{'} ⟩ + (0.71 \pm 0.02) \exp [(0.47 \pm 0.02) i π] | V^{'} ⟩$ , while numerically $0.72 | H^{'} ⟩ + 0.69 \exp (0.50 i π) | V^{'} ⟩$ .
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