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arXiv:2103.10924 · Optics Letters

Probing the topology of the two-photon bands via time-dependent quantum walks

Andrei A. Stepanenko, Maxim A. Gorlach

Published May 10, 2021• 1 citation
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Abstract

Topological protection of quantum correlations opens new horizons and opportunities in quantum technologies. A variety of topological effects has recently been observed in qubit networks. However, the experimental identification of the topological phase still remains challenging, especially in the entangled many-body case. Here, we propose an approach to independently probe single- and two-photon topological invariants from the time evolution of the two-photon state in a one-dimensional array of qubits. Extending the bulk-boundary correspondence to the two-photon scenario, we show that an appropriate choice of the initial state enables the retrieval of the topological invariant for the different types of the two-photon states in the interacting Su-Schrieffer-Heeger model. Our analysis of the Zak phase reveals additional facets of topological protection in the case of collapse of bound photon pairs.

DOI
10.1364/ol.425841

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•Oct 20, 2016•4 citations
Topological protection of quantum correlations opens new horizons and opportunities in quantum technologies. A variety of topological effects has recently been observed in qubit networks. However, the experimental identification of the topological phase still remains challenging, especially in the entangled many-body case. Here, we propose an approach to independently probe single- and two-photon topological invariants from the time evolution of the two-photon state in a one-dimensional array of qubits. Extending the bulk-boundary correspondence to the two-photon scenario, we show that an appropriate choice of the initial state enables the retrieval of the topological invariant for the different types of the two-photon states in the interacting Su-Schrieffer-Heeger model. Our analysis of the Zak phase reveals additional facets of topological protection in the case of collapse of bound photon pairs.
Topological phases of matter offer a new approach to quantum technologies based on disorder-robust quantum states. Here we present the topological characterization of the two-photon bands in a dissipative array of qubits described by the extended Bose-Hubbard model which includes density-dependent coupling. While density-dependent coupling forms the nontrivial two-photon topological phase, single-photon states remain topologically trivial. We discuss the connection between the topological invariant of the band and the number of states in this band that allows us to determine the topological phase of the system with reflection symmetry using only the information on the number of eigenstates retrieved from the experiment in the case of small dissipation. We also discuss another method of topological order probing based on the analysis of a time evolution that can be used in a regime with stronger dissipation.
We investigate two-photon topological states in the array of nearest-neighbor coupled qubits for two model systems based on the extended version of the Bose-Hubbard Hamiltonian, one of which includes direct two-photon tunneling and an- other one incorporates density-dependent hopping. For both models, we demonstrate that the effective photon-photon interaction gives rise to the topological bandgap and topologically protected edge states of bound photon pairs. We design effective density- dependent hopping by inserting auxiliary frequency-detuned qubits in the appropriate positions in the array, which is feasible to realize, for instance, in existing arrays of transmon qubits. Furthermore, we map our system onto the paradigmatic Su-Schrieffer- Heeger model in the limiting cases illustrating our conclusions by the results of numerical simulations.
In the presence of symmetries, one-dimensional quantum systems can exhibit topological order, which in many cases can be characterized by a quantized value of the many-body geometric Zak or Berry phase. We establish that this topological Zak phase is directly related to the Zak phase of an elementary quasiparticle excitation in the system. By considering various systems, we establish this connection for a number of different interacting phases including the Su-Schrieffer-Heeger model, ppp-wave topological superconductors, and the Haldane chain. Crucially, in contrast to the bulk many-body Zak phase associated with the ground state of such systems, the topological invariant associated with quasiparticle excitations (above this ground state) exhibits a more natural route for direct experimental detection. To this end, we build upon recent work [F. Grusdt, et al., Nat. Commun. 7, 11994 (2016)] and demonstrate that mobile quantum impurities can be used, in combination with Ramsey interferometry and Bloch oscillations, to directly measure these quasiparticle topological invariants. Finally, a concrete experimental realization of our protocol for dimerized Mott insulators in ultracold atomic systems is discussed and analyzed.
Identifying and measuring topological invariants remains a fundamental challenge due to the non-local nature of momentum eigenstates in real space. Established approaches based on Bloch oscillations, quantum walks, charge pumping and quench dynamics, typically fail to provide direct momentum-dependent geometric information of each Bloch state. Here, we propose a novel framework for implementing direct band-topology tomography via momentum-selective measurements, and experimentally demonstrate it on a superconducting processor featuring a highly connected coupling architecture. Utilizing a ``star-like" geometry in which an auxiliary probe couples to a one-dimensional topological chain with spatially modulated coupling strengths, we selectively address the desired momentum states of the chain. This approach enables the direct and robust extraction of the winding phase that contributes to the associated topological invariant. Leveraging non-local couplers bridged by long tantalum airbridges, we realize an extended Su-Schrieffer-Heeger (SSH) model and map its full topological phase diagram. Through a nested Fourier analysis over time and space, we directly extract the winding phase across Brillouin zone and resolve quantized winding numbers of ν={0,±1,2}ν=\{0,\pm1,2\}ν={0,±1,2} characterized by phase winding {0,±2π,4π}\{0, \pm 2π,4π\}{0,±2π,4π}. Our work establishes a versatile tool for probing topological characteristics and opens a new avenue for robust quantum computing architectures based on topological protection.
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.