Direct Probe of Topological Invariants Using Bloch Oscillating Quantum Walks

V. V. Ramasesh, E. Flurin, M. Rudner, I. Siddiqi, and N. Y. Yao

Phys. Rev. Lett. 118, 130501 – Published 27 March 2017

Abstract

The topology of a single-particle band structure plays a fundamental role in understanding a multitude of physical phenomena. Motivated by the connection between quantum walks and such topological band structures, we demonstrate that a simple time-dependent, Bloch-oscillating quantum walk enables the direct measurement of topological invariants. We consider two classes of one-dimensional quantum walks and connect the global phase imprinted on the walker with its refocusing behavior. By disentangling the dynamical and geometric contributions to this phase, we describe a general strategy to measure the topological invariant in these quantum walks. As an example, we propose an experimental protocol in a circuit QED architecture where a superconducting transmon qubit plays the role of the coin, while the quantum walk takes place in the phase space of a cavity.

Received 18 November 2016

DOI:https://doi.org/10.1103/PhysRevLett.118.130501

Physics Subject Headings (PhySH)

Geometric & topological phases Quantum information with atoms & light Quantum simulation Random walks Spin-orbit coupling Superconducting qubits Topological insulators Topological phases of matter

General PhysicsQuantum Information, Science & TechnologyCondensed Matter, Materials & Applied Physics

Authors & Affiliations

V. V. Ramasesh¹, E. Flurin¹, M. Rudner², I. Siddiqi¹, and N. Y. Yao¹

¹Department of Physics, University of California, Berkeley California 94720, USA
²Niels Bohr International Academy and Center for Quantum Devices, University of Copenhagen, 2100 Copenhagen, Denmark

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Vol. 118, Iss. 13 — 31 March 2017

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Images

Figure 1
The sequence of unitary operations associated with a single step of (a) the split-step quantum walk and (b) the Bloch-oscillating quantum walk. (c) The band structure and spin texture (arrows accompanying the band) for a trivial split-step quantum walk [ $U_{SS} (3 π / 4, π / 4)$ ] with $W = 0$ and (d) for a topological split-step quantum walk [ $U_{SS} (π / 4, 3 π / 4)$ ] with $W = 1$ . (e),(f) Schematic evolution of the spin eigenvectors in (c),(d) as one traverses the Brillouin zone. In the topological phase, the spin texture fully winds around the origin as $k$ varies from $[- π, π]$ . (g),(h) Analogous band structures for the trivial and topological Bloch-oscillating quantum walk. The shift in the effective momentum induced by the $z$ rotations (by $ϕ$ ) are shown explicitly.
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Figure 2
(a) Refocusing fidelity $F = | ⟨ ψ_{0} | ψ_{f} ⟩ |^{2}$ computed from numerical simulations of an $N = 30$ step Bloch-oscillating quantum walk as a function of ${θ_{1}, θ_{2}}$ , where $ϕ = 2 π / N$ . Two topologically distinct regions $W = 0$ ( $θ_{1} > θ_{2}$ ) and $W = 1$ ( $θ_{1} < θ_{2}$ ) are separated by a gapless line. In the vicinity of this gap closure, the refocusing fidelity drops dramatically owing to nonadiabatic transitions. The observed stripe pattern in the refocusing fidelity follows the contour lines for which the accumulated dynamical phase is a multiple of $π$ . (b)–(d) Three specific time evolutions associated with various ${θ_{1}, θ_{2}}$ : (b) a perfectly refocusing walk, (c) a nonrefocusing walk due to nonadiabatic transitions, and (d) a nonrefocusing walk due to an accumulated dynamical phase of $π / 2$ .
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Figure 3
(a) Nonadiabatic transition probability as a function of $θ_{1}$ for the single-step quantum walk ( $θ_{2} = 0$ ). Solid lines correspond to analytic formulas derived in Ref. [31], which capture the deviations from ideal refocusing behavior. Points correspond to $P_{↑ ↓}$ as computed from Eq. (9), demonstrating that the physical origin of such deviations is nonadiabatic Landau-Zener transitions. (b) Schematic band structures for various $θ_{1}$ . The band gap increases as $θ_{1}$ varies from 0 to $π$ leading to a smaller nonadiabatic transition probability.
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Figure 4
(a) Schematic of the proposed cQED setup for realizing Bloch-oscillating quantum walks, utilizing a superconducting cavity mode coupled to a transmon qubit. The levels $| g ⟩, | e ⟩$ form the internal spin states of the walker, while $| f ⟩$ is used as a shelving state. (b) The qubit and cavity couple dispersively, realizing a state-dependent shift of the bare cavity transition frequency $ω_{c}$ that naturally enables spin-dependent translations. (c) The quantum walk takes place on a circular lattice in the phase space of the cavity. Each coherent state depicted in the figure represents a particular lattice site of the walk. Spin-down (-up) corresponds to state $| g ⟩$ ( $| e ⟩$ ). (d) Wigner tomography $W (α)$ of the cavity following a refocused Bloch-oscillating quantum walk in the topological and trivial band structures reveals the underlying winding number in the phase of the interference fringes.
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