Direct Probe of Topological Invariants Using Bloch Oscillating Quantum Walks

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.Much like their classical stochastic counterparts, discrete-time quantum walks [1] have stimulated activity across a broad range of disciplines. In the context of computation, they provide exponential speedup for certain oracular problems and represent a universal platform for quantum information processing [2][3][4]. Quantum walks also exhibit features characteristic of a diverse set of physical phenomena, ranging from localization to molecule formation [5,6]. At their core, discrete-time quantum walks (DTQW) are dynamical protocols associated with spinful particles on a lattice, where the internal spin state controls the direction of motion [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. Motivated by this intrinsic spin-orbit coupling, a tremendous body of recent work has focused on exploring the topological features of DTQWs both theoretically and experimentally [5,[11][12][13][14][15].A connection between quantum walks and topology has been made by mapping the unitary evolution of the DTQW protocol to stroboscopic evolution under an effective Hamiltonian. In certain cases-distinguished by a combination of symmetry and dimensionality-the effective Hamiltonian's bandstructure exhibits a quantized invariant, analogous to those found in topological insulators [5,11,12,14]. On one hand, these invariants have helped to enable a sharp classification of non-interacting topological phases, which unlike conventional symmetry breaking phases, do not exhibit any local order parameters [20,21]. On the other, they underly a multitude of exotic physical phenomena ranging from protected edge modes and quantized conductance to fractional charges and magnetic monopoles [22,23]. Despite their importance and owing to their non-locality, bulk topological invariants have been directly probed in only a handful of quantum optical systems [24][25][26][27] and a generic blueprint for their measurement remains an outstanding challenge.In this Letter, we demonstrate that the simulation platform associated with discrete-time quantum walks is naturally suited for the direct extraction of topolog- ical invariants. We analyze a time-dependent,"Blochoscillating" generalization of t...

Topological polarons, quasiparticle invariants, and their detection in one-dimensional symmetry-protected phases

“…Here |Φ h (q) denotes the wavefunction of a hole excitation at quasimomentum q while |ψ I describes the wavefunction of the impurity bound to the hole. If the strong-coupling wavefunction (27) applies, where |ψ I is independent of q, it follows that the TP Zak phase is equivalent to the Zak phase of the hole. Being in the strong-coupling regime requires the impurity to be sufficiently mobile, J t 1,2 (see also Eq.…”

Detecting Topological Invariants in Nonunitary Discrete-Time Quantum Walks

“…While the experimental system in [12] features continuous-time non-Hermitian dynamics, an important question is whether topological invariants can also be defined for discrete-time non-unitary dynamics. This is particularly interesting in light of the recent demonstration of topological properties in unitary discrete-time QWs [9,10,26,29,30].…”

Simulation of the multiphase configuration and phase transitions with quantum walks utilizing a step-dependent coin

“…Since each phase is characterized by a topological invariants, this step dependency of bands of energy could be used as a method to measure topological invariants, similar to Ref. [23]. Later, in our calculations of topological invariants, we address this possibility.…”

“…In contrast, in Ref. [23], it was shown that if the protocol of the quantum walks is modified to be time dependent (more precisely, position dependent at each step), it will result in a Bloch-oscillating quantum walk which can be used to directly measure topological invariants imprinted as a Berry phase on the quantum state of the particle. The considered protocol for quantum walks in this paper is also Bloch-oscillating [32].…”

“…[21,22]. It was shown that quantum walks are suitable for direct extraction of topological invariants [23]. Photonic quantum walks were also explored experimentally in order to observe topologically protected bound states [24] and topological quantum transitions [25,26], to simulate Zak phases [27] or to detect topological invariants [28][29][30][31].…”