Observing Topological Invariants Using Quantum Walks in Superconducting Circuits

Abstract

The direct measurement of topological invariants in both engineered and naturally occurring quantum materials is a key step in classifying quantum phases of matter. Here we motivate a toolbox based on time-dependent quantum walks as a method to digitally simulate single-particle topological band structures. Using a superconducting qubit dispersively coupled to a microwave cavity, we implement two classes of split-step quantum walks and directly measure the topological invariant (winding number) associated with each. The measurement relies upon interference between two components of a cavity Schrödinger cat state and highlights a novel refocusing technique which allows for the direct implementation of a digital version of Bloch oscillations. Our scheme can readily be extended to higher dimensions, whereby quantum walk-based simulations can probe topological phases ranging from the quantum spin Hall effect to the Hopf insulator.

Topological characterization of chiral models through their long time dynamics

“…The scheme proposed above effectively realizes a discrete-time quantum walk with a four-dimensional (4D) coin, a generalization of the usual topological quantum walk with two-dimensional coin [21][22][23][24][25]27]. In order to completely characterize the topology of this driven model, we follow the method proposed in [27], and recently implemented in [24].…”

“…The topological invariant of the bulk, the winding number  , allows one to predict the number of zero energy edge states. 1D chiral topological insulators have been realized in numerous platforms as ultracold atoms [6,11], photonic crystals [15], photonic quantum walks [21][22][23][24][25]. Let us notice that the 1D chiral Hamiltonian can be static or the effective Hamiltonian of a Floquet system.…”

“…The observation of edge states [11,21,30,31] and the measurement of the winding number through the mean chiral displacement [24] belong to the first category. The measurement of the winding number by interferometric architectures [6,25], by introducing losses [15,32,33], and by scattering measurements [22] belong to the second category. Here we generalize the notion of mean chiral displacement introduced in [24], and we present an intrinsic measurement of the topology for 1D chiral Hamiltonians with an arbitrary (even) number of sites per unit cell.…”

Eigenvalue measurement of topologically protected edge states in split-step quantum walks

“…Phenomena such as the quantum Hall effect [1] and topological insulators [2,3] aroused vivid interest in the study of the topological properties of physical systems. While these effects have been originally observed in semiconductor systems, experimental studies have been conducted on systems such as ultra cold atoms [4][5][6][7], photonic model systems [8][9][10][11][12], solid-state systems [13,14], superconducting circuits [15], mechanical oscillators [16] and microwave networks [17][18][19]. In photonic systems, topological phenomena can be accessed by implementing a split-step quantum walk on a 1D optical lattice [20][21][22].…”

Simulating Dirac Hamiltonian in curved space-time by split-step quantum walk

“…This motivates us to generalize the SS-DQW operation and study the consequences of it.In this paper starting from a slightly modified version of the single-step split-step DQW (SS-DQW) [26] whose coin operators are time and position-step dependent (inhomogeneous both in time and space), we derive a SS-DQW version of the (1+1) dimensional massive Dirac particle Hamiltonian under the influence of the U(1) gauge potential in curved space-time. This scheme is realizable in various physical table-top system as the SS-DQW has been proposed and successfully implemented in various systems like cold atoms [27], superconducting qubits [28,29], photonic systems [30,31]. Our scheme can also describe the (2+1) dimensional Dirac Hamiltonian in curved space-time when one component of momentum of the particle remains fixed.…”