Measuring a Topological Transition in an Artificial Spin-1/2System

Abstract

We present measurements of a topological property, the Chern number (C1), of a closed manifold in the space of two-level system Hamiltonians, where the two-level system is formed from a superconducting qubit. We manipulate the parameters of the Hamiltonian of the superconducting qubit along paths in the manifold and extract C1 from the nonadiabitic response of the qubit. By adjusting the manifold such that a degeneracy in the Hamiltonian passes from inside to outside the manifold, we observe a topological transition C1 = 1 → 0. Our measurement of C1 is quantized to within 2 percent on either side of the transition.

Quantum systems under frequency modulation

“…This has been demonstrated in two experiments with superconducting qubits [267,268]. Differently from the work on geometric phases, as we show below, the measurement technique in these works is not an interferometric scheme.…”

“…In a) the vector wraps the sphere, and this phase is topologically nontrivial with the first Chern number Ch 1 = 1. In b) the phase is topologically trivial, Ch 1 = 0. c) Phase diagram showing the expected transition between the two phases at δ 0 = δ r , obtained experimentally in references [267,268] by measuring the Berry curvature and using equation (168). Now, the possibility of controlling the detuning δ d in these systems enables the adiabatic exploration of this ellipsoidal manifold, see figure 13.…”

“…In the diagrams below, the Bloch vector is represented as an arrow for each point on the surface of the ellipsoid in the coordinates ξ x , ξ y , and ξ z . For a perfect adiabatic transport, the angle ϕ from the σ x axis on the Bloch sphere should equal φ (both are zero in the experiments in references [267,268]). Nonadiabaticity produces shifts in the Bloch vectors away from the plane ϕ = 0, and this leads to a measurable response function which, in the first order in the speed v χ = dχ/dt, is precisely the Berry curvature.…”

“…The quantity σ y is obtained by performing quantum tomography at each time, that is, at each point in the parameter space [268]. From the values of the Berry curvature, the Chern number can be calculated by integration, using the symmetry with respect to φ,…”

Superconducting circuit probe for analog quantum simulators

“…These circuit elements can be used to emulate fermionic and bosonic degrees of freedom in a broad range of many-body problems, including quantum spin systems [7][8][9][10][11][12][13][14][15][16], coupled cavity array models [17][18][19], topological phases [20,21], and electron-phonon physics [22,23]. Recent experiments have demonstrated arrays of quantum spins coupled simultaneously to one resonator [24], switching of the Chern number on one or two qubits [25,26] and weak localization of superconducting qubits [27].…”