Extracting the quantum metric tensor through periodic driving

Abstract

We propose a generic protocol to experimentally measure the quantum metric tensor, a fundamental geometric property of quantum states. Our method is based on the observation that the excitation rate of a quantum state directly relates to components of the quantum metric upon applying a proper time-periodic modulation. We discuss the applicability of this scheme to generic two-level systems, where the Hamiltonian's parameters can be externally tuned, and also to the context of Bloch bands associated with lattice systems. As an illustration, we extract the quantum metric of the multi-band Hofstadter model. Moreover, we demonstrate how this method can be used to directly probe the spread functional, a quantity which sets the lower bound on the spread of Wannier functions and signals phase transitions. Our proposal offers a universal probe for quantum geometry, which could be readily applied in a wide range of physical settings, ranging from circuit-QED systems to ultracold atomic gases. arXiv:1803.05818v1 [cond-mat.mes-hall]

Floquet-engineering of nodal rings and nodal spheres and their characterization using the quantum metric

“…when q 2 y + q 2 z = ρ 2 . Consequently, performing a quantummetric tomography in the projected parameter spacefor instance, by measuring dissipative responses upon parametric modulations [45] -would offer a direct signature of the topological nodal ring. As a technical remark, we point out that such quantum-geometric measurements rely on preparing the system in a gapped state, i.e.…”

Experimental Measurement of the Quantum Metric Tensor and Related Topological Phase Transition with a Superconducting Qubit

“…Recently, it was shown that the quantum metric is closely related to the quantum phase transition [26], superfluidity in flat bands [27], and topological matter [28][29][30]. Several schemes have been proposed to extract the quantum metric [25,[31][32][33][34][35][36], but its direct experimental measurement remains elusive. Moreover, to the best of our knowledge, a topological phase transition characterized by the ECN [24], which can be obtained from the quantum-metric measurements, is still lacking in experiments.…”

“…The second method is to detect the excitation rate under weak periodic driving on a quantum state, as proposed in Ref. [32]. Furthermore, based on quantum-metric and Berrycurvature measurements, we explore a phase transition in a simulated time-reversal-symmetric (TRS) system, which is characterized by the ECN instead of the (vanishing) Chern number.…”

“…To extract the off-diagonal components g µν (µ = ν), we apply a sudden quench to λ 0 +δλe µ +δλe ν along the e µ +e ν direction and then measure the probability P + µν , which has the relation P + µν −P + µµ −P + νν = 2g µν δλ 2 +O(δλ 3 ). Another scheme we used to extract the components of the quantum metric is via the excitation rate of a quantum state upon applying proper time-periodic modulations [32,37].…”

Localization anisotropy and complex geometry in two-dimensional insulators

“…Numerous proposals for extracting the quantum metric have been presented (see Refs. [21][22][23][24][25][26][27] ). The first successful measurement of the quantum metric and, actually, the full quantum geometric tensor, whose real part is the quantum metric and whose imaginary part is the Berry curvature, was performed using nitrogen vacancy centers in diamond, see Ref.…”