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Extracting the quantum metric tensor through periodic driving

Tomoki Ozawa, Nathan Goldman

Published Aug 20, 2020• 0 citations
Publisher

Abstract

A quantum state, such as a qubit state, defined in a parameter space is endowed with a property called the geometrical property. The geometrical property tells how much the quantum state changes in a parameter space; this property is a gauge invariant, physical, quantity, but it has not been experimentally observed until very recently. We give a simple experimental protocol to measure the geometrical property by periodically modulating the system and looking at the system response. The method can be easily applied to various qubit systems such as diamond NV centers and superconducting qubits. We will also review the current experimental situations regarding the measurement of geometrical structures.

DOI
10.1117/12.2567905

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A quantum state, such as a qubit state, defined in a parameter space is endowed with a property called the geometrical property. The geometrical property tells how much the quantum state changes in a parameter space; this property is a gauge invariant, physical, quantity, but it has not been experimentally observed until very recently. We give a simple experimental protocol to measure the geometrical property by periodically modulating the system and looking at the system response. The method can be easily applied to various qubit systems such as diamond NV centers and superconducting qubits. We will also review the current experimental situations regarding the measurement of geometrical structures.
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 multiband 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 superconducting quantum circuits to ultracold atomic gases.
Abstract Geometry and topology are fundamental concepts, which underlie a wide range of fascinating physical phenomena such as topological states of matter and topological defects. In quantum mechanics, the geometry of quantum states is fully captured by the quantum geometric tensor. Using a qubit formed by an NV center in diamond, we perform the first experimental measurement of the complete quantum geometric tensor. Our approach builds on a strong connection between coherent Rabi oscillations upon parametric modulations and the quantum geometry of the underlying states. We then apply our method to a system of two interacting qubits, by exploiting the coupling between the NV center spin and a neighboring 13C nuclear spin. Our results establish coherent dynamical responses as a versatile probe for quantum geometry, and they pave the way for the detection of novel topological phenomena in solid state.
Topology played an important role in physics research during the last few decades. In particular, the quantum geometric tensor that provides local information about topological properties has attracted much attention. It will reveal interesting topological properties but have not been measured in non-Abelian systems. Here, we use a four-qubit quantum system in superconducting circuits to construct a degenerate Hamiltonian with parametric modulation. By manipulating the Hamiltonian with periodic drivings, we simulate the Bernevig–Hughes–Zhang model and obtain the quantum geometric tensor from interference oscillation. In addition, we reveal its topological feature by extracting the topological invariant, demonstrating an effective protocol for quantum simulation of a non-Abelian system.
A Berry curvature is an imaginary component of the quantum geometric tensor (QGT) and is well studied in many branches of modern physics; however, the quantum metric as a real component of the QGT is less explored. Here, by using tunable superconducting circuits, we experimentally demonstrate two methods to directly measure the quantum metric tensor for characterizing the geometry and topology of underlying quantum states in parameter space. The first method is to probe the transition probability after a sudden quench, and the second one is to detect the excitation rate under weak periodic driving. Furthermore, based on quantum metric and Berry-curvature measurements, we explore a topological phase transition in a simulated time-reversal-symmetric system. The work opens up a unique approach to explore the topology of quantum states with the QGT.
We explore the relation between quantum geometry in non-Hermitian systems and physically measurable phenomena. We highlight various situations in which the behavior of a non-Hermitian system is best understood in terms of quantum geometry, namely the notion of adiabatic potentials in non-Hermitian systems and the localization of Wannier states in periodic non-Hermitian systems. Further, we show that the non-Hermitian quantum metric appears in the response of the system upon time-periodic modulation, which one can use to experimentally measure the non-Hermitian quantum metric. We validate our results by providing numerical simulations of concrete exemplary systems.
Quantum mechanics traditionally describes the state of a system by a vector in Hilbert space and the dynamics by a unitary evolution governed by a Hamiltonian. Recent advances in differential geometry applied to quantum theory have revealed that quantum states can carry geometric structures such as Berry phases and curvature tensors. In this paper we introduce the notion of a quantum state curvature field, denoted Kψ(x)K_\psi(x)Kψ​(x), which is defined pointwise on the parameter space of a family of states {ψ(x)}\{\psi(x)\}{ψ(x)}. We propose a protocol called Quantum State Curvature Measurement (QSCM) that reconstructs KψK_\psiKψ​ by observing continuous quantum evolution along a path in parameter space. The method relies on measuring the infinitesimal phase accumulation and the deviation from parallel transport, which together encode the curvature. We discuss the mathematical foundation of KψK_\psiKψ​, its relation to the Berry curvature, and the experimental feasibility of implementing QSCM on current quantum platforms. Finally, we outline the potential implications of discovering new quantum geometric degrees of freedom, including possible modifications to the foundational structure of quantum theory.