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arXiv:1803.05818 · Physical review. B./Physical review. B

Extracting the quantum metric tensor through periodic driving

Tomoki Ozawa, Nathan Goldman

Published May 31, 2018• 127 citations
arXivHTMLPDF

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 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.

DOI
10.1103/physrevb.97.201117

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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.
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.
Hyperbolic band theory (HBT) has recently unveiled a wealth of exotic physical phenomena in negatively curved spaces. Concurrently, the quantum metric has emerged as a fundamental tensor governing quantum geometry and topological phases. In this Letter, we bridge these two frontiers by investigating the quantum metric in HBT. Because conventional Euclidean metric extraction protocols fundamentally fail in these non-commutative geometries, we introduce holonomy shaking, a novel, experimentally viable technique utilizing periodic lattice driving to dynamically extract the metric. Using the {8,3} regular hyperbolic and kagome-like lattices as concrete examples, we demonstrate the high-fidelity dynamical extraction of the quantum metric. Our results establish a robust blueprint for probing hyperbolic quantum geometries, paving the way for the discovery of exotic topological phenomena in hyperbolic quantum matter.
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.
As a fundamental concept, geometry is widely used in understanding physical phenomena. In quantum mechanics, geometry is related to the system’s quantum state and can be characterized by the quantum geometric tensor (QGT), whose real part is referred to as the quantum metric tensor (QMT), which defines the distance between two neighboring quantum states in the projected Hilbert space. Several pieces of research based on discrete variables have been proposed to extract the QMT, but research with the use of continuous variables is lacking. Here, we propose a method to extract the QMT of a continuous variable system, specified here as a cat-qubit. The method is developed by constructing the Kerr nonlinear parametric oscillator (KNPO) and by modulating it with external drives to induce adiabatic dynamics process within the state subspace spanned by the even and odd Scho¨dinger cat states. The method paves the way for exploring the geometry for continuous variable systems.
The quantum geometric tensor sets the local relation between variations of external parameters and distances between states in Hilbert space. Its real and imaginary parts are, respectively, the quantum Fisher information matrix, which determines the ultimate precision bound in parameter estimation, and the Berry curvature. We study the geometry of SU(1,1) Hamiltonian eigenstates and propose two dynamical methods for extracting the components of the SU(1,1) quantum geometric tensor. A time-separated quench method extracts both the real and imaginary parts of the quantum geometric tensor from transition probabilities following two sudden parameter changes separated by free evolution, while a weak resonant modulation method relates quasiparticle production to the same geometric tensor. These results provide an operational method for measuring the local geometry of single-mode squeezed states and, in particular, for extracting the quantum Fisher information matrix directly from experimental observables.
The complete geometry of quantum states in parameter space is characterized by the quantum geometric tensor, which contains the quantum metric and Berry curvature as the real and imaginary parts, respectively. When the quantum states are degenerate, the quantum metric and Berry curvature take non-Abelian forms. The non-Abelian (Abelian) Berry curvature and Abelian quantum metric have been experimentally measured. However, an experimentally feasible scheme to extract all the components of the non-Abelian quantum metric tensor is still lacking. Here we propose a generic protocol to directly extract the non-Abelian quantum metric tensor in arbitrary degenerate quantum states in any dimensional parameter space based on measuring the transition probabilities after parameter quenches. Furthermore, we show that the non-Abelian quantum metric can be measured to obtain the real Chern number of a generalized Dirac monopole and the second Chern number of a Yang monopole, which can be simulated in three- and five-dimensional parameter space of artificial quantum systems, respectively. We also demonstrate the feasibility of our quench scheme for these two applications with numerical simulations.