Classical analog of the quantum metric tensor

The geometry of the parameter space is encoded by the quantum geometric tensor, which captures fundamental information about quantum states and contains both the quantum metric tensor and the curvature of the Berry connection. We present a formulation of the Berry connection and the quantum geometric tensor in the framework of the phase space or Wigner function formalism. This formulation is obtained through the direct application of the Weyl correspondence to the geometric structure under consideration. In particular, we show that the quantum metric tensor can be computed using only the Wigner functions, which opens an alternative way to experimentally measure the components of this tensor. We also address the non-Abelian generalization and obtain the phase space formulation of the Wilczek-Zee connection and the non-Abelian quantum geometric tensor. In this case, the non-Abelian quantum metric tensor involves only the non-diagonal Wigner functions. Then, we verify our approach with examples and apply it to a system of N coupled harmonic oscillators, showing that the associated Berry connection vanishes and obtaining the analytic expression for the quantum metric tensor. Our results indicate that the developed approach is well adapted to study the parameter space associated with quantum many-body systems.

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“…This result coincides with the usual expression found in the literature (see Ref [26], for instance). Notice that as a consequence of Eq.…”

Section: Illustrative Examplesupporting

confidence: 93%

Phase space formulation of the Abelian and non-Abelian quantum geometric tensor

González¹,

Gutiérrez‐Ruiz²,

Vergara³

2020

J. Phys. A: Math. Theor.

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“…The advantage of the metric over the metric introduced in ref. [] is that it does not require for its calculation the knowledge of the generating functions

G_{i} false(q, I; x false) : = - {(\partial_{i} S^{(α)})}_{q, I}

, which are not always easy to determine. Notice that Equation provides the semiclassical relation between the quantum metric tensor and the classical metric, and its validity is limited to the case of classical integrable systems.…”

Section: Classical Counterparts Of the Quantum Metric Tensor And The mentioning

confidence: 99%

“…Inserting Equations – into Equation and using

\begin{matrix} true \\ left \int_{- \infty}^{0} normald t_{1} \int_{0}^{\infty} normald t_{2} \cos 2 ω t_{12} = - \frac{1}{4 ω^{2}}, \end{matrix}

\begin{matrix} true \\ left \int_{- \infty}^{0} normald t_{1} \int_{0}^{\infty} normald t_{2} \sin 2 ω t_{12} = 0 \end{matrix}

we find the classical metric

g_{i j} false(I; x false)

g_{i j} false(I; x false) = \frac{I^{2}}{32 ω^{4}} (\begin{matrix} Z^{2} & - 2 Y Z & 2 Y^{2} - X Z \\ - 2 Y Z & 4 X Z & - 2 X Y \\ 2 Y^{2} - X Z & - 2 X Y & X^{2} \end{matrix})

which is exactly the same as that obtained in ref. [] by using the classical metric that involves generating functions. This corroborates our claim above that Equation yields the same results of the metric introduced in ref.…”

Section: Illustrative Examplesmentioning

confidence: 99%

“…Then, plugging these expressions into Equation and using Equations and , we arrive at the components of the classical curvature

F_{12} false(I; x false) = - \frac{Z I}{4 ω^{3}}, F_{13} false(I; x false) = \frac{Y I}{4 ω^{3}}, F_{23} false(I; x false) = - \frac{X I}{4 ω^{3}}

The components are precisely those found in ref. [] via the use of generating functions (see also ref. []).…”

Section: Illustrative Examplesmentioning

confidence: 99%

“…In the case of the quantum metric tensor, its classical analog was recently introduced in ref. [28] and corresponds to a classical metric in the www.advancedsciencenews.com www.ann-phys.org parameter space that measures the distance between two points in phase space with infinitesimally different parameters. The essential element involved in the classical metric is the generating function of the canonical transformation that infinitesimally shifts the phase space variables with respect to the parameters of the Hamiltonian.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Geometry of the Parameter Space of a Quantum System: Classical Point of View

Alvarez‐Jimenez¹,

González²,

Gutiérrez‐Ruiz³

et al. 2019

Annalen der Physik

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The local geometry of the parameter space of a quantum system is described by the quantum metric tensor and the Berry curvature, which are two fundamental objects that play a crucial role in understanding geometrical aspects of condensed matter physics. Classical integrable systems are considered and a new approach is reported to obtain the classical analogs of the quantum metric tensor and the Berry curvature. An advantage of this approach is that it can be applied to a wide variety of classical systems corresponding to quantum systems with bosonic and fermionic degrees of freedom. The approach used arises from the semiclassical approximation of the Berry curvature and the quantum metric tensor in the Lagrangian formalism. This semiclassical approximation is exploited to establish, for the first time, the relation between the quantum metric tensor and its classical counterpart. The approach described is illustrated and validated by applying it to five systems: the generalized harmonic oscillator, the symmetric and linearly coupled harmonic oscillators, the singular Euclidean oscillator, and a spin‐half particle in a magnetic field. Finally, some potential applications of this approach and possible generalizations that can be of interest in the field of condensed matter physics are mentioned.

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Adiabatic driving, geometric phases, and the geometric tensor for classical states

Bermúdez Manjarres

2024

Annals of Physics

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Classical analog of the quantum metric tensor

Cited by 15 publications

References 44 publications

Phase space formulation of the Abelian and non-Abelian quantum geometric tensor

Phase space formulation of the Abelian and non-Abelian quantum geometric tensor

Geometry of the Parameter Space of a Quantum System: Classical Point of View

Adiabatic driving, geometric phases, and the geometric tensor for classical states

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