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

TL;DR

This paper develops a phase space (Wigner function) formulation of the quantum geometric tensor, including non-Abelian cases, enabling new experimental and theoretical insights into quantum state geometry.

Contribution

It introduces a phase space approach to compute the quantum geometric tensor and Berry connection, including non-Abelian generalizations, using Wigner functions.

Findings

Quantum metric tensor can be computed from Wigner functions.

The approach is applicable to quantum many-body systems.

Analytic expressions are obtained for coupled harmonic oscillators.

Abstract

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.