Time-dependent quantum geometric tensor and some applications

TL;DR

This paper introduces a time-dependent quantum geometric tensor to analyze the geometry of quantum states evolving over time, revealing new insights into energy dispersion, curvature, and entanglement in time-dependent systems.

Contribution

It extends the quantum geometric tensor to include temporal components, enabling analysis of explicitly time-dependent quantum states and their geometric properties.

Findings

Time-time component relates to energy dispersion.

Scalar curvature analysis reveals geometric features.

Purity analysis shows entanglement behavior in oscillator chains.

Abstract

We define a time-dependent extension of the quantum geometric tensor to describe the geometry of the time-parameter space for a quantum state, by considering small variations in both time and wave function parameters. Compared to the standard quantum geometric tensor, this tensor introduces new temporal components, enabling the analysis of systems with non-time-separable or explicitly time-dependent quantum states and encoding new information about these systems. In particular, the time-time component of this tensor is related to the energy dispersion of the system. We applied this framework to a harmonic/inverted oscillator, a time-dependent harmonic oscillator, and a chain of generalized harmonic/inverted oscillators. We show some results on the scalar curvature associated with the time-dependent quantum geometric tensor and the generalized Berry curvature behavior on the transition from harmonic oscillators to inverted ones. Furthermore, we analyze the entanglement for the chain through purity analysis, obtaining that the purity for any excited state is zero in the mentioned transitions.