Time-dependent quantum geometric tensor and some applications

TL;DR

This work introduces a time-dependent quantum geometric tensor (tQGT) that extends the Provost–Valle quantum geometric tensor by treating time as a coordinate on the same manifold as state-parameters. The tensor $Q_{IJ}$ decomposes into a real time-dependent metric $g_{IJ}$ and an imaginary time-dependent curvature $F_{IJ}$ (the tBerry curvature), with the notable result that $Q_{00}=(\Delta E)^2/\hbar^2$ under Schrödinger dynamics and stationary states recovering the standard QGT. The authors apply the formalism to Gaussian, non-time-separable states in harmonic/inverted oscillators, a time-dependent harmonic oscillator, and a chain of coupled oscillators, revealing ground-state signatures, bifurcations at oscillator transitions, hyperbolic geometry (e.g., $\mathcal{R}=-16$ in 2D regions), and entanglement behavior via purity that vanishes at bifurcations. The framework provides a unified geometric lens for time-dependent quantum dynamics, with potential applications to non-adiabatic, driven, and open systems and to exploring dynamical phase-like structures through curvature and metric singularities.

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.

Figures (21)

• Figure 1: Some components of the tQMT, as a function of the parameter $B$. Except for $g_{03}$, the tQMT components are singular at $B=0$. The remaining parameters were fixed as follows: $X=W=1$ and $t=10$. (a) These components have a minimum at around $B=1$. (b) These components show a monotone behavior.
• Figure 2: Component $g_{11}$ of the tQMT in the region 1 as a function of time. The parameters were fixed as $X=W=1$.
• Figure 3: Non-zero components of the tBerry curvature in the region 1 by fixing $X=W=1$. As a function of the parameter $B$, taking $t=5$. They cross zero around $B=1$.
• Figure 4: Non-zero components of the tBerry curvature for the harmonic oscillator as a function of time, taking $B=2$.
• Figure 5: Distribution density probability as a function of $q$ and $t$, with $X=W=1$ and $B=1/2,2$. With this choice of parameters, $\rho(q,t;1/2)$ and $\rho(q,t;2)$ crosses at $\tau=\pi/4,3\pi/4$