Geometric Diagnostics of Scrambling-Related Sensitivity in a Bohmian Preparation Space

TL;DR

This paper introduces a geometric diagnostic for quantum scrambling based on Bohmian trajectories and Lagrangian Descriptors, providing a new perspective on sensitivity growth in quantum systems, especially for the inverted harmonic oscillator.

Contribution

It develops a Bohmian, trajectory-based geometric framework for diagnosing quantum scrambling, connecting stability analysis with OTOC growth in a preparation space.

Findings

Exponential sensitivity growth in preparation space for inverted harmonic oscillator

Analytical tractability of wavepacket dynamics and stability matrix

Proposes a geometric indicator of scrambling-related sensitivity

Abstract

The Out-of-Time-Order Correlator (OTOC) is a standard algebraic diagnostic of quantum information scrambling, but it offers limited direct geometric intuition. In this note, we propose a Bohmian, trajectory-based framework for constructing a geometric diagnostic of scrambling-related sensitivity using Lagrangian Descriptors (LDs). To avoid the uncertainty-principle obstruction to assigning independent initial position and momentum within a single wave function, we evaluate Bohmian dynamics over a two-dimensional preparation space of localized Gaussian wavepackets labeled by their initial center and momentum kick. For the inverted harmonic oscillator, this construction is analytically tractable: the wavepacket-center dynamics and their dependence on preparation parameters can be written explicitly. In particular, away from the equilibrium origin, the exponential growth of the associated preparation-space stability matrix yields an $\mathcal{O}(e^{\omega T})$ bound on the sensitivity of the wavepacket-center LDs, motivating a semiclassical comparison with sensitivity structures associated with OTOC growth. In this sense, the LD provides a geometric indicator of scrambling-related sensitivity. We conclude by discussing how this preparation-space picture suggests a program for future work regarding the distinct microcanonical regimes previously reported for the inverted harmonic oscillator.