Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Stephen Wiggins

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 $\mathcal{O}(e^{ωT})$ 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.

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

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 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.
Paper Structure (6 sections, 21 equations, 1 figure)

This paper contains 6 sections, 21 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Theoretical Framework
  3. Scrambling at the Quantum Saddle: Explicit Trajectories
  4. Semiclassical Heuristics for the OTOC
  5. Outlook: Conjectural Extension to Microcanonical Regimes
  6. Conclusions

Figures (1)

  • Figure 1: Wavepacket-center Lagrangian Descriptors for the preparation-space dynamics of the inverted harmonic oscillator ($m=1, \omega=1$). The diagnostic is evaluated over a $1000 \times 1000$ grid of initial preparation parameters $(q_0, p_0)$ with an integration horizon of $T=5$. (Left and Center) The forward (Eq. 5) and backward (Eq. 6) wavepacket-center LDs, plotted on a logarithmic scale. Because LD arc-lengths are minimized on the invariant manifolds of a saddle, the stable and unstable manifolds appear as dark trenches. (Right) The product diagnostic $\mathcal{M}^{\text{wpc}}(\mathbf{x}_0)$ (Eq. 7). The negative logarithm compresses the exponential sensitivity and inverts the LD minima, revealing the underlying classical wavepacket-center skeleton as bright ridges in preparation space.