Quantum Mechanical Out-Of-Time-Ordered-Correlators for the Anharmonic (Quartic) Oscillator

TL;DR

This paper calculates out-of-time-ordered correlators (OTOCs) for the quantum anharmonic oscillator, revealing temperature-dependent behaviors that relate to quantum chaos and spectral properties, with implications for understanding quantum integrable systems.

Contribution

It provides the first detailed analysis of OTOCs in a quantum anharmonic oscillator, highlighting their temperature-dependent dynamics and spectral form factor behavior.

Findings

OTOCs are periodic at low temperature.

At high temperature, OTOCs rise rapidly then saturate.

Spectral form factor shows initial decay, bounce, and plateau with fluctuations.

Abstract

Out-of-time-ordered correlators (OTOCs) have been suggested as a means to study quantum chaotic behavior in various systems. In this work, I calculate OTOCs for the quantum mechanical anharmonic oscillator with quartic potential, which is classically integrable and has a Poisson-like energy-level distribution. For low temperature, OTOCs are periodic in time, similar to results for the harmonic oscillator and the particle in a box. For high temperature, OTOCs exhibit a rapid (but power-like) rise at early times, followed by saturation consistent with $2\langle x^2\rangle_T \langle p^2\rangle_T$ at late times. At high temperature, the spectral form factor decreases at early times, bounces back and then reaches a plateau with strong fluctuations.