Third-Order Perturbative OTOC of the Harmonic Oscillator with Quartic Interaction and Quantum Chaos

TL;DR

This paper analytically and numerically investigates the third-order out-of-time-order correlator of a harmonic oscillator with quartic interaction, revealing quantum chaos signatures through exponential growth and saturation behaviors.

Contribution

It provides the first analytic relations for the spectrum and matrix elements of the system, enabling detailed numerical evaluation of the OTOC and identifying chaos indicators.

Findings

OTOC saturates after initial growth, indicating quantum scrambling.

Exponential growth in OTOC observed with strong quartic interactions.

Numerical results support the presence of quantum chaos signatures.

Abstract

We calculate the third-order out-of-time-order correlator (OTOC) of a simple harmonic oscillator with an additional quartic interaction using the second quantization method. We obtain analytic relations for the spectrum, Fock space states, and matrix elements of the coordinate, which are then used to numerically evaluate the OTOC. We observe that after the scrambling, the OTOC becomes a fluctuation around a saturation point at later times, which is associated with quantum chaotic behavior in systems that exhibit chaos. We analyze the early-time properties of the OTOC and find that in systems with sufficiently strong quartic interactions, an exponential growth curve fitting over a long time window clearly emerges in the third-order perturbation