The Multi-faceted Inverted Harmonic Oscillator: Chaos and Complexity

TL;DR

This paper investigates the quantum chaos and complexity of the inverted harmonic oscillator using advanced diagnostics like OTOC, Lyapunov spectrum, and circuit complexity, revealing chaotic behavior and new spectral structures.

Contribution

It introduces comprehensive chaos diagnostics for the inverted harmonic oscillator, including the full Lyapunov spectrum and complexity analysis, extending to multiple oscillators and different dynamical regimes.

Findings

OTOC reveals genuine and quasi scrambling in the IHO.

Lyapunov exponents form a paired spectral structure.

Complexity exhibits chaotic behavior across regimes.

Abstract

The harmonic oscillator is the paragon of physical models; conceptually and computationally simple, yet rich enough to teach us about physics on scales that span classical mechanics to quantum field theory. This multifaceted nature extends also to its inverted counterpart, in which the oscillator frequency is analytically continued to pure imaginary values. In this article we probe the inverted harmonic oscillator (IHO) with recently developed quantum chaos diagnostics such as the out-of-time-order correlator (OTOC) and the circuit complexity. In particular, we study the OTOC for the displacement operator of the IHO with and without a non-Gaussian cubic perturbation to explore genuine and quasi scrambling respectively. In addition, we compute the full quantum Lyapunov spectrum for the inverted oscillator, finding a paired structure among the Lyapunov exponents. We also use the Heisenberg group to compute the complexity for the time evolved displacement operator, which displays chaotic behaviour. Finally, we extended our analysis to N-inverted harmonic oscillators to study the behaviour of complexity at the different timescales encoded in dissipation, scrambling and asymptotic regimes.