Quantum instability and Ehrenfest time for an inverted harmonic oscillator

TL;DR

This paper explores the quantum-classical correspondence in an inverted harmonic oscillator, revealing that out-of-time-order correlators exhibit exponential growth rates related to classical Lyapunov exponents, influenced by initial conditions.

Contribution

It demonstrates that OTOCs in the IHO system have exponential growth rates twice the classical Lyapunov exponent and visualizes quantum wave packet evolution during this growth.

Findings

OTOCs have exponential growth rates twice the classical Lyapunov exponent.

Growth rates depend on initial photon number and phase space position.

Husimi Q function visualizes quantum wave packet dynamics during exponential growth.

Abstract

We investigate the classical-quantum correspondence in the inverted harmonic oscillator (IHO) system. It is shown that the out-of-time-order correlators (OTOCs) which the initial states are located at any position in the IHO system possess the same exponential growth rates (EGRs) as that at the saddle point, and their EGRs are twice the classical lyapunov exponent (CLE) of the saddle point. Through the time evolution of mean photon number and the OTOCs, we exhibit that the classical-quantum correspondence in the IHO system not only depends on the initial system photon number, but also on the central positions of the initial states in the phase space. Moreover, we use the Husimi Q function to visualize the quantum wave packets during the OTOCs grow exponentially.