# Out-of-time-order correlators in quantum mechanics

## TL;DR

This paper explores the calculation of out-of-time-order correlators (OTOCs) in quantum mechanics, demonstrating their behavior in different systems and discussing their classical limit, with implications for understanding quantum chaos.

## Contribution

It provides a general formulation for calculating OTOCs in quantum systems and analyzes their behavior in various models, including chaotic and integrable systems.

## Key findings

- OTOCs are periodic in harmonic oscillator and particle in a box due to energy spectrum
- In billiard systems, OTOCs saturate to temperature-dependent constants
- No exponential growth of OTOC observed in stadium billiards, despite classical chaos

## Abstract

The out-of-time-order correlator (OTOC) is considered as a measure of quantum chaos. We formulate how to calculate the OTOC for quantum mechanics with a general Hamiltonian. We demonstrate explicit calculations of OTOCs for a harmonic oscillator, a particle in a one-dimensional box, a circle billiard and stadium billiards. For the first two cases, OTOCs are periodic in time because of their commensurable energy spectra. For the circle and stadium billiards, they are not recursive but saturate to constant values which are linear in temperature. Although the stadium billiard is a typical example of the classical chaos, an expected exponential growth of the OTOC is not found. We also discuss the classical limit of the OTOC. Analysis of a time evolution of a wavepacket in a box shows that the OTOC can deviate from its classical value at a time much earlier than the Ehrenfest time.