Out-of-time-ordered correlator in the quantum bakers map and truncated unitary matrices

TL;DR

This paper analytically studies the out-of-time-ordered correlator in the quantum bakers map, revealing exponential growth at the Lyapunov rate and saturation behavior linked to random matrix theory, using semiquantum approximations and truncated unitaries.

Contribution

It provides an exact analytical solution for the OTOC in the quantum bakers map and connects the growth and saturation of correlators to classical chaos and random matrix theory.

Findings

OTOC grows exponentially at Lyapunov rate

Semiquantum approximation tracks correlator growth till Ehrenfest time

Saturation of correlators aligns with random matrix predictions

Abstract

The out-of-time-ordered correlator (OTOC) is a measure of quantum chaos that is being vigorously investigated. Analytically accessible simple models that have long been studied in other contexts could provide insights into such measures. This paper investigates the OTOC in the quantum bakers map which is the quantum version of a simple and exactly solvable model of deterministic chaos that caricatures the action of kneading dough. Exact solutions based on the semiquantum approximation are derived that tracks very well the correlators till the Ehrenfest time. The growth occurs at the exponential rate of the classical Lyapunov exponent, but modulated by slowly changing coefficients. Beyond this time saturation occurs as a value close to that of random matrices. Using projectors for observables naturally leads to truncations of the unitary time-$t$ propagator and the growth of their singular values is shown to be intimately related to the growth of the out-of-time-ordered correlators.