Quantum bounds on the generalized Lyapunov exponents

TL;DR

This paper explores quantum bounds on generalized Lyapunov exponents, linking them to chaos limits via fluctuation-dissipation relations, and demonstrates these concepts through numerical analysis of the kicked top model.

Contribution

It introduces a generalized bound on quantum chaos using the spectrum of the commutator and verifies these bounds with numerical simulations of a quantum chaotic system.

Findings

Generalized exponents obey a chaos bound derived from fluctuation-dissipation theorem.

Stronger bounds for larger q restrict large deviations in quantum chaos.

Numerical results from the kicked top model support the theoretical bounds.

Abstract

We discuss the generalized quantum Lyapunov exponents $L_q$, defined from the growth rate of the powers of the square commutator. They may be related to an appropriately defined thermodynamic limit of the spectrum of the commutator, which plays the role of a large deviation function, obtained from the exponents $L_q$ via a Legendre transform. We show that such exponents obey a generalized bound to chaos due to the fluctuation-dissipation theorem, as already discussed in the literature. The bounds for larger $q$ are actually stronger, placing a limit on the large deviations of chaotic properties. Our findings at infinite temperature are exemplified by a numerical study of the kicked top, a paradigmatic model of quantum chaos.