# A quantum Kolmogorov-Arnold-Moser theorem in the anisotropic Dicke model and its possible implications in the hybrid Sachdev-Ye-Kitaev models

## TL;DR

This paper establishes a quantum KAM theorem for the anisotropic Dicke model, linking quantum integrability and chaos, and explores implications for hybrid Sachdev-Ye-Kitaev models using large N expansion and Random Matrix Theory.

## Contribution

It provides the first quantum KAM theorem for the anisotropic Dicke model, connecting quantum stability and chaos with potential implications for SYK models.

## Key findings

- Quantum KAM theorem characterizes transition from integrability to chaos.
- Agreement between large N expansion and Random Matrix Theory approaches.
- Connections to and differences from Sachdev-Ye-Kitaev models are analyzed.

## Abstract

The classical Kolmogorov-Arnold-Moser (KAM) theorem provides the underlying mechanism for the stability of the solar system under some small chaotic perturbations. Despite many previous efforts, any quantum version of the KAM theorem remains elusive In this work, we provide a quantum KAM theorem in the context of the anisotropic Dicke model which is the most important quantum optics model. It describes a single mode of photons coupled to $ N $ qubits with both a rotating wave (RW) term and a counter-RW (CRW) term. As the ratio of the CRW over the RW term increases from zero to one, the systems evolves from quantum integrable to quantum chaotic. We establish a quantum KAM theorem to characterize such a evolution quantitatively by both large $ N $ expansion and Random Matrix Theory and find agreement from the two complementary approaches. Connections and differences between the Dicke models and Sachdev-Ye-Kitaev (SYK) or hybrid SYK models are examined. Possible Quantum KAM theorem in terms of other quantum chaos criterion such as quantum Lyapunov exponent is also discussed.