Quantum resonant systems, integrable and chaotic

TL;DR

This paper introduces quantum infinite-dimensional resonant systems, revealing their surprisingly simple structure despite complex classical dynamics, and explores their spectral properties and potential applications.

Contribution

It presents a detailed analysis of quantum resonant systems, showing their block-diagonal Hamiltonian structure and solvability, bridging classical chaos and quantum simplicity.

Findings

Quantum resonant systems have a block-diagonal Hamiltonian in the Fock basis.

Spectral statistics distinguish integrable from chaotic cases.

Quantum systems are simpler than their classical counterparts despite complex dynamics.

Abstract

Resonant systems emerge as weakly nonlinear approximations to problems with highly resonant linearized perturbations. Examples include nonlinear Schroedinger equations in harmonic potentials and nonlinear dynamics in Anti-de Sitter spacetime. The classical dynamics within this class of systems can be very rich, ranging from fully integrable to chaotic as one changes the values of the mode coupling coefficients. Here, we initiate a study of quantum infinite-dimensional resonant systems, which are mathematically a highly special case of two-body interaction Hamiltonians (extensively researched in condensed matter, nuclear and high-energy physics). Despite the complexity of the corresponding classical dynamics, the quantum version turns out to be remarkably simple: the Hamiltonian is block-diagonal in the Fock basis, with all blocks of varying finite sizes. Being solvable in terms of diagonalizing finite numerical matrices, these systems are thus arguably the simplest interacting quantum field theories known to man. We demonstrate how to perform the diagonalization in practice, and study both numerical patterns emerging for the integrable cases, and the spectral statistics, which efficiently distinguishes the special integrable cases from generic (chaotic) points in the parameter space. We discuss a range of potential applications in view of the computational simplicity and dynamical richness of quantum resonant systems.