Integrability in time-dependent systems with one degree of freedom

TL;DR

This paper explores the integrability of classical nonautonomous one-degree-of-freedom systems, proving the existence of invariants that prevent chaos and establishing conditions for Liouville integrability, with applications to quantum systems.

Contribution

It introduces a method to prove integrability in time-dependent systems using Lie algebra structures and extends the analysis to quantum analogs.

Findings

Existence of two invariants in involution for these systems.

Chaotic motion cannot occur in systems with these invariants.

Quantum structures satisfying similar integrability conditions are identified.

Abstract

The notion of integrability is discussed for classical nonautonomous systems with one degree of freedom. The analysis is focused on models which are linearly spanned by finite Lie algebras. By constructing the autonomous extension of the time-dependent Hamiltonian we prove the existence of two invariants in involution which are shown to obey the criterion of functional independence. The implication of this result is that chaotic motion cannot exist in these systems. In addition, if the invariant manifold is compact, then the system is Liouville integrable. As an application, we discuss regimes of integrability in models of dynamical tunneling and parametric resonance, and in the dynamics of two-level systems under generic classical fields. A corresponding quantum algebraic structure is shown to exist which satisfies analog conditions of Liouville integrability and reproduces the classical dynamics in an appropriate limit within the Weyl-Wigner formalism. The quantum analog is then conjectured to be integrable as well.