Classical and Quantum Dynamics of a Periodically Driven Particle in a Triangular Well

TL;DR

This paper explores the quantum-classical correspondence in a periodically driven particle within a triangular well, highlighting the role of dynamical tunneling in resonance states and spectral properties.

Contribution

It provides new insights into quantum tunneling effects and spectral transitions in a complex, time-dependent quantum system with unresolved mathematical challenges.

Findings

Quantum tunneling influences the spectral nature of the system.

Resonance zones are connected through dynamical tunneling.

Spectral transition depends on tunneling between resonance zones.

Abstract

We investigate the correspondence between classical and quantum mechanics for periodically time dependent Hamiltonian systems, using the example of a periodically forced particle in a one-dimensional triangular well potential. In particular, we consider quantum mechanical Floquet states associated with resonances in the classical phase space. When the classical motion exhibits {\it sub}harmonic resonances, the corresponding Floquet states maintain the driving field's periodicity through dynamical tunneling. This principle applies both to Floquet states associated with classical invariant vortex tubes surrounding stable, elliptic periodic orbits and to Floquet states that are associated with unstable, hyperbolic periodic orbits. The triangular well model also poses a yet unsolved mathematical problem, related to perturbation theory for systems with a dense pure point spectrum. The present approximate analytical and numerical results indicate that quantum tunneling between different resonance zones is of crucial importance for the question whether the driven triangular well has a dense point or an absolutely continuous quasienergy spectrum, or whether there is a transition from the one to the other.