Dynamical tunneling versus fast diffusion for a non-convex Hamiltonian

Abstract

This paper attempts to resolve the issue of the nature of the 0.01-0.1 cm(-1) peak splittings observed in high-resolution IR spectra of polyatomic molecules. One hypothesis is that these splittings are caused by dynamical tunneling, a quantum-mechanical phenomenon whereby energy flows between two disconnected regions of phase-space across dynamical barriers. However, a competing classical mechanism for energy flow is Arnol'd diffusion, which connects different regions of phase-space by a resonance network known as the Arnol'd web. The speed of diffusion is bounded by the Nekhoroshev theorem, which guarantees stability on exponentially long time scales if the Hamiltonian is steep. Here we consider a non-convex Hamiltonian that contains the characteristics of a molecular Hamiltonian, but does not satisfy the Nekhoroshev theorem. The diffusion along the Arnol'd web is expected to be fast for a non-convex Hamiltonian. While fast diffusion is an unlikely competitor for longtime energy flow in molecules, we show how dynamical tunneling dominates compared to fast diffusion in the nearly integrable regime for a non-convex Hamiltonian, as well as present a new kind of dynamical tunneling.

Relevance of the Resonance Junctions on the Arnold Web to Dynamical Tunneling and Eigenstate Delocalization

“…A complete understanding of the classical-quantum correspondence requires [86] variations in the effective and the influence on the IVR dynamics and dynamical tunneling. This is particularly important to establish whether or not chaos-assisted tunneling near the junctions needs to be considered.…”

“…Consequently, it is expected that the competition between classical and quantum IVR pathways will be subtle and, at the least, the classical-quantum correspondence will be tested severely near the resonance junctions. In this context it is relevant to note that a recent study [86] explored the competition between dynamical tunneling and fast classical diffusion along resonance zones in a certain class of Hamiltonians with DOF ≥ 3.…”

Stable chaos and delayed onset of statisticality in unimolecular dissociation reactions

“…We expect [42,44] that the dynamical stabilisation will survive quantisation. However, the extent to which quantum effects like dynamical tunnelling [65,66] can lead to enhanced localisation or de-trapping from the junctions [44,67] requires a systematic study of the classical and quantum dynamics near the junctions, particularly those with multiplicities greater than two. Such studies, given the modest effective dimensionality of the vibrational state space even for large molecules [68], may prove important towards the possibility of control by nudging the system to the regions of stable chaos using weak external fields [69][70][71].…”

Arnold web and dynamical tunneling in a four-site Bose-Hubbard model

“…Since the Arnold web is a network of nonlinear resonances, one expects on rather general grounds that the phenomenon of resonance-assisted [32] and chaos-assisted [33] tunneling (RAT and CAT) will also be sensitive to the features on the web. In particular, due to the substantial degeneracy of the quantum states near the junctions one can have enhanced quantum transport due to the availability of multiple pathways [34][35][36]. Therefore, combined with the fact that chaotic dynamics occurs in the vicinity of the junctions, there is an intriguing possibility of subtle competition between the classical and quantum transport.…”