Non-trivial dynamic regimes of small (nano-scale) quantum systems

TL;DR

This paper investigates the complex transition behaviors between regular and chaotic dynamics in small quantum systems, using exactly solvable models to understand how these regimes manifest and evolve.

Contribution

It introduces generalized Zwanzig models to analyze non-trivial dynamic regimes and transitions in nano-scale quantum systems, including temperature effects and multi-level couplings.

Findings

Dynamic transitions occur when Loschmidt echo time exceeds recurrence cycle.

Generalized models show temperature-dependent broadening and decay of echo components.

Applicable to various nano-systems like molecular chains and exchange reactions.

Abstract

Small (but still containing many atoms) quantum systems (traditionally termed nano-systems) are dramatically different from their macroscopic or genuine microscopic (atomic) cousins. Microscopic molecular systems (with a few atoms) obey a regular quantum dynamics (described by time dependent Schrodinger equation), whereas in macroscopic systems with continuous energy spectra, one can expect, also regular, although typically relaxation, dynamic behavior. The topic of our paper is in-between these limits. System behavior becomes non-trivial and manifests a sort of transitions between regular and chaotic dynamics. We show that such dynamic transitions occur when the Loschmidt echo time of life exceeds the typical recurrence cycle period. We illustrate this behavior in the frame work of a few versions of the exactly solvable quantum problem, proposed long ago by Zwanzig. It is based on the study of time evolution of the initially prepared vibrational state coupled to a reservoir with dense spectrum of its vibrational states. In the simplest version of the Zwanzig model, the reservoir has an equidistant spectrum, and the system - reservoir coupling matrix elements are independent of the reservoir states. We generalize the model to include into consideration the coupling of the initially prepared single state to system phonon excitations. The coupling results to temperature dependent broadening and decay of the echo components. Another generalization is to replace a single level by two states coupled to the Zwanzig reservoir. We anticipate that the basic ideas inspiring our work can be applied to a large variety of interesting for the applications nano-systems (e.g., dissipative free propagation of excitations along molecular chains, or as a model for exchange reactions).