Duck hunting with quantum mechanics
nlin.PS, cond-mat.stat-mech, math.DS, quant-ph
Submitted: 2026-08-01
Updated: 2026-09-21
Comments: Quantization condition derivation was revised; two figure were updated
License: http://creativecommons.org/licenses/by/4.0/
The gist: We bridge two sides of singular perturbation theory: the classical theory of slow-fast systems and the semi-classical approach to quantum mechanical systems.
Terminology
Abstract
We bridge two sides of singular perturbation theory: the classical theory of slow-fast systems and the semi-classical approach to quantum mechanical systems. For a specific but physically important class of dynamical systems, we show that purely classical and exotic objects, so-called canard solutions, are shadows of instantons in the corresponding quantum system. We demonstrate that canard solutions exist in a domain of parameter space whose boundaries are determined by an instanton action. We illustrate our statements analytically for the relevant example, the overdamped Josephson junction, and confirm them numerically. For the Josephson junction, the canard window is the exponentially narrow gap between consecutive Shapiro steps.
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