Hamiltonian theory of the crossing of the $2 Q_x -2 Q_y=0$ nonlinear coupling resonance

TL;DR

This paper extends Hamiltonian adiabatic theory to analyze the crossing of nonlinear coupling resonance $2 Q_x - 2 Q_y=0$, revealing phase-space dynamics, emittance exchange, and deriving scaling laws relevant for applications.

Contribution

It introduces a Hamiltonian model for the nonlinear coupling resonance crossing and analyzes phase-space topology and emittance exchange, providing new insights and scaling laws.

Findings

Detailed phase-space topology evolution during resonance crossing

Characterization of emittance exchange phenomena

Derived scaling laws for nonlinear coupling resonance crossing

Abstract

In a recent paper, the adiabatic theory of Hamiltonian systems was successfully applied to study the crossing of the linear coupling resonance, $Q_x-Q_y=0$. A detailed explanation of the well-known phenomena that occur during the resonance-crossing process, such as emittance exchange and its dependence on the adiabaticity of the process was obtained. In this paper, we consider the crossing of the resonance of nonlinear coupling $2 Q_x -2 Q_y = 0$ using the same theoretical framework. We perform the analysis using a Hamiltonian model in which the nonlinear coupling resonance is excited and the corresponding dynamics is studied in detail, in particular looking at the phase-space topology and its evolution, in view of characterizing the emittance exchange phenomena. The theoretical results are then tested using a symplectic map. Thanks to this approach, scaling laws of general interest for applications are derived.