Observation of fixed lines induced by a nonlinear resonance in the CERN Super Proton Synchrotron

The motion of systems with linear restoring forces and recurring nonlinear perturbations is of central importance in physics. When a system's natural oscillation frequencies and the frequency of the nonlinear restoring forces satisfy certain algebraic relations, the dynamics become resonant. In accelerator physics, an understanding of resonances and nonlinear dynamics is crucial for avoiding the loss of beam particles. Here we confirm the theoretical prediction of the dynamics for a single two-dimensional coupled resonance by observing so-called fixed lines. Specifically, we use the CERN Super Proton Synchrotron to measure the position of a particle beam at discrete locations around the accelerator. These measurements allow us to construct the Poincare surface of section, which captures the main features of the dynamics in a periodic system. In our setting, any resonant particle passing through the Poincare surface of section lies on a curve embedded in a four-dimensional phase space, the fixed line. These findings are relevant for mitigating beam degradation and thus for achieving high-intensity and high-brightness beams, as required for both current and future accelerator projects. Nonlinear resonances can cause particle loss in accelerators. Experiments confirm that a coupled nonlinear resonance traps beam particles on a four-dimensional closed curve. This finding allows the development of mitigation strategies.