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

Resonant dynamics arise when natural oscillation frequencies satisfy algebraic relations with the frequencies of nonlinear perturbations. In periodic systems, the Poincaré surface of section reveals key structures such as fixed points, islands, and separatrices in 1D; with two degrees of freedom, these structures extend into four-dimensional phase space and can form closed curves (fixed lines) visible in mixed-coordinate projections. In circular accelerators, transverse motion has two degrees of freedom and nonlinearities from magnet imperfections can drive resonances, impacting dynamic aperture and beam lifetime. The Large Hadron Collider highlighted the importance of accurately modeling nonlinearities to ensure sufficient dynamic aperture, with experiments confirming agreement between simulated and measured dynamic aperture. For future colliders, improved predictive power could reduce safety margins and costs. At lower energies and high intensities, space-charge-driven resonance crossing contributes to halo formation and beam loss; recent work suggested fixed lines can drive asymmetric halos. However, fixed lines had not been observed experimentally. This study aims to provide the first experimental confirmation of fixed lines for a single two-dimensional coupled (third-order) resonance in the CERN SPS, by reconstructing Poincaré surfaces of section from turn-by-turn beam position data and testing the predicted fixed-line topology and phase orientation.

Prior accelerator studies emphasized avoiding resonances and maintaining sufficient dynamic aperture to ensure beam lifetime, with the LHC serving as a notable example where nonlinear field components required careful modeling and validation of the dynamic aperture against measurements. For future superconducting colliders, similarly detailed control of nonlinearities is envisaged to maintain performance with reduced safety margins. In lower-energy, high-intensity regimes (e.g., SIS100 at FAIR and CERN injector complex), space-charge-driven resonance crossing has been identified as a key mechanism behind halo formation and associated losses. Analytical and simulation studies proposed that fixed lines in coupled resonances can generate asymmetric halos, but direct experimental evidence for fixed lines was lacking. This work addresses that gap by measuring and characterizing fixed lines in the SPS.

- Facility and operating point: Experiments were conducted in the CERN SPS with a single proton bunch (≈4×10^10 protons), r.m.s. normalized transverse emittances εx≈0.5 μm and εy≈0.5 μm, bunch length ≈2.5 ns, at a momentum of 100 GeV/c to mitigate collective effects and power converter ripple sensitivity. The revolution period was ≈23 μs. - Resonance and optics: A third-order coupled resonance with condition Qx+2Qy=N (N=79) was excited using two sextupoles (strengths K21=-0.12 m^-3, K22=-0.21 m^-3). Machine tunes were set near resonance: Qx=26.104, Qy=26.448; chromaticities corrected to Q′x≈0, Q′y≈0 using dedicated sextupoles; a family of weak octupoles provided small amplitude detuning for stabilization. Beta-beating at the few-percent level (≈5%) was anticipated and accounted for in analysis. - Beam excitation and constraints: Transverse oscillations were induced by one vertical and one horizontal kicker. Due to the kicker configuration and phase advances, only a unique fixed-line orientation could be accessed for a given optics (Methods: Constraint on the fixed-line orientation). During the campaign the horizontal kicker fired one turn after the vertical, fixing the sequence. The accessible fixed-line orientation depended on the sextupole driving term angle α and lattice phases between kickers and BPMs; α was scanned by coordinated changes of K21 and K22 while keeping driving term amplitude constant. - Measurements: Turn-by-turn beam positions were recorded from the available beam position monitors (BPMs). Groups of four consecutive BPMs (two per plane) were used to reconstruct scaled Courant–Snyder coordinates (x,px,y,py) at a reference horizontal BPM location, exploiting the ≈π/2 phase advance between consecutive BPMs and an additional rotation to transport vertical coordinates to the reference. - Scaled coordinates and invariants: Using βBPM≈103 m, scaled CS coordinates were defined so that x=√βBPM xn, px=√βBPM pxn, etc., enabling direct retrieval from BPM readings. Action–angle variables were obtained via x=√āx cosφx, px=−√āx sinφx, y=√āy cosφy, py=−√āy sinφy. The resonance phase variable Ω=Δφx+2Δφy was computed turn-by-turn. A 100-turn moving average filter reduced instrumental noise. - Theoretical fixed-line model and fitting: For the third-order coupled resonance, a perturbative model predicts 2āx=āy+C (C determined by initial conditions) and fixed-line solutions with stationary āx, āy and a constant Ω=−α+πM. The 1D closed curve in 4D phase space was parameterized (equation (1)) by t∈(0,2π]. Least-squares fits of equation (1) to measured Poincaré sections tested consistency with fixed-line topology and yielded the experimental fixed-line orientation ψEXP (in units of 2π). - Experimental protocol: For each shot, ≈3,000 turns after the initial 1,000-turn transient were analyzed (early turns excluded due to particle losses and unreliable BPM readings). Systematic scans varied sextupole angle α, distance from resonance, and kicker strengths θx, θy to identify conditions with beam trapping on the resonance. Approximately 400 shots were recorded under fixed machine tunes and resonance excitation; ≈150 were on resonance. - Data selection and metrics: Locking to the fixed line was assessed by the oscillatory behavior of Ω; shots with σφ≤10% (φ≡Ω/(2π)) were considered trapped. Stability near the fixed line was quantified by D=√[(σαx^2+σαy^2)/(āx^2+āy^2)]. BPM random errors were estimated as σx≈0.66 mm and σy≈0.5 mm from ring thickness in phase-space projections; typical σΩ≈0.0032 (in units of 2π). Effects of beta-beating and sextupole-scan granularity on fixed-line orientation were estimated (each contributing ≈0.016 rms in units of 2π). Simulations with MAD-X of the ideal lattice provided reference orientations and validated topology. - Analysis outputs: Projections of Poincaré sections in (x,px) and (y,py) showed (near-)circular orbits (constant āx, āy); mixed planes (x,y) and (px,py) showed Lissajous patterns characteristic of fixed lines. Tune diagrams (Qx,Qy), (⟨Ω⟩,σΩ) plots, and (āx,āy) charts summarized resonance proximity, trapping, and stationarity.

- First experimental observation of fixed lines for a single two-dimensional coupled resonance in a circular accelerator (CERN SPS), confirming theoretical predictions. - Poincaré surface-of-section projections display: (a) circular orbits in (x,px) and (y,py), consistent with stationary āx and āy when trapped; (b) Lissajous curves in mixed planes (x,y) and (px,py). Least-squares fits of the theoretical fixed-line parameterization (equation (1)) match the data, confirming the fixed-line topology. - Resonance condition and trapping: The excited third-order resonance Qx+2Qy=79 was achieved; from ~400 shots, ~150 lay on resonance. In (⟨Ω⟩,σΩ) space, trapped shots cluster near the excited fixed-line orientation ⟨Ω⟩r. - Fixed-line orientation: For a representative stable dataset, the beam revolves around a fixed point in (āx,āy) with ψEXP≈0.275 (in units of 2π), while the expected ⟨Ω⟩r from sextupole settings is ≈0.30. The discrepancy is consistent with ≈5% beta-beating and the sextupole scan granularity, each contributing ≈0.016 rms to orientation uncertainty. Ideal-lattice simulations yield orientation ≈0.290; the ideal kicker-constrained orientation is ψr≈0.34. - Robustness under machine drifts: Even with unwanted tune drifts, the beam’s Ω oscillates around a specific value (the fixed-line orientation) while āy spans a larger range, indicating resonance trapping despite machine parameter variations. - Measurement precision and noise: BPM random errors estimated as σx≈0.66 mm and σy≈0.5 mm; σΩ≈0.0032 (in units of 2π). The observed thickness of the (Ω,āy) ring and phase-space rings is consistent with these errors. The D metric distinguishes shots close to stationary fixed lines (e.g., D≈0.11) from those affected by drift or offset (e.g., D≈0.36). - Practical implications: Demonstrating fixed lines and beam trapping provides a direct handle on nonlinear resonance dynamics, supporting improved prediction of dynamic aperture and mitigation strategies for high-intensity/high-brightness beams and future collider designs.

The study directly addresses the longstanding hypothesis that two-dimensional coupled resonances generate fixed lines in 4D phase space. By reconstructing the Poincaré surface of section from turn-by-turn BPM data and fitting a theoretical fixed-line parameterization, the authors provide clear experimental evidence of fixed lines and resonance trapping in the SPS. The constant actions in uncoupled planes and the specific correlations in mixed planes align with predictions for a third-order coupled resonance. The measured fixed-line orientation agrees with expectations within uncertainties dominated by beta-beating and scan granularity. Moreover, persistence of trapping under tune drifts underscores the dynamical robustness of the resonant structure. These results validate analytical and simulation frameworks that incorporate fixed lines, strengthening their use in predicting dynamic aperture and managing nonlinearities. Consequently, accelerator design and operation can leverage such validated models to optimize magnet specifications, reduce conservative safety margins, and mitigate beam degradation, benefiting both high-energy collider projects and high-intensity synchrotrons where resonance-driven halo formation is critical.

This work presents the first experimental observation and characterization of fixed lines induced by a third-order coupled resonance in the CERN SPS. By exciting the resonance with sextupoles, steering the beam via synchronized kickers, and reconstructing scaled Courant–Snyder Poincaré sections from BPM data, the authors verified the fixed-line topology and quantified its orientation and trapping properties. Agreement with theory and simulations, within uncertainties from beta-beating and instrumentation, validates the fixed-line framework. The findings support more accurate modeling of nonlinear dynamics and dynamic aperture, with implications for reducing design safety margins and mitigating beam degradation in current and future accelerators. Future research could explore unstable fixed lines, broaden resonance types and orientations (e.g., with more flexible kicker configurations), refine control of beta-beating, and investigate multi-particle and space-charge effects on fixed-line dynamics and halo formation.

- The study focuses on stable fixed lines; unstable fixed lines and associated dynamics are not addressed. - Kicker configuration imposes a unique accessible fixed-line orientation, limiting the explored phase-space orientations. - Experimental conditions are sensitive to tune modulation and power converter ripple; mitigation required high beam energy and careful settings. - Beta-beating (~5%) introduces systematic uncertainty in fixed-line orientation; sextupole scan granularity adds comparable uncertainty. - Instrumental noise from BPMs (σx≈0.66 mm, σy≈0.5 mm) limits resolution; initial 1,000-turn transient was excluded due to losses and unreliable readings. - Generalization to different lattices, resonance orders, or strong collective effects was not tested in this single-bunch, weak-collective regime.