Threshold for loss of Landau damping in double-harmonic rf systems

The loss of Landau damping (LLD) can lead to beam quality degradation in single- and multi-bunch beams in synchrotrons. In the longitudinal plane, the LLD threshold is increased by reducing the beam-coupling impedance, applying controlled emittance blowup, or employing higher-harmonic rf systems. This work focused on the significant threshold modifications when moving from the single- to the double-harmonic rf system case, depending on the relative phase, the rf voltages, and harmonic ratios. Moreover, the impact of the impedance cutoff frequency is discussed. Special attention is given to the configuration where both rf systems are in phase bunch shorting mode (BSM) or in counterphase bunch lengthening mode at the bunch position. Although in BSM, an approximate analytical expression can still be derived, numerical solutions of the linearized Vlasov equation are required in most cases. For the first time, we demonstrate that nonmonotonic amplitude dependence of the synchrotron frequency does not always lead to a vanishing LLD threshold. In BSM, this nonmonotonicity also results in counterintuitive beam evolution after a phase perturbation. The theoretical and semianalytical predictions are validated by an unambiguous beam measurement technique in the Proton Synchrotron and Super Proton Synchrotron at CERN and supported by macroparticle tracking simulations.