Phase-space Analysis by Multiple Resonance-Frequency Identification: Applications to the LHC and LEP

Many beam-dynamical phenomena are studied, experimentally or computationally, by means of spectral analysis of a time-series of values of a dynamical variable. When the underlying dynamics is regular, the frequencies appearing in the spectrum are integer combinations of a small set of basic frequencies, e.g., the three tunes in the case of single-particle orbital dynamics. For well-known reasons, identification of the frequencies can be ambiguous or subjective in practice. We present an algorithm that overcomes these difficulties by exploiting theoretical bounds on the spectral power density to transform time series into sets of labelled resonance lines. In our examples, the time series are orbits obtained by tracking single particles from many initial conditions. The method has been applied to off-momentum LHC injection optics. This is a deterministic Hamiltonian system. A second application, to orbits with strong quantum fluctuations in LEP2, shows that it also works well in a noisy, dissipative system.