Measurement and analysis of longitudinal bunched beam echoes in the Fermilab Tevatron

Bunched beam echoes have been generated in the Fermilab Tevatron using a simple impulse kick technique involving the rf system. The echoes occur as a result of the nonlinear mixing of quadrupole and dipole components of successive impulse kicks, and may be readily observed using a bunch phase measurement system. The echo contains information on the synchrotron tune spread, as well as on various nonlinear phenomena, including the nonlinear tune spread and the effects of noise on the incoherent synchrotron motion. By measuring the detailed form of the echo response as well as its overall envelope as a function of the echo delay time, it is possible, in principle, to extract some of this information. In this paper we present experimental results along with a perturbation model which includes the effects of a nonlinear tune spread. In particular, we wish to explain the observed rapid decay of the echo envelope in terms of relevant beam parameters.