Fokker-Planck approach to stochastic momentum cooling with a notch filter

A Fokker-Planck equation that describes the stochastic momentum cooling with a notch filter is formulated. The Fokker-Planck coefficients are calculated for an idealized linear notch filter. The coherent energy correction is expressed explicitly with a parameter representing the difference of the particle's time-of-flight and the signal's transmission time from the pickup to the kicker. This formulation provides us a guide to determine the system passband and a setting accuracy of the signal's transmission time for a given momentum spread of the beam. From the Fokker-Planck equation, a differential equation for time variation of a standard deviation of energy error is derived to obtain the initial cooling time and the final momentum spread. This formulation is useful for analysis of experimental data and for design of the cooling system, i.e. optimization of the system passband and gain, specification of accuracy of the signal's transmission time, etc.. (author)