Lawrence Berkeley National Laboratory

A theoretical formalism for stochastic phase-space cooling of bunched beams in storage rings is developed on the dual basis of classical fluctuation theory and kinetic theory of many-body systems in phase-space. The physics is that of a collection of three-dimensional oscillators coupled via retarded nonconservative interactions determined by an electronic feedback loop. At the heart of the formulation is the existence of several disparate time-scales characterizing the cooling process. Both theoretical approaches describe the cooling process in the form of a Fokker-Planck transport equation in phase-space valid up to second order in the strength and first order in the auto-correlation of the cooling signal. With neglect of the collective correlations induced by the feedback loop, identical expressions are obtained in both cases for the coherent damping and Schottky noise diffusion coefficients. These are expressed in terms of Fourier coefficients in a harmonic decomposition in angle of the generalized nonconservative cooling force written in canonical action-angle variables of the particles in six-dimensional phase-space. The formulation includes nonlinear pick-ups and kickers, multi-dimensional cooling with coupled degrees of freedom and instrinsic electronic noise of the feedback system. The effect of dynamic signal suppression arising from feedback loop induced collective correlations manifests naturally n a consistent solution of kinetic theoretic hierarchy for simple cases. For general situations, the existence of disparate time-scales allows one to use simple fluctuation theoretic results but with transport coefficients dynamically suppressed by factors determined independently from the well-known Vlasov theory. The general coupled-mode matrix for the longitudinal and transverse signal suppression for bunched beams is derived and solved in the limit of no synchrotron band overlap. The distinctive feature of synchrotron band overlap in the bunched beam Schottky signal for a higher bandwidth-system is discussed. The signal suppression matrix describing the tensorial collective response of a coasting beam with coupled transverse cooling is also derived. Comparison of analytic results to a numerical simulation study with 90 pseudo-particles in a model cooling system is presented. Estimates of transverse cooling rates for bunches in a prototype high-energy storage ring with typical large bandwidth feedback systems are provided.