Courant-Snyder formalism of longitudinal dynamics

Abstract

Originating from the stochastic nature of the photon emission process, the diffusion of the electron longitudinal coordinate exists even if the global phase slippage of a storage ring is zero, as we cannot zero all the local phase slippages simultaneously. This quantum diffusion is viewed as the most fundamental limit of the lowest bunch length realizable in an electron storage ring from the single-particle dynamics perspective. Here, we present an analysis of this effect using the Courant-Snyder parametrization in the longitudinal dimension. Analytical formulas for the longitudinal emittance, energy spread, and bunch length are derived. The same formalism is used to discuss the application of multiple radio frequency systems for longitudinal strong focusing. The presented work is expected to be useful in the development of novel light source mechanisms like steady-state microbunching, where a short bunch length or small longitudinal emittance is needed.