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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.

Low longitudinal emittance means small energy spread and short bunch length, which makes high-power short-wavelength coherent radiation possible. In a storage ring, due to quantum excitation, the equilibrium longitudinal emittance is mainly the overall contribution of all bend-related elements, such as bends, undulators and laser modulators. By introducing longitudinal Twiss function and analyzing the 6D one-turn map in 3D Twiss form, the longitudinal emittance contribution of all these elements is theoretically studied in this paper, and a general method for minimizing the storage ring equilibrium longitudinal emittance is proposed. An integrated optimization of longitudinal beta function shows, the longitudinal emittance scales as the third power of bending angle, and can be as low as subpicometer with a beam energy of 400 MeV in an ultimate state.