Effects of multiple nonlinear Compton scatterings on electrons in ultra-strong fields are described analytically. Based on this analytic theory, a new pure quantum effect of multiple nonlinear Compton scattering called quantum peak splitting is identified: the electron peak splits into two when the average number of nonlinear Compton scatterings per electron passes the threshold of ~5.1 and is below ~9. One of the split peak is formed by electrons emit 0-3 times and the other is formed by electrons emit ≥4 times. This new effect provides a new mechanism to form electron peaks, puts a new beamstrahlung limit on future colliders and corrects the picture of quantum radiation reaction. Experiment can be performed on lasers with intensity ≥10²¹W/cm², which is reachable on PW scale facilities. This new phenomenon can be employed as an in-situ, prepulse insensitive and comparatively precise method to measure the intensity of 0.3-10 PW ultra-strong lasers.

Nonlinear Compton scattering,stochastic radiation reaction