Stochasticity effects in the spin (de)polarization of an ultrarelativistic electron beam during photon emissions in a counterpropoagating ultrastrong focused laser pulse in the quantum radiation reaction regime are investigated. We employ a Monte Carlo method to describe the electron dynamics semiclassically, and photon emissions as well as the electron radiative polarization quantum mechanically. While in the latter the photon emission is inherently stochastic, we were able to identify its imprints in comparison with the new developed semiclassical stochasticity-free method of radiative polarization applicable in the quantum regime. With an initially spin-polarized electron beam, the stochastic spin effects are seen in the dependence of the depolarization degree on the electron scattering angle and the electron final energy (spin stochastic diffusion). With an initially unpolarized electron beam, the spin stochasticity is exhibited in enhancing the known effect of splitting of the electron beam along the propagation direction into two oppositely polarized parts by an elliptically polarized laser pulse. The considered stochasticity effects for the spin are observable with currently achievable laser and electron beam parameters.
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