Radiative polarization of electrons and positrons through the Sokolov-Ternov effect is important for applications in high-energy physics. Radiative spin-polarization is a manifestation of quantum radiation reaction affecting the spin-dynamics of electrons. We recently proposed that an analogue of the Sokolov-Ternov effect could occur in the strong electromagnetic fields of ultra-high-intensity lasers, which would result in a build-up of spin-polarization in femtoseconds. In this paper we develop a density matrix formalism for describing beam polarization in strong electromagnetic fields. We start by using the density matrix formalism to study spin-flips in non-linear Compton scattering and its dependence on the initial polarization state of the electrons. Numerical calculations show a radial polarization of the scattered electron beam in a circularly polarized laser, and we find azimuthal asymmetries in the polarization patterns for ultra-short laser pulses. A degree of polarization approaching 9 % is achieved after emitting just a single photon. We develop the theory by deriving a local constant crossed field approximation (LCFA) for the polarization density matrix, which is a generalization of the well known LCFA scattering rates. We find spin-dependent expressions that may be included in electromagnetic charged-particle simulation codes, such as particle-in-cell plasma simulation codes, using Monte-Carlo modules. In particular, these expressions include the spin-flip rates for arbitrary initial polarization of the electrons. The validity of the LCFA is confirmed by explicit comparison with an exact QED calculation of electron polarization in an ultrashort laser pulse.
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