  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.
