Magneto-optical Kerr effect in pump-probe setups

We develop a general theoretical framework for computing the time-resolved magneto-optical Kerr effect in ultrafast pump-probe setups, formulated within the dynamical projective operatorial approach (DPOA) and its application to the generalized linear-response theory for pumped systems. Furthermore, we exploit this formalism to express the postpump optical conductivity—and consequently the Kerr rotation—in terms of the time-evolved single-particle density matrix (SPDM), providing a transparent and computationally efficient description of photoexcited multiband systems. This extension, in addition to its lower computational cost, has the advantage of allowing the inclusion of phenomenological damping. We illustrate the formalism using both (1) a two-band tight-binding model, which captures the essential physics of ultrafast spin-charge dynamics and the Kerr rotation and (2) weakly spin-polarized germanium, as a realistic playground with a complex band structure. The results demonstrate that, by exploiting DPOA and/or its SPDM extension, one can reliably reproduce both the short-time features under the pump-pulse envelope and the long-time dynamics after excitation, offering a versatile framework for analyzing time-resolved magneto-optical Kerr effect experiments in complex materials. Moreover, this analysis clearly shows that the Kerr rotation can be used to deduce experimentally the relevant <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mi>n</a:mi> </a:math> -photon resonances for a given specific material.