Equation of motion for the full spin density matrix with two-phonon (fourth-order) spin-phonon processes

Develop an equation of motion for the full spin density matrix that explicitly incorporates two-phonon spin-phonon processes derived from fourth-order time-dependent perturbation theory, enabling simulation of Raman relaxation with coupled population and coherence dynamics without resorting to symmetry-breaking approximations.

Background

The authors simulate spin relaxation in Kramers systems. For one-phonon processes, they use the non-diagonal secular approximation, which requires evolving the full density matrix to capture coupling between populations and coherences, and the corresponding formalism is available in the literature. However, for Raman (two-phonon) processes—which arise at fourth order in time-dependent perturbation theory—an analogous equation for the full density matrix is missing.

To circumvent this methodological gap, they break Kramers degeneracy by applying a small magnetic field, which removes population–coherence coupling and allows them to compute Raman relaxation rates. This workaround highlights a fundamental theoretical deficit: the absence of a rigorous fourth-order master equation that treats two-phonon processes with full density-matrix dynamics.