Secular non-secular master equation

Redfield non-secular master equation governing relaxation of a spin in weak interaction with a thermal bath is studied. Using the fact that the relaxation follows the exponential law, we prove that in most cases the semi-secular approximation is sufficient to find the system relaxation rate. Based on this, a "secular" form of the non-secular master equation is for the first time developed which correctly set up one of most fundamental equations in relaxation investigation. This key secular form allows us to derive a general formula of the phonon-induced quantum tunneling rate which is valid for the entire range of temperature regardless of the basis. In incoherent tunneling regime and localized basis, this formula reduces to the ubiquitous incoherent tunneling rate. Meanwhile, in eigenstates basis, this tunneling rate is demonstrated to be equal to zero. From this secular form, we end the controversy surrounding the selection of basis for the secular approximation by figuring out the conditions for using this approximation in localized and eigenstates basis. Particularly, secular approximation in localized basis is justified in the regime of high temperature and small tunnel splittings. In contrast, a large ground doublet's tunnel splitting is required for the secular approximation in eigenstates basis. With these findings, this research lays a sound foundation for any treatments of the spin-phonon relaxation under any conditions provided that the non-secular master equation is relevant.Comment: 9 pages, 0 figure