Polarization dynamics for the muon-nuclear Breit-Rabi interaction

We study the polarization dynamics due to the Breit-Rabi interaction of a muon in the 1S atomic state, with a nucleus of spin I having arbitrary initial polarization and orientation. We derive the equations of motion for the polarization in the commutator approach in two different bases for the density matrix. We find that the commonly used Cartesian basis is not very convenient for the present problem. In it, the 4(2I${+1)}^2$-1 polarization equations break only into two closed coupled subsets, the longitudinal and the transverse one, with a modest additional decoupling in the transverse sector, in presence of an external magnetic field. We have found an optimal basis in which these equations reduce to systems of four coupled differential equations, independent of the value of I, which can be readily solved for arbitrary boundary conditions. We determine the degree of muon repolarization, derive the equations for the longitudinal and transverse muon-spin-rotation spectroscopy, and point out the advantage of nuclear target polarizations in these applications.