3D Larmor clock and barrier interaction times for tunneling

We study a three-dimensional spin dynamics of tunneling particles with spin 1/2 through a barrier in the presence of a static and a small time-dependent magnetic fields, perpendicular to each other. The spin dynamics is adopted as a 3D Larmor clock to derive the traversal and reflection times for tunneling (Büttiker, 1983) 14]. We show that the two times can be determined by a real three-dimensional rotation of spin, without requiring the effective rotation due to probability imbalance, and are directly related to the change of spin along its initial polarization direction. We also discuss the experimental feasibility of the present model.