Exact fixed-angle spinor evolutions via the rotating-frame formalism

Abstract

We examine spinor evolutions in a general time-dependent magnetic field B. We derive a condition the B variation must satisfy in order that the effective field direction ${\mathbf{r}\mathbf{⁁}}_{\mathrm{B}}^{\prime }$ for a frame rotating with B becomes stationary in that frame. This condition is both sufficient and necessary for B to bring about a spinor evolution which exactly maintains a fixed angle between the spin and B. The adiabatic evolution used by Berry is just a special case, albeit only approximate, of these fixed-angle evolutions. To the other extreme of the fixed-angle evolutions, the nontrivial parallel transportation occurs, which is exact, necessarily nonadiabatic, and yields a pure geometric phase. Hence an experiment aimed at observing the geometric phase should seek to effect not an adiabatic evolution but a parallel transportation.