Frequency shift between coherent superposition states induced by the Berry phase evolving linearly in time

Abstract

The frequency shift induced by the Berry phase between two coherent superposition states with ${m}^{\ensuremath{'}}=\ensuremath{-}1$ and $m=1$ was demonstrated under constant rotation of a magnetic field with frequency $f$ for angle $\ensuremath{\theta}$ from the rotation axis. It was found that the frequency shift is $\mathit{v}=2f\ensuremath{-}2fcos\ensuremath{\theta}$ for $0\ensuremath{\le}\ensuremath{\theta}\ensuremath{\le}\ensuremath{\pi}/3$, $\mathit{v}=\ensuremath{-}2fcos\ensuremath{\theta}$ for $\ensuremath{\pi}/3<\ensuremath{\theta}<2\ensuremath{\pi}/3$, and $\mathit{v}=\ensuremath{-}2f\ensuremath{-}2fcos\ensuremath{\theta}$ for $2\ensuremath{\pi}/3\ensuremath{\le}\ensuremath{\theta}\ensuremath{\le}\ensuremath{\pi}$ in the case of the right-handed rotation. For the left-handed rotation, the frequency changes in the opposite sign. The frequency shift is zero at $\ensuremath{\theta}=0,\phantom{\rule{0.16em}{0ex}}\ensuremath{\pi}/2$, and $\ensuremath{\pi}$, and it jumps by $2f$ in the vicinity of $\ensuremath{\theta}=\ensuremath{\pi}/3\phantom{\rule{0.28em}{0ex}}\mathrm{and}\phantom{\rule{0.28em}{0ex}}2\ensuremath{\pi}/3$. We confirm that the frequency shift is given by the time derivative of the Berry phase which does not depend on the sign of the $g$ factor.