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Physical Review A

Dependence of Berry's phase on the sign of the g factor for conical rotation of a magnetic field, measured without any dynamical phase shift

Atsuo Morinaga, Koichi Toriyama, Hirotaka Narui, T. Aoki, Hiromitsu Imai

Published May 9, 2011• 6 citations
Publisher

Abstract

Berry's phase for a whole turn in a conical rotation of the magnetic field with a semiangle \ensuremath{\theta} has been clearly manifested free from the dynamical phase shift using the magnetic-field-insensitive two-photon transitions between sodium-ground hyperfine states having different signs of the $g$ factors. The solid angles for states with a positive $g$ factor and with a negative $g$ factor are verified to be $2\ensuremath{\pi}(1\ensuremath{-}\mathrm{cos}\ensuremath{\theta})$ and $\ensuremath{-}2\ensuremath{\pi}(1+\mathrm{cos}\ensuremath{\theta})$, respectively, for a right-handed rotation of a magnetic field and a semiangle of $0\ensuremath{\leqslant}\ensuremath{\theta}\ensuremath{\leqslant}\ensuremath{\pi}/2$.

DOI
10.1103/physreva.83.052109

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Berry's phase for a whole turn in a conical rotation of the magnetic field with a semiangle \ensuremath{\theta} has been clearly manifested free from the dynamical phase shift using the magnetic-field-insensitive two-photon transitions between sodium-ground hyperfine states having different signs of the ggg factors. The solid angles for states with a positive ggg factor and with a negative ggg factor are verified to be 2\ensuremathπ(1\ensuremath−cos\ensuremathθ)2\ensuremath{\pi}(1\ensuremath{-}\mathrm{cos}\ensuremath{\theta})2\ensuremathπ(1\ensuremath−cos\ensuremathθ) and \ensuremath−2\ensuremathπ(1+cos\ensuremathθ)\ensuremath{-}2\ensuremath{\pi}(1+\mathrm{cos}\ensuremath{\theta})\ensuremath−2\ensuremathπ(1+cos\ensuremathθ), respectively, for a right-handed rotation of a magnetic field and a semiangle of 0\ensuremath⩽\ensuremathθ\ensuremath⩽\ensuremathπ/20\ensuremath{\leqslant}\ensuremath{\theta}\ensuremath{\leqslant}\ensuremath{\pi}/20\ensuremath⩽\ensuremathθ\ensuremath⩽\ensuremathπ/2.
Berry's phase of the atom in the state with a positive or negative ggg factor for partial cycles of a rotating magnetic field was determined free from the dynamical phase shift using a time-domain atom interferometer. The experimental phase shift is in good agreement with the prediction of Berry's phase for partial cycles. It was found that the sense of Berry's phase depends on the sign of the magnetic quantum number, the sense of the rotating magnetic field, and the sign of the ggg factor of the state.
The frequency shift induced by the Berry phase between two coherent superposition states with ${m}^{\ensuremath{'}}=\ensuremath{-}1$ and m=1m=1m=1 was demonstrated under constant rotation of a magnetic field with frequency fff 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 2f2f2f 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 ggg factor.
The $\ensuremath{\pi}$-phase shift between ∣F=1,mF=0⟩|F=1,{m}_{F}=0⟩∣F=1,mF​=0⟩ and ∣F=2,mF=0⟩|F=2,{m}_{F}=0⟩∣F=2,mF​=0⟩ states for an adiabatic rotation of a magnetic field observed in Ramsey atom interferometry arises from the negative sign of the transition amplitude between the ∣1,0⟩|1,0⟩∣1,0⟩ and ∣2,0⟩|2,0⟩∣2,0⟩ states when the wave functions are rotated to the opposite direction. Using a two-photon Raman atom interferometer with a cold ensemble of sodium atoms, the phase shift for a half-rotation of the magnetic field was confirmed to be $\ensuremath{\pi}\text{ }\text{rad}$ with an uncertainty of 4%, and the phase shift for a noncyclic adiabatic rotation of the magnetic field was investigated.
We investigate the phenomenon that a Berry phase evolving linearly in time induces a frequency shift of the resonance transition between two eigenstates, regardless of whether or not they are superposed. Using the magnetic-field-insensitive two-photon microwave---radio-frequency transition, which is free of any other dynamical frequency shift, we demonstrate that the frequency shift caused by a uniform rotation of the magnetic field corresponds to the derivative of the Berry phase with respect to time and depends on the direction of rotation of the magnetic field.
According to Berry, quantum states of a hamiltonian which varies adiabatically through a circuit C in parameter space may acquire geometrical phase factors exp (iγ(C)) in addition to the normal dynamical phase factors exp ((-i/ħ) ∝ E(t) dt). We present N.M.R. experiments in the rotating frame which bear out these predictions for simple conical circuits, and point out that they are related to familiar behaviour based on the classical Bloch equations and on Haeberlen-Waugh coherent averaging theory. Extensions to coupled spins and electric quadrupolar effects are discussed.