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Physical Review A

Dependence of Berry’s phase for atom on a sign of the g factor in the rotating magnetic field

Atsuo Morinaga, T. Aoki, Makoto Yasuhara

Published May 27, 2005• 10 citations
Publisher

Abstract

Berry's phase of the atom in the state with a positive or negative $g$ 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 $g$ factor of the state.

DOI
10.1103/physreva.71.054101

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Physical Review A•Jul 6, 2015•4 citations
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.
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 $2\ensuremath{\pi}(1\ensuremath{-}\mathrm{cos}\ensuremath{\theta})$ and $\ensuremath{-}2\ensuremath{\pi}(1+\mathrm{cos}\ensuremath{\theta})$, respectively, for a right-handed rotation of a magnetic field and a semiangle of $0\ensuremath{\leqslant}\ensuremath{\theta}\ensuremath{\leqslant}\ensuremath{\pi}/2$.
We investigate the Berry phase in an ensemble of thermal 87Rb{}^{87}\mathrm{Rb}87Rb atoms which we prepare in a superposition state under conditions near and at electromagnetically induced transparency. The geometric phase is imprinted in the atoms by rotating the laboratory magnetic field. Phase-stabilized light fields permit us to monitor phase changes of the atomic sample in a Ramsey-type interferometer by instant probing of the absorptive response of the atoms as well as by monitoring the free-induction decay of the coherent superposition. The absolute sign of the phase is determined by reference to controllable phase shifts imposed by the experimenter. We prove that the geometric phase is independent of the rotational frequency of the magnetic field in the adiabatic regime, that the phase is additive in multiple rotations, and it is independent of the Land\'e factor of the atomic magnetic moment, as predicted in Berry's seminal paper. We show that the absolute sign of the phase encodes the sign of the observable angular momentum in relation to laboratory coordinates.
Using a magnetic-field-insensitive two-photon transition, the phase shift of atomic interference fringes and the frequency shift of the resonance spectrum were examined for a noncyclic rotation of a magnetic field during the interrogation time of an atom interferometer. We observed a phase shift proportional to the rotation angle of the magnetic field when the initial state and the final state were subjected to interference. We confirmed that the phase shift is equivalent to the scalar Aharonov-Bohm phase due to the geometrical potential. On the other hand, we demonstrated that the noncyclic Berry phase defined by the geodesic gauge is measurable using a strategy that the magnetic field rapidly returns to the initial direction during the interrogation time of the atom interferometer.
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.