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Physical Review A

Dynamical manifestation of an evolving Berry phase as a frequency shift of the resonance transition between two eigenstates

Koichi Toriyama, Akihide Oguchi, Atsuo Morinaga

Published Dec 2, 2011• 5 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.84.062103

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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.

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AIP Advances•Oct 31, 2016•0 citations
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.
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.
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 $0\ensuremath{\leqslant}\ensuremath{\theta}\ensuremath{\leqslant}\ensuremath{\pi}/2$.
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.
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 geometric phase in the dynamics of a spin qubit driven by transverse microwave (MW) and longitudinal radiofrequency (RF) fields is studied. The phase acquired by the qubit during the full period of the "slow" RF field manifests in the shift of Rabi frequency ω_{1} of a spin qubit in the MW field. We find out that, for a linearly polarized RF field, this shift does not vanish at the second and higher even orders in the adiabaticity parameter ω_{rf} / ω_{1}, where ω_{rf} is the RF frequency. As a result, the adiabatic (Berry) phases for the rotating and counter-rotating RF components compensate each other, and only the higher-order geometric phase is observed. We experimentally identify that phase in the frequency shift of the Rabi oscillations detected by a time-resolved electron paramagnetic resonance.
The Berry phase has universal applications in various fields. Here, we explore the coupling effect on the Berry phase of a two-level system adiabatically driven by a rotating classical field and interacting with a single quantized mode. Our simulations clearly reveal that the Berry phase change is quadratic proportional to the coupling constant if it is less than the level spacing between neighboring instantaneous eigenstates. Remarkably, if the nearest neighbouring level spacing is comparable with the coupling constant, this simple quadratic dependence is lost. Around this resonance, the Berry phase can be significantly tuned by slightly adjusting the parameters, such as the coupling constant, the frequency of the quantized mode, and the transition frequency. These numerical results, agreeing well with the perturbation theory calculations, provide an alternative approach to tune the Berry phase near the resonance, which is useful in quantum information science, i.e. designing quantum logic gates.