We present a concept for measuring a Berry phase using an atom interferometer in an optical Ramsey arrangement. The adiabatic cyclic process needed for the appearance of a Berry phase consists of the adiabatic change of population in one of the interferometer arms. A closed circuit in parameter space is realized by the time-dependent Doppler detuning of two additional strongly focused Gaussian beams, which couple selectively to one of the interferometer arms. It is shown that the resulting additional atomic phase consists of a Berry phase corresponding to the phase difference of the two Gaussian beams and a dynamical phase caused by the dynamical Stark effect. By recombining the two interferometer arms this total phase gives rise to a shift of the interference pattern. In case of a Ramsey arrangement with magnesium atoms the parameters of an experimental realization have been calculated. Further improvements using laser-cooled and trapped atoms are discussed.
In a recent paper we have studied the peculiar features of the Berry and Aharonov-Anandan geometric phases for isolated spins $S\ensuremath{\geqslant}1.$ We have assumed that they are submitted to a dipole and quadrupole coupling to external
E and
B fields with the mild restriction $\mathbf{E}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{B}=0$. This implies discrete symmetries leading to remarkable simplifications of the geometry and algebra involved. The aim of the present work is to describe realistic proposals, within the realm of atomic physics, for the verification of some of our most significant theoretical predictions. There are several challenges to be overcome. For alkali-metal atoms, most commonly used in atomic interferometers, the only practical way to generate quadrupole coupling, with a strength comparable to the dipole one, is the ac Stark effect induced by a nearly resonant light beam. One then has to face the instability of the``dressed'' atom hyperfine (hf) level, a candidate for our isolated spin. One deleterious effect is the apparition of an imaginary part in the quadrupole to dipole coupling strength ratio, $\ensuremath{\lambda}$. Fortunately we have found a simple way to get rid of $\mathfrak{I}(\ensuremath{\lambda})$ by an appropriate detuning. We are left with an unstable isolated spin. This implies an upper bound to the quantum cycle duration
Tc. In the case of the Berry's phase,
Tc has a lower bound due to the necessity of keeping the nonadiabatic corrections below a predefined level. We have found a compromise in the case of the
F=2,m=0 87Rb ground-state hf level. This is our candidate for the measurement of the somewhat ``exotic'' Berry's phase acquired by the
S=2,m=0 state at the end of a quantum cycle involving a rotation of $\ensuremath{\pi}$ of the
E field---in practice, the linear polarization of the dressing beam ---about the
B field direction. We have found a way to implement, in a Ramsey-type interferometric measurement, the procedures aimed at controlling the nonadiabatic corrections, as described in detail in our previous theoretical paper. A numerical simulation of our experimental proposal shows that a
0.1 accurate determination of Berry's phase, free of nonadiabatic corrections, can be achieved. Measurements could also be considered for cold
52Cr chromium atoms with
S=3, where values of $\ensuremath{\lambda}\ensuremath{\simeq}1$ can be obtained with an instability smaller than in the
87Rb case, due to a more favorable spectroscopic structure. The
F=1,m=1 hf level of the
87Rb ground state offers the opportunity to extend the measurement of Aharonov-Anandan's phases beyond the case
S=21. We construct, using ``light shift,'' the Hamiltonian ${H}_{\ensuremath{\parallel}}(t)$ generating a closed circuit in the density-matrix space which satisfies at any time the ``parallel transport'' condition, thus making the quantum cycle free from the adiabaticity condition. We also consider the case of half-integer spins (e.g.,
201Hg,
135Ba, and
137Ba), with their own specific features. We show how the difference of Berry's phases for states
S=23 and
S=21, with
m=21, can be exploited to achieve a holonomic maximum entanglement of three qubits.
In 1950, Ramsey and Hahn separately described a method using separated oscillatory magnetic fields and electronic levels to measure interference in atomic states. Today, similar physics is exploited to make atom interferometer sensors utilizing what is called the spin-echo sequence. In 1956, Pancharatnam described the geometric phase difference between two polarized states of light. In 1984 Berry described the phase accrued in cyclic and adiabatic quantum systems. A Pancharatnam-Berry phase experiment involving a quarter-wave – half-wave – quarter-wave plate stack of optics and an experiment using the spin-echo sequence are based on seemingly different phenomena in optics versus atomic optics, respectively. However, the mathematical representations used to describe the Pancharatnam-Berry phase in optics are identical to those used to describe the spin-echo sequence. Therefore, the relationship between the spin-echo sequence and the Pancharatnam-Berry phase was explored. Experiments to test the comparisons using a pre-existing cold atom experiment, ~150 μK, were performed. The mathematical connection between the two types of experiments was explained more deeply.