Atomic interferometer measurements of Berry and Aharonov-Anandan phases for isolated spins S>12 nonlinearly coupled to external fields

Abstract

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 $\mathbf{E}$ and $\mathbf{B}$ 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 ${T}_{c}$. In the case of the Berry's phase, ${T}_{c}$ 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 $F=2,m=0$ $^{87}\mathrm{Rb}$ 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 $\mathbf{E}$ field---in practice, the linear polarization of the dressing beam ---about the $\mathbf{B}$ 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 $^{52}\mathrm{Cr}$ chromium atoms with $S=3$, where values of $\ensuremath{\lambda}\ensuremath{\simeq}1$ can be obtained with an instability smaller than in the $^{87}\mathrm{Rb}$ case, due to a more favorable spectroscopic structure. The $F=1,m=1$ hf level of the $^{87}\mathrm{Rb}$ ground state offers the opportunity to extend the measurement of Aharonov-Anandan's phases beyond the case $S=\frac{1}{2}$. 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., $^{201}\mathrm{Hg}$, $^{135}\mathrm{Ba}$, and $^{137}\mathrm{Ba}$), with their own specific features. We show how the difference of Berry's phases for states $S=\frac{3}{2}$ and $S=\frac{1}{2}$, with $m=\frac{1}{2}$, can be exploited to achieve a holonomic maximum entanglement of three qubits.