The Berry phase acquired by an electromagnetic field undergoing an adiabatic and cyclic evolution in phase space is a purely quantum-mechanical effect of the field. However, this phase is usually accompanied by a dynamical contribution and cannot be manifested in any light-beam interference experiment because it is independent of the field state. We here show that such a phase can be produced using an atom coupled to a quantized field and driven by a slowly changing classical field, and it is manifested in atomic Ramsey interference oscillations. We also show how this effect may be applied to one-step implementation of multiqubit geometric phase gates, which is impossible by previous geometric methods. The effects of dissipation and fluctuations in the parameters of the pump field on the Berry phase and visibility of the Ramsey interference fringes are analyzed.
Summary form only given. The field of atom interferometry is experiencing rapid progress. Atom interferometers have been used to determine topological phase shifts such as the Aharonov-Bohm or Aharonov-Casher effects. Both two beam and more recently multiple beam atom interferometers have been realized. We report on an investigation of the electric (or scalar) Aharonov-Bohm effect in a multiple beam Ramsey experiment. The topological phase shift leads to collapse and revival of the fringe pattern. In a first interaction region cesium atoms are optically pumped into a dark coherent superposition of the magnetic quantum numbers m/sub F/=-4, -2, 0, 2 and 4 of the 6S/sub 1/2/ (F=4) ground state. The coherent superposition is probed after a time T in a second interaction region. When the atoms are irradiated in between the Ramsey interactions by an additional linearly polarized light field 400 MHz blue detuned to the F=4/spl rarr/F'=4 transition of the cesium D1 line, the induced AC Stark shift increases quadratically with the magnetic quantum number. The laser field coupling to the induced atomic dipole moment causes-in the absence of any classical forces-an electric Aharonov-Bohm phase shift.
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.
The aim of the present paper is to propose experiments for an observation of the significant features of Berry's phases for S>1, studied in our previous theoretical work. They are generated by Hamiltonians endowed with a magnetic dipole together with an electric quadrupole coupling with external B and E fields, assumed orthogonal, this mild restriction leading to geometric and algebraic simplifications. Alkali atoms appear as good candidates for interferometric measurements but there are challenges to be overcome. The only practical way to generate a suitable E-field is to use the ac Stark effect which induces an instability of the dressed atom. Besides atom losses, this might invalidate Berry's phase derivation but this latter problem can be solved by an appropriate detuning. The former puts an upper limit to the cycle duration, which is bounded below by the adiabatic condition. By relying upon our previous analysis of the non-adiabatic corrections, we have been able to reach a compromise for the
87Rb hf level F=2, m=0 state, which is our candidate for an interferometric measurement of the exotic Berry's phase generated by a rotation of the E-field around the fixed B-field. By a numerical simulation we have shown that the non-adiabatic corrections can be kept below the 0.1% level. As an alternative candidate, we discuss the chromium ground state J=S=3, where the instability problem is easily solved. We make a proposal to extend the measurement of Aharonov-Anandan's phase beyond S=1/2 to the
87Rb hf level F=m=1, by constructing, with the help of light-shifts, a Hamiltonian able to perform a parallel transport along a closed circuit upon the density matrix space, without any adiabatic constraint. In Appendix A, Berry's phase difference for S=3/2 and 1/2, m=1/2 states is used to perform an entanglement of 3 Qbits.