We present a nonrelativistic theory for the quantum mechanical description of longitudinal atomic beam spin echo experiments, where a beam of neutral atoms is subjected to static electric and magnetic fields. The atomic wave function is the solution of a matrix-valued Schroedinger equation and can be written as superposition of local (atomic) eigenstates of the potential matrix. The position- and time-dependent amplitude function of each eigenstate represents an atomic wave packet and can be calculated in a series expansion with a master formula that we derive. The zeroth order of this series expansion describes the adiabatic limit, whereas the higher order contributions contain the mixing of the eigenstates and the corresponding amplitude functions. We give a tutorial for the theoretical description of longitudinal atomic beam spin echo experiments and for the so-called Fahrplan model, which is a visualisation tool for the propagation of wave packets of different atomic eigenstates. As an example for the application of our theory, we study parity violating geometric (Berry-)phases. In this context, we define geometric flux densities, which for certain field configurations can be used to illustrate geometric phases in a vector diagram. Considering an example with a specific field configuration, we prove the existence of a parity violating geometric phase.
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.
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.
We study the stationary interference of the 2S1/2 or 2P1/2 states due to the passage of metastable hydrogen-like helium ions 4He+ through static electric fields. The basic theory of atomic state interference is presented and experimental conditions allowing observation of the interference pattern with a beam of 4He+ ions are analyzed. We show that optimal conditions are within the reach of modern day experimental techniques. We present theoretical interference curves for two types of atomic interferometers and discuss in detail approaches for the precise measurement of the Lamb shift δn=2 = 2S1/2 2P1/2 in 4He+. PACS Nos.: 31.30.Jv, 32.70.Fw, 39.10.+j, 39.20.+q
We report on interference studies in the internal and external degrees of freedom of metastable triplet helium atoms trapped near quantum degeneracy in a $$1.5 \,\upmu {\mathrm{m}}$$ optical dipole trap. Applying a single
π/2 rf pulse we demonstrate that 50% of the atoms initially in the
m=+1 state can be transferred to the magnetic field insensitive
m=0 state. Two
π/2 pulses with varying time delay allow a Ramsey-type measurement of the Zeeman shift for a high precision measurement of the
23S1 –
21S0 transition frequency. We show that this method also allows strong suppression of mean-field effects on the measurement of the Zeeman shift, which is necessary to reach the accuracy goal of 0.1 kHz on the absolute transition frequencies. Theoretically the feasibility of using metastable triplet helium atoms in the
m=0 state for atom interferometry is studied demonstrating favorable conditions, compared to the alkali atoms that are used traditionally, for a non-QED determination of the fine structure constant.