### a) Effect of a magnetic field gradient on the interference signals

We have studied the effect of a magnetic field gradient on the interference signals. In the simplest theory, the Zeeman effect can be treated as linear and each F, MF sublevel receives a phase-shift which is proportional to the difference of the magnetic field integrals along the two atomic paths in the interferometer. This phase-shift is also proportional to MF and to the Landé factor of the F hyperfine level. It is easy to verify that this phase-shift is also proportional to the diffraction order and to the inverse of the square of the atom velocity v.

During the experiment, the magnetic field is created by a current I circulating in a small coil located close to the atomic beams and the phase-shift is proportional to I (provided that we can neglect the Earth magnetic field). We have recorded interference fringes for various values of the current I and we have measured the fringe visibility V. When the current I increases, the fringe visibility first decreases rapidly and it presents a series of recurrences, corresponding to the cases where the phase-shifts are integer multiples of 2 Because, the phase-shift is velocity dependent, the recurrence intensities decrease when the current I increases. This experiment is therefore very sensitive to the velocity distribution of the atoms contributing to the interference signals.

These experiments are very interesting as they give a very direct test of the operation of our atom interferometer :

The sensitivity to the F,MF sublevels and to the Landé factors gF gives a way to test the isotopic selectivity of our interferometer : we can choose the laser frequency close to a transition of lithium 7 or of lithium 6 and, in this way, one isotope dominates the interferometer signal. This experiment is thus able to measure the relative weight of the two isotopes in the interference signal because the recurrences do not occur for the same current in the coil (because the Landé factors gF are different for lithium 6 and 7) and the recurrence pattern has not the same shape for the two isotopes (because of the different distribution of MF values).

The number of observed recurrences depends on the velocity distribution, which itself is a function of the chosen diffraction order, because Bragg diffraction is velocity selective and this selectivity is strongly dependent of the diffraction order. One can thus measure the relative width of the velocity distribution and give a direct proof of the velocity selective character of Bragg diffraction.

These two effects (velocity selection and isotopic selection) are easy to predict on general grounds but a quantitative prediction require a very complete analysis of the Bragg diffraction process and a model in which the laser beams are treated as having a top-hat profile reveal too rough for a quantitative analysis. We are developing a complete model of our interferometer, adequate to predict accurately these effects.

An exact theoretical treatment must take into account the quadratic Zeeman effect due to hyperfine uncoupling : the non-linear terms are not negligible, even for the low magnetic field we use, near 10-3 Tesla. As the role of the non-linear Zeeman effect is different for the two hyperfine levels F=1 et F=2 of lithium 7, we have made a complementary experiment in which lithium 7 is optically pumped in its F=1 level and we have thus been able to make a detailed verification of these predictions.

### b) Measurement of the electric static polarisability of Lithium atoms

The measurement of the static electric polarizability of an atom is a particularly interesting application of atom interferometry for two main reasons :

this measurement is not possible by spectroscopy, which can access only to polarizability differences ;

the measurements of the electric polarizability of an atom generally results from the detection of a usually very small modification of the atom trajectory, due to an intense electric field, with a strong gradient. The extreme sensitivity of atom interferometry is perfectly suited to detect a weak modification of the atom propagation due to an electric field and we can work with considerably lower fields. THis possibility was recognized in 1995 by D. Pritchard and co-workers (C. R. Ekstrom, et al, Phys. Rev. A 51, 3883 (1995)) who thus measured the electric polarizability of sodium atom ;

the most accurate measurement (by Bederson and co-workers in 1974) of the electric polarizability of lithium is in fact an indirect measurement. The polarizability of lithium is compared to the polarizability of metastable helium, which is calculated. With atom interferometry, we can make a direct absolute measurement, as shown below.

Our experiment is very similar to the one performed on sodium atom by D. Pritchard and co-workers. We have built a precision capacitor, with a thin electrode (a " septum ") which can be inserted between the two atomic beams, so that we can apply an electric field to only one of these two beams inside the atom interferometer. This electric field creates a phase-shift proportional the electric polarizability, to the square of the applied electric field (or more precisely to the integral of this quantity along the atomic path) and to the inverse of the atom velocity.

Figure 3 : Measurement of the electric polarizability of lithium atom by atom interferometry. Interference fringes are recorded successively with a zero electric field (V=0 on the capacitor ; black dots) and a non-zero electric field (V = 260 Volts, grey dots) The phase shift due to the electric field is close to 3 . Each data points correspond to a counting time equal to 0.36 second.

The capacitor must be such that one can describe the electric field inside with a good accuracy and we have developed an original geometry in which the integral of the square of the electric field can be calculated analytically. We must take into account of the atom velocity distribution, as the phase-shift is proportional to v-1 and we have verified that this is absolutely necessary for an accurate representation of experimental results. The phase-shift measurements are very well represented by our calculations and we are able to extract the proportionality constant relating the phase-shift to the square of the applied voltage with a very small uncertainty, smaller than 0.1%. The main uncertainty on the electric polarizability is due to our measurement of the mean velocity of the lithium atoms and we get :

alpha = (24.33 +/- 0.16) 10^-30 m^3 = 164.2 1.1 atomic units

in very good agreement with the previous measurements, alpha = (22 +/- 2) 10^-30 m^3 by Chamberlain et Zorn in 1963 and alpha = (24.3 +/- 0.5) 10^-30 m^3 by Bederson and co-workers in 1974. We hope to further reduce the uncertainty of the present measurement.

There are numerous calculations of the lithium atom polarizability, including many ab initio calculations. The Hartree-Fock values are close to = 170. a.u., while the calculations taking into account electron correlations are almost all in the 162.-166. a.u. range. In 1994, Kassimi and Thakkar have made a series of Moller-Plesset calculations of order 2, 3 and 4 and, from the convergence of this series of calculations, they predicted alpha = 164.2 +/- 0.1 u.a.. The most accurate value, alpha = 164.111 +/- 0.002 a.u., has been obtained in 1996 by G. F. Drake and co-workers, through an Hylleraas calculation. These very accurate works still neglect the finite nuclear mass effect and relativistic effects, but the associated corrections are very small, as they surely do not exceed a few 10^-2 u.a.. Our experimental result is in excellent agreement with these two very accurate calculations.

A. MIFFRE, M. JACQUEY, M. BÜCHNER, G. TRÉNEC AND J. VIGUÉ, " Measurement of the electric polarizability of lithium by atom interferometry ", Phys. Rev A 73, 011603(R) (2006) ; preprint available on https://hal.ccsd.cnrs.fr/ccsd-00005359.

A. MIFFRE, M. JACQUEY, M. BÜCHNER, G. TRÉNEC AND J. VIGUÉ, " Atom interferometry measurement of the electric polarizability of lithium ", Eur. Phys. J. D DOI : 10.1140/epjd/e2006-00015-5 ; preprint available on https://hal.ccsd.cnrs.fr/ccsd-00007737.

### c) Measurements of the refraction index of rare gases for atom waves

We are presently starting these measurements. We insert a windowless cell, containing a low pressure gas (about 10-3 millibar) on one of the two atomic beams inside the interferometer. We will deduce the phase-shift and the attenuation of the atomic wave which has crossed this gas cell from the interference fringes. Up-to-now, such experiments have been done only with sodium by the research group of D. Prichard (J. Schmiedmayer et al., Phys. Rev. Lett. 74, 1043 (1995) ; T. D. Roberts et al., Phys. Rev. Lett. 89, 200406 (2002)).

We hope to make very complete and accurate measurements in the case of lithium and to compare our results with calculations : the index of refraction is related to the real and imaginary parts of the forward scattering amplitude and the interaction potentials of lithium with several other atoms are already well known thanks to collision studies (total and differential cross-sections) as well as by laser spectroscopy of van der Waals complexes.

### d) Related works : optimization of the hot wire detector

We have used a hot-wire detector to ionize and detect the lithium atom coming out of the interferometer. Because our lithium atoms have a large velocity, close to 1000 m/s, laser induced fluorescennce is not very sensitive and a hot-wire detector remains the most sensitive detector for fast alkali atoms. However, the optimization of such a detector is rather delicate in the case of lithium, because lithium has the highest ionization potential (5.3 eV) of the alkali atoms. We have thoroughly analyzed the operation of this detector as a function of the wire temperature in the case of a rhenium wire.

R. DELHUILLE, A. MIFFRE, E. LAVALLETTE, M. BÜCHNER, C. RIZZO, G. TRÉNEC, J. VIGUÉ, H. J. LOESCH and J. P. GAUYACQ, " Optimization of a Langmuir-Taylor detector for lithium ", Rev. Scient. Instrum. 73, 2249-58 (2002)

### e) Related works : velocity distribution of the atomic beam

We have measured by Doppler effect the velocity distribution of our lithium beam, which is obtained by seeding lithium in a supersonic expansion of argon. The result was surprising : the parallel velocity distribution is well characterized by a temperature, close to 6.6 K in our usual operating conditions for which the calculated parallel temperature of argon is 19 K.

This ultra-cooling of a light atom (here lithium of atomic mass number A=7) in a heavier carrier gas (here argon with an atomic mass number A = 40) was not well known (a unique previous case had been reported by the group of R. Campargue, corresponding to atomic oxygen seeded in argon) and it was not explained. We have shown that the existing theory of supersonic expansion can be extended to give a quantitative explanation of this effect in the case of a light atom seeded (in the high dilution limit) in a heavier carrier gas. The parallel and perpendicular temperatures of the seeded gas are coupled by collisions one to the other as well as to parallel and perpendicular temperatures of the carrier gas and the key effect is that the parallel and perpendicular temperatures of the seeded gas remain coupled together while the parallel and perpendicular temperatures of the carrier gas are already uncoupled. As the perpendicular temperatures decrease indefinitely, the parallel temperature of the seeded gas may reach a value considerably lower than the same quantity for the carrier gas.

In atom interferometers, the parallel velocity distribution of the atomic beam is very important as it defines the coherence length and the maximum observable phase-shift (at least for the dispersive phase-shift which are usually proportional to v-1 where v is the atom velocity).

A. MIFFRE, M. JACQUEY, M. BÜCHNER, G. TRENEC and J. VIGUÉ, " Anomalous cooling of the parallel velocity in seeded beams ", Phys. Rev. A 70, 030701(R) (2004) preprint available on http://hal.ccsd.cnrs.fr/ccsd-00001006

A. MIFFRE, M. JACQUEY, M. BÜCHNER, G. TRENEC and J. VIGUÉ, " Parallel temperatures in supersonic beams : Ultracooling of light atoms seeded in a heavier carrier gas ", J. Chem. Phys. 122, 094308 (2005) ; preprint available on http://hal.ccsd.cnrs.fr/ccsd-00002148

### f) Related works : analysis of an three-grating Mach-Zehnder optical interferometer

In order to control the positions of the three mirrors serving to produce the laser standing waves, we use an optical Mach-Zehnder interferometer with three diffraction gratings. This apparatus is rather easy to construct and it is the perfect optical analogue of the neutron and atom interferometers. It is therefore quite interesting to use this apparatus for atutorial introduction to matter wave interferometers and we have written a paper describing this interferometer and its operation.

A. MIFFRE, R. DELHUILLE, B. VIARIS DE LESEGNO, M. BÜCHNER, C. RIZZO and J. VIGUÉ, " The three-grating Mach-Zehnder optical interferometer : a tutorial approach using particle optics ", European Journal of Physics, 23, 623-635 (2002)

### g) Construction of a powerful cw laser at 671 nm

We are presently using a single frequency dye laser pumped by an argon ion laser to produce a laser power close to about 500 milliwatts at 671 nm, which is necessary to produce the laser standing waves for lithium diffraction.

We are building a solid-state laser pumped by laser diodes at 808 nm to replace this laser system. We hope to get at the same time a larger laser power at 671 nm, a better frequency stability, a considerably improved ease of operation and a reduced operating cost (the annual operating cost of an argon ion laser plus a dye system is important, while diode lasers at 808 nm are getting cheaper every year). We have received the help of several other research groups during this project (INM/CNAM in Paris, LCFIO/IOTA in Orsay, LGE in Grenoble) and we have already achieved interesting results.