STATE OF THE ART The methods for the measurement of the acceleration of gravity g have a broad interest, and several methods are know nowadays. In geophysics and geology, the possibility of performing an accurate measurement of the acceleration of gravity is of great interest, since it can allow the detection of minerals or oil.

The instruments currently used to measure gravity are based on mechanical or superconductive devices, and also techniques based on free-falling atoms have been demonstrated. Moreover, the U. S. patent # 6,314, 809 describes a method and an apparatus for the measurement of the acceleration of gravity based of Bose-Einstein condensates composed by bosonic atoms confined in microscopic traps or in periodic potentials. The measurement technique is based on the study of the temporal evolution of the quantum interference between the atoms. The apparatus and the method described in the patent above, despite the rather high prospected sensitivity, shows some intrinsic limitations; indeed, the use of bosonic atoms leads actually to a rapid damping of the temporal evolution of the interference, due to the presence of atom-atom interactions, that results in a low sensitivity in the measurement of the interference period, hence a low sensitivity in the measurement of gravity.

It is therefore clear the interest in developing methods and apparatus that could allow to overcome the limitations above, allowing a measurement of the acceleration of gravity with ultimate sensitivity and accuracy, as requested in the fields interested in this kind of measurements.

SUMMARY A method and an apparatus based on ultracold fermionic atoms to measure the acceleration of gravity are here described.

SHORT DESCRIPTION OF FIGURES

Figure 1: Shows schematically an apparatus to measure accelerations according to the invention.

Figure 2: Shows the typical images of the atomic cloud performing Bloch oscillations after release form the periodic potential.

Figure 3: Shows the plot of a measurement that follows the first 50 periods of Bloch oscillations of potassium atoms.

DETAILED DESCRIPTION OF THE INVENTION The present invention allows to obtain an apparatus and a measurement method that allows to measure the acceleration of gravity with high sensitivity using a sample of ultracold atoms, and overcoming the limitations of the already known apparatus and methods.

As it is known, when a sample of atoms is confined in a periodic potential aligned along the direction of a force acting on the atoms, their quantum interference evolves in time under the effect of the force. This evolution consists in oscillations of the quasi-momentum of the atoms, which is accompanied by oscillations in space over a distance of a few microns, also known as Bloch oscillations, whose period is proportional to the applied force according to: T = h/Fd where: T is the period, F is the applied force h is the Planck's constant d e is the spatial period of the periodic potential..

If the atoms are subjected to the acceleration of gravity, since F=mg, where F is the force of gravity, m is the atom mass, they perform Bloch oscillations as above, with a period: T=h/Fd=h/mgd. The measurement of the Bloch oscillations period therefore allows in this case to measure the acceleration of gravity.

Figure 1 shows the apparatus according to the invention, which is composed by a ultra-high vacuum cell 10, which is traversed by two counterpropagating laser beams 11, and where a sample of ultracold, trapped sample of fermionic atoms 12 is present. The counterpropagating laser beams are off-resonance with respect to the atomic sample, and are generated by an appropriate laser source, not shown in the figure. The apparatus consists also in a laser source (not shown) to generate a resonant laser beam 13, a CCD sensor 14 and, possibly, of a mirror

15.

The sample of fermionic atoms 12 is prepared with known techniques, such as a combination of laser and evaporative cooling in a magnetic trap. Both the necessary laser beams and the magnetic coils are not shown in figure. At the end of the cooling stage, the sample has a width of the velocity distribution below h/mX, where m is the atom mass, h is the one defined above and ? is the wavelength of the laser beams 11. This corresponds in a temperature below h2/2MX2 kb, where kB is Boltzmann constant and the other quantities have already been defined above.

One should note that the ultracold atoms of fermionic nature have a negligible probability of interacting between themselves via collisions. This avoids possible interaction-induced shifts of the Bloch frequency and allows to follow Bloch oscillations for a very long time interval, hence to measure the Bloch period with a precision higher than that available with atoms of bosonic nature.

As for the atomic species, one can employ any species of fermionic nature, e. g. alkali atoms as the fermionic isotope of potassium (40K) or of lithium (6Li), or the fermionic isotopes of strontium and ytterbium. Any atomic or molecular species which is coolable to ultralow temperatures can be however used, with preference to those with larger mass.

The atomic sample is typically constituted by an atom number ranging from 1,000 to 10,000, 000, depending on the particular atomic species. A larger atom number clearly results in a larger signal-to-noise ratio. The typical temperatures are of the order of 100 nK.

The two counterpropagating laser beams 11 are aligned along the direction of the force to be measured, therefore along the vertical direction to measure the acceleration of gravity. The can be obtained, for example, by a single laser beam which is reflected by the mirror 15 after a first crossing of the cell. The two beams have the same wavelength X and are off-resonant, which means that X is larger than the typical absorption wavelength of the atoms, typically B<1 micron. The interference between the two beams creates a sinusoidal periodic potential with a spatial periodicity d=X/2.

The laser beam 13 crosses the cell along the horizontal direction, typically a few

millimetres below the position of the atomic sample. Its wavelength is resonant with the atoms and the light is therefore absorbed by the atoms. After crossing the cell, the beam 13 impinges on the CCD sensor 14, which measures its spatial intensity profile at precise instants of time.

The sample of ultracold fermionic atoms is trapped in the periodic potential created by the interference of the two laser beams aligned along gravity. The atoms evolve under the action of gravity and perform Bloch oscillations as describe above, with a period T=h/Fd=2h/mgS.

The sample is then released from the periodic potential, and the distribution of positions within the sample is measured after a ballistic expansion via absorption imaging, using the resonant laser beam 13 and the CCD sensor 14. According to the invention, the release of the sample from the periodic potential is preferentially performed at successive time intervals, by using an adiabatic release procedure, where the intensity of the laser beams forming the periodic potential is gradually reduced to zero. This gradual reduction happens on a timescale shorter than the Bloch oscillation period T, but longer than the characteristic period mk2/h of the optical potential, which is typically 100 microseconds. This method allows to transform adiabatically the quasimomentum in the lattice into real momentum, which can in turn be measured by letting the sample expand for a sufficiently long time. This method in particular allows to follow the oscillations exploiting a zero- background signal, hence with high efficiency (large signal-to-noise ratio). The intensity reduction of the laser beams above is performed with a standard technique for the control of the intensity, for example using an acusto-optic or electro-optic modulator.

According to the method of this invention, a series of identical measurements is then performed, in which only the holding time of the sample in the periodic potential is varied, each time by a time interval smaller than the Bloch oscillations period. In such way it is possible to reconstruct the temporal evolution of the Bloch oscillations and to determine its period with high precision. From the measured period T=2h/mgA, it is then possible to extract the acceleration of gravity g with high precision, since the other quantities involved are either known with high precision, as in the case of h and m, or can however be determined with high

precision with known techniques, as in the case of X. o have the largest accuracy with the smallest number of repetitions of the measurements, it is advisable to perform a measurement of the first Bloch oscillation, and then of a complete period of oscillation every 10xN Bloch oscillations, where N is an integer number. In such way it possible to determine in a gradual way the successive digits of the Bloch oscillations period.

One should note that the instrument has no intrinsic limitations to the sensitivity originating from practical limitations of its size, since the sample is trapped during the measurement, and not in free fall. The key feature of the instrument is indeed to allow high precision measurement of the acceleration while keeping the sample confined over a few tens of microns. The theoretical lower limit to the spatial resolution stems just from the spatial amplitude of the Bloch oscillations, which is indeed of the order of a few microns. This feature represents a relevant limitation to the use of bosonic atoms, which indeed cannot be compressed to this degree in the periodic potential. In that case the compression over a few tens of microns would imply a high probability of having interatomic collisions, which would rapidly destroy the coherence of Bloch oscillations and would not allow an accurate determination of the acceleration of gravity.

To better illustrate the invention, it is now reported an example of the measurement of gravity with the method and apparatus described above.

EXAMPLE The apparatus is based on a sample of potassium 40K (m = 40 atomic masses) composed of about 10,000 toms at a temperature of 100nK, which are prepared inside a ultra-high-vacuum cell. The sample is prepared in a magnetic trap using a standard procedure which consists of a laser cooling stage followed by an evaporative cooling stage (e. g. as described in G. Roati, F. Riboli, G. Modugno, M.

Inguscio, Phys. Rev. Lett. 89, (2002) ). To measure the acceleration of gravity, the sample is trapped in a periodic potential created with a continuous laser beam that crosses the cell along the vertical direction and is then retroreflected by a mirror.

The laser beam is in the near infrared, B=870nm, and has a power of about 300mW, a diameter of about 100 micron, and is created by a titanium: sapphire laser source. The adiabatic release of the sample, after an appropriate holding

time, is performed by reducing the laser power to zero via an acusto-optic modulator on the laser beam path. After a ballistic expansion lasting 8 ms a resonant laser beam (, =767nm) that crosses the cell in an horizontal direction is pulsed on for about 100 microseconds and collected by a CCD sensor. The intensity profile of the laser beam acquired by the CCD is then analyzed to extract the density profile of the atoms in free fall. In Figure 2 the typical images of the atoms performing Bloch oscillations are shown. The time values shown represent the holding times in the periodic potential.

Following the vertical position of the main peak in the density distribution we reconstruct the Bloch oscillations. With a simple fitting procedure using a sawtooth function, it is possible to determine with high precision the oscillation period and therefore the acceleration of gravity. Figure 3 shows a measurement performed by following the first 110 period of the oscillations, from which it is possible to extract a period T=2. 32792 ms.. Note how after just 250ms of measurement the sensitivity in the determination of the period is already below 100 parts per million.

This corresponds to an identical sensitivity in the determination of the local acceleration of gravity, which turns out to be g = 9. 73729 m/s2.

The sensitivity of the instrument can be improved by various orders of magnitude by increasing both the signal-to-noise ratio and the number of measured Bloch periods. To the first purpose it is sufficient to increase the number of atoms in the sample. To the second purpose, it is instead necessary to increase the total measurement time, and at the same time to decrease the Bloch oscillation period, e. g. by choosing an atomic species with larger mass and a laser radiation with longer wavelength. For example, it is likely to be possible to use a sample of 100,000, 000 atoms in a periodic potential realized with a gas laser in the mid infrared, B=10 micron, with a total measurement time of 100 seconds. For this example we estimate an accuracy of 3 10-11 g, which is calculated as follows. In 100 seconds an acceleration of 3 10-7 g is able to produce a phase shift of the Bloch oscillations of one radian; assuming the measurement sensitivity limited just by the quantum noise, the accuracy must then be scaled by the square root of the atom number, hence by 10,000, obtaining the result above. This accuracy is larger by over one order of magnitude than the one demonstrated with other gravimeters (mechanical, superconductive or the ones based of atoms in free fall), which is limited to about 10-9 g.