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arXiv:1907.03352

Optimization of the atom interferometer phase produced by the set of cylindrical source masses to measure the Newtonian gravity constant

B. Dubetsky

Published Jul 7, 2019• 1 citation
arXivHTMLPDF

Abstract

An analytical expression for the gravitational field of a homogeneous cylinder is derived. The phase of the atom interferometer produced by the gravity field of the set of cylinders has been calculated. The optimal values of the initial positions and velocities of atomic clouds were obtained. It is shown that at equal sizes of the atomic cloud in the vertical and transverse directions, as well as at equal atomic vertical and transverse temperatures, systematic errors due to the finite size and temperature of the cloud disappear. To overcome the influence of the Earth gravitational field on the accuracy of the phase double difference measurement, it is proposed to use the technique of eliminating gravity-gradient terms. After eliminating, one can use extreme values of the atomic positions and velocities. Nonlinear dependences of the phase on the uncertainties of atomic positions and velocities near those extreme values required us to modify the expression for the standard phase deviation. Moreover, such dependences lead to a phase shift, which was also calculated. The relative accuracy of measurements of Newtonian gravitational constant 10^-4 and 2*10^-5 is predicted for sets of 24 and 630 cylinders, respectively.

DOI
10.48550/arxiv.1907.03352

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An analytical expression for the gravitational field of a homogeneous cylinder is derived. The phase of the atom interferometer produced by the gravity field of the set of cylinders has been calculated. The optimal values of the initial positions and velocities of atomic clouds were obtained. It is shown that at equal sizes of the atomic cloud in the vertical and transverse directions, as well as at equal atomic vertical and transverse temperatures, systematic errors due to the finite size and temperature of the cloud disappear. To overcome the influence of the Earth gravitational field on the accuracy of the phase double difference measurement, it is proposed to use the technique of eliminating gravity-gradient terms. After eliminating, one can use extreme values of the atomic positions and velocities. Nonlinear dependences of the phase on the uncertainties of atomic positions and velocities near those extreme values required us to modify the expression for the standard phase deviation. Moreover, such dependences lead to a phase shift, which was also calculated. The relative accuracy of measurements of Newtonian gravitational constant 10^-4 and 2*10^-5 is predicted for sets of 24 and 630 cylinders, respectively.
Gravity gradients represent a major obstacle in high-precision measurements by atom interferometry. Controlling their effects to the required stability and accuracy imposes very stringent requirements on the relative positioning of freely falling atomic clouds, as in the case of precise tests of Einstein's equivalence principle. We demonstrate a new method to exactly compensate the effects introduced by gravity gradients in a Raman-pulse atom interferometer. By shifting the frequency of the Raman lasers during the central π pulse, it is possible to cancel the initial position- and velocity-dependent phase shift produced by gravity gradients. We apply this technique to simultaneous interferometers positioned along the vertical direction and demonstrate a new method for measuring local gravity gradients that does not require precise knowledge of the relative position between the atomic clouds. Based on this method, we also propose an improved scheme to determine the Newtonian gravitational constant G towards the 10 ppm relative uncertainty.
The contribution to the phase of the atom interferometer caused by the gravity field of a massive test mass is considered. This contribution can be extracted by applying the double difference technique to measure the Newtonian gravitational constant G. Estimates and further calculations showed that after choosing the largest (given the current state of the art) multiphoton wave vector, the time delay between pulses, the mass of the test body and the signal optimization in respect to atomic positions and velocities, one should be able to obtain an accurate G measurement of 200 ppb, which is 2-3 orders of magnitude more accurate than what can currently be obtained. Calculated variations of the phases under the small deviations of atomic variables made it clear that atom clouds' radii and temperatures have to be as small as 100 micron and 100 pK, which has also been achieved already.
Accurately measuring the three-dimensional position of an atomic cloud is essential for improving the precision of atom interferometers. In the work presented here, the three-dimensional position of an atomic cloud in an atom interferometer is measured with only one laser beam, generated by a phase modulator. The vertical position is determined from the change in the additional-sideband effect with the time interval TTT between adjacent Raman pulses in the atom interferometer. Based on this method, the vertical position of the atomic cloud is measured with an uncertainty of 0.2 mm; therefore, the uncertainty in a gravity measurement contributed by the gravity gradient is not limited by the uncertainty in the measurement of the vertical position at the level of $0.1\phantom{\rule{0.2em}{0ex}}\ensuremath{\mu}\mathrm{Gal}$. Furthermore, a method of measuring Rabi oscillations at different horizontal positions to determine the horizontal position of the atomic cloud is proposed and is demonstrated experimentally. With this method, the horizontal position can be determined with an uncertainty of less than 0.1 mm, and the horizontal velocity of the atomic cloud can also be deduced. The methods demonstrated in this paper are helpful for evaluating systematic errors in atom interferometers, and also useful in precision measurements based on an atomic fountain.
We describe two experiments where atom interferometry is applied for precision measurements of gravitational effects. In the first, we measure the Newtonian gravitational constant G using an atom interferometry gravity-gradiometer which combines a rubidium fountain, a juggling scheme for fast launch of two atomic clouds, and Raman interferometry. We show that the sensor is able to detect the gravitational field produced by source masses and G is measured with better than 10-2 accuracy. In the second experiment, using ultra-cold strontium atoms in a vertical optical lattice and observing persistent Bloch oscillations for several seconds, we measure gravity acceleration with micromet-ric spatial resolution. We discuss the prospects for the study of gravitational forces at short distances and show that unexplored regions can be investigated in the search for deviations from Newtonian gravity.
We present the first direct measurement of the gravity-field curvature based on three conjugated atom interferometers. Three atomic clouds launched in the vertical direction are simultaneously interrogated by the same atom interferometry sequence and used to probe the gravity field at three equally spaced positions. The vertical component of the gravity-field curvature generated by nearby source masses is measured from the difference between adjacent gravity gradient values. Curvature measurements are of interest in geodesy studies and for the validation of gravitational models of the surrounding environment. The possibility of using such a scheme for a new determination of the Newtonian constant of gravity is also discussed.
We investigated the Raman-pulse-duration effect in a gravity gradiometer with two atom interferometers. Since the two atom clouds in the gradiometer experience different gravitational fields, it is hard to compensate for the Doppler shifts of the two clouds simultaneously by chirping the frequency of a common Raman laser. This leads to an appreciable phase shift. The magnitude of the phase shift relative to the differential phase shift of the two interferometers is in an order of $\ensuremath{\tau}/T$, and cannot be neglected in the precision measurements such as measuring the gravity gradient and the Newtonian gravitational constant.