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Fundamental limitations to high-precision tests of the universality of free fall by dropping atoms

Tests of the universality of free fall and the weak equivalence principle probe the foundations of General Relativity. Evidence of a violation may lead to the discovery of a new force. The best torsion balance experiments have ruled it out to 10^-13. Cold-atom drop tests have reached 10^-7 and promise to do 7 to 10 orders of magnitude better, on the ground or in space. They are limited by the random shot noise, which depends on the number N of atoms in the clouds. As mass-dropping experiments in the non-uniform gravitational field of Earth, they are sensitive to the initial conditions. Random accelerations due to initial condition errors of the clouds are designed to be at the same level as shot noise, so that they can be reduced with the number of drops along with it. This sets the requirements for the initial position and velocity spreads of the clouds with given N. In the STE-QUEST space mission proposal aiming at 2x10^-15 they must be about a factor 8 above Heisenberg's principle limit, and the integration time required to reduce both errors is 3 years, with a mission duration of 5 years. Instead, offset errors at release between different atom clouds are systematic and give rise to a systematic effect which mimics a violation. Such offsets must be demonstrated to be as small as required in all drops, must be small by design and must be measured. For STE-QUEST to meet its goal they must be several orders of magnitude smaller than the size of each individual cloud, which in its turn must be at most 8 times larger than the uncertainty principle limit. Even if all technical problems are solved and the clouds are released with negligible systematic errors, still they must be measured. Then, Heisenberg's principle dictates that the measurement lasts as long as the experiment and the systematic nature of the effect requires many measurements for it to be ruled out as a source of violation.