Fluctuations of the number of particles within a given volume in cold quantum gases

In ultracold gases many experiments use atom imaging as a basic observable. The resulting image is averaged over a number of realizations and mostly only this average is used. Only recently the noise has been measured to extract physical information. In the present paper we investigate the quantum noise arising in these gases at zero temperature. We restrict ourselves to the homogeneous situation and study the fluctuations in particle number found within a given volume in the gas, and more specifically inside a sphere of radius R. We show that zero-temperature fluctuations are not extensive and the leading term scales with sphere radius R as R² ln R (or ln R) in three- (or one-) dimensional systems. We calculate systematically the next term beyond this leading order. We consider first the generic case of a compressible superfluid. Then we investigate the whole Bose-Einstein-condensation (BEC) --BCS crossover, and in particular the limiting cases of the weakly interacting Bose gas and of the free Fermi gas.