We report on the single-atom-resolved measurement of the distribution of momentahk in a weaklyinteracting Bose gas after a 330 ms time-of-flight. We investigate it for various temperatures and clearly separate two contributions to the depletion of the condensate by their k-dependence. The first one is the thermal depletion. The second contribution falls off as k −4 , and its magnitude increases with the in-trap condensate density as predicted by the Bogoliubov theory at zero temperature. These observations suggest associating it with the quantum depletion. How this contribution can survive the expansion of the released interacting condensate is an intriguing open question.In quantum systems, intriguing many-body phenomena emerge from the interplay between quantum fluctuations and interactions. Quantum depletion is an emblematic example of such an effect, occurring in one of the simplest many-body systems: a gas of interacting bosons at zero temperature. In the absence of interactions, the ground state corresponds to all particles occupying the same single-particle state. Taking into account inter-particle repulsive interactions at the meanfield level leads to a similar solution where all particles are condensed in the same one-particle state whose shape is determined by the trapping potential and interactions. In a beyond mean-field approach, which can be interpreted as taking into account quantum fluctuations and twobody interactions, the description is dramatically different. The many-body ground state consists of several components: a macroscopically occupied single-particle state, the condensate, and a population of single-particle states different from the condensate, the depletion.This many-body description applies to diverse bosonic systems such as superfluid Helium [1], exciton-polaritons [2] and degenerate Bose gases [3]; it has also found analogies in phenomena such as Hawking radiation from a black-hole [4] and spontaneous parametric down conversion in optics [5]. The fraction of atoms not in the condensate at zero temperature, the quantum depletion, increases with the strength of inter-particle interactions and with the density, rising up to 90% in liquid 4 He [1]. In ultracold gases, where the density is significantly smaller, the quantum depletion usually represents a small fraction (less than 1%) of the total population. At non-zero temperature there is an additional contribution to the population of single-particle states above the condensate, originating from the presence of thermal fluctuations.For weakly interacting systems, Bogoliubov theory describes quantum and thermal contributions to the condensate depletion [6,7]. This approach shows that the elementary, low-energy excitations are collective quasiparticle (phonon) modes, as has been verified in experimental studies with liquid 4 He [8], degenerate quantum gases [9] and exciton-polaritons [2]. At zero temperature, the many-body ground state is defined as a vacuum of these quasi-particle modes. When projected onto a basis of single-particl...

Measuring the full distribution of individual particles is of fundamental importance to characterize many-body quantum systems through correlation functions at any order. Here we demonstrate the possibility to reconstruct the momentum-space distribution of three-dimensional interacting lattice gases atom-by-atom. This is achieved by detecting individual metastable Helium atoms in the farfield regime of expansion, when released from an optical lattice. We benchmark our technique with Quantum Monte-Carlo calculations, demonstrating the ability to resolve momentum distributions of superfluids occupying 10 5 lattice sites. It permits a direct measure of the condensed fraction across phase transitions, as we illustrate on the superfluid-to-normal transition. Our single-atom-resolved approach opens a new route to investigate interacting lattice gases through momentum correlations. arXiv:1710.08392v2 [cond-mat.quant-gas]