Abstract : In this thesis, an experiment of correlated atom pairs production through four-wave mixing in an optical lattice is described. The twin atoms are analogous to the twin photons produced by parametric down conversion, used in many fondamental quantum optics experiments, and applied in interferometry and quantum information. Because of the dispersion relation, phase matching can be obtained when atoms move in a periodic potential. Four-wave mixing then spontaneously occurs and is a special case of dynamical instability. We performed the experiment with a degenerate metastable helium gas, obtained in a very elongated optical trap. A moving optical lattice, whose characterisation can also be found in the manuscript, was applied on the atoms. The resulting four-wave mixing was studied using a 3D-resolved single atom detector. The phase-matching conditions of this process and the populated modes were investigated. We showed that with our method atoms are preferentially scattered into two narrow classes with tunable velocities and populations. This versatility should be an advantage when using the pairs in future experiments. For each of these velocity classes, we mesured a Hanbury Brown and Twiss local correlation. Furthermore, we demonstrated relative number squeezing between both classes. These two simultaneous effects indicate the non-classicality of the generated pairs, which can be used in quantum atom optics experiments, for example to observe the Hong-Ou-Mandel effect with atoms.