We measure the population distribution in one of the atomic twin beams generated by four-wave mixing in an optical lattice. Although the produced two-mode squeezed vacuum state is pure, each individual mode is described as a statistical mixture. We confirm the prediction that the particle number follows an exponential distribution when only one spatio-temporal mode is selected. We also show that this distribution accounts well for the contrast of an atomic Hong-Ou-Mandel experiment. These experiments constitute an important validation of our twin beam source in view of a future test of a Bell inequalities.1 By "thermal state", it is meant that the population distribution of the mode under consideration follows an exponential law characterized by a single parameter, which plays the role of an effective temperature.

We propose a minimalistic model to account for the main properties of a continuous superradiant laser, in which a beam of atoms crosses the mode of a high-finesse Fabry-Perot cavity, and collectively emits light into the cavity mode. We focus on the case of weak single atom - cavity cooperativity, and highlight the relevant regime where decoherence due to the finite transit time dominates over spontaneous emission. We propose an original approach where the dynamics of atoms entering and leaving the cavity is described by a Hamiltonian process. This allows deriving the main dynamical equations for the superradiant laser, without the need for a stochastic approach. We derive analytical conditions for a sustained emission and show that the ultimate linewidth is set by the fundamental quantum fluctuations of the collective atomic dipole. We calculate steady-state values of the two-body correlators and show that the continuous superradiant regime is tied to the growth of atom-atom correlations, although these correlations only have a small impact on the laser linewidth.