Phonon distributions of a single-bath mode coupled to a quantum dot

The properties of an unconventional, single-mode phonon bath coupled to a quantum dot are investigated within the rotating wave approximation. The electron current through the dot induces an out of equilibrium bath, with a phonon distribution qualitatively different from the thermal one. In selected transport regimes, such a distribution is characterized by a peculiar selective population of few phonon modes and can exhibit sub-Poissonian behavior. It is shown that such sub-Poissonian behavior is favored by a double occupancy of the dot. The crossover from an unequilibrated to a conventional thermal bath is explored, and the limitations of the rotating wave approximation are discussed.