Nonlinear narrow Doppler-free resonances for optical transitions and annihilation radiation of a positronium atom

The possibilities of obtaining narrow resonances without the Doppler broadening for transition between the fine structure levels of the ground and first excited states of a positronium atom are considered. An analysis is carried out of the conditions required for observation of the narrow resonances of saturation of single quantum absorption in the 1S-2P transitions and observation of narrow two-photon absorption resonances in the 1S-2S transitions. It is shown that narrow 2γ annihilation radiation lines of a positronium atom may be obtained with a width much smaller than the Doppler one