A scheme of an intense coherent gamma-ray source based on the spontaneous radiation of positronium atoms in a Bose-Einstein condensate (BEC) due to two-photon collective annihilation decay is investigated analytically arising from the second quantized formalism. It is shown that because of the intrinsic instability of annihilation decay of BEC, the spontaneously emitted entangled photon pairs are amplified, leading to an exponential buildup of a macroscopic population into end-fire modes at a certain shape of the elongated condensate. The considered scheme may also be applied to a BEC of atoms or quasiparticles as a laser mechanism with double coherence to create entangled photonic beams with a macroscopic number of photons.

The nonlinear process of electron-positron pair production in the field of a strong electromagnetic wave in a plasma on the basis of the Dirac model is considered. The formulas for angular distribution and total number of electron-positron pairs are obtained. The latter, in the center-of-mass frame of the produced particles, also describe the process of pair production in vacuum by a uniform periodic electric field. The obtained approximate nonlinear solution of the Dirac equation is applicable in the field when the energy of interaction with the Dirac vacuum is comparable to the electron rest mass.

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