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AVETISSIAN, AVETISSIAN, AND MKRTCHIAN PHYSICAL REVIEW A 92, 023820 (2015)

Because of the ongoing progress in physics for the produc-tion and manipulation of positronium atoms, one can expectthe realization of BEC of Ps atoms in the near future. Thelatter is a very challenging project and could lead to variousfundamental applications. In particular, it is of great interest toinvestigate the process of collective annihilation decay of Psatoms from the BEC state, as a coherent ensemble of lasingatoms, towards the generation of intense coherent γ rays inthe MeV domain of energies. The creation of a γ -ray laserhas been the subject of extreme interest since the realizationof the first lasers. The annihilation of electron-positron pairshas been considered as one of the basic processes for theintense γ -ray sources. The induced annihilation of a pair wasalready considered by Dirac [32]. For the observed γ -raylines from the astrophysical objects, the radiation through thespontaneous [33] and stimulated annihilation [34,35] of anelectron-positron plasma was considered. Then the ideas of PsBEC and subsequent annihilation in the context of a γ -ray laserwere considered in Refs. [19,36– 38]. In these papers, the lasinggain coefficient has been obtained from the rate equations. Thelatter is applicable to lasing systems with drivers (initial seed)and resonators and cannot be extrapolated to the exponentialgain regime [39]. Meanwhile, because there are no driversor mirrors operable at γ -ray frequencies, one should realizesingle pass lasers operating in the so-called self-amplifiedspontaneous-emission regime. A mechanism of a γ -ray laserat the collective annihilation of Ps atoms in a BEC statein the self-amplified spontaneous-emission regime has beenproposed in Ref. [40]. It has been shown that at the couplingof two macroscopic coherent ensembles of bosons—the BECof Ps atoms and photons—there is an instability at which,starting from the vacuum state of the photonic field, theexpectation value of the photon’s mode occupation growsexponentially for a narrow interval of frequencies aroundthe 511 keV line. In the present paper, a more detailedand thorough study of the γ -ray generation at the collectiveannihilation of Ps atoms in a BEC state in the self-amplifiedspontaneous-emission regime is presented. Here we utilizethe more general Hamiltonian including the stimulated o-Psto p-Ps transition. For the elongated shape of the BEC, itshows a laserlike action, i.e., directional radiation when thespontaneously emitted entangled and the oppositely directedphoton pairs are amplified, leading to an exponential buildupof a macroscopic population into the end-fire modes. We alsoinvestigate the influence of an external potential and interactionbetween the Ps atoms on the γ -ray self-amplification process.

The paper is organized as follows. In Sec. II, the mainHamiltonian is constructed. In Sec. III, two-photon decayof a Ps atom is analyzed. In Sec. IV, we consider theintrinsic instability of recoilless collective two-photon decayand present the setup for a γ -ray laser. In Sec. V, weconsider the influence of confinement and interaction betweenthe positronium atoms on the considered process. Finally,conclusions are given in Sec. VI.

II. BASIC HAMILTONIAN

We begin our study with construction of the Hamiltonianwhich governs the quantum dynamics of the consideredprocess. Here and below, except where it is stated otherwise,

FIG. 1. (Color online) The energy levels of interest. The uppertwo levels represent hyperfine splitting of the ground state of Ps. Theapplied electromagnetic field drives the o-Ps =⇒ p-Ps transition.The annihilation decay of the p-Ps into two entangled photons of thesame helicity is shown.

we employ natural units (c = ! = 1). Since o-Ps has arelatively long lifetime, in a laboratory-based experiment itwill be more suitable to obtain a Bose-Einstein condensate foro-Ps. As was proposed in Ref. [7], the use of spin-polarizedpositrons will eventually lead to a gas of spin-polarized Ps,which does not undergo the mutual spin-conversion reaction.Thus, in the ensemble of Ps atoms, rapid annihilation of thesinglet states and collisions among the various triplet substateswill cause the Ps atoms to become completely polarizedinto a pure m = 1 triplet state. Then, to trigger two-photonannihilation, one should induce the triplet-to-singlet transition.The latter can be realized via the ground-state hyperfinetransition either by the resonant sub-THz radiation (0.2 THz)or strong off-resonant electromagnetic field. Thus, in Fig. 1, theenergy levels of interest are schematically shown. The uppertwo levels represent hyperfine splitting of the ground state ofa Ps atom. The applied electromagnetic wave field drives theo-Ps =⇒ p-Ps transition. Then annihilation decay of the p-Psinto the two entangled photons of the same helicity are shown.

To obtain dynamic equations, we will utilize the secondquantized formalism. For this purpose, let us introduce the cre-ation and annihilation operators for p-Ps and o-Ps. The opera-tor describing the creation of p-Ps in the internal ground statewith the total center-of-mass momentum p can be written as

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