1. Bogolyubov transformations in the theory of Positronium Bose-Einstein condensate gamma laser V.S.Vanyashin Dnepropetrosk national university Gomel School-Seminar 24 July 2009

2. Bogolyubov transformations N. N. Bogoliubov, J. Phys. (USSR) 9, 23 (1947) To the Theory of Superfluidity Let us switch to some slides from D.V.Shirkov presentation on A.D.Sakharov Conference 20 May 2009

6. Positronium Bose-Einstein condensate gamma laser This is an old and ingenious idea. Some people even attribute it to P. A. M. Dirac, On the Annihilation of Electrons and Protons, Proc. Cambridge Phil. Soc. 26, 361 (1930)

7. Ortho- and Para-Positronium

9. 9 Letters in Mathematical Physics 31: 143-149, 1994. © 1994 Kluwer Academic Publishers. Printed in the Netherlands. Coherent Decay of Positronium Bose Condensate VLADIMIR VANYASHIN International Centre for Theoretical Physics, Trieste 34100, Italy and Dnepropetrovsk State University, Dnepropetrovsk 320625, Ukraine. (Received: 2 December 1993) Abstract. The rate of self-stimulated emission of photon pairs by pseudoscalar particles from Bose condensate is calculated. Growing with density, this rate exceeds the density-independent rate of spontaneous two-photon decay at plausible density values of positronium gas, thus opening, in principle, the way to the annihilation gamma ray laser realization. Mathematics Subject Classifications (1991). 81V80, 81V10, 47D45.

10. Phys. Rev. B 49, 454 - 458 (1994) Possibilities for Bose condensation of positronium P. M. Platzman and A. P. Mills, Jr. P. M. PlatzmanA. P. Mills, Jr. AT&T Bell Laboratories, 600 Mountain Avenue, Murray Hill, New Jersey 07974 Received 2 September 1993 In this paper we consider the possibilities for producing a dense gas of N ≃ 10^5 triplet positronium (Ps) atoms in vacuum contained in small cavity V ≃ 10^-13 cm^3. We then consider the scenario where this dense gas of polarized Ps atoms may cool through a weakly interacting Bose transition. The rates for thermalization with the wall, equilibrium in the bulk, and loss of polarization by exchange collisions are calculated. A method for observing the transition is also discussed. ©1994 The American Physical Society http://link.aps.org/abstract/PRB/v49/p454

11. Define r= R/R Bohr, where the closed packed spheres radius R is determined by N/V = 1/(6 R 3 ) Then for a dilute enough gas with r= 50, N/V = 9 10 18 cm -3. According to the LMP 1994 paper this value is good for the target density of a Positronium condensate gamma laser. Bose-Einstein condensate with this density comprise 80 % of all Positronium atoms at the liquid hydrogen temperature T= 20 K.

12. Maser stimulated ortho-para transition of Positronium BEC To implement the idea of stimulated transition the theoretical background is needed. Let us introduce the phenomenological local fields for para- and ortho-positronium ground state atoms: φ(x) and ψ (x). The transition Hamiltonian is trilinear function of φ(x), ψ (x) and the magnetic component of the 200 GHz maser field H (x): 4 e φ(x) ψ (x) H (x). From this expression the time increment of para-positronium BEC density follows  12

13. Maser stimulated ortho-para transition of Positronium BEC This is definitely bigger than So r=50 and N/V = 9 10 18 cm -3 is also the appropriate target density for ortho-para induced transition. The critical laser length is 

14. Coherent Decay of Positronium Bose Condensate LMP, 1994 Let us see LMP.pdf

18. Conclusion The coherence of emitters, contrary the case of non-coherent emitters in common lasers, is the main point in the Physics of Positronium condensate gamma laser. Just the time-dependent Bogolyubov transformations provide an adequate mathematical tool for related coherent phenomena.

19. Acknowledgments The author would like to thank organizers of the 2009 Gomel School-Seminar for the invitation

20. Thank you for your attention The End