Bogolyubov transformations in the theory

of Positronium Bose-Einsteincondensate gamma laser

V.S.VanyashinDnepropetrosk national university

Gomel School-Seminar 24 July 2009

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 onA.D.Sakharov Conference 20 May 2009

Positronium Bose-Einsteincondensate 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)

Ortho- and Para-Positronium

;521

epara m61

92( )ortho em

4 5712 4

ie em m 1 0

20,1 0,1(2 / 4) em

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.

Phys. Rev. B 49, 454 - 458 (1994)Possibilities for Bose condensation of positronium

P. M. Platzman and A. 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

Define r= R/RBohr ,

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.

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-positroniumBEC density follows

12

332

6741reminduced

33

26741reminduced

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

3,2 3

241remcohdecay

cmL 33.31037.1 1050

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

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.

Acknowledgments

The author would like to thank organizers of the 2009 Gomel School-Seminar

for the invitation

Thank you for your attention

The End