Dynamics of the Bose-Einstein condensation of a particle-antiparticle gas

1. Dynamics of the Bose-Einstein condensation of a particle-antiparticle gas O. Morandi : Institut de Physique et Chimie des Matériaux de Strasbourg, CNRS, 23 rue du Loess, Strasbourg, France. and Dipartimento di Matematica e Informatica “U.Dini” University of Florence, via S. Marta 3, Florence. morandi@dipmat.univpm.it

2. BEC condensation dynamics Main interest: estimation of the time necessary to achieve BEC of Positronium in a realistic experimental set-up metastable boson

3. energy BEC condensation dynamics Bose-Einstein condensate : Boson Condensation T < Tc energy space BEC: Phase transition where a macroscopic number of bosons occupy a single quantum state

4. BEC condensation dynamics: general considerations Condensation time depends on various factors Cooling process : evaporation (cold atoms: Rb) particle scattering (phonon-polariton) 87 Rb Polariton [2] [1] [1] M. Köhl, M. J. Davis, C.W. Gardiner, T.W. Hänsch, T. Esslinger, Phys. Rev. Lett. 88, 080402 (2002). [2] H. T. Cao, T. D. Doan, D. B. Tran Thoai, Phys Rev. B 69, 245325 (2004).

5. BEC condensation dynamics in porous silica Porous silica and positronium [1]-[2] : BEC of positronium trapped in a cylindrical cavity could be used as a source of coherent radiation of gamma ray [1] silica [1] D. B. Cassidy, S. H. M. Deng, R. G. Greaves, T. Maruo, N. Nishiyama, J. B. Snyder, H. K. M. Tanaka and A. P. Mills, Phys. Rev. Lett. 95, 195006 (2005). [2] D. B. Cassidy, P. Crivelli, T. H. Hisakado, L. Liszkay, V. E. Meligne, P. Perez, H. W. K. Tom, A. P. Mills Jr Phys Rev A 81, 012715 (2010). [3] A. P. Mills, Nuclear Instruments and Methods in Physics Research B, 192, 107 (2002).

6. Storing and cooling Ps Porous silica Creation of Ps : Diffusion : an electron can be captured Too much energy

7. Physical model external potential Hamiltonian of the system Free positronium “Real boson” : we discard the internal structure of the positronium

8. Theoretical model : Ps scattering What is special in B-E condensation? « Normal » system: Standard many-body theory is based on the hypothesis Bogoliubov showed that in the presence of the condensate this property is not longer true and where is the density of particles in the ground state

9. non-condensed cloud energy condensed particles Physical quantities : Condensate (order parameter) We describe the non-condensed particles by the Green-Wigner formalism Full quantum evolution equation: Extremely complicated ! Classical limit

10. Ps dynamics in the presence of a condensate Phonon_like behaviours

11. Ps dynamics in the presence of a condensate Quantum effects enter only in the particles scattering kernel Takes into account processes where particles enter or leave the condensate

12. Condensed particles: evolution equation (Gross Pitaevskii) Condensate (order parameter)

13. Numerical solution : Spherical pore • Spherical symmetry Numerical discretization • Transport (Hyperbolic conservation law) • 5th order WENO (weighted essentially non-oscillatory) scheme • Time integration • 3rd order Runge-Kutta • Total Variation Diminishing (TVD)‏ [1] • Collision operators • Deterministic approach

14. Numerical solution : Spherical pore • Descrpition of the dynamics • For t = 0 a hot gas is present in the cavity porous silica

15. Numerical solution : Spherical pore • Description of the dynamics • For t = 0 a hot gas is present in the cavity • When a particle impacts the wall is reemitted with lower energy porous silica

16. Numerical solution : Spherical pore • Description of the dynamics • For t = 0 a hot gas is present in the cavity • When a particle impacts the wall is emitted with lower energy • Since TL is lessen than the critical temperature • the condensate is formed • Hot and cold particles interact porous silica porous silica

17. Gas dynamics : Variables Equations Hot Ps dynamics f1,1 BE : • - Spin polarization : 4 Ps pop. 1 single pop. • Thermalization : Thot TL • ~ 6000 K ~100 K f1,0 f1,-1 Time f0,0 Coarse grid in the Energy variable f1,1 BE : Cold Ps dynamics Fine grid in the Energy variable - Condensation G-P : F

18. - Spin polarization : F = 0 - Many scattering channels: Example (without production of p-Ps) o-Ps lifetime t = 142 ns Example (with production of p-Ps) lifetime t = 125 ps

19. - Spin polarization : F = 0 - Numerical simulation - Many scattering channels: o-Ps lifetime t = 142 ns lifetime t = 125 ps

20. Numerical solution : Spherical pore t = 7.6 ns current current current current

21. Numerical solution : Spherical pore t =7.6 ns t = 8 ns

22. Numerical solution : Condensate Condensate particles

23. Numerical solution : Condensate Condensate particles R=100 nm

24. Numerical solution : Condensate Condensate particles R=100 nm

25. Numerical solution : Condensate R=300 nm Condensate particles t =7.6 ns The contact with the cold pore surface leads to an efficient condensate production Particles in the center of the condensate evaporate

26. Modeling condensation via evaporation Trap energy energy space

27. energy Modeling condensation via evaporation V cutoff t = 50 fs t = 30 ps t = 500 ps

28. Modeling condensation via evaporation non condensed particles condensed particles

29. Conclusions • Mathematical model for the study of the condensation dynamic of Ps • Numerical results: • reproduction of the chemical reactions and condensation dynamics • Study of the confinement effects • - Strong out-of-equilibrium dynamics (condensation via evaporation)