POLARITON CONDENSATION IN TRAP MICROCAVITIES: AN ANALYTICAL APPROACH

1. POLARITON CONDENSATION IN TRAP MICROCAVITIES: AN ANALYTICAL APPROACH C. Trallero-Giner, A. V. Kavokin and T. C. H. Liew Havana University and CINVESTAV-DF University of Southampton Ecole Polytechnique Fédérale de Lausanne

2. OUTLINE • Introduction • Analytical approaches • Results • Conclusions

3. Introduction

5. Peter Littlewood SCIENCE VOL 316 -Photons from a laser create electron-hole pairs or excitons. -The excitons and photons interaction form a new quantum state= polariton.

6. 2 dimensional GaAs-based microcavity structure. Spatial strep trap ( R. Balili, et al. Science 316, 1007 (2007))

8. The description of the linearly polarized exciton polariton condensate formed in a lateral trap semiconductor microcavity: two dimensional Gross-Pitaievskii equation α1 and α2 – self-interaction parameter ω – trap frequency m – exciton-polariton mass

9. The main goal -Explicit analytical representations for the whole range of the self-interaction parameter α1+α2. -To show the range of validity.

10. Analytical approaches Thomas-Fermi approach Experimentally it is not always the case

11. Variational method For non-linear differential equation the variational method is not well establish.

12. Green function formalism Gross-Pitaievskii integral equation -Green function

13. -spectral representation -harmonic oscillator wavefunctions -Integral representation

14. Perturbative method It is useful to get simple expressions for μ0 and Φ0 through a perturbation approach. ∫|Φ0(r)|2dr=N

15. Ψ0=Φ0/√N ∫| Ψ0|2dr=1 -small term

16. -must fulfill the non-linear equation system T is a fourth-range tensor

17. Energy Λ/2

19. The normalized order parameter Ψ0 Hn(z) the Hermite polynomial Ei(z)-the exponential integral; γ-the Euler constant

20. Ψ(r)= Φ(r)/√N r→r/l

21. The effect of a magnetic field The polaritons have two allowed spin projections If the absence of external magnetic field the ‘‘parallel spins’’ and ‘‘anti-parallelspin’’ states of noninteracting polaritons are degenerate. We are in presence of two independent circular polarized statesΦ± To find the order parameter in a magnetic field we start with the spinor GPE:

22. -Ωis the magnetic field splitting -α1 the interaction of excitons with parallel spin -α2the interaction of excitons with anti-parallel spin -two coupled spinor GPEs for the two circularly polarized components Φ± The normalization ∫|Φ±|dr = N± Ψ±(r)= Φ±(r)/√N ±

23. η=N+/N- Λ1=α1N+ /(2l2ћω) Energies Λ12=α2N- /(2l2ћω)

24. μ +=(E+-Ω))/ ћω =1+0.159*(Λ1+Λ12)+ 0.0036*F+(Λ1,Λ12) F+=(3Λ1+2Λ12)(Λ1/η+ηΛ12)+Λ12(Λ1+Λ12) μ -=(E-+Ω))/ ћω =1+0.159*(Λ1/η+Λ12 η)+ 0.0036*F-(Λ1/ η,Λ12η) F-=(3Λ1/η+2Λ12η)(Λ1+Λ12)+(Λ1/η+ηΛ12)Λ12η

25. μ +=(E+-Ω))/ ћω μ -=(E-+Ω))/ ћω Λ1=α1N+ /(2l2ћω) Λ12=α2N- /(2l2ћω)

26. μ +=1+0.159*(Λ1+Λ12) +0.0036*F+(Λ1,Λ12) μ -=1+0.159* (Λ1/ η+Λ12η)+0.0036* F-(Λ1, Λ12)

27. Order parameter for the two circularly polarized Ψ± components.

28. Λ1=1 Λ12=0.4 Ψ±= Φ±/√N± η=N+/N- =1 =0.6 =0.4

29. Conclusions -We have provided analytical solution for the exciton-polariton condensate formed in a lateral trap semiconductor microcavity. -An absolute estimation of the accuracy ofthe method −3 < Λ < 3

30. Λ versus the detuning parameter δ Typical Values GaAs N~105-106

31. We extended the method to find the ground state of the condensate in a magnetic field

32. -Validity of the method

33. THANKS