1. Polarization of exciton polariton condensates in lateral traps 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 Scalar BEC in a two dimensional trap Spinor condensates of exciton- polaritons Conclusions

3. Introduction

6. Es posible obtener alta densidad de un gas de “átomos” ligeros. La coherencia cuántica debe ser a las altas tempertaturas Los polaritones cuya masa es 0.0001 m e POLARITON CONDENSATION IN TRAP MICROCAVITIES

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

9. 2 dimensional GaAs-based microcavity structure. Spatial strep trap ( R. Balili, et al. Science 316, 1007 (2007))

11. REVIEWS OF MODERN PHYSICS, VOL. 82, APRIL–JUNE 2010

12. two dimensional Gross-Pitaievskii equation The description of the linearly polarized exciton polariton condensate formed in a lateral trap semiconductor microcavity : α 1 and α 2 – self-interaction parameter ω – trap frequency m – exciton-polariton mass Scalar BEC

13. - Explicit analytical representations for the whole range of the self-interaction parameter α 1 +α 2. The main goal -To show the range of validity.

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

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

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

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

18. Perturbative method It is useful to get simple expressions for μ 0 and Φ 0 through a perturbation approach. ∫|Φ 0 (r)| 2 dr=N

19. Ψ 0 =Φ 0 / √N -small term ∫| Ψ 0 | 2 dr=1

20. Using the integral representation for the 2D GPE The general solution for the order parameter Ψ 0 has an explicit representation as {φ n1;n2 (r)} -2D harmonic oscillator wave functions

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

22. The eigenvector C is sought in the form of a series of the nonlinear interaction parameter Λ -small term

23. Energy Λ/2

25. The normalized order parameter Ψ 0 H n (z) the Hermite polynomial Ei(z)-the exponential integral; γ-the Euler constant

26. Ψ(r)= Φ(r)/√N r→r/l

27. In typical microcavities the values of the interaction constants can change with the exciton-photon detuning, δ E b -the exciton binding energy, a b -the exciton Bohr radius X -the excitonic Hopfield coefficient V the exciton-photon coupling energy GaAs

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

30. -Ω is the magnetic field splitting -two coupled spinor GPEs for the two circularly polarized components Φ ± -α 1 the interaction of excitons with parallel spin -α 2 the interaction of excitons with anti-parallel spin The normalization ∫ |Φ ± |dr = N ± Ψ ± (r)= Φ ± (r)/√N ±

31. Λ 1 =α 1 N + /(2l 2 ћω) Λ 12 =α 2 N - /(2l 2 ћω) η=N + /N - Energies

32. μ + =(E + -Ω))/ ћω =1+0.159*(Λ 1 +Λ 12 )+ 0.0036*F + (Λ 1,Λ 12 ) μ - =(E - +Ω))/ ћω =1+0.159*(Λ 1 / η +Λ 12 η)+ 0.0036*F - (Λ 1 / η, Λ 12 η) F + =(3Λ 1 +2Λ 12 )(Λ 1 /η+ηΛ 12 )+Λ 12 (Λ 1 +Λ 12 ) F - =(3Λ 1 /η+2Λ 12 η)(Λ 1 +Λ 12 )+(Λ 1 /η+ηΛ 12 )Λ 12 η

33. μ + =(E + -Ω))/ ћω μ - =(E - +Ω))/ ћω Λ 1 =α 1 N + /(2l 2 ћω) Λ 12 =α 2 N - /(2l 2 ћω)

34. μ + =1+0.159*(Λ 1 +Λ 12 ) +0.0036*F + (Λ 1,Λ 12 ) μ - =1+0.159* (Λ 1 / η +Λ 12 η)+0.0036* F - (Λ 1, Λ 12 )

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

36. Λ 1 =1 Λ 12 =0.4 Ψ ± = Φ ± /√N ± η=N + /N - =1 =0.6 =0.4

37. The circular polarization degree If the condensate is elliptically polarized we find a nonuniform distribution of the Polarization in space.

38. The circular polarization degree at r = 0 Polariton number The polarization changes from circular to elliptical and approaches a linear polarization asymptotically at high polariton number.

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

40. Λ versus the detuning parameter δ Typical Values GaAs N~10 5 -10 6

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

42. -Validity of the method

43. THANKS

45. PRL. 86, 4447 (2001)