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

OUTLINE Introduction Analytical approaches Results Conclusions

Introduction

-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

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

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

- 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.

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

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

Gross-Pitaievskii integral equation - Green function Green function formalism

-spectral representation -Integral representation -harmonic oscillator wavefunctions

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

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

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

Energy Λ/2

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

Ψ(r)= Φ(r)/√N r→r/l

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±

-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 ±

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

μ + =(E + -Ω))/ ћω = *(Λ 1 +Λ 12 ) *F + (Λ 1,Λ 12 ) μ - =(E - +Ω))/ ћω = *(Λ 1 / η +Λ 12 η) *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

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

μ + = *(Λ 1 +Λ 12 ) *F + (Λ 1,Λ 12 ) μ - = * (Λ 1 / η +Λ 12 η) * F - (Λ 1, Λ 12 )

Order parameter for the two circularly polarized Ψ ± components.

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

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

Λ versus the detuning parameter δ Typical Values GaAs N~

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

-Validity of the method

THANKS