POLARITON CONDENSATION POLARITON CONDENSATION IN TRAP MICROCAVITIES: AN IN TRAP MICROCAVITIES: AN

ANALYTICAL APPROACHANALYTICAL 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

OUTLINEOUTLINE

• Introduction

• Analytical approaches

• Results

• Conclusions

Introduction

-Photons from a laser create electron-hole pairs or excitons.

-The excitons and photons interaction form a new quantum state= polaritonpolariton.

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:

α1 and α2 – self-interaction parameter ω – trap frequencym – exciton-polariton mass

-Explicit analytical representations for the whole range of the self-interactionparameter α1+α2.

The main goal

-To show the range of validity.

Thomas-Fermi approach

Experimentally it is not always the case

Analytical approaches

Variational methodFor non-linear differential equation the variationalmethod is not well establish.

-5 -4 -3 -2 -1 0 1 2 3 4 5

0.0

0.1

0.2

0.3

0.4

0.5

0.6

Numeric solution

ThomasFermi

VariationalMethod

x / l0

Norm

ali

zed

ord

er p

aram

ete

r (l 0)1

/2 x/l 0)

a)

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 μIt is useful to get simple expressions for μ00

and Φand Φ00 through a perturbation approach. through a perturbation approach.

∫|Φ0(r)|2dr=N

Ψ0=Φ0/√N

-small term

∫| Ψ0|2dr=1

-must fulfill the non-linear equation system

T is a fourth-range tensor

Energy Λ/2

-3 -2 -1 0 1 2 3 4 5

-0,5

0,0

0,5

1,0

1,5Numerical solution Analytical solution

ner

gy/

Universal result

The normalized order parameter Ψ0

Hn(z) the Hermite polynomial

Ei(z)-the exponential integral; γ-the Euler constant

Ψ(r)= Φ(r)/√N

r→r/l

0.4 0.8 1.2 1.6 2.0 2.4 2.8

0.1

0.2

0.3

0.4

0.5

r

Norm

alized o

der para

mete

r

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 fieldThe 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±

-α1 the interaction of excitons with parallel spin-α2 the interaction of excitons with anti-parallel spin

The normalization ∫|Φ±|dr = N±Ψ± (r)= Φ± (r)/√N ±

Λ1=α1N+ /(2l2ћω)

Λ12=α2N- /(2l2ћω)

η=N+/N-

EnergiesEnergies

μ +=(E+-Ω))/ ћω =1+0.159*(Λ1+Λ12)+ 0.0036*F+(Λ1,Λ12)

μ -=(E-+Ω))/ ћω =1+0.159*(Λ1/ η +Λ12 η)+ 0.0036*F-(Λ1/ η , Λ12 η)

F+=(3Λ1+2Λ12)(Λ1/η+ηΛ12)+Λ12(Λ1+Λ12)

F-=(3Λ1/η+2Λ12η)(Λ1+Λ12)+(Λ1/η+ηΛ12)Λ12η

0.5 1.0 1.5 2.0 2.5

1.2

1.3

1.4

1.5

μ +=(E+-Ω))/ ћω

μ -=(E-+Ω))/ ћω

Λ1=α1N+ /(2l2ћω)

Λ12=α2N- /(2l2ћω)

0.4 0.8 1.2 1.6

1.1

1.2

1.3

1.4

+= ( E+-

-= ( E-+

μ +=1+0.159*(Λ1+Λ12)+0.0036*F+(Λ1,Λ12)

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

Order parameter for the two circularly Order parameter for the two circularly polarized polarized ΨΨ±± components. components.

Λ1=1Λ12=0.4

Ψ± = Φ±/√N±

η=N+/N- =1

=0.6 =0.40.5 1.0 1.5 2.0 2.5

0.1

0.2

0.3

0.4

0.5_(r):N+=0.6N--

r

Norm

alized o

der para

mete

r

_(r):N+=0.4N-

r)

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

ΛΛ versus versus the detuning parameter the detuning parameter δδTypical Values GaAs

N~105-106

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

3/

N+/N

-<1

3

--Validity of the methodValidity of the method

THANKSTHANKS