Condensed exciton-polaritons in microcavity traps

C. Trallero-Giner

Centro Latinoamericano de Fisica, Rio de Janeiro, Brazil

Quito/Encuentro de Fisica/2013

OutlineI. Introduction

II. Mean field description of EPC

III. Bogoliubov excitationsIV. EPC coupled to uncondensed

polaritonsV. Conclusions

Satyendra Nath Bose y Satyendra Nath Bose y Albert EinsteinAlbert Einstein

Boson: Statistics Boson: Statistics

I. INTRODUCTIONA. Einstein, Sitzungsber. K. Preuss. Akad. Wiss. Phys. Math., 261, (1924).

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are needed to see this picture.

Eric Cornell and Carl Wieman

Bose-Einstein Condensation of Rb 87

Phase transition

O. Morsch and M. Oberthaler, Reviews of Modern Physics, Vol. 78, (2006), 179.

5

Gross-Pitaievskii equation

μ-the chemical potentialω-trap frequencym-the alkaline massλ-self-interaction parameter

L.K. Pitaevskii, Sov. Phys. JETP, 13, (1961), 451

6

Bose-Einstein condensation in an optical lattice

REVIEWS OF MODERN PHYSICS, VOLUME 78, JANUARY 2006

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•Bloch oscillationsPhys. Rev. Lett. 82, 2022 (1999)

•Superfluidity

•Dispersion and effective mass Phys. Rev. Lett. 86, 4447 (2001)

•Josephson physics in optical lattices

•Mott-insulator transition

8

Superfluidity

Estabilidad de la ecuación de GP y de las soluciones

S. Burger, et al., Phys. Rev. Lett. 86, 4447 (2001).

X t( )−center of mass

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Phys. Rev. Lett. 86, 4447 (2001).

C.Trallero-Giner et alEur. Phys. J. D 66, 177 (2012).

Atoms Tc ~ 10−9K

polaritons-----m is 0.0001 electron mass

Science V. 316

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

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

Tc ~ 300K

Exciton-polariton condensates

Excitations in a Nonequilibrium of Exciton Polaritons PRL 99, 140402 (2007)

G. Tosi et. al., Nature Physics 8, 190 (2012).

Spatially-mapped polariton condensate wavefunctions

Expt. scheme with two 1μm-diameter pump spots of separation 20μm. The effective potential V (red)

Tomographic images of polariton emission (repulsive potential seen as dark circles around pump spots).

G. Tosi et. al., Nature Physics 8, 190 (2012).

Real space spectra along line between pump spots

Spatially-resolved polariton energies on a line between pump spots (white arrows).

PRL 106, 126401 (2011)

Interactions in Confined Polariton Condensates

I. MEAN FIELD DESCRIPTION OF EPC

The description of the linearly polarized exciton polariton condensate formed in a lateral trap semiconductor microcavity:

two dimensional time dependent Gross-Pitaievskii equation

g – self-interaction parameter m – exciton-polariton mass

R – loss F – generationVres – interaction with the reservoir

V(r) - confinedpotential

Assuming R, F constant and using the transformation

we get

Y. Núñez Fernandez et al (to be published)

Two limit cases

A) Under stationary conditions, R ≈ F and the number of polaritons in the reservoir is small enough Nr Np≪

Stationary GPE

Bogoliubov-type elementary excitations

B) Under the conditions Nr N≫ p R < F, tthe stability condition of the condensate

after a certain time, t ∼ 1/[R-F] .

Text

Linear differential equation

III. BOGOLIUBOV EXCITATIONSThe collective excitations with frequencies ω

We linearized in terms of the amplitudes u and v

Axial symmetry: z-component of the angular momentum, mz and the

principal quantum number, N

sPerturbation theory approach

-some numbers

Partial conclusions-The excitation modes are weakly dependent Λ. -The total energy of the excited state, shows almost the same blue-shift dependence on Λ as the ground state energy.-The spectrum of the Bogolyubov-type excitations is nearly equidistant.

IV. EPC COUPLED TO UNCONDENSED POLARITONS

Vres – interaction with the reservoir

Assuming that the interaction with the reservoir is proportional to the pump profile

vres

=gres is the coupling constant describing the repulsive interaction with uncondensed excitons.

Employing the Bubnov-Galerkin method we solved the above linear equation.

=a/l0

Normalized energy levels, EN;0, of the EPC coupled to uncondensed polaritons created at the center of the trap as a function of the laser excitation power (Λres). Solid lines are for a = 0.2l0 and solid lines with full circles for a = l0.

Two limiting cases can be distinguished. -If a = 0 the energy levels tend to the harmonic oscillator eigenvalues, EN = N + 1

-If a = ∞, EN = N + 1+ Λres.

The level spacings, ∆EN = EN+2-EN, show a strong dependence on the laser spot size.

Laser pumping setup.a)The pumping scheme.

-if a increases, ∆EN,M=0 ≠ 2 and it depends on the number of polaritons in the reservoir.

For example, -if a = 0.2l0,∆EN = EN+2-EN 2, as for the 2D harmonic potential∼ .

Figure shows the influence of the uncondensed excitons on the condensate-The position of the density maximum is pushed

away from the origin as Λres increases.

-It is linked to the repulsive interactions produced bythe Gaussian density profile of uncondensed polaritonscreated in the trap.

-The condensate is repelled from the origin as the number of uncondensed excitons Nr (proportionalto the pumping beam intensity) increases..

PRL 106, 126401 (2011)

Infuence of the laser spot on the EPC density.a=l0

ρ1,2 = a ln2vresa2

ρ0 = 0;

Veff =12ρ2 +vres exp(−

ρ2

a2 )

The dependence of the condensate density profile on

Λres for the excited states with N = 1, 2 and 3.

Y. Núñez-Fernández Havana University,M. Vasilevskiy

Universidade do Minho, A. I. Kavokin

University of Southampton

Acknowledgments

I1.- We obtained convenient analytical description of the Bogolyubov-type elementary excitations. This can be used to describe the dynamics of the polariton BEC.

V. CONCLUSIONS

2.- The spectrum of these Bogolyubov-type excitations is almost equidistant even for rather larger values of thepolariton-polariton interaction parameter.

∆EN = EN+2-EN = 2

G. Tosi et. al., Nature Physics 8, 190 (2012).

3.- We obtained a semi-analytical solution for the ground and excited states of the condensate consider when the interaction with the reservoir of uncondensed polaritons is the most important one. It is shown that the states are "reshaped" by the repulsive interaction with the reservoir. Our results are in agreement with recent experiments

4.- It is shown that the level spacings between the condensate states increase with the pump power in correspondence with the recent experimental observation. We conclude that the experimentally observed emission patterns in confined condensates, pumped through polariton reservoir are not due to Bogolyubov-type elementary excitations in the condensate itself, rather they are determined by the repulsive condensate-reservoir interaction reshaping the density profile.

Nature Physics 8, 190 (2012).

∆EN = EN+2-EN ≠ 2

5.- We point out that the spectrum of these Bogolyubov-type excitations in a condensate whose interaction with uncondensed polaritons can be neglected, is almost equidistant even for rather larger values of the polariton-polariton interaction parameter inside the condensate. This makes polariton parabolic traps promising candidates for realization of bosonic cascade lasers. [T.C. H. Liew, et. al., Phys. Rev. Lett. 110 , 047402 (2013).]

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