1. Condensed exciton-polaritons in microcavity traps C. Trallero-Giner Centro Latinoamericano de Fisica, Rio de Janeiro, Brazil Quito/Encuentro de Fisica/2013

2. Outline I. Introduction II. Mean field description of EPC III. Bogoliubov excitations IV. EPC coupled to uncondensed polaritons V. Conclusions

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6. 6 Bose-Einstein condensation in an optical lattice REVIEWS OF MODERN PHYSICS, VOLUME 78, JANUARY 2006

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9. 9 Phys. Rev. Lett. 86, 4447 (2001). C.Trallero-Giner et al Eur. Phys. J. D 66, 177 (2012).

10. Atoms polaritons -----m is 0.0001 electron mass Science V. 316 1.- Photons from a laser create electron-hole pairs or excitons. polariton 2.- The excitons and photons interaction form a new quantum state= polariton. Exciton-polariton condensates

13. Excitations in a Nonequilibrium of Exciton Polaritons PRL 99, 140402 (2007) G. Tosi et. al., Nature Physics 8, 190 (2012).

14. 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).

15. Real space spectra along line between pump spots Spatially-resolved polariton energies on a line between pump spots (white arrows).

16. PRL 106, 126401 (2011) Interactions in Confined Polariton Condensates

17. 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 C. Trallero-Giner, et al. Phys. Rev. B 82, 165421 (2010) R – loss F – generation V res – interaction with the reservoir V ( r ) - confined potential

18. Assuming R, F constant and using the transformation we get Y. Núñez Fernandez et al (to be published)

19. 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

20. B) Under the conditions N r ≫ N p R < F, t t he stability condition of the condensate after a certain time, t ∼ 1/[R-F]. Text Linear differential equation

21. III. BOGOLIUBOV EXCITATIONS The collective excitations with frequencies ω We linearized in terms of the amplitudes u and v Axial symmetry: z-component of the angular momentum, m z and the principal quantum number, N s Perturbation theory approach -some numbers

24. 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.

25. IV. EPC COUPLED TO UNCONDENSED POLARITONS V res – interaction with the reservoir Assuming that the interaction with the reservoir is proportional to the pump profile v res = g res is the coupling constant describing the repulsive interaction with uncondensed excitons. Employing the Bubnov-Galerkin method we solved the above linear equation. =a/l 0

26. Normalized energy levels, E N;0, of the EPC coupled to uncondensed polaritons created at the center of the trap as a function of the laser excitation power (Λ res ). Solid lines are for a = 0.2l 0 and solid lines with full circles for a = l 0. Two limiting cases can be distinguished. -If a = 0 the energy levels tend to the harmonic oscillator eigenvalues, E N = N + 1 -If a = ∞, E N = N + 1+ Λ res. The level spacings, ∆E N = E N+2 -E N, show a strong dependence on the laser spot size.

27. Laser pumping setup. a)The pumping scheme.

28. -if a increases, ∆E N,M=0 ≠ 2 and it depends on the number of polaritons in the reservoir. For example, -if a = 0.2l 0, ∆E N = E N+2 -E N ∼ 2, as for the 2D harmonic potential.

29. Figure shows the influence of the uncondensed excitons on the condensate -The position of the density maximum is pushed away from the origin as Λ res increases. -It is linked to the repulsive interactions produced by the Gaussian density profile of uncondensed polaritons created in the trap. -The condensate is repelled from the origin as the number of uncondensed excitons N r (proportional to the pumping beam intensity) increases.. PRL 106, 126401 (2011)

30. Infuence of the laser spot on the EPC density. a=l 0

31. The dependence of the condensate density profile on Λ res for the excited states with N = 1, 2 and 3.

32. Y. Núñez-Fernández Havana University, M. Vasilevskiy Universidade do Minho, A. I. Kavokin University of Southampton Acknowledgments

33. I 1.- 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 the polariton-polariton interaction parameter. ∆E N = E N+2 -E N = 2 G. Tosi et. al., Nature Physics 8, 190 (2012).

34. 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

35. 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). ∆E N = E N+2 -E N ≠ 2

36. 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).] Light THANKS