Bose-Einstein Condensation of Exciton-Polaritons in a Two-Dimensional Trap

1. Bose-Einstein Condensation of Exciton-Polaritons in a Two-Dimensional Trap D.W. Snoke R. Balili V. Hartwell University of Pittsburgh L. Pfeiffer K. West Bell Labs, Lucent Technologies Supported by the U.S. National Science Foundation under Grant 0404912 and by DARPA/ARO Grant W911NF-04-1-0075

2. Outline 1. What is an exciton-polariton? 2. Are the exciton-polaritons really a delocalized gas? Can we trap them like atoms? 3. Recent evidence for quasiequibrium Bose- Einstein condensation of exciton-polaritons 4. Some quibbles

3. What is an exciton-polariton? A) What is an exciton? Coulomb attraction between electron and hole gives bound state net lower energy for pair than for free electron and hole  states below single-particle gap “Wannier” limit: electron and hole form atom like positronium Excitonic Rydberg: Excitonic radius:

4. B) What is a cavity polariton? “microcavity” J. Kasprzak et al., Nature 443, 409 (2006). cavity photon: quantum well exciton:

5. || 2 2 ( E ( k ) E ( k )) 4 | | E ( k ) E ( k ) h - + W + c x R c x E m = LP , UP 2 2 Tune Eex(0) to equal Ephot(0): Mixing leads to “upper polariton” (UP) and “lower polariton” (LP) LP effective mass ~ 10-4me r r r r

6. Light effective mass ideal for Bose quantum effects: Why not use bare cavity photons? ...photons are non-interacting. Excitons have strong short-range interaction Lifetime of polariton ~ 5-10 ps Scattering time ~ 4 ps at 109 cm-2 (shorter as density increases)

7. Nozieres’ argument on the stability of the condensate: Interaction energy of condensate: Interaction energy of two condensates in nearly equal states, N1+N2=N: 1 1 E V N ( N 1 ) V N ( N 1 ) 2 V N N = - + - + 0 1 1 0 2 2 0 1 2 2 2 1 2 ~ V N V N N + 0 0 1 2 2 Exchange energy in interactions drives the phase transition! --Noninteracting gas is pathological-- unstable to fracture

8. hydrostatic compression = higher energy symmetry change E state splitting s Trapping Polaritons How to put a force on neutral particles? hydrostatic stress: shear stress:

9. Bending free-standing sample gives hydrostatic expansion: finite-element analysis of stress: strain (arb. units) x (mm) hydrostatic strain shear strain

10. Using inhomogenous stress to shift exciton states: GaAs quantum well excitons Relative Energy (meV) x (mm) Negoita, Snoke and Eberl, Appl. Phys. Lett. 75, 2059 (1999)

11. Typical wafer properties • Wedge in the layer thickness • Cavity photon shifts in energy due layer thickness • Only a tiny region in the wafer is in strong coupling! Reflectivity spectrum around point of strong coupling

12. Sample Photoluminescence and Reflectivity Photoluminescence Reflectivity

13. Reflectivity and luminescence spectra vs. position on wafer false color: luminescence grayscale: reflectivity increasing stress trap Balili et al., Appl. Phys. Lett. 88, 031110 (2006).

14. bare exciton bare photon resonance (ring) Motion of polaritons into trap unstressed positive detuning resonant creation accumulation in trap

15. 40 m 1.608 1.606 1.604 1.602 1.600 Energy [meV] Do the polaritons really move? Drift and trapping of polaritons in trap Images of polariton luminescence as laser spot is moved

16. , rs ~ n-1/2 (in 2D) E trap implies spatial condensation x Toward Bose-Einstein Condensation of Cavity Polaritons  superfluid at low T, high n log T normal superfluid log n

17. Critical threshold of pump intensity Luminescence intensity at k|| =0 vs. pump power Nonresonant, circular polarized pump Pump here! 115 meV excess energy

18. Spatial profiles of polariton luminescence

19. Spatial narrowing cannot be simply result of nonlinear emission model of gain and saturation

20. Spatial profiles of polariton luminescence- creation at side of trap

21. General property of condensates: spontaneous coherence Andrews et al., Science 275, 637 (1997).

22. L R L R L R L R L R Measurement of coherence: Spatially imaging Michelson interferometer

23. Michelson interferometer results Below threshold Above threshold

24. Spontaneous linear polarization --symmetry breaking kBT small splitting of ground state aligned along [110] cystal axis Cf. F.P. Laussy, I.A. Shelykh, G. Malpuech, and A. Kavokin, PRB 73, 035315 (2006), G. Malpuech et al, Appl. Phys. Lett. 88, 111118 (2006).

25. Degree of polarization vs. pump power Note: Circular Polarized Pumping!

26. Threshold behavior k||=0 intensity k||=0 spectral width degree of polarization

27. In-plane k|| is conserved  angle-resolved luminescence gives momentum distribution of polaritons.

28. Angle-resolved luminescence spectra 50 mW 400 mW 600 mW 800 mW

29. Intensity profile of momentum distribution of polaritons 0.4 mW 0.6 mW 0.8 mW

30. Occupation number Nk vs. Energy Maxwell-Boltzmann fit Ae-E/kBT min

31.  = -.001 kBT  = -.1 kBT Bose-Einstein Nk Maxwell-Boltzmann E/kBT Can the polariton gas be treated as an equilibrium system? Does lack of equilibrium destroy the concept of a condensate? lifetime larger, but not much larger, than collision time continuous pumping Ideal equilibrium Bose-Einstein distribution

32. Occupation number vs. Energy MB 80 K BE 80 K

33. Kinetic simulations of equilibration Exciton distribution function in Cu2O: D.W. Snoke and J.P. Wolfe, Physical Review B 39, 4030 (1989). - collisional time scale for BEC “Quantum Boltzmann equation” “Fokker-Planck equation” Maxwell-Boltzmann distribution Snoke, Braun and Cardona, Phys. Rev. B 44, 2991 (1991).

34. The square of the interaction matrix element between two states • Polariton-polariton scattering or • polariton-phonon scattering • Accounts for the particle statistics, bosons in this case Kinetic simulations of polariton equilibration Tassone, et al , Phys Rev B 56, 7554 (1997). Tassone and Yamamoto, Phys Rev B 59, 10830 (1999). Porras et al., Phys. Rev. B 66, 085304 (2002). Haug et al., Phys Rev B 72, 085301 (2005). Sarchi and Savona, Solid State Comm 144, 371 (2007).

35. Full kinetic model for interacting polaritons V. Hartwell, unpublished

36. Angle-resolved data Unstressed-- weakly coupled “bottleneck” Weakly stressed Resonant-- strongly coupled

37. Power dependence

38. Fit to experimental data for normal but highly degenerate state

39. Strong condensate component: far above threshold above threshold below threshold logarithmic intensity scale thermal particles condensate (ground state wave function in k-space) linear intensity scale

40. threshold Quibbles and other philosophical questions Are the polaritons still in the strong coupling limit when the threshold effects occur? i.e., are the polaritons still polaritons? (phase space filling can reduce coupling, close gap between LP and UP) mean-field shift: blue shift for both LP, UP phase-space filling LP, UP shift opposite

41. 40 m 1.608 1.606 1.604 1.602 1.600 Energy [meV] Power dependence of trapped population Images of polariton luminescence as laser power is increased

42. 2. Does the trap really play a role, or is this essentially the same as a 2D Kosterlitz-Thouless transition?

43. Spatially resolved spectra below threshold above threshold at threshold Flat potential Trapped

44. 3. Optical pump, coherent emission: Is this a laser? normal laser “lasing without inversion” “stimulated scattering” “stimulated emission” exciton-exciton interaction coupling (inversion can be negligible) radiative coupling (oscillators can be isolated)

45. Two thresholds in same sample Deng, Weihs, Snoke, Bloch, and Yamamoto, Proc. Nat. Acad. Sci. 100, 15318 (2003).

46. Conclusions 1. Cavity polaritons really do move from place to place and act as a gas, and can be trapped 2. Multiple evidences of Bose-Einstein condensation of exciton-polaritons in a trap in two dimensions 3. Bimodal momentum distribution is consistent with steady-state kinetic models 4. “Coherent light emission without lasing” “Lasing in the strongly coupled regime” or, “Lasing without inversion”