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 and by DARPA/ARO Grant W911NF

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

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: What is an exciton-polariton? A) What is an exciton?

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

Mixing leads to “upper polariton” (UP) and “lower polariton” (LP) LP effective mass ~ m e Tune E ex (0) to equal E phot (0): ||  2 ||4))()(( 2 )()( 22, Rxc xc UPLP kEkE kEkE E      

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 10 9 cm -2 (shorter as density increases)

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

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

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

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

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

Sample Photoluminescence and Reflectivity Photoluminescence Reflectivity

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

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

40  m Do the polaritons really move? Drift and trapping of polaritons in trap Images of polariton luminescence as laser spot is moved Energy [meV]

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

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

Spatial profiles of polariton luminescence

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

Spatial profiles of polariton luminescence- creation at side of trap

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

Measurement of coherence: Spatially imaging Michelson interferometer L R

Below threshold Above threshold Michelson interferometer results

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

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

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

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

Angle-resolved luminescence spectra 50  W400  W 600  W800  W

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

Maxwell- Boltzmann fit Ae -E/k B T Occupation number N k vs. Energy min

Ideal equilibrium Bose-Einstein distribution E/k B T Maxwell-Boltzmann Bose-Einstein NkNk  = k B T  = -.1 k B T 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

Occupation number vs. Energy MB 80 K BE 80 K

Exciton distribution function in Cu 2 O: Snoke, Braun and Cardona, Phys. Rev. B 44, 2991 (1991). Maxwell-Boltzmann distribution D.W. Snoke and J.P. Wolfe, Physical Review B 39, 4030 (1989). - collisional time scale for BEC Kinetic simulations of equilibration “Quantum Boltzmann equation” “Fokker-Planck equation”

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 Tassone, et al, Phys Rev B 56, 7554 (1997). Tassone and Yamamoto, Phys Rev B 59, (1999). Porras et al., Phys. Rev. B 66, (2002). Haug et al., Phys Rev B 72, (2005). Sarchi and Savona, Solid State Comm 144, 371 (2007). Kinetic simulations of polariton equilibration

V. Hartwell, unpublished Full kinetic model for interacting polaritons

Unstressed-- weakly coupled “bottleneck” Weakly stressedResonant-- strongly coupled Angle-resolved data

Power dependence

Fit to experimental data for normal but highly degenerate state

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

1.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) threshold mean-field shift: blue shift for both LP, UP phase-space filling LP, UP shift opposite Quibbles and other philosophical questions

40  m Power dependence of trapped population Images of polariton luminescence as laser power is increased Energy [meV]

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

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

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

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

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”