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
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 BoseEinstein condensation of exciton-polaritons
4. Some quibbles
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:

 Ps
2
Excitonic radius:
a   aPs
B) What is a cavity polariton?
“microcavity”
J. Kasprzak et al., Nature 443, 409 (2006).
cavity photon:
E  c kz2  k||2  c ( / L)2  k||2
quantum well exciton:
E  Egap   bind
2 2
h2 N 2
k||


2
2mr (2L)
2m
Tune Eex(0) to equal Ephot(0):
||
Mixing leads to “upper polariton” (UP) and “lower polariton” (LP)
LP effective mass ~ 10-4 me
ELP ,UP
r
r 2
r
r
2
(
E
(
k
)
E
(
k
Ec (k )  Ex (k )
c
 x ))  4 | hW R |
m

2
2
Light effective mass ideal for Bose quantum effects:
rs ~ dB
n1/ d ~ h / mkB T


h 2n 2 / d
T~
m
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)
Nozieres’ argument on the stability of the condensate:
Interaction energy of condensate:
1
1
1
E  V0 a0†a0†a0 a0  V0 N(N  1) ~ V0 N 2
2
2
2
Interaction energy of two condensates in nearly equal states, N1+N2=N:
1
1
E  V0 N1 (N1  1)  V0 N 2 (N 2  1)  2V0 N1 N 2
2
2
1
2
~ V0 N  V0 N1 N 2
2
Exchange energy in interactions drives the phase transition!
--Noninteracting gas is pathological-- unstable to fracture
Trapping Polaritons
How to put a force on neutral particles?
hydrostatic stress:
hydrostatic compression
= higher energy
h2
E
2m2
shear stress:
symmetry change
state splitting
E
s
Bending free-standing sample gives hydrostatic expansion:
finite-element analysis of stress:
3 x10
strain (arb. units)
0 x10
-3 x10
-6 x10
-9 10
x
-1.2 10
x
-5
0
-5
-5
F
-5
-4
-1
-0.5
0
0.5
1
x (mm)
hydrostatic strain
shear strain
Using inhomogenous stress to shift exciton states:
Relative Energy (meV)
GaAs quantum well excitons
5
0
-5
-10
-15
-0.8
-0.6
-0.4 -0.2
0
0.2
0.4
0.6
x (mm)
Negoita, Snoke and Eberl, Appl. Phys. Lett. 75, 2059 (1999)
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
increasing stress
trap
Balili et al., Appl. Phys. Lett. 88, 031110 (2006).
Motion of polaritons into trap
unstressed
bare exciton
bare photon
positive detuning
resonance (ring)
resonant creation
accumulation in trap
Do the polaritons really move?
Drift and trapping of polaritons in trap
Images of polariton luminescence
as laser spot is moved
40 m
Energy [meV]
1.608
1.606
1.604
1.602
1.600
Toward Bose-Einstein Condensation of Cavity Polaritons
  h / 2mkBT,
rs ~ n-1/2 (in 2D)
 superfluid at low T, high n
trap implies spatial
condensation
log T
normal
E
superfluid
x
log n
Critical threshold of pump intensity
Nonresonant,
circular polarized pump
Pump here! 115 meV excess energy
Luminescence intensity
at k|| =0 vs. pump power
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 RL R L RL RL R
Michelson interferometer results
Below threshold
Above threshold
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).
Degree of polarization vs. pump power
Note: Circular Polarized Pumping!
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 W
400 W
600 W
800 W
Intensity profile of momentum distribution of polaritons
0.4 mW
0.6 mW
0.8 mW
Occupation number Nk vs. Energy
MaxwellBoltzmann fit
Ae-E/kBT
min
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
 = -.001 kBT
Nk 
 = -.1 kBT
1
e(Ek   )/ kB T  1
Nk
Bose-Einstein
Maxwell-Boltzmann
E/kBT
Occupation number vs. Energy
3 106
MB 80 K
BE 80 K
6
N(k)
6
10
5
8 105
6 105
4 105
2 105
0
0.001
0.002
E-E
min
(eV)
0.003
0.004
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).
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).
r
n(k1 ) 2

t
h
r r
r
r
r
r
M (| k1  k1' |) n(k1 )n(k2 )[1  n(k1' )][1  n(k2' )] ( E1  E2  E1'  E2' )

r r
2
k 2 k1'
r r
2
M (| k1  k1' |)
r
r
r
r
n(k1 )n(k 2 )[1  n(k1' )][1  n(k 2' )]
•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
Cavity lifetime = 5 ps
Lattice Temperature = 20 K
Polariton-phonon scattering only
Polariton-polariton scattering without Bose terms and full polariton-phonon scattering
Full polariton-polariton scattering and full polariton-phonon scattering
Full kinetic model
for interacting
polaritons
100
Simulated Occupation
10
1
0.1
0.01
0.001
0
2
4
6
E-E
min
8
(meV)
V. Hartwell, unpublished
10
12
Unstressed-- weakly coupled
Angle-resolved data
“bottleneck”
Weakly stressed
Resonant-- strongly coupled
Power dependence
Cavity lifetime = 10 ps
Lattice T emperature = 20 K
Simulated Occupation
P=L
P=1.5L
P=2L
P=3L
10
1
0
0.5
1
1.5
2
E-E
min
(meV)
2.5
3
3.5
Fit to experimental data for normal but highly degenerate state
10
9
Occupation Number N
k
8
7
6
5
4
3
0
0.5
1
1.5
E-Emin (meV)
2
2.5
3
Strong condensate component:
below threshold
above threshold
far above threshold
logarithmic
intensity scale
thermal particles
linear
intensity scale
condensate
(ground state wave function in k-space)
Quibbles and other philosophical questions
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)
mean-field shift:
blue shift for both LP, UP
phase-space filling
LP, UP shift opposite
threshold
Power dependence of trapped population
Images of polariton luminescence
as laser power is increased
40 m
Energy [meV]
1.608
1.606
1.604
1.602
1.600
2. Does the trap really play a role, or is this essentially the same as
a 2D Kosterlitz-Thouless transition?
Spatially resolved spectra
below threshold
Flat potential
Trapped
at threshold
above threshold
3. Optical pump, coherent emission: Is this a laser?
normal laser
“lasing without inversion”
“stimulated scattering”
“stimulated emission”
radiative coupling
exciton-exciton interaction
coupling
(oscillators can be isolated)
(inversion can be negligible)
Two thresholds in same sample
Deng, Weihs, Snoke, Bloch, and Yamamoto,
Proc. Nat. Acad. Sci. 100, 15318 (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”