bose-einstein condensation of exciton-polaritons in a two-dimensional trap
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
Bose-Einstein Condensation of Exciton-Polaritonsin a Two-Dimensional Trap
D.W. SnokeR. BaliliV. Hartwell
University of Pittsburgh
L. PfeifferK. 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 Bose- Einstein condensation of exciton-polaritons
4. Some quibbles
Coulomb attraction between electron and hole givesbound 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?
Δ =ΔPs
ε 2a =εaPs
B) What is a cavity polariton?
“microcavity”
J. Kasprzak et al., Nature 443, 409 (2006).
cavity photon:
E =hc kz2 + k||
2 =hc (π / L)2 + k||2
quantum well exciton:
E =Egaπ −Δbind +
h2N2
2m r(2L)2 +
h2k||2
2m
Mixing leads to “upper polariton” (UP) and “lower polariton” (LP)
LP effective mass ~ 10-4 me
Tune Eex(0) to equal Ephot(0):
||
rr
2||4))()((
2)()( 22
,Rxcxc
UPLPkEkEkEkEE ++=
hmrr
Light effective mass ideal for Bose quantum effects:
€
rs ~ λ dB
€
n−1/ d ~ h / mkBT
€
T ~ h2n2 / d
m
Why not use bare cavity photons?
...photons are non-interacting.
Excitons have strong short-range interactionLifetime of polariton ~ 5-10 psScattering time ~ 4 ps at 109 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, N1+N2=N:
E =12V 0 a0
†a0†a0a0 =
12V 0N(N −1)~
12V 0N
2
E=12
V0N1(N1 1)+12
V0N2 (N2 1)+ 2V0N1N2
~12
V0N 2 +V0N1N2
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:
E= h2
2mλ2hydrostatic compression = higher energy
symmetry changestate splitting
hydrostatic stress:
s
E
Trapping Polaritons
stra
in (a
rb. u
nits
)
x (mm)
hydrostatic strain
shear strain
-1.2 10 -4
-9 10 -5
-6 10 -5
-3 10 -5
0 100
3 10 -5
-1 -0.5 0 0.5 1
F
x
x
x
x
x
x
finite-element analysis of stress:
Bending free-standing sample gives hydrostatic expansion:
Negoita, Snoke and Eberl, Appl. Phys. Lett. 75, 2059 (1999)
-15
-10
-5
0
5
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6R
elat
ive
Ene
rgy
(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, 031110 (2006).
Motion of polaritons into trap unstressed
positive detuning
resonant creation
accumulation in trap
bare photon
bare exciton
resonance (ring)
40 mm
Do the polaritons really move? Drift and trapping of polaritons in trap
Images of polariton luminescence as laser spot is moved
1.608
1.606
1.604
1.602
1.600
Ene
rgy
[meV
]
Toward Bose-Einstein Condensation of Cavity Polaritons
superfluid at low T, high n
λ =h / 2mkBT , rs ~ n-1/2 (in 2D)
log n
log T
superfluid
normal
E
x
trap implies spatialcondensation
Critical threshold of pump intensity
Nonresonant, circular polarized pump
Luminescence intensityat 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 RL RL RL RL R
Below threshold Above threshold
Michelson interferometer results
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).
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 mW 400 mW
600 mW 800 mW
Intensity profile of momentum distribution of polaritons
0.4 mW
0.6 mW
0.8 mW
Maxwell-Boltzmann fit Ae-E/k
BT
Occupation number Nk vs. Energy
min
Ideal equilibrium Bose-Einstein distribution
Nk =1
ε(Ek−m)/kBT −1
E/kBT
Maxwell-Boltzmann
Bose-EinsteinNk
m = -.001 kBT
m = -.1 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
2 105
4 105
6 105
8 105
106
3 106
0 0.001 0.002 0.003 0.004
N(k)
E-Emin
(eV)
2 105
4 105
6 105
8 105
106
3 106
0 0.001 0.002 0.003 0.004
N(k)
E-Emin
(eV)
Occupation number vs. Energy
MB 80 KBE 80 K
Exciton distribution function in Cu2O:
Snoke, Braun and Cardona, Phys. Rev. B 44, 2991 (1991).
Maxwell-Boltzmann distribution
D.W. Snoke and J.P. Wolfe, PhysicalReview 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
|)(| '112 kkM
rr−
)](1)][(1)[()( '2'121 knknknknrrrr
++
∑ −−+++−=∂
∂
'12
)()](1)][(1)[()(|)(|2)('2'121'2'121'11
21
kk
EEEEknknknknkkMtkn
rr
rrrrrrh
rδπ
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).
Kinetic simulations of polariton equilibration
0.001
0.01
0.1
1
10
100
0 2 4 6 8 10 12
Cavity lifetime = 5 psLattice Temperature = 20 K
Polariton-phonon scattering onlyPolariton-polariton scattering without Bose terms and full polariton-phonon scatteringFull polariton-polariton scattering and full polariton-phonon scattering
Simulated Occupation
E-Emin
(meV)
V. Hartwell, unpublished
Full kinetic modelfor interactingpolaritons
Unstressed-- weakly coupled
“bottleneck”
Weakly stressed Resonant-- strongly coupled
Angle-resolved data
1
10
0 0.5 1 1.5 2 2.5 3 3.5
Cavity lifetime = 10 psLattice Temperature = 20 K
P=LP=1.5LP=2LP=3L
E - Emin
(meV)
Power dependence
3
4
5
6
7
8
9
10
0 0.5 1 1.5 2 2.5 3
E-Emin
(meV)
Fit to experimental data for normal but highly degenerate state
logarithmicintensity scale
linearintensity 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 fillingLP, UP shift opposite
Quibbles and other philosophical questions
40 mm
Power dependence of trapped population
Images of polariton luminescence as laser power is increased
1.608
1.606
1.604
1.602
1.600
Ene
rgy
[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, 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 withsteady-state kinetic models
4. “Coherent light emission without lasing”“Lasing in the strongly coupled regime”or, “Lasing without inversion”