Bose-Einstein condensation of excitons and polaritons

Exciton and Polariton Condensates
Outline
I. Historical review of exciton BEC and general issues
II. BEC of excitons in coupled quantum wells (bilayers)
III. BEC of polaritons in microcavities
Supported by the U.S. National Science Foundation
under Grant DMR-0706331 and the Department of
Energy under grant DE-FG02-99ER45780.

General concepts of excitons
quasi number conservation. Cf. proton decay and H atom BEC
crossover: quasiequilibrium BEC ! nonequilibrium BEC ! immediate decay
incoherent
phonon
emission
photon emission: weak probe of instantaneous energy and momentum
distribution
conduction states
valence states
“Wannier” limit: electron and hole form atom
like positronium that has center of mass velocity
Excitonic Rydberg: Excitonic radius:
" = " Ps
! 2
a = !a Ps

Prehistory of exciton BEC
early, independent predictions: Moskalenko 1959, Blatt 1962
early mean-field theory: Keldysh and Kozlov, 1967
Nozieres and Comte, 1982
Early focus was on bulk semiconductors with deeply bound excitons
(Rydberg ~ 100 meV)
•nearly impossible to exceed Mott density with 3D tightly bound (~ 5Å)
excitons
•direct gap semiconductors: no electron-hole liquid (EHL) phase
(analogous to liquid hydrogen)
• in principle can have BEC in indirect-gap semiconductors on short
time scales (ca. diamond, Gonokami)

CuCl biexcitons
Peyghambarian et al., 1983
• sharp spectral peak at low energy
• superradiance? lasing?
• no evidence of coherence or Bose-
Einstein thermal distribution
• no trapping

Cu 2 O excitons
Snoke et al, 1987
• very good fits to Bose-Einstein
(not condensed) distribution
• later interpreted as due to spatial
integration in 3D light collection,
inhomogeneous spatial profile
(no trapping, surface excitation)
• estimates of exciton density (very difficult)
need for trap!
2D systems: direct imaging
(and also momentum distribution by angle resolution)

Coupled Quantum Wells
+ +
-
-
large spatial separation
gives long exciton
lifetime.
(Lozovik and Yudson 1975)
Lozovik and Berman,
JETP 84, 1027 (1997):
overall repulsion
when D > 3a B

Tradeoff:
binding energy decreases
as well width increases
100 Å GaAs wells:
total D = 100 + 40 + 100 = 240 Å
need Ry >> E disorder, k BT
binding energy 4 meV
lifetime ~ 10 µs!
M. Szymanska (Cambridge)

But mobility increases strongly with increasing well width
µ ~ L 6
H. Sakaki et al., APL 51, 1934 (1987)

Measurements of exciton drift/diffusion in flat 2D potential
(with disorder)
Spectral imaging system
x
#
CCD
imager
imaging spectrometer
laser
imaging lens

Rise time of luminescence vs. x
(lifetime = 6.5 µs)
Intensity (arb. units)
0
60
120
180
240
400
300
200
0 500 1000 1500 2000 2500 3000 3500
t (ns)

strong dipole-dipole repulsion of indirect excitons:
+ +
H = H0 + !
+ +
Ua1 a2 a3a4 -
blue shift at high density
Voros et al, PRL 2005
-
E

Early claims of exciton BEC in QCW’s
Fukuzawa, Kash (IBM group)
PRL 1990 PRL 1991
quasi-Fermi distribution: hard core repulsion in disordered potential
mapping: one spatial location $one energy level

Timofeev et al. (many JETP)
micro-PL from disorder minima

Butov et al.
exciton luminescence at interface of electron-hole gases
in-plane coherence length (µm)

strain (arb. units)
Trapping excitons:
bending free-standing sample gives hydrostatic expansion:
3 10 -5
x
0 10 0
x
-3 10 -5
x
-6 10 -5
x
-9 10 -5
x
finite-element analysis of stress:
-1.2 10 -4
x
-1 -0.5 0 0.5 1
x (mm)
hydrostatic strain
shear strain
F
fit to experimental exciton line position
using known deformation potentials:

Energy
position

E%
Equilibrium in trap:
blue shift and broadening
-- trap is flattened at high density

The “black hole”
substrate luminescence
at laser spot
dark spot appears at stress where lh-exciton crosses hh-exciton.
disappears at low density.
center spot is not completely dark.
T=2K

Temperature series-- spatial images

polarization of
bright luminescence
low stress
high stress,
with dark spot

General considerations for coupled quantum wells
spin structure: lowest exciton has two J=1 (“bright”) and two J=1 (“dark”)
states.
fourfold degeneracy broken by stress.
exchange effects greatly suppressed by low overlap of electron-hole
wave functions.
brightness/lifetime affected by lh-hh mixing with stress
Combescot: for degenerate spin states, BEC should always occur in
dark state.
limited available density range:
localization in disorder at low density,
low T.
interactions: flattening of trap at
high density.
Effective Exciton
temperature (K)
Bath temperature (K)

interactions: generally a difficult problem for all types of excitons.
e-e exchange
h-h exchange
e-h exchange
Like Ps, but gap is only ~100X greater than binding energy.
Not a generally solved problem after 50 years!
Correlations play a huge role. mean-field theory differs from
experiment by over an order of magnitude.
ratio of width to shift

Cavity Polaritons
!
cavity photon:
E = hc kz 2 + k||
quantum well exciton:
E = E gap " # bind +
2 = hc (" /L) 2 + k||
h 2 N 2
2mr(2L) 2 + h2 2
k ||
2m
2

Light effective mass ideal
for Bose quantum effects:
(Imamoglu 1996?) !
||
!
LP effective mass ~ 10 -4 m e
r s ~ " dB
n "1/ d ~ h / mk BT
T ~ h 2 2 / d
n
m
Why not use bare cavity photons?
paid for by reduced lifetime ~ 10 ps
!
...photons are (nearly) non-interacting.
Excitons have strong short-range interaction
polariton = “photon dressed with mass and hard core repulsion”

Typical wafer
properties
• Wedge in the layer
thickness
• Cavity photon
shifts in energy
due layer
thickness
Reflectivity spectrum
around point of strong
coupling
GaAs MBE 70-Å QW’s

Trapping as stress is increased
false color:
luminescence
grayscale:
reflectivity
increasing stress
trap
Balili et al., Appl. Phys. Lett. 88, 031110 (2006).

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

!
Results at Critical Threshold in Trap
General condition for exchange to be important:
E'
, rs ~ n-1/2 " = h / 2mkBT (in 2D)
& superfluid at low T or high density
trap implies spatial
condensation
excited thermal particles
x!
log T
normal
log n
superfluid

Four types of pumping:
E
k
electron-hole continuum
exciton states
k
||
1) direct resonant (normal incidence)
coherence of laser directly mapped
2) “magic angle”: single elastic scatt
populates ground state
coherence of laser mapped?
Whitakker: yes. Kavakin: no
3) large angle: many scatterings
need intense laser pulse since
reflected by cavity
4) high excess energy, must emit
phonons

Threshold behavior with incoherent pump:
Pump:
115 meV excess energy
circular polarized
E
k
electron-hole continuum
exciton states
k
Luminescence intensity
at k || =0 vs. pump power

Spatial profiles of polariton luminescence

Spatial profiles of polariton luminescence- creation at side of trap

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

Angle-resolved luminescence spectra
50 µW 400 µW
600 µW 800 µW
!
"x "p (
h

Momentum distribution of polaritons
0.4 mW
0.6 mW
0.8 mW
Energy distribution of polaritons
Maxwell-
Boltzmann fit
Ae -E/k B T

Issues of equilibration
Does lack of equilibrium destroy the concept of a condensate?
N k =
lifetime longer, but not much longer, than collision time
continuous pumping
1
e (Ek )µ)/kBT )1
Not seen in any
polariton experiments
for E >> k BT
Ideal equilibrium Bose-Einstein distribution
N k
µ = -.001 k B T
µ = -.1 k B T
Bose-Einstein
Maxwell-Boltzmann
E/k B T

Occupation number vs. Energy
Tail is too hot!
N(k)
3 10 6
3 10 6
10 6
10 6
8 10 5
8 10 5
6 10 5
6 10 5
4 10 5
4 10 5
2 10 5
2 10 5
0 0.001 0.002 0.003 0.004
E-E (eV)
min MB
BE

Kinetic simulations of polariton equilibration
r
# n(
k1)
# t
=
2%
h
!
r r
k k
2 1'
M
2
r
(| k
1
r
" k
1'
r r r r
|) n(
k ) n(
k )[ 1+
n(
k )][ 1+
n(
k )] $ ( E
1
!"##$%&'(!"#$%#&#)*+#(,&-(.(!"'(/001(2344/56
!"##$%&("%7(8"9"9$:$'()*+#(,&-(.(!#'(3;>"#(&:("?6'()*+#6(,&-6(.(""'(;

Three regimes:
polaritons near equilibrium
tail to high energy in “bottleneck” region
excitons at lattice T

Issues with low-density data
universal curve
at low density

phonon scattering
totally inadequate
to populate
polariton states.
but pol-pol
scattering should
by density dependent!
Malpeuch and
coworkers:
ad hoc energy broadening
(associated with
disorder)
alternative:
assume small number
of permanent free electrons
Simulated Occupation
100
10
1
0.1
0.01
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
0.001
0 2 4 6 8 10 12
E-E min (meV)
V. Hartwell, Ph.D. thesis (2008)

Spontaneous linear polarization
-a type of symmetry breaking
k B T
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).

Splitting of bright
exciton states seen in
microcavity
different linear polarization
due to anisotropic exchange
splitting of exciton states

Other recent experiments:
vortices
superfluid wavepacket motion, suppression of scattering
spontaneous symmetry breaking (random polarization)
first- and second-order coherence
onset time of coherence
BEC in microtraps with discrete states
room temperature polaritons
“driven” resonant coherence with highly nonlinear effects
...applications in optical communications (nonlinear modulation,
low-threshold lasing, cw OPO, optical spin-Hall effect, etc.)

phase locking via interactions
E
-1.5 -1 -0.5 0 0.5 1 1.5
k
spatial: Deveaud et al.
momentum space: Yamamoto et al.
“pi band” via in-plane periodic pattern

Is there a difference between polariton BEC and lasing?
coherent light emitted
• spectral narrowing
• linear polarization
• beamlike emission
(Is current emitted from a superconductor different?
output = probe of state of matter)
“phase space filling” ! “weak coupling”
lasing
A
conduction band
valence band

Weak coupling turns off Rabi splitting
“Photon Lasing”
“BEC”

Two ways to tune through resonance
!=0
!=0

Our results for
resonant, non-trapped
polaritons
(“magic spot” on sample)
PL intensity
at low density
pump power
at threshold

Results in stress trap
line position
blue triangles: 2.5x threshold
emission follows exciton shift
PL intensity
at low density
pump power
at threshold

Resonant, non-trapped
Intensity vs.
pump power
energy shift (dots)
FWHM (solid line)
lasing

With stress trap
Intensity vs.
pump power
energy shift (dots)
FWHM (solid line)
-two thresholds!
BEC
lasing

Angle-resolved spectra
Stress trap, resonant

Spatially-resolved spectra
laser spot center
Stress trap, resonant

Angle-resolved spectrum
Unstressed, resonant

Summary
1. big advantage of 2D structures over 3D bulk
spatial imaging, k-space imaging
tailor for lifetime, effective mass
2. coupled quantum wells: no good evidence for BEC
why not? interactions too strong? dark excitons?
3. polaritons: good evidence for nonequilibrium BEC
“polariton laser”? terminology battle, but clearly different from
normal lasing

Botao Zhang
Nick Sinclair
Annie Wang
Bryan Nelson
Bridget Bertoni
Vince Hartwell
David Snoke
Chuan Yang
Jeff Wuenschell
Ryan Balili
Zoltan Vörös

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