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”