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Dynamical Condensation of Exciton-Polaritons
kξ=1
-15o 15o
1.613 eV
8 m
eV
kξ=1-15o 15o
1.613 eV
S. Utsunomiya, C.W. Lai, G. Roumpos and Y.YamamotoStanford University, National Institute of Informatics
A. Loeffler, S. Hoefling, and A. ForchelTechnische Physik, Universität Wurzburg
International School of Physics “Enrico Fermi”:Quantum Coherence in Solid State PhysicsVarenna (Italy) July 1 – 11, 2008
1
Lecture 3 Lecture 3 BogoliubovBogoliubov excitation and excitation and superfluiditysuperfluidity
2
Outline
Bose-Einstein condensation and superfluidityBogoliubov theory for weakly interacting Bose gasInteraction energy of exciton-polaritonsEnergy-momentum dispersion relation: revisited - Condensates and excitation spectrum - Sound velocity - Free particle energyQuantum depletion vs. thermal depletion
Array of exciton-polariton condensates - bonded s-wave and anti-bonded p-wave condensation -Future prospects
3
Interpretation of superfluid He interms of BEC of ideal gas (1938)
Phenomenological model based onstrongly interacting systems (1941)
London, Tisza
L. D. Landau
Discovery of superfluid 4He (1937)
P.L. Kapitsa
Excitation spectrum of superfluid He
BEC of non-interacting ideal gas (1925)
A. Einstein
Atomic theory of Landau’s two fluid model(1955)
R. P. FeynmanL.D.Landa
u
P.L. Kapitsa
A. Einstein
R.D. Feynman
Bose-Einstein Condensation and Superfluidity
4
Weakly interacting Bose gas : Bogoliubov transformation (1947)
• Diagonalization of the Hamiltonianof interacting bosonic particles
Diagonalized Hamiltonian
p1 p2
p2-qp1+q
Mutual interactionof condensate
Bogoliubov dispersion law for elementary excitations
N.N. Bogoliubov
Diagonalization using the linear transformation
Energy of acondensate Energy of excitations
0)(g ˆˆˆˆ2
1ˆˆ
2 ,,
2
21
2121>+= !! +
"+
+
+
qpp
ppqpqp
p
pp aaaaV
gaa
m
pH
,ˆˆˆ ,ˆˆˆ **
pppppppp bvbuabvbua pp !
+++
! !! +=+=
gnm
pp +=
2)(
2
!cpp =)(!
linearat small k
quadraticat large k
kinetic term interacting term
5
70 years after Einstein
E. Cornell
C.E. Wieman
W. Ketterle
Bogoliubov excitation spectrum
Bose-Einstein Condensation of Neutral Atoms(1995)
6
Outline
Bose-Einstein condensation and superfluidity
Bogoliubov theory for weakly interacting Bose gas
Interaction energy of exciton-polaritonsEnergy-momentum dispersion relation: revisited - Condensates and excitation spectrum
- Sound velocity
- Free particle energy
Quantum depletion vs. thermal depletion
Array of exciton-polariton condensates - bonded s-wave and anti-bonded p-wave condensation -
Future prospects
7
Polariton-Polariton Interaction: Recap
s
Bexc
n
nE=!"h
• M. Kuwata-Gonokami et al., Phys. Rev. Lett. 79, 1341 (1997)• S. Schmitt-Rink, et al., Phys. Rev. B 32, 6601 (1985)
Exciton energy blue shift due to fermionic exchange interaction
Dipole moment reduction due to phase space filling and fermionic exchange interactionbased on Pauli’s exclusion principle based on Coulomb interaction
where
•C. Ciuti et al., Phys. Rev. B58, 7926 (1998)•P.R. Eastham et al., Phys. Rev. B68, 075324 (2003)
'0
sn
ngg !=" where
22*2.2 Xa
SNn
B
QW
s
!=
22*4
'
Xa
SNn
B
QW
s
!=
22*2.2 Xa
SNn
B
QW
s
!=
-3 -2 -1 0 1 2 30
0.5
1
1.5
2
2.5
3
3.5
ExcitonLP UP
smaller spitting
Blue shift
Δωexc
g!
exc!"h
g!
Origin of strong mutual interaction because oflarge Bohr radius (~100A) .
22 )(42
)(2
phexkphexLPUP gE !!!! "+±+=
hh
!"
#$%
&'+'+= ]4/1[
2
1 2
0
2gX
exciton fraction at k=0
exciton binding energy
initial state
final state
Total number of polaritons
Inte
ract
ion
ener
gy (µ
eV)
10
100
1000
104
105
Numericalsimulation
TheoryExperiment
Interaction energy vs. total number of polaritons
gndN
dEnU
NV
gE
==
=
0
2
0
)(
2(Total energy of the condensate)
(Blue shift of the photon from the condensate at k=0 LP)
(S:fixed)
(nonlinear increase of S isincluded in Gross-Pitaevskiiequation)
4
)()(
42
1 22
0
22
0exc
excggg
!!
"#"+"+#
"++"=
hh
8
9
Outline
Bose-Einstein condensation and superfluidity
Bogoliubov theory for weakly interacting Bose gas
Interaction energy of exciton-polaritons
Energy-momentum dispersion relation: revisited - Condensates and excitation spectrum - Sound velocity - Free particle energyQuantum depletion vs. thermal depletion
Array of exciton-polariton condensates - bonded s-wave and anti-bonded p-wave condensation -
Future prospects
10-15o 15o-15o 15o -15o 15o
1.613 eV1.613 eV
kξ=1
-15o 15o
1.613 eVCo-polarization detection
8 m
eV
kξ=1-15o 15o
1.613 eVLinear scale
Cross-polarization detection Mixed-polarization detection
Dispersion relation for untrapped condensatewith circularly polarized pump wave
11
-15o 15o
1.614 eV
-15o 15o
1.614 eV
kξ=1
-15o 15o-15o 15o
1.614 eV
8 m
eV1.614 eV
-15o 15o
P=1.2PthP=0.05Pth
Dispersion relation for trapped condensatewith different pump levels
P=6PthP=4Pth
12
Bogoliubov dispersion for trapped and untrappedcondensates
Linear dispersion at low momentum regime c=dE(p)/dp~108 cm/s(c~1cm/s for atomic BEC, c~104cm/s for 4He)
Nature Physics (in press)
Trapped
2
1
0
-1
E/U
-1.0 0.0 1.0
k!
BogoliubovA B
C D
quadratic
2
1
0
-1
E/µ
-1 0 1
k!
Untrapped
E: d=7um, Δ=3.3 (meV), P=2.9PthF: d=7um, Δ=2.9 (meV), P=3.8PthG: d=8um, Δ=1.6 (meV), P=2PthH: d=8um, Δ=2.5 (meV), P=3.8Pth
A: Δ=1.41 (meV), P=4PthB: Δ=0.82 (meV), P=8PthC: Δ=4.2 (meV), P=4PthD: Δ=-0.23 (meV), P=24Pth
Sound Velocity vs. Pump Power / Interaction Energy
13
Far Blue Detuning Zero Detuning
14
Energy shift Ek(P>Pth)-Ek(P<<Pth) vs. Interaction energy Ufor free particle regime (at kξ=1)
Ek(P>Pth)-Ek(PaPth)=2U
Interaction energy U (meV)
A B
C D
E B-E
LP (m
eV)
Interaction energy U (meV)
E F
G H
E B-E
LP (m
eV)
0.1
1
0.1 1
0.1
1
0.1 1
Trapped system Untrapped system
EK(P>Pth)-EK(P<<Pth)=2U
Energy shift for condensate particleU=gn0
Energy shift for non-condensed particleU=gn0
( )!!"
#+
#+++ ++++=
0
2
0
2
ˆˆˆˆˆˆ22
1
2ˆˆ
2
ˆ
p
pppppppp
p
aaaaaagnNV
gaa
m
pH
15
Quantum depletion vs. thermal depletion
Quantum depletion (proportional to 1/k)
Thermal depletion (proportional to 1/k2)
pppppppp bbubbuvn !
+
!!
+
! ++= ˆˆ||ˆˆ|||| 222
!!"
#$$%
&
'=
+
1)](exp[
1ˆˆp
bb pp()
00ˆˆˆˆ aaaa
kk
+
!
+
k
mUnk
h2!
2)( k
Tmkn
B
k
h!
=0 at T=0
phk baa ˆˆˆ0
+
phonon
Quantumdepletion
Thermaldepletion
Quantum depletion Thermal depletion
16
Outline
Bose-Einstein condensation and superfluidity
Bogoliubov theory for weakly interacting Bose gas
Interaction energy of exciton-polaritons
Energy-momentum dispersion relation: revisited - Condensates and excitation spectrum
- Sound velocity
- Free particle energy
Quantum depletion vs. thermal depletion
Array of exciton-polariton condensates - bonded s-wave and anti-bonded p-wave condensation -Future prospects
17
Cold atoms in 3D Optical lattice Experimental evidence forquantum phase transition from
BEC, superfluid to Mott insulator
M. Greiner et al., Nature 419, 6901 (2002)
Bose – Hubbard Hamiltonian
M.P.A. Fisher et al., PRB 40, 546 (1989)D. Jaksch et al., PRL 81, 3108 (1998)
Exciton-polaritons in 2D lattice 2D physicsMass can be varied over four orders of magnitudeOptical input and readout
Quantum EmulationQuantum Emulation- - SuperfluidSuperfluid to Mott Insulator Phase Transition in 3D-Optical Lattice - to Mott Insulator Phase Transition in 3D-Optical Lattice -
180 1 2 30
50
100
X/a
δE0 (
µeV
)
Au/Ti
12 mm
SEM
1.4µm line & space
Theory & Independent Measurements under a uniform layer: ~200µeVLimited by imaging resolution
Spatial modulation of LP energy
akzθ
substrate
k
DBR
DBR
λ/2
AlA
s ca
vity
3 st
acks
of 4
GaA
s Q
Ws
GaAlAsAlAs
Au/Ti strips
k||=k×sin(θ)
Au/Ti
Air Air
GaAlAs
GaAs
E
cavity only
k||
QW exciton
LPU0
cavity + Aucavityphoton
0
~2U0
a = 2.8 µm
LP energy can be spatially modulated by periodic metallicLP energy can be spatially modulated by periodic metallicfilmsfilms
19
on top of metal gate
bright = gap region bright = metal region bright = gap region
above thresholdbelow threshold well above threshold
Real Space and Momentum Space Distributions in 1DReal Space and Momentum Space Distributions in 1DLatticeLattice
19
20
“0-state” (s-wave)
π-state (s-wave)
”π-state” (p-wave)
Periodic potential
Nature 450, 529 (2007)
Below threshold
Above threshold
Anti-Phased p-wave and In-Phased s-wave inAnti-Phased p-wave and In-Phased s-wave inOne-Dimensional One-Dimensional Exciton-PolaritonExciton-Polariton Condensate Array Condensate Array
(π state) (0 state)
Excited state condensation Ground state condensation
Diffraction Patterns of the Diffraction Patterns of the ““Zero-StateZero-State”” and and ””ππ-State-State””
21
22
n2
Pump: n3
Γ3<< Γ1, Γ2
Γ2
Γ1
Γ31<< Γ32Γ21n1
π-state
0-state
Γ32
0 20 40 60 80 1000
2
4
6
8
10
PL
In
ten
sit
y (
arb
. u
nit
s)
Time (ps)
zero-state
!-state
0 20 40 60 80 1000.0
0.2
0.4
0.6
0.8
1.0
1.2
zero-state
Inte
ns
ity
(a
rb.
un
its
)
Time (ps)
!-state
MetastableMetastable ππ-State-State (p-wave) Forms before Stable Zero-State (s-wave)(p-wave) Forms before Stable Zero-State (s-wave)
23
Future Prospect
• Quantum states of the ground state (phase-locking, quantum correlation)• Superfluidity (second sound wave, quantized vortices)• Exotic excitation spectrum (maxon, roton, negative-energy branch)•Orbital physics (p-wave, d-wave and f-wave superfluid states)
Physics of Bose-Einstein Condensation
Quantum emulation of many body systems
• Quantum annealing machine (bosonic stimulated cooling)Ground state searching
• Polariton BEC as laser without inversion• Electrically driven polariton laser
New coherent light source
BEC-BKT cross-over, BEC-BCS cross-over
Generation of single photons (repulsive interaction) andentangled photon-pairs (attractive interaction) from MI state
Phys. Rev. A 77, 031803(R) (2008).
Q/104
Goal: Construction of Bose-Hubbard phase diagram and validationcheck with cold-atom-based quantum emulator
PolaritonicPolaritonic Quantum Emulator: Quantum Emulator: SuperfluidSuperfluid to Mott- Insulator to Mott- InsulatorQuantum Phase TransitionQuantum Phase Transition
24
arXiv:0804.1829 (2008)
Generation of Indistinguishable Single Photons andGeneration of Indistinguishable Single Photons andPolarization Entangled Photon PairsPolarization Entangled Photon Pairs
25