Yoan Léger
Laboratory of Quantum Opto-electronics
Ecole Polytechnique Fédérale de Lausanne
Switzerland
Polariton Superfluidity
Heterodyne four wave mixing
From standard fluid to superfluidity
2d fourier spectroscopy
Polariton Superfluidity
Heterodyne four wave mixing
From standard fluid to superfluidity
2d fourier spectroscopy
Superfluidity & sound wave excitations
Striking properties of superfluids
Zero viscosity, Rollin film, foutain effect
Quantized vortices….
Bogoliubov theory
of the weakly interacting Bose gas
Elementary excitation are collective excitations!
with sound wave behavior
Woods et al. Rep. Prog. Phys. 36 1135 (1973)
Superfluidity in the solid state
Microcavity polaritons
Momentum dispersion
Cavity
field
DBR
Polariton
Exciton
QW
UP
Energy
Ph.
X
UP
Spacing layer
DBR
LP
LP
In-plane momentum
~ Emission angle
Superfluidity in the solid state
Microcavity polaritons
Bose Einstein condensation
Cavity
field
DBR
Polariton
Exciton
QW
UP
Ph.
X
Spacing layer
Kasprzak et al. Nature 443, 409 (2006)
DBR
LP
Coulomb interactions
Polaritons should be
superfluid!!
Amo et al. Nat. Phys. 5, 805 (2009)
The superfluid dispersion
Injecting polaritons at k=0
Linearization comes
from the coupling of
counter-propagating modes
by interactions
Appearance of a ghost branch
Naive picture of the ghost branch
Diluted polariton gas
Sound wave in superfluid
Particle-hole superposition
Gross-Pitaevskii formalism
Weakly interacting bosons:
H    k0 ak† ak 
k
1
2V
g a
†
†
k1  q k 2  q k1
a
a ak2
Normal branch
k1 , k 2 , q
Ghost branch
Mean field theory:
2

 ,

   0 1  uk ei ( k .r  t )  vk*e i ( k .r  t ) 
B
Linearization of interaction term:
Energy (meV)
2.5
gn=1meV
2.0
ωB
εk 0
1.5
1.0
0.5
0.0
uk2
1.5
vk2
1.0
0.5
0.0
0.0 0.5 1.0 1.5 2.0 2.5
-1
k (m )
B
B   k0 ( k0  2 gn0 )
Contribution
   2 2
i
  
 g
t  2m
k=1μm-1
0.0 0.5 1.0 1.5 2.0 2.5
gn (meV)
Looking for the Ghost branch
PL measurements
Kind of linearization
No ghost branch
Utsunomiya et al. Nat. Phys. 4, 700 (2008)
Accessing the ghost branch with FWM
In the proposal: non-resonant condensate
Wouters et al. Phys. Rev. B 79, 125311 (2009)
Polariton Superfluidity
Heterodyne four wave mixing
From standard fluid to superfluidity
2d fourier spectroscopy
Polariton FWM
Four wave mixing and selection rules

  *
(3)
EFWM   E3 E2 E1
Angular selection rule
Energy selection rule
Third order nonlinearity

  
kFWM  k3  k2  k1
FWM  3  2  1
Polariton FWM
Four wave mixing and selection rules

 2 *
( 3)
EFWM   E2 E1
Angular selection rule
Energy selection rule
Third order nonlinearity

 
kFWM  2 k2  k1
FWM  2 2  1
Polariton FWM
2 fields : condensate field and probe field
Stimulated parametric scattering of 2
polaritons from the condensate
Heterodyne FWM
Problem: Condensate emission should largely dominate the spectrum
How to extract useful signal when angular selection is not enough?
Heterodyne FWM
• Based on spectral interferometry
Excitation fields
Linear emission
FWM
Heterodyne
setup
• requires :
• a full control of the excitation fields
• Pulsed excitation to cover the full emission spectrum
• provides:
• best sensitivity, and selectivity
• access to amplitude and phase of the nonlinear emission
FWM
Coherent excitation
Pulsed resonant
excitation
Spectral interferometry
Energy selection
Pump
Spectro
Balanced
detection
FWM
FWM 71 MHz
Ref. Pump 75 MHz
Ref. Trigger 79MHz
75 MHz
AOM
Trigger
79 MHz
AOM
ω0
AOM
Trigger
Pump
71 MHz
Local
Osc..
Sample
Intensity
Heterodyning
Excitation
Pulses
LP
UP
FWM 71 MHz
Ref. Pump 75 MHz
Ref. Trigger 79MHz
transmission
79 MHz
75 MHz
AOM
ω0
Intensity
Pump
+ local osc.
@ 71MHz
Frequency comb:
80MHz
NB
GB
Energy
extracted
FWM
Sample
pump
FWM
trigger
75MHz
79MHz
71 MHz
Local
Osc..
Trigger
Energy
Intensity
AOM
AOM
Energy
Spectro
Balanced
detection
71MHz
Energy
Polariton Superfluidity
Heterodyne four wave mixing
From standard fluid to superfluidity
2d fourier spectroscopy
Dispersion & dissipation…
Normal & ghost branch
Damping of polariton density!
Low density
K=0
t1
NB
E-E0 (meV)
E-E0 (meV)
GB
0.6
0.6
0.4
0.4
t2
0.2
0.2
t3
0.0
0.0
-0.2
-0.2
-0.4
-0.4
-0.6
-0.6
1.2 1.2
0.0
0.4 0.60.80.8 1.0
0.0 0.2 0.4
-1
-1
(m
Wavevector
)
Wavevector
Stating on the ghost branch?
OPO experiment
 Linear dispersion
but off-resonances can
always exist in FWM
we have to go further!
Savvidis et al.
Phys. Rev. B. 64, 075311 (2001)



 LP (r , t )   0  u e i k.r  v *e i k.r
Nature of the excitations
1/2
Off-resonance or “real” ghost?
Dissipative Gross-Pitaevskii equation with:



i k .r
 i k.r
 LP (r , t )  0  u e  v * e
pump
FWM
trigger
Always 2 energy modes:
Ghost and normal branch
Change of intensity and linewidth
With polariton density
1/3
1
Standard fluid
Single particle
excitations
Superfluid
Sound waves
Nature of the excitations
2/2
Intensity dependence
Redistribution of intensity
Density of state on the ghost!
GB k=0 NB
Normal
Ghost
12
Exc. Power (mW)
10
8
Threshold!
6
4
Normalized Intensity
0.8
0.6
0.4
0.2
2
1.482
1.483
1.484
1.485
Energy (eV)
1.486
1.487
2
4
6
8
10
Exc. Power (mW)
12
Polariton Superfluidity
Heterodyne four wave mixing
From standard fluid to superfluidity
2d fourier spectroscopy
Investigating the processes
Delay dependence
Energy (eV)
1.4860
Exp.
1.5
1.4855
1.0
1.4850
0.5
1.4845
0.0
1.4840
-0.5
1.4835
-1.0
-5
0
Th.
5
-5
0
Delay between pulses (ps)
pump
pump
Trig.
Trig.
FWM
Delay<0
t
FWM
Delay>0
t
5
2D fourier transform spectroscopy
Energy (eV)
Delay dependence
LP
ΩR
1.4860
1.5
1.4855
1.0
1.4850
0.5
1.4845
0.0
1.4840
-0.5
1.4835
-1.0
-5
0
5
Delay between pulses (ps)
delay
Trig.
UP
pump
-2
-5
0
02
4 5
Trigger energy (meV)
Fourier transform on delay
E(ωdet ,τ)
E(ωdet , ωexc)
FWM
t
Conclusions & perspectives
Ghost branch of a superfluid
• In solid state, for the first time
• Transformation of the excitations
Sound like dispersion
• Linear for the normal branch
• Assymmetry due to dissipation
2D fourier transform spectroscopy
• Highly powerful method
• Starting the process investigation…
acknowledgements
To the audience!
To my collaborators:
Verena Kohnle, Michiel Wouters, Maxime Richard,
Marcia Portella-Oberli, Benoit Deveaud-Plédran