Pattern formation, Superfluidity and Coherence of Polariton Condensates. Jonathan Keeling

Pattern formation, Superfluidity and Coherence of
Polariton Condensates.
Jonathan Keeling
UMass Amherst, January 2012
Funding:
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
1 / 32
Bose-Einstein condensation: macroscopic occupation
Polaritons. ∼ 20K
Atoms. ∼ 10−7 K
[Kasprzak et al. Nature, ’06]
[Anderson et al. Science ’95]
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
2 / 32
Macroscopic coherence: vortices
Polaritons:
Atoms:
[Lagoudakis et al. Nat. Phys. ’08]
[Abo-Shaeer et al. Science ’01]
But also, nonlinear optics:
[Arecchi et al. PRL ’91]
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
3 / 32
Excitons
Energy
Electronic spectrum:
Momentum
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
4 / 32
Excitons
Energy
Electronic spectrum:
Holes
Momentum
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
4 / 32
Excitons
Energy
Electronic spectrum:
H=
X
X
Tie +Tih +
Vijee +Vijhh −Vijeh
i
ij
Holes
Ti =
pi2
2mj
Vij =
e2
r |ri − rj |
Momentum
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
4 / 32
Excitons
Energy
Electronic spectrum:
H=
X
X
Tie +Tih +
Vijee +Vijhh −Vijeh
i
ij
Holes
Ti =
pi2
2mj
Vij =
e2
r |ri − rj |
Momentum
Bound state: Exciton,
M ∼ me + mh
Approximate Bose statistics:
†
[cexciton,k , cexciton,k
0 ] ' δk,k 0
D
If ρ(aB,exciton ) 1
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
4 / 32
Excitons
Energy
Electronic spectrum:
H=
X
X
Tie +Tih +
Vijee +Vijhh −Vijeh
i
ij
Holes
Ti =
pi2
2mj
Vij =
e2
r |ri − rj |
Momentum
Optical spectrum
e−h continuum
Jonathan Keeling
1s exciton
Energy
Approximate Bose statistics:
†
[cexciton,k , cexciton,k
0 ] ' δk,k 0
D
If ρ(aB,exciton ) 1
Momentum
Polariton condensation
Absorption
Bound state: Exciton,
M ∼ me + mh
Excitons
e−h continuum
Energy
UMass Amherst, January 2012
4 / 32
Microcavity polaritons
Cavity
Jonathan Keeling
Quantum Wells
Polariton condensation
UMass Amherst, January 2012
5 / 32
Microcavity polaritons
Quantum Wells
n=3
n=2
n=1
m∗ ∼ 10−4 me
lk
' ω0 + k 2 /2m∗
n=4
Bu
Cavity photons:
q
ωk = ω02 + c 2 k 2
Energy
Cavity
Momentum
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
5 / 32
Microcavity polaritons
Cavity
Quantum Wells
' ω0 + k /2m
Pho
Exciton
Energy
2
ton
Cavity photons:
q
ωk = ω02 + c 2 k 2
∗
∗
m ∼ 10−4 me
In−plane momentum
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
5 / 32
Microcavity polaritons
Cavity
Quantum Wells
' ω0 + k /2m
Pho
Exciton
Energy
2
ton
Cavity photons:
q
ωk = ω02 + c 2 k 2
∗
∗
m ∼ 10−4 me
In−plane momentum
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
5 / 32
Microcavity polaritons
θ
Cavity
Quantum Wells
Cavity
' ω0 + k /2m
Pho
Exciton
Energy
2
ton
Cavity photons:
q
ωk = ω02 + c 2 k 2
∗
∗
m ∼ 10−4 me
In−plane momentum
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
5 / 32
Polariton experiments: occupation and coherence
[Kasprzak, et al. Nature, ’06]
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
6 / 32
Polariton experiments: occupation and coherence
Sample
Coherence map:
Beam
Splitter
+
=
Tunable
Delay
CCD
Retroreflector
[Kasprzak, et al. Nature, ’06]
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
6 / 32
(Some) other polariton condensation experiments
Polariton traps
[Balili et al. Science ’07)]
Multimode condensate
and sharp lines
[Love et al. PRL ’08]
Wavepacket propagation
[Amo et al. Nature ’09]
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
7 / 32
(Some) other polariton condensation experiments
Quantised vortices
[Lagoudakis et al. Nat. Phys. ’08. Science ’09,
PRL ’10; Sanvitto et al. Nat. Phys. ’10; Roumpos
et al. Nat. Phys. ’10 ]
+
=
Josephson oscillations
[Lagoudakis et al. PRL ’10]
Pattern formation/Hydrodynamics
[Amo et al. Science ’11, Nature ’09;
Wertz et al. Nat. Phys ’10]
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
8 / 32
1
Introduction to polariton condensation
What are excitons and polaritons
Experimental features
Approaches to modelling
2
Pattern formation
Experiments
Modelling pattern formation
3
Superfluidity
Non-equilibrium condensate spectrum
Experiments and aspects of superfluidity
Current-current response function
4
Coherence
Experiments
Power law decay of coherence
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
9 / 32
1
Introduction to polariton condensation
What are excitons and polaritons
Experimental features
Approaches to modelling
2
Pattern formation
Experiments
Modelling pattern formation
3
Superfluidity
Non-equilibrium condensate spectrum
Experiments and aspects of superfluidity
Current-current response function
4
Coherence
Experiments
Power law decay of coherence
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
10 / 32
Pump bath
H = Hsys + Hsys,bath + Hbath ,
X
X
Hsys =
ωk ψk ψk† +
gα (φ†α ψk + H.c.)
α
k
+
System
Hex [φα , φ†α ]
Jonathan Keeling
Energy P
hoton
Non-equilibrium approach: Steady state, and
fluctuations
Exciton
In−plane momentum
Decay
bath
Polariton condensation
UMass Amherst, January 2012
11 / 32
Pump bath
H = Hsys + Hsys,bath + Hbath ,
X
X
Hsys =
ωk ψk ψk† +
gα (φ†α ψk + H.c.)
α
k
+
System
Hex [φα , φ†α ]
Energy P
hoton
Non-equilibrium approach: Steady state, and
fluctuations
Exciton
In−plane momentum
Decay
bath
Steady state, ψ(r, t) = ψ0 e−iµS t .
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
11 / 32
Pump bath
H = Hsys + Hsys,bath + Hbath ,
X
X
Hsys =
ωk ψk ψk† +
gα (φ†α ψk + H.c.)
α
k
+
System
Hex [φα , φ†α ]
Energy P
hoton
Non-equilibrium approach: Steady state, and
fluctuations
Exciton
In−plane momentum
Decay
bath
Steady state, ψ(r, t) = ψ0 e−iµS t .
X
gα hφα i
Self-consistent equation: (i∂t − ω0 + iκ) ψ =
α
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
11 / 32
Pump bath
H = Hsys + Hsys,bath + Hbath ,
X
X
Hsys =
ωk ψk ψk† +
gα (φ†α ψk + H.c.)
α
k
+
Energy P
hoton
Non-equilibrium approach: Steady state, and
fluctuations
System
Hex [φα , φ†α ]
Exciton
In−plane momentum
Decay
bath
Steady state, ψ(r, t) = ψ0 e−iµS t .
Self-consistent equation: (µs − ω0 + iκ) ψ0 = χ(ψ0 , µs )ψ0
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
11 / 32
Pump bath
H = Hsys + Hsys,bath + Hbath ,
X
X
Hsys =
ωk ψk ψk† +
gα (φ†α ψk + H.c.)
α
k
+
Energy P
hoton
Non-equilibrium approach: Steady state, and
fluctuations
System
Hex [φα , φ†α ]
Exciton
In−plane momentum
Decay
bath
Steady state, ψ(r, t) = ψ0 e−iµS t .
Self-consistent equation: (µs − ω0 + iκ) ψ0 = χ(ψ0 , µs )ψ0
Fluctuations
h
i † 0
[D − D ](t, t ) = −i ψ(t), ψ (t )
R
A
0
−
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
11 / 32
Pump bath
H = Hsys + Hsys,bath + Hbath ,
X
X
Hsys =
ωk ψk ψk† +
gα (φ†α ψk + H.c.)
α
k
+
Energy P
hoton
Non-equilibrium approach: Steady state, and
fluctuations
System
Hex [φα , φ†α ]
Exciton
In−plane momentum
Decay
bath
Steady state, ψ(r, t) = ψ0 e−iµS t .
Self-consistent equation: (µs − ω0 + iκ) ψ0 = χ(ψ0 , µs )ψ0
Fluctuations
h
i † 0
[D − D ](t, t ) = −i ψ(t), ψ (t )
R
A
0
−
Jonathan Keeling
Polariton condensation
[D R − D A ](ω) = DoS(ω)
UMass Amherst, January 2012
11 / 32
Pump bath
H = Hsys + Hsys,bath + Hbath ,
X
X
Hsys =
ωk ψk ψk† +
gα (φ†α ψk + H.c.)
α
k
+
Energy P
hoton
Non-equilibrium approach: Steady state, and
fluctuations
System
Hex [φα , φ†α ]
Exciton
In−plane momentum
Decay
bath
Steady state, ψ(r, t) = ψ0 e−iµS t .
Self-consistent equation: (µs − ω0 + iκ) ψ0 = χ(ψ0 , µs )ψ0
Fluctuations
h
i † 0
[D − D ](t, t ) = −i ψ(t), ψ (t )
−
h
i D K (t, t 0 ) = −i ψ(t), ψ † (t 0 )
R
A
0
[D R − D A ](ω) = DoS(ω)
+
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
11 / 32
Pump bath
H = Hsys + Hsys,bath + Hbath ,
X
X
Hsys =
ωk ψk ψk† +
gα (φ†α ψk + H.c.)
α
k
+
Energy P
hoton
Non-equilibrium approach: Steady state, and
fluctuations
System
Hex [φα , φ†α ]
Exciton
In−plane momentum
Decay
bath
Steady state, ψ(r, t) = ψ0 e−iµS t .
Self-consistent equation: (µs − ω0 + iκ) ψ0 = χ(ψ0 , µs )ψ0
Fluctuations
h
i † 0
[D − D ](t, t ) = −i ψ(t), ψ (t )
−
h
i D K (t, t 0 ) = −i ψ(t), ψ † (t 0 )
R
A
0
+
Jonathan Keeling
Polariton condensation
[D R − D A ](ω) = DoS(ω)
D K (ω) = (2n(ω) + 1)DoS(ω)
UMass Amherst, January 2012
11 / 32
Pattern formation:
1
Introduction to polariton condensation
What are excitons and polaritons
Experimental features
Approaches to modelling
2
Pattern formation
Experiments
Modelling pattern formation
3
Superfluidity
Non-equilibrium condensate spectrum
Experiments and aspects of superfluidity
Current-current response function
4
Coherence
Experiments
Power law decay of coherence
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
12 / 32
Pattern formation in experiments
Polariton Traps
Vortex formation
Elliptical ring pump
[Lagoudakis et al. Nat.
Phys ’08]
[Manni et al. PRL ’11]
[Balili et al. Science ’07]
Patterned lattice: Momentum space image
[Kim et al. Nat. Phys ’11]
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
13 / 32
Non-equilibrium features in experiment
Flow from pumping spot
[Wertz et al. Nat. Phys. ’10]
|ψ(k)|2 6= |ψ(−k)|2 :
Broken time-reversal symmetry.
[Krizhanovskii et al. PRB ’09]
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
14 / 32
Complex Gross-Pitaevskii equation
Steady state equation:
(µs − ω0 + iκ) ψ = χ(ψ, µs )ψ
Local density limit:
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
15 / 32
Complex Gross-Pitaevskii equation
Steady state equation:
(µs − ω0 + iκ) ψ = χ(ψ, µs )ψ
Local density limit: Gross-Pitaevskii equation
∇2
ψ(r ) = χ(ψ(r , t))ψ(r , t)
i∂t + iκ − V (r ) −
2m
Nonlinear, complex susceptibility
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
15 / 32
Complex Gross-Pitaevskii equation
Steady state equation:
(µs − ω0 + iκ) ψ = χ(ψ, µs )ψ
Local density limit: Gross-Pitaevskii equation
∇2
ψ(r ) = χ(ψ(r , t))ψ(r , t)
i∂t + iκ − V (r ) −
2m
Nonlinear, complex susceptibility
i∂t ψ|nlin = U|ψ|2 ψ
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
15 / 32
Complex Gross-Pitaevskii equation
Steady state equation:
(µs − ω0 + iκ) ψ = χ(ψ, µs )ψ
Local density limit: Gross-Pitaevskii equation
∇2
ψ(r ) = χ(ψ(r , t))ψ(r , t)
i∂t + iκ − V (r ) −
2m
Nonlinear, complex susceptibility
i∂t ψ|nlin = U|ψ|2 ψ
i∂t ψ|loss = −iκψ
Jonathan Keeling
i∂t ψ|gain = iγeff ψ
Polariton condensation
UMass Amherst, January 2012
15 / 32
Complex Gross-Pitaevskii equation
Steady state equation:
(µs − ω0 + iκ) ψ = χ(ψ, µs )ψ
Local density limit: Gross-Pitaevskii equation
∇2
ψ(r ) = χ(ψ(r , t))ψ(r , t)
i∂t + iκ − V (r ) −
2m
Nonlinear, complex susceptibility
i∂t ψ|nlin = U|ψ|2 ψ
i∂t ψ|loss = −iκψ
Jonathan Keeling
i∂t ψ|gain = iγeff ψ − iΓ|ψ|2 ψ
Polariton condensation
UMass Amherst, January 2012
15 / 32
Complex Gross-Pitaevskii equation
Steady state equation:
(µs − ω0 + iκ) ψ = χ(ψ, µs )ψ
Local density limit: Gross-Pitaevskii equation
∇2
ψ(r ) = χ(ψ(r , t))ψ(r , t)
i∂t + iκ − V (r ) −
2m
Nonlinear, complex susceptibility
i∂t ψ|nlin = U|ψ|2 ψ
i∂t ψ|loss = −iκψ
i∂t ψ|gain = iγeff ψ − iΓ|ψ|2 ψ
∇2
2
2
i∂t ψ = −
+ V (r ) + U|ψ| + i γeff − κ − Γ|ψ|
ψ
2m
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
15 / 32
Gross-Pitaevskii equation: Harmonic trap
mω 2 2
∇2
2
2
+
r + U|ψ| + i γeff − κ − Γ|ψ|
i∂t ψ = −
ψ
2m
2
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
16 / 32
Gross-Pitaevskii equation: Harmonic trap
mω 2 2
∇2
2
2
+
r + U|ψ| + i γeff − κ − Γ|ψ|
i∂t ψ = −
ψ
2m
2
Density
30
25
20
15
10
5
0
0
Jonathan Keeling
Polariton condensation
2
4
6
8
Radius
UMass Amherst, January 2012
16 / 32
Stability of Thomas-Fermi solution
1
∂t ρ+∇·(ρv) = (γnet − Γρ)ρ
2
3γnet
2Γ
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
17 / 32
Stability of Thomas-Fermi solution
High m modes: δρn,m ' eimθ r m . . .
1
∂t ρ+∇·(ρv) = (γnet − Γρ)ρ
2
Unstable growth
3γnet
2Γ
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
17 / 32
Stability of Thomas-Fermi solution
High m modes: δρn,m ' eimθ r m . . .
1
∂t ρ+∇·(ρv) = (γnet Θ(r0 −r )−Γρ)ρ
2
Stabilised
3γnet
2Γ
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
17 / 32
Stability of Thomas-Fermi solution
High m modes: δρn,m ' eimθ r m . . .
1
∂t ρ+∇·(ρv) = (γnet Θ(r0 −r )−Γρ)ρ
2
????
3γnet
2Γ
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
17 / 32
Time evolution:
t=0
t=2
t=22
t=30
t=35
t=40
t=45
t=56
[Keeling & Berloff PRL ’08]
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
18 / 32
Why vortices
25
Density profile
Thomas-Fermi in flattened trap
Density
20
15
10
5
0
-15
-10
-5
0
5
Cross Section
10
15
∇ · [ρ(v − Ω × r)] = (γnet Θ(r0 − r ) − Γρ) ρ,
√
∇2 ρ
m
m 2 2
2
2
µ=
|v − Ω × r| + r (ω − Ω ) + Uρ −
√
2
2
2m ρ
γnet
µ
v = Ω × r, Ω = ω, ρ =
Θ(r0 − r ) =
Γ
U
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
19 / 32
Why vortices
25
Density profile
Thomas-Fermi in flattened trap
Density
20
15
10
5
0
-15
-10
-5
0
5
Cross Section
10
15
Rotating solution: i∂t ψ = (µ − 2ΩLz )ψ
∇ · [ρ(v − Ω × r)] = (γnet Θ(r0 − r ) − Γρ) ρ,
√
∇2 ρ
m
m 2 2
2
2
µ=
|v − Ω × r| + r (ω − Ω ) + Uρ −
√
2
2
2m ρ
γnet
µ
v = Ω × r, Ω = ω, ρ =
Θ(r0 − r ) =
Γ
U
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
19 / 32
Why vortices
25
Density profile
Thomas-Fermi in flattened trap
Density
20
15
10
5
0
-15
-10
-5
0
5
Cross Section
10
15
Rotating solution: i∂t ψ = (µ − 2ΩLz )ψ
∇ · [ρ(v − Ω × r)] = (γnet Θ(r0 − r ) − Γρ) ρ,
√
∇2 ρ
m
m 2 2
2
2
µ=
|v − Ω × r| + r (ω − Ω ) + Uρ −
√
2
2
2m ρ
γnet
µ
v = Ω × r, Ω = ω, ρ =
Θ(r0 − r ) =
Γ
U
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
19 / 32
Why vortices
25
Density profile
Thomas-Fermi in flattened trap
Density
20
15
10
5
0
-15
-10
-5
0
5
Cross Section
10
15
Rotating solution: i∂t ψ = (µ − 2ΩLz )ψ
∇ · [ρ(v − Ω × r)] = (γnet Θ(r0 − r ) − Γρ) ρ,
√
∇2 ρ
m
m 2 2
2
2
µ=
|v − Ω × r| + r (ω − Ω ) + Uρ −
√
2
2
2m ρ
γnet
µ
v = Ω × r, Ω = ω, ρ =
Θ(r0 − r ) =
Γ
U
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
19 / 32
Why vortices
25
Density profile
Thomas-Fermi in flattened trap
Density
20
15
10
5
0
-15
-10
-5
0
5
Cross Section
10
15
Rotating solution: i∂t ψ = (µ − 2ΩLz )ψ
∇ · [ρ(v − Ω × r)] = (γnet Θ(r0 − r ) − Γρ) ρ,
√
∇2 ρ
m
m 2 2
2
2
µ=
|v − Ω × r| + r (ω − Ω ) + Uρ −
√
2
2
2m ρ
γnet
µ
v = Ω × r, Ω = ω, ρ =
Θ(r0 − r ) =
Γ
U
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
19 / 32
Why vortices
25
Density profile
Thomas-Fermi in flattened trap
Density
20
15
10
5
0
-15
-10
-5
0
5
Cross Section
10
15
Rotating solution: i∂t ψ = (µ − 2ΩLz )ψ
∇ · [ρ(v − Ω × r)] = (γnet Θ(r0 − r ) − Γρ) ρ,
√
∇2 ρ
m
m 2 2
2
2
µ=
|v − Ω × r| + r (ω − Ω ) + Uρ −
√
2
2
2m ρ
γnet
µ
v = Ω × r, Ω = ω, ρ =
Θ(r0 − r ) =
Γ
U
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
19 / 32
Why vortices
25
Density profile
Thomas-Fermi in flattened trap
Density
20
15
10
5
0
-15
-10
-5
0
5
Cross Section
10
15
Rotating solution: i∂t ψ = (µ − 2ΩLz )ψ
∇ · [ρ(v − Ω × r)] = (γnet Θ(r0 − r ) − Γρ) ρ,
√
∇2 ρ
m
m 2 2
2
2
µ=
|v − Ω × r| + r (ω − Ω ) + Uρ −
√
2
2
2m ρ
γnet
µ
v = Ω × r, Ω = ω, ρ =
Θ(r0 − r ) =
Γ
U
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
19 / 32
Why vortices
25
Density profile
Thomas-Fermi in flattened trap
Density
20
15
10
5
0
-15
-10
-5
0
5
Cross Section
10
15
Rotating solution: i∂t ψ = (µ − 2ΩLz )ψ
∇ · [ρ(v − Ω × r)] = (γnet Θ(r0 − r ) − Γρ) ρ,
√
∇2 ρ
m
m 2 2
2
2
µ=
|v − Ω × r| + r (ω − Ω ) + Uρ −
√
2
2
2m ρ
γnet
µ
v = Ω × r, Ω = ω, ρ =
Θ(r0 − r ) =
Γ
U
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
19 / 32
Why vortices
25
Density profile
Thomas-Fermi in flattened trap
Density
20
15
10
5
0
-15
-10
-5
0
5
Cross Section
10
15
Rotating solution: i∂t ψ = (µ − 2ΩLz )ψ
∇ · [ρ(v − Ω × r)] = (γnet Θ(r0 − r ) − Γρ) ρ,
√
∇2 ρ
m
m 2 2
2
2
µ=
|v − Ω × r| + r (ω − Ω ) + Uρ −
√
2
2
2m ρ
γnet
µ
v = Ω × r, Ω = ω, ρ =
Θ(r0 − r ) =
Γ
U
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
19 / 32
Why vortices
25
Density profile
Thomas-Fermi in flattened trap
Density
20
15
10
5
0
-15
-10
-5
0
5
Cross Section
10
15
Rotating solution: i∂t ψ = (µ − 2ΩLz )ψ
∇ · [ρ(v − Ω × r)] = (γnet Θ(r0 − r ) − Γρ) ρ,
√
∇2 ρ
m
m 2 2
2
2
µ=
|v − Ω × r| + r (ω − Ω ) + Uρ −
√
2
2
2m ρ
γnet
µ
v = Ω × r, Ω = ω, ρ =
Θ(r0 − r ) =
Γ
U
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
19 / 32
Superfluidity
1
Introduction to polariton condensation
What are excitons and polaritons
Experimental features
Approaches to modelling
2
Pattern formation
Experiments
Modelling pattern formation
3
Superfluidity
Non-equilibrium condensate spectrum
Experiments and aspects of superfluidity
Current-current response function
4
Coherence
Experiments
Power law decay of coherence
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
20 / 32
Fluctuations above transition
When condensed
h
i−1
Det D R (ω, k)
= ω 2 − ξk2
frequency
Sound mode
With ξk ' ck
Poles:
momentum
ω ∗ = ξk
Generic structure
of Green’s function:
ω + iγnet − k − µ
iγnet − µ
R −1
[D ] =
−iγnet − µ
−ω − iγnet − k − µ
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
21 / 32
When condensed
h
i−1
2
Det D R (ω, k)
= (ω+iγnet )2 +γnet
−ξk2
With ξk ' ck
Poles:
ω ∗ = − iγnet ±
frequency
Fluctuations above transition
Real
Imaginary
momentum
q
2
ξk2 − γnet
Generic structure
of Green’s function:
ω + iγnet − k − µ
iγnet − µ
R −1
[D ] =
−iγnet − µ
−ω − iγnet − k − µ
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
21 / 32
When condensed
h
i−1
2
Det D R (ω, k)
= (ω+iγnet )2 +γnet
−ξk2
With ξk ' ck
Poles:
ω ∗ = − iγnet ±
frequency
Fluctuations above transition
Real
Imaginary
momentum
q
2
ξk2 − γnet
Generic structure
of Green’s function:
ω + iγnet − k − µ
iγnet − µ
R −1
[D ] =
−iγnet − µ
−ω − iγnet − k − µ
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
21 / 32
Polariton “superfluidity” experiments
Quantised vortices in disorder potential
[Lagoudakis et al. Nature Phys. ’08]
Changes to excitation spectrum
[Utsunomiya et al. Nature Phys. ’08]
Wavepacket propagation
[Amo et al. Nature ’09]
Driven superfluidity
[Amo et al. Nature Phys. (’09)
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
22 / 32
Aspects of superfluidity
Quantised Landau
vortices critical
velocity
Superfluid 4 He/cold atom
Bose-Einstein condensate
Non-interacting
Bose-Einstein condensate
Classical irrotational fluid
Incoherently pumped
polariton condensates
"
"
%
"
"
%
"
%
Metastable Two-fluid Local
persistent hydrody- thermal
flow
namics
equilibrium
"
%
%
?
"
"
"
?
"
"
"
%
Solitary
waves
"
%
"
?
Lagoudakis et al. Nat. Phys. ’08. Utsunomiya et al. Nat. Phys. ’08. Amo et al. Nature
’09; Nat. Phys. ’09
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
23 / 32
Aspects of superfluidity
Quantised Landau
vortices critical
velocity
Superfluid 4 He/cold atom
Bose-Einstein condensate
Non-interacting
Bose-Einstein condensate
Classical irrotational fluid
Incoherently pumped
polariton condensates
"
"
%
"
"
%
"
%
Metastable Two-fluid Local
persistent hydrody- thermal
flow
namics
equilibrium
"
%
%
?
"
"
"
?
"
"
"
%
Solitary
waves
"
%
"
?
Lagoudakis et al. Nat. Phys. ’08. Utsunomiya et al. Nat. Phys. ’08. Amo et al. Nature
’09; Nat. Phys. ’09
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
23 / 32
Superfluid density
Two-fluid hydrodynamics
Experimentally, rotation:
ρ/ρtotal
1
ρnormal
ρsuperfluid
0
0
1
To calculate,
transverse/longitudinal:
T/Tc
ρs , ρn distinguished by slow
rotation
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
24 / 32
Superfluid density
Two-fluid hydrodynamics
Experimentally, rotation:
ρ/ρtotal
1
ρnormal
ρsuperfluid
0
0
1
To calculate,
transverse/longitudinal:
T/Tc
ρs , ρn distinguished by slow
rotation
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
24 / 32
Superfluid density
Two-fluid hydrodynamics
Experimentally, rotation:
ρ/ρtotal
1
ρnormal
ρsuperfluid
0
0
1
To calculate,
transverse/longitudinal:
T/Tc
ρs , ρn distinguished by slow
rotation
Jonathan Keeling
Polariton condensation
00000000000
11111111111
00000000000
11111111111
11111111111
00000000000
00000000000
11111111111
00000000000
11111111111
00000000000
11111111111
00000000000
11111111111
00000000000
11111111111
00000000000
11111111111
00000000000
11111111111
111111
000000
111111
000000
UMass Amherst, January 2012
24 / 32
Superfluid density
Currrent:
J = ρv = ψ † i∇ψ = |ψ|2 ∇φ
2ki + qi
†
ψk
Ji (q) = ψk+q
2m
Response function:
X
H→H−
f(q) · Ji (q) Ji (q) = χij (q)fj (q)
q
Vertex corrections essential for superfluid part.
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
25 / 32
Superfluid density
Currrent:
J = ρv = ψ † i∇ψ = |ψ|2 ∇φ
2ki + qi
†
ψk
Ji (q) = ψk+q
2m
Response function:
X
H→H−
f(q) · Ji (q) Ji (q) = χij (q)fj (q)
q
Vertex corrections essential for superfluid part.
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
25 / 32
Superfluid density
00000000000
11111111111
00000000000
11111111111
11111111111
00000000000
00000000000
11111111111
00000000000
11111111111
00000000000
11111111111
00000000000
11111111111
00000000000
11111111111
00000000000
11111111111
00000000000
11111111111
Currrent:
111111
000000
111111
000000
J = ρv = ψ † i∇ψ = |ψ|2 ∇φ
2ki + qi
†
ψk
Ji (q) = ψk+q
2m
Response function:
X
H→H−
f(q) · Ji (q) Ji (q) = χij (q)fj (q)
q
χij (ω = 0, q → 0) = h[Ji (q), Jj (−q)]i =
ρS qi qj
ρ
+ N δij
2
m q
m
Vertex corrections essential for superfluid part.
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
25 / 32
Superfluid density
00000000000
11111111111
00000000000
11111111111
11111111111
00000000000
00000000000
11111111111
00000000000
11111111111
00000000000
11111111111
00000000000
11111111111
00000000000
11111111111
00000000000
11111111111
00000000000
11111111111
Currrent:
111111
000000
111111
000000
J = ρv = ψ † i∇ψ = |ψ|2 ∇φ
2ki + qi
†
ψk
Ji (q) = ψk+q
2m
Response function:
X
H→H−
f(q) · Ji (q) Ji (q) = χij (q)fj (q)
q
χij (ω = 0, q → 0) = h[Ji (q), Jj (−q)]i =
ρS qi qj
ρ
+ N δij
2
m q
m
Vertex corrections essential for superfluid part.
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
25 / 32
Calculating superfluid response function
Using Keldysh generating functional
d 2 Z[f , θ]
i
,
χij (q) = −
2 dfi (q)dθj (−q)
Z
Z[f , θ] =
f , θ couple as force/response current.
X
θi
S[f , θ] = S +
ψ̄cl ψ̄q k+q
fi − θi
k,q
fi + θi
−θi
Dψ exp(iS[f , θ])
q
2ki + qi
2m
ψcl
ψq
k
Saddle point + fluctuations:
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
26 / 32
Calculating superfluid response function
Using Keldysh generating functional
d 2 Z[f , θ]
i
,
χij (q) = −
2 dfi (q)dθj (−q)
Z
Z[f , θ] =
f , θ couple as force/response current.
X
θi
S[f , θ] = S +
ψ̄cl ψ̄q k+q
fi − θi
k,q
fi + θi
−θi
Dψ exp(iS[f , θ])
q
2ki + qi
2m
ψcl
ψq
k
Saddle point + fluctuations:
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
26 / 32
Calculating superfluid response function
Using Keldysh generating functional
d 2 Z[f , θ]
i
,
χij (q) = −
2 dfi (q)dθj (−q)
Z
Z[f , θ] =
f , θ couple as force/response current.
X
θi
S[f , θ] = S +
ψ̄cl ψ̄q k+q
fi − θi
fi + θi
−θi
k,q
Dψ exp(iS[f , θ])
q
2ki + qi
2m
ψcl
ψq
k
Saddle point + fluctuations:
+
+
Jonathan Keeling
+
+ ... +
Polariton condensation
UMass Amherst, January 2012
26 / 32
Calculating superfluid response function
Using Keldysh generating functional
d 2 Z[f , θ]
i
,
χij (q) = −
2 dfi (q)dθj (−q)
Z
Z[f , θ] =
f , θ couple as force/response current.
X
θi
S[f , θ] = S +
ψ̄cl ψ̄q k+q
fi − θi
fi + θi
−θi
k,q
Dψ exp(iS[f , θ])
q
2ki + qi
2m
ψcl
ψq
k
Saddle point + fluctuations: Only one diagram for χN
+
+
Jonathan Keeling
+
+ ... +
Polariton condensation
UMass Amherst, January 2012
26 / 32
Non-equilibrium superfluid response
Superfluid response exists because:
iψ0 qi
1 iψ0 qj
R
=
(1, −1) D (q, ω = 0)
−1
2m
2m
R
D (ω = 0) ∝ 1/q despite pumping/decay — superfluid response
exists.
Normal density: Z
Z
i
dω h
d
ρN = d k k
Tr σz D K σz (D R + D A )
2π
Is affected by pump/decay:
Does not vanish at T → 0.
[JK PRL ’11]
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
27 / 32
Non-equilibrium superfluid response
Superfluid response exists because:
iψ0 qi
1 iψ0 qj
R
=
(1, −1) D (q, ω = 0)
−1
2m
2m
R
D (ω = 0) ∝ 1/q despite pumping/decay — superfluid response
exists.
Normal density: Z
Z
i
dω h
d
ρN = d k k
Tr σz D K σz (D R + D A )
2π
Is affected by pump/decay:
Does not vanish at T → 0.
[JK PRL ’11]
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
27 / 32
Non-equilibrium superfluid response
Superfluid response exists because:
iψ0 qi
1 iψ0 qj
R
=
(1, −1) D (q, ω = 0)
−1
2m
2m
R
D (ω = 0) ∝ 1/q despite pumping/decay — superfluid response
exists.
Normal density: Z
Z
i
dω h
d
ρN = d k k
Tr σz D K σz (D R + D A )
2π
Is affected by pump/decay:
Does not vanish at T → 0.
[JK PRL ’11]
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
27 / 32
Non-equilibrium superfluid response
Superfluid response exists because:
iψ0 qi
1 iψ0 qj
R
=
(1, −1) D (q, ω = 0)
−1
2m
2m
R
D (ω = 0) ∝ 1/q despite pumping/decay — superfluid response
exists.
Normal density: Z
Z
i
dω h
d
ρN = d k k
Tr σz D K σz (D R + D A )
2π
Is affected by pump/decay:
Does not vanish at T → 0.
ρN / m µ
3
γnet/µ = 0.0
γnet/µ = 0.1
γnet/µ = 0.5
2
1
0
0
1
2
3
4
5
T/µ
[JK PRL ’11]
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
27 / 32
Coherence:
1
Introduction to polariton condensation
What are excitons and polaritons
Experimental features
Approaches to modelling
2
Pattern formation
Experiments
Modelling pattern formation
3
Superfluidity
Non-equilibrium condensate spectrum
Experiments and aspects of superfluidity
Current-current response function
4
Coherence
Experiments
Power law decay of coherence
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
28 / 32
Correlations in a 2D Gas
+
Correlations:
=
D
E
g1 (r, r0 , t) = ψ † (r, t)ψ(r0 , 0)
D< = DK − DR + DA
Generally, get:
"
(
D
E
ln(r /r0 )
†
2
ψ (r, t)ψ(0, 0) ' |ψ0 | exp −ap 1
2
2
2 ln(c t/γnet r0 )
t '0
r '0
#
[Szymańska et al. PRL ’06; PRB ’07]
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
29 / 32
Correlations in a 2D Gas
Correlations: (in 2D)
D
E
g1 (r, r0 , t) = ψ † (r, t)ψ(r0 , 0)
h
i
<
' |ψ0 |2 exp −Dφφ
(r, r0 , t)
+
=
D< = DK − DR + DA
Generally, get:
"
(
D
E
ln(r /r0 )
†
2
ψ (r, t)ψ(0, 0) ' |ψ0 | exp −ap 1
2
2
2 ln(c t/γnet r0 )
t '0
r '0
#
[Szymańska et al. PRL ’06; PRB ’07]
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
29 / 32
Correlations in a 2D Gas
Correlations: (in 2D)
D
E
g1 (r, r0 , t) = ψ † (r, t)ψ(r0 , 0)
h
i
<
' |ψ0 |2 exp −Dφφ
(r, r0 , t)
+
=
D< = DK − DR + DA
Generally, get:
"
(
D
E
ln(r /r0 )
†
2
ψ (r, t)ψ(0, 0) ' |ψ0 | exp −ap 1
2
2
2 ln(c t/γnet r0 )
t '0
r '0
#
[Szymańska et al. PRL ’06; PRB ’07]
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
29 / 32
Experimental observation of power-law decay
G. Rompos, Y. Yamamoto et al. submitted
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
30 / 32
Experimental observation of power-law decay
g1 (r, −r) ∝
r
r0
−ap
G. Rompos, Y. Yamamoto et al. submitted
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
30 / 32
Exponent in a non-equilibrium 2D gas
D
E
h
i
2r
†
2
<
lim ψ (r, 0)ψ(−r, 0) = |ψ0 | exp −Dφφ (r, −r) ∝ exp −ap ln
r →∞
r0
Experimentally, aP ' 1.2
mkB T
1
In equilibrium ap =
< (BKT transition)
4
2π~2 ns
Non-equilibrium theory depends on
thermalisation.
I
I
Thermalised (yet diffusive modes)
mkB T
ap =
2π~2 ns
Non-thermalised,
Pumping noise
aP ∝
.
ns
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
31 / 32
Exponent in a non-equilibrium 2D gas
D
E
h
i
2r
†
2
<
lim ψ (r, 0)ψ(−r, 0) = |ψ0 | exp −Dφφ (r, −r) ∝ exp −ap ln
r →∞
r0
Experimentally, aP ' 1.2
mkB T
1
In equilibrium ap =
< (BKT transition)
4
2π~2 ns
Non-equilibrium theory depends on
thermalisation.
I
I
Thermalised (yet diffusive modes)
mkB T
ap =
2π~2 ns
Non-thermalised,
Pumping noise
aP ∝
.
ns
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
31 / 32
Exponent in a non-equilibrium 2D gas
D
E
h
i
2r
†
2
<
lim ψ (r, 0)ψ(−r, 0) = |ψ0 | exp −Dφφ (r, −r) ∝ exp −ap ln
r →∞
r0
Experimentally, aP ' 1.2
mkB T
1
In equilibrium ap =
< (BKT transition)
4
2π~2 ns
Non-equilibrium theory depends on
thermalisation.
I
I
Thermalised (yet diffusive modes)
mkB T
ap =
2π~2 ns
Non-thermalised,
Pumping noise
aP ∝
.
ns
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
31 / 32
Exponent in a non-equilibrium 2D gas
D
E
h
i
2r
†
2
<
lim ψ (r, 0)ψ(−r, 0) = |ψ0 | exp −Dφφ (r, −r) ∝ exp −ap ln
r →∞
r0
Experimentally, aP ' 1.2
mkB T
1
In equilibrium ap =
< (BKT transition)
4
2π~2 ns
Non-equilibrium theory depends on
thermalisation.
I
I
Thermalised (yet diffusive modes)
mkB T
ap =
2π~2 ns
Non-thermalised,
Pumping noise
aP ∝
.
ns
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
31 / 32
Exponent in a non-equilibrium 2D gas
D
E
h
i
2r
†
2
<
lim ψ (r, 0)ψ(−r, 0) = |ψ0 | exp −Dφφ (r, −r) ∝ exp −ap ln
r →∞
r0
Experimentally, aP ' 1.2
mkB T
1
In equilibrium ap =
< (BKT transition)
4
2π~2 ns
Non-equilibrium theory depends on
thermalisation.
I
I
Thermalised (yet diffusive modes)
mkB T
ap =
2π~2 ns
Non-thermalised,
Pumping noise
aP ∝
.
ns
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
31 / 32
Conclusion
Instability of Thomas-Fermi and spontaneous rotation
t=22
t=35
t=56
Survival of superfluid response
momentum
ρN / m µ
frequency
3
Real
Imaginary
γnet/µ = 0.0
γnet/µ = 0.1
γnet/µ = 0.5
2
1
0
0
1
2
3
4
5
T/µ
Power law decay of correlations
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
32 / 32
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
33 / 38
Extra slides
5
GPE stability
6
Detecting vortex lattice
7
Measuring superfluid density
8
Coherence Finite size and Schawlow-Townes
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
34 / 38
Instability of Thomas-Fermi: details
1
∂t ρ + ∇ · (ρv) = (γnet − Γρ)ρ
2
mω 2 2 m 2
∂t v + ∇(Uρ +
r + |v| ) = 0
2
2
Jonathan Keeling
3γnet
2Γ
Polariton condensation
UMass Amherst, January 2012
35 / 38
Instability of Thomas-Fermi: details
1
∂t ρ + ∇ · (ρv) = (γnet − Γρ)ρ
2
mω 2 2 m 2
∂t v + ∇(Uρ +
r + |v| ) = 0
2
2
Normal modes for γnet , Γ → 0:
3γnet
2Γ
δρn,m (r , θ, t) = eimθ hn,m (r )eiωn,m t
p
ωn,m = ω2 m(1 + 2n) + 2n(n + 1)
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
35 / 38
Instability of Thomas-Fermi: details
1
∂t ρ + ∇ · (ρv) = (γnet − Γρ)ρ
2
mω 2 2 m 2
∂t v + ∇(Uρ +
r + |v| ) = 0
2
2
Normal modes for γnet , Γ → 0:
3γnet
2Γ
δρn,m (r , θ, t) = eimθ hn,m (r )eiωn,m t
p
ωn,m = ω2 m(1 + 2n) + 2n(n + 1)
Add weak pumping/decay:
ωn,n → ωn,m + iγnet
Jonathan Keeling
m(1 + 2n) + 2n(n + 1)−m2
2m(1 + 2n) + 4n(n + 1) + m2
Polariton condensation
UMass Amherst, January 2012
35 / 38
Instability of Thomas-Fermi: details
1
∂t ρ + ∇ · (ρv) = (γnet − Γρ)ρ
2
mω 2 2 m 2
∂t v + ∇(Uρ +
r + |v| ) = 0
2
2
Normal modes for γnet , Γ → 0:
Unstable growth
3γnet
2Γ
δρn,m (r , θ, t) = eimθ hn,m (r )eiωn,m t
p
ωn,m = ω2 m(1 + 2n) + 2n(n + 1)
Add weak pumping/decay:
ωn,n → ωn,m + iγnet
Jonathan Keeling
m(1 + 2n) + 2n(n + 1)−m2
2m(1 + 2n) + 4n(n + 1) + m2
Polariton condensation
UMass Amherst, January 2012
35 / 38
Instability of Thomas-Fermi: details
1
∂t ρ + ∇ · (ρv) = (γnet − Γρ)ρ
2
mω 2 2 m 2
∂t v + ∇(Uρ +
r + |v| ) = 0
2
2
Normal modes for γnet , Γ → 0:
Stabilised
3γnet
2Γ
δρn,m (r , θ, t) = eimθ hn,m (r )eiωn,m t
p
ωn,m = ω2 m(1 + 2n) + 2n(n + 1)
Add weak pumping/decay:
ωn,n → ωn,m + iγnet
Jonathan Keeling
m(1 + 2n) + 2n(n + 1)−m2
2m(1 + 2n) + 4n(n + 1) + m2
Polariton condensation
UMass Amherst, January 2012
35 / 38
Detecting vortex lattices
30
25
20
ω
Snapshot
Spectrum:
15
10
5
0
−5
0
kx
5
Defocussed homodyne intereference:
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
36 / 38
Measuring superfluid density
1. Effect rotating frame
Polariton polarization: (ψ , ψ )
`2
r 2 e2iφ
H=λ
r 2 e−2iφ −`2
Jonathan Keeling
Polariton condensation
(a)
µc
UMass Amherst, January 2012
37 / 38
Measuring superfluid density
Polariton condensation
mℓ2ω
`2
φ̂
1− √
= mω × r =
r
r 4 + `4
Jonathan Keeling
µc
(b)0.3
Ground state Berry phase:
qAeff
(a)
0.4
0.3
0.2
0.2
0.1
0.1
0
0
1
2
r/ℓ
3
qAφℓ = mvℓ
1. Effect rotating frame
Polariton polarization: (ψ , ψ )
`2
r 2 e2iφ
H=λ
r 2 e−2iφ −`2
0
4
UMass Amherst, January 2012
37 / 38
Measuring superfluid density
µc
`2
φ̂
1− √
= mω × r =
r
r 4 + `4
mℓ2ω
(b)0.3
Ground state Berry phase:
qAeff
(a)
0.4
0.3
0.2
0.2
0.1
0.1
0
0
1
2
r/ℓ
3
qAφℓ = mvℓ
1. Effect rotating frame
Polariton polarization: (ψ , ψ )
`2
r 2 e2iφ
H=λ
r 2 e−2iφ −`2
0
4
2. Measure resulting current
Energy shift of normal state:
∆E = (1/2)mv 2 = 0.08/m`2 ' 0.1meV
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
37 / 38
Finite size effects: Single mode vs many mode
D
E
h
i
<
ψ † (r, t)ψ(r0 , 0) ' |ψ0 |2 exp −Dφφ
(r, r0 , t)
Jonathan Keeling
Polariton condensation
UMass Amherst, January 2012
38 / 38
Finite size effects: Single mode vs many mode
D
E
h
i
<
ψ † (r, t)ψ(r0 , 0) ' |ψ0 |2 exp −Dφφ
(r, r0 , t)
<
Dφφ
(r, r0 , t) from sum of phase modes. Study ct r limit:
<
Dφφ
(r, r, t)
∝
nX
max Z
n
Jonathan Keeling
|ϕn (r)|2 (1 − eiωt )
dω
2π (ω + iγnet )2 + γ 2 − ξn2 2
net
Polariton condensation
UMass Amherst, January 2012
38 / 38
Finite size effects: Single mode vs many mode
D
E
h
i
<
ψ † (r, t)ψ(r0 , 0) ' |ψ0 |2 exp −Dφφ
(r, r0 , t)
<
Dφφ
(r, r0 , t) from sum of phase modes. Study ct r limit:
<
Dφφ
(r, r, t)
∝
nX
max Z
n
r
∆ξ γnet
Emax
t
Jonathan Keeling
|ϕn (r)|2 (1 − eiωt )
dω
2π (ω + iγnet )2 + γ 2 − ξn2 2
net
Emax
∆
<
Dφφ
∼ 1 + ln(Emax
Energy
Polariton condensation
UMass Amherst, January 2012
q
t
γnet )
38 / 38
Finite size effects: Single mode vs many mode
D
E
h
i
<
ψ † (r, t)ψ(r0 , 0) ' |ψ0 |2 exp −Dφφ
(r, r0 , t)
<
Dφφ
(r, r0 , t) from sum of phase modes. Study ct r limit:
<
Dφφ
(r, r, t)
∝
nX
max Z
n
r
|ϕn (r)|2 (1 − eiωt )
dω
2π (ω + iγnet )2 + γ 2 − ξn2 2
net
E
γnet
∆ξ Emax
t
Energy
∆
r
E
γnet
∆ξ Emax
t
Energy
∆
(Recovers Schawlow-Townes laser linewidth)
max
max
Jonathan Keeling
Polariton condensation
<
Dφφ
∼ 1 + ln(Emax
<
Dφφ
∼
πC
2γnet
q
t
γnet )
t
2γnet
UMass Amherst, January 2012
38 / 38