lecture 3: dissipative systems and non-equilibrium bose...
TRANSCRIPT
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Lecture 3: Dissipative Systems and Non-EquilibriumBose-Einstein Condensation
Marzena Hanna Szymanska
Department of PhysicsUniversity of Warwick
March 2011 / Universite Libre de Bruxelles
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 1 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Acknowledgements
Theory
J. Keeling
P. B. Littlewood
F. M. Marchetti
Experiment
G. Roumpos
Y. Yamamoto
D. Sanvitto
L. Vina
Funding:
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 2 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Macroscopic Quantum Coherence
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 3 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Macroscopic Quantum Coherence
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 3 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Macroscopic Quantum Coherence
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 3 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
1 Non-equilibrium condensation in dissipative environmentPolariton model and Keldysh field theoryMean-field equationsConnections of mean-field equations to other limitsPhase transition and fluctuationsFinite size effects: single vs many modesSuperfluidity
2 Perspectives
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 4 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Overview: Polaritons
Experiments
[Kasprzak, et al., Nature2006]
[Lagoudakis et al.,Nature Physics 2008]
[Utsunomiya et al., NaturePhysics 2008]
[Amo et al., Nature 2009]
Theoretically complexextended size, internal structure, stronginteractions and the BCS-BEC crossover
2D vs finite size
excitonic and photonic disorder
non-equilibrium and dissipation
[Keeling, Eastham, Szymanska, et al., PRL 2004, PRB 2005]
[Marchetti, Keeling, Szymanska, et al., PRL 2006, PRB 2007]
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 5 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Overview: Polaritons
Experiments
[Kasprzak, et al., Nature2006]
[Lagoudakis et al.,Nature Physics 2008]
[Utsunomiya et al., NaturePhysics 2008]
[Amo et al., Nature 2009]
Theoretically complexextended size, internal structure, stronginteractions and the BCS-BEC crossover
2D vs finite size
excitonic and photonic disorder
non-equilibrium and dissipation
[Keeling, Eastham, Szymanska, et al., PRL 2004, PRB 2005]
[Marchetti, Keeling, Szymanska, et al., PRL 2006, PRB 2007]
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 5 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Overview: Polaritons
Experiments
[Kasprzak, et al., Nature2006]
[Lagoudakis et al.,Nature Physics 2008]
[Utsunomiya et al., NaturePhysics 2008]
[Amo et al., Nature 2009]
Theoretically complexextended size, internal structure, stronginteractions and the BCS-BEC crossover
2D vs finite size
excitonic and photonic disorder
non-equilibrium and dissipation
[Keeling, Eastham, Szymanska, et al., PRL 2004, PRB 2005]
[Marchetti, Keeling, Szymanska, et al., PRL 2006, PRB 2007]
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 5 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Overview: Polaritons
Experiments
[Kasprzak, et al., Nature2006]
[Lagoudakis et al.,Nature Physics 2008]
[Utsunomiya et al., NaturePhysics 2008]
[Amo et al., Nature 2009]
Theoretically complexextended size, internal structure, stronginteractions and the BCS-BEC crossover
2D vs finite size
excitonic and photonic disorder
non-equilibrium and dissipation
[Keeling, Eastham, Szymanska, et al., PRL 2004, PRB 2005]
[Marchetti, Keeling, Szymanska, et al., PRL 2006, PRB 2007]
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 5 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Overview: Polaritons
Experiments
[Kasprzak, et al., Nature2006]
[Lagoudakis et al.,Nature Physics 2008]
[Utsunomiya et al., NaturePhysics 2008]
[Amo et al., Nature 2009]
Theoretically complexextended size, internal structure, stronginteractions and the BCS-BEC crossover
2D vs finite size
excitonic and photonic disorder
non-equilibrium and dissipation
[Keeling, Eastham, Szymanska, et al., PRL 2004, PRB 2005]
[Marchetti, Keeling, Szymanska, et al., PRL 2006, PRB 2007]
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 5 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Overview: Polaritons
Experiments
[Kasprzak, et al., Nature2006]
[Lagoudakis et al.,Nature Physics 2008]
[Utsunomiya et al., NaturePhysics 2008]
[Amo et al., Nature 2009]
Theoretically complex
extended size, internal structure, stronginteractions and the BCS-BEC crossover
2D vs finite size
excitonic and photonic disorder
non-equilibrium and dissipation
[Keeling, Eastham, Szymanska, et al., PRL 2004, PRB 2005]
[Marchetti, Keeling, Szymanska, et al., PRL 2006, PRB 2007]
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 5 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Overview: Polaritons
Experiments
[Kasprzak, et al., Nature2006]
[Lagoudakis et al.,Nature Physics 2008]
[Utsunomiya et al., NaturePhysics 2008]
[Amo et al., Nature 2009]
Theoretically complexextended size, internal structure, stronginteractions and the BCS-BEC crossover
2D vs finite size
excitonic and photonic disorder
non-equilibrium and dissipation
[Keeling, Eastham, Szymanska, et al., PRL 2004, PRB 2005]
[Marchetti, Keeling, Szymanska, et al., PRL 2006, PRB 2007]
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 5 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Overview: Polaritons
Experiments
[Kasprzak, et al., Nature2006]
[Lagoudakis et al.,Nature Physics 2008]
[Utsunomiya et al., NaturePhysics 2008]
[Amo et al., Nature 2009]
Theoretically complexextended size, internal structure, stronginteractions and the BCS-BEC crossover
2D vs finite size
excitonic and photonic disorder
non-equilibrium and dissipation
[Keeling, Eastham, Szymanska, et al., PRL 2004, PRB 2005]
[Marchetti, Keeling, Szymanska, et al., PRL 2006, PRB 2007]
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 5 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Overview: Polaritons
Experiments
[Kasprzak, et al., Nature2006]
[Lagoudakis et al.,Nature Physics 2008]
[Utsunomiya et al., NaturePhysics 2008]
[Amo et al., Nature 2009]
Theoretically complexextended size, internal structure, stronginteractions and the BCS-BEC crossover
2D vs finite size
excitonic and photonic disorder
non-equilibrium and dissipation
[Keeling, Eastham, Szymanska, et al., PRL 2004, PRB 2005]
[Marchetti, Keeling, Szymanska, et al., PRL 2006, PRB 2007]
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 5 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Overview: Polaritons
Experiments
[Kasprzak, et al., Nature2006]
[Lagoudakis et al.,Nature Physics 2008]
[Utsunomiya et al., NaturePhysics 2008]
[Amo et al., Nature 2009]
Theoretically complexextended size, internal structure, stronginteractions and the BCS-BEC crossover
2D vs finite size
excitonic and photonic disorder
non-equilibrium and dissipation
[Keeling, Eastham, Szymanska, et al., PRL 2004, PRB 2005]
[Marchetti, Keeling, Szymanska, et al., PRL 2006, PRB 2007]
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 5 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Polariton model and Keldysh field theoryMean-field equationsConnections of mean-field equations to other limitsPhase transition and fluctuationsFinite size effects: single vs many modesSuperfluidity
Non-equilibrium Condensation: model and method
Model: Hsys =X
pωpψ
†pψp +
Xα
εα“
b†αbα − a†αaα”
+Xα,p
“gα,pψpb†αaα + h.c.
”
H = Hsys
Schematically: pump γ, decay κ
Hsys,bath 'Xp,k
√κψpΨ†k+
Xα,k
√γ“
a†αAk + b†αBk
”+h.c.
Method: Keldysh path integral techniques adopted for broken symmetry systemswith strong dissipation
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 6 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Polariton model and Keldysh field theoryMean-field equationsConnections of mean-field equations to other limitsPhase transition and fluctuationsFinite size effects: single vs many modesSuperfluidity
Non-equilibrium Condensation: model and method
Model: Hsys =X
pωpψ
†pψp
+Xα
εα“
b†αbα − a†αaα”
+Xα,p
“gα,pψpb†αaα + h.c.
”
H = Hsys
Schematically: pump γ, decay κ
Hsys,bath 'Xp,k
√κψpΨ†k+
Xα,k
√γ“
a†αAk + b†αBk
”+h.c.
Method: Keldysh path integral techniques adopted for broken symmetry systemswith strong dissipation
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 6 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Polariton model and Keldysh field theoryMean-field equationsConnections of mean-field equations to other limitsPhase transition and fluctuationsFinite size effects: single vs many modesSuperfluidity
Non-equilibrium Condensation: model and method
Model: Hsys =X
pωpψ
†pψp +
Xα
εα“
b†αbα − a†αaα”
+Xα,p
“gα,pψpb†αaα + h.c.
”
H = Hsys
Schematically: pump γ, decay κ
Hsys,bath 'Xp,k
√κψpΨ†k+
Xα,k
√γ“
a†αAk + b†αBk
”+h.c.
Method: Keldysh path integral techniques adopted for broken symmetry systemswith strong dissipation
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 6 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Polariton model and Keldysh field theoryMean-field equationsConnections of mean-field equations to other limitsPhase transition and fluctuationsFinite size effects: single vs many modesSuperfluidity
Non-equilibrium Condensation: model and method
Model: Hsys =X
pωpψ
†pψp +
Xα
εα“
b†αbα − a†αaα”
+Xα,p
“gα,pψpb†αaα + h.c.
”
H = Hsys
Schematically: pump γ, decay κ
Hsys,bath 'Xp,k
√κψpΨ†k+
Xα,k
√γ“
a†αAk + b†αBk
”+h.c.
Method: Keldysh path integral techniques adopted for broken symmetry systemswith strong dissipation
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 6 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Polariton model and Keldysh field theoryMean-field equationsConnections of mean-field equations to other limitsPhase transition and fluctuationsFinite size effects: single vs many modesSuperfluidity
Non-equilibrium Condensation: model and method
Model: Hsys =X
pωpψ
†pψp +
Xα
εα“
b†αbα − a†αaα”
+Xα,p
“gα,pψpb†αaα + h.c.
”
H = Hsys + Hsys,bath + Hbath
Schematically: pump γ, decay κ
Hsys,bath 'Xp,k
√κψpΨ†k+
Xα,k
√γ“
a†αAk + b†αBk
”+h.c.
Method: Keldysh path integral techniques adopted for broken symmetry systemswith strong dissipation
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 6 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Polariton model and Keldysh field theoryMean-field equationsConnections of mean-field equations to other limitsPhase transition and fluctuationsFinite size effects: single vs many modesSuperfluidity
Non-equilibrium Condensation: model and method
Model: Hsys =X
pωpψ
†pψp +
Xα
εα“
b†αbα − a†αaα”
+Xα,p
“gα,pψpb†αaα + h.c.
”
H = Hsys + Hsys,bath + Hbath
Schematically: pump γ, decay κ
Hsys,bath 'Xp,k
√κψpΨ†k+
Xα,k
√γ“
a†αAk + b†αBk
”+h.c.
Method: Keldysh path integral techniques adopted for broken symmetry systemswith strong dissipation
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 6 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Polariton model and Keldysh field theoryMean-field equationsConnections of mean-field equations to other limitsPhase transition and fluctuationsFinite size effects: single vs many modesSuperfluidity
Non-equilibrium Condensation: model and method
Model: Hsys =X
pωpψ
†pψp +
Xα
εα“
b†αbα − a†αaα”
+Xα,p
“gα,pψpb†αaα + h.c.
”
H = Hsys + Hsys,bath + Hbath
Schematically: pump γ, decay κ
Hsys,bath 'Xp,k
√κψpΨ†k+
Xα,k
√γ“
a†αAk + b†αBk
”+h.c.
Method: Keldysh path integral techniques adopted for broken symmetry systemswith strong dissipation
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 6 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Polariton model and Keldysh field theoryMean-field equationsConnections of mean-field equations to other limitsPhase transition and fluctuationsFinite size effects: single vs many modesSuperfluidity
Keldysh non-equilibrium field theory
f
b
Green’s functions on Keldysh contour (D<, D>, DT , DT ) for example
D>(t) = −iD
Tc
“ψ(t , b)ψ†(0, f )
”E.
It is convenient to move to ψcl/q = ψ(t,f )±ψ(t,b)√2
, then
D = −ifi
Tc
„ψcl (t)ψq(t)
«“ψ†cl (t) ψ†q(t)
”fl=
„DK DR
DA 0
«.
In practice we usually know D−1 but 0
ˆDA˜−1ˆ
DR˜−1 ˆD−1˜K
!„DK DR
DA 0
«=
„1 00 1
«,
andˆD−1˜K = −
ˆDR˜−1 DK ˆDA˜−1 so DK = −DR ˆD−1˜K DA.
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 7 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Polariton model and Keldysh field theoryMean-field equationsConnections of mean-field equations to other limitsPhase transition and fluctuationsFinite size effects: single vs many modesSuperfluidity
Keldysh non-equilibrium field theory
f
b
Green’s functions on Keldysh contour (D<, D>, DT , DT ) for example
D>(t) = −iD
Tc
“ψ(t , b)ψ†(0, f )
”E.
It is convenient to move to ψcl/q = ψ(t,f )±ψ(t,b)√2
, then
D = −ifi
Tc
„ψcl (t)ψq(t)
«“ψ†cl (t) ψ†q(t)
”fl=
„DK DR
DA 0
«.
In practice we usually know D−1 but 0
ˆDA˜−1ˆ
DR˜−1 ˆD−1˜K
!„DK DR
DA 0
«=
„1 00 1
«,
andˆD−1˜K = −
ˆDR˜−1 DK ˆDA˜−1 so DK = −DR ˆD−1˜K DA.
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 7 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Polariton model and Keldysh field theoryMean-field equationsConnections of mean-field equations to other limitsPhase transition and fluctuationsFinite size effects: single vs many modesSuperfluidity
Keldysh non-equilibrium field theory
f
b
Green’s functions on Keldysh contour (D<, D>, DT , DT ) for example
D>(t) = −iD
Tc
“ψ(t , b)ψ†(0, f )
”E.
It is convenient to move to ψcl/q = ψ(t,f )±ψ(t,b)√2
, then
D = −ifi
Tc
„ψcl (t)ψq(t)
«“ψ†cl (t) ψ†q(t)
”fl=
„DK DR
DA 0
«.
In practice we usually know D−1 but 0
ˆDA˜−1ˆ
DR˜−1 ˆD−1˜K
!„DK DR
DA 0
«=
„1 00 1
«,
andˆD−1˜K = −
ˆDR˜−1 DK ˆDA˜−1 so DK = −DR ˆD−1˜K DA.
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 7 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Polariton model and Keldysh field theoryMean-field equationsConnections of mean-field equations to other limitsPhase transition and fluctuationsFinite size effects: single vs many modesSuperfluidity
Keldysh non-equilibrium field theoryFor example for free bosons we have
DR/A = (ω − ωk ± i0)−1, DK = −2πi(2nB(ω) + 1)δ(ω − ωk ).
The partition function
Z = NZ Y
pD[ψp, ψp]
Yα
D[φα, φα]Y
k
D[Ak ,Ak , Bk ,Bk , Ψk ,Ψk ]eiS ,
where φ = (b, a) and φ = (b, a)T.
Lets look in detail at photons and their decaying bath
Sψ+Sbathψ = −Z ∞
−∞dωX
pψp(ω)
„0 ω + ωp + dA(k,−ω)
ω + ωp + dR(k,−ω) dK (k,−ω)
«ψp(−ω).
assume flat bath’s density of states and constant coupling to the system
Sψ + Sbathψ =
Z ∞
−∞dωX
pψp(ω)
„0 −ω − ωp − iκ
−ω − ωp + iκ 2iκFψ(−ω)
«ψp(−ω).
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 8 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Polariton model and Keldysh field theoryMean-field equationsConnections of mean-field equations to other limitsPhase transition and fluctuationsFinite size effects: single vs many modesSuperfluidity
Keldysh non-equilibrium field theoryFor example for free bosons we have
DR/A = (ω − ωk ± i0)−1, DK = −2πi(2nB(ω) + 1)δ(ω − ωk ).
The partition function
Z = NZ Y
pD[ψp, ψp]
Yα
D[φα, φα]Y
k
D[Ak ,Ak , Bk ,Bk , Ψk ,Ψk ]eiS ,
where φ = (b, a) and φ = (b, a)T.
Lets look in detail at photons and their decaying bath
Sψ+Sbathψ = −Z ∞
−∞dωX
pψp(ω)
„0 ω + ωp + dA(k,−ω)
ω + ωp + dR(k,−ω) dK (k,−ω)
«ψp(−ω).
assume flat bath’s density of states and constant coupling to the system
Sψ + Sbathψ =
Z ∞
−∞dωX
pψp(ω)
„0 −ω − ωp − iκ
−ω − ωp + iκ 2iκFψ(−ω)
«ψp(−ω).
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 8 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Polariton model and Keldysh field theoryMean-field equationsConnections of mean-field equations to other limitsPhase transition and fluctuationsFinite size effects: single vs many modesSuperfluidity
Keldysh non-equilibrium field theoryFor example for free bosons we have
DR/A = (ω − ωk ± i0)−1, DK = −2πi(2nB(ω) + 1)δ(ω − ωk ).
The partition function
Z = NZ Y
pD[ψp, ψp]
Yα
D[φα, φα]Y
k
D[Ak ,Ak , Bk ,Bk , Ψk ,Ψk ]eiS ,
where φ = (b, a) and φ = (b, a)T.
Lets look in detail at photons and their decaying bath
Sψ+Sbathψ = −Z ∞
−∞dωX
pψp(ω)
„0 ω + ωp + dA(k,−ω)
ω + ωp + dR(k,−ω) dK (k,−ω)
«ψp(−ω).
assume flat bath’s density of states and constant coupling to the system
Sψ + Sbathψ =
Z ∞
−∞dωX
pψp(ω)
„0 −ω − ωp − iκ
−ω − ωp + iκ 2iκFψ(−ω)
«ψp(−ω).
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 8 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Polariton model and Keldysh field theoryMean-field equationsConnections of mean-field equations to other limitsPhase transition and fluctuationsFinite size effects: single vs many modesSuperfluidity
Keldysh non-equilibrium field theoryFor example for free bosons we have
DR/A = (ω − ωk ± i0)−1, DK = −2πi(2nB(ω) + 1)δ(ω − ωk ).
The partition function
Z = NZ Y
pD[ψp, ψp]
Yα
D[φα, φα]Y
k
D[Ak ,Ak , Bk ,Bk , Ψk ,Ψk ]eiS ,
where φ = (b, a) and φ = (b, a)T.
Lets look in detail at photons and their decaying bath
Sψ+Sbathψ = −Z ∞
−∞dωX
pψp(ω)
„0 ω + ωp + dA(k,−ω)
ω + ωp + dR(k,−ω) dK (k,−ω)
«ψp(−ω).
assume flat bath’s density of states and constant coupling to the system
Sψ + Sbathψ =
Z ∞
−∞dωX
pψp(ω)
„0 −ω − ωp − iκ
−ω − ωp + iκ 2iκFψ(−ω)
«ψp(−ω).
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 8 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Polariton model and Keldysh field theoryMean-field equationsConnections of mean-field equations to other limitsPhase transition and fluctuationsFinite size effects: single vs many modesSuperfluidity
Keldysh non-equilibrium field theoryThe final form
S = −iXα
TrlnG−1α,α +
Z ∞
−∞dtdt ′
Xpψp(t)
„0 i∂t − ωp−iκ
i∂t − ωp + iκ 2iκFΨ(t − t ′)
«ψp(t ′)
Introducing the abbreviations λcl =P
pgp,α√
2ψp,cl and λq =
Pp
gp,α√2ψp,q
G−1(t , t ′) =„λq(t)σ+ + λq(t)σ− ∂tσ0 − εασ3 − λcl (t)σ+ − λcl (t)σ−−iγσ0
i∂tσ0 − εασ3 − λcl (t)σ+ − λcl (t)σ−+iγσ0 −λq(t)σ+ + λq(t)σ− + 2iγ(Fbσ↑ + Faσ↓)
«
For steady-state solution ψ(r, t) = ψ0e−iµS t we can invert and get usingE2 = (ε− 1
2µS)2 + g2|ψ|2
GK (ν) = −2iγ
[(ν+iγ)2 − E2][(ν−iγ)2 − E2]
ׄ
ν + ε+iγ gψ0gψ0 ν − ε+ iγ
«„FB(ν) 0
0 FA(ν)
«„ν + ε−iγ gψ0
gψ0 ν − ε−iγ
«,
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 9 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Polariton model and Keldysh field theoryMean-field equationsConnections of mean-field equations to other limitsPhase transition and fluctuationsFinite size effects: single vs many modesSuperfluidity
Keldysh non-equilibrium field theoryThe final form
S = −iXα
TrlnG−1α,α +
Z ∞
−∞dtdt ′
Xpψp(t)
„0 i∂t − ωp−iκ
i∂t − ωp + iκ 2iκFΨ(t − t ′)
«ψp(t ′)
Introducing the abbreviations λcl =P
pgp,α√
2ψp,cl and λq =
Pp
gp,α√2ψp,q
G−1(t , t ′) =„λq(t)σ+ + λq(t)σ− ∂tσ0 − εασ3 − λcl (t)σ+ − λcl (t)σ−−iγσ0
i∂tσ0 − εασ3 − λcl (t)σ+ − λcl (t)σ−+iγσ0 −λq(t)σ+ + λq(t)σ− + 2iγ(Fbσ↑ + Faσ↓)
«
For steady-state solution ψ(r, t) = ψ0e−iµS t we can invert and get usingE2 = (ε− 1
2µS)2 + g2|ψ|2
GK (ν) = −2iγ
[(ν+iγ)2 − E2][(ν−iγ)2 − E2]
ׄ
ν + ε+iγ gψ0gψ0 ν − ε+ iγ
«„FB(ν) 0
0 FA(ν)
«„ν + ε−iγ gψ0
gψ0 ν − ε−iγ
«,
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 9 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Polariton model and Keldysh field theoryMean-field equationsConnections of mean-field equations to other limitsPhase transition and fluctuationsFinite size effects: single vs many modesSuperfluidity
Saddle-point of non-equilibrium action: mean-fieldEquilibrium BEC: described by Gross-Pitaevskii equation (mean-field equation for thecondensate)
i∂t +∇2
2m− V (r)
!ψ = U|ψ|2ψ
Non-equilibrium generalisation of Gross-Pitaevskii in BEC and gap equation in BCSregimes (note: now it is complex)
(i∂t − ωc + iκ)ψ = χ(ψ)ψ
For steady-state solution ψ(r, t) = ψe−iµS t
Susceptibility:
χ(ψ, µS) = −g2γX
excitons
Zdν2π
(Fa + Fb)ν + (Fb − Fa)(iγ + εα − 12µS)
[(ν − Eα)2 + γ2][(ν + Eα)2 + γ2]
E2α = (εα − 1
2µS)2 + g2|ψ|2
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 10 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Polariton model and Keldysh field theoryMean-field equationsConnections of mean-field equations to other limitsPhase transition and fluctuationsFinite size effects: single vs many modesSuperfluidity
Saddle-point of non-equilibrium action: mean-fieldEquilibrium BEC: described by Gross-Pitaevskii equation (mean-field equation for thecondensate)
i∂t +∇2
2m− V (r)
!ψ = U|ψ|2ψ
Non-equilibrium generalisation of Gross-Pitaevskii in BEC and gap equation in BCSregimes (note: now it is complex)
(i∂t − ωc + iκ)ψ = χ(ψ)ψ
For steady-state solution ψ(r, t) = ψe−iµS t
Susceptibility:
χ(ψ, µS) = −g2γX
excitons
Zdν2π
(Fa + Fb)ν + (Fb − Fa)(iγ + εα − 12µS)
[(ν − Eα)2 + γ2][(ν + Eα)2 + γ2]
E2α = (εα − 1
2µS)2 + g2|ψ|2
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 10 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Polariton model and Keldysh field theoryMean-field equationsConnections of mean-field equations to other limitsPhase transition and fluctuationsFinite size effects: single vs many modesSuperfluidity
Saddle-point of non-equilibrium action: mean-fieldEquilibrium BEC: described by Gross-Pitaevskii equation (mean-field equation for thecondensate)
i∂t +∇2
2m− V (r)
!ψ = U|ψ|2ψ
Non-equilibrium generalisation of Gross-Pitaevskii in BEC and gap equation in BCSregimes (note: now it is complex)
(i∂t − ωc + iκ)ψ = χ(ψ)ψ
For steady-state solution ψ(r, t) = ψe−iµS t
Susceptibility:
χ(ψ, µS) = −g2γX
excitons
Zdν2π
(Fa + Fb)ν + (Fb − Fa)(iγ + εα − 12µS)
[(ν − Eα)2 + γ2][(ν + Eα)2 + γ2]
E2α = (εα − 1
2µS)2 + g2|ψ|2
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 10 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Polariton model and Keldysh field theoryMean-field equationsConnections of mean-field equations to other limitsPhase transition and fluctuationsFinite size effects: single vs many modesSuperfluidity
Limits of mean-field equations
Mean-field equations
µs − ωc + iκ = −g2γX
excitons
Zdν2π
(Fa + Fb)ν + (Fb − Fa)(iγ + εα − 12µS)
[(ν − Eα)2 + γ2][(ν + Eα)2 + γ2]
Large temperature T :
laser limit, only imaginary part, gain balances loss
κ = −g2γX
excitons
Fb − Fa
4E2α + 4γ2
→g2
2γ×Inversion
Equilibrium limit κ, γ → 0: well known gap equation
ωc − µs =X
excitons
g2
2Eαtanh
„βEα
2
«
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 11 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Polariton model and Keldysh field theoryMean-field equationsConnections of mean-field equations to other limitsPhase transition and fluctuationsFinite size effects: single vs many modesSuperfluidity
Limits of mean-field equations
Mean-field equations
µs − ωc + iκ = −g2γX
excitons
Zdν2π
(Fa + Fb)ν + (Fb − Fa)(iγ + εα − 12µS)
[(ν − Eα)2 + γ2][(ν + Eα)2 + γ2]
Large temperature T :
laser limit, only imaginary part, gain balances loss
κ = −g2γX
excitons
Fb − Fa
4E2α + 4γ2
→g2
2γ×Inversion
Equilibrium limit κ, γ → 0: well known gap equation
ωc − µs =X
excitons
g2
2Eαtanh
„βEα
2
«
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 11 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Polariton model and Keldysh field theoryMean-field equationsConnections of mean-field equations to other limitsPhase transition and fluctuationsFinite size effects: single vs many modesSuperfluidity
Limits of mean-field equations
Mean-field equations
µs − ωc + iκ = −g2γX
excitons
Zdν2π
(Fa + Fb)ν + (Fb − Fa)(iγ + εα − 12µS)
[(ν − Eα)2 + γ2][(ν + Eα)2 + γ2]
Large temperature T : laser limit, only imaginary part, gain balances loss
κ = −g2γX
excitons
Fb − Fa
4E2α + 4γ2
→g2
2γ×Inversion
Equilibrium limit κ, γ → 0: well known gap equation
ωc − µs =X
excitons
g2
2Eαtanh
„βEα
2
«
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 11 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Polariton model and Keldysh field theoryMean-field equationsConnections of mean-field equations to other limitsPhase transition and fluctuationsFinite size effects: single vs many modesSuperfluidity
Limits of mean-field equations
Mean-field equations
µs − ωc + iκ = −g2γX
excitons
Zdν2π
(Fa + Fb)ν + (Fb − Fa)(iγ + εα − 12µS)
[(ν − Eα)2 + γ2][(ν + Eα)2 + γ2]
Large temperature T : laser limit, only imaginary part, gain balances loss
κ = −g2γX
excitons
Fb − Fa
4E2α + 4γ2
→g2
2γ×Inversion
Equilibrium limit κ, γ → 0: well known gap equation
ωc − µs =X
excitons
g2
2Eαtanh
„βEα
2
«
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 11 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Polariton model and Keldysh field theoryMean-field equationsConnections of mean-field equations to other limitsPhase transition and fluctuationsFinite size effects: single vs many modesSuperfluidity
Limits of mean-field equations
Mean-field equations
µs − ωc + iκ = −g2γX
excitons
Zdν2π
(Fa + Fb)ν + (Fb − Fa)(iγ + εα − 12µS)
[(ν − Eα)2 + γ2][(ν + Eα)2 + γ2]
Large temperature T : laser limit, only imaginary part, gain balances loss
κ = −g2γX
excitons
Fb − Fa
4E2α + 4γ2
→g2
2γ×Inversion
Equilibrium limit κ, γ → 0:
well known gap equation
ωc − µs =X
excitons
g2
2Eαtanh
„βEα
2
«
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 11 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Polariton model and Keldysh field theoryMean-field equationsConnections of mean-field equations to other limitsPhase transition and fluctuationsFinite size effects: single vs many modesSuperfluidity
Limits of mean-field equations
Mean-field equations
µs − ωc + iκ = −g2γX
excitons
Zdν2π
(Fa + Fb)ν + (Fb − Fa)(iγ + εα − 12µS)
[(ν − Eα)2 + γ2][(ν + Eα)2 + γ2]
Large temperature T : laser limit, only imaginary part, gain balances loss
κ = −g2γX
excitons
Fb − Fa
4E2α + 4γ2
→g2
2γ×Inversion
Equilibrium limit κ, γ → 0: well known gap equation
ωc − µs =X
excitons
g2
2Eαtanh
„βEα
2
«
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 11 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Polariton model and Keldysh field theoryMean-field equationsConnections of mean-field equations to other limitsPhase transition and fluctuationsFinite size effects: single vs many modesSuperfluidity
Limits of mean-field equationsMean-field equation
(i∂t − ωc + iκ)ψ = χ(ψ, µs)ψ
Local density approximation:"i∂t + iκ−
V (r)−
∇2
2m
!#ψ(r) = χ(ψ(r , t))ψ(r , t)
At low density the nonlinear, complex susceptibility can be simplified: ComplexGross-Pitaevskii equation
i∂tψ =
"−∇2
2m+ V (r) + U|ψ|2 + i
“γeff − κ− Γ|ψ|2
”#ψ
Vortex lattices, Keeling & Berloff PRL 2008, Disordered pined vortices, Carusotto et al., 2008Persistent currents and quantised vortices, D. Sanvitto, F. M. Marchetti, M. H. Szymanska et al, Nature Physics, July 2010
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 12 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Polariton model and Keldysh field theoryMean-field equationsConnections of mean-field equations to other limitsPhase transition and fluctuationsFinite size effects: single vs many modesSuperfluidity
Limits of mean-field equationsMean-field equation
(i∂t − ωc + iκ)ψ = χ(ψ, µs)ψ
Local density approximation:"i∂t + iκ−
V (r)−
∇2
2m
!#ψ(r) = χ(ψ(r , t))ψ(r , t)
At low density the nonlinear, complex susceptibility can be simplified: ComplexGross-Pitaevskii equation
i∂tψ =
"−∇2
2m+ V (r) + U|ψ|2 + i
“γeff − κ− Γ|ψ|2
”#ψ
Vortex lattices, Keeling & Berloff PRL 2008, Disordered pined vortices, Carusotto et al., 2008Persistent currents and quantised vortices, D. Sanvitto, F. M. Marchetti, M. H. Szymanska et al, Nature Physics, July 2010
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 12 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Polariton model and Keldysh field theoryMean-field equationsConnections of mean-field equations to other limitsPhase transition and fluctuationsFinite size effects: single vs many modesSuperfluidity
Limits of mean-field equationsMean-field equation
(i∂t − ωc + iκ)ψ = χ(ψ, µs)ψ
Local density approximation:"i∂t + iκ−
V (r)−
∇2
2m
!#ψ(r) = χ(ψ(r , t))ψ(r , t)
At low density the nonlinear, complex susceptibility can be simplified: ComplexGross-Pitaevskii equation
i∂tψ =
"−∇2
2m+ V (r) + U|ψ|2 + i
“γeff − κ− Γ|ψ|2
”#ψ
Vortex lattices, Keeling & Berloff PRL 2008, Disordered pined vortices, Carusotto et al., 2008Persistent currents and quantised vortices, D. Sanvitto, F. M. Marchetti, M. H. Szymanska et al, Nature Physics, July 2010
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 12 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Polariton model and Keldysh field theoryMean-field equationsConnections of mean-field equations to other limitsPhase transition and fluctuationsFinite size effects: single vs many modesSuperfluidity
Fluctuations: phase transition
Keldysh approach:GS = GR − GA = −i
fihψ†, ψ
i−
flGK = −i
fihψ†, ψ
i+
fl= (2n(ω) + 1)GS
L =Dψ†ψ
E= i(n(ω) + 1)GS
GS =2Im[GR−1]
Im[GR−1]2 + Re[GR−1]2
n(ω) =−GK−1
4Im[GR−1]2
Re[GR−1(ω∗p)]= 0 ω∗p normal modes
Im[GR−1(µeff)]= 0 n(µeff) diverges
At transition gap Equation is:
G−1R (ω = µS ,p = 0) = 0
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 13 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Polariton model and Keldysh field theoryMean-field equationsConnections of mean-field equations to other limitsPhase transition and fluctuationsFinite size effects: single vs many modesSuperfluidity
Fluctuations: phase transition
Keldysh approach:GS = GR − GA = −i
fihψ†, ψ
i−
flGK = −i
fihψ†, ψ
i+
fl= (2n(ω) + 1)GS
L =Dψ†ψ
E= i(n(ω) + 1)GS
GS =2Im[GR−1]
Im[GR−1]2 + Re[GR−1]2
n(ω) =−GK−1
4Im[GR−1]2
Re[GR−1(ω∗p)]= 0 ω∗p normal modes
Im[GR−1(µeff)]= 0 n(µeff) diverges
At transition gap Equation is:
G−1R (ω = µS ,p = 0) = 0
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 13 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Polariton model and Keldysh field theoryMean-field equationsConnections of mean-field equations to other limitsPhase transition and fluctuationsFinite size effects: single vs many modesSuperfluidity
Fluctuations: phase transition
Keldysh approach:GS = GR − GA = −i
fihψ†, ψ
i−
flGK = −i
fihψ†, ψ
i+
fl= (2n(ω) + 1)GS
L =Dψ†ψ
E= i(n(ω) + 1)GS
GS =2Im[GR−1]
Im[GR−1]2 + Re[GR−1]2
n(ω) =−GK−1
4Im[GR−1]2
Re[GR−1(ω∗p)]= 0 ω∗p normal modes
Im[GR−1(µeff)]= 0 n(µeff) diverges
At transition gap Equation is:
G−1R (ω = µS ,p = 0) = 0
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 13 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Polariton model and Keldysh field theoryMean-field equationsConnections of mean-field equations to other limitsPhase transition and fluctuationsFinite size effects: single vs many modesSuperfluidity
Fluctuations: phase transition
Keldysh approach:GS = GR − GA = −i
fihψ†, ψ
i−
flGK = −i
fihψ†, ψ
i+
fl= (2n(ω) + 1)GS
L =Dψ†ψ
E= i(n(ω) + 1)GS
GS =2Im[GR−1]
Im[GR−1]2 + Re[GR−1]2
n(ω) =−GK−1
4Im[GR−1]2
Re[GR−1(ω∗p)]= 0 ω∗p normal modes
Im[GR−1(µeff)]= 0 n(µeff) diverges
At transition gap Equation is:
G−1R (ω = µS ,p = 0) = 0
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 13 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Polariton model and Keldysh field theoryMean-field equationsConnections of mean-field equations to other limitsPhase transition and fluctuationsFinite size effects: single vs many modesSuperfluidity
Fluctuations: phase transition
Keldysh approach:GS = GR − GA = −i
fihψ†, ψ
i−
flGK = −i
fihψ†, ψ
i+
fl= (2n(ω) + 1)GS
L =Dψ†ψ
E= i(n(ω) + 1)GS
GS =2Im[GR−1]
Im[GR−1]2 + Re[GR−1]2
n(ω) =−GK−1
4Im[GR−1]2
Re[GR−1(ω∗p)]= 0 ω∗p normal modes
Im[GR−1(µeff)]= 0 n(µeff) diverges
Equilibrium BECn(ω) = nB(ω) i.e n(µ) diverges
At transition gap Equation is:
G−1R (ω = µS ,p = 0) = 0
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 13 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Polariton model and Keldysh field theoryMean-field equationsConnections of mean-field equations to other limitsPhase transition and fluctuationsFinite size effects: single vs many modesSuperfluidity
Fluctuations: phase transition
Keldysh approach:GS = GR − GA = −i
fihψ†, ψ
i−
flGK = −i
fihψ†, ψ
i+
fl= (2n(ω) + 1)GS
L =Dψ†ψ
E= i(n(ω) + 1)GS
GS =2Im[GR−1]
Im[GR−1]2 + Re[GR−1]2
n(ω) =−GK−1
4Im[GR−1]2
Re[GR−1(ω∗p)]= 0 ω∗p normal modes
Im[GR−1(µeff)]= 0 n(µeff) diverges
At transition gap Equation is:
G−1R (ω = µS ,p = 0) = 0
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 13 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Polariton model and Keldysh field theoryMean-field equationsConnections of mean-field equations to other limitsPhase transition and fluctuationsFinite size effects: single vs many modesSuperfluidity
Fluctuations: phase transition
Keldysh approach:GS = GR − GA = −i
fihψ†, ψ
i−
flGK = −i
fihψ†, ψ
i+
fl= (2n(ω) + 1)GS
L =Dψ†ψ
E= i(n(ω) + 1)GS
GS =2Im[GR−1]
Im[GR−1]2 + Re[GR−1]2
n(ω) =−GK−1
4Im[GR−1]2
Re[GR−1(ω∗p)]= 0 ω∗p normal modes
Im[GR−1(µeff)]= 0 n(µeff) diverges
At transition gap Equation is:
G−1R (ω = µS ,p = 0) = 0
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 13 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Polariton model and Keldysh field theoryMean-field equationsConnections of mean-field equations to other limitsPhase transition and fluctuationsFinite size effects: single vs many modesSuperfluidity
Fluctuations: phase transition
Keldysh approach:GS = GR − GA = −i
fihψ†, ψ
i−
flGK = −i
fihψ†, ψ
i+
fl= (2n(ω) + 1)GS
L =Dψ†ψ
E= i(n(ω) + 1)GS
GS =2Im[GR−1]
Im[GR−1]2 + Re[GR−1]2
n(ω) =−GK−1
4Im[GR−1]2
Re[GR−1(ω∗p)]= 0 ω∗p normal modes
Im[GR−1(µeff)]= 0 n(µeff) diverges[Szymanska et al., PRL ’06; PRB ’07]
At transition gap Equation is:
G−1R (ω = µS ,p = 0) = 0
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 13 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Polariton model and Keldysh field theoryMean-field equationsConnections of mean-field equations to other limitsPhase transition and fluctuationsFinite size effects: single vs many modesSuperfluidity
Fluctuations: phase transition
Keldysh approach:GS = GR − GA = −i
fihψ†, ψ
i−
flGK = −i
fihψ†, ψ
i+
fl= (2n(ω) + 1)GS
L =Dψ†ψ
E= i(n(ω) + 1)GS
GS =2Im[GR−1]
Im[GR−1]2 + Re[GR−1]2
n(ω) =−GK−1
4Im[GR−1]2
Re[GR−1(ω∗p)]= 0 ω∗p normal modes
Im[GR−1(µeff)]= 0 n(µeff) diverges[Szymanska et al., PRL ’06; PRB ’07]
At transition gap Equation is:
G−1R (ω = µS ,p = 0) = 0
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 13 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Polariton model and Keldysh field theoryMean-field equationsConnections of mean-field equations to other limitsPhase transition and fluctuationsFinite size effects: single vs many modesSuperfluidity
Laser Limit
Maxwell-Bloch equations
∂tψ = −iω0ψ − κψ + gP,
∂t P = −2iεP − ηP + gψN
∂t N = λ(N0 − N)− 2g(ψ∗P + P∗ψ),
where P = −in〈a†b〉,N = n〈b†b − a†a〉.
Green’s function
ψ = iDR(ω)Fe−iωt
thus
[DR ]−1 = ω − ω0 + iκ+g2N0
ω − 2ε+ i2γ. Keeling et al. arXiv:1001.3338 (2010)
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 14 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Polariton model and Keldysh field theoryMean-field equationsConnections of mean-field equations to other limitsPhase transition and fluctuationsFinite size effects: single vs many modesSuperfluidity
Laser Limit
Polariton BEC Laser
Keeling et al. arXiv:1001.3338 (2010)
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 15 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Polariton model and Keldysh field theoryMean-field equationsConnections of mean-field equations to other limitsPhase transition and fluctuationsFinite size effects: single vs many modesSuperfluidity
Fluctuations: collective modes and decay of correlations
When condensed
DethG−1
R (ω,p)i
= ω2 − c2p2
(ω+ix)2+x2 − c2p2
Poles:ω∗ = c|p|
−ix ±q
c2p2−x2
Landau criterion? vc = minpω(p)
p[also using different model Wouters & Carusotto PRL’07]
Correlations (in 2D):˙ψ†(r, t)ψ(0, r ′)
¸' |ψ|2 exp
ˆ−Dφφ(t , r , r ′)
˜Dψ†(r, t)ψ(0, 0)
E' |ψ|2 exp
"−η(
ln(r/ξ) r →∞, t ' 012 ln(c2t/xξ2) r ' 0, t →∞
#η(pump,decay,density) and not η(T,density) as in equilibrium
[Szymanska et al., PRL ’06; PRB ’07]
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 16 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Polariton model and Keldysh field theoryMean-field equationsConnections of mean-field equations to other limitsPhase transition and fluctuationsFinite size effects: single vs many modesSuperfluidity
Fluctuations: collective modes and decay of correlations
When condensed
DethG−1
R (ω,p)i
=
ω2 − c2p2
(ω+ix)2+x2 − c2p2
Poles:ω∗ =
c|p|
−ix ±q
c2p2−x2
Landau criterion? vc = minpω(p)
p[also using different model Wouters & Carusotto PRL’07]
Correlations (in 2D):˙ψ†(r, t)ψ(0, r ′)
¸' |ψ|2 exp
ˆ−Dφφ(t , r , r ′)
˜Dψ†(r, t)ψ(0, 0)
E' |ψ|2 exp
"−η(
ln(r/ξ) r →∞, t ' 012 ln(c2t/xξ2) r ' 0, t →∞
#η(pump,decay,density) and not η(T,density) as in equilibrium
[Szymanska et al., PRL ’06; PRB ’07]
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 16 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Polariton model and Keldysh field theoryMean-field equationsConnections of mean-field equations to other limitsPhase transition and fluctuationsFinite size effects: single vs many modesSuperfluidity
Fluctuations: collective modes and decay of correlations
When condensed
DethG−1
R (ω,p)i
=
ω2 − c2p2
(ω+ix)2+x2 − c2p2
Poles:ω∗ =
c|p|
−ix ±q
c2p2−x2
Landau criterion? vc = minpω(p)
p[also using different model Wouters & Carusotto PRL’07]
Correlations (in 2D):˙ψ†(r, t)ψ(0, r ′)
¸' |ψ|2 exp
ˆ−Dφφ(t , r , r ′)
˜
Dψ†(r, t)ψ(0, 0)
E' |ψ|2 exp
"−η(
ln(r/ξ) r →∞, t ' 012 ln(c2t/xξ2) r ' 0, t →∞
#η(pump,decay,density) and not η(T,density) as in equilibrium
[Szymanska et al., PRL ’06; PRB ’07]
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 16 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Polariton model and Keldysh field theoryMean-field equationsConnections of mean-field equations to other limitsPhase transition and fluctuationsFinite size effects: single vs many modesSuperfluidity
Fluctuations: collective modes and decay of correlations
When condensed
DethG−1
R (ω,p)i
=
ω2 − c2p2
(ω+ix)2+x2 − c2p2
Poles:ω∗ =
c|p|
−ix ±q
c2p2−x2
Landau criterion? vc = minpω(p)
p[also using different model Wouters & Carusotto PRL’07]
Correlations (in 2D):˙ψ†(r, t)ψ(0, r ′)
¸' |ψ|2 exp
ˆ−Dφφ(t , r , r ′)
˜Dψ†(r, t)ψ(0, 0)
E' |ψ|2 exp
"−η(
ln(r/ξ) r →∞, t ' 012 ln(c2t/xξ2) r ' 0, t →∞
#η(pump,decay,density) and not η(T,density) as in equilibrium
[Szymanska et al., PRL ’06; PRB ’07]
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 16 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Polariton model and Keldysh field theoryMean-field equationsConnections of mean-field equations to other limitsPhase transition and fluctuationsFinite size effects: single vs many modesSuperfluidity
Decay of spatial correlations
g1(∆x , t = 0) = nS exp−Z
kdk2π
Zdω2π
[1− J0(k∆x)] iD<φφ(ω, k)
ff∆x→∞'
„∆xl0
«−ap
D<φφ = DKφφ − DR
φφ + DAφφ iDφφ =
i8nS
(1 − 1)D„
1−1
«
i∂tψ =
"−
~2∇2
2m+ U|ψ|2 + i
“γnet − Γ|ψ|2
”#ψ
DR =1
ω2 + 2iγnetω − ξ2k
„µ+ εk + ω + iγnet −µ+ iγnet
−µ− iγnet µ+ εk − ω − iγnet
«Thermal distribution
[D−1]K = 2iγnet coth(βω/2)
„1 00 −1
«→ aP = mkBT/2πns~2 as for the closed system
Non-equilibrium flat distribution
[D−1]K = 2iζ„
1 00 1
«→ aP ∝ ζ/nS
G. Rompos, Y. Yamamoto et al. submitted
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 17 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Polariton model and Keldysh field theoryMean-field equationsConnections of mean-field equations to other limitsPhase transition and fluctuationsFinite size effects: single vs many modesSuperfluidity
Decay of spatial correlations
g1(∆x , t = 0) = nS exp−Z
kdk2π
Zdω2π
[1− J0(k∆x)] iD<φφ(ω, k)
ff∆x→∞'
„∆xl0
«−ap
D<φφ = DKφφ − DR
φφ + DAφφ iDφφ =
i8nS
(1 − 1)D„
1−1
«
i∂tψ =
"−
~2∇2
2m+ U|ψ|2 + i
“γnet − Γ|ψ|2
”#ψ
DR =1
ω2 + 2iγnetω − ξ2k
„µ+ εk + ω + iγnet −µ+ iγnet
−µ− iγnet µ+ εk − ω − iγnet
«Thermal distribution
[D−1]K = 2iγnet coth(βω/2)
„1 00 −1
«→ aP = mkBT/2πns~2 as for the closed system
Non-equilibrium flat distribution
[D−1]K = 2iζ„
1 00 1
«→ aP ∝ ζ/nS
G. Rompos, Y. Yamamoto et al. submitted
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 17 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Polariton model and Keldysh field theoryMean-field equationsConnections of mean-field equations to other limitsPhase transition and fluctuationsFinite size effects: single vs many modesSuperfluidity
Decay of spatial correlations
g1(∆x , t = 0) = nS exp−Z
kdk2π
Zdω2π
[1− J0(k∆x)] iD<φφ(ω, k)
ff∆x→∞'
„∆xl0
«−ap
D<φφ = DKφφ − DR
φφ + DAφφ iDφφ =
i8nS
(1 − 1)D„
1−1
«
i∂tψ =
"−
~2∇2
2m+ U|ψ|2 + i
“γnet − Γ|ψ|2
”#ψ
DR =1
ω2 + 2iγnetω − ξ2k
„µ+ εk + ω + iγnet −µ+ iγnet
−µ− iγnet µ+ εk − ω − iγnet
«Thermal distribution
[D−1]K = 2iγnet coth(βω/2)
„1 00 −1
«→ aP = mkBT/2πns~2 as for the closed system
Non-equilibrium flat distribution
[D−1]K = 2iζ„
1 00 1
«→ aP ∝ ζ/nS
G. Rompos, Y. Yamamoto et al. submitted
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 17 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Polariton model and Keldysh field theoryMean-field equationsConnections of mean-field equations to other limitsPhase transition and fluctuationsFinite size effects: single vs many modesSuperfluidity
Decay of spatial correlations
g1(∆x , t = 0) = nS exp−Z
kdk2π
Zdω2π
[1− J0(k∆x)] iD<φφ(ω, k)
ff∆x→∞'
„∆xl0
«−ap
D<φφ = DKφφ − DR
φφ + DAφφ iDφφ =
i8nS
(1 − 1)D„
1−1
«
i∂tψ =
"−
~2∇2
2m+ U|ψ|2 + i
“γnet − Γ|ψ|2
”#ψ
DR =1
ω2 + 2iγnetω − ξ2k
„µ+ εk + ω + iγnet −µ+ iγnet
−µ− iγnet µ+ εk − ω − iγnet
«
Thermal distribution
[D−1]K = 2iγnet coth(βω/2)
„1 00 −1
«→ aP = mkBT/2πns~2 as for the closed system
Non-equilibrium flat distribution
[D−1]K = 2iζ„
1 00 1
«→ aP ∝ ζ/nS
G. Rompos, Y. Yamamoto et al. submitted
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 17 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Polariton model and Keldysh field theoryMean-field equationsConnections of mean-field equations to other limitsPhase transition and fluctuationsFinite size effects: single vs many modesSuperfluidity
Decay of spatial correlations
g1(∆x , t = 0) = nS exp−Z
kdk2π
Zdω2π
[1− J0(k∆x)] iD<φφ(ω, k)
ff∆x→∞'
„∆xl0
«−ap
D<φφ = DKφφ − DR
φφ + DAφφ iDφφ =
i8nS
(1 − 1)D„
1−1
«
i∂tψ =
"−
~2∇2
2m+ U|ψ|2 + i
“γnet − Γ|ψ|2
”#ψ
DR =1
ω2 + 2iγnetω − ξ2k
„µ+ εk + ω + iγnet −µ+ iγnet
−µ− iγnet µ+ εk − ω − iγnet
«Thermal distribution
[D−1]K = 2iγnet coth(βω/2)
„1 00 −1
«→ aP = mkBT/2πns~2 as for the closed system
Non-equilibrium flat distribution
[D−1]K = 2iζ„
1 00 1
«→ aP ∝ ζ/nS
G. Rompos, Y. Yamamoto et al. submitted
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 17 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Polariton model and Keldysh field theoryMean-field equationsConnections of mean-field equations to other limitsPhase transition and fluctuationsFinite size effects: single vs many modesSuperfluidity
Decay of spatial correlations
g1(∆x , t = 0) = nS exp−Z
kdk2π
Zdω2π
[1− J0(k∆x)] iD<φφ(ω, k)
ff∆x→∞'
„∆xl0
«−ap
D<φφ = DKφφ − DR
φφ + DAφφ iDφφ =
i8nS
(1 − 1)D„
1−1
«
i∂tψ =
"−
~2∇2
2m+ U|ψ|2 + i
“γnet − Γ|ψ|2
”#ψ
DR =1
ω2 + 2iγnetω − ξ2k
„µ+ εk + ω + iγnet −µ+ iγnet
−µ− iγnet µ+ εk − ω − iγnet
«Thermal distribution
[D−1]K = 2iγnet coth(βω/2)
„1 00 −1
«→ aP = mkBT/2πns~2 as for the closed system
Non-equilibrium flat distribution
[D−1]K = 2iζ„
1 00 1
«→ aP ∝ ζ/nS
G. Rompos, Y. Yamamoto et al. submittedMarzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 17 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Polariton model and Keldysh field theoryMean-field equationsConnections of mean-field equations to other limitsPhase transition and fluctuationsFinite size effects: single vs many modesSuperfluidity
Finite size effects: Single mode vs many modeDψ†(r , t)ψ(0, r ′)
E' |ψ0|2 exp
ˆ−Dφφ(t , r , r ′)
˜
Dφφ(t , r , r ′) from sum of phase modes. Study ct � r limit:
Dφφ(t , r , r) ∝nmaxX
n
Zdω2π
|ϕn(r)|2(1− eiωt )˛(ω + ix)2 + x2 − (n∆)2
˛2
∆ �p
x/t � Emax Dφφ ∼ 1 + ln(Emaxp
t/x)
px/t � ∆ � Emax Dφφ ∼
“πC2x
”“t
2x
”Keeling et al. arXiv:1001.3338 (2010)
Atom Laser
[Atom Laser, E.W. Hagley et al., Science (1999)]
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 18 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Polariton model and Keldysh field theoryMean-field equationsConnections of mean-field equations to other limitsPhase transition and fluctuationsFinite size effects: single vs many modesSuperfluidity
Finite size effects: Single mode vs many modeDψ†(r , t)ψ(0, r ′)
E' |ψ0|2 exp
ˆ−Dφφ(t , r , r ′)
˜Dφφ(t , r , r ′) from sum of phase modes. Study ct � r limit:
Dφφ(t , r , r) ∝nmaxX
n
Zdω2π
|ϕn(r)|2(1− eiωt )˛(ω + ix)2 + x2 − (n∆)2
˛2
∆ �p
x/t � Emax Dφφ ∼ 1 + ln(Emaxp
t/x)
px/t � ∆ � Emax Dφφ ∼
“πC2x
”“t
2x
”Keeling et al. arXiv:1001.3338 (2010)
Atom Laser
[Atom Laser, E.W. Hagley et al., Science (1999)]
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 18 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Polariton model and Keldysh field theoryMean-field equationsConnections of mean-field equations to other limitsPhase transition and fluctuationsFinite size effects: single vs many modesSuperfluidity
Finite size effects: Single mode vs many modeDψ†(r , t)ψ(0, r ′)
E' |ψ0|2 exp
ˆ−Dφφ(t , r , r ′)
˜Dφφ(t , r , r ′) from sum of phase modes. Study ct � r limit:
Dφφ(t , r , r) ∝nmaxX
n
Zdω2π
|ϕn(r)|2(1− eiωt )˛(ω + ix)2 + x2 − (n∆)2
˛2
∆ �p
x/t � Emax Dφφ ∼ 1 + ln(Emaxp
t/x)
px/t � ∆ � Emax Dφφ ∼
“πC2x
”“t
2x
”
Keeling et al. arXiv:1001.3338 (2010)
Atom Laser
[Atom Laser, E.W. Hagley et al., Science (1999)]
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 18 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Polariton model and Keldysh field theoryMean-field equationsConnections of mean-field equations to other limitsPhase transition and fluctuationsFinite size effects: single vs many modesSuperfluidity
Finite size effects: Single mode vs many modeDψ†(r , t)ψ(0, r ′)
E' |ψ0|2 exp
ˆ−Dφφ(t , r , r ′)
˜Dφφ(t , r , r ′) from sum of phase modes. Study ct � r limit:
Dφφ(t , r , r) ∝nmaxX
n
Zdω2π
|ϕn(r)|2(1− eiωt )˛(ω + ix)2 + x2 − (n∆)2
˛2
∆ �p
x/t � Emax Dφφ ∼ 1 + ln(Emaxp
t/x)
px/t � ∆ � Emax Dφφ ∼
“πC2x
”“t
2x
”Keeling et al. arXiv:1001.3338 (2010)
Atom Laser
[Atom Laser, E.W. Hagley et al., Science (1999)]
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 18 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Polariton model and Keldysh field theoryMean-field equationsConnections of mean-field equations to other limitsPhase transition and fluctuationsFinite size effects: single vs many modesSuperfluidity
Finite size effects: Single mode vs many modeDψ†(r , t)ψ(0, r ′)
E' |ψ0|2 exp
ˆ−Dφφ(t , r , r ′)
˜Dφφ(t , r , r ′) from sum of phase modes. Study ct � r limit:
Dφφ(t , r , r) ∝nmaxX
n
Zdω2π
|ϕn(r)|2(1− eiωt )˛(ω + ix)2 + x2 − (n∆)2
˛2
∆ �p
x/t � Emax Dφφ ∼ 1 + ln(Emaxp
t/x)
px/t � ∆ � Emax Dφφ ∼
“πC2x
”“t
2x
”Keeling et al. arXiv:1001.3338 (2010)
Atom Laser
[Atom Laser, E.W. Hagley et al., Science (1999)]
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 18 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Polariton model and Keldysh field theoryMean-field equationsConnections of mean-field equations to other limitsPhase transition and fluctuationsFinite size effects: single vs many modesSuperfluidity
Polariton superfluidity
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 19 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Polariton model and Keldysh field theoryMean-field equationsConnections of mean-field equations to other limitsPhase transition and fluctuationsFinite size effects: single vs many modesSuperfluidity
Conclusion
Unified description of condensates (BEC and BCS) and lasers
Bose-Einstein distribution not special - BEC possible for any “bosonic” distribution(i.e divergent at some frequency)
However, dissipation and driving changes spectrum of excitationsdifferent collective modes
affected correlations i.e spatial and temporal coherence
spontaneous vortices
changed superfluid properties - still under experimental investigation
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 20 / 20
OutlineNon-equilibrium condensation in dissipative environment
Perspectives
Polariton model and Keldysh field theoryMean-field equationsConnections of mean-field equations to other limitsPhase transition and fluctuationsFinite size effects: single vs many modesSuperfluidity
Conclusion
Unified description of condensates (BEC and BCS) and lasers
Bose-Einstein distribution not special - BEC possible for any “bosonic” distribution(i.e divergent at some frequency)
However, dissipation and driving changes spectrum of excitationsdifferent collective modes
affected correlations i.e spatial and temporal coherence
spontaneous vortices
changed superfluid properties - still under experimental investigation
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 20 / 20
Extra slides
Marzena Hanna Szymanska Dissipative Systems and Non-Equilibrium BEC 21 / 21