lecture 3: dissipative systems and non-equilibrium bose...

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

Perspectives

Acknowledgements

Theory

J. Keeling

P. B. Littlewood

F. M. Marchetti

Experiment

G. Roumpos

Y. Yamamoto

D. Sanvitto

L. Vina

Funding:

Perspectives

Macroscopic Quantum Coherence

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

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]

Perspectives

Experiments

2D vs finite size

Perspectives

Experiments

2D vs finite size

Perspectives

Experiments

2D vs finite size

Perspectives

Experiments

2D vs finite size

Perspectives

Experiments

Theoretically complex

extended size, internal structure, stronginteractions and the BCS-BEC crossover

2D vs finite size

Perspectives

Experiments

2D vs finite size

Perspectives

Experiments

2D vs finite size

Perspectives

Experiments

2D vs finite size

Perspectives

Experiments

2D vs finite size

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 +

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+

√γ“

a†αAk + b†αBk

”+h.c.

Method: Keldysh path integral techniques adopted for broken symmetry systemswith strong dissipation

Perspectives

Model: Hsys =X

pωpψ

†pψp

εα“

+Xα,p

H = Hsys

Hsys,bath 'Xp,k

√κψpΨ†k+

√γ“

a†αAk + b†αBk

”+h.c.

Perspectives

Model: Hsys =X

pωpψ

†pψp +

εα“

+Xα,p

H = Hsys

Hsys,bath 'Xp,k

√κψpΨ†k+

√γ“

a†αAk + b†αBk

”+h.c.

Perspectives

Model: Hsys =X

pωpψ

†pψp +

εα“

+Xα,p

H = Hsys

Hsys,bath 'Xp,k

√κψpΨ†k+

√γ“

a†αAk + b†αBk

”+h.c.

Perspectives

Model: Hsys =X

pωpψ

†pψp +

εα“

+Xα,p

H = Hsys + Hsys,bath + Hbath

Hsys,bath 'Xp,k

√κψpΨ†k+

√γ“

a†αAk + b†αBk

”+h.c.

Perspectives

Model: Hsys =X

pωpψ

†pψp +

εα“

+Xα,p

Hsys,bath 'Xp,k

√κψpΨ†k+

√γ“

a†αAk + b†αBk

”+h.c.

Perspectives

Model: Hsys =X

pωpψ

†pψp +

εα“

+Xα,p

Hsys,bath 'Xp,k

√κψpΨ†k+

√γ“

a†αAk + b†αBk

”+h.c.

Perspectives

Keldysh non-equilibrium field theory

Green’s functions on Keldysh contour (D<, D>, DT , DT ) for example

D>(t) = −iD

“ψ(t , b)ψ†(0, f )

It is convenient to move to ψcl/q = ψ(t,f )±ψ(t,b)√2

, then

D = −ifi

„ψcl (t)ψq(t)

«“ψ†cl (t) ψ†q(t)

”fl=

„DK DR

In practice we usually know D−1 but 0

ˆDA˜−1ˆ

DR˜−1 ˆD−1˜K

!„DK DR

„1 00 1

andˆD−1˜K = −

ˆDR˜−1 DK ˆDA˜−1 so DK = −DR ˆD−1˜K DA.

Perspectives

D>(t) = −iD

“ψ(t , b)ψ†(0, f )

, then

D = −ifi

„ψcl (t)ψq(t)

”fl=

„DK DR

ˆDA˜−1ˆ

DR˜−1 ˆD−1˜K

!„DK DR

„1 00 1

andˆD−1˜K = −

Perspectives

D>(t) = −iD

“ψ(t , b)ψ†(0, f )

, then

D = −ifi

„ψcl (t)ψq(t)

”fl=

„DK DR

ˆDA˜−1ˆ

DR˜−1 ˆD−1˜K

!„DK DR

„1 00 1

andˆD−1˜K = −

Perspectives

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]

D[φα, φα]Y

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 =

−∞dωX

pψp(ω)

„0 −ω − ωp − iκ

−ω − ωp + iκ 2iκFψ(−ω)

«ψp(−ω).

Perspectives

Z = NZ Y

pD[ψp, ψp]

D[φα, φα]Y

−∞dωX

pψp(ω)

„0 ω + ωp + dA(k,−ω)

«ψp(−ω).

Sψ + Sbathψ =

−∞dωX

pψp(ω)

„0 −ω − ωp − iκ

«ψp(−ω).

Perspectives

Z = NZ Y

pD[ψp, ψp]

D[φα, φα]Y

−∞dωX

pψp(ω)

„0 ω + ωp + dA(k,−ω)

«ψp(−ω).

Sψ + Sbathψ =

−∞dωX

pψp(ω)

„0 −ω − ωp − iκ

«ψp(−ω).

Perspectives

Z = NZ Y

pD[ψp, ψp]

D[φα, φα]Y

−∞dωX

pψp(ω)

„0 ω + ωp + dA(k,−ω)

«ψp(−ω).

Sψ + Sbathψ =

−∞dωX

pψp(ω)

„0 −ω − ωp − iκ

«ψp(−ω).

Perspectives

Keldysh non-equilibrium field theoryThe final form

S = −iXα

TrlnG−1α,α +

−∞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 =

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

Perspectives

Keldysh non-equilibrium field theoryThe final form

S = −iXα

TrlnG−1α,α +

−∞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 =

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

Perspectives

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

Perspectives

i∂t +∇2

2m− V (r)

!ψ = U|ψ|2ψ

Susceptibility:

excitons

Zdν2π

[(ν − Eα)2 + γ2][(ν + Eα)2 + γ2]

E2α = (εα − 1

2µS)2 + g2|ψ|2

Perspectives

i∂t +∇2

2m− V (r)

!ψ = U|ψ|2ψ

Susceptibility:

excitons

Zdν2π

[(ν − Eα)2 + γ2][(ν + Eα)2 + γ2]

E2α = (εα − 1

2µS)2 + g2|ψ|2

Perspectives

Limits of mean-field equations

Mean-field equations

µs − ωc + iκ = −g2γX

excitons

Zdν2π

[(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

2γ×Inversion

Equilibrium limit κ, γ → 0: well known gap equation

ωc − µs =X

excitons

2Eαtanh

„βEα

Perspectives

excitons

Zdν2π

[(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

2γ×Inversion

ωc − µs =X

excitons

2Eαtanh

„βEα

Perspectives

excitons

Zdν2π

[(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

2γ×Inversion

ωc − µs =X

excitons

2Eαtanh

„βEα

Perspectives

excitons

Zdν2π

[(ν − Eα)2 + γ2][(ν + Eα)2 + γ2]

κ = −g2γX

excitons

Fb − Fa

4E2α + 4γ2

2γ×Inversion

ωc − µs =X

excitons

2Eαtanh

„βEα

Perspectives

excitons

Zdν2π

[(ν − Eα)2 + γ2][(ν + Eα)2 + γ2]

κ = −g2γX

excitons

Fb − Fa

4E2α + 4γ2

2γ×Inversion

Equilibrium limit κ, γ → 0:

well known gap equation

ωc − µs =X

excitons

2Eαtanh

„βEα

Perspectives

excitons

Zdν2π

[(ν − Eα)2 + γ2][(ν + Eα)2 + γ2]

κ = −g2γX

excitons

Fb − Fa

4E2α + 4γ2

2γ×Inversion

ωc − µs =X

excitons

2Eαtanh

„βEα

Perspectives

Limits of mean-field equationsMean-field equation

(i∂t − ωc + iκ)ψ = χ(ψ, µs)ψ

Local density approximation:"i∂t + iκ−

V (r)−

!#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

Perspectives

V (r)−

!#ψ(r) = χ(ψ(r , t))ψ(r , t)

i∂tψ =

"−∇2

2m+ V (r) + U|ψ|2 + i

”#ψ

Perspectives

V (r)−

!#ψ(r) = χ(ψ(r , t))ψ(r , t)

i∂tψ =

"−∇2

2m+ V (r) + U|ψ|2 + i

”#ψ

Perspectives

Fluctuations: phase transition

Keldysh approach:GS = GR − GA = −i

fihψ†, ψ

flGK = −i

fihψ†, ψ

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

Perspectives

fihψ†, ψ

flGK = −i

fihψ†, ψ

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

G−1R (ω = µS ,p = 0) = 0

Perspectives

fihψ†, ψ

flGK = −i

fihψ†, ψ

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

G−1R (ω = µS ,p = 0) = 0

Perspectives

fihψ†, ψ

flGK = −i

fihψ†, ψ

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

G−1R (ω = µS ,p = 0) = 0

Perspectives

fihψ†, ψ

flGK = −i

fihψ†, ψ

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

Equilibrium BECn(ω) = nB(ω) i.e n(µ) diverges

G−1R (ω = µS ,p = 0) = 0

Perspectives

fihψ†, ψ

flGK = −i

fihψ†, ψ

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

G−1R (ω = µS ,p = 0) = 0

Perspectives

fihψ†, ψ

flGK = −i

fihψ†, ψ

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

G−1R (ω = µS ,p = 0) = 0

Perspectives

fihψ†, ψ

flGK = −i

fihψ†, ψ

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

Im[GR−1(µeff)]= 0 n(µeff) diverges[Szymanska et al., PRL ’06; PRB ’07]

G−1R (ω = µS ,p = 0) = 0

Perspectives

fihψ†, ψ

flGK = −i

fihψ†, ψ

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

Im[GR−1(µeff)]= 0 n(µeff) diverges[Szymanska et al., PRL ’06; PRB ’07]

G−1R (ω = µS ,p = 0) = 0

Perspectives

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

[DR ]−1 = ω − ω0 + iκ+g2N0

ω − 2ε+ i2γ. Keeling et al. arXiv:1001.3338 (2010)

Perspectives

Laser Limit

Polariton BEC Laser

Keeling et al. arXiv:1001.3338 (2010)

Perspectives

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]

Perspectives

When condensed

DethG−1

R (ω,p)i

ω2 − c2p2

(ω+ix)2+x2 − c2p2

Poles:ω∗ =

−ix ±q

c2p2−x2

¸' |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 →∞

Perspectives

When condensed

DethG−1

R (ω,p)i

ω2 − c2p2

(ω+ix)2+x2 − c2p2

Poles:ω∗ =

−ix ±q

c2p2−x2

¸' |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 →∞

Perspectives

When condensed

DethG−1

R (ω,p)i

ω2 − c2p2

(ω+ix)2+x2 − c2p2

Poles:ω∗ =

−ix ±q

c2p2−x2

¸' |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 →∞

Perspectives

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 =

(1 − 1)D„

i∂tψ =

~2∇2

2m+ U|ψ|2 + i

“γnet − Γ|ψ|2

”#ψ

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

Perspectives

g1(∆x , t = 0) = nS exp−Z

kdk2π

Zdω2π

[1− J0(k∆x)] iD<φφ(ω, k)

ff∆x→∞'

„∆xl0

«−ap

(1 − 1)D„

i∂tψ =

~2∇2

2m+ U|ψ|2 + i

”#ψ

„1 00 −1

[D−1]K = 2iζ„

1 00 1

«→ aP ∝ ζ/nS

Perspectives

g1(∆x , t = 0) = nS exp−Z

kdk2π

Zdω2π

[1− J0(k∆x)] iD<φφ(ω, k)

ff∆x→∞'

„∆xl0

«−ap

(1 − 1)D„

i∂tψ =

~2∇2

2m+ U|ψ|2 + i

”#ψ

„1 00 −1

[D−1]K = 2iζ„

1 00 1

«→ aP ∝ ζ/nS

Perspectives

g1(∆x , t = 0) = nS exp−Z

kdk2π

Zdω2π

[1− J0(k∆x)] iD<φφ(ω, k)

ff∆x→∞'

„∆xl0

«−ap

(1 − 1)D„

i∂tψ =

~2∇2

2m+ U|ψ|2 + i

”#ψ

Thermal distribution

„1 00 −1

[D−1]K = 2iζ„

1 00 1

«→ aP ∝ ζ/nS

Perspectives

g1(∆x , t = 0) = nS exp−Z

kdk2π

Zdω2π

[1− J0(k∆x)] iD<φφ(ω, k)

ff∆x→∞'

„∆xl0

«−ap

(1 − 1)D„

i∂tψ =

~2∇2

2m+ U|ψ|2 + i

”#ψ

„1 00 −1

[D−1]K = 2iζ„

1 00 1

«→ aP ∝ ζ/nS

Perspectives

g1(∆x , t = 0) = nS exp−Z

kdk2π

Zdω2π

[1− J0(k∆x)] iD<φφ(ω, k)

ff∆x→∞'

„∆xl0

«−ap

(1 − 1)D„

i∂tψ =

~2∇2

2m+ U|ψ|2 + i

”#ψ

„1 00 −1

[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

Perspectives

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

Zdω2π

|ϕn(r)|2(1− eiωt )˛(ω + ix)2 + x2 − (n∆)2

∆ �p

x/t � Emax Dφφ ∼ 1 + ln(Emaxp

px/t � ∆ � Emax Dφφ ∼

“πC2x

”“t

”Keeling et al. arXiv:1001.3338 (2010)

Atom Laser

[Atom Laser, E.W. Hagley et al., Science (1999)]

Perspectives

E' |ψ0|2 exp

ˆ−Dφφ(t , r , r ′)

˜Dφφ(t , r , r ′) from sum of phase modes. Study ct � r limit:

Zdω2π

|ϕn(r)|2(1− eiωt )˛(ω + ix)2 + x2 − (n∆)2

∆ �p

“πC2x

”“t

Atom Laser

Perspectives

E' |ψ0|2 exp

ˆ−Dφφ(t , r , r ′)

Zdω2π

|ϕn(r)|2(1− eiωt )˛(ω + ix)2 + x2 − (n∆)2

∆ �p

“πC2x

”“t

Keeling et al. arXiv:1001.3338 (2010)

Atom Laser

Perspectives

E' |ψ0|2 exp

ˆ−Dφφ(t , r , r ′)

Zdω2π

|ϕn(r)|2(1− eiωt )˛(ω + ix)2 + x2 − (n∆)2

∆ �p

“πC2x

”“t

Atom Laser

Perspectives

E' |ψ0|2 exp

ˆ−Dφφ(t , r , r ′)

Zdω2π

|ϕn(r)|2(1− eiωt )˛(ω + ix)2 + x2 − (n∆)2

∆ �p

“πC2x

”“t

Atom Laser

Perspectives

Polariton superfluidity

Perspectives

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

Perspectives

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

Extra slides

lecture 3: dissipative systems and non-equilibrium bose...

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