SUPA University of St Andrews 600 YEARS RETUNE, Heidelberg, June - - PowerPoint PPT Presentation

Condensation, superfluidity and lasing of coupled light-matter systems.

Jonathan Keeling

University of St Andrews

600

YEARS



SUPA

RETUNE, Heidelberg, June 2012

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 1 / 22

Outline

1

Introduction to polariton condensation Approaches to modelling

2

Pattern formation Non-equilibrium pattern formation Spontaneous vortex lattices

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 Condensation, superfluidity and lasing RETUNE, June 2012 2 / 22

Acknowledgements

People: Funding:

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 3 / 22

Microcavity polaritons — incoherent pumping

Quantum Wells Cavity

In−plane momentum Exciton Energy Photon Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 4 / 22

Non-equilibrium features in experiment

Flow from pumping spot

[Wertz et al. Nat. Phys. ’10]

|ψ(k)|2 = |ψ(−k)|2: Broken time-reversal symmetry.

[Krizhanovskii et al. PRB ’09]

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 5 / 22

Non-equilibrium approach: Steady state, and fluctuations

H = Hsys + Hsys,bath + Hbath, Hsys =

• k

ωkψkψ†

k +

• α

gα(φ†

αψk + H.c.)

+ Hex[φα, φ†

α]

In−plane momentum Exciton Photon Energy System Decay bath Pump bath Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 6 / 22

Non-equilibrium approach: Steady state, and fluctuations

H = Hsys + Hsys,bath + Hbath, Hsys =

• k

ωkψkψ†

k +

• α

gα(φ†

αψk + H.c.)

+ Hex[φα, φ†

α]

In−plane momentum Exciton Photon Energy System Decay bath Pump bath

Steady state, ψ(r, t) = ψ0e−iµSt.

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 6 / 22

Non-equilibrium approach: Steady state, and fluctuations

H = Hsys + Hsys,bath + Hbath, Hsys =

• k

ωkψkψ†

k +

• α

gα(φ†

αψk + H.c.)

+ Hex[φα, φ†

α]

In−plane momentum Exciton Photon Energy System Decay bath Pump bath

Steady state, ψ(r, t) = ψ0e−iµSt. Self-consistent equation: (i∂t − ω0 + iκ) ψ =

• α

gαφα

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 6 / 22

Non-equilibrium approach: Steady state, and fluctuations

H = Hsys + Hsys,bath + Hbath, Hsys =

• k

ωkψkψ†

k +

• α

gα(φ†

αψk + H.c.)

+ Hex[φα, φ†

α]

In−plane momentum Exciton Photon Energy System Decay bath Pump bath

Steady state, ψ(r, t) = ψ0e−iµSt. Self-consistent equation: (µs − ω0 + iκ) ψ0 = χ(ψ0, µs)ψ0

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 6 / 22

Non-equilibrium approach: Steady state, and fluctuations

H = Hsys + Hsys,bath + Hbath, Hsys =

• k

ωkψkψ†

k +

• α

gα(φ†

αψk + H.c.)

+ Hex[φα, φ†

α]

In−plane momentum Exciton Photon Energy System Decay bath Pump bath

Steady state, ψ(r, t) = ψ0e−iµSt. Self-consistent equation: (µs − ω0 + iκ) ψ0 = χ(ψ0, µs)ψ0 Fluctuations [DR − DA](t, t′) = −i

• ψ(t), ψ†(t′)
• −
• Jonathan Keeling

Condensation, superfluidity and lasing RETUNE, June 2012 6 / 22

Non-equilibrium approach: Steady state, and fluctuations

H = Hsys + Hsys,bath + Hbath, Hsys =

• k

ωkψkψ†

k +

• α

gα(φ†

αψk + H.c.)

+ Hex[φα, φ†

α]

In−plane momentum Exciton Photon Energy System Decay bath Pump bath

Steady state, ψ(r, t) = ψ0e−iµSt. Self-consistent equation: (µs − ω0 + iκ) ψ0 = χ(ψ0, µs)ψ0 Fluctuations [DR − DA](t, t′) = −i

• ψ(t), ψ†(t′)
• −
• [DR − DA](ω) = DoS(ω)

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 6 / 22

Non-equilibrium approach: Steady state, and fluctuations

H = Hsys + Hsys,bath + Hbath, Hsys =

• k

ωkψkψ†

k +

• α

gα(φ†

αψk + H.c.)

+ Hex[φα, φ†

α]

In−plane momentum Exciton Photon Energy System Decay bath Pump bath

Steady state, ψ(r, t) = ψ0e−iµSt. Self-consistent equation: (µs − ω0 + iκ) ψ0 = χ(ψ0, µs)ψ0 Fluctuations [DR − DA](t, t′) = −i

• ψ(t), ψ†(t′)
• −
• [DR − DA](ω) = DoS(ω)

DK(t, t′) = −i

• ψ(t), ψ†(t′)
• +
• Jonathan Keeling

Condensation, superfluidity and lasing RETUNE, June 2012 6 / 22

Non-equilibrium approach: Steady state, and fluctuations

H = Hsys + Hsys,bath + Hbath, Hsys =

• k

ωkψkψ†

k +

• α

gα(φ†

αψk + H.c.)

+ Hex[φα, φ†

α]

In−plane momentum Exciton Photon Energy System Decay bath Pump bath

Steady state, ψ(r, t) = ψ0e−iµSt. Self-consistent equation: (µs − ω0 + iκ) ψ0 = χ(ψ0, µs)ψ0 Fluctuations [DR − DA](t, t′) = −i

• ψ(t), ψ†(t′)
• −
• [DR − DA](ω) = DoS(ω)

DK(t, t′) = −i

• ψ(t), ψ†(t′)
• +
• DK(ω) = (2n(ω) + 1)DoS(ω)

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 6 / 22

Pattern formation:

1

Introduction to polariton condensation Approaches to modelling

2

Pattern formation Non-equilibrium pattern formation Spontaneous vortex lattices

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 Condensation, superfluidity and lasing RETUNE, June 2012 7 / 22

Complex Gross-Pitaevskii equation

Steady state equation: (µs − ω0 + iκ) ψ = χ(ψ, µs)ψ Local density limit:

See also [Wouters and Carusotto, PRL ’07]

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 8 / 22

Complex Gross-Pitaevskii equation

Steady state equation: (µs − ω0 + iκ) ψ = χ(ψ, µs)ψ Local density limit: Gross-Pitaevskii equation

• i∂t + iκ −
• V(r) − ∇2

2m

• ψ(r) = χ(ψ(r, t))ψ(r, t)

Nonlinear, complex susceptibility (incoherent pump)

See also [Wouters and Carusotto, PRL ’07]

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 8 / 22

Complex Gross-Pitaevskii equation

Steady state equation: (µs − ω0 + iκ) ψ = χ(ψ, µs)ψ Local density limit: Gross-Pitaevskii equation

• i∂t + iκ −
• V(r) − ∇2

2m

• ψ(r) = χ(ψ(r, t))ψ(r, t)

Nonlinear, complex susceptibility (incoherent pump) i∂tψ|nlin = U|ψ|2ψ

See also [Wouters and Carusotto, PRL ’07]

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 8 / 22

Complex Gross-Pitaevskii equation

Steady state equation: (µs − ω0 + iκ) ψ = χ(ψ, µs)ψ Local density limit: Gross-Pitaevskii equation

• i∂t + iκ −
• V(r) − ∇2

2m

• ψ(r) = χ(ψ(r, t))ψ(r, t)

Nonlinear, complex susceptibility (incoherent pump) i∂tψ|nlin = U|ψ|2ψ i∂tψ|loss = −iκψ i∂tψ|gain = iγeffψ

See also [Wouters and Carusotto, PRL ’07]

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 8 / 22

Complex Gross-Pitaevskii equation

Steady state equation: (µs − ω0 + iκ) ψ = χ(ψ, µs)ψ Local density limit: Gross-Pitaevskii equation

• i∂t + iκ −
• V(r) − ∇2

2m

• ψ(r) = χ(ψ(r, t))ψ(r, t)

Nonlinear, complex susceptibility (incoherent pump) i∂tψ|nlin = U|ψ|2ψ i∂tψ|loss = −iκψ i∂tψ|gain = iγeffψ − iΓ|ψ|2ψ

See also [Wouters and Carusotto, PRL ’07]

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 8 / 22

Complex Gross-Pitaevskii equation

Steady state equation: (µs − ω0 + iκ) ψ = χ(ψ, µs)ψ Local density limit: Gross-Pitaevskii equation

• i∂t + iκ −
• V(r) − ∇2

2m

• ψ(r) = χ(ψ(r, t))ψ(r, t)

Nonlinear, complex susceptibility (incoherent pump) i∂tψ|nlin = U|ψ|2ψ i∂tψ|loss = −iκψ i∂tψ|gain = iγeffψ − iΓ|ψ|2ψ i∂tψ =

• − ∇2

2m + V(r) + U|ψ|2 + i

• γeff − κ − Γ|ψ|2

ψ

See also [Wouters and Carusotto, PRL ’07]

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 8 / 22

Stability of Thomas-Fermi solution

i∂tψ =

• − ∇2

2m + mω2 2 r 2 + U|ψ|2 + i

• γeff − κ − Γ|ψ|2

ψ

3γnet 2Γ

2 4 6 8 Radius

Density

5 10 15 20 25 30

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 9 / 22

Stability of Thomas-Fermi solution

i∂tψ =

• − ∇2

2m + mω2 2 r 2 + U|ψ|2 + i

• γeff − κ − Γ|ψ|2

ψ

3γnet 2Γ Unstable growth

2 4 6 8 Radius

Density

5 10 15 20 25 30

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 9 / 22

Stability of Thomas-Fermi solution

i∂tψ =

• − ∇2

2m + mω2 2 r 2 + U|ψ|2 + i

• γeff − κ − Γ|ψ|2

ψ

3γnet 2Γ Unstable growth

2 4 6 8 Radius

Density

5 10 15 20 25 30

High m modes: δρn,m ≃ eimθr m . . . 1 2∂tρ+∇·(ρv) = (γnet − Γρ)ρ

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 9 / 22

Stability of Thomas-Fermi solution

i∂tψ =

• − ∇2

2m + mω2 2 r 2 + U|ψ|2 + i

• γeff − κ − Γ|ψ|2

ψ

3γnet 2Γ Stabilised

2 4 6 8 Radius

Density

5 10 15 20 25 30

High m modes: δρn,m ≃ eimθr m . . . 1 2∂tρ+∇·(ρv) = (γnetΘ(r0−r)−Γρ)ρ

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 9 / 22

Stability of Thomas-Fermi solution

i∂tψ =

• − ∇2

2m + mω2 2 r 2 + U|ψ|2 + i

• γeff − κ − Γ|ψ|2

ψ

3γnet 2Γ ????

2 4 6 8 Radius

Density

5 10 15 20 25 30

High m modes: δρn,m ≃ eimθr m . . . 1 2∂tρ+∇·(ρv) = (γnetΘ(r0−r)−Γρ)ρ

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 9 / 22

Time evolution:

t=0 t=2 t=22 t=30 t=35 t=40 t=45 t=56

[Keeling & Berloff PRL ’08]

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 10 / 22

Time evolution:

t=0 t=2 t=22 t=30 t=35 t=40 t=45 t=56

Why? i∂tψ = (µ − 2ΩLz)ψ Ω = ω, cancels trap.

[Keeling & Berloff PRL ’08]

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 10 / 22

Observability of vortex lattices

Not seen experimentally (yet?) Observation: Fast rotation Stability: Disorder, ellipticity Relaxation, thermalisation?

[Borgh et al. PRB ’12 in press]

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 11 / 22

Observability of vortex lattices

Not seen experimentally (yet?) Observation: Fast rotation

Spectrum:

Stability: Disorder, ellipticity Relaxation, thermalisation?

[Borgh et al. PRB ’12 in press]

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 11 / 22

Observability of vortex lattices

Not seen experimentally (yet?) Observation: Fast rotation

Spectrum:

Interference:

Stability: Disorder, ellipticity Relaxation, thermalisation?

[Borgh et al. PRB ’12 in press]

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 11 / 22

Observability of vortex lattices

Not seen experimentally (yet?) Observation: Fast rotation

Spectrum:

Interference:

Stability: Disorder, ellipticity Relaxation, thermalisation?

[Borgh et al. PRB ’12 in press]

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 11 / 22

Observability of vortex lattices

Not seen experimentally (yet?) Observation: Fast rotation

Spectrum:

Interference:

Stability: Disorder, ellipticity Relaxation, thermalisation?

[Borgh et al. PRB ’12 in press]

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 11 / 22

Superfluidity

1

Introduction to polariton condensation Approaches to modelling

2

Pattern formation Non-equilibrium pattern formation Spontaneous vortex lattices

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 Condensation, superfluidity and lasing RETUNE, June 2012 12 / 22

Fluctuations above transition

When condensed Det

• DR(ω, k)

−1 = ω2 − ξ2

k

With ξk ≃ ck Poles: ω∗ = ξk

frequency momentum Sound mode

Generic structure of Green’s function: [DR]−1 = ω + iγnet − ǫk − µ iγnet − µ −iγnet − µ −ω − iγnet − ǫk − µ

• Jonathan Keeling

Condensation, superfluidity and lasing RETUNE, June 2012 13 / 22

Fluctuations above transition

When condensed Det

• DR(ω, k)

−1 = (ω+iγnet)2+γ2

net−ξ2 k

With ξk ≃ ck Poles: ω∗ = − iγnet ±

• ξ2

k − γ2 net

frequency momentum Real Imaginary

Generic structure of Green’s function: [DR]−1 = ω + iγnet − ǫk − µ iγnet − µ −iγnet − µ −ω − iγnet − ǫk − µ

• Jonathan Keeling

Condensation, superfluidity and lasing RETUNE, June 2012 13 / 22

Fluctuations above transition

When condensed Det

• DR(ω, k)

−1 = (ω+iγnet)2+γ2

net−ξ2 k

With ξk ≃ ck Poles: ω∗ = − iγnet ±

• ξ2

k − γ2 net

frequency momentum Real Imaginary

Generic structure of Green’s function: [DR]−1 = ω + iγnet − ǫk − µ iγnet − µ −iγnet − µ −ω − iγnet − ǫk − µ

• Jonathan Keeling

Condensation, superfluidity and lasing RETUNE, June 2012 13 / 22

Aspects of superfluidity

Quantised vortices Landau critical velocity Metastable persistent flow Two-fluid hydrody- namics Local thermal equilib- rium Solitary waves Superfluid 4He/cold atom Bose-Einstein condensate

✧ ✧ ✧ ✧ ✧ ✧

Non-interacting Bose-Einstein condensate

✧ ✪ ✪ ✧ ✧ ✪

Classical irrotational fluid

✪ ✧ ✪ ✧ ✧ ✧

Incoherently pumped polariton condensates

✧ ✪ ? ? ✪ ?

Lagoudakis et al. Nat. Phys. ’08. Utsunomiya et al. Nat. Phys. ’08. Amo et al. Nature ’09; Nat. Phys. ’09

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 14 / 22

Aspects of superfluidity

Quantised vortices Landau critical velocity Metastable persistent flow Two-fluid hydrody- namics Local thermal equilib- rium Solitary waves Superfluid 4He/cold atom Bose-Einstein condensate

✧ ✧ ✧ ✧ ✧ ✧

Non-interacting Bose-Einstein condensate

✧ ✪ ✪ ✧ ✧ ✪

Classical irrotational fluid

✪ ✧ ✪ ✧ ✧ ✧

Incoherently pumped polariton condensates

✧ ✪ ? ? ✪ ?

Lagoudakis et al. Nat. Phys. ’08. Utsunomiya et al. Nat. Phys. ’08. Amo et al. Nature ’09; Nat. Phys. ’09

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 14 / 22

Superfluid density

Two-fluid hydrodynamics

1 1 ρ/ρtotal T/Tc ρnormal ρsuperfluid

ρs, ρn distinguished by slow rotation Experimentally, rotation: To calculate, transverse/longitudinal:

• Jonathan Keeling

Condensation, superfluidity and lasing RETUNE, June 2012 15 / 22

Superfluid density

Two-fluid hydrodynamics

1 1 ρ/ρtotal T/Tc ρnormal ρsuperfluid

ρs, ρn distinguished by slow rotation Experimentally, rotation: To calculate, transverse/longitudinal:

• Jonathan Keeling

Condensation, superfluidity and lasing RETUNE, June 2012 15 / 22

Superfluid density

Two-fluid hydrodynamics

1 1 ρ/ρtotal T/Tc ρnormal ρsuperfluid

ρs, ρn distinguished by slow rotation Experimentally, rotation: To calculate, transverse/longitudinal:

• Jonathan Keeling

Condensation, superfluidity and lasing RETUNE, June 2012 15 / 22

Superfluid density

Currrent: J = ρv = ψ†i∇ψ = |ψ|2∇φ Ji(q) = ψ†

k+q

2ki + qi 2m ψk

• Response function:

H → H −

• q

f(q) · Ji(q) Ji(q) = χij(q)fj(q) Vertex corrections essential for superfluid part.

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 16 / 22

Superfluid density

Currrent: J = ρv = ψ†i∇ψ = |ψ|2∇φ Ji(q) = ψ†

k+q

2ki + qi 2m ψk

• Response function:

H → H −

• q

f(q) · Ji(q) Ji(q) = χij(q)fj(q) Vertex corrections essential for superfluid part.

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 16 / 22

Superfluid density

Currrent: J = ρv = ψ†i∇ψ = |ψ|2∇φ Ji(q) = ψ†

k+q

2ki + qi 2m ψk

• Response function:

H → H −

• q

f(q) · Ji(q) Ji(q) = χij(q)fj(q) χij(ω = 0, q → 0) = [Ji(q), Jj(−q)] = ρS m qiqj q2 + ρN m δij Vertex corrections essential for superfluid part.

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 16 / 22

Superfluid density

Currrent: J = ρv = ψ†i∇ψ = |ψ|2∇φ Ji(q) = ψ†

k+q

2ki + qi 2m ψk

• Response function:

H → H −

• q

f(q) · Ji(q) Ji(q) = χij(q)fj(q) χij(ω = 0, q → 0) = [Ji(q), Jj(−q)] = ρS m qiqj q2 + ρN m δij Vertex corrections essential for superfluid part.

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 16 / 22

Non-equilibrium superfluid response

Superfluid response exists because: = iψ0qi 2m (1, −1) DR(q, ω = 0)

• 1

−1 iψ0qj 2m DR(ω = 0) ∝ 1/ǫq despite pumping/decay — superfluid response exists. Normal density: ρN =

• ddkǫk

dω 2π Tr

• σzDKσz(DR + DA)
• Is affected by pump/decay:

Does not vanish at T → 0.

[JK PRL ’11]

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 17 / 22

Non-equilibrium superfluid response

Superfluid response exists because: = iψ0qi 2m (1, −1) DR(q, ω = 0)

• 1

−1 iψ0qj 2m DR(ω = 0) ∝ 1/ǫq despite pumping/decay — superfluid response exists. Normal density: ρN =

• ddkǫk

dω 2π Tr

• σzDKσz(DR + DA)
• Is affected by pump/decay:

Does not vanish at T → 0.

[JK PRL ’11]

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 17 / 22

Non-equilibrium superfluid response

Superfluid response exists because: = iψ0qi 2m (1, −1) DR(q, ω = 0)

• 1

−1 iψ0qj 2m DR(ω = 0) ∝ 1/ǫq despite pumping/decay — superfluid response exists. Normal density: ρN =

• ddkǫk

dω 2π Tr

• σzDKσz(DR + DA)
• Is affected by pump/decay:

Does not vanish at T → 0.

[JK PRL ’11]

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 17 / 22

Non-equilibrium superfluid response

Superfluid response exists because: = iψ0qi 2m (1, −1) DR(q, ω = 0)

• 1

−1 iψ0qj 2m DR(ω = 0) ∝ 1/ǫq despite pumping/decay — superfluid response exists. Normal density: ρN =

• ddkǫk

dω 2π Tr

• σzDKσz(DR + DA)
• Is affected by pump/decay:

Does not vanish at T → 0.

[JK PRL ’11]

1 2 3 1 2 3 4 5 ρN / m µ T/µ γnet/µ = 0.0 γnet/µ = 0.1 γnet/µ = 0.5

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 17 / 22

Coherence:

1

Introduction to polariton condensation Approaches to modelling

2

Pattern formation Non-equilibrium pattern formation Spontaneous vortex lattices

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 Condensation, superfluidity and lasing RETUNE, June 2012 18 / 22

Correlations in a 2D Gas

Correlations: g1(r, r′, t) =

• ψ†(r, t)ψ(r′, 0)
• +

= D< = DK − DR + DA Generally, get:

• ψ†(r, t)ψ(0, 0)
• ≃ |ψ0|2 exp
• −ap
• ln(r/r0)

t ≃ 0

1 2 ln(c2t/γnetr 2 0 )

r ≃ 0

• [Szyma´

nska et al. PRL ’06; PRB ’07] [Wouters and Savona PRB ’09]

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 19 / 22

Correlations in a 2D Gas

Correlations: (in 2D) g1(r, r′, t) =

• ψ†(r, t)ψ(r′, 0)
• ≃ |ψ0|2 exp
• −D<

φφ(r, r′, t)

• +

= D< = DK − DR + DA Generally, get:

• ψ†(r, t)ψ(0, 0)
• ≃ |ψ0|2 exp
• −ap
• ln(r/r0)

t ≃ 0

1 2 ln(c2t/γnetr 2 0 )

r ≃ 0

• [Szyma´

nska et al. PRL ’06; PRB ’07] [Wouters and Savona PRB ’09]

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 19 / 22

Correlations in a 2D Gas

Correlations: (in 2D) g1(r, r′, t) =

• ψ†(r, t)ψ(r′, 0)
• ≃ |ψ0|2 exp
• −D<

φφ(r, r′, t)

• +

= D< = DK − DR + DA Generally, get:

• ψ†(r, t)ψ(0, 0)
• ≃ |ψ0|2 exp
• −ap
• ln(r/r0)

t ≃ 0

1 2 ln(c2t/γnetr 2 0 )

r ≃ 0

• [Szyma´

nska et al. PRL ’06; PRB ’07] [Wouters and Savona PRB ’09]

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 19 / 22

Experimental observation of power-law decay

• G. Rompos, Y. Yamamoto et al. submitted

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 20 / 22

Experimental observation of power-law decay

g1(r, −r) ∝ r r0 −ap

• G. Rompos, Y. Yamamoto et al. submitted

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 20 / 22

Exponent in a non-equilibrium 2D gas

lim

r→∞

• ψ†(r, 0)ψ(−r, 0)
• = |ψ0|2 exp
• −D<

φφ(r, −r)

• ∝ exp
• −ap ln

2r r0

• Experimentally, aP ≃ 1.2

In equilibrium ap = mkBT 2π2ns < 1 4 (BKT transition) Non-equilibrium theory depends on thermalisation.

◮ Thermalised (yet diffusive modes)

ap = mkBT 2π2ns

◮ Non-thermalised,

aP ∝ Pumping noise ns .

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 21 / 22

Exponent in a non-equilibrium 2D gas

lim

r→∞

• ψ†(r, 0)ψ(−r, 0)
• = |ψ0|2 exp
• −D<

φφ(r, −r)

• ∝ exp
• −ap ln

2r r0

• Experimentally, aP ≃ 1.2

In equilibrium ap = mkBT 2π2ns < 1 4 (BKT transition) Non-equilibrium theory depends on thermalisation.

◮ Thermalised (yet diffusive modes)

ap = mkBT 2π2ns

◮ Non-thermalised,

aP ∝ Pumping noise ns .

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 21 / 22

Exponent in a non-equilibrium 2D gas

lim

r→∞

• ψ†(r, 0)ψ(−r, 0)
• = |ψ0|2 exp
• −D<

φφ(r, −r)

• ∝ exp
• −ap ln

2r r0

• Experimentally, aP ≃ 1.2

In equilibrium ap = mkBT 2π2ns < 1 4 (BKT transition) Non-equilibrium theory depends on thermalisation.

◮ Thermalised (yet diffusive modes)

ap = mkBT 2π2ns

◮ Non-thermalised,

aP ∝ Pumping noise ns .

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 21 / 22

Exponent in a non-equilibrium 2D gas

lim

r→∞

• ψ†(r, 0)ψ(−r, 0)
• = |ψ0|2 exp
• −D<

φφ(r, −r)

• ∝ exp
• −ap ln

2r r0

• Experimentally, aP ≃ 1.2

In equilibrium ap = mkBT 2π2ns < 1 4 (BKT transition) Non-equilibrium theory depends on thermalisation.

◮ Thermalised (yet diffusive modes)

ap = mkBT 2π2ns

◮ Non-thermalised,

aP ∝ Pumping noise ns .

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 21 / 22

Exponent in a non-equilibrium 2D gas

lim

r→∞

• ψ†(r, 0)ψ(−r, 0)
• = |ψ0|2 exp
• −D<

φφ(r, −r)

• ∝ exp
• −ap ln

2r r0

• Experimentally, aP ≃ 1.2

In equilibrium ap = mkBT 2π2ns < 1 4 (BKT transition) Non-equilibrium theory depends on thermalisation.

◮ Thermalised (yet diffusive modes)

ap = mkBT 2π2ns

◮ Non-thermalised,

aP ∝ Pumping noise ns .

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 21 / 22

Conclusion

Instability of Thomas-Fermi and spontaneous rotation

t=22 t=35 t=56

Survival of superfluid response

Power law decay of correlations

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 22 / 22

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 23 / 35

Extra slides

5

Condensation vs Lasing

6

GPE stability

7

Detecting vortex lattice

8

Calculating superfluid density

9

Measuring superfluid density

10

Finite size coherence and Schawlow-Townes

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 24 / 35

Simple Laser: Maxwell Bloch equations

H = ω0ψ†ψ +

• α

ǫαSz

α + gα,k

√ A ψS+

α + H.c.

Maxwell-Bloch eqns: P = −iS−, N = 2Sz ∂tψ = −iω0ψ − κψ +

α gαPα

∂tPα = −2iǫαPα − 2γP + gαψNα ∂tNα = 2γ(N0 − Nα) − 2gα(ψ∗Pα + P∗

αψ)

κ N g γN0 γ

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 25 / 35

Simple Laser: Maxwell Bloch equations

H = ω0ψ†ψ +

• α

ǫαSz

α + gα,k

√ A ψS+

α + H.c.

Maxwell-Bloch eqns: P = −iS−, N = 2Sz ∂tψ = −iω0ψ − κψ +

α gαPα

∂tPα = −2iǫαPα − 2γP + gαψNα ∂tNα = 2γ(N0 − Nα) − 2gα(ψ∗Pα + P∗

αψ)

κ N g γN0 γ

2 4 6

• 1

1 Absorption ω/g

• 0.6
• 0.4
• 0.2

0.2 N0

Strong coupling. κ, γ < g√n Inversion causes collapse before lasing

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 25 / 35

Simple Laser: Maxwell Bloch equations

H = ω0ψ†ψ +

• α

ǫαSz

α + gα,k

√ A ψS+

α + H.c.

Maxwell-Bloch eqns: P = −iS−, N = 2Sz ∂tψ = −iω0ψ − κψ +

α gαPα

∂tPα = −2iǫαPα − 2γP + gαψNα ∂tNα = 2γ(N0 − Nα) − 2gα(ψ∗Pα + P∗

αψ)

κ N g γN0 γ

2 4 6

• 1

1 Absorption ω/g

• 0.6
• 0.4
• 0.2

0.2 N0

Strong coupling. κ, γ < g√n Inversion causes collapse before lasing

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 25 / 35

Simple Laser: Maxwell Bloch equations

H = ω0ψ†ψ +

• α

ǫαSz

α + gα,k

√ A ψS+

α + H.c.

Maxwell-Bloch eqns: P = −iS−, N = 2Sz ∂tψ = −iω0ψ − κψ +

α gαPα

∂tPα = −2iǫαPα − 2γP + gαψNα ∂tNα = 2γ(N0 − Nα) − 2gα(ψ∗Pα + P∗

αψ)

κ N g γN0 γ

2 4 6

• 1

1 Absorption ω/g

• 0.6
• 0.4
• 0.2

0.2 N0

Strong coupling. κ, γ < g√n Inversion causes collapse before lasing

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 25 / 35

Maxwell-Bloch Equations: Retarded Green’s function

2 4 6

• 1

1 Absorption ω/g

• 0.6
• 0.4
• 0.2

0.2 N0

Introduce DR(ω): Response to perturbation Absorption = −2ℑ[DR(ω)]

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 26 / 35

Maxwell-Bloch Equations: Retarded Green’s function

2 4 6

• 1

1 Absorption ω/g

• 0.6
• 0.4
• 0.2

0.2 N0

Introduce DR(ω): Response to perturbation ∂tψ = −iω0ψ − κψ +

α gαPα

∂tPα = −2iǫαPα − 2γP + gαψNα ∂tNα = 2γ(N0 − Nα) − 2gα(ψ∗Pα + P∗

αψ)

Absorption = −2ℑ[DR(ω)]

• DR(ω)

−1 = ω − ωk + iκ + g2N0 ω − 2ǫ + i2γ

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 26 / 35

Maxwell-Bloch Equations: Retarded Green’s function

2 4 6

• 1

1 Absorption ω/g

• 0.6
• 0.4
• 0.2

0.2 N0

Introduce DR(ω): Response to perturbation ∂tψ = −iω0ψ − κψ +

α gαPα

∂tPα = −2iǫαPα − 2γP + gαψNα ∂tNα = 2γ(N0 − Nα) − 2gα(ψ∗Pα + P∗

αψ)

Absorption = −2ℑ[DR(ω)] = 2B(ω) A(ω)2 + B(ω)2

• DR(ω)

−1 = ω − ωk + iκ + g2N0 ω − 2ǫ + i2γ = A(ω) + iB(ω)

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 26 / 35

Maxwell-Bloch Equations: Retarded Green’s function

• 1

1

• 1

1 ω/g (a) A(ω) 2 4 6

• 1

1 Absorption ω/g

• 0.6
• 0.4
• 0.2

0.2 N0

Introduce DR(ω): Response to perturbation ∂tψ = −iω0ψ − κψ +

α gαPα

∂tPα = −2iǫαPα − 2γP + gαψNα ∂tNα = 2γ(N0 − Nα) − 2gα(ψ∗Pα + P∗

αψ)

Absorption = −2ℑ[DR(ω)] = 2B(ω) A(ω)2 + B(ω)2

• DR(ω)

−1 = ω − ωk + iκ + g2N0 ω − 2ǫ + i2γ = A(ω) + iB(ω)

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 26 / 35

Maxwell-Bloch Equations: Retarded Green’s function

• 1

1

• 1

1 ω/g (a) A(ω) B(ω) 2 4 6

• 1

1 Absorption ω/g

• 0.6
• 0.4
• 0.2

0.2 N0

Introduce DR(ω): Response to perturbation ∂tψ = −iω0ψ − κψ +

α gαPα

∂tPα = −2iǫαPα − 2γP + gαψNα ∂tNα = 2γ(N0 − Nα) − 2gα(ψ∗Pα + P∗

αψ)

Absorption = −2ℑ[DR(ω)] = 2B(ω) A(ω)2 + B(ω)2

• DR(ω)

−1 = ω − ωk + iκ + g2N0 ω − 2ǫ + i2γ = A(ω) + iB(ω)

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 26 / 35

Maxwell-Bloch Equations: Retarded Green’s function

• 1

1

• 1

1 ω/g (a) A(ω) B(ω) 2 4 6

• 1

1 Absorption ω/g

• 0.6
• 0.4
• 0.2

0.2 N0

• 1

1

• 1

1 ω/g (b) A(ω) B(ω)

Introduce DR(ω): Response to perturbation ∂tψ = −iω0ψ − κψ +

α gαPα

∂tPα = −2iǫαPα − 2γP + gαψNα ∂tNα = 2γ(N0 − Nα) − 2gα(ψ∗Pα + P∗

αψ)

Absorption = −2ℑ[DR(ω)] = 2B(ω) A(ω)2 + B(ω)2

• DR(ω)

−1 = ω − ωk + iκ + g2N0 ω − 2ǫ + i2γ = A(ω) + iB(ω)

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 26 / 35

Evolution of poles with Inversion

• 1

1

• 1

1 ω/g (a) A(ω) B(ω) 2 4 6

• 1

1 Absorption ω/g

• 0.6
• 0.4
• 0.2

0.2 N0

• 1

1

• 1

1 ω/g (b) A(ω) B(ω)

• 2
• 1

1 2

• (2γ/g)2

2γκ/g2 ω/g Inversion, N0 (a) (b) Zero of Re Zero of Im

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 27 / 35

Evolution of poles with Inversion

• 1

1

• 1

1 ω/g (a) A(ω) B(ω) 2 4 6

• 1

1 Absorption ω/g

• 0.6
• 0.4
• 0.2

0.2 N0

• 1

1

• 1

1 ω/g (b) A(ω) B(ω)

Laser:

• 2
• 1

1 2

• (2γ/g)2

2γκ/g2 ω/g Inversion, N0 (a) (b) Zero of Re Zero of Im

Equilibrium:

• 4
• 2

2

• 2
• 1.5
• 1
• 0.5

ω/g µ/g

UP LP µ non-condensed condensed

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 27 / 35

Luminescence spectrum and Green’s functions

− 2ℑ[DR(ω)] = DoS(ω)

• DR(ω)

−1 = A(ω) + iB(ω) DoS(ω) = 2B(ω) B(ω)2 + A(ω)2

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 28 / 35

Luminescence spectrum and Green’s functions

− 2ℑ[DR(ω)] = DoS(ω) iDK(ω) = (2n(ω) + 1)DoS(ω)

• DR(ω)

−1 = A(ω) + iB(ω) DoS(ω) = 2B(ω) B(ω)2 + A(ω)2

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 28 / 35

Luminescence spectrum and Green’s functions

− 2ℑ[DR(ω)] = DoS(ω) iDK(ω) = (2n(ω) + 1)DoS(ω)

• DR(ω)

−1 = A(ω) + iB(ω) DoS(ω) = 2B(ω) B(ω)2 + A(ω)2

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 28 / 35

Luminescence spectrum and Green’s functions

− 2ℑ[DR(ω)] = DoS(ω) iDK(ω) = (2n(ω) + 1)DoS(ω)

• DR(ω)

−1 = A(ω) + iB(ω) DoS(ω) = 2B(ω) B(ω)2 + A(ω)2

• 1.5
• 1
• 0.5

0.5 1 1 A(ω)

• 1.5
• 1
• 0.5

0.5 1

Energy (units of g)

1 2 3

Intensity (a.u.)

Density of states, 2 Im[D

R]

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 28 / 35

Luminescence spectrum and Green’s functions

− 2ℑ[DR(ω)] = DoS(ω) iDK(ω) = (2n(ω) + 1)DoS(ω)

• DR(ω)

−1 = A(ω) + iB(ω) DoS(ω) = 2B(ω) B(ω)2 + A(ω)2

• 1.5
• 1
• 0.5

0.5 1 1 A(ω) B(ω)

• 1.5
• 1
• 0.5

0.5 1

Energy (units of g)

1 2 3

Intensity (a.u.)

Density of states, 2 Im[D

R]

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 28 / 35

Luminescence spectrum and Green’s functions

− 2ℑ[DR(ω)] = DoS(ω) iDK(ω) = (2n(ω) + 1)DoS(ω)

• DR(ω)

−1 = A(ω) + iB(ω) DoS(ω) = 2B(ω) B(ω)2 + A(ω)2 iDK(ω) = C(ω) B(ω)2 + A(ω)2

• 1.5
• 1
• 0.5

0.5 1 1 A(ω) B(ω)

• 1.5
• 1
• 0.5

0.5 1

Energy (units of g)

1 2 3

Intensity (a.u.)

Density of states, 2 Im[D

R]

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 28 / 35

Luminescence spectrum and Green’s functions

− 2ℑ[DR(ω)] = DoS(ω) iDK(ω) = (2n(ω) + 1)DoS(ω)

• DR(ω)

−1 = A(ω) + iB(ω) DoS(ω) = 2B(ω) B(ω)2 + A(ω)2 iDK(ω) = C(ω) B(ω)2 + A(ω)2

• 1.5
• 1
• 0.5

0.5 1 1 A(ω) B(ω) C(ω)

• 1.5
• 1
• 0.5

0.5 1

Energy (units of g)

1 2 3

Intensity (a.u.)

Density of states, 2 Im[D

R]

Occupation, n(ω)

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 28 / 35

Stability and evolution with pumping

• 1.5
• 1
• 0.5

0.5 1 1 A(ω) B(ω) C(ω)

• 1.5
• 1
• 0.5

0.5 1

Energy (units of g)

1 2 3

Intensity (a.u.)

Density of states, 2 Im[D

R]

Occupation, n(ω)

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 29 / 35

Stability and evolution with pumping

• 1.5
• 1
• 0.5

0.5 1 1 A(ω) B(ω) C(ω)

• 1.5
• 1
• 0.5

0.5 1

Energy (units of g)

1 2 3

Intensity (a.u.)

Density of states, 2 Im[D

R]

Occupation, n(ω)

• DR(ω)

−1 = (ω − ξk)+iα(ω − µeff)

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 29 / 35

Stability and evolution with pumping

• 1.5
• 1
• 0.5

0.5 1 1 A(ω) B(ω) C(ω)

• 1.5
• 1
• 0.5

0.5 1

Energy (units of g)

1 2 3

Intensity (a.u.)

Density of states, 2 Im[D

R]

Occupation, n(ω)

• DR(ω)

−1 = (ω − ξk)+iα(ω − µeff)

• DR(ω∗

k)

−1 = 0 → ℑ(ω∗) ∝ µeff−ξk

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 29 / 35

Stability and evolution with pumping

• 1.5
• 1
• 0.5

0.5 1 1 A(ω) B(ω) C(ω)

• 1.5
• 1
• 0.5

0.5 1

Energy (units of g)

1 2 3

Intensity (a.u.)

Density of states, 2 Im[D

R]

Occupation, n(ω)

• DR(ω)

−1 = (ω − ξk)+iα(ω − µeff)

• DR(ω∗

k)

−1 = 0 → ℑ(ω∗) ∝ µeff−ξk

• 0.6
• 0.5
• 0.4
• 0.3

Bath occupation, µB/g

• 3
• 2
• 1

1

Energy of zero (units of g)

Zero of Re Zero of Im

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 29 / 35

Strong coupling and lasing — low temperature phenomenon

ω/g µ/g ξ µeff

• 2
• 1

1

• 2
• 1

non- condensed condensed

• Eqbm. polariton

µB/g

• 2
• 1

non- condensed condensed Non-eqbm. polariton Inversion, N0

• 1

1 non- condensed condensed Laser

Laser: Uniformly invert TLS Non-equilibrium polaritons: Cold bath If TB ≫ γ → Laser limit

• 1.5
• 1
• 0.5

0.5 1

Energy (units of g)

1 A(ω) B(ω) Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 30 / 35

Strong coupling and lasing — low temperature phenomenon

ω/g µ/g ξ µeff

• 2
• 1

1

• 2
• 1

non- condensed condensed

• Eqbm. polariton

µB/g

• 2
• 1

non- condensed condensed Non-eqbm. polariton Inversion, N0

• 1

1 non- condensed condensed Laser

Laser: Uniformly invert TLS Non-equilibrium polaritons: Cold bath If TB ≫ γ → Laser limit

• 1.5
• 1
• 0.5

0.5 1

Energy (units of g)

1 A(ω) B(ω) Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 30 / 35

Strong coupling and lasing — low temperature phenomenon

ω/g µ/g ξ µeff

• 2
• 1

1

• 2
• 1

non- condensed condensed

• Eqbm. polariton

µB/g

• 2
• 1

non- condensed condensed Non-eqbm. polariton Inversion, N0

• 1

1 non- condensed condensed Laser

Laser: Uniformly invert TLS Non-equilibrium polaritons: Cold bath If TB ≫ γ → Laser limit

• 1.5
• 1
• 0.5

0.5 1

Energy (units of g)

1 A(ω) B(ω) Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 30 / 35

Instability of Thomas-Fermi: details

1 2∂tρ + ∇ · (ρv) = (γnet − Γρ)ρ ∂tv + ∇(Uρ + mω2 2 r 2 + m 2 |v|2) = 0

3γnet 2Γ

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 31 / 35

Instability of Thomas-Fermi: details

1 2∂tρ + ∇ · (ρv) = (γnet − Γρ)ρ ∂tv + ∇(Uρ + mω2 2 r 2 + m 2 |v|2) = 0 Normal modes for γnet, Γ → 0: δρn,m(r, θ, t) = eimθhn,m(r)eiωn,mt ωn,m = ω2

• m(1 + 2n) + 2n(n + 1)

3γnet 2Γ

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 31 / 35

Instability of Thomas-Fermi: details

1 2∂tρ + ∇ · (ρv) = (γnet − Γρ)ρ ∂tv + ∇(Uρ + mω2 2 r 2 + m 2 |v|2) = 0 Normal modes for γnet, Γ → 0: δρn,m(r, θ, t) = eimθhn,m(r)eiωn,mt ωn,m = ω2

• m(1 + 2n) + 2n(n + 1)

3γnet 2Γ

Add weak pumping/decay: ωn,n → ωn,m + iγnet m(1 + 2n) + 2n(n + 1)−m2 2m(1 + 2n) + 4n(n + 1) + m2

• Jonathan Keeling

Condensation, superfluidity and lasing RETUNE, June 2012 31 / 35

Instability of Thomas-Fermi: details

1 2∂tρ + ∇ · (ρv) = (γnet − Γρ)ρ ∂tv + ∇(Uρ + mω2 2 r 2 + m 2 |v|2) = 0 Normal modes for γnet, Γ → 0: δρn,m(r, θ, t) = eimθhn,m(r)eiωn,mt ωn,m = ω2

• m(1 + 2n) + 2n(n + 1)

3γnet 2Γ Unstable growth

Add weak pumping/decay: ωn,n → ωn,m + iγnet m(1 + 2n) + 2n(n + 1)−m2 2m(1 + 2n) + 4n(n + 1) + m2

• Jonathan Keeling

Condensation, superfluidity and lasing RETUNE, June 2012 31 / 35

Instability of Thomas-Fermi: details

1 2∂tρ + ∇ · (ρv) = (γnet − Γρ)ρ ∂tv + ∇(Uρ + mω2 2 r 2 + m 2 |v|2) = 0 Normal modes for γnet, Γ → 0: δρn,m(r, θ, t) = eimθhn,m(r)eiωn,mt ωn,m = ω2

• m(1 + 2n) + 2n(n + 1)

3γnet 2Γ Stabilised

Add weak pumping/decay: ωn,n → ωn,m + iγnet m(1 + 2n) + 2n(n + 1)−m2 2m(1 + 2n) + 4n(n + 1) + m2

• Jonathan Keeling

Condensation, superfluidity and lasing RETUNE, June 2012 31 / 35

Detecting vortex lattices

Snapshot Spectrum:

kx ω −5 5 5 10 15 20 25 30

Defocussed homodyne intereference:

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 32 / 35

Calculating superfluid response function

Using Keldysh generating functional χij(q) = − i 2 d2Z[f, θ] dfi(q)dθj(−q), Z[f, θ] =

• Dψ exp(iS[f, θ])

f, θ couple as force/response current. S[f, θ] = S +

• k,q

¯ ψcl ¯ ψq

• k+q
• θi

fi + θi fi − θi −θi

• q

2ki + qi 2m ψcl ψq

• k

Saddle point + fluctuations:

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 33 / 35

Calculating superfluid response function

Using Keldysh generating functional χij(q) = − i 2 d2Z[f, θ] dfi(q)dθj(−q), Z[f, θ] =

• Dψ exp(iS[f, θ])

f, θ couple as force/response current. S[f, θ] = S +

• k,q

¯ ψcl ¯ ψq

• k+q
• θi

fi + θi fi − θi −θi

• q

2ki + qi 2m ψcl ψq

• k

Saddle point + fluctuations:

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 33 / 35

Calculating superfluid response function

Using Keldysh generating functional χij(q) = − i 2 d2Z[f, θ] dfi(q)dθj(−q), Z[f, θ] =

• Dψ exp(iS[f, θ])

f, θ couple as force/response current. S[f, θ] = S +

• k,q

¯ ψcl ¯ ψq

• k+q
• θi

fi + θi fi − θi −θi

• q

2ki + qi 2m ψcl ψq

• k

Saddle point + fluctuations:

+ + + + . . . +

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 33 / 35

Calculating superfluid response function

Using Keldysh generating functional χij(q) = − i 2 d2Z[f, θ] dfi(q)dθj(−q), Z[f, θ] =

• Dψ exp(iS[f, θ])

f, θ couple as force/response current. S[f, θ] = S +

• k,q

¯ ψcl ¯ ψq

• k+q
• θi

fi + θi fi − θi −θi

• q

2ki + qi 2m ψcl ψq

• k

Saddle point + fluctuations: Only one diagram for χN

+ + + + . . . +

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 33 / 35

Measuring superfluid density

• 1. Effect rotating frame

Polariton polarization: (ψ, ψ) H = λ

• ℓ2

r 2e2iφ r 2e−2iφ −ℓ2

• (a)

µc Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 34 / 35

Measuring superfluid density

• 1. Effect rotating frame

Polariton polarization: (ψ, ψ) H = λ

• ℓ2

r 2e2iφ r 2e−2iφ −ℓ2

• Ground state Berry phase:

qAeff = mω × r = ˆ φ r

• 1 −

ℓ2 √ r 4 + ℓ4

• (a)

µc 0.1 0.2 0.3 1 2 3 4 0.1 0.2 0.3 0.4 mℓ2ω qAφℓ = mvℓ r/ℓ

(b)

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 34 / 35

Measuring superfluid density

• 1. Effect rotating frame

Polariton polarization: (ψ, ψ) H = λ

• ℓ2

r 2e2iφ r 2e−2iφ −ℓ2

• Ground state Berry phase:

qAeff = mω × r = ˆ φ r

• 1 −

ℓ2 √ r 4 + ℓ4

• (a)

µc 0.1 0.2 0.3 1 2 3 4 0.1 0.2 0.3 0.4 mℓ2ω qAφℓ = mvℓ r/ℓ

(b)

• 2. Measure resulting current

Energy shift of normal state: ∆E = (1/2)mv2 = 0.08/mℓ2 ≃ 0.1meV

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 34 / 35

Finite size effects: Single mode vs many mode

• ψ†(r, t)ψ(r′, 0)
• ≃ |ψ0|2 exp
• −D<

φφ(r, r′, t)

• Jonathan Keeling

Condensation, superfluidity and lasing RETUNE, June 2012 35 / 35

Finite size effects: Single mode vs many mode

• ψ†(r, t)ψ(r′, 0)
• ≃ |ψ0|2 exp
• −D<

φφ(r, r′, t)

• D<

φφ(r, r′, t) from sum of phase modes. Study ct ≫ r limit:

D<

φφ(r, r, t) ∝ nmax

• n

dω 2π |ϕn(r)|2(1 − eiωt)

• (ω + iγnet)2 + γ2

net − ξ2 n

• 2

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 35 / 35

Finite size effects: Single mode vs many mode

• ψ†(r, t)ψ(r′, 0)
• ≃ |ψ0|2 exp
• −D<

φφ(r, r′, t)

• D<

φφ(r, r′, t) from sum of phase modes. Study ct ≫ r limit:

D<

φφ(r, r, t) ∝ nmax

• n

dω 2π |ϕn(r)|2(1 − eiωt)

• (ω + iγnet)2 + γ2

net − ξ2 n

• 2

∆ξ ≪ γnet t ≪ Emax

Emax

∆ Energy

D<

φφ ∼ 1 + ln(Emax

• t

γnet )

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 35 / 35

Finite size effects: Single mode vs many mode

• ψ†(r, t)ψ(r′, 0)
• ≃ |ψ0|2 exp
• −D<

φφ(r, r′, t)

• D<

φφ(r, r′, t) from sum of phase modes. Study ct ≫ r limit:

D<

φφ(r, r, t) ∝ nmax

• n

dω 2π |ϕn(r)|2(1 − eiωt)

• (ω + iγnet)2 + γ2

net − ξ2 n

• 2

∆ξ ≪ γnet t ≪ Emax

Emax

∆ Energy

D<

φφ ∼ 1 + ln(Emax

• t

γnet )

γnet t ≪ ∆ξ ≪ Emax

Emax

∆ Energy

D<

φφ ∼

πC 2γnet t 2γnet

• (Recovers Schawlow-Townes laser linewidth)

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 35 / 35