Non-equilibrium coherence in light-matter systems Condensation, lasing, superradiance and more Jonathan Keeling University of St Andrews 600 YEARS SUPA FQCMP2013, NY, March 2013 Jonathan Keeling Condensation lasing & superradiance FQCMP2013 1 / 40
Non-equilibrium coherence in light-matter systemsCondensation, lasing, superradiance and more
Jonathan Keeling
University ofSt Andrews
600YEARS
SUPA
FQCMP2013, NY, March 2013
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 1 / 40
Acknowledgements
GROUP:
COLLABORATORS: Szymanska, Littlewood, Simons, Bhaseen,Schmidt, Blatter, Tureci, KrugerEXPERIMENT: Houck, Wallraff, Fink, Mylnek
FUNDING:
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 2 / 40
Coupling many atoms to lightOld question: What happens to radiation when many atoms interact“collectively” with light.Superradiance — dynamical and steady state.
New relevanceSuperconducting qubits
Quantum dots & NV centres
Ultra-cold atomsκ
Pump
κCavity
Pump
Rydberg atoms/polaritons
Microcavity Polaritons
Photon condensation
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 3 / 40
Coupling many atoms to lightOld question: What happens to radiation when many atoms interact“collectively” with light.Superradiance — dynamical and steady state.New relevance
Superconducting qubits
Quantum dots & NV centres
Ultra-cold atomsκ
Pump
κCavity
Pump
Rydberg atoms/polaritons
Microcavity Polaritons
Photon condensation
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 3 / 40
Dicke effect: Enhanced emission
Hint =∑k ,i
gk
(ψ†kS−i e−ik·ri + H.c.
)
If |ri − rj | λ, use∑
i Si → SCollective decay:
dρdt
= −Γ
2[S+S−ρ− S−ρS+ + ρS+S−
]
If Sz = |S| = N/2 initially:
I ∝ −Γd〈Sz〉
dt=
ΓN2
4sech2
[ΓN2
t]
-N/2
0
N/2
tD
⟨Sz⟩
tD
0
ΓN2/2
I=-Γ
d⟨S
z⟩/
dt
Problem: dipole interactions dephase. [Friedberg et al, Phys. Lett. 1972]
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 4 / 40
Dicke effect: Enhanced emission
Hint =∑k ,i
gk
(ψ†kS−i e−ik·ri + H.c.
)If |ri − rj | λ, use
∑i Si → S
Collective decay:dρdt
= −Γ
2[S+S−ρ− S−ρS+ + ρS+S−
]
If Sz = |S| = N/2 initially:
I ∝ −Γd〈Sz〉
dt=
ΓN2
4sech2
[ΓN2
t]
-N/2
0
N/2
tD
⟨Sz⟩
tD
0
ΓN2/2
I=-Γ
d⟨S
z⟩/
dt
Problem: dipole interactions dephase. [Friedberg et al, Phys. Lett. 1972]
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 4 / 40
Dicke effect: Enhanced emission
Hint =∑k ,i
gk
(ψ†kS−i e−ik·ri + H.c.
)If |ri − rj | λ, use
∑i Si → S
Collective decay:dρdt
= −Γ
2[S+S−ρ− S−ρS+ + ρS+S−
]
If Sz = |S| = N/2 initially:
I ∝ −Γd〈Sz〉
dt=
ΓN2
4sech2
[ΓN2
t]
-N/2
0
N/2
tD
⟨Sz⟩
tD
0
ΓN2/2
I=-Γ
d⟨S
z⟩/
dt
Problem: dipole interactions dephase. [Friedberg et al, Phys. Lett. 1972]
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 4 / 40
Dicke effect: Enhanced emission
Hint =∑k ,i
gk
(ψ†kS−i e−ik·ri + H.c.
)If |ri − rj | λ, use
∑i Si → S
Collective decay:dρdt
= −Γ
2[S+S−ρ− S−ρS+ + ρS+S−
]
If Sz = |S| = N/2 initially:
I ∝ −Γd〈Sz〉
dt=
ΓN2
4sech2
[ΓN2
t]
-N/2
0
N/2
tD
⟨Sz⟩
tD
0
ΓN2/2
I=-Γ
d⟨S
z⟩/
dt
Problem: dipole interactions dephase. [Friedberg et al, Phys. Lett. 1972]
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 4 / 40
Collective radiation with a cavity: Dynamics
Hint =∑
i
(ψ†S−i + ψS+
i
)
0
200
400
600
800
1000
1200
1400
1600
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
|ψ(t
)|2
Time
T=2ln(√N__
)/√N__
1/√N__
Single cavity mode: oscillations
If Sz = |S| = N/2 initially
[Bonifacio and Preparata PRA ’70]
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 5 / 40
Collective radiation with a cavity: Dynamics
Hint =∑
i
(ψ†S−i + ψS+
i
)
0
200
400
600
800
1000
1200
1400
1600
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
|ψ(t
)|2
Time
T=2ln(√N__
)/√N__
1/√N__
Single cavity mode: oscillationsIf Sz = |S| = N/2 initially
[Bonifacio and Preparata PRA ’70]
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 5 / 40
Dicke model: Equilibrium superradiance transition
H = ωψ†ψ + ω0Sz + g(ψ†S− + ψS+
).
Coherent state: |Ψ〉 → eλψ†+ηS+ |Ω〉
Small g, min at λ, η = 0
Spontaneous polarisation if: Ng2 > ωω0
00
ω
g-√N
⇓ SR
[Hepp, Lieb, Ann. Phys. ’73]
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 6 / 40
Dicke model: Equilibrium superradiance transition
H = ωψ†ψ + ω0Sz + g(ψ†S− + ψS+
).
Coherent state: |Ψ〉 → eλψ†+ηS+ |Ω〉
Small g, min at λ, η = 0
Spontaneous polarisation if: Ng2 > ωω0
00
ω
g-√N
⇓ SR
[Hepp, Lieb, Ann. Phys. ’73]
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 6 / 40
Dicke model: Equilibrium superradiance transition
H = ωψ†ψ + ω0Sz + g(ψ†S− + ψS+
).
Coherent state: |Ψ〉 → eλψ†+ηS+ |Ω〉
Small g, min at λ, η = 0
Spontaneous polarisation if: Ng2 > ωω0
00
ω
g-√N
⇓ SR
[Hepp, Lieb, Ann. Phys. ’73]
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 6 / 40
Dicke model: Equilibrium superradiance transition
H = ωψ†ψ + ω0Sz + g(ψ†S− + ψS+
).
Coherent state: |Ψ〉 → eλψ†+ηS+ |Ω〉
Small g, min at λ, η = 0
Spontaneous polarisation if: Ng2 > ωω0
00
ω
g-√N
⇓ SR
[Hepp, Lieb, Ann. Phys. ’73]
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 6 / 40
Dicke model: Superradiance at T 6= 0
H = ωψ†ψ + ω0Sz + g(ψ†S− + ψS+
).
T = 0 ground state if:
Ng2 > ωω0 00
ω
g-√N
⇓ SR
T > 0, minimum free energy if
Ng2 tanh(βω0)
ω0> ω
0
T
g-√N
⇓ SR
[Hepp, Lieb, Ann. Phys. ’73]
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 7 / 40
Dicke model: Superradiance at T 6= 0
H = ωψ†ψ + ω0Sz + g(ψ†S− + ψS+
).
T = 0 ground state if:
Ng2 > ωω0 00
ω
g-√N
⇓ SR
T > 0, minimum free energy if
Ng2 tanh(βω0)
ω0> ω
0
T
g-√N
⇓ SR
[Hepp, Lieb, Ann. Phys. ’73]
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 7 / 40
No go theorem and transition
Spontaneous polarisation if: Ng2 > ωω0
No go theorem:. Minimal coupling (p − eA)2/2m
−∑
i
em
A · pi ⇔ g(ψ†S− + ψS+),∑
i
A2
2m⇔ Nζ(ψ + ψ†)2
For large N, ω → ω + 2Nζ. (RWA)
Need Ng2 > ω0(ω+ 2Nζ).
But Thomas-Reiche-Kuhn sum rule states: g2/ω0 < 2ζ. No transition
[Rzazewski et al PRL ’75]Jonathan Keeling Condensation lasing & superradiance FQCMP2013 8 / 40
No go theorem and transition
Spontaneous polarisation if: Ng2 > ωω0
No go theorem:. Minimal coupling (p − eA)2/2m
−∑
i
em
A · pi ⇔ g(ψ†S− + ψS+),∑
i
A2
2m⇔ Nζ(ψ + ψ†)2
For large N, ω → ω + 2Nζ. (RWA)
Need Ng2 > ω0(ω+ 2Nζ).
But Thomas-Reiche-Kuhn sum rule states: g2/ω0 < 2ζ. No transition
[Rzazewski et al PRL ’75]Jonathan Keeling Condensation lasing & superradiance FQCMP2013 8 / 40
No go theorem and transition
Spontaneous polarisation if: Ng2 > ωω0
No go theorem:. Minimal coupling (p − eA)2/2m
−∑
i
em
A · pi ⇔ g(ψ†S− + ψS+),∑
i
A2
2m⇔ Nζ(ψ + ψ†)2
For large N, ω → ω + 2Nζ. (RWA)
Need Ng2 > ω0(ω+ 2Nζ).
But Thomas-Reiche-Kuhn sum rule states: g2/ω0 < 2ζ. No transition
[Rzazewski et al PRL ’75]Jonathan Keeling Condensation lasing & superradiance FQCMP2013 8 / 40
No go theorem and transition
Spontaneous polarisation if: Ng2 > ωω0
No go theorem:. Minimal coupling (p − eA)2/2m
−∑
i
em
A · pi ⇔ g(ψ†S− + ψS+),∑
i
A2
2m⇔ Nζ(ψ + ψ†)2
For large N, ω → ω + 2Nζ. (RWA)
Need Ng2 > ω0(ω+ 2Nζ).
But Thomas-Reiche-Kuhn sum rule states: g2/ω0 < 2ζ. No transition
[Rzazewski et al PRL ’75]Jonathan Keeling Condensation lasing & superradiance FQCMP2013 8 / 40
No go theorem and transition
Spontaneous polarisation if: Ng2 > ωω0
No go theorem:. Minimal coupling (p − eA)2/2m
−∑
i
em
A · pi ⇔ g(ψ†S− + ψS+),∑
i
A2
2m⇔ Nζ(ψ + ψ†)2
For large N, ω → ω + 2Nζ. (RWA)
Need Ng2 > ω0(ω+ 2Nζ).
But Thomas-Reiche-Kuhn sum rule states: g2/ω0 < 2ζ. No transition[Rzazewski et al PRL ’75]
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 8 / 40
Dicke phase transition: ways out
Problem: g2/ω0 < 2ζ for intrinsic parameters. Solutions:InterpretationFerroelectric transition in D · r gauge.[JK JPCM ’07, Vukics & Domokos PRA 2012 ]
Circuit QED [Nataf and Ciuti, Nat. Comm. ’10; Viehmann et al. PRL ’11]
Grand canonical ensemble:I If H → H − µ(Sz + ψ†ψ), need only:
g2N > (ω − µ)(ω0 − µ)I Incoherent pumping — polariton
condensation.
Dissociate g, ω0,e.g. Raman scheme: ω0 ω.[Dimer et al. PRA ’07; Baumann et al. Nature’10. Also, Black et al. PRL ’03 ]
κ
Pump
κCavity
Pump
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 9 / 40
Dicke phase transition: ways out
Problem: g2/ω0 < 2ζ for intrinsic parameters. Solutions:InterpretationFerroelectric transition in D · r gauge.[JK JPCM ’07, Vukics & Domokos PRA 2012 ]
Circuit QED [Nataf and Ciuti, Nat. Comm. ’10; Viehmann et al. PRL ’11]
Grand canonical ensemble:I If H → H − µ(Sz + ψ†ψ), need only:
g2N > (ω − µ)(ω0 − µ)I Incoherent pumping — polariton
condensation.
Dissociate g, ω0,e.g. Raman scheme: ω0 ω.[Dimer et al. PRA ’07; Baumann et al. Nature’10. Also, Black et al. PRL ’03 ]
κ
Pump
κCavity
Pump
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 9 / 40
Dicke phase transition: ways out
Problem: g2/ω0 < 2ζ for intrinsic parameters. Solutions:InterpretationFerroelectric transition in D · r gauge.[JK JPCM ’07, Vukics & Domokos PRA 2012 ]
Circuit QED [Nataf and Ciuti, Nat. Comm. ’10; Viehmann et al. PRL ’11]
Grand canonical ensemble:I If H → H − µ(Sz + ψ†ψ), need only:
g2N > (ω − µ)(ω0 − µ)I Incoherent pumping — polariton
condensation.
Dissociate g, ω0,e.g. Raman scheme: ω0 ω.[Dimer et al. PRA ’07; Baumann et al. Nature’10. Also, Black et al. PRL ’03 ]
κ
Pump
κCavity
Pump
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 9 / 40
Dicke phase transition: ways out
Problem: g2/ω0 < 2ζ for intrinsic parameters. Solutions:InterpretationFerroelectric transition in D · r gauge.[JK JPCM ’07, Vukics & Domokos PRA 2012 ]
Circuit QED [Nataf and Ciuti, Nat. Comm. ’10; Viehmann et al. PRL ’11]
Grand canonical ensemble:I If H → H − µ(Sz + ψ†ψ), need only:
g2N > (ω − µ)(ω0 − µ)I Incoherent pumping — polariton
condensation.
Dissociate g, ω0,e.g. Raman scheme: ω0 ω.[Dimer et al. PRA ’07; Baumann et al. Nature’10. Also, Black et al. PRL ’03 ]
κ
Pump
κCavity
Pump
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 9 / 40
Dicke phase transition: ways out
Problem: g2/ω0 < 2ζ for intrinsic parameters. Solutions:InterpretationFerroelectric transition in D · r gauge.[JK JPCM ’07, Vukics & Domokos PRA 2012 ]
Circuit QED [Nataf and Ciuti, Nat. Comm. ’10; Viehmann et al. PRL ’11]
Grand canonical ensemble:I If H → H − µ(Sz + ψ†ψ), need only:
g2N > (ω − µ)(ω0 − µ)I Incoherent pumping — polariton
condensation.
Dissociate g, ω0,e.g. Raman scheme: ω0 ω.[Dimer et al. PRA ’07; Baumann et al. Nature’10. Also, Black et al. PRL ’03 ]
κ
Pump
κCavity
Pump
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 9 / 40
1 Dicke model and superradiance
2 Polariton and photon condensationPolaritonsNon-equilibrium condensation vs lasingPhoton condensation
3 Jaynes Cummings Hubbard modelJCHM vv DickeCoherently driven arrayDisorder
4 Phase transitions with SC qubitsPumping without symmetry breakingCollective dephasing
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 10 / 40
Polariton and photon Condensation
1 Dicke model and superradiance
2 Polariton and photon condensationPolaritonsNon-equilibrium condensation vs lasingPhoton condensation
3 Jaynes Cummings Hubbard modelJCHM vv DickeCoherently driven arrayDisorder
4 Phase transitions with SC qubitsPumping without symmetry breakingCollective dephasing
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 11 / 40
Microcavity polaritons
Quantum WellsCavity
Cavity
θ
Cavity photons:
ωk =√ω2
0 + c2k2
' ω0 + k2/2m∗
m∗ ∼ 10−4me
Ene
rgy
Momentum
n=1
n=2
n=3
n=4
Bulk
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 12 / 40
Microcavity polaritons
Quantum WellsCavity
Cavity
θ
Cavity photons:
ωk =√ω2
0 + c2k2
' ω0 + k2/2m∗
m∗ ∼ 10−4me
Ene
rgy
Momentum
n=1
n=2
n=3
n=4
Bulk
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 12 / 40
Microcavity polaritons
Quantum WellsCavity
Cavity
θ
Cavity photons:
ωk =√ω2
0 + c2k2
' ω0 + k2/2m∗
m∗ ∼ 10−4meIn−plane momentum
Exciton
Ene
rgy
Phot
onJonathan Keeling Condensation lasing & superradiance FQCMP2013 12 / 40
Microcavity polaritons
Quantum WellsCavity
Cavity
θ
Cavity photons:
ωk =√ω2
0 + c2k2
' ω0 + k2/2m∗
m∗ ∼ 10−4meIn−plane momentum
Exciton
Ene
rgy
Phot
onJonathan Keeling Condensation lasing & superradiance FQCMP2013 12 / 40
Microcavity polaritons
Quantum WellsCavity
Cavity
θ
Cavity photons:
ωk =√ω2
0 + c2k2
' ω0 + k2/2m∗
m∗ ∼ 10−4meIn−plane momentum
Exciton
Ene
rgy
Phot
onJonathan Keeling Condensation lasing & superradiance FQCMP2013 12 / 40
Lasing-condensation crossover model
Use model that can show lasing and condensation:
κ
N
g
γN0γ
⇔In−plane momentum
Exciton
Ene
rgy
Phot
on
Dicke model:
Hsys =∑
k
ωkψ†kψk +
∑α
[εSz
α +
(gα,kψkS+
α + H.c.)
√A
]
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 13 / 40
Lasing-condensation crossover model
Use model that can show lasing and condensation:
κ
N
g
γN0γ
⇔In−plane momentum
Exciton
Ene
rgy
Phot
on
Dicke model:
Hsys =∑
k
ωkψ†kψk +
∑α
[εSz
α +
(gα,kψkS+
α + H.c.)
√A
]
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 13 / 40
Polariton model and equilibrium resultsLocalised excitons, propagating photons
H − µN =∑
k
(ωk − µ)ψ†kψk +∑α
(εα − µ)Szα +
gα,k√AψkS+
α + H.c.
Self-consistent polarisation and field
(ω − µ)ψ =1A
∑α
g2αψ
2Eαtanh (βEα) , Eα2 =
(εα − µ
2
)2
+ g2α|ψ|2
Phase diagram:
0
10
20
30
40
0 1×109
2×109
T (
K)
n (cm)-2
non-condensed
condensed
Modes (at k = 0)
-4
-2
0
2
-2 -1.5 -1 -0.5 0
ω/g
µ/g
UP
LP
µ
non-condensed condensed
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 14 / 40
Polariton model and equilibrium resultsLocalised excitons, propagating photons
H − µN =∑
k
(ωk − µ)ψ†kψk +∑α
(εα − µ)Szα +
gα,k√AψkS+
α + H.c.
Self-consistent polarisation and field
(ω − µ)ψ =1A
∑α
g2αψ
2Eαtanh (βEα) , Eα2 =
(εα − µ
2
)2
+ g2α|ψ|2
Phase diagram:
0
10
20
30
40
0 1×109
2×109
T (
K)
n (cm)-2
non-condensed
condensed
Modes (at k = 0)
-4
-2
0
2
-2 -1.5 -1 -0.5 0
ω/g
µ/g
UP
LP
µ
non-condensed condensed
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 14 / 40
Polariton model and equilibrium resultsLocalised excitons, propagating photons
H − µN =∑
k
(ωk − µ)ψ†kψk +∑α
(εα − µ)Szα +
gα,k√AψkS+
α + H.c.
Self-consistent polarisation and field
(ω − µ)ψ =1A
∑α
g2αψ
2Eαtanh (βEα) , Eα2 =
(εα − µ
2
)2
+ g2α|ψ|2
Phase diagram:
0
10
20
30
40
0 1×109
2×109
T (
K)
n (cm)-2
non-condensed
condensed
Modes (at k = 0)
-4
-2
0
2
-2 -1.5 -1 -0.5 0
ω/g
µ/g
UP
LP
µ
non-condensed condensed
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 14 / 40
Polariton model and equilibrium resultsLocalised excitons, propagating photons
H − µN =∑
k
(ωk − µ)ψ†kψk +∑α
(εα − µ)Szα +
gα,k√AψkS+
α + H.c.
Self-consistent polarisation and field
(ω − µ)ψ =1A
∑α
g2αψ
2Eαtanh (βEα) , Eα2 =
(εα − µ
2
)2
+ g2α|ψ|2
Phase diagram:
0
10
20
30
40
0 1×109
2×109
T (
K)
n (cm)-2
non-condensed
condensed
Modes (at k = 0)
-4
-2
0
2
-2 -1.5 -1 -0.5 0
ω/g
µ/g
UP
LP
µ
non-condensed condensed
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 14 / 40
Polariton model and equilibrium resultsLocalised excitons, propagating photons
H − µN =∑
k
(ωk − µ)ψ†kψk +∑α
(εα − µ)Szα +
gα,k√AψkS+
α + H.c.
Self-consistent polarisation and field
(ω − µ)ψ =1A
∑α
g2αψ
2Eαtanh (βEα) , Eα2 =
(εα − µ
2
)2
+ g2α|ψ|2
Phase diagram:
0
10
20
30
40
0 1×109
2×109
T (
K)
n (cm)-2
non-condensed
condensed
Modes (at k = 0)
-4
-2
0
2
-2 -1.5 -1 -0.5 0
ω/g
µ/g
UP
LP
µ
non-condensed condensed
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 14 / 40
Simple Laser: Maxwell Bloch equations
H = ωψ†ψ +∑α
εαSzα +
gα,k√AψS+
α + H.c.
Maxwell-Bloch eqns: P = −i〈S−〉,N = 2〈Sz〉∂tψ = −iωψ − κψ +
∑α gαPα
∂tPα = −2iεαPα − 2γP + gαψNα
∂tNα = 2γ(N0 − Nα)− 2gα(ψ∗Pα + P∗αψ)
κ
N
g
γN0γ
0
2
4
6
-1 0 1
Abso
rpti
on
ω/g
-0.6
-0.4
-0.2
0
0.2
N0 Strong coupling. κ, γ < g
√n
Inversion causes collapsebefore lasing
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 15 / 40
Simple Laser: Maxwell Bloch equations
H = ωψ†ψ +∑α
εαSzα +
gα,k√AψS+
α + H.c.
Maxwell-Bloch eqns: P = −i〈S−〉,N = 2〈Sz〉∂tψ = −iωψ − κψ +
∑α gαPα
∂tPα = −2iεαPα − 2γP + gαψNα
∂tNα = 2γ(N0 − Nα)− 2gα(ψ∗Pα + P∗αψ)
κ
N
g
γN0γ
0
2
4
6
-1 0 1
Abso
rpti
on
ω/g
-0.6
-0.4
-0.2
0
0.2
N0 Strong coupling. κ, γ < g
√n
Inversion causes collapsebefore lasing
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 15 / 40
Simple Laser: Maxwell Bloch equations
H = ωψ†ψ +∑α
εαSzα +
gα,k√AψS+
α + H.c.
Maxwell-Bloch eqns: P = −i〈S−〉,N = 2〈Sz〉∂tψ = −iωψ − κψ +
∑α gαPα
∂tPα = −2iεαPα − 2γP + gαψNα
∂tNα = 2γ(N0 − Nα)− 2gα(ψ∗Pα + P∗αψ)
κ
N
g
γN0γ
0
2
4
6
-1 0 1
Abso
rpti
on
ω/g
-0.6
-0.4
-0.2
0
0.2
N0 Strong coupling. κ, γ < g
√n
Inversion causes collapsebefore lasing
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 15 / 40
Simple Laser: Maxwell Bloch equations
H = ωψ†ψ +∑α
εαSzα +
gα,k√AψS+
α + H.c.
Maxwell-Bloch eqns: P = −i〈S−〉,N = 2〈Sz〉∂tψ = −iωψ − κψ +
∑α gαPα
∂tPα = −2iεαPα − 2γP + gαψNα
∂tNα = 2γ(N0 − Nα)− 2gα(ψ∗Pα + P∗αψ)
κ
N
g
γN0γ
0
2
4
6
-1 0 1
Abso
rpti
on
ω/g
-0.6
-0.4
-0.2
0
0.2
N0 Strong coupling. κ, γ < g
√n
Inversion causes collapsebefore lasing
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 15 / 40
Poles of Retarded Green’s function and gain[DR(ν)
]−1= ν − ωk + iκ+
g2N0
ν − 2ε+ i2γ
= A(ν) + iB(ν)
-1
0
1
-1 0 1
ω/g
(a)
A(ω)B(ω) -1
0
1
-1 0 1
ω/g
(b)
A(ω)B(ω)
Laser:
-2
-1
0
1
2
-(2γ/g)2
2γκ/g2
ω/g
Inversion, N0
(a) (b)
Zero of Re Zero of Im
Equilibrium:
-4
-2
0
2
-2 -1.5 -1 -0.5 0
ω/g
µ/g
UP
LP
µ
non-condensed condensed
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 16 / 40
Poles of Retarded Green’s function and gain[DR(ν)
]−1= ν − ωk + iκ+
g2N0
ν − 2ε+ i2γ= A(ν) + iB(ν)
-1
0
1
-1 0 1
ω/g
(a)
A(ω)B(ω)
0
2
4
6
-1 0 1
Abso
rpti
on
ω/g
-0.6
-0.4
-0.2
0
0.2
N0
-1
0
1
-1 0 1
ω/g
(b)
A(ω)B(ω)
Laser:
-2
-1
0
1
2
-(2γ/g)2
2γκ/g2
ω/g
Inversion, N0
(a) (b)
Zero of Re Zero of Im
Equilibrium:
-4
-2
0
2
-2 -1.5 -1 -0.5 0
ω/g
µ/g
UP
LP
µ
non-condensed condensed
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 16 / 40
Poles of Retarded Green’s function and gain[DR(ν)
]−1= ν − ωk + iκ+
g2N0
ν − 2ε+ i2γ= A(ν) + iB(ν)
-1
0
1
-1 0 1
ω/g
(a)
A(ω)B(ω)
0
2
4
6
-1 0 1
Abso
rpti
on
ω/g
-0.6
-0.4
-0.2
0
0.2
N0
-1
0
1
-1 0 1
ω/g
(b)
A(ω)B(ω)
Laser:
-2
-1
0
1
2
-(2γ/g)2
2γκ/g2
ω/g
Inversion, N0
(a) (b)
Zero of Re Zero of Im
Equilibrium:
-4
-2
0
2
-2 -1.5 -1 -0.5 0
ω/g
µ/g
UP
LP
µ
non-condensed condensed
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 16 / 40
Poles of Retarded Green’s function and gain[DR(ν)
]−1= ν − ωk + iκ+
g2N0
ν − 2ε+ i2γ= A(ν) + iB(ν)
-1
0
1
-1 0 1
ω/g
(a)
A(ω)B(ω)
0
2
4
6
-1 0 1
Abso
rpti
on
ω/g
-0.6
-0.4
-0.2
0
0.2
N0
-1
0
1
-1 0 1
ω/g
(b)
A(ω)B(ω)
Laser:
-2
-1
0
1
2
-(2γ/g)2
2γκ/g2
ω/g
Inversion, N0
(a) (b)
Zero of Re Zero of Im
Equilibrium:
-4
-2
0
2
-2 -1.5 -1 -0.5 0
ω/g
µ/g
UP
LP
µ
non-condensed condensed
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 16 / 40
Poles of Retarded Green’s function and gain[DR(ν)
]−1= ν − ωk + iκ+
g2N0
ν − 2ε+ i2γ= A(ν) + iB(ν)
-1
0
1
-1 0 1
ω/g
(a)
A(ω)B(ω)
0
2
4
6
-1 0 1
Abso
rpti
on
ω/g
-0.6
-0.4
-0.2
0
0.2
N0
-1
0
1
-1 0 1
ω/g
(b)
A(ω)B(ω)
Laser:
-2
-1
0
1
2
-(2γ/g)2
2γκ/g2
ω/g
Inversion, N0
(a) (b)
Zero of Re Zero of Im
Equilibrium:
-4
-2
0
2
-2 -1.5 -1 -0.5 0
ω/g
µ/g
UP
LP
µ
non-condensed condensed
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 16 / 40
Non-equilibrium description: baths
Excitons Cavity mode
System
γκ
Bulk photon modesPumping Bath
In−plane momentum
ExcitonPhoton
Energ
y
System
Decaybath
Pump bath
H = Hsys + Hsys,bath + Hbath
Decay bath: Empty (µ→ −∞)Pump bath: Thermal µB,TB
Mean field theory
0
0.1
0.2
0.3
0 0.1 0.2 0.3
TB
ath/g
Density n/n0
Increasing γ
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 17 / 40
Non-equilibrium description: baths
Excitons Cavity mode
System
γκ
Bulk photon modesPumping Bath
In−plane momentum
ExcitonPhoton
Energ
y
System
Decaybath
Pump bath
H = Hsys + Hsys,bath + Hbath
Decay bath: Empty (µ→ −∞)Pump bath: Thermal µB,TB
Mean field theory
0
0.1
0.2
0.3
0 0.1 0.2 0.3
TB
ath/g
Density n/n0
Increasing γ
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 17 / 40
Non-equilibrium description: baths
Excitons Cavity mode
System
γκ
Bulk photon modesPumping Bath
In−plane momentum
ExcitonPhoton
Energ
y
System
Decaybath
Pump bath
H = Hsys + Hsys,bath + Hbath
Decay bath: Empty (µ→ −∞)Pump bath: Thermal µB,TB
Mean field theory
0
0.1
0.2
0.3
0 0.1 0.2 0.3
TB
ath/g
Density n/n0
Increasing γ
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 17 / 40
Stability and evolution with pumping-1.5 -1 -0.5 0 0.5 1
0
1
A(ω)B(ω)
-1.5 -1 -0.5 0 0.5 1Energy (units of g)
0
1
2
3
Inte
nsi
ty (
a.u
.) Density of states, 2 Im[DR]
[DR(ν)
]−1= A(ν) + iB(ν)
2n(ν) + 1 =iDK (ν)
−2=[DR(ν)]=
C(ν)
2B(ν)
-0.6 -0.5 -0.4 -0.3
Bath occupation, µB/g
-3
-2
-1
0
1
Ener
gy o
f ze
ro (
unit
s of
g)
Zero of ReZero of Im
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 18 / 40
Stability and evolution with pumping-1.5 -1 -0.5 0 0.5 1
0
1
A(ω)B(ω)C(ω)
-1.5 -1 -0.5 0 0.5 1Energy (units of g)
0
1
2
3
Inte
nsi
ty (
a.u
.) Density of states, 2 Im[DR]
Occupation, n(ω)
[DR(ν)
]−1= A(ν) + iB(ν)
2n(ν) + 1 =iDK (ν)
−2=[DR(ν)]=
C(ν)
2B(ν)
-0.6 -0.5 -0.4 -0.3
Bath occupation, µB/g
-3
-2
-1
0
1
Ener
gy o
f ze
ro (
unit
s of
g)
Zero of ReZero of Im
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 18 / 40
Stability and evolution with pumping-1.5 -1 -0.5 0 0.5 1
0
1
A(ω)B(ω)C(ω)
-1.5 -1 -0.5 0 0.5 1Energy (units of g)
0
1
2
3
Inte
nsi
ty (
a.u
.) Density of states, 2 Im[DR]
Occupation, n(ω)
[DR(ν)
]−1= A(ν) + iB(ν)
2n(ν) + 1 =iDK (ν)
−2=[DR(ν)]=
C(ν)
2B(ν)
-0.6 -0.5 -0.4 -0.3
Bath occupation, µB/g
-3
-2
-1
0
1
Ener
gy o
f ze
ro (
unit
s of
g)
Zero of ReZero of Im
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 18 / 40
Strong coupling and lasing — low temperaturephenomenon
ω/g
µ/g
ξµeff
-2
-1
0
1
-2 -1 0
non-condensed
condensed
Eqbm. polariton
µB/g-2 -1 0
non-condensed
condensed
Non-eqbm. polariton
Inversion, N0
-1 0 1
non-condensed
condensed
Laser
Laser: Uniformly invert TLSNon-equilibrium polaritons: Cold bathIf TB γ → Laser limit -1.5 -1 -0.5 0 0.5 1
Energy (units of g)
0
1
A(ω)B(ω)
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 19 / 40
Strong coupling and lasing — low temperaturephenomenon
ω/g
µ/g
ξµeff
-2
-1
0
1
-2 -1 0
non-condensed
condensed
Eqbm. polariton
µB/g-2 -1 0
non-condensed
condensed
Non-eqbm. polariton
Inversion, N0
-1 0 1
non-condensed
condensed
Laser
Laser: Uniformly invert TLSNon-equilibrium polaritons: Cold bathIf TB γ → Laser limit -1.5 -1 -0.5 0 0.5 1
Energy (units of g)
0
1
A(ω)B(ω)
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 19 / 40
Strong coupling and lasing — low temperaturephenomenon
ω/g
µ/g
ξµeff
-2
-1
0
1
-2 -1 0
non-condensed
condensed
Eqbm. polariton
µB/g-2 -1 0
non-condensed
condensed
Non-eqbm. polariton
Inversion, N0
-1 0 1
non-condensed
condensed
Laser
Laser: Uniformly invert TLSNon-equilibrium polaritons: Cold bathIf TB γ → Laser limit -1.5 -1 -0.5 0 0.5 1
Energy (units of g)
0
1
A(ω)B(ω)
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 19 / 40
Coherence, inversion, strong-coupling
Polariton condensation:
Inversionlessallows strong couplingrequires low T ↔ condensation
ω/g
µ/g
ξµeff
-2
-1
0
1
-2 -1 0
non-condensed
condensed
Eqbm. polariton
µB/g-2 -1 0
non-condensed
condensed
Non-eqbm. polariton
Inversion, N0
-1 0 1
non-condensed
condensed
Laser
NB NOT thresholdless/single atom lasing.
Related weak-coupling inversionless lasing:
Circuit QED [Marthaler et al. PRL ’11]
ωTLSω
Cavity
Noiseassisted
I Noise-assistedI Off-resonant cavityI Emission/absorption Γ± ∼ 2nB(±δω) + 1I Low T → inversionless threshold
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 20 / 40
Coherence, inversion, strong-coupling
Polariton condensation:
Inversionlessallows strong couplingrequires low T ↔ condensation
ω/g
µ/g
ξµeff
-2
-1
0
1
-2 -1 0
non-condensed
condensed
Eqbm. polariton
µB/g-2 -1 0
non-condensed
condensed
Non-eqbm. polariton
Inversion, N0
-1 0 1
non-condensed
condensed
Laser
NB NOT thresholdless/single atom lasing.
Related weak-coupling inversionless lasing:
Circuit QED [Marthaler et al. PRL ’11]
ωTLSω
Cavity
Noiseassisted I Noise-assisted
I Off-resonant cavityI Emission/absorption Γ± ∼ 2nB(±δω) + 1I Low T → inversionless threshold
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 20 / 40
Polariton and photon Condensation
1 Dicke model and superradiance
2 Polariton and photon condensationPolaritonsNon-equilibrium condensation vs lasingPhoton condensation
3 Jaynes Cummings Hubbard modelJCHM vv DickeCoherently driven arrayDisorder
4 Phase transitions with SC qubitsPumping without symmetry breakingCollective dephasing
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 21 / 40
Photon BEC experiments
Dye filled microcavityPump at angleNo strong couplingCondensation:
I Far below inversionI Thermalised emission spectrum
[Klaers et al, Nature, 2010]
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 22 / 40
Photon BEC experiments
Dye filled microcavityPump at angleNo strong couplingCondensation:
I Far below inversionI Thermalised emission spectrum
[Klaers et al, Nature, 2010]
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 22 / 40
Photon BEC experiments
Dye filled microcavityPump at angleNo strong couplingCondensation:
I Far below inversionI Thermalised emission spectrum
[Klaers et al, Nature, 2010]
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 22 / 40
Photon BEC experiments
Dye filled microcavityPump at angleNo strong couplingCondensation:
I Far below inversionI Thermalised emission spectrum
[Klaers et al, Nature, 2010]
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 22 / 40
Modelling
Hsys =∑
m
ωmψ†mψm +
∑α
[εSz
α + g(ψmS+
α + H.c.)
+ Ω(
b†αbα + 2√
SSzα
(b†α + bα
))
]Consider harmonic cavity modesωm = ωcutoff + mωH.O.
Add local vibrational modeIntegrate out phonon effects
I Polaron transformI Perturbation theory in g
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 23 / 40
Modelling
Hsys =∑
m
ωmψ†mψm +
∑α
[εSz
α + g(ψmS+
α + H.c.)
+ Ω(
b†αbα + 2√
SSzα
(b†α + bα
)) ]Consider harmonic cavity modesωm = ωcutoff + mωH.O.
Add local vibrational modeIntegrate out phonon effects
I Polaron transformI Perturbation theory in g
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 23 / 40
Modelling
Hsys =∑
m
ωmψ†mψm +
∑α
[εSz
α + g(ψmS+
α + H.c.)
+ Ω(
b†αbα + 2√
SSzα
(b†α + bα
)) ]Consider harmonic cavity modesωm = ωcutoff + mωH.O.
Add local vibrational modeIntegrate out phonon effects
I Polaron transformI Perturbation theory in g
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 23 / 40
ModellingRate equation
ρ = −i[H0, ρ]−∑
m
κ
2L[ψm]−
∑α
[Γ↑2L[S+
α ] +Γ↓2L[S−α ]
]
−∑m,α
[Γ(δm = ωm − ε)
2L[S+
α ψm] +Γ(−δm = ε− ωm)
2L[S−α ψ
†m]
]
0
0.2
0.4
0.6
0.8
1
−200 −100 0 100 200δ
Γ(−δ) Γ(δ)
Γ(+δ) ' Γ(−δ)e−βδ
Γ→ 0 at large δ
[Marthaler et al PRL ’11, Kirton & JK arXiv:1303.3459]
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 24 / 40
ModellingRate equation
ρ = −i[H0, ρ]−∑
m
κ
2L[ψm]−
∑α
[Γ↑2L[S+
α ] +Γ↓2L[S−α ]
]−∑m,α
[Γ(δm = ωm − ε)
2L[S+
α ψm] +Γ(−δm = ε− ωm)
2L[S−α ψ
†m]
]
0
0.2
0.4
0.6
0.8
1
−200 −100 0 100 200δ
Γ(−δ) Γ(δ)Γ(+δ) ' Γ(−δ)e−βδ
Γ→ 0 at large δ
[Marthaler et al PRL ’11, Kirton & JK arXiv:1303.3459]
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 24 / 40
ModellingRate equation
ρ = −i[H0, ρ]−∑
m
κ
2L[ψm]−
∑α
[Γ↑2L[S+
α ] +Γ↓2L[S−α ]
]−∑m,α
[Γ(δm = ωm − ε)
2L[S+
α ψm] +Γ(−δm = ε− ωm)
2L[S−α ψ
†m]
]
0
0.2
0.4
0.6
0.8
1
−200 −100 0 100 200δ
Γ(−δ) Γ(δ)Γ(+δ) ' Γ(−δ)e−βδ
Γ→ 0 at large δ
[Marthaler et al PRL ’11, Kirton & JK arXiv:1303.3459]
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 24 / 40
ModellingRate equation
ρ = −i[H0, ρ]−∑
m
κ
2L[ψm]−
∑α
[Γ↑2L[S+
α ] +Γ↓2L[S−α ]
]−∑m,α
[Γ(δm = ωm − ε)
2L[S+
α ψm] +Γ(−δm = ε− ωm)
2L[S−α ψ
†m]
]
0
0.2
0.4
0.6
0.8
1
−200 −100 0 100 200δ
Γ(−δ) Γ(δ)Γ(+δ) ' Γ(−δ)e−βδ
Γ→ 0 at large δ
[Marthaler et al PRL ’11, Kirton & JK arXiv:1303.3459]
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 24 / 40
Distribution gmnm
Rate equation — include spontaneous emissionBose-Einstein distribution without losses
Low loss: Thermal High loss→ Laser
[Kirton & JK arXiv:1303.3459]
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 25 / 40
Distribution gmnm
Rate equation — include spontaneous emissionBose-Einstein distribution without losses
Low loss: Thermal
High loss→ Laser
[Kirton & JK arXiv:1303.3459]
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 25 / 40
Distribution gmnm
Rate equation — include spontaneous emissionBose-Einstein distribution without losses
Low loss: Thermal High loss→ Laser[Kirton & JK arXiv:1303.3459]
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 25 / 40
Threshold condition
200
300
400
500
600
10 5 10 4 10 3 0 01 0 1
(b) (c)κ κ κ
Increasing loss
Compare threshold:Pump rate (Laser)Critical density(condensate)
Thermal at low κ/high temperatureHigh loss, κ competes with Γ(±δ0)
Low temperature, Γ(±δ0) shrinksHigh temperature, thermal, but inversion
[Kirton & JK arXiv:1303.3459]
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 26 / 40
Threshold condition
200
300
400
500
600
10 5 10 4 10 3 0 01 0 1
(b) (c)κ κ κ
Increasing loss
Compare threshold:Pump rate (Laser)Critical density(condensate)
Thermal at low κ/high temperatureHigh loss, κ competes with Γ(±δ0)
Low temperature, Γ(±δ0) shrinksHigh temperature, thermal, but inversion
[Kirton & JK arXiv:1303.3459]
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 26 / 40
Threshold condition
200
300
400
500
600
10 5 10 4 10 3 0 01 0 1
(b) (c)κ κ κ
Increasing loss
Compare threshold:Pump rate (Laser)Critical density(condensate)
Thermal at low κ/high temperatureHigh loss, κ competes with Γ(±δ0)
Low temperature, Γ(±δ0) shrinksHigh temperature, thermal, but inversion
[Kirton & JK arXiv:1303.3459]
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 26 / 40
Threshold condition
200
300
400
500
600
10 5 10 4 10 3 0 01 0 1
(b) (c)κ κ κ
Increasing loss
Compare threshold:Pump rate (Laser)Critical density(condensate)
Thermal at low κ/high temperatureHigh loss, κ competes with Γ(±δ0)
Low temperature, Γ(±δ0) shrinksHigh temperature, thermal, but inversion
[Kirton & JK arXiv:1303.3459]
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 26 / 40
Threshold condition
200
300
400
500
600
10 5 10 4 10 3 0 01 0 1
(b) (c)κ κ κ
Increasing loss
Compare threshold:Pump rate (Laser)Critical density(condensate)
Thermal at low κ/high temperatureHigh loss, κ competes with Γ(±δ0)
Low temperature, Γ(±δ0) shrinksHigh temperature, thermal, but inversion
[Kirton & JK arXiv:1303.3459]
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 26 / 40
Jaynes Cummings Hubbard model
1 Dicke model and superradiance
2 Polariton and photon condensationPolaritonsNon-equilibrium condensation vs lasingPhoton condensation
3 Jaynes Cummings Hubbard modelJCHM vv DickeCoherently driven arrayDisorder
4 Phase transitions with SC qubitsPumping without symmetry breakingCollective dephasing
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 27 / 40
Equilibrium: Dicke model with chemical potential
H − µN = (ω − µ)ψ†ψ + (ω0 − µ)Sz + g(ψ†S− + ψS+
)
-5
-4
-3
-2
-1
0
1
2
-4 -3 -2 -1 0 1 2
(µ-ω
)/g
(ω0 - ω)/g
SRTransition at:g2N > (ω − µ)(ω0 − µ)
Reduce critical gUnstable if µ > ω
Inverted if µ > ω0
[Eastham and Littlewood, PRB ’01]
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 28 / 40
Equilibrium: Dicke model with chemical potential
H − µN = (ω − µ)ψ†ψ + (ω0 − µ)Sz + g(ψ†S− + ψS+
)
-5
-4
-3
-2
-1
0
1
2
-4 -3 -2 -1 0 1 2
(µ-ω
)/g
(ω0 - ω)/g
unstable
SRTransition at:g2N > (ω − µ)(ω0 − µ)
Reduce critical gUnstable if µ > ω
Inverted if µ > ω0
[Eastham and Littlewood, PRB ’01]
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 28 / 40
Equilibrium: Dicke model with chemical potential
H − µN = (ω − µ)ψ†ψ + (ω0 − µ)Sz + g(ψ†S− + ψS+
)
-5
-4
-3
-2
-1
0
1
2
-4 -3 -2 -1 0 1 2
(µ-ω
)/g
(ω0 - ω)/g
⇑
⇓
unstable
SRTransition at:g2N > (ω − µ)(ω0 − µ)
Reduce critical gUnstable if µ > ω
Inverted if µ > ω0
[Eastham and Littlewood, PRB ’01]
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 28 / 40
Jaynes-Cummings Hubbard model
H = −Jz
∑ij
ψ†i ψj +∑
i
∆
2σz
i + g(ψ†i σ−i + H.c.)
-2
-1
0
0.001 0.01 0.1 1
µ/g
J/g
Unstable
Normal
∆/g=1
-10
-8
-6
-4
-2
0
0 1 2 3 4 5 6 7 8 9 10
µ/g
J/g
Unstable
Normal
∆/g=-6
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 29 / 40
Jaynes-Cummings Hubbard model
H = −Jz
∑ij
ψ†i ψj +∑
i
∆
2σz
i + g(ψ†i σ−i + H.c.)
-2
-1
0
0.001 0.01 0.1 1
µ/g
J/g
Unstable
Normal
∆/g=1
-10
-8
-6
-4
-2
0
0 1 2 3 4 5 6 7 8 9 10
µ/g
J/g
Unstable
Normal
∆/g=-6
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 29 / 40
Jaynes-Cummings Hubbard model
H = −Jz
∑ij
ψ†i ψj +∑
i
∆
2σz
i + g(ψ†i σ−i + H.c.)
-2
-1
0
0.001 0.01 0.1 1
µ/g
J/g
Unstable
Normal
∆/g=1
-10
-8
-6
-4
-2
0
0 1 2 3 4 5 6 7 8 9 10
µ/g
J/g
Unstable
Normal
∆/g=-6
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 29 / 40
Dicke vs JCHM
JCHM
-10
-8
-6
-4
-2
0
0 1 2 3 4 5 6 7 8 9 10
µ/g
J/g
Unstable
Normal
∆/g=-6
E
k
UP
Photon
LP
2LS
∆JCHM
∆Dicke
Dicke
-5
-4
-3
-2
-1
0
1
2
-4 -3 -2 -1 0 1 2
(µ-ω
)/g
(ω0 - ω)/g
⇑
⇓
unstable
SR
k = 0 mode of JCHM ↔ Dicke photon mode⇑ ↔ n = 1 Mott lobe
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 30 / 40
Dicke vs JCHM
JCHM
-10
-8
-6
-4
-2
0
0 1 2 3 4 5 6 7 8 9 10
µ/g
J/g
Unstable
Normal
∆/g=-6
E
k
UP
Photon
LP
2LS
∆JCHM
∆Dicke
Dicke
-5
-4
-3
-2
-1
0
1
2
-4 -3 -2 -1 0 1 2
(µ-ω
)/g
(ω0 - ω)/g
⇑
⇓
unstable
SR
k = 0 mode of JCHM ↔ Dicke photon mode⇑ ↔ n = 1 Mott lobe
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 30 / 40
Dicke vs JCHM
JCHM
-10
-8
-6
-4
-2
0
0 1 2 3 4 5 6 7 8 9 10
µ/g
J/g
Unstable
Normal
∆/g=-6
E
k
UP
Photon
LP
2LS
∆JCHM
∆Dicke
Dicke
-5
-4
-3
-2
-1
0
1
2
-4 -3 -2 -1 0 1 2
(µ-ω
)/g
(ω0 - ω)/g
⇑
⇓
unstable
SR
k = 0 mode of JCHM ↔ Dicke photon mode⇑ ↔ n = 1 Mott lobe
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 30 / 40
Dicke vs JCHM
JCHM
-10
-8
-6
-4
-2
0
0 1 2 3 4 5 6 7 8 9 10
µ/g
J/g
Unstable
Normal
∆/g=-6 E
k
UP
Photon
LP
2LS
∆JCHM
∆Dicke
Dicke
-5
-4
-3
-2
-1
0
1
2
-4 -3 -2 -1 0 1 2
(µ-ω
)/g
(ω0 - ω)/g
⇑
⇓
unstable
SR
k = 0 mode of JCHM ↔ Dicke photon mode⇑ ↔ n = 1 Mott lobe
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 30 / 40
Jaynes Cummings Hubbard model
1 Dicke model and superradiance
2 Polariton and photon condensationPolaritonsNon-equilibrium condensation vs lasingPhoton condensation
3 Jaynes Cummings Hubbard modelJCHM vv DickeCoherently driven arrayDisorder
4 Phase transitions with SC qubitsPumping without symmetry breakingCollective dephasing
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 31 / 40
Coherently pumped JCHM
H = −Jz
∑ij
ψ†i ψj +∑
i
∆
2σz
i + g(ψ†i σ−i + H.c.)+f (ψieiωLt + H.c.)
∂tρ = −i[H, ρ]−κ2
Lψ[ρ]− γ
2Lσ− [ρ]
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 32 / 40
Coherently pumped single cavity [Bishop et al. Nat. Phys ’09]
g H =∆
2σz + g(ψ†σ− + H.c.)+f (ψeiωpump t + H.c.)
∂tρ = −i[H, ρ]−κ2
Lψ[ρ]− γ
2Lσ− [ρ]
Anti-resonance in |〈ψ〉|.Effective 2LS:|Empty〉, |1 polariton〉
Increasing Pum
ping
Mollow triplet fluorescence
[Lang et al. PRL ’11]
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 33 / 40
Coherently pumped single cavity [Bishop et al. Nat. Phys ’09]
g H =∆
2σz + g(ψ†σ− + H.c.)+f (ψeiωpump t + H.c.)
∂tρ = −i[H, ρ]−κ2
Lψ[ρ]− γ
2Lσ− [ρ]
Anti-resonance in |〈ψ〉|.Effective 2LS:|Empty〉, |1 polariton〉
Increasing Pum
ping
Mollow triplet fluorescence
[Lang et al. PRL ’11]
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 33 / 40
Coherently pumped single cavity [Bishop et al. Nat. Phys ’09]
g H =∆
2σz + g(ψ†σ− + H.c.)+f (ψeiωpump t + H.c.)
∂tρ = −i[H, ρ]−κ2
Lψ[ρ]− γ
2Lσ− [ρ]
Anti-resonance in |〈ψ〉|.Effective 2LS:|Empty〉, |1 polariton〉
Increasing Pum
ping
Mollow triplet fluorescence
[Lang et al. PRL ’11]
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 33 / 40
Coherently pumped dimer & arrayChose detuning a la Dicke model
ωpump
ωpump
LP
UP
2gCavityQubit
Single cavity
2J
LP
LP
UP
E
k
Photon
Qubit
Array
2g
Evolution of anti-resonance vs J.
0
0.1
0.2
-1.06 -1.04 -1.02 -1
|<a
>|
ωpump/g
Bistability at intermediate JI More/less localised statesI Connects to Dicke limit
[Nissen et al. PRL ’12]
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 34 / 40
Coherently pumped dimer & arrayChose detuning a la Dicke model
ωpump
ωpump
LP
UP
2gCavityQubit
Single cavity
2J
LP
LP
UP
E
k
Photon
Qubit
Array
2g
Evolution of anti-resonance vs J.
0
0.1
0.2
-1.06 -1.04 -1.02 -1
|<a
>|
ωpump/g
Bistability at intermediate JI More/less localised statesI Connects to Dicke limit
[Nissen et al. PRL ’12]
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 34 / 40
Coherently pumped dimer & arrayChose detuning a la Dicke model
ωpump
ωpump
LP
UP
2gCavityQubit
Single cavity
2J
LP
LP
UP
E
k
Photon
Qubit
Array
2g
Evolution of anti-resonance vs J.
0
0.1
0.2
-1.06 -1.04 -1.02 -1
|<a
>|
ωpump/g
Bistability at intermediate JI More/less localised statesI Connects to Dicke limit
[Nissen et al. PRL ’12]
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 34 / 40
Coherently pumped dimer & arrayChose detuning a la Dicke model
ωpump
ωpump
LP
UP
2gCavityQubit
Single cavity
2J
LP
LP
UP
E
k
Photon
Qubit
Array
2g
Evolution of anti-resonance vs J.
0
0.1
0.2
-1.06 -1.04 -1.02 -1
|<a>
|
ωpump/g
Bistability at intermediate JI More/less localised statesI Connects to Dicke limit
[Nissen et al. PRL ’12]
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 34 / 40
Coherently pumped dimer & arrayChose detuning a la Dicke model
ωpump
ωpump
LP
UP
2gCavityQubit
Single cavity
2J
LP
LP
UP
E
k
Photon
Qubit
Array
2g
Evolution of anti-resonance vs J.
0
0.1
0.2
-1.06 -1.04 -1.02 -1
|<a>
|
ωpump/g
Bistability at intermediate JI More/less localised statesI Connects to Dicke limit
[Nissen et al. PRL ’12]
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 34 / 40
Photon blockade picture J . g
Polariton basisNonlinearity |ε2 − 2ε1| ∝ g.
H =∑
i
( ε2τ z
i + f τ xi
)
− Jz
∑〈ij〉
τ+i τ−j
Decouple hopping:τ+i τ
−j → ψτ+ + ψ∗τ−
Bistability for
J > Jc =4f 2
(2f ” + (κ/2)2
3
)3/2
0
0.1
0.2
-1.06 -1.04 -1.02 -1
|<a>
|
ωpump/g
[Nissen et al. PRL ’12]
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 35 / 40
Photon blockade picture J . g
Polariton basisNonlinearity |ε2 − 2ε1| ∝ g.
H =∑
i
( ε2τ z
i + f τ xi
)− J
z
∑〈ij〉
τ+i τ−j
Decouple hopping:τ+i τ
−j → ψτ+ + ψ∗τ−
Bistability for
J > Jc =4f 2
(2f ” + (κ/2)2
3
)3/2
0
0.1
0.2
-1.06 -1.04 -1.02 -1
|<a>
|
ωpump/g
[Nissen et al. PRL ’12]
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 35 / 40
Photon blockade picture J . g
Polariton basisNonlinearity |ε2 − 2ε1| ∝ g.
H =∑
i
( ε2τ z
i + f τ xi
)− J
z
∑〈ij〉
τ+i τ−j
Decouple hopping:τ+i τ
−j → ψτ+ + ψ∗τ−
Bistability for
J > Jc =4f 2
(2f ” + (κ/2)2
3
)3/2
0
0.1
0.2
-1.06 -1.04 -1.02 -1
|<a>
|
ωpump/g
[Nissen et al. PRL ’12]
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 35 / 40
Photon blockade picture J . g
Polariton basisNonlinearity |ε2 − 2ε1| ∝ g.
H =∑
i
( ε2τ z
i + f τ xi
)− J
z
∑〈ij〉
τ+i τ−j
Decouple hopping:τ+i τ
−j → ψτ+ + ψ∗τ−
Bistability for
J > Jc =4f 2
(2f ” + (κ/2)2
3
)3/2
0
0.1
0.2
-1.06 -1.04 -1.02 -1
|<a>
|
ωpump/g
[Nissen et al. PRL ’12]
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 35 / 40
Coherent pumped array – disorder
Effect of disorder, ∆→ ∆iI Distribution of ψ – Washes out bistable jump
Bistability near resonance — phase of ψ depends on ∆i
Complex ψ distributionSuperfluid phases in driven system?
-1.02 -1.01 -1 -0.99 -0.98 -0.97 -0.96 -0.95
Pump frequency
0
0.1
0.2
0.3
ψ
0
20
40
60
80
100
| |
-0.2
0
0.2(a) ωp=-0.988 (b) ωp=-0.987 (c) ωp=-0.986
-0.2
0
0.2(d) ωp=-0.985 (e) ωp=-0.982 (f) ωp=-0.978
-0.2 0 0.2
-0.2
0
0.2
0
20
40
60
80
100
(g) ωp=-0.975
-0.2 0 0.2
(h) ωp=-0.971
-0.2 0 0.2
(i) ωp=-0.968
Re( )
Im(
)
[Kulaitis et al. PRA, ’13]
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 36 / 40
Coherent pumped array – disorder
Effect of disorder, ∆→ ∆iI Distribution of ψ – Washes out bistable jump
Bistability near resonance — phase of ψ depends on ∆i
Complex ψ distributionSuperfluid phases in driven system?
-1.02 -1.01 -1 -0.99 -0.98 -0.97 -0.96 -0.95
Pump frequency
0
0.1
0.2
0.3
ψ
0
20
40
60
80
100
| |
-0.2
0
0.2(a) ωp=-0.988 (b) ωp=-0.987 (c) ωp=-0.986
-0.2
0
0.2(d) ωp=-0.985 (e) ωp=-0.982 (f) ωp=-0.978
-0.2 0 0.2
-0.2
0
0.2
0
20
40
60
80
100
(g) ωp=-0.975
-0.2 0 0.2
(h) ωp=-0.971
-0.2 0 0.2
(i) ωp=-0.968
Re( )
Im(
)
[Kulaitis et al. PRA, ’13]
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 36 / 40
Coherent pumped array – disorder
Effect of disorder, ∆→ ∆iI Distribution of ψ – Washes out bistable jump
Bistability near resonance — phase of ψ depends on ∆i
Complex ψ distributionSuperfluid phases in driven system?
-1.02 -1.01 -1 -0.99 -0.98 -0.97 -0.96 -0.95
Pump frequency
0
0.1
0.2
0.3
ψ
0
20
40
60
80
100
| |
-0.2
0
0.2(a) ωp=-0.988 (b) ωp=-0.987 (c) ωp=-0.986
-0.2
0
0.2(d) ωp=-0.985 (e) ωp=-0.982 (f) ωp=-0.978
-0.2 0 0.2
-0.2
0
0.2
0
20
40
60
80
100
(g) ωp=-0.975
-0.2 0 0.2
(h) ωp=-0.971
-0.2 0 0.2
(i) ωp=-0.968
Re( )
Im(
)
[Kulaitis et al. PRA, ’13]
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 36 / 40
Coherent pumped array – disorder
Effect of disorder, ∆→ ∆iI Distribution of ψ – Washes out bistable jump
Bistability near resonance — phase of ψ depends on ∆i
Complex ψ distributionSuperfluid phases in driven system?
-1.02 -1.01 -1 -0.99 -0.98 -0.97 -0.96 -0.95
Pump frequency
0
0.1
0.2
0.3
ψ
0
20
40
60
80
100
| |
-0.2
0
0.2(a) ωp=-0.988 (b) ωp=-0.987 (c) ωp=-0.986
-0.2
0
0.2(d) ωp=-0.985 (e) ωp=-0.982 (f) ωp=-0.978
-0.2 0 0.2
-0.2
0
0.2
0
20
40
60
80
100
(g) ωp=-0.975
-0.2 0 0.2
(h) ωp=-0.971
-0.2 0 0.2
(i) ωp=-0.968
Re( )
Im(
)
[Kulaitis et al. PRA, ’13]
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 36 / 40
Phase transitions with SC qubits
1 Dicke model and superradiance
2 Polariton and photon condensationPolaritonsNon-equilibrium condensation vs lasingPhoton condensation
3 Jaynes Cummings Hubbard modelJCHM vv DickeCoherently driven arrayDisorder
4 Phase transitions with SC qubitsPumping without symmetry breakingCollective dephasing
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 37 / 40
Raman pumpingHow to pump without breaking symmetryCounter-rotating terms — Raman pumping
I Atom proposal [Dimer et al. PRA ’07]I Atom experiment [Baumann et al. Nature ’10]
Qubit — allowed transitions ∆n = 1Qubit dephasing much bigger than atom
Tunable-coupling-qubit
00
01
10
11
02
20
g
g
0
1
Ω
Ω a
b
PumpCavity
0 0.5 1
g0
0
1
2
3
4
Ωa
= Ω
b=
Ω
⇓
SR?
JK, Tureci, Houck in progressJonathan Keeling Condensation lasing & superradiance FQCMP2013 38 / 40
Raman pumpingHow to pump without breaking symmetryCounter-rotating terms — Raman pumping
I Atom proposal [Dimer et al. PRA ’07]I Atom experiment [Baumann et al. Nature ’10]
Qubit — allowed transitions ∆n = 1Qubit dephasing much bigger than atom
Tunable-coupling-qubit
00
01
10
11
02
20
g
g
0
1
Ω
Ω a
b
PumpCavity
0 0.5 1
g0
0
1
2
3
4
Ωa
= Ω
b=
Ω
⇓
SR?
JK, Tureci, Houck in progressJonathan Keeling Condensation lasing & superradiance FQCMP2013 38 / 40
Raman pumpingHow to pump without breaking symmetryCounter-rotating terms — Raman pumping
I Atom proposal [Dimer et al. PRA ’07]I Atom experiment [Baumann et al. Nature ’10]
Qubit — allowed transitions ∆n = 1Qubit dephasing much bigger than atom
Tunable-coupling-qubit
00
01
10
11
02
20
g
g
0
1
Ω
Ω a
b
PumpCavity
0 0.5 1
g0
0
1
2
3
4
Ωa
= Ω
b=
Ω
⇓
SR?
JK, Tureci, Houck in progressJonathan Keeling Condensation lasing & superradiance FQCMP2013 38 / 40
Raman pumpingHow to pump without breaking symmetryCounter-rotating terms — Raman pumping
I Atom proposal [Dimer et al. PRA ’07]I Atom experiment [Baumann et al. Nature ’10]
Qubit — allowed transitions ∆n = 1Qubit dephasing much bigger than atom
Tunable-coupling-qubit
00
01
10
11
02
20
g
g
0
1
Ω
Ω a
b
PumpCavity
0 0.5 1
g0
0
1
2
3
4
Ωa
= Ω
b=
Ω
⇓
SR?
JK, Tureci, Houck in progressJonathan Keeling Condensation lasing & superradiance FQCMP2013 38 / 40
Raman pumpingHow to pump without breaking symmetryCounter-rotating terms — Raman pumping
I Atom proposal [Dimer et al. PRA ’07]I Atom experiment [Baumann et al. Nature ’10]
Qubit — allowed transitions ∆n = 1Qubit dephasing much bigger than atom
Tunable-coupling-qubit
00
01
10
11
02
20
g
g
0
1
Ω
Ω a
b
PumpCavity
0 0.5 1
g0
0
1
2
3
4
Ωa
= Ω
b=
Ω
⇓
SR?
JK, Tureci, Houck in progressJonathan Keeling Condensation lasing & superradiance FQCMP2013 38 / 40
Collective dephasing
Real environment is not MarkovianI [Carmichael & Walls JPA ’73] Requirements for correct equilibriumI [Ciuti & Carusotto PRA ’09] Dicke SR and emission
Cannot assume fixed κ, γPhase transition→ soft modesStrong coupling→ varying decay
Dicke model linewidth:
H = ωψ†ψ+N∑
i=1
εi2σz
i +g(σ+i ψ + h.c.
)+∑
i
σzi
∑q
γq
(b†q + bq
)+∑
q
βqb†iqbq.
0.008
0.01
0.012
0.014
1 2 3 4 5
linew
idth
/g
number of qubits, N
experimenttheory
⟨a⟩
2 (
a.u
.)
frequency (a.u.)
123
[Nissen, Fink et al. arXiv:1302.0665]
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 39 / 40
Collective dephasing
Real environment is not MarkovianI [Carmichael & Walls JPA ’73] Requirements for correct equilibriumI [Ciuti & Carusotto PRA ’09] Dicke SR and emission
Cannot assume fixed κ, γPhase transition→ soft modesStrong coupling→ varying decay
Dicke model linewidth:
H = ωψ†ψ+N∑
i=1
εi2σz
i +g(σ+i ψ + h.c.
)+∑
i
σzi
∑q
γq
(b†q + bq
)+∑
q
βqb†iqbq.
0.008
0.01
0.012
0.014
1 2 3 4 5
linew
idth
/g
number of qubits, N
experimenttheory
⟨a⟩
2 (
a.u
.)
frequency (a.u.)
123
[Nissen, Fink et al. arXiv:1302.0665]
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 39 / 40
Collective dephasing
Real environment is not MarkovianI [Carmichael & Walls JPA ’73] Requirements for correct equilibriumI [Ciuti & Carusotto PRA ’09] Dicke SR and emission
Cannot assume fixed κ, γPhase transition→ soft modesStrong coupling→ varying decay
Dicke model linewidth:
H = ωψ†ψ+N∑
i=1
εi2σz
i +g(σ+i ψ + h.c.
)+∑
i
σzi
∑q
γq
(b†q + bq
)+∑
q
βqb†iqbq.
0.008
0.01
0.012
0.014
1 2 3 4 5
linew
idth
/g
number of qubits, N
experimenttheory
⟨a⟩
2 (
a.u
.)
frequency (a.u.)
123
[Nissen, Fink et al. arXiv:1302.0665]
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 39 / 40
Collective dephasing
Real environment is not MarkovianI [Carmichael & Walls JPA ’73] Requirements for correct equilibriumI [Ciuti & Carusotto PRA ’09] Dicke SR and emission
Cannot assume fixed κ, γPhase transition→ soft modesStrong coupling→ varying decay
Dicke model linewidth:
H = ωψ†ψ+N∑
i=1
εi2σz
i +g(σ+i ψ + h.c.
)+∑
i
σzi
∑q
γq
(b†q + bq
)+∑
q
βqb†iqbq.
0.008
0.01
0.012
0.014
1 2 3 4 5
linew
idth
/g
number of qubits, N
experimenttheory
⟨a⟩
2 (
a.u
.)
frequency (a.u.)
123
[Nissen, Fink et al. arXiv:1302.0665]
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 39 / 40
Collective dephasing
Real environment is not MarkovianI [Carmichael & Walls JPA ’73] Requirements for correct equilibriumI [Ciuti & Carusotto PRA ’09] Dicke SR and emission
Cannot assume fixed κ, γPhase transition→ soft modesStrong coupling→ varying decay
Dicke model linewidth:
H = ωψ†ψ+N∑
i=1
εi2σz
i +g(σ+i ψ + h.c.
)+∑
i
σzi
∑q
γq
(b†q + bq
)+∑
q
βqb†iqbq.
0.008
0.01
0.012
0.014
1 2 3 4 5
linew
idth
/g
number of qubits, N
experimenttheory
⟨a⟩
2 (
a.u
.)
frequency (a.u.)
123
[Nissen, Fink et al. arXiv:1302.0665]
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 39 / 40
Collective dephasing
Real environment is not MarkovianI [Carmichael & Walls JPA ’73] Requirements for correct equilibriumI [Ciuti & Carusotto PRA ’09] Dicke SR and emission
Cannot assume fixed κ, γPhase transition→ soft modesStrong coupling→ varying decay
Dicke model linewidth:
H = ωψ†ψ+N∑
i=1
εi2σz
i +g(σ+i ψ + h.c.
)+∑
i
σzi
∑q
γq
(b†q + bq
)+∑
q
βqb†iqbq.
0.008
0.01
0.012
0.014
1 2 3 4 5
linew
idth
/g
number of qubits, N
experimenttheory
⟨a⟩
2 (
a.u.)
frequency (a.u.)
123
[Nissen, Fink et al. arXiv:1302.0665]
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 39 / 40
SummaryNon-equilibrium Dicke relevant to increasing number of systems
Quantum WellsCavity
κ
Pump
κ
x
z
0 0.5 1
g0
0
1
2
3
4
Ωa
= Ω
b=
Ω
⇓
SR?
Polariton condensation vs lasing
ω/g
µ/g
ξµeff
-2
-1
0
1
-2 -1 0
non-condensed
condensed
Eqbm. polariton
µB/g-2 -1 0
non-condensed
condensed
Non-eqbm. polariton
Inversion, N0
-1 0 1
non-condensed
condensed
Laser
Photon condensation and thermalisation
Pumped coupled cavity array — bistability and disorder
0
0.1
0.2
-1.06 -1.04 -1.02 -1
|<a
>|
ωpump/g-1.02 -1.01 -1 -0.99 -0.98 -0.97 -0.96 -0.95
Pump frequency
0
0.1
0.2
0.3
ψ
0
20
40
60
80
100
| |
-0.2
0
0.2(a) ωp=-0.988 (b) ωp=-0.987 (c) ωp=-0.986
-0.2
0
0.2(d) ωp=-0.985 (e) ωp=-0.982 (f) ωp=-0.978
-0.2 0 0.2
-0.2
0
0.2
0
20
40
60
80
100
(g) ωp=-0.975
-0.2 0 0.2
(h) ωp=-0.971
-0.2 0 0.2
(i) ωp=-0.968
Re( )
Im(
)
Future prospects – SC cavity array transitions
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 40 / 40
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 41 / 53
Extra slides
5 Ferroelectric transition
6 Pumped JCHM correlations
7 Retarded Green’s function for laser
8 Timescales for Raman pumped experiment
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 42 / 53
Ferroelectric transitionAtoms in Coulomb gauge
H =∑
ωka†kak +∑
i
[pi − eA(ri)]2 + Vcoul
Two-level systems — dipole-dipole coupling
H = ω0Sz + ωψ†ψ + g(S+ + S−)(ψ +ψ†) + Nζ(ψ +ψ†)2−η(S+ − S−)2
(nb g2, ζ, η ∝ 1/V ).Ferroelectric polarisation if ω0 < 2ηN
Gauge transform to dipole gauge D · r
H = ω0Sz + ωψ†ψ + g(S+ − S−)(ψ − ψ†)
“Dicke” transition at ω0 < Ng2/ω ≡ 2ηN
But, ψ describes electric displacement
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 43 / 53
Ferroelectric transitionAtoms in Coulomb gauge
H =∑
ωka†kak +∑
i
[pi − eA(ri)]2 + Vcoul
Two-level systems — dipole-dipole coupling
H = ω0Sz + ωψ†ψ + g(S+ + S−)(ψ +ψ†) + Nζ(ψ +ψ†)2−η(S+ − S−)2
(nb g2, ζ, η ∝ 1/V ).
Ferroelectric polarisation if ω0 < 2ηN
Gauge transform to dipole gauge D · r
H = ω0Sz + ωψ†ψ + g(S+ − S−)(ψ − ψ†)
“Dicke” transition at ω0 < Ng2/ω ≡ 2ηN
But, ψ describes electric displacement
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 43 / 53
Ferroelectric transitionAtoms in Coulomb gauge
H =∑
ωka†kak +∑
i
[pi − eA(ri)]2 + Vcoul
Two-level systems — dipole-dipole coupling
H = ω0Sz + ωψ†ψ + g(S+ + S−)(ψ +ψ†) + Nζ(ψ +ψ†)2−η(S+ − S−)2
(nb g2, ζ, η ∝ 1/V ).Ferroelectric polarisation if ω0 < 2ηN
Gauge transform to dipole gauge D · r
H = ω0Sz + ωψ†ψ + g(S+ − S−)(ψ − ψ†)
“Dicke” transition at ω0 < Ng2/ω ≡ 2ηN
But, ψ describes electric displacement
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 43 / 53
Ferroelectric transitionAtoms in Coulomb gauge
H =∑
ωka†kak +∑
i
[pi − eA(ri)]2 + Vcoul
Two-level systems — dipole-dipole coupling
H = ω0Sz + ωψ†ψ + g(S+ + S−)(ψ +ψ†) + Nζ(ψ +ψ†)2−η(S+ − S−)2
(nb g2, ζ, η ∝ 1/V ).Ferroelectric polarisation if ω0 < 2ηN
Gauge transform to dipole gauge D · r
H = ω0Sz + ωψ†ψ + g(S+ − S−)(ψ − ψ†)
“Dicke” transition at ω0 < Ng2/ω ≡ 2ηN
But, ψ describes electric displacement
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 43 / 53
Extra slides
5 Ferroelectric transition
6 Pumped JCHM correlations
7 Retarded Green’s function for laser
8 Timescales for Raman pumped experiment
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 44 / 53
Coherently pumped array: correlations & fluorescence
0
0.5
1
g2(t
=0
)
0.1
0.2
0.3
0.001 0.01 0.1 1 10
|⟨a⟩|
Hopping zJ/g
Correlationsg2 : 0→ 1 crossover.
Fluorescence
Small J: Mollow tripletLarge J: Off resonancefluorescenceI Pump at collective
resonanceI Mismatch if J 6= 0.
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 45 / 53
Coherently pumped array: correlations & fluorescence
0
0.5
1
g2(t
=0
)
0.1
0.2
0.3
0.001 0.01 0.1 1 10
|⟨a⟩|
Hopping zJ/g
Correlationsg2 : 0→ 1 crossover.
Fluorescence
Small J: Mollow tripletLarge J: Off resonancefluorescenceI Pump at collective
resonanceI Mismatch if J 6= 0.
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 45 / 53
Coherently pumped array: correlations & fluorescence
0
0.5
1
g2(t
=0
)
0.1
0.2
0.3
0.001 0.01 0.1 1 10
|⟨a⟩|
Hopping zJ/g
Correlationsg2 : 0→ 1 crossover.
FluorescenceSmall J: Mollow tripletLarge J: Off resonancefluorescenceI Pump at collective
resonanceI Mismatch if J 6= 0.
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 45 / 53
Coherently pumped array: correlations & fluorescence
0
0.5
1
g2(t
=0
)
0.1
0.2
0.3
0.001 0.01 0.1 1 10
|⟨a⟩|
Hopping zJ/g
Correlationsg2 : 0→ 1 crossover.
FluorescenceSmall J: Mollow tripletLarge J: Off resonancefluorescenceI Pump at collective
resonanceI Mismatch if J 6= 0.
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 45 / 53
Coherently pumped array: correlations & fluorescence
0
0.5
1
g2(t
=0
)
0.1
0.2
0.3
0.001 0.01 0.1 1 10
|⟨a⟩|
Hopping zJ/g
Correlationsg2 : 0→ 1 crossover.
FluorescenceSmall J: Mollow tripletLarge J: Off resonancefluorescenceI Pump at collective
resonanceI Mismatch if J 6= 0.
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 45 / 53
Coherently pumped array: correlations & fluorescence
0
0.5
1
g2(t
=0
)
0.1
0.2
0.3
0.001 0.01 0.1 1 10
|⟨a⟩|
Hopping zJ/g
Correlationsg2 : 0→ 1 crossover.
FluorescenceSmall J: Mollow tripletLarge J: Off resonancefluorescenceI Pump at collective
resonanceI Mismatch if J 6= 0.
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 45 / 53
Extra slides
5 Ferroelectric transition
6 Pumped JCHM correlations
7 Retarded Green’s function for laser
8 Timescales for Raman pumped experiment
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 46 / 53
Maxwell-Bloch Equations: Retarded Green’s function
-1
0
1
-1 0 1
ω/g
(a)
A(ω)
0
2
4
6
-1 0 1
Abso
rpti
on
ω/g
-0.6
-0.4
-0.2
0
0.2
N0
-1
0
1
-1 0 1
ω/g
(b)
A(ω)B(ω)
Introduce DR(ω):Response to perturbation
∂tψ = −iω0ψ − κψ +∑
α gαPα∂tPα = −2iεαPα − 2γP + gαψNα
∂tNα = 2γ(N0 − Nα)− 2gα(ψ∗Pα + P∗αψ)
Absorption = −2=[DR(ω)]
=2B(ω)
A(ω)2 + B(ω)2[DR(ω)
]−1= ω − ωk + iκ+
g2N0
ω − 2ε+ i2γ
= A(ω) + iB(ω)
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 47 / 53
Maxwell-Bloch Equations: Retarded Green’s function
-1
0
1
-1 0 1
ω/g
(a)
A(ω)
0
2
4
6
-1 0 1
Abso
rpti
on
ω/g
-0.6
-0.4
-0.2
0
0.2
N0
-1
0
1
-1 0 1
ω/g
(b)
A(ω)B(ω)
Introduce DR(ω):Response to perturbation
∂tψ = −iω0ψ − κψ +∑
α gαPα∂tPα = −2iεαPα − 2γP + gαψNα
∂tNα = 2γ(N0 − Nα)− 2gα(ψ∗Pα + P∗αψ)
Absorption = −2=[DR(ω)]
=2B(ω)
A(ω)2 + B(ω)2
[DR(ω)
]−1= ω − ωk + iκ+
g2N0
ω − 2ε+ i2γ
= A(ω) + iB(ω)
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 47 / 53
Maxwell-Bloch Equations: Retarded Green’s function
-1
0
1
-1 0 1
ω/g
(a)
A(ω)
0
2
4
6
-1 0 1
Abso
rpti
on
ω/g
-0.6
-0.4
-0.2
0
0.2
N0
-1
0
1
-1 0 1
ω/g
(b)
A(ω)B(ω)
Introduce DR(ω):Response to perturbation
∂tψ = −iω0ψ − κψ +∑
α gαPα∂tPα = −2iεαPα − 2γP + gαψNα
∂tNα = 2γ(N0 − Nα)− 2gα(ψ∗Pα + P∗αψ)
Absorption = −2=[DR(ω)] =2B(ω)
A(ω)2 + B(ω)2[DR(ω)
]−1= ω − ωk + iκ+
g2N0
ω − 2ε+ i2γ= A(ω) + iB(ω)
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 47 / 53
Maxwell-Bloch Equations: Retarded Green’s function
-1
0
1
-1 0 1
ω/g
(a)
A(ω)0
2
4
6
-1 0 1
Abso
rpti
on
ω/g
-0.6
-0.4
-0.2
0
0.2
N0
-1
0
1
-1 0 1
ω/g
(b)
A(ω)B(ω)
Introduce DR(ω):Response to perturbation
∂tψ = −iω0ψ − κψ +∑
α gαPα∂tPα = −2iεαPα − 2γP + gαψNα
∂tNα = 2γ(N0 − Nα)− 2gα(ψ∗Pα + P∗αψ)
Absorption = −2=[DR(ω)] =2B(ω)
A(ω)2 + B(ω)2[DR(ω)
]−1= ω − ωk + iκ+
g2N0
ω − 2ε+ i2γ= A(ω) + iB(ω)
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 47 / 53
Maxwell-Bloch Equations: Retarded Green’s function
-1
0
1
-1 0 1
ω/g
(a)
A(ω)B(ω)
0
2
4
6
-1 0 1
Abso
rpti
on
ω/g
-0.6
-0.4
-0.2
0
0.2
N0
-1
0
1
-1 0 1
ω/g
(b)
A(ω)B(ω)
Introduce DR(ω):Response to perturbation
∂tψ = −iω0ψ − κψ +∑
α gαPα∂tPα = −2iεαPα − 2γP + gαψNα
∂tNα = 2γ(N0 − Nα)− 2gα(ψ∗Pα + P∗αψ)
Absorption = −2=[DR(ω)] =2B(ω)
A(ω)2 + B(ω)2[DR(ω)
]−1= ω − ωk + iκ+
g2N0
ω − 2ε+ i2γ= A(ω) + iB(ω)
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 47 / 53
Maxwell-Bloch Equations: Retarded Green’s function
-1
0
1
-1 0 1
ω/g
(a)
A(ω)B(ω)
0
2
4
6
-1 0 1
Abso
rpti
on
ω/g
-0.6
-0.4
-0.2
0
0.2
N0
-1
0
1
-1 0 1
ω/g
(b)
A(ω)B(ω)
Introduce DR(ω):Response to perturbation
∂tψ = −iω0ψ − κψ +∑
α gαPα∂tPα = −2iεαPα − 2γP + gαψNα
∂tNα = 2γ(N0 − Nα)− 2gα(ψ∗Pα + P∗αψ)
Absorption = −2=[DR(ω)] =2B(ω)
A(ω)2 + B(ω)2[DR(ω)
]−1= ω − ωk + iκ+
g2N0
ω − 2ε+ i2γ= A(ω) + iB(ω)
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 47 / 53
Extra slides
5 Ferroelectric transition
6 Pumped JCHM correlations
7 Retarded Green’s function for laser
8 Timescales for Raman pumped experiment
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 48 / 53
Dynamics: Evolution from normal state
Gray: S = (√
N,√
N,−N/2)Black: Wigner distribution of S, ψ
-40
-20
0
20
40
0 0.5 1 1.5
ω (
MH
z)
g√N (MHz)
(i)
(ii)
(iii)
⇓
⇑
⇓+⇑
SRA
SRA
SRB
UN=-40
Oscillations: ∼ 0.1msDecay: 20ms, 0.1ms, 20ms
(i) SR(A)
0 20 40 60 80t (ms)
0
40
80
|ψ|2 0 1 2
0
100
(ii) SR(B)
0 0.1 0.2 0.3 0.4t (ms)
0
100
200
|ψ|2
(iii) SR(A)
0 100 200t (ms)
0
40
80
120
|ψ|2 150 151
40
50
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 49 / 53
Asymptotic state: Evolution from normal state
(Near to experimental UN = −13MHz).
All stable attractors:
-40
-20
0
20
40
0 0.5 1 1.5
ω (
MH
z)
g√N (MHz)
⇑
⇓ SRA
SRA
⇓+⇑ SRB
UN=-10
Starting from ⇓
-40
-20
0
20
40
0 0.5 1 1.5
ω (
MH
z)
g√N (MHz)
⇑
⇓
SRA
SRB
SRA
10-1
100
101
102
103
|ψ|2
Asymptotic state
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 50 / 53
Asymptotic state: Evolution from normal state
(Near to experimental UN = −13MHz).
All stable attractors:
-40
-20
0
20
40
0 0.5 1 1.5
ω (
MH
z)
g√N (MHz)
⇑
⇓ SRA
SRA
⇓+⇑ SRB
UN=-10
Starting from ⇓
-40
-20
0
20
40
0 0.5 1 1.5ω
(M
Hz)
g√N (MHz)
⇑
⇓
SRA
SRB
SRA
10-1
100
101
102
103
|ψ|2
Asymptotic state
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 50 / 53
Timescales for dynamics: What are they?
-40
-20
0
20
40
0 0.5 1 1.5
ω (
MH
z)
g√N (MHz)
⇑
⇓
SRA
SRB
SRA
10-1
100
101
102
103
|ψ|2
Asymptotic state
Growth Most unstable eigenvaluesnear S = (0,0,−N/2)
Decay Slowest stable eigenvaluesnear final state
-40
-20
0
20
40
0 0.5 1 1.5
ω (
MH
z)
g√N (MHz)
No unstable
directions
Two unstable directions
One unstable direction
10µs
100µs
1ms
10ms
100ms
1s
10sInitial growth time
-40
-20
0
20
40
0 0.5 1 1.5
ω (
MH
z)
g√N (MHz)
⇑
⇓
SRA
SRB
SRA
10µs
100µs
1ms
10ms
100ms
1s
10sAsymptotic decay time
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 51 / 53
Timescales for dynamics: What are they?
-40
-20
0
20
40
0 0.5 1 1.5
ω (
MH
z)
g√N (MHz)
⇑
⇓
SRA
SRB
SRA
10-1
100
101
102
103
|ψ|2
Asymptotic state
Growth Most unstable eigenvaluesnear S = (0,0,−N/2)
Decay Slowest stable eigenvaluesnear final state
-40
-20
0
20
40
0 0.5 1 1.5
ω (
MH
z)
g√N (MHz)
No unstable
directions
Two unstable directions
One unstable direction
10µs
100µs
1ms
10ms
100ms
1s
10sInitial growth time
-40
-20
0
20
40
0 0.5 1 1.5
ω (
MH
z)
g√N (MHz)
⇑
⇓
SRA
SRB
SRA
10µs
100µs
1ms
10ms
100ms
1s
10sAsymptotic decay time
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 51 / 53
Timescales for dynamics: Consequences forexperiment
-40
-20
0
20
40
0 0.5 1 1.5
ω (
MH
z)
g√N (MHz)
⇑
⇓
SRA
SRB
SRA
10-1
100
101
102
103
|ψ|2
Asymptotic state
-40
-20
0
20
40
60
0.0 0.5 1.0 1.5 2.0 2.5
ω (
MH
z)
g2 N (MHz
2)
10-1
100
101
102
10310ms sweep
-40
-20
0
20
40
60
0.0 0.5 1.0 1.5 2.0 2.5
ω (
MH
z)
g2 N (MHz
2)
10-1
100
101
102
103200ms sweep
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 52 / 53
Timescales for dynamics: Consequences forexperiment
-40
-20
0
20
40
0 0.5 1 1.5
ω (
MH
z)
g√N (MHz)
⇑
⇓
SRA
SRB
SRA
10-1
100
101
102
103
|ψ|2
Asymptotic state
-40
-20
0
20
40
60
0.0 0.5 1.0 1.5 2.0 2.5
ω (
MH
z)
g2 N (MHz
2)
10-1
100
101
102
10310ms sweep
-40
-20
0
20
40
60
0.0 0.5 1.0 1.5 2.0 2.5
ω (
MH
z)
g2 N (MHz
2)
10-1
100
101
102
103200ms sweep
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 52 / 53
Timescales for dynamics: Consequences forexperiment
-40
-20
0
20
40
0 0.5 1 1.5
ω (
MH
z)
g√N (MHz)
⇑
⇓
SRA
SRB
SRA
10-1
100
101
102
103
|ψ|2
Asymptotic state
-40
-20
0
20
40
60
0.0 0.5 1.0 1.5 2.0 2.5
ω (
MH
z)
g2 N (MHz
2)
10-1
100
101
102
10310ms sweep
-40
-20
0
20
40
60
0.0 0.5 1.0 1.5 2.0 2.5
ω (
MH
z)
g2 N (MHz
2)
10-1
100
101
102
103200ms sweep
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 52 / 53
Timescales for dynamics: Why so slow and varied?Suppose co- and counter-rotating terms differ
2 Level System
Ω
∆
Ω
ψb
a b
a
∆
ψg0
g0
H = . . .+ g(ψ†S− + ψS+) + g′(ψ†S+ + ψS−) + . . .
δg = g′ − g, 2g = g′ + g
-40
-20
0
20
40
0 0.5 1 1.5
ω (
MH
z)
g√N (MHz)
⇑
⇓ SRA
SRA
⇓+⇑ SRB
UN=-10
-40
-20
0
20
40
-0.01 -0.005 0 0.005 0.01
ω (
MH
z)
δg/g-
g-√N=1
SR(A) near phase boundary at small δg → Critical slowing downSR(A), SR(B) continuously connect
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 53 / 53
Timescales for dynamics: Why so slow and varied?Suppose co- and counter-rotating terms differ
2 Level System
Ω
∆
Ω
ψb
a b
a
∆
ψg0
g0
H = . . .+ g(ψ†S− + ψS+) + g′(ψ†S+ + ψS−) + . . .
δg = g′ − g, 2g = g′ + g
-40
-20
0
20
40
0 0.5 1 1.5
ω (
MH
z)
g√N (MHz)
⇑
⇓ SRA
SRA
⇓+⇑ SRB
UN=-10
-40
-20
0
20
40
-0.01 -0.005 0 0.005 0.01
ω (
MH
z)
δg/g-
g-√N=1
SR(A) near phase boundary at small δg → Critical slowing downSR(A), SR(B) continuously connect
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 53 / 53
Timescales for dynamics: Why so slow and varied?Suppose co- and counter-rotating terms differ
2 Level System
Ω
∆
Ω
ψb
a b
a
∆
ψg0
g0
H = . . .+ g(ψ†S− + ψS+) + g′(ψ†S+ + ψS−) + . . .
δg = g′ − g, 2g = g′ + g
-40
-20
0
20
40
0 0.5 1 1.5
ω (
MH
z)
g√N (MHz)
⇑
⇓ SRA
SRA
⇓+⇑ SRB
UN=-10
-40
-20
0
20
40
-0.01 -0.005 0 0.005 0.01
ω (
MH
z)
δg/g-
g-√N=1
SR(A) near phase boundary at small δg → Critical slowing downSR(A), SR(B) continuously connect
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 53 / 53
Timescales for dynamics: Why so slow and varied?Suppose co- and counter-rotating terms differ
2 Level System
Ω
∆
Ω
ψb
a b
a
∆
ψg0
g0
H = . . .+ g(ψ†S− + ψS+) + g′(ψ†S+ + ψS−) + . . .
δg = g′ − g, 2g = g′ + g
-40
-20
0
20
40
0 0.5 1 1.5
ω (
MH
z)
g√N (MHz)
⇑
⇓ SRA
SRA
⇓+⇑ SRB
UN=-10
-40
-20
0
20
40
-0.01 -0.005 0 0.005 0.01
ω (
MH
z)
δg/g-
g-√N=1
SR(A) near phase boundary at small δg → Critical slowing downSR(A), SR(B) continuously connect
Jonathan Keeling Condensation lasing & superradiance FQCMP2013 53 / 53