Jonathan Keeling - University of St Andrewsjmjk/talks/2013-03-polari...Non-equilibrium coherence in...

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:


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


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


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]



Hint =∑k ,i

gk


)If |ri − rj | λ, use

∑i Si → S

Collective decay:dρdt

= −Γ


]



dt=

ΓN2

4sech2

[ΓN2

t]

-N/2

0

N/2

tD

⟨Sz⟩

tD

0

ΓN2/2

I=-Γ

d⟨S

z⟩/

dt




Hint =∑k ,i

gk



∑i Si → S


= −Γ


]



dt=

ΓN2

4sech2

[ΓN2

t]

-N/2

0

N/2

tD

⟨Sz⟩

tD

0

ΓN2/2

I=-Γ

d⟨S

z⟩/

dt




Hint =∑k ,i

gk



∑i Si → S


= −Γ


]



dt=

ΓN2

4sech2

[ΓN2

t]

-N/2

0

N/2

tD

⟨Sz⟩

tD

0

ΓN2/2

I=-Γ

d⟨S

z⟩/

dt



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]


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]


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]




).




00

ω

g-√N

⇓ SR





).




00

ω

g-√N

⇓ SR





).




00

ω

g-√N

⇓ SR



Dicke model: Superradiance at T 6= 0


).

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



Dicke model: Superradiance at T 6= 0


).

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



No go theorem and transition


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




−∑

i

em

A · pi ⇔ g(ψ†S− + ψS+),∑

i

A2

2m⇔ Nζ(ψ + ψ†)2








−∑

i

em

A · pi ⇔ g(ψ†S− + ψS+),∑

i

A2

2m⇔ Nζ(ψ + ψ†)2








−∑

i

em

A · pi ⇔ g(ψ†S− + ψS+),∑

i

A2

2m⇔ Nζ(ψ + ψ†)2








−∑

i

em

A · pi ⇔ g(ψ†S− + ψS+),∑

i

A2

2m⇔ Nζ(ψ + ψ†)2



But Thomas-Reiche-Kuhn sum rule states: g2/ω0 < 2ζ. No transition[Rzazewski et al PRL ’75]


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







condensation.


κ

Pump

κCavity

Pump







condensation.


κ

Pump

κCavity

Pump







condensation.


κ

Pump

κCavity

Pump







condensation.


κ

Pump

κCavity

Pump


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


Polariton and photon Condensation






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



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



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


Quantum WellsCavity

Cavity

θ

Cavity photons:

ωk =√ω2

0 + c2k2

' ω0 + k2/2m∗


Exciton

Ene

rgy

Phot



Quantum WellsCavity

Cavity

θ

Cavity photons:

ωk =√ω2

0 + c2k2

' ω0 + k2/2m∗


Exciton

Ene

rgy

Phot


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

]


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

]


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



H − µN =∑

k


(εα − µ)Szα +

gα,k√AψkS+

α + H.c.


(ω − µ)ψ =1A

∑α

g2αψ


(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




H − µN =∑

k


(εα − µ)Szα +

gα,k√AψkS+

α + H.c.


(ω − µ)ψ =1A

∑α

g2αψ


(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




H − µN =∑

k


(εα − µ)Szα +

gα,k√AψkS+

α + H.c.


(ω − µ)ψ =1A

∑α

g2αψ


(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




H − µN =∑

k


(εα − µ)Szα +

gα,k√AψkS+

α + H.c.


(ω − µ)ψ =1A

∑α

g2αψ


(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



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




εαSzα +

gα,k√AψS+

α + H.c.


∑α gαPα



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


√n





εαSzα +

gα,k√AψS+

α + H.c.


∑α gαPα



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


√n





εαSzα +

gα,k√AψS+

α + H.c.


∑α gαPα



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


√n



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




]−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)


Equilibrium:

-4

-2

0

2

-2 -1.5 -1 -0.5 0

ω/g

µ/g

UP

LP

µ




]−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)


Equilibrium:

-4

-2

0

2

-2 -1.5 -1 -0.5 0

ω/g

µ/g

UP

LP

µ




]−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)


Equilibrium:

-4

-2

0

2

-2 -1.5 -1 -0.5 0

ω/g

µ/g

UP

LP

µ




]−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)


Equilibrium:

-4

-2

0

2

-2 -1.5 -1 -0.5 0

ω/g

µ/g

UP

LP

µ



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




System

γκ


In−plane momentum

ExcitonPhoton

Energ

y

System

Decaybath

Pump bath



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




System

γκ


In−plane momentum

ExcitonPhoton

Energ

y

System

Decaybath

Pump bath



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


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



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


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


-3

-2

-1

0

1

Ener

gy o

f ze

ro (

unit

s of

g)




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


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


-3

-2

-1

0

1

Ener

gy o

f ze

ro (

unit

s of

g)



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)



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


Energy (units of g)

0

1

A(ω)B(ω)



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


Energy (units of g)

0

1

A(ω)B(ω)


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


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


Polariton and photon Condensation






Photon BEC experiments

Dye filled microcavityPump at angleNo strong couplingCondensation:

I Far below inversionI Thermalised emission spectrum

[Klaers et al, Nature, 2010]

















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


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.




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.




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]



ρ = −i[H0, ρ]−∑

m

κ

2L[ψm]−

∑α

[Γ↑2L[S+

α ] +Γ↓2L[S−α ]

]−∑m,α


2L[S+


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




ρ = −i[H0, ρ]−∑

m

κ

2L[ψm]−

∑α

[Γ↑2L[S+

α ] +Γ↓2L[S−α ]

]−∑m,α


2L[S+


2L[S−α ψ

†m]

]

0

0.2

0.4

0.6

0.8

1

−200 −100 0 100 200δ


Γ→ 0 at large δ




ρ = −i[H0, ρ]−∑

m

κ

2L[ψm]−

∑α

[Γ↑2L[S+

α ] +Γ↓2L[S−α ]

]−∑m,α


2L[S+


2L[S−α ψ

†m]

]

0

0.2

0.4

0.6

0.8

1

−200 −100 0 100 200δ


Γ→ 0 at large δ



Distribution gmnm

Rate equation — include spontaneous emissionBose-Einstein distribution without losses

Low loss: Thermal High loss→ Laser

[Kirton & JK arXiv:1303.3459]


Distribution gmnm


Low loss: Thermal

High loss→ Laser



Distribution gmnm


Low loss: Thermal High loss→ Laser[Kirton & JK arXiv:1303.3459]


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



Threshold condition

200

300

400

500

600

10 5 10 4 10 3 0 01 0 1

(b) (c)κ κ κ

Increasing loss






Threshold condition

200

300

400

500

600

10 5 10 4 10 3 0 01 0 1

(b) (c)κ κ κ

Increasing loss






Threshold condition

200

300

400

500

600

10 5 10 4 10 3 0 01 0 1

(b) (c)κ κ κ

Increasing loss






Threshold condition

200

300

400

500

600

10 5 10 4 10 3 0 01 0 1

(b) (c)κ κ κ

Increasing loss






Jaynes Cummings Hubbard model






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]




)

-5

-4

-3

-2

-1

0

1

2

-4 -3 -2 -1 0 1 2

(µ-ω

)/g

(ω0 - ω)/g

unstable








)

-5

-4

-3

-2

-1

0

1

2

-4 -3 -2 -1 0 1 2

(µ-ω

)/g

(ω0 - ω)/g

⇑

⇓

unstable






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



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



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


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


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



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



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



Jaynes Cummings Hubbard model






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]


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]



g H =∆


∂tρ = −i[H, ρ]−κ2

Lψ[ρ]− γ

2Lσ− [ρ]


Increasing Pum

ping





g H =∆


∂tρ = −i[H, ρ]−κ2

Lψ[ρ]− γ

2Lσ− [ρ]


Increasing Pum

ping




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]



ωpump

ωpump

LP

UP

2gCavityQubit

Single cavity

2J

LP

LP

UP

E

k

Photon

Qubit

Array

2g


0

0.1

0.2

-1.06 -1.04 -1.02 -1

|<a

>|

ωpump/g





ωpump

ωpump

LP

UP

2gCavityQubit

Single cavity

2J

LP

LP

UP

E

k

Photon

Qubit

Array

2g


0

0.1

0.2

-1.06 -1.04 -1.02 -1

|<a

>|

ωpump/g





ωpump

ωpump

LP

UP

2gCavityQubit

Single cavity

2J

LP

LP

UP

E

k

Photon

Qubit

Array

2g


0

0.1

0.2

-1.06 -1.04 -1.02 -1

|<a>

|

ωpump/g





ωpump

ωpump

LP

UP

2gCavityQubit

Single cavity

2J

LP

LP

UP

E

k

Photon

Qubit

Array

2g


0

0.1

0.2

-1.06 -1.04 -1.02 -1

|<a>

|

ωpump/g




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





H =∑

i

( ε2τ z

i + f τ xi

)− J

z

∑〈ij〉

τ+i τ−j


−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





H =∑

i

( ε2τ z

i + f τ xi

)− J

z

∑〈ij〉

τ+i τ−j


−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





H =∑

i

( ε2τ z

i + f τ xi

)− J

z

∑〈ij〉

τ+i τ−j


−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



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]






-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(

)







-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(

)







-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(

)



Phase transitions with SC qubits






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





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?






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?






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?






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?


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]






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







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







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







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







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



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



Extra slides

5 Ferroelectric transition

6 Pumped JCHM correlations

7 Retarded Green’s function for laser

8 Timescales for Raman pumped experiment


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



H =∑

ωka†kak +∑

i




(nb g2, ζ, η ∝ 1/V ).

Ferroelectric polarisation if ω0 < 2ηN


H = ω0Sz + ωψ†ψ + g(S+ − S−)(ψ − ψ†)





H =∑

ωka†kak +∑

i






H = ω0Sz + ωψ†ψ + g(S+ − S−)(ψ − ψ†)





H =∑

ωka†kak +∑

i






H = ω0Sz + ωψ†ψ + g(S+ − S−)(ψ − ψ†)




Extra slides






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.



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


Fluorescence

Small J: Mollow tripletLarge J: Off resonancefluorescenceI Pump at collective




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


FluorescenceSmall J: Mollow tripletLarge J: Off resonancefluorescenceI Pump at collective




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






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






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





Extra slides






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


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)



-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)


∂tψ = −iω0ψ − κψ +∑



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)



-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)


∂tψ = −iω0ψ − κψ +∑



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)



-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)


∂tψ = −iω0ψ − κψ +∑




A(ω)2 + B(ω)2[DR(ω)

]−1= ω − ωk + iκ+

g2N0

ω − 2ε+ i2γ= A(ω) + iB(ω)



-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)


∂tψ = −iω0ψ − κψ +∑




A(ω)2 + B(ω)2[DR(ω)

]−1= ω − ωk + iκ+

g2N0

ω − 2ε+ i2γ= A(ω) + iB(ω)



-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)


∂tψ = −iω0ψ − κψ +∑




A(ω)2 + B(ω)2[DR(ω)

]−1= ω − ωk + iκ+

g2N0

ω − 2ε+ i2γ= A(ω) + iB(ω)


Extra slides






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


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


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


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


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


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



-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



-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


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



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




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




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



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