Jonathan Keeling - University of St Andrewsjmjk/talks/2017-04-trieste.pdf · Modelling organic...

Modelling organic condensates from weak tostrong coupling

Jonathan Keeling

University of

St Andrews

FOUNDED

1413

SUPA

SISSA, April 2017

Jonathan Keeling Polariton and photon condensates SISSA, April 2017 1

Condensation, Lasing, Superradiance

Atomic BEC T ∼ 10−7K

[Anderson et al. Science ’95]

Polariton Condensate T ∼ 20K

Quantum Wells"Cavity"

[Kasprzak et al. Nature, ’06]

Photon CondensateT ∼ 300K

[Klaers et al. Nature, ’10]

LaserT ∼?, < 0,∞

Superradiance transitionT ∼ 0

κ

Pump

κ

x

z

[Baumann et al. Nature ’10]


Organic polaritons: What & Why

Why: Polariton splitting∼ 1eV � kBTRoom

In−plane momentum

Exciton

Ene

rgy

Phot

on

Examples:

Anthracene Polariton Lasing

[Kena Cohen and Forrest, Nat. Photon ’10]

Polymers (MeLPPP, TDAF)

[Plumhoff et al. Nat. Materials ’14, Daskalakis etal. ibid]

Biologically produced materials (GFP)[Dietrich et al. Sci. Adv. ’16]




In−plane momentum

Exciton

Ene

rgy

Phot

on

Examples:

Anthracene Polariton Lasing








In−plane momentum

Exciton

Ene

rgy

Phot

on

Examples:Anthracene Polariton Lasing






Motivation: vacuum-state strong coupling

Linear response (no pump, nocondensate): effects ofmatter-light coupling alone.

[Canaguier-Durand et al. Angew. Chem. ’13;Baumberg group]

Q1. Can ultra-strong couplingto light change:I charge distribution?I vibrational configuration?I molecular orientation?I crystal structure?

Q2. Are changes collective(√

N factor) or not?


Motivation: vacuum-state strong coupling

Linear response (no pump, nocondensate): effects ofmatter-light coupling alone.

[Canaguier-Durand et al. Angew. Chem. ’13;Baumberg group]

Q1. Can ultra-strong couplingto light change:I charge distribution?I vibrational configuration?I molecular orientation?I crystal structure?

Q2. Are changes collective(√

N factor) or not?


Motivation: Bose-Einstein condensation of photons

(Curved) microcavityOrganic R6G dye (in solvent)Thermalisation of lightCondensation at P > Pth

[Klaers et al, Nature, 2010]










Paradigms & Models

Weakly interacting dilute Bose gas

H =

∫dd r ψ†(−µ−∇2)ψ + Uψ†ψ†ψψ

I Single field — assumes strong coupling

I Continuum model, hard to include molecular physics

7

Laser rate equationsI Emission, absorption — assumes weak coupling, lasing.

7

Complex Gross-Pitaevskii/Ginzburg Landau equationsi∂tψ =

(−∇2ψ + V (r) + U|ψ|2

)ψ + i (P(ψ,n, r)− κ)ψ

I Applies to laser, condensate — fluids of light

I Continuum theory

7


Paradigms & Models


H =




7


7


(−∇2ψ + V (r) + U|ψ|2

)ψ + i (P(ψ,n, r)− κ)ψ


I Continuum theory

7


Paradigms & Models


H =




7


7


(−∇2ψ + V (r) + U|ψ|2

)ψ + i (P(ψ,n, r)− κ)ψ


I Continuum theory

7


Paradigms & Models


H =




7


7Complex Gross-Pitaevskii/Ginzburg Landau equationsi∂tψ =

(−∇2ψ + V (r) + U|ψ|2

)ψ + i (P(ψ,n, r)− κ)ψ


I Continuum theory

7


Paradigms & Models


H =




7



(−∇2ψ + V (r) + U|ψ|2

)ψ + i (P(ψ,n, r)− κ)ψ


I Continuum theory

7


Paradigms & Models


H =




7



(−∇2ψ + V (r) + U|ψ|2

)ψ + i (P(ψ,n, r)− κ)ψ


I Continuum theory

7


What kinds of modelling

Top-downI Equilibrium stat. mech.I (complex/stochastic/. . . )GPE (+

Boltzmann)→ condensateI Rate equations→ laser

Tractable microscopic toy modelsBottom up

I DFT (or quantum chemistry)→ electronic structure

I Time-dependent DFT /MD→ vibrational spectra

I FDTD/transfer-matrix→ cavity modes


















Toy models1 Full molecular spectra electronic

structure & Raman spectrum

2 Focus on low-energy effective theory

Two-level system, HOMO/LUMOSingle DoF PES

⇑

En

erg

y

nuclear coordinate⇓

See also [Galego, Garcia-Vidal,Feist. PRX ’15]

3 Simplified archetypal model: Dicke-Holstein

Each molecule: two DoFI Electronic state: 2LSI Vibrational state: harmonic oscillator






⇑

En

erg

y










⇑

En

erg

y


Photon









⇑

En

erg

y


Photon









⇑

En

erg

y


Photon





Holstein-Tavis-Cummings modelModel capable of lasing & condensation

H = ωa†a +N∑

i=1

[ωXσ

+i σ−i + g

(σ+i a + H.c.

)

+ ωv

(b†i bi − λ0σ

+i σ−i (b†i + bi)

)

]

⇑

En

erg

y


Photon

Cwik et al. EPL 105 ’14; Spano, J. Chem. Phys ’15; Galego et al. PRX ’15; Cwik etal. PRA ’16; Herrera & Spano PRL ’16; Wu et al. PRB ’16; Zeb etal. arXiv:1608.08929; Herrera & Spano arXiv:1610.04252; . . .


Holstein-Tavis-Cummings modelModel capable of lasing & condensation

H = ωa†a +N∑

i=1

[ωXσ

+i σ−i + g

(σ+i a + H.c.

)+ ωv

(b†i bi − λ0σ


)]

⇑

En

erg

ynuclear coordinate

⇓

Photon

Cwik et al. EPL 105 ’14; Spano, J. Chem. Phys ’15; Galego et al. PRX ’15; Cwik etal. PRA ’16; Herrera & Spano PRL ’16; Wu et al. PRB ’16; Zeb etal. arXiv:1608.08929; Herrera & Spano arXiv:1610.04252; . . .


Introduction and models

1 Introduction and modelsHolstein-Dicke model

2 Weak couplingPhoton BECSpatial profile

3 Strong couplingExact eigenstatesSpectrum

4 Ultra strong couplingVibrational reconfigurationVibrations and disorder


Weak coupling






Photon: Microscopic Model

H =∑

m

ωma†mam +N∑

i=1

[ωXσ

+i σ−i + g

(σ+i am + H.c.

)+ ωv

(b†i bi − λ0σ


)]2D harmonic oscillatorωm = ωcutoff + mωH.O.

Incoherent processes: excitation,decay, loss, vibrationalthermalisation.Weak coupling, perturbative in g



H =∑

m

ωma†mam +N∑

i=1

[ωXσ

+i σ−i + g

(σ+i am + H.c.

)+ ωv

(b†i bi − λ0σ






H =∑

m

ωma†mam +N∑

i=1

[ωXσ

+i σ−i + g

(σ+i am + H.c.

)+ ωv

(b†i bi − λ0σ





Steady state populations and equilibriumRate equation: ∂tnm = −κnm + Γ(−δm)(nm + 1)N↑ − Γ(δm)nmN↓

Steady state distribution:

nm

nm + 1=

Γ(−δm)N↑Γ(δm)N↓

Microscopic conditions for equilibrium:I Emission/absorption rate:

Γ(δ) ' 2g2 Re[∫

dte−iδt〈D†α(t)Dα(0)〉]

Dα = exp(

2λ0(bα − b†α))

I Equilibrium,→ Kubo-Martin-Schwingercondition:

〈D†α(t)Dα(0)〉 = 〈D†α(−t − iβ)Dα(0)〉 ↔ Γ(+δ) = Γ(−δ)eβδ

0

0.2

0.4

0.6

0.8

1

-400 -200 0 200 400

Spec

tru

m

δ=ω - ωZPL

Γ(-δ)Γ(δ)

[Kirton & JK PRL ’13]Jonathan Keeling Polariton and photon condensates SISSA, April 2017 13



nm

nm + 1=

Γ(−δm)N↑κ+ Γ(δm)N↓


Γ(δ) ' 2g2 Re[∫


Dα = exp(

2λ0(bα − b†α))



0

0.2

0.4

0.6

0.8

1

-400 -200 0 200 400

Spec

tru

m

δ=ω - ωZPL

Γ(-δ)Γ(δ)




nm

nm + 1=



Γ(δ) ' 2g2 Re[∫


Dα = exp(

2λ0(bα − b†α))



0

0.2

0.4

0.6

0.8

1

-400 -200 0 200 400

Spec

trum

δ=ω - ωZPL

Γ(-δ)Γ(δ)




nm

nm + 1=



Γ(δ) ' 2g2 Re[∫


Dα = exp(

2λ0(bα − b†α))



0

0.2

0.4

0.6

0.8

1

-400 -200 0 200 400

Spec

trum

δ=ω - ωZPL

Γ(-δ)Γ(δ)




nm

nm + 1=



Γ(δ) ' 2g2 Re[∫


Dα = exp(

2λ0(bα − b†α))



0

0.2

0.4

0.6

0.8

1

-400 -200 0 200 400

Spec

trum

δ=ω - ωZPL

Γ(-δ)Γ(δ)




nm

nm + 1=



Γ(δ) ' 2g2 Re[∫


Dα = exp(

2λ0(bα − b†α))



0

0.2

0.4

0.6

0.8

1

-400 -200 0 200 400

Spec

trum

δ=ω - ωZPL

Γ(-δ)Γ(δ)


Chemical potential?

Steady state, thermalised:

nm

nm + 1=


' e−βδm+βµ,

eβµ ≡N↑N↓

=Γ↑ +

∑m Γ(δm)nm

Γ↓ +∑

m Γ(−δm)(nm + 1)

Below threshold,

µ = kBT ln[Γ↑/Γ↓]

At/above threshold, µ→ δ0

[Kirton & JK, PRA ’15]


Chemical potential?


nm

nm + 1=


' e−βδm+βµ,

eβµ ≡N↑N↓

=Γ↑ +

∑m Γ(δm)nm

Γ↓ +∑

m Γ(−δm)(nm + 1)

Below threshold,





Chemical potential?


nm

nm + 1=


' e−βδm+βµ,

eβµ ≡N↑N↓

=Γ↑ +

∑m Γ(δm)nm

Γ↓ +∑

m Γ(−δm)(nm + 1)

Below threshold,





Chemical potential?


nm

nm + 1=


' e−βδm+βµ,

eβµ ≡N↑N↓

=Γ↑ +

∑m Γ(δm)nm

Γ↓ +∑

m Γ(−δm)(nm + 1)

Below threshold,





Weak coupling






Spatially varying pump intensity

Consider effects of pump profile, Γ↑(r) =Γ↑ exp

(−r2/2σ2

p)

(2πσ2p)d/2

Experiments: [Marelic & Nyman, PRA ’15]


Spatially varying pump intensity

Consider effects of pump profile, Γ↑(r) =Γ↑ exp

(−r2/2σ2

p)

(2πσ2p)d/2

Experiments: [Marelic & Nyman, PRA ’15]


Modelling spatial profile.

Gauss-Hermite modes:I(r) =

∑m nm|ψm(r)|2

Use exact R6G spectrum

0

0.2

0.4

0.6

0.8

1

-400 -200 0 200 400

Sp

ectr

um

δ=ω - ωZPL

Γ(-δ)Γ(δ)

Varying excited density – differential coupling to modes

∂tnm = −κnm + Γ(−δm)Om(nm + 1)− Γ(δm)(ρM −Om)nm

Om =

∫drρ↑(r)|ψm(r)|2, ρ↑ + ρ↓ = ρM

∂tρ↑(r) = −Γ↓(r)ρ↑(r) + Γ↑(r)ρ↓(r))




∑m nm|ψm(r)|2


0

0.2

0.4

0.6

0.8

1

-400 -200 0 200 400

Sp

ectr

um

δ=ω - ωZPL

Γ(-δ)Γ(δ)



Om =

∫drρ↑(r)|ψm(r)|2, ρ↑ + ρ↓ = ρM

∂tρ↑(r) = −Γ↓(r)ρ↑(r) + Γ↑(r)ρ↓(r))




∑m nm|ψm(r)|2


0

0.2

0.4

0.6

0.8

1

-400 -200 0 200 400

Sp

ectr

um

δ=ω - ωZPL

Γ(-δ)Γ(δ)



Om =

∫drρ↑(r)|ψm(r)|2, ρ↑ + ρ↓ = ρM

∂tρ↑(r) = −Γ↓(r)ρ↑(r) + Γ↑(r)ρ↓(r))




∑m nm|ψm(r)|2


0

0.2

0.4

0.6

0.8

1

-400 -200 0 200 400

Sp

ectr

um

δ=ω - ωZPL

Γ(-δ)Γ(δ)



Om =

∫drρ↑(r)|ψm(r)|2, ρ↑ + ρ↓ = ρM

∂tρ↑(r) = −Γ↓(r)ρ↑(r) + Γ↑(r)ρ↓(r))


Spatially varying pump: below threshold

Far below threshold:I If κ� ρMΓ(δm),

nm

nm + 1' e−βδm ×

∫dr

1ρM

ρ↑(r)|ψm(r)|2

Resulting profile, I(r) =∑

m nm|ψm(r)|2

0

0.5

1

0 5 10 15

I(r)

/I(0

)

r/lHO

I(r)Boltzmann

0

0.5

1

Γ↑(r

)/Γ

↑(0

)

Pump shape

0

1

2

3

4

5

6

0 2 4 6 8 10 12 14 16

σcl

ou

d/l

HO

σpump/lHO

σcloudσpump

σT




nm


∫dr

1ρM

ρ↑(r)|ψm(r)|2


m nm|ψm(r)|2

0

0.5

1

0 5 10 15

I(r)

/I(0

)

r/lHO

I(r)Boltzmann

0

0.5

1

Γ↑(r

)/Γ

↑(0

)

Pump shape

0

1

2

3

4

5

6

0 2 4 6 8 10 12 14 16

σcl

ou

d/l

HO

σpump/lHO

σcloudσpump

σT




nm


∫dr

1ρM

ρ↑(r)|ψm(r)|2


m nm|ψm(r)|2

0

0.5

1

0 5 10 15

I(r)

/I(0

)

r/lHO

I(r)Boltzmann

0

0.5

1

Γ↑(r

)/Γ

↑(0

)

Pump shape

0

1

2

3

4

5

6

0 2 4 6 8 10 12 14 16

σcl

ou

d/l

HO

σpump/lHO

σcloudσpump

σT




nm


∫dr

1ρM

ρ↑(r)|ψm(r)|2


m nm|ψm(r)|2

0

0.5

1

0 5 10 15

I(r)

/I(0

)

r/lHO

I(r)Boltzmann

0

0.5

1

Γ↑(r

)/Γ

↑(0

)

Pump shape

0

1

2

3

4

5

6

0 2 4 6 8 10 12 14 16

σcl

ou

d/l

HO

σpump/lHO

σcloudσpump

σT




nm


∫dr

1ρM

ρ↑(r)|ψm(r)|2


m nm|ψm(r)|2

0

0.5

1

0 5 10 15

I(r)

/I(0

)

r/lHO

I(r)Boltzmann

0

0.5

1

Γ↑(r

)/Γ

↑(0

)

Pump shape

0

1

2

3

4

5

6

0 2 4 6 8 10 12 14 16σ

clo

ud/l

HO

σpump/lHO

σcloudσpump

σT


Near threshold behaviour

0

0.002

0.004

Exci

ted m

ole

cule

s, f

f(r)PumpEqbm

0

0.5

1

-20 -15 -10 -5 0 5 10 15 20

Photo

ns

r/lHO

I(r)Boltz.

Large spot, σp � lHO

“Gain saturation” at centreSaturation of f (r) = 1/(1 + e−βµ) — spatial equilibriation



0

0.002

0.004

Exci

ted m

ole

cule

s, f

f(r)PumpEqbm

0

0.5

1

-20 -15 -10 -5 0 5 10 15 20

Photo

ns

r/lHO

I(r)Boltz.





0

0.002

0.004

Exci

ted m

ole

cule

s, f

f(r)PumpEqbm

0

0.5

1

-20 -15 -10 -5 0 5 10 15 20

Photo

ns

r/lHO

I(r)Boltz.





0

0.002

0.004

Exci

ted m

ole

cule

s, f

f(r)PumpEqbm

0

0.5

1

-20 -15 -10 -5 0 5 10 15 20

Photo

ns

r/lHO

I(r)Boltz.





0

0.002

0.004

Exci

ted m

ole

cule

s, f

f(r)PumpEqbm

0

0.5

1

-20 -15 -10 -5 0 5 10 15 20

Photo

ns

r/lHO

I(r)Boltz.




Strong coupling






Strong coupling: One excitation subspace

H = ωa†a +N∑

i=1

[ωXσ

+i σ−i + g

(σ+i a + H.c.

)+ ωv

(b†i bi − λ0σ


)]Strong coupling: fate of spectrum

0

0.2

0.4

0.6

0.8

1

-400 -200 0 200 400

Spec

trum

δ=ω - ωZPL

Γ(-δ)Γ(δ)

Photon

Exciton

Restrict, a†a +∑

i σ+i σ−i = 1.

Questions:I Competition of g

√N vs ωv ,

ωvλ20

I Scaling with N



H = ωa†a +N∑

i=1

[ωXσ

+i σ−i + g

(σ+i a + H.c.

)+ ωv

(b†i bi − λ0σ



0

0.2

0.4

0.6

0.8

1

-400 -200 0 200 400

Spec

trum

δ=ω - ωZPL

Γ(-δ)Γ(δ)

Photon

Exciton Restrict, a†a +∑

i σ+i σ−i = 1.


√N vs ωv ,

ωvλ20

I Scaling with N



H = ωa†a +N∑

i=1

[ωXσ

+i σ−i + g

(σ+i a + H.c.

)+ ωv

(b†i bi − λ0σ



0

0.2

0.4

0.6

0.8

1

-400 -200 0 200 400

Spec

trum

δ=ω - ωZPL

Γ(-δ)Γ(δ)

Photon


i σ+i σ−i = 1.


√N vs ωv ,

ωvλ20

I Scaling with N



H = ωa†a +N∑

i=1

[ωXσ

+i σ−i + g

(σ+i a + H.c.

)+ ωv

(b†i bi − λ0σ



0

0.2

0.4

0.6

0.8

1

-400 -200 0 200 400

Spec

trum

δ=ω - ωZPL

Γ(-δ)Γ(δ)

Photon


i σ+i σ−i = 1.


√N vs ωv ,

ωvλ20

I Scaling with N



H = ωa†a +N∑

i=1

[ωXσ

+i σ−i + g

(σ+i a + H.c.

)+ ωv

(b†i bi − λ0σ



0

0.2

0.4

0.6

0.8

1

-400 -200 0 200 400

Spec

trum

δ=ω - ωZPL

Γ(-δ)Γ(δ)

Photon


i σ+i σ−i = 1.


√N vs ωv ,

ωvλ20

I Scaling with N


Exact solution

Vibrational Wigner function:

W (x ,p) =

∫dy〈x + y/2|ρ|x − y/2〉ieiyp,

(bi + b†i√

2

)|x〉i = x |x〉i

Conditioned on Photon |P〉/Exciton at i , |X 〉i /Other site |X 〉j 6=i

N = 2, ω = ωX , ωR ≡ g/√

N = 1

-1 0 1 2 3 4 5-1

0 1

2 3

4 5

-4

-3

-2

-1

0

1

2

V/ωv

x1

x2

V/ωv


Exact solution


W (x ,p) =


(bi + b†i√

2

)|x〉i = x |x〉i


N = 2, ω = ωX , ωR ≡ g/√

N = 1

-1 0 1 2 3 4 5-1

0 1

2 3

4 5

-4

-3

-2

-1

0

1

2

V/ωv

x1

x2

V/ωv


Exact solution


W (x ,p) =


(bi + b†i√

2

)|x〉i = x |x〉i


N = 2, ω = ωX , ωR ≡ g/√

N = 1

-1 0 1 2 3 4 5-1

0 1

2 3

4 5

-4

-3

-2

-1

0

1

2

V/ωv

x1

x2

V/ωv


Exact solution


W (x ,p) =


(bi + b†i√

2

)|x〉i = x |x〉i


N = 2, ω = ωX , ωR ≡ g/√

N = 1

-1 0 1 2 3 4 5-1

0 1

2 3

4 5

-4

-3

-2

-1

0

1

2

V/ωv

x1

x2

V/ωv


Exact solution


W (x ,p) =


(bi + b†i√

2

)|x〉i = x |x〉i


N = 2, ω = ωX , ωR ≡ g/√

N = 1

-1 0 1 2 3 4 5-1

0 1

2 3

4 5

-4

-3

-2

-1

0

1

2

V/ωv

x1

x2

V/ωv


Exact solution


W (x ,p) =


(bi + b†i√

2

)|x〉i = x |x〉i


N = 2, ω = ωX , ωR ≡ g/√

N = 1

-1 0 1 2 3 4 5-1

0 1

2 3

4 5

-4

-3

-2

-1

0

1

2

V/ωv

x1

x2

V/ωv


Exact solution, larger N

Brute force approach, N sites, b†b < M , DHilbert = MN

Permutation symmetry. DHilbert ∼ NM , typical M ∼ 5− 6

N = 20, ω = ωX , ωR ≡ g/√

N = 1

Increasing N, suppressW|P〉(x 6= 0)

Exact energy and statevs ωR, λ0 for validation





N = 20, ω = ωX , ωR ≡ g/√

N = 1







N = 20, ω = ωX , ωR ≡ g/√

N = 1







N = 20, ω = ωX , ωR ≡ g/√

N = 1




Extending to arbitrary N, polaron ansatz

Polaron transform, Di(λ) = exp(λ(b†i − bi)

)N site polaron ansatz

|Ψ〉 =

α |P〉∏j

Dj(λa) +β√N

∑i

|X 〉i Di(λb)∏j 6=i

Dj(λc)

|0〉V[Wu et al. PRB ’16, Zeb et al. arXiv:1608.08929]

I Allows distinct Wigner functions |P〉 , |X 〉i , |X 〉j 6=i

I Polaron energy: ELP =ωX + ωP

2−

√(ωX + ωP

2

)2

+ ω2R

ωX = ωX + ωv (λb2 − 2λ0λb + (N − 1)λc

2), ωP = ω + ωv Nλa2

ω2R = ω2

R exp[−(λa − λb)2 − (N − 1)(λa − λc)2]





|Ψ〉 =

α |P〉∏j

Dj(λa) +β√N

∑i


Dj(λc)




2−

√(ωX + ωP

2

)2

+ ω2R

ωX = ωX + ωv (λb2 − 2λ0λb + (N − 1)λc


ω2R = ω2

R exp[−(λa − λb)2 − (N − 1)(λa − λc)2]





|Ψ〉 =

α |P〉∏j

Dj(λa) +β√N

∑i


Dj(λc)




2−

√(ωX + ωP

2

)2

+ ω2R

ωX = ωX + ωv (λb2 − 2λ0λb + (N − 1)λc


ω2R = ω2

R exp[−(λa − λb)2 − (N − 1)(λa − λc)2]





|Ψ〉 =

α |P〉∏j

Dj(λa) +β√N

∑i


Dj(λc)




2−

√(ωX + ωP

2

)2

+ ω2R

ωX = ωX + ωv (λb2 − 2λ0λb + (N − 1)λc


ω2R = ω2

R exp[−(λa − λb)2 − (N − 1)(λa − λc)2]


Polaron ansatz energy

Polaron energy:

ωX = ωX + ωv (λb2 − 2λ0λb + (N − 1)λc


ω2R = ω2

R exp[−(λa − λb)2 − (N − 1)(λa − λc)2

]EP: At N →∞ Suggests λa = λc ∼ 1/

√N → 0

PP: If ωR � ωv , suggests λa = λb = λc ∼ 1/√

N — factorisationMinimisation:

Multipolaron ansatz: bimodal Wigner

-0.2

-0.1

0

0 0.5 1 1.5 2 2.5

EL

P

λ0

N=106, ωR=1

PP: λa=λb=λcEP: λa=λc=0

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.5 1

ωR

N=106, λ0=1



Polaron energy:

ωX = ωX + ωv (λb2 − 2λ0λb + (N − 1)λc


ω2R = ω2

R exp[−(λa − λb)2 − (N − 1)(λa − λc)2


√N → 0




-0.2

-0.1

0

0 0.5 1 1.5 2 2.5

EL

P

λ0

N=106, ωR=1


-0.2

0

0.2

0.4

0.6

0.8

1

0 0.5 1

ωR

N=106, λ0=1



Polaron energy:

ωX = ωX + ωv (λb2 − 2λ0λb + (N − 1)λc


ω2R = ω2

R exp[−(λa − λb)2 − (N − 1)(λa − λc)2


√N → 0




-0.2

-0.1

0

0 0.5 1 1.5 2 2.5

EL

P

λ0

N=106, ωR=1


-0.2

0

0.2

0.4

0.6

0.8

1

0 0.5 1

ωR

N=106, λ0=1



Polaron energy:

ωX = ωX + ωv (λb2 − 2λ0λb + (N − 1)λc


ω2R = ω2

R exp[−(λa − λb)2 − (N − 1)(λa − λc)2


√N → 0




-0.2

-0.1

0

0 0.5 1 1.5 2 2.5

EL

P

λ0

N=106, ωR=1


-0.2

0

0.2

0.4

0.6

0.8

1

0 0.5 1

ωR

N=106, λ0=1



Polaron energy:

ωX = ωX + ωv (λb2 − 2λ0λb + (N − 1)λc


ω2R = ω2

R exp[−(λa − λb)2 − (N − 1)(λa − λc)2


√N → 0




-0.2

-0.1

0

0 0.5 1 1.5 2 2.5

EL

P

λ0

N=20, ωR=1


-0.2

0

0.2

0.4

0.6

0.8

1

0 0.5 1

ωR

N=20, λ0=1



Polaron energy:

ωX = ωX + ωv (λb2 − 2λ0λb + (N − 1)λc


ω2R = ω2

R exp[−(λa − λb)2 − (N − 1)(λa − λc)2


√N → 0




-0.2

-0.1

0

0 0.5 1 1.5 2 2.5

EL

P

λ0

N=20, ωR=1


Exact

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.5 1

ωR

N=20, λ0=1



Polaron energy:

ωX = ωX + ωv (λb2 − 2λ0λb + (N − 1)λc


ω2R = ω2

R exp[−(λa − λb)2 − (N − 1)(λa − λc)2


√N → 0


N — factorisationMinimisation: Multipolaron ansatz: bimodal Wigner

-0.2

-0.1

0

0 0.5 1 1.5 2 2.5

EL

P

λ0

N=20, ωR=1


2PS

Exact

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.5 1

ωR

N=20, λ0=1


Strong coupling






Calculating spectra: Input-Output formalism

Obsevable features: absorption spectrum, A(ν) = 1− T (ν)−R(ν)

Scattering matrix gives:

A(ν) = −κt

[2 Im[DR(ν)] + (κt + κb)|DR(ν)|2

]Green’s function:

DR(t) = −i⟨

0∣∣∣[a(t), a†(0)

]∣∣∣0⟩ θ(t)





A(ν) = −κt

[2 Im[DR(ν)] + (κt + κb)|DR(ν)|2


DR(t) = −i⟨

0∣∣∣[a(t), a†(0)

]∣∣∣0⟩ θ(t)





A(ν) = −κt

[2 Im[DR(ν)] + (κt + κb)|DR(ν)|2


DR(t) = −i⟨

0∣∣∣[a(t), a†(0)

]∣∣∣0⟩ θ(t)





A(ν) = −κt

[2 Im[DR(ν)] + (κt + κb)|DR(ν)|2


DR(t) = −i⟨

0∣∣∣[a(t), a†(0)

]∣∣∣0⟩ θ(t)


Tavis-Cummings-Holstein spectrum

Direct calculationDR(t) = −i

⟨0∣∣∣[a(t), a†(0)

]∣∣∣0⟩ θ(t)

Time-evolve |ψ0〉 = a†|0〉.Mean-field Green’s function

DR(ν) =1

ν + iκ/2− ωP + ΣX (ν)

ΣX (ν) = −∑

m

ω2R|fm(λ0)|2

ν + iγ/2− ωm

-4 -2 0 2 4ν ωv

0.2

0.4

0.6

0.8

Spect

ral w

eig

ht

M =3ωR =2.5ωvλ0 =1

N =10

/

Photon

Exciton

Multiple excitation ∼ 1/N,




⟨0∣∣∣[a(t), a†(0)

]∣∣∣0⟩ θ(t)


DR(ν) =1

ν + iκ/2− ωP + ΣX (ν)

ΣX (ν) = −∑

m

ω2R|fm(λ0)|2

ν + iγ/2− ωm

-4 -2 0 2 4ν ωv

0.2

0.4

0.6

0.8

Spect

ral w

eig

ht

M =3ωR =2.5ωvλ0 =1

N =10

/

Photon

Exciton





⟨0∣∣∣[a(t), a†(0)

]∣∣∣0⟩ θ(t)


DR(ν) =1

ν + iκ/2− ωP + ΣX (ν)

ΣX (ν) = −∑

m

ω2R|fm(λ0)|2

ν + iγ/2− ωm

-4 -2 0 2 4ν ωv

0.2

0.4

0.6

0.8

Spect

ral w

eig

ht

M =3ωR =2.5ωvλ0 =1

N =10

/

Photon

Exciton





⟨0∣∣∣[a(t), a†(0)

]∣∣∣0⟩ θ(t)


DR(ν) =1

ν + iκ/2− ωP + ΣX (ν)

ΣX (ν) = −∑

m

ω2R|fm(λ0)|2

ν + iγ/2− ωm

-4 -2 0 2 4ν ωv

0.2

0.4

0.6

0.8

Spect

ral w

eig

ht

M =3ωR =2.5ωvλ0 =1

N =10

GR, N =∞

/

Photon

Exciton





⟨0∣∣∣[a(t), a†(0)

]∣∣∣0⟩ θ(t)


DR(ν) =1

ν + iκ/2− ωP + ΣX (ν)

ΣX (ν) = −∑

m

ω2R|fm(λ0)|2

ν + iγ/2− ωm

-4 -2 0 2 4ν ωv

0.2

0.4

0.6

0.8

Spect

ral w

eig

ht

M =3ωR =2.5ωvλ0 =1

N =10

N =20

GR, N =∞

/

Photon

Exciton





⟨0∣∣∣[a(t), a†(0)

]∣∣∣0⟩ θ(t)


DR(ν) =1

ν + iκ/2− ωP + ΣX (ν)

ΣX (ν) = −∑

m

ω2R|fm(λ0)|2

ν + iγ/2− ωm

-4 -2 0 2 4ν ωv

0.2

0.4

0.6

0.8

Spect

ral w

eig

ht

M =3ωR =2.5ωvλ0 =1

N =10

N =20N =100

GR, N =∞

/

Photon

Exciton





⟨0∣∣∣[a(t), a†(0)

]∣∣∣0⟩ θ(t)


DR(ν) =1

ν + iκ/2− ωP + ΣX (ν)

ΣX (ν) = −∑

m

ω2R|fm(λ0)|2

ν + iγ/2− ωm

-4 -2 0 2 4ν ωv

0.2

0.4

0.6

0.8

Spect

ral w

eig

ht

M =3ωR =2.5ωvλ0 =1

N =10

N =20N =100

GR, N =∞

/

Photon

Exciton



Ultra strong coupling






Ultra strong coupling experimental featuresUltra-strong coupling: ω, ωX ∼ g

√N ∝

√concentration

Normal state: configuration of molecules

[Canaguier-Durand et al. Angew. Chem. ’13 ]I Polariton vs molecular spectral weight – chemical eqbmI (Weakly) temperature dependent

Questions:I Can USC change ground state configurationI Disorder + vibrations + USC



√N ∝

√concentration






√N ∝

√concentration





Vibrational reconfiguration

Many photon modes, beyond RWA perturbatively

H =∑

k

ωk a†k ak +N∑

i=1

[ωXσ

+i σ−i +

∑k

gk

(σ+i (ak + a†k ) + H.c.

)+ ωv

(b†i bi − λ0σ


)]Reduced vibrational offset

λ0 → λ0(1− K1), K1 =∑

k

g2k

(ωk + ωX )2

I Increased effective coupling:g2

eff = g2 exp(−λ2

eff

)I But, no collective effect: δH ' K1N

[Cwik et al. PRA ’16]

⇑

⇑

lHOλ0

lHOλeff




H =∑

k

ωk a†k ak +N∑

i=1

[ωXσ

+i σ−i +

∑k

gk

(σ+i (ak + a†k ) + H.c.

)+ ωv

(b†i bi − λ0σ



λ0 → λ0(1− K1), K1 =∑

k

g2k

(ωk + ωX )2


eff = g2 exp(−λ2

eff



⇑

⇑

lHOλ0

lHOλeff




H =∑

k

ωk a†k ak +N∑

i=1

[ωXσ

+i σ−i +

∑k

gk

(σ+i (ak + a†k ) + H.c.

)+ ωv

(b†i bi − λ0σ



λ0 → λ0(1− K1), K1 =∑

k

g2k

(ωk + ωX )2


eff = g2 exp(−λ2

eff



⇑

⇑

lHOλ0

lHOλeff




H =∑

k

ωk a†k ak +N∑

i=1

[ωXσ

+i σ−i +

∑k

gk

(σ+i (ak + a†k ) + H.c.

)+ ωv

(b†i bi − λ0σ



λ0 → λ0(1− K1), K1 =∑

k

g2k

(ωk + ωX )2


eff = g2 exp(−λ2

eff



⇑

⇑

lHOλ0

lHOλeff


Ultra strong coupling






Bumps in the middle of the spectrum

Orgin of bumps in middle of spectrum: Disorder

0

0.1

0.2

0.3

0.4

1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2

Spe

ctra

l wei

ght

ω [eV]

ωRωRωR

=0.3 eV=0.5 eV=0.7 eV

Central peak:

DR(ν) =1

ν + iκ/2− ωk + ΣX (ν)

ΣX (ν) = −∫

dxρ(x)ω2

Rν + iγ/2− x

Gaussian ρ(x), variance σx[Houdré et al. , PRA ’96]


Bumps in the middle of the spectrum

Orgin of bumps in middle of spectrum: Disorder

0

0.1

0.2

0.3

0.4

1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2

Spe

ctra

l wei

ght

ω [eV]

ωRωRωR

=0.3 eV=0.5 eV=0.7 eV

Central peak:

DR(ν) =1

ν + iκ/2− ωk + ΣX (ν)

ΣX (ν) = −∫

dxρ(x)ω2

Rν + iγ/2− x

Gaussian ρ(x), variance σx[Houdré et al. , PRA ’96]


Disorder + Vibrations + Strong coupling

Disordered spectrum +vibrations,λ2

0 = 0.02� 1, σx = 0.01eV

0

0.1

0.2

0.3

0.4

1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2

Spe

ctra

l wei

ght

ω

=0.3eV=0.5eV=0.7eV

ωRωRωR

Stronger disorder,λ2

0 = 0.5, σ = 0.025eV

0

0.2

0.4

0.6

0.8

1

1.2

1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5

Spec

tral

wei

ght

ω [eV]

0.03

0.04

0.05

kBT[eV]





0 = 0.02� 1, σx = 0.01eV

0

0.1

0.2

0.3

0.4

1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2

Spe

ctra

l wei

ght

ω [eV]

1.9 2 2.1

Bar

e m

olec

ule

ωRωRωR

=0.3 eV=0.5 eV=0.7 eV


0 = 0.5, σ = 0.025eV

0

0.2

0.4

0.6

0.8

1

1.2

1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5

Spec

tral

wei

ght

ω [eV]

0.03

0.04

0.05

kBT[eV]





0 = 0.02� 1, σx = 0.01eV

0

0.1

0.2

0.3

0.4

1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2

Spe

ctra

l wei

ght

ω [eV]

1.9 2 2.1

Bar

e m

olec

ule

ωRωRωR

=0.3 eV=0.5 eV=0.7 eV


0 = 0.5, σ = 0.025eV

0

0.2

0.4

0.6

0.8

1

1.2

1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5S

pec

tral

wei

ght

ω [eV]

0.03

0.04

0.05

kBT[eV]



Acknowledgements

GROUP:

COLLABORATION: S. De Liberato (Southhampton).

FUNDING:


Summary

Photon BEC and thermalisation

0

1

2

3

4

5

6

0 2 4 6 8 10 12 14 16

σcl

oud/l

HO

σpump/lHO

σcloud, η=10-5

σcloud, η=10-3

σpumpσT

0

0.002

0.004

Exci

ted m

ole

cule

s, f

f(r)PumpEqbm

0

0.5

1

-20 -15 -10 -5 0 5 10 15 20

Photo

ns

r/lHO

I(r)Boltz.

[Kirton & JK, PRL ’13, PRA ’15, JK & Kirton, PRA ’16]

Single polariton state, Exact solution vs Polaron ansatz

-0.2

-0.1

0

0 0.5 1 1.5 2 2.5

EL

P

λ0

N=20, ωR=1


2PS

Exact

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.5 1

ωR

N=20, λ0=1

-4 -2 0 2 4ν ωv

0.2

0.4

0.6

0.8

Sp

ect

ral w

eig

ht

M =3ωR =2.5ωvλ0 =1

N =10

N =20N =100

GR, N =∞

/

[Zeb, Kirton, JK, arXiv:1608.08929]

Vibrations + disorder + USC

0

0.1

0.2

0.3

0.4

1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2

Spe

ctra

l wei

ght

ω [eV]

1.9 2 2.1

Bar

e m

olec

ule

ωRωRωR

=0.3 eV=0.5 eV=0.7 eV

0

0.1

1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5

Spec

tral

wei

ght

ω [eV]

0.03

0.04

0.05

kBT[eV]

0

0.2

0.4

0.6

0.8

1

1.2

1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5

Spec

tral

wei

ght

ω [eV]

0.03

0.04

0.05

kBT[eV]



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