SUPA From weak to ultra-strong matter-light coupling with organic materials Jonathan Keeling

From weak to ultra-strong matter-light coupling with
organic materials
Jonathan Keeling

SUPA
University of
St Andrews
1413-2013
Snowbird, January 2015
Jonathan Keeling
Weak, strong, ultra-strong
Snowbird, January 2015
1
Matter-Light coupling with organic molecules
What & why?
[Kena Cohen and Forrest, Nat. Photon ’10; Plumhoff et al. Nat. Materials ’14,
Daskalakis et al. ibid ’14] [Klaers et al. Nature ’10]
I
I
Wide variety of systems:
polymers, fluorenes, J-aggregrates,
√ molecular crystals.
Often large polariton splitting, g N ∼ 0.1 eV ↔ 1000K
Theory questions/challenges
I
I
I
Ultrastrong coupling
Vibrational modes
(Partial) thermalisation
Jonathan Keeling
Weak, strong, ultra-strong
Snowbird, January 2015
2
Matter-Light coupling with organic molecules
What & why?
[Kena Cohen and Forrest, Nat. Photon ’10; Plumhoff et al. Nat. Materials ’14,
Daskalakis et al. ibid ’14] [Klaers et al. Nature ’10]
I
I
Wide variety of systems:
polymers, fluorenes, J-aggregrates,
√ molecular crystals.
Often large polariton splitting, g N ∼ 0.1 eV ↔ 1000K
Theory questions/challenges
I
I
I
Ultrastrong coupling
Vibrational modes
(Partial) thermalisation
Jonathan Keeling
Weak, strong, ultra-strong
Snowbird, January 2015
2
Matter-Light coupling with organic molecules
What & why?
[Kena Cohen and Forrest, Nat. Photon ’10; Plumhoff et al. Nat. Materials ’14,
Daskalakis et al. ibid ’14] [Klaers et al. Nature ’10]
I
I
Wide variety of systems:
polymers, fluorenes, J-aggregrates,
√ molecular crystals.
Often large polariton splitting, g N ∼ 0.1 eV ↔ 1000K
Theory questions/challenges
I
I
I
Ultrastrong coupling
Vibrational modes
(Partial) thermalisation
Jonathan Keeling
Weak, strong, ultra-strong
Snowbird, January 2015
2
Dicke Holstein Model
Dicke model: 2LS ↔ photons
Molecular vibrational mode
I
I
Phonon frequency Ω
Huang-Rhys parameter S —
coupling strength
Hsys = ωψ † ψ +
Xh
α
Jonathan Keeling
2
i
σαz + g ψ + ψ † σα+ + σα−
Weak, strong, ultra-strong
Snowbird, January 2015
3
Dicke Holstein Model
Dicke model: 2LS ↔ photons
Molecular vibrational mode
I
I
Phonon frequency Ω
Huang-Rhys parameter S —
coupling strength
Hsys = ωψ † ψ +
Xh
α
2
i
σαz + g ψ + ψ † σα+ + σα−
+
X
n
o
√
Ω bα† bα + Sσαz bα† + bα
α
Jonathan Keeling
Weak, strong, ultra-strong
Snowbird, January 2015
3
Three stories
1
Weak coupling: photon condensation
2
Strong coupling: polaritons
3
Ultra strong coupling: vibrational reconfiguration
Jonathan Keeling
Weak, strong, ultra-strong
Snowbird, January 2015
4
Modelling
Hsys =
X
†
ωm ψ m
ψm +
m
Xh
α
2
i
σαz + g ψm σα+ + H.c.
+
X
n
o
√
Ω bα† bα + Sσαz bα† + bα
α
2D harmonic oscillator
ωm = ωcutoff + mωH.O.
Incoherent processes: excitation,
decay, loss, vibrational
thermalisation.
Weak coupling, perturbative in g
Jonathan Keeling
Weak, strong, ultra-strong
Snowbird, January 2015
5
Modelling
Hsys =
X
†
ωm ψ m
ψm +
m
Xh
α
2
i
σαz + g ψm σα+ + H.c.
+
X
n
o
√
Ω bα† bα + Sσαz bα† + bα
α
2D harmonic oscillator
ωm = ωcutoff + mωH.O.
Incoherent processes: excitation,
decay, loss, vibrational
thermalisation.
Weak coupling, perturbative in g
Jonathan Keeling
Weak, strong, ultra-strong
Snowbird, January 2015
5
Modelling
Hsys =
X
†
ωm ψ m
ψm +
m
Xh
α
2
i
σαz + g ψm σα+ + H.c.
+
X
n
o
√
Ω bα† bα + Sσαz bα† + bα
α
2D harmonic oscillator
ωm = ωcutoff + mωH.O.
Incoherent processes: excitation,
decay, loss, vibrational
thermalisation.
Weak coupling, perturbative in g
Jonathan Keeling
Weak, strong, ultra-strong
Snowbird, January 2015
5
Modelling
Master equation
ρ̇ = −i[H0 , ρ] −
Xκ
m
−
X
m,α
2
L[ψm ] −
X Γ↑
α
2
L[σα+ ]
Γ↓
+ L[σα− ]
2
Γ(δm = ωm − )
Γ(−δm = − ωm )
+
− †
L[σα ψm ] +
L[σα ψm ]
2
2
1
0.8
Kennard-Stepanov
Γ(+δ) ' Γ(−δ)eβδ
0.6
Γ(δ)
Γ(−δ)
0.4
Expt: ω0 < 0.2
0
−200
Γ → 0 at large δ
−100
0
100
δ [THz]
200
[Marthaler et al PRL ’11, Kirton & JK PRL ’13]
Jonathan Keeling
Weak, strong, ultra-strong
Snowbird, January 2015
6
Modelling
Master equation
ρ̇ = −i[H0 , ρ] −
Xκ
m
−
X
m,α
2
L[ψm ] −
X Γ↑
α
2
L[σα+ ]
Γ↓
+ L[σα− ]
2
Γ(δm = ωm − )
Γ(−δm = − ωm )
+
− †
L[σα ψm ] +
L[σα ψm ]
2
2
1
0.8
Kennard-Stepanov
Γ(+δ) ' Γ(−δ)eβδ
0.6
Γ(δ)
Γ(−δ)
0.4
Expt: ω0 < 0.2
0
−200
Γ → 0 at large δ
−100
0
100
δ [THz]
200
[Marthaler et al PRL ’11, Kirton & JK PRL ’13]
Jonathan Keeling
Weak, strong, ultra-strong
Snowbird, January 2015
6
Distribution gm nm
Master equation → Rate equation
∂t nm = −κnm + N Γ(−δm )(nm + 1)hσ ee i − Γ(δm )nM hσ gg i
Bose-Einstein distribution without losses
Low loss: Thermal
[Kirton & JK PRL ’13]
Jonathan Keeling
Weak, strong, ultra-strong
Snowbird, January 2015
7
Distribution gm nm
Master equation → Rate equation
∂t nm = −κnm + N Γ(−δm )(nm + 1)hσ ee i − Γ(δm )nM hσ gg i
Bose-Einstein distribution without losses
Low loss: Thermal
High loss → Laser
[Kirton & JK PRL ’13]
Jonathan Keeling
Weak, strong, ultra-strong
Snowbird, January 2015
7
Time evolution
Initial state: excited molecules
Initial emission, follows gain peak
Thermalisation by repeated absorption
[Kirton & JK arXiv:1410.6632]
Jonathan Keeling
Weak, strong, ultra-strong
Snowbird, January 2015
8
Time evolution
1
Initial state: excited molecules
Initial emission, follows gain peak
0.8
0.6
Γ(δ)
Γ(−δ)
0.4
Thermalisation by repeated absorption
0.2
0
−200
−100
0
100
δ [THz]
200
[Kirton & JK arXiv:1410.6632]
Jonathan Keeling
Weak, strong, ultra-strong
Snowbird, January 2015
8
Time evolution
1
Initial state: excited molecules
Initial emission, follows gain peak
0.8
0.6
Γ(δ)
Γ(−δ)
0.4
Thermalisation by repeated absorption
0.2
0
−200
−100
0
100
δ [THz]
200
[Kirton & JK arXiv:1410.6632]
Jonathan Keeling
Weak, strong, ultra-strong
Snowbird, January 2015
8
Strong coupling: polaritons
1
Weak coupling: photon condensation
2
Strong coupling: polaritons
3
Ultra strong coupling: vibrational reconfiguration
Jonathan Keeling
Weak, strong, ultra-strong
Snowbird, January 2015
9
Strong coupling phase diagram — mean field
Mean field — single photon mode
n
oi
Xh
√ H = ωψ † ψ+
Sαz + g ψSα+ + ψ † Sα− +Ω bα† bα + S bα† + bα Sαz
α
= ω − ∆,
Mott lobes if < ω − 2g
S reduces geff
Reentrant behaviour — Min µ at kB T ∼ 0.1Ω
Jonathan Keeling
Weak, strong, ultra-strong
Snowbird, January 2015
10
Strong coupling phase diagram — mean field
Mean field — single photon mode
n
oi
Xh
√ H = ωψ † ψ+
Sαz + g ψSα+ + ψ † Sα− +Ω bα† bα + S bα† + bα Sαz
α
-x*y
S=0
0.8
T
0.4
0.6
0.3
0.4
0.2
0.2
0.1
0
S reduces geff
0
-6
-5
-4
-3
µ-ωc
-2
-1
Energy
g=2. S=2, ∆=4, Ω=0.1
0.5
= ω − ∆,
Mott lobes if < ω − 2g
1
Coherent field 〈ψ〉
0.6
⇑
Photon
0
nuclear coordinate
⇓
Reentrant behaviour — Min µ at kB T ∼ 0.1Ω
Jonathan Keeling
Weak, strong, ultra-strong
Snowbird, January 2015
10
Strong coupling phase diagram — mean field
Mean field — single photon mode
n
oi
Xh
√ H = ωψ † ψ+
Sαz + g ψSα+ + ψ † Sα− +Ω bα† bα + S bα† + bα Sαz
α
-x*y
S=0
0.8
T
0.4
0.6
0.3
0.4
0.2
0.2
0.1
0
S reduces geff
0
-6
-5
-4
-3
µ-ωc
-2
-1
Energy
g=2. S=2, ∆=4, Ω=0.1
0.5
= ω − ∆,
Mott lobes if < ω − 2g
1
Coherent field 〈ψ〉
0.6
⇑
Photon
0
nuclear coordinate
⇓
Reentrant behaviour — Min µ at kB T ∼ 0.1Ω
Jonathan Keeling
Weak, strong, ultra-strong
Snowbird, January 2015
10
Polariton spectrum: photon weight
0.3
T=0.4, S=2, ∆=4, Ω=0.1, g=2
0.2
-4.5
0.1
-4.6
0
-4.7
-0.1
Photon weight, Zn
Energy
-4.4
-0.2
-4.8
µ
0
0.1
Density ρ
-0.3
0.2
2 ∼ g 2 (1 − 2ρ)
Saturating 2LS: geff
What is nature of polariton mode?
GR (t) = −ihψ † (t)ψ(0)i,
GR (ν) =
X
n
Zn
ν − ωn
[Cwik et al. EPL ’14]
Jonathan Keeling
Weak, strong, ultra-strong
Snowbird, January 2015
11
Polariton spectrum: photon weight
0.3
T=0.4, S=2, ∆=4, Ω=0.1, g=2
0.2
-4.5
0.1
-4.6
0
-4.7
-0.1
Photon weight, Zn
Energy
-4.4
-0.2
-4.8
µ
0
0.1
Density ρ
-0.3
0.2
2 ∼ g 2 (1 − 2ρ)
Saturating 2LS: geff
What is nature of polariton mode?
GR (t) = −ihψ † (t)ψ(0)i,
GR (ν) =
X
n
Zn
ν − ωn
[Cwik et al. EPL ’14]
Jonathan Keeling
Weak, strong, ultra-strong
Snowbird, January 2015
11
Polariton spectrum: photon weight
0.3
T=0.4, S=2, ∆=4, Ω=0.1, g=2
0.2
-4.5
0.1
-4.6
0
-4.7
-0.1
Photon weight, Zn
Energy
-4.4
-0.2
-4.8
µ
0
0.1
Density ρ
-0.3
0.2
2 ∼ g 2 (1 − 2ρ)
Saturating 2LS: geff
What is nature of polariton mode?
GR (t) = −ihψ † (t)ψ(0)i,
GR (ν) =
X
n
Zn
ν − ωn
[Cwik et al. EPL ’14]
Jonathan Keeling
Weak, strong, ultra-strong
Snowbird, January 2015
11
Ultra strong coupling: vibrational reconfiguration
1
Weak coupling: photon condensation
2
Strong coupling: polaritons
3
Ultra strong coupling: vibrational reconfiguration
Jonathan Keeling
Weak, strong, ultra-strong
Snowbird, January 2015
12
Ultra-strong coupling, changing configuration
√
√
Ultra-strong coupling: ω, ∼ g N ∝ concentration
Normal state: configuration of molecules
[Canaguier-Durand et al. Angew. Chem. ’13 ]
I
I
Polariton vs molecular spectral weight – chemical eqbm
Temperature dependent
Questions:
I
I
Vibrationally dressed spectrum + disorder
Microscopic theory – changing configuration
Jonathan Keeling
Weak, strong, ultra-strong
Snowbird, January 2015
13
Ultra-strong coupling, changing configuration
√
√
Ultra-strong coupling: ω, ∼ g N ∝ concentration
Normal state: configuration of molecules
[Canaguier-Durand et al. Angew. Chem. ’13 ]
I
I
Polariton vs molecular spectral weight – chemical eqbm
Temperature dependent
Questions:
I
I
Vibrationally dressed spectrum + disorder
Microscopic theory – changing configuration
Jonathan Keeling
Weak, strong, ultra-strong
Snowbird, January 2015
13
Ultra-strong coupling, changing configuration
√
√
Ultra-strong coupling: ω, ∼ g N ∝ concentration
Normal state: configuration of molecules
[Canaguier-Durand et al. Angew. Chem. ’13 ]
I
I
Polariton vs molecular spectral weight – chemical eqbm
Temperature dependent
Questions:
I
I
Vibrationally dressed spectrum + disorder
Microscopic theory – changing configuration
Jonathan Keeling
Weak, strong, ultra-strong
Snowbird, January 2015
13
Disordered molecules — spectrum
Calculate Green’s function GR (ν):
T (ν) ∝ |GR (ν)|2 ,
A(ν) ∝ −=[GR (ν)] + (interference)
Ultra-strong coupling — renormalised photon
50
Central peak:
g√N=0.3 eV
Spectral weight
40
30
20
10
0
1.6
1.8
2
2.2
2.4 2.6
ω [eV]
2.8
3
3.2
√
Temperature independent (for kB T g N)
Jonathan Keeling
Weak, strong, ultra-strong
Snowbird, January 2015
14
Disordered molecules — spectrum
Calculate Green’s function GR (ν):
T (ν) ∝ |GR (ν)|2 ,
A(ν) ∝ −=[GR (ν)] + (interference)
Ultra-strong coupling — renormalised photon
50
Spectral weight
Central peak:
g√N=0.3 eV
g√N=0.5 eV
40
30
20
10
0
1.6
1.8
2
2.2
2.4 2.6
ω [eV]
2.8
3
3.2
√
Temperature independent (for kB T g N)
Jonathan Keeling
Weak, strong, ultra-strong
Snowbird, January 2015
14
Disordered molecules — spectrum
Calculate Green’s function GR (ν):
T (ν) ∝ |GR (ν)|2 ,
A(ν) ∝ −=[GR (ν)] + (interference)
Ultra-strong coupling — renormalised photon
50
Spectral weight
Central peak:
g√N=0.3 eV
g√N=0.5 eV
g√N=0.7 eV
40
30
20
10
0
1.6
1.8
2
2.2
2.4 2.6
ω [eV]
2.8
3
3.2
√
Temperature independent (for kB T g N)
Jonathan Keeling
Weak, strong, ultra-strong
Snowbird, January 2015
14
Disordered molecules — spectrum
Calculate Green’s function GR (ν):
T (ν) ∝ |GR (ν)|2 ,
A(ν) ∝ −=[GR (ν)] + (interference)
Ultra-strong coupling — renormalised photon
50
Spectral weight
Central peak:
g√N=0.3 eV
g√N=0.5 eV
g√N=0.7 eV
40
30
20
10
0
1.6
1.8
2
2.2
2.4 2.6
ω [eV]
2.8
3
3.2
√
Temperature independent (for kB T g N)
Jonathan Keeling
Weak, strong, ultra-strong
Snowbird, January 2015
14
Disordered molecules — spectrum
Calculate Green’s function GR (ν):
T (ν) ∝ |GR (ν)|2 ,
A(ν) ∝ −=[GR (ν)] + (interference)
Ultra-strong coupling — renormalised photon
Spectral weight
0.4
Central peak:
g√N=0.3 eV
g√N=0.5 eV
g√N=0.7 eV
0.3
0.2
0.1
0
1.6
1.8
2
2.2
2.4 2.6
ω [eV]
2.8
3
3.2
√
Temperature independent (for kB T g N)
Jonathan Keeling
Weak, strong, ultra-strong
Snowbird, January 2015
14
Disordered molecules — spectrum
Calculate Green’s function GR (ν):
T (ν) ∝ |GR (ν)|2 ,
A(ν) ∝ −=[GR (ν)] + (interference)
Ultra-strong coupling — renormalised photon
Spectral weight
0.4
Central peak:
g√N=0.3 eV
g√N=0.5 eV
g√N=0.7 eV
0.3
GR (ν) =
1
R (ν)
ν + iκ/2 − ωk − g 2 GExc.
2
κ
R
A(ν) ∼
− =[GExc.
] GR (ν)
2
0.2
0.1
0
1.6
1.8
2
2.2
2.4 2.6
ω [eV]
2.8
3
3.2
[Houdré et al. , PRA ’96]
√
Temperature independent (for kB T g N)
Jonathan Keeling
Weak, strong, ultra-strong
Snowbird, January 2015
14
Disordered molecules — spectrum
Calculate Green’s function GR (ν):
T (ν) ∝ |GR (ν)|2 ,
A(ν) ∝ −=[GR (ν)] + (interference)
Ultra-strong coupling — renormalised photon
Spectral weight
0.4
Central peak:
g√N=0.3 eV
g√N=0.5 eV
g√N=0.7 eV
0.3
GR (ν) =
1
R (ν)
ν + iκ/2 − ωk − g 2 GExc.
2
κ
R
A(ν) ∼
− =[GExc.
] GR (ν)
2
0.2
0.1
0
1.6
1.8
2
2.2
2.4 2.6
ω [eV]
2.8
3
3.2
[Houdré et al. , PRA ’96]
√
Temperature independent (for kB T g N)
Jonathan Keeling
Weak, strong, ultra-strong
Snowbird, January 2015
14
Disordered molecules — vibrational mode
With vibrational sidebands, S = 0.02
Spectral weight
0.4
g√N=0.3eV
g√N=0.5eV
g√N=0.7eV
0.3
0.2
0.1
0
1.6
Jonathan Keeling
1.8
2
2.2
2.4 2.6
ω [eV]
Weak, strong, ultra-strong
2.8
3
3.2
Snowbird, January 2015
15
Disordered molecules — vibrational mode
Spectral weight
0.4
g√N=0.3eV
g√N=0.5eV
g√N=0.7eV
0.3
0.2
0.1
Bare molecule
With vibrational sidebands, S = 0.02
1.9
2
2.4 2.6
ω [eV]
2.8
2.1
0
1.6
Jonathan Keeling
1.8
2
2.2
Weak, strong, ultra-strong
3
3.2
Snowbird, January 2015
15
Disordered molecules + vibrations – vs temperature
Stronger disorder &
S = 0.5, σ = 0.025eV
vs vs temperature
Spectral weight
0.05
0.04
kBT
[eV]
0.1
0.03
0
1.7
1.8
1.9
2
Jonathan Keeling
2.1 2.2
ω [eV]
2.3
2.4
2.5
Weak, strong, ultra-strong
Snowbird, January 2015
16
Disordered molecules + vibrations – vs temperature
Stronger disorder &
S = 0.5, σ = 0.025eV
vs vs temperature
Spectral weight
0.05
0.04
kBT
[eV]
0.1
0.03
0
1.7
1.8
1.9
2
Jonathan Keeling
2.1 2.2
ω [eV]
2.3
2.4
2.5
Weak, strong, ultra-strong
Snowbird, January 2015
16
Disordered molecules + vibrations – vs temperature
Stronger disorder &
S = 0.5, σ = 0.025eV
vs vs temperature
Spectral weight
0.05
0.04
kBT
[eV]
0.1
0.03
0
1.7
1.8
1.9
2
Jonathan Keeling
2.1 2.2
ω [eV]
2.3
2.4
2.5
Weak, strong, ultra-strong
Snowbird, January 2015
16
Disordered molecules + vibrations – vs temperature
Stronger disorder &
S = 0.5, σ = 0.025eV
vs vs temperature
Spectral weight
0.05
0.04
kBT
[eV]
0.1
0.03
0
1.7
1.8
1.9
2
Jonathan Keeling
2.1 2.2
ω [eV]
2.3
2.4
2.5
Weak, strong, ultra-strong
Snowbird, January 2015
16
Disordered molecules + vibrations – vs temperature
Stronger disorder &
S = 0.5, σ = 0.025eV
vs vs temperature
S = 0.02, σ = 0.01eV
1.2
0.05
0.05
Spectral weight
Spectral weight
1
0.04
kBT
[eV]
0.1
0.03
0.8
0.6
0.04
kBT
[eV]
0.4
0.03
0.2
0
0
1.7
1.8
1.9
2
Jonathan Keeling
2.1 2.2
ω [eV]
2.3
2.4
2.5
1.7
Weak, strong, ultra-strong
1.8
1.9
2
2.1 2.2
ω [eV]
2.3
2.4
Snowbird, January 2015
2.5
16
Acknowledgements
G ROUP :
C OLLABORATORS : Reja (MPI-PKS), Littlewood (ANL & Chicago), De
Liberato (Southhampton)
F UNDING :
Jonathan Keeling
Weak, strong, ultra-strong
Snowbird, January 2015
17
Summary
Photon condensation and thermalisation
[Kirton & Keeling, PRL ’13, arXiv:1410.6632]
Reentrance, phonon assisted transition, 1st order at S 1
0.3
T
0.4
1
-4.4
0.8
-4.5
0.6
0.3
0.4
0.2
0.2
0.1
0
0
-6
-5
-4
-3
µ-ωc
-2
-1
T=0.4, S=2, ∆=4, Ω=0.1, g=2
0.2
0.1
-4.6
0
-4.7
-0.1
Photon weight, Zn
S=0
Energy
-x*y
g=2. S=2, ∆=4, Ω=0.1
0.5
Coherent field 〈ψ〉
0.6
-0.2
-4.8
µ
0
0
0.1
Density ρ
-0.3
0.2
[Cwik et al. EPL ’14]
Vibrational configuration
0.1
0.05
1
1.9
2
2.4 2.6
ω [eV]
2.8
Spectral weight
0.2
1.2
0.05
Spectral weight
g√N=0.3eV
g√N=0.5eV
g√N=0.7eV
0.3
Bare molecule
Spectral weight
0.4
0.04
kBT
[eV]
0.1
0.03
0.8
0.04
kBT
[eV]
0.6
0.4
0.03
2.1
0.2
0
0
1.6
1.8
2
2.2
3
3.2
0
1.7
1.8
1.9
2
2.1 2.2
ω [eV]
2.3
2.4
2.5
1.7
1.8
1.9
2
2.1 2.2
ω [eV]
2.3
2.4
2.5
[Cwik et al. in preparation]
Jonathan Keeling
Weak, strong, ultra-strong
Snowbird, January 2015
18
Extra Slides
4
Dye laser
5
Photon BEC threshold
6
Photon BEC with spatial profile
7
Ultra-strong phonon coupling?
8
Anticrossing vs ρ
9
Polariton spectrum nature
10
Vibrational reconfiguration
Jonathan Keeling
Weak, strong, ultra-strong
Snowbird, January 2015
19
Dicke-Holstein model: dye laser
Energy
4 Level Dye Laser
Multiple photon modes
⇑
⇓
I
Pump
Cavity
vibrational
coordinate
I
I
Condensate mode is not
maximum gain
Gain/Absorption in balance
Thermalisation
(Ultra)strong matter-light
coupling
No strong coupling
No electronic inversion —
vibrational inversion.
Jonathan Keeling
Weak, strong, ultra-strong
Snowbird, January 2015
20
Dicke-Holstein model: dye laser
Energy
4 Level Dye Laser
Multiple photon modes
⇑
⇓
I
Pump
Cavity
vibrational
coordinate
I
I
Typical operation
Condensate mode is not
maximum gain
Gain/Absorption in balance
Thermalisation
(Ultra)strong matter-light
coupling
No strong coupling
No electronic inversion —
vibrational inversion.
Jonathan Keeling
Weak, strong, ultra-strong
Snowbird, January 2015
20
Dicke-Holstein model: dye laser
Energy
4 Level Dye Laser
In this talk:
Multiple photon modes
⇑
⇓
I
Pump
Cavity
vibrational
coordinate
I
I
Typical operation
Condensate mode is not
maximum gain
Gain/Absorption in balance
Thermalisation
(Ultra)strong matter-light
coupling
No strong coupling
No electronic inversion —
vibrational inversion.
Jonathan Keeling
Weak, strong, ultra-strong
Snowbird, January 2015
20
Dicke-Holstein model: dye laser
Energy
4 Level Dye Laser
In this talk:
Multiple photon modes
⇑
⇓
I
Pump
Cavity
vibrational
coordinate
I
I
Typical operation
Condensate mode is not
maximum gain
Gain/Absorption in balance
Thermalisation
(Ultra)strong matter-light
coupling
No strong coupling
No electronic inversion —
vibrational inversion.
Jonathan Keeling
Weak, strong, ultra-strong
Snowbird, January 2015
20
Threshold condition
κ =10 MHz
κ
κ
κ =5 GHz
κ
κ=0.5 GHz
600
500
400
Compare threshold:
In
cr
ea
si
ng
300
Pump rate (Laser)
lo
ss
Critical density
(condensate)
200
10−5 10−4 10−3 10−2 10−1 2
3
4
5
6
7
[Kirton & JK PRL ’13]
Jonathan Keeling
Weak, strong, ultra-strong
Snowbird, January 2015
21
Spatially varying pump intensity
7
Jonathan Keeling
Weak, strong, ultra-strong
Cloud width/lHO
6
∂t ρ↑ = −Γ̃↓ (r )ρ↑ + Γ̃↑ (r )(ρm − ρ↑ )
Z
∂t nm = Γ(δm ) d~r ρ↑ |ψm (r )|2 (nm + 1)
Z
2
− κ + Γ(δm ) d~r (ρm − ρ↑ )|ψm (r )| nm
5
4
3
2
Cloud
Pump
Thermal
1
0
0
2
4
6
Pump spot width/lHO
8
Snowbird, January 2015
10
22
Spatially varying pump intensity
7
Photons
Thermal
20
0.06
0.04
10
0.02
0
0
r/l
Jonathan Keeling
10
20
Photons
f, pump
0.08
Peak pump power (au)
30
-10
4
3
2
Cloud
Pump
Thermal
1
0
0
2
4
6
Pump spot width/lHO
0.3
0.1
0
-20
5
8
10
5
4
0.2
3
2
0.1
1
Peak power
Integrated power
0
0
extHO
Weak, strong, ultra-strong
5
Integrated power (au)
Pump
ρ↑
Cloud width/lHO
6
∂t ρ↑ = −Γ̃↓ (r )ρ↑ + Γ̃↑ (r )(ρm − ρ↑ )
Z
∂t nm = Γ(δm ) d~r ρ↑ |ψm (r )|2 (nm + 1)
Z
2
− κ + Γ(δm ) d~r (ρm − ρ↑ )|ψm (r )| nm
0
10
15
Spot size (lHO)
20
Snowbird, January 2015
22
Critical coupling with increasing S
ε - ω=-4
4
3
Re-orient phase diagram
gc√Ν
2
g vs µ, T
1
Colors → Jump of hψi
0-5
Jonathan Keeling
Weak, strong, ultra-strong
-4 -3
µ-ω
-2
-1
0 0
1
0.6 0.8
0.2 0.4 T
Snowbird, January 2015
23
Critical coupling with increasing S
ε - ω=-4
4
3
Re-orient phase diagram
gc√Ν
2
g vs µ, T
1
Colors → Jump of hψi
0-5
-4 -3
µ-ω
-2
-1
0 0
1
0.6 0.8
0.2 0.4 T
S=3
-4
µ-ω-3 -2 0
Jonathan Keeling
0.1
T
7
6
5
4
3
2
1
0.2 -50
S=6
0.4
gc√ Ν
7
6
5
4
3
2
1
0.2 -50
gc√ Ν
gc√ Ν
S=0
∆=4, Ω=1
-4
µ-ω-3 -2 0
0.1
T
Weak, strong, ultra-strong
0.3
0.2
0.1
0.2
-4
µ-ω-3 -2 0
0.1
T
Jump in 〈 ψ 〉
0.5
7
6
5
4
3
2
1
0
-5
0
Snowbird, January 2015
23
Explanation: Polaron formation
Unitary transform
√
Hα → H̃α = eKα Hα e−Kα
K =
SSαz (bα† − bα )
Coupling moves to S ±
h
i
√
†
H̃α = const. + Sαz + Ωbα† bα + g ψSα+ e S(bα −bα ) + H.c.
Optimal phonon displacements, ∼
√
S
Reduced geff ∼ g × exp(−S/2)
For ψ 6= 0, competition
Variational MFT |ψiα ∼ exp(−ηKα − ζbα† )|0, Siα
Jonathan Keeling
Weak, strong, ultra-strong
Snowbird, January 2015
24
Explanation: Polaron formation
Unitary transform
√
Hα → H̃α = eKα Hα e−Kα
K =
SSαz (bα† − bα )
Coupling moves to S ±
h
i
√
†
H̃α = const. + Sαz + Ωbα† bα + g ψSα+ e S(bα −bα ) + H.c.
Optimal phonon displacements, ∼
√
S
Reduced geff ∼ g × exp(−S/2)
For ψ 6= 0, competition
Variational MFT |ψiα ∼ exp(−ηKα − ζbα† )|0, Siα
Jonathan Keeling
Weak, strong, ultra-strong
Snowbird, January 2015
24
Explanation: Polaron formation
Unitary transform
√
Hα → H̃α = eKα Hα e−Kα
K =
SSαz (bα† − bα )
Coupling moves to S ±
h
i
√
†
H̃α = const. + Sαz + Ωbα† bα + g ψSα+ e S(bα −bα ) + H.c.
Optimal phonon displacements, ∼
√
S
Reduced geff ∼ g × exp(−S/2)
For ψ 6= 0, competition
Variational MFT |ψiα ∼ exp(−ηKα − ζbα† )|0, Siα
Jonathan Keeling
Weak, strong, ultra-strong
Snowbird, January 2015
24
Explanation: Polaron formation
Unitary transform
√
Hα → H̃α = eKα Hα e−Kα
K =
SSαz (bα† − bα )
Coupling moves to S ±
h
i
√
†
H̃α = const. + Sαz + Ωbα† bα + g ψSα+ e S(bα −bα ) + H.c.
Optimal phonon displacements, ∼
√
S
Reduced geff ∼ g × exp(−S/2)
For ψ 6= 0, competition
Variational MFT |ψiα ∼ exp(−ηKα − ζbα† )|0, Siα
Jonathan Keeling
Weak, strong, ultra-strong
Snowbird, January 2015
24
Explanation: Polaron formation
Unitary transform
√
Hα → H̃α = eKα Hα e−Kα
K =
SSαz (bα† − bα )
Coupling moves to S ±
h
i
√
†
H̃α = const. + Sαz + Ωbα† bα + g ψSα+ e S(bα −bα ) + H.c.
Optimal phonon displacements, ∼
√
S
Reduced geff ∼ g × exp(−S/2)
For ψ 6= 0, competition
Variational MFT |ψiα ∼ exp(−ηKα − ζbα† )|0, Siα
Jonathan Keeling
Weak, strong, ultra-strong
Snowbird, January 2015
24
Collective polaron formation
(a) Exact diagonalization
Compares well at S 1
Coherent bosonic state
6
5
5
4
g√ Ν 3
4
g√ Ν 3
2
0
-5
S=6, ∆=4, Ω=1
2
1
0.2
-4
µ-ωc
-3
-2
0
0.1
T
1
0
-5
-4
µ-ωc
-3
-2
0
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
λ
(b) Ansatz
6
0.2
0.1
T
Feedback: Large/small geff ↔ λ = hψi
Variational free energy
η(2 − η)
ξ
F = (ωc − µ)λ2 + N Ω ζ 2 − S
− T ln 2 cosh
4
T
Effective 2LS energy in field:
2
√
−µ
2
2
ξ =
+ Ω S(1 − η)ζ + g 2 λ2 e−Sη
2
[Cwik et al. EPL ’14]
Jonathan Keeling
Weak, strong, ultra-strong
Snowbird, January 2015
25
Collective polaron formation
(a) Exact diagonalization
Compares well at S 1
Coherent bosonic state
6
5
5
4
g√ Ν 3
4
g√ Ν 3
2
0
-5
S=6, ∆=4, Ω=1
2
1
0.2
-4
µ-ωc
-3
-2
0
0.1
T
1
0
-5
-4
µ-ωc
-3
-2
0
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
λ
(b) Ansatz
6
0.2
0.1
T
Feedback: Large/small geff ↔ λ = hψi
Variational free energy
η(2 − η)
ξ
F = (ωc − µ)λ2 + N Ω ζ 2 − S
− T ln 2 cosh
4
T
Effective 2LS energy in field:
2
√
−µ
2
2
ξ =
+ Ω S(1 − η)ζ + g 2 λ2 e−Sη
2
[Cwik et al. EPL ’14]
Jonathan Keeling
Weak, strong, ultra-strong
Snowbird, January 2015
25
Collective polaron formation
(a) Exact diagonalization
Compares well at S 1
Coherent bosonic state
6
5
5
4
g√ Ν 3
4
g√ Ν 3
2
0
-5
S=6, ∆=4, Ω=1
2
1
0.2
-4
µ-ωc
-3
-2
0
0.1
T
1
0
-5
-4
µ-ωc
-3
-2
0
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
λ
(b) Ansatz
6
0.2
0.1
T
Feedback: Large/small geff ↔ λ = hψi
Variational free energy
η(2 − η)
ξ
F = (ωc − µ)λ2 + N Ω ζ 2 − S
− T ln 2 cosh
4
T
Effective 2LS energy in field:
2
√
−µ
2
2
ξ =
+ Ω S(1 − η)ζ + g 2 λ2 e−Sη
2
[Cwik et al. EPL ’14]
Jonathan Keeling
Weak, strong, ultra-strong
Snowbird, January 2015
25
Polariton spectrum — coupled oscillators
Jonathan Keeling
Weak, strong, ultra-strong
Snowbird, January 2015
26
Polariton spectrum — coupled oscillators
Photon
Exciton
1
UP
Energy
0
-1
-2
LP
-3
0
Jonathan Keeling
0.2
0.4
0.6
0.8
1
1.2
Coupling, g
Weak, strong, ultra-strong
1.4
1.6
1.8
2
Snowbird, January 2015
26
Polariton spectrum — coupled oscillators
Photon
Exciton
Exciton-Ω
1
UP
Energy
0
-1
-2
LP
-3
0
Jonathan Keeling
0.2
0.4
0.6
0.8
1
1.2
Coupling, g
Weak, strong, ultra-strong
1.4
1.6
1.8
2
Snowbird, January 2015
26
Polariton spectrum — coupled oscillators
Photon
Exciton-nΩ
1
UP
Energy
0
-1
-2
-3
0
Jonathan Keeling
0.2
0.4
0.6
0.8
1
1.2
Coupling, g
Weak, strong, ultra-strong
1.4
1.6
1.8
2
Snowbird, January 2015
26
Polariton spectrum — coupled oscillators
Photon
Exciton-nΩ
1
UP
Energy
0
-1
-2
-3
0
Jonathan Keeling
0.2
0.4
0.6
0.8
1
1.2
Coupling, g
Weak, strong, ultra-strong
1.4
1.6
1.8
2
Snowbird, January 2015
26
Polariton spectrum: what condensed
Repeat weight for n-phonon channel
Eigenvector that is macroscopically occupied
Optimal T ∼ 2Ω
[Cwik et al. EPL ’14]
Jonathan Keeling
Weak, strong, ultra-strong
Snowbird, January 2015
27
Polariton spectrum: what condensed
Repeat weight for n-phonon channel
Eigenvector that is macroscopically occupied
Optimal T ∼ 2Ω
1
Sideband spectral weight
S=2, ∆=4, Ω=0.1, g=2
T=0.00
0.8
0.6
0.4
0.2
0
-6 -5 -4 -3 -2 -1 0 1 2 3
Absorbed phonons: q-p
4
5
6
[Cwik et al. EPL ’14]
Jonathan Keeling
Weak, strong, ultra-strong
Snowbird, January 2015
27
Polariton spectrum: what condensed
Repeat weight for n-phonon channel
Eigenvector that is macroscopically occupied
Optimal T ∼ 2Ω
1
Sideband spectral weight
S=2, ∆=4, Ω=0.1, g=2
0.8
T=0.00
T=0.05
T=0.15
0.6
0.4
0.2
0
-6 -5 -4 -3 -2 -1 0 1 2 3
Absorbed phonons: q-p
4
5
6
[Cwik et al. EPL ’14]
Jonathan Keeling
Weak, strong, ultra-strong
Snowbird, January 2015
27
Polariton spectrum: what condensed
Repeat weight for n-phonon channel
Eigenvector that is macroscopically occupied
Optimal T ∼ 2Ω
1
Sideband spectral weight
S=2, ∆=4, Ω=0.1, g=2
0.8
0.6
T=0.00
T=0.05
T=0.15
T=0.20
T=0.30
T=0.40
T=0.45
0.4
0.2
0
-6 -5 -4 -3 -2 -1 0 1 2 3
Absorbed phonons: q-p
4
5
6
[Cwik et al. EPL ’14]
Jonathan Keeling
Weak, strong, ultra-strong
Snowbird, January 2015
27
Polariton spectrum: what condensed
Repeat weight for n-phonon channel
Eigenvector that is macroscopically occupied
Optimal T ∼ 2Ω
0.8
0.6
T=0.00
T=0.05
T=0.15
T=0.20
T=0.30
T=0.40
T=0.45
0.6
g=2. S=2, ∆=4, Ω=0.1
1
-x*y
0.5
0.8
0.4
0.4
T
Sideband spectral weight
S=2, ∆=4, Ω=0.1, g=2
0.6
0.3
0.4
0.2
0.2
0.2
0.1
0
-6 -5 -4 -3 -2 -1 0 1 2 3
Absorbed phonons: q-p
4
5
6
0
Coherent field 〈ψ〉
1
0
-6
-5
-4
-3
µ-ωc
-2
-1
0
[Cwik et al. EPL ’14]
Jonathan Keeling
Weak, strong, ultra-strong
Snowbird, January 2015
27
Vibrational reconfiguration
H = H0 + H1 , H1 =
P
n,k
gn,k (ψk† σn+ + H.c.)
Schrieffer-Wolff: admixture of excited state
(
" √ #)
√
g2N
Ω S(b + b† )
Ω 2 g N
,
Heff,vacuum = H0 −
1−
+O
2( + ω)
+ω
Reduced
√
√ vibrational offset
S → S(1 − g 2 N/( + ω))
Increased effective coupling:
2
geff
= g 2 exp(−S)
Numerically tiny effect, Ω Jonathan Keeling
Weak, strong, ultra-strong
Snowbird, January 2015
28
Vibrational reconfiguration
H = H0 + H1 , H1 =
P
n,k
gn,k (ψk† σn+ + H.c.)
Energy
Schrieffer-Wolff: admixture of excited state
(
" √ #)
√
g2N
Ω S(b + b† )
Ω 2 g N
,
Heff,vacuum = H0 −
1−
+O
2( + ω)
+ω
Reduced
√
√ vibrational offset
S → S(1 − g 2 N/( + ω))
⇑
Photon
nuclear coordinate
⇓
l
S ×l
Increased effective coupling:
2
geff
= g 2 exp(−S)
Numerically tiny effect, Ω Jonathan Keeling
Weak, strong, ultra-strong
Snowbird, January 2015
28
Vibrational reconfiguration
H = H0 + H1 , H1 =
P
n,k
gn,k (ψk† σn+ + H.c.)
Energy
Schrieffer-Wolff: admixture of excited state
(
" √ #)
√
g2N
Ω S(b + b† )
Ω 2 g N
,
Heff,vacuum = H0 −
1−
+O
2( + ω)
+ω
Reduced
√
√ vibrational offset
S → S(1 − g 2 N/( + ω))
⇑
Photon
nuclear coordinate
⇓
l
S ×l
Increased effective coupling:
2
geff
= g 2 exp(−S)
Numerically tiny effect, Ω Jonathan Keeling
Weak, strong, ultra-strong
Snowbird, January 2015
28