SUPA Modelling weak and strong matter-light coupling with organic molecules Jonathan Keeling
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SUPA Modelling weak and strong matter-light coupling with organic molecules Jonathan Keeling
Modelling weak and strong matter-light coupling with organic molecules Jonathan Keeling SUPA University of St Andrews 1413-2013 Telluride, July 2015 Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 1 Motivation: polariton condensates CdTe Polariton Condensate T ∼ 20K. [Kasprzak et al. Nature, ’06] "Cavity" Quantum Wells Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 2 Motivation: polariton condensates CdTe Polariton Condensate T ∼ 20K. [Kasprzak et al. Nature, ’06] Models: WIDBG I I Statistical mechanics Boltzmann/cGPE Hybrids Saturable excitons "Cavity" Quantum Wells Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 2 Motivation: polariton condensates CdTe Polariton Condensate T ∼ 20K. [Kasprzak et al. Nature, ’06] Models: WIDBG I I Statistical mechanics Boltzmann/cGPE Hybrids Saturable excitons "Cavity" Quantum Wells Q1. Lasing crossover? Q2. Energetics vs dynamics (esp. spin state). Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 2 Motivation: polariton condensates CdTe Polariton Condensate T ∼ 20K. [Kasprzak et al. Nature, ’06] Models: WIDBG I I Statistical mechanics Boltzmann/cGPE Hybrids Saturable excitons "Cavity" Q1. Lasing crossover? Quantum Wells Q2. Energetics vs dynamics (esp. spin state). Anthracene Polariton Lasing T ∼ 300K [Kena Cohen and Forrest, Nat. Photon ’10] Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 2 Motivation: polariton condensates CdTe Polariton Condensate T ∼ 20K. [Kasprzak et al. Nature, ’06] Models: WIDBG I I Statistical mechanics Boltzmann/cGPE Hybrids Saturable excitons "Cavity" Q1. Lasing crossover? Quantum Wells Q2. Energetics vs dynamics (esp. spin state). Anthracene Polariton Lasing T ∼ 300K Q1. Vibrational replicas? Q2. Relevance of disorder? Q3. Lasing vs condensation? [Kena Cohen and Forrest, Nat. Photon ’10] Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 2 Motivation: photon condensates Photon Condensate T ∼ 300K [Klaers et al. Nature, ’10] Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 3 Motivation: photon condensates Photon Condensate T ∼ 300K Q1. Relation to dye laser? Q2. Relation to polaritons? Q3. Thermalisation breakdown? [Klaers et al. Nature, ’10] Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 3 Motivation: vacuum-state strong coupling Linear response (no pump, no condensate): effects of matter-light coupling alone. [Canaguier-Durand et al. Angew. Chem. ’13] Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 4 Motivation: vacuum-state strong coupling Linear response (no pump, no condensate): effects of matter-light coupling alone. Q1. Can ultra-strong coupling to light change: I I I I charge distribution? vibrational configuration? molecular orientation? crystal structure? Q2. Are √ changes collective ( N factor) or not? [Canaguier-Durand et al. Angew. Chem. ’13] Jonathan Keeling Q3. If not, what is data showing? Modelling matter-light coupling Telluride, July 2015 4 1 Modelling photon BEC & organic polaritons 2 (Ultra-)strong coupling, weak pumping Ultra strong coupling & reconfiguration Vibrational sidebands in spectrum 3 Driven dissipative systems Limitations of rate equation model Toy problem – two bosonic modes Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 5 Modelling photon BEC & organic polaritons 1 Modelling photon BEC & organic polaritons 2 (Ultra-)strong coupling, weak pumping Ultra strong coupling & reconfiguration Vibrational sidebands in spectrum 3 Driven dissipative systems Limitations of rate equation model Toy problem – two bosonic modes Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 6 What kinds of modelling Bottom up I I I DFT (or quantum chemistry) → electronic structure Time-dependent DFT /MD → vibrational spectra FDTD/transfer-matrix → cavity modes Tractable microscopic toy models Top-down I I I Jonathan Keeling Equilibrium stat. mech. (complex/stochastic/. . . )GPE (+ Boltzmann) Rate equations Modelling matter-light coupling Telluride, July 2015 7 What kinds of modelling Bottom up I I I DFT (or quantum chemistry) → electronic structure Time-dependent DFT /MD → vibrational spectra FDTD/transfer-matrix → cavity modes Tractable microscopic toy models Top-down I I I Jonathan Keeling Equilibrium stat. mech. (complex/stochastic/. . . )GPE (+ Boltzmann) Rate equations Modelling matter-light coupling Telluride, July 2015 7 What kinds of modelling Bottom up I I I DFT (or quantum chemistry) → electronic structure Time-dependent DFT /MD → vibrational spectra FDTD/transfer-matrix → cavity modes Tractable microscopic toy models Top-down I I I Jonathan Keeling Equilibrium stat. mech. (complex/stochastic/. . . )GPE (+ Boltzmann) → condensate Rate equations → laser Modelling matter-light coupling Telluride, July 2015 7 What kinds of modelling Bottom up I I I DFT (or quantum chemistry) → electronic structure Time-dependent DFT /MD → vibrational spectra FDTD/transfer-matrix → cavity modes Tractable microscopic toy models Top-down I I [From Auerbach, Interacting I Equilibrium stat. mech. (complex/stochastic/. . . )GPE (+ Boltzmann) → condensate Rate equations → laser electrons and quantum magnetism] Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 7 Toy models 1 Full molecular spectra electronic structure & Raman spectrum 3 Simplified archetypal model: Dicke-Holstein Each molecule: two DoF I I Electronic state: 2LS Vibrational state: harmonic oscillator Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 8 Toy models Full molecular spectra electronic structure & Raman spectrum Energy 1 ⇑ 2 3 nuclear coordinate Focus on low-energy effective theory ⇓ Two-level system, HOMO/LUMO See also [Galego, Garcia-Vidal, Single DoF PES Feist. arXiv:1506:03331] Simplified archetypal model: Dicke-Holstein Each molecule: two DoF I I Electronic state: 2LS Vibrational state: harmonic oscillator Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 8 Toy models Full molecular spectra electronic structure & Raman spectrum Energy 1 ⇑ Photon 2 3 nuclear coordinate Focus on low-energy effective theory ⇓ Two-level system, HOMO/LUMO See also [Galego, Garcia-Vidal, Single DoF PES Feist. arXiv:1506:03331] Simplified archetypal model: Dicke-Holstein Each molecule: two DoF I I Electronic state: 2LS Vibrational state: harmonic oscillator Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 8 Toy models Full molecular spectra electronic structure & Raman spectrum Energy 1 ⇑ Photon 2 3 nuclear coordinate Focus on low-energy effective theory ⇓ Two-level system, HOMO/LUMO See also [Galego, Garcia-Vidal, Single DoF PES Feist. arXiv:1506:03331] Simplified archetypal model: Dicke-Holstein Each molecule: two DoF I I Electronic state: 2LS Vibrational state: harmonic oscillator Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 8 Toy models Full molecular spectra electronic structure & Raman spectrum Energy 1 ⇑ Photon 2 3 nuclear coordinate Focus on low-energy effective theory ⇓ Two-level system, HOMO/LUMO See also [Galego, Garcia-Vidal, Single DoF PES Feist. arXiv:1506:03331] Simplified archetypal model: Dicke-Holstein Each molecule: two DoF I I Electronic state: 2LS Vibrational state: harmonic oscillator Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 8 Dicke Holstein Model Dicke model: 2LS ↔ photons Molecular vibrational mode I I Phonon frequency Ω Huang-Rhys parameter S — coupling strength Collective coupling to light Hsys = ωψ † ψ + Xh α Jonathan Keeling 2 i σαz + g ψ + ψ † σα+ + σα− Modelling matter-light coupling Telluride, July 2015 9 Dicke Holstein Model Dicke model: 2LS ↔ photons Molecular vibrational mode I I Phonon frequency Ω Huang-Rhys parameter S — coupling strength Collective coupling to light 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 Modelling matter-light coupling Telluride, July 2015 9 Dicke Holstein Model Dicke model: 2LS ↔ photons Molecular vibrational mode I I Phonon frequency Ω Huang-Rhys parameter S — coupling strength Collective coupling to light 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 Modelling matter-light coupling Telluride, July 2015 9 (Ultra-)strong coupling, weak pumping 1 Modelling photon BEC & organic polaritons 2 (Ultra-)strong coupling, weak pumping Ultra strong coupling & reconfiguration Vibrational sidebands in spectrum 3 Driven dissipative systems Limitations of rate equation model Toy problem – two bosonic modes Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 10 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 Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 11 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 Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 11 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 Modelling matter-light coupling Telluride, July 2015 12 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 Modelling matter-light coupling Telluride, July 2015 12 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 Modelling matter-light coupling Telluride, July 2015 12 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 Modelling matter-light coupling Telluride, July 2015 12 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 Modelling matter-light coupling Telluride, July 2015 12 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 Modelling matter-light coupling Telluride, July 2015 12 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 Modelling matter-light coupling Telluride, July 2015 12 Molecular adaptation Central peak — depends on g, not T . Can geff depend on T ? Rotational degrees of freedom i Xh H = ... + . . . + gα,k cos(θα )(ψk† + ψ−k )σαx + E0 (θα ) α Schrieffer-Wolff, δH = Heff = . . . + P α,k gα,k (ψk† σα+ + H.c.): i Xh −K0 cos2 (θα ) + E0 (θ) , α I No √ K0 = X k gk2 ωk + N enhancement — K0 small, independent of density [Cwik et al. , arXiv:1506.08974] Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 13 Molecular adaptation Central peak — depends on g, not T . Can geff depend on T ? Rotational degrees of freedom i Xh H = ... + . . . + gα,k cos(θα )(ψk† + ψ−k )σαx + E0 (θα ) α Schrieffer-Wolff, δH = Heff = . . . + P α,k gα,k (ψk† σα+ + H.c.): i Xh −K0 cos2 (θα ) + E0 (θ) , α I No √ K0 = X k gk2 ωk + N enhancement — K0 small, independent of density [Cwik et al. , arXiv:1506.08974] Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 13 Molecular adaptation Central peak — depends on g, not T . Can geff depend on T ? θ ld E fie Rotational degrees of freedom i Xh H = ... + . . . + gα,k cos(θα )(ψk† + ψ−k )σαx + E0 (θα ) α Schrieffer-Wolff, δH = Heff = . . . + P α,k gα,k (ψk† σα+ + H.c.): i Xh −K0 cos2 (θα ) + E0 (θ) , α I No √ K0 = X k gk2 ωk + N enhancement — K0 small, independent of density [Cwik et al. , arXiv:1506.08974] Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 13 Molecular adaptation Central peak — depends on g, not T . Can geff depend on T ? θ ld E fie Rotational degrees of freedom i Xh H = ... + . . . + gα,k cos(θα )(ψk† + ψ−k )σαx + E0 (θα ) α Schrieffer-Wolff, δH = Heff = . . . + P α,k gα,k (ψk† σα+ + H.c.): i Xh −K0 cos2 (θα ) + E0 (θ) , α I No √ K0 = X k gk2 ωk + N enhancement — K0 small, independent of density [Cwik et al. , arXiv:1506.08974] Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 13 Molecular adaptation Central peak — depends on g, not T . Can geff depend on T ? θ ld E fie Rotational degrees of freedom i Xh H = ... + . . . + gα,k cos(θα )(ψk† + ψ−k )σαx + E0 (θα ) α Schrieffer-Wolff, δH = Heff = . . . + P α,k gα,k (ψk† σα+ + H.c.): i Xh −K0 cos2 (θα ) + E0 (θ) , α I No √ K0 = X k gk2 ωk + N enhancement — K0 small, independent of density [Cwik et al. , arXiv:1506.08974] Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 13 Vibrational adaptation Schrieffer-Wolff – mixes vibrational states ( " √ #) √ X gk2 Ω S(b + b† ) Ω 2 g N , Heff = H0 − 1− +O 2( + ωk ) + ωk k Reduced vibrational offset S → S(1 − 2K1 ), K1 = X k I I gk2 (ωk + )2 2 Increased effective coupling: geff = g 2 exp(−S) Again, K1 1, independent of density. [Cwik et al. , arXiv:1506.08974] Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 14 Vibrational adaptation Schrieffer-Wolff – mixes vibrational states ( " √ #) √ X gk2 Ω S(b + b† ) Ω 2 g N , Heff = H0 − 1− +O 2( + ωk ) + ωk k Energy Reduced vibrational offset S → S(1 − 2K1 ), K1 = X k gk2 (ωk + )2 ⇑ Photon nuclear coordinate ⇓ l S ×l I I 2 Increased effective coupling: geff = g 2 exp(−S) Again, K1 1, independent of density. [Cwik et al. , arXiv:1506.08974] Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 14 Vibrational adaptation Schrieffer-Wolff – mixes vibrational states ( " √ #) √ X gk2 Ω S(b + b† ) Ω 2 g N , Heff = H0 − 1− +O 2( + ωk ) + ωk k Energy Reduced vibrational offset S → S(1 − 2K1 ), K1 = X k gk2 (ωk + )2 ⇑ Photon nuclear coordinate ⇓ l S ×l I I 2 Increased effective coupling: geff = g 2 exp(−S) Again, K1 1, independent of density. [Cwik et al. , arXiv:1506.08974] Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 14 Vibrational adaptation Schrieffer-Wolff – mixes vibrational states ( " √ #) √ X gk2 Ω S(b + b† ) Ω 2 g N , Heff = H0 − 1− +O 2( + ωk ) + ωk k Energy Reduced vibrational offset S → S(1 − 2K1 ), K1 = X k gk2 (ωk + )2 ⇑ Photon nuclear coordinate ⇓ l S ×l I I 2 Increased effective coupling: geff = g 2 exp(−S) Again, K1 1, independent of density. [Cwik et al. , arXiv:1506.08974] Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 14 Any other kind of adaptation Generic (classical) DoF (solvation, charge distribution . . . ) i Xh H = ... + . . . + gα,k f (xα )(ψk† + ψ−k )σαx + E0 (xα ) α Schrieffer-Wolff: Heff = . . . + Xh i E0 (xα ) − K0 f 2 (xα ) , K0 = α I I X k gk2 ωk + Ground state energy: −K0 Nf (x0 ), collective. Extent of reconfiguration, δx0 ∼ g 2 , not collective. Why: I I R 2 dxf (x)e−βN [E0 (x)−K0 f (x)] geff If xα = x, then = R g dxe−βN[E0 (x)−K0 f 2 (x)] But xα are all individuals: P Q R −β α E0 (xα )−K0 f 2 (xα ) geff,α0 α dxα f (xα0 )eP = Q R −β α E0 (xα )−K0 f 2 (xα ) g α dxα e Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 15 Any other kind of adaptation Generic (classical) DoF (solvation, charge distribution . . . ) i Xh H = ... + . . . + gα,k f (xα )(ψk† + ψ−k )σαx + E0 (xα ) α Schrieffer-Wolff: Heff = . . . + Xh i E0 (xα ) − K0 f 2 (xα ) , K0 = α I I X k gk2 ωk + Ground state energy: −K0 Nf (x0 ), collective. Extent of reconfiguration, δx0 ∼ g 2 , not collective. Why: I I R 2 dxf (x)e−βN [E0 (x)−K0 f (x)] geff If xα = x, then = R g dxe−βN[E0 (x)−K0 f 2 (x)] But xα are all individuals: P Q R −β α E0 (xα )−K0 f 2 (xα ) geff,α0 α dxα f (xα0 )eP = Q R −β α E0 (xα )−K0 f 2 (xα ) g α dxα e Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 15 Any other kind of adaptation Generic (classical) DoF (solvation, charge distribution . . . ) i Xh H = ... + . . . + gα,k f (xα )(ψk† + ψ−k )σαx + E0 (xα ) α Schrieffer-Wolff: Heff = . . . + Xh i E0 (xα ) − K0 f 2 (xα ) , K0 = α I I X k gk2 ωk + Ground state energy: −K0 Nf (x0 ), collective. Extent of reconfiguration, δx0 ∼ g 2 , not collective. Why: I I R 2 dxf (x)e−βN [E0 (x)−K0 f (x)] geff If xα = x, then = R g dxe−βN[E0 (x)−K0 f 2 (x)] But xα are all individuals: P Q R −β α E0 (xα )−K0 f 2 (xα ) geff,α0 α dxα f (xα0 )eP = Q R −β α E0 (xα )−K0 f 2 (xα ) g α dxα e Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 15 Any other kind of adaptation Generic (classical) DoF (solvation, charge distribution . . . ) i Xh H = ... + . . . + gα,k f (xα )(ψk† + ψ−k )σαx + E0 (xα ) α Schrieffer-Wolff: Heff = . . . + Xh i E0 (xα ) − K0 f 2 (xα ) , K0 = α I I X k gk2 ωk + Ground state energy: −K0 Nf (x0 ), collective. Extent of reconfiguration, δx0 ∼ g 2 , not collective. Why: I I R 2 dxf (x)e−βN [E0 (x)−K0 f (x)] geff If xα = x, then = R g dxe−βN[E0 (x)−K0 f 2 (x)] But xα are all individuals: P Q R −β α E0 (xα )−K0 f 2 (xα ) geff,α0 α dxα f (xα0 )eP = Q R −β α E0 (xα )−K0 f 2 (xα ) g α dxα e Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 15 Any other kind of adaptation Generic (classical) DoF (solvation, charge distribution . . . ) i Xh H = ... + . . . + gα,k f (xα )(ψk† + ψ−k )σαx + E0 (xα ) α Schrieffer-Wolff: Heff = . . . + Xh i E0 (xα ) − K0 f 2 (xα ) , K0 = α I I X k gk2 ωk + Ground state energy: −K0 Nf (x0 ), collective. Extent of reconfiguration, δx0 ∼ g 2 , not collective. Why: I I R 2 dxf (x)e−βN [E0 (x)−K0 f (x)] geff If xα = x, then = R g dxe−βN[E0 (x)−K0 f 2 (x)] But xα are all individuals: P Q R −β α E0 (xα )−K0 f 2 (xα ) geff,α0 α dxα f (xα0 )eP = Q R −β α E0 (xα )−K0 f 2 (xα ) g α dxα e Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 15 Any other kind of adaptation Generic (classical) DoF (solvation, charge distribution . . . ) i Xh H = ... + . . . + gα,k f (xα )(ψk† + ψ−k )σαx + E0 (xα ) α Schrieffer-Wolff: Heff = . . . + Xh i E0 (xα ) − K0 f 2 (xα ) , K0 = α I I X k gk2 ωk + Ground state energy: −K0 Nf (x0 ), collective. Extent of reconfiguration, δx0 ∼ g 2 , not collective. Why: I I R 2 dxf (x)e−βN [E0 (x)−K0 f (x)] geff If xα = x, then = R g dxe−βN[E0 (x)−K0 f 2 (x)] But xα are all individuals: P Q R −β α E0 (xα )−K0 f 2 (xα ) geff,α0 α dxα f (xα0 )eP = Q R −β α E0 (xα )−K0 f 2 (xα ) g α dxα e Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 15 Any other kind of adaptation Generic (classical) DoF (solvation, charge distribution . . . ) i Xh H = ... + . . . + gα,k f (xα )(ψk† + ψ−k )σαx + E0 (xα ) α Schrieffer-Wolff: Heff = . . . + Xh i E0 (xα ) − K0 f 2 (xα ) , K0 = α I I X k gk2 ωk + Ground state energy: −K0 Nf (x0 ), collective. Extent of reconfiguration, δx0 ∼ g 2 , not collective. Why: Entropy – this is why BEC matters!! I I R 2 dxf (x)e−βN [E0 (x)−K0 f (x)] geff If xα = x, then = R g dxe−βN[E0 (x)−K0 f 2 (x)] But xα are all individuals: R 2 dxα0 f (xα0 )e−β[E0 (xα0 )−K0 f (xα0 )] geff,α0 = R 2 g dxα0 e−β[E0 (xα0 )−K0 f (xα0 )] Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 15 Disordered molecules — vibrational mode But: spectrum with vibrational sidebands, S = 0.02 Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 16 Disordered molecules — vibrational mode But: spectrum 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] Modelling matter-light coupling 2.8 3 3.2 Telluride, July 2015 16 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 But: spectrum 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 Modelling matter-light coupling 3 3.2 Telluride, July 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 2.1 2.2 ω [eV] 2.3 2.4 2.5 [Cwik et al. , arXiv:1506.08974] Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 17 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 2.1 2.2 ω [eV] 2.3 2.4 2.5 [Cwik et al. , arXiv:1506.08974] Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 17 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 2.1 2.2 ω [eV] 2.3 2.4 2.5 [Cwik et al. , arXiv:1506.08974] Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 17 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 2.1 2.2 ω [eV] 2.3 2.4 2.5 [Cwik et al. , arXiv:1506.08974] Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 17 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 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. , arXiv:1506.08974] Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 17 Driven dissipative systems 1 Modelling photon BEC & organic polaritons 2 (Ultra-)strong coupling, weak pumping Ultra strong coupling & reconfiguration Vibrational sidebands in spectrum 3 Driven dissipative systems Limitations of rate equation model Toy problem – two bosonic modes Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 18 Photon BEC, rate equations & oscillations Photon BEC: can derive photon rate equation [See Kirton talk] 1 ∂t nm = −κnm + Γ(−δm )(nm + 1)N↑ − Γ(δm )nm N↓ Spectrum 0.8 Γ(-δ) Γ(δ) 0.6 0.4 0.2 0 -400 -200 0 200 δ=ω - ωZPL 400 Experiments [Schmitt et al. PRA ’15 Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 19 Photon BEC, rate equations & oscillations Photon BEC: can derive photon rate equation [See Kirton talk] 1 ∂t nm = −κnm + Γ(−δm )(nm + 1)N↑ − Γ(δm )nm N↓ Spectrum 0.8 Γ(-δ) Γ(δ) 0.6 0.4 0.2 0 -400 -200 0 200 δ=ω - ωZPL 400 Experiments [Schmitt et al. PRA ’15 Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 19 Photon BEC rate equations ∂t nm = −κnm + Γ(−δm )(nm + 1)N↑ − Γ(δm )nm N↓ Describes emission into Gauss-Hermite mode m X I(x) = nm |ψm (x)|2 m Oscillations: beating of modes. P Need I(x) = m,m0 nm,m0 ψm (x)ψm0 (x) Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 20 Photon BEC rate equations ∂t nm = −κnm + Γ(−δm )(nm + 1)N↑ − Γ(δm )nm N↓ Describes emission into Gauss-Hermite mode m X I(x) = nm |ψm (x)|2 m Oscillations: beating of modes. P Need I(x) = m,m0 nm,m0 ψm (x)ψm0 (x) Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 20 Photon BEC rate equations ∂t nm = −κnm + Γ(−δm )(nm + 1)N↑ − Γ(δm )nm N↓ Describes emission into Gauss-Hermite mode m X I(x) = nm |ψm (x)|2 m Oscillations: beating of modes. P Need I(x) = m,m0 nm,m0 ψm (x)ψm0 (x) Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 20 Photon BEC rate equations ∂t nm = −κnm + Γ(−δm )(nm + 1)N↑ − Γ(δm )nm N↓ Describes emission into Gauss-Hermite mode m X I(x) = nm |ψm (x)|2 m Oscillations: beating of modes. P Need I(x) = m,m0 nm,m0 ψm (x)ψm0 (x) Emission must create coherence between non-degenerate modes. Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 20 Toy problem: two bosonic modes Basic problem: Emission from thermal bath Mode b H = ωa ψ̂a† ψ̂a + ωb ψ̂b† ψ̂b + HBath X + (ϕ∗a ψ̂a† + ϕ∗b ψ̂b† ) gi ĉi + H.c. Bath J( ν) Mode a Jonathan Keeling i ν Modelling matter-light coupling Telluride, July 2015 21 Toy problem: naı̈ve solutions Two “expected” behaviours: At resonance: “weak lasing” — coupling to bath dominates d ρ = Γ↓ L[ϕa ψ̂a + ϕb ψ̂b ] + Γ↑ L[ϕ∗a ψ̂a† + ϕ∗b ψ̂b† ] dt Far from resonance: pointer states are eigenstates X ↓ d ρ= Γi L[ψ̂i ] + Γ↑i L[ψ̂i† ] dt i=a,b Questions: I I How does crossover work? Are these actually right? Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 22 Toy problem: naı̈ve solutions Two “expected” behaviours: At resonance: “weak lasing” — coupling to bath dominates d ρ = Γ↓ L[ϕa ψ̂a + ϕb ψ̂b ] + Γ↑ L[ϕ∗a ψ̂a† + ϕ∗b ψ̂b† ] dt Far from resonance: pointer states are eigenstates X ↓ d ρ= Γi L[ψ̂i ] + Γ↑i L[ψ̂i† ] dt i=a,b Questions: I I How does crossover work? Are these actually right? Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 22 Toy problem: naı̈ve solutions Two “expected” behaviours: At resonance: “weak lasing” — coupling to bath dominates d ρ = Γ↓ L[ϕa ψ̂a + ϕb ψ̂b ] + Γ↑ L[ϕ∗a ψ̂a† + ϕ∗b ψ̂b† ] dt Far from resonance: pointer states are eigenstates X ↓ d ρ= Γi L[ψ̂i ] + Γ↑i L[ψ̂i† ] dt i=a,b Questions: I I How does crossover work? Are these actually right? Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 22 Toy problem: exact solution Solve via Laplace transform. Find Fij (t) = hψ̂i† (t)ψ̂j (t)i Steady state: I Singular at ∆ = 0 Time evolution — Fab (t) ∼ exp(−α∆2 t) Always some coherence I (individual always wrong) Fab ∼ Faa , Fbb only at ∆ = 0 I Jonathan Keeling (collective almost always wrong) Modelling matter-light coupling Telluride, July 2015 23 Toy problem: exact solution I Singular at ∆ = 0 1×10-2 0 Faa, Fbb Time evolution — Fab (t) ∼ exp(−α∆2 t) Re[Fab] Solve via Laplace transform. Find Fij (t) = hψ̂i† (t)ψ̂j (t)i Steady state: 2×10-2 Fab 0.2 Faa Fbb 0 -0.4 -0.2 0 0.2 -∆=2(ωb-ωa) 0.4 Always some coherence I (individual always wrong) Fab ∼ Faa , Fbb only at ∆ = 0 I Jonathan Keeling (collective almost always wrong) Modelling matter-light coupling Telluride, July 2015 23 Toy problem: exact solution I Singular at ∆ = 0 1×10-2 0 Faa, Fbb Time evolution — Fab (t) ∼ exp(−α∆2 t) Re[Fab] Solve via Laplace transform. Find Fij (t) = hψ̂i† (t)ψ̂j (t)i Steady state: 2×10-2 Fab 0.2 Faa Fbb 0 -0.4 -0.2 0 0.2 -∆=2(ωb-ωa) 0.4 Always some coherence I (individual always wrong) Fab ∼ Faa , Fbb only at ∆ = 0 I Jonathan Keeling (collective almost always wrong) Modelling matter-light coupling Telluride, July 2015 23 Toy problem: exact solution Singular at ∆ = 0 Time evolution — Fab (t) ∼ exp(−α∆2 t) Fab 0 1×10-2 0 0.05 0.1 2000 Faa, Fbb I Re[Fab] Solve via Laplace transform. Find Fij (t) = hψ̂i† (t)ψ̂j (t)i Steady state: 2×10-2 Fab 0.2 Faa Fbb 0 -0.4 Time 1500 0 0.2 -∆=2(ωb-ωa) 0.4 Always some coherence 1000 I (individual always wrong) Fab ∼ Faa , Fbb only at ∆ = 0 500 I 0 -0.4 -0.2 -0.2 Jonathan Keeling 0 0.2 ∆=2(ωa-ωb) (collective almost always wrong) 0.4 Modelling matter-light coupling Telluride, July 2015 23 Toy problem: exact solution Singular at ∆ = 0 Time evolution — Fab (t) ∼ exp(−α∆2 t) Fab 0 1×10-2 0 0.05 0.1 2000 Faa, Fbb I Re[Fab] Solve via Laplace transform. Find Fij (t) = hψ̂i† (t)ψ̂j (t)i Steady state: 2×10-2 Fab 0.2 Faa Fbb 0 -0.4 Time 1500 0 0.2 -∆=2(ωb-ωa) 0.4 Always some coherence 1000 I (individual always wrong) Fab ∼ Faa , Fbb only at ∆ = 0 500 I 0 -0.4 -0.2 -0.2 Jonathan Keeling 0 0.2 ∆=2(ωa-ωb) (collective almost always wrong) 0.4 Modelling matter-light coupling Telluride, July 2015 23 Toy problem: exact solution Singular at ∆ = 0 Time evolution — Fab (t) ∼ exp(−α∆2 t) Fab 0 1×10-2 0 0.05 0.1 2000 Faa, Fbb I Re[Fab] Solve via Laplace transform. Find Fij (t) = hψ̂i† (t)ψ̂j (t)i Steady state: 2×10-2 Fab 0.2 Faa Fbb 0 -0.4 Time 1500 0 0.2 -∆=2(ωb-ωa) 0.4 Always some coherence 1000 I (individual always wrong) Fab ∼ Faa , Fbb only at ∆ = 0 500 I 0 -0.4 -0.2 -0.2 Jonathan Keeling 0 0.2 ∆=2(ωa-ωb) (collective almost always wrong) 0.4 Modelling matter-light coupling Telluride, July 2015 23 Toy problem: Bloch-Redfield theory Unsecularised Bloch-Redfield theory: ∂t ρ = −i[Ĥ, ρ] + X L↓ij ϕ∗i ϕj 2ψ̂j ρψ̂i† − [ρ, ψ̂i† ψ̂j ]+ ij + X L↑ij ϕ∗i ϕj 2ψ̂j† ρψ̂i − [ρ, ψ̂i ψ̂j† ]+ . ij Compare to exact solution: Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 24 Toy problem: Bloch-Redfield theory Unsecularised Bloch-Redfield theory: ∂t ρ = −i[Ĥ, ρ] + X L↓ij ϕ∗i ϕj 2ψ̂j ρψ̂i† − [ρ, ψ̂i† ψ̂j ]+ ij + L↑ij ϕ∗i ϕj 2ψ̂j† ρψ̂i − [ρ, ψ̂i ψ̂j† ]+ . X ij Compare to exact solution: 0.1 0.01 ∆=0.2 Re[Fab] Re[Fab] t=200 0.05 0 0 Exact Redfield -0.01 -0.4 -0.2 0 0.2 ∆=2(ωa-ωb) Jonathan Keeling 0.4 0 Modelling matter-light coupling 200 400 Time 600 800 Telluride, July 2015 24 Toy problem: Secularisation Leads to Fab (t → ∞) = 0. Exact: Re[Fab] ↑,↓ Secularisation (in eigenbasis of Ĥ): L↑,↓ ij → Lii δij 2×10-2 1×10-2 Fab 0 -0.4 -0.2 0 0.2 -∆=2(ωb-ωa) 0.4 Secularisation often invoked to cure negative eigenvalues of Lij↑,↓ . → Non-positivity of density matrix, → Unstable (unbounded growth). Check stability: consider f = (Faa , Fbb , <[Fab ], =[Fab ]) ∂t f = −Mf + f0 Eigenvalues of M exist in closed form: I Unstable (negative only if dJ(ν)/dν 1 — Markov breakdown) Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 25 Toy problem: Secularisation Leads to Fab (t → ∞) = 0. Exact: Re[Fab] ↑,↓ Secularisation (in eigenbasis of Ĥ): L↑,↓ ij → Lii δij 2×10-2 1×10-2 Fab 0 -0.4 -0.2 0 0.2 -∆=2(ωb-ωa) 0.4 Secularisation often invoked to cure negative eigenvalues of Lij↑,↓ . → Non-positivity of density matrix, → Unstable (unbounded growth). Check stability: consider f = (Faa , Fbb , <[Fab ], =[Fab ]) ∂t f = −Mf + f0 Eigenvalues of M exist in closed form: I Unstable (negative only if dJ(ν)/dν 1 — Markov breakdown) Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 25 Toy problem: Secularisation Leads to Fab (t → ∞) = 0. Exact: Re[Fab] ↑,↓ Secularisation (in eigenbasis of Ĥ): L↑,↓ ij → Lii δij 2×10-2 1×10-2 Fab 0 -0.4 -0.2 0 0.2 -∆=2(ωb-ωa) 0.4 Secularisation often invoked to cure negative eigenvalues of Lij↑,↓ . → Non-positivity of density matrix, → Unstable (unbounded growth). Check stability: consider f = (Faa , Fbb , <[Fab ], =[Fab ]) ∂t f = −Mf + f0 Eigenvalues of M exist in closed form: I Unstable (negative only if dJ(ν)/dν 1 — Markov breakdown) Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 25 Toy problem: Secularisation Leads to Fab (t → ∞) = 0. Exact: Re[Fab] ↑,↓ Secularisation (in eigenbasis of Ĥ): L↑,↓ ij → Lii δij 2×10-2 1×10-2 Fab 0 -0.4 -0.2 0 0.2 -∆=2(ωb-ωa) 0.4 Secularisation often invoked to cure negative eigenvalues of Lij↑,↓ . → Non-positivity of density matrix, → Unstable (unbounded growth). Check stability: consider f = (Faa , Fbb , <[Fab ], =[Fab ]) ∂t f = −Mf + f0 Eigenvalues of M exist in closed form: I Unstable (negative only if dJ(ν)/dν 1 — Markov breakdown) Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 25 Toy problem: Secularisation Leads to Fab (t → ∞) = 0. Exact: Re[Fab] ↑,↓ Secularisation (in eigenbasis of Ĥ): L↑,↓ ij → Lii δij 2×10-2 1×10-2 Fab 0 -0.4 -0.2 0 0.2 -∆=2(ωb-ωa) 0.4 Secularisation often invoked to cure negative eigenvalues of Lij↑,↓ . → Non-positivity of density matrix, → Unstable (unbounded growth). Check stability: consider f = (Faa , Fbb , <[Fab ], =[Fab ]) ∂t f = −Mf + f0 Eigenvalues of M exist in closed form: I Unstable (negative only if dJ(ν)/dν 1 — Markov breakdown) Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 25 Toy problem: Secularisation Leads to Fab (t → ∞) = 0. Exact: Re[Fab] ↑,↓ Secularisation (in eigenbasis of Ĥ): L↑,↓ ij → Lii δij 2×10-2 1×10-2 Fab 0 -0.4 -0.2 0 0.2 -∆=2(ωb-ωa) 0.4 Secularisation often invoked to cure negative eigenvalues of Lij↑,↓ . → Non-positivity of density matrix, → Unstable (unbounded growth). Check stability: consider f = (Faa , Fbb , <[Fab ], =[Fab ]) ∂t f = −Mf + f0 Eigenvalues of M exist in closed form: I Unstable (negative only if dJ(ν)/dν 1 — Markov breakdown) Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 25 Toy problem: Secularisation Leads to Fab (t → ∞) = 0. Exact: Re[Fab] ↑,↓ Secularisation (in eigenbasis of Ĥ): L↑,↓ ij → Lii δij 2×10-2 1×10-2 Fab 0 -0.4 -0.2 0 0.2 -∆=2(ωb-ωa) 0.4 Secularisation often invoked to cure negative eigenvalues of Lij↑,↓ . → Non-positivity of density matrix, → Unstable (unbounded growth). Check stability: consider f = (Faa , Fbb , <[Fab ], =[Fab ]) ∂t f = −Mf + f0 0.02 I Unstable (negative only if dJ(ν)/dν 1 — Markov breakdown) J(ν) Eigenvalues of M exist in closed form: 0.01 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 ν Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 25 Toy problem: Secularisation Leads to Fab (t → ∞) = 0. Exact: Re[Fab] ↑,↓ Secularisation (in eigenbasis of Ĥ): L↑,↓ ij → Lii δij 2×10-2 1×10-2 Fab 0 -0.4 -0.2 0 0.2 -∆=2(ωb-ωa) 0.4 Secularisation often invoked to cure negative eigenvalues of Lij↑,↓ . → Non-positivity of density matrix, → Unstable (unbounded growth). Check stability: consider f = (Faa , Fbb , <[Fab ], =[Fab ]) ∂t f = −Mf + f0 0.02 I Unstable (negative only if dJ(ν)/dν 1 — Markov breakdown) J(ν) Eigenvalues of M exist in closed form: 0.01 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 ν Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 25 Beyond Redfield: Schrödinger picture Bloch Redfield Is BR the best (time-local) theory we can find? Hints it is not: I I Eigenvalues of M vs exact sol’n near ∆ = 0. Sum rule [Salmilehto et al. PRA ’12; Hell et al. PRB ’14]: “For X̂ s.t. [X̂ , Ĥsystem-bath ] = 0, then ∂t hX̂ i should match closed system.” I 0 Here, hX̂ i = ϕ2b Faa + ϕ2a Fbb − 2ϕa ϕb Fab . Fails Alternate approach: I I I BR assumes ρ̃(t) is “slow” in interaction picture Asymptotically ρ(t) is steady in Schrödinger picture Assume instead ρ(t) is slow in Schrödinger picture “Schrödinger picture Bloch Redfield.” I I Correct ∆2 expansion Satisfies sum rule Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 26 Beyond Redfield: Schrödinger picture Bloch Redfield Is BR the best (time-local) theory we can find? Hints it is not: I I Eigenvalues of M vs exact sol’n near ∆ = 0. Sum rule [Salmilehto et al. PRA ’12; Hell et al. PRB ’14]: “For X̂ s.t. [X̂ , Ĥsystem-bath ] = 0, then ∂t hX̂ i should match closed system.” I 0 Here, hX̂ i = ϕ2b Faa + ϕ2a Fbb − 2ϕa ϕb Fab . Fails Alternate approach: I I I BR assumes ρ̃(t) is “slow” in interaction picture Asymptotically ρ(t) is steady in Schrödinger picture Assume instead ρ(t) is slow in Schrödinger picture “Schrödinger picture Bloch Redfield.” I I Correct ∆2 expansion Satisfies sum rule Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 26 Beyond Redfield: Schrödinger picture Bloch Redfield Is BR the best (time-local) theory we can find? Hints it is not: I I Eigenvalues of M vs exact sol’n near ∆ = 0. Sum rule [Salmilehto et al. PRA ’12; Hell et al. PRB ’14]: “For X̂ s.t. [X̂ , Ĥsystem-bath ] = 0, then ∂t hX̂ i should match closed system.” I 0 Here, hX̂ i = ϕ2b Faa + ϕ2a Fbb − 2ϕa ϕb Fab . Fails Alternate approach: I I I BR assumes ρ̃(t) is “slow” in interaction picture Asymptotically ρ(t) is steady in Schrödinger picture Assume instead ρ(t) is slow in Schrödinger picture “Schrödinger picture Bloch Redfield.” I I Correct ∆2 expansion Satisfies sum rule Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 26 Beyond Redfield: Schrödinger picture Bloch Redfield Is BR the best (time-local) theory we can find? Hints it is not: I I Eigenvalues of M vs exact sol’n near ∆ = 0. Sum rule [Salmilehto et al. PRA ’12; Hell et al. PRB ’14]: “For X̂ s.t. [X̂ , Ĥsystem-bath ] = 0, then ∂t hX̂ i should match closed system.” I 0 Here, hX̂ i = ϕ2b Faa + ϕ2a Fbb − 2ϕa ϕb Fab . Fails Alternate approach: I I I BR assumes ρ̃(t) is “slow” in interaction picture Asymptotically ρ(t) is steady in Schrödinger picture Assume instead ρ(t) is slow in Schrödinger picture “Schrödinger picture Bloch Redfield.” I I Correct ∆2 expansion Satisfies sum rule Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 26 Beyond Redfield: Schrödinger picture Bloch Redfield Is BR the best (time-local) theory we can find? Hints it is not: I I Eigenvalues of M vs exact sol’n near ∆ = 0. Sum rule [Salmilehto et al. PRA ’12; Hell et al. PRB ’14]: “For X̂ s.t. [X̂ , Ĥsystem-bath ] = 0, then ∂t hX̂ i should match closed system.” I 0 Here, hX̂ i = ϕ2b Faa + ϕ2a Fbb − 2ϕa ϕb Fab . Fails Alternate approach: I I I BR assumes ρ̃(t) is “slow” in interaction picture Asymptotically ρ(t) is steady in Schrödinger picture Assume instead ρ(t) is slow in Schrödinger picture “Schrödinger picture Bloch Redfield.” I I Correct ∆2 expansion Satisfies sum rule Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 26 Beyond Redfield: Schrödinger picture Bloch Redfield Is BR the best (time-local) theory we can find? Hints it is not: I I Eigenvalues of M vs exact sol’n near ∆ = 0. Sum rule [Salmilehto et al. PRA ’12; Hell et al. PRB ’14]: “For X̂ s.t. [X̂ , Ĥsystem-bath ] = 0, then ∂t hX̂ i should match closed system.” I 0 Here, hX̂ i = ϕ2b Faa + ϕ2a Fbb − 2ϕa ϕb Fab . Fails Alternate approach: I I I BR assumes ρ̃(t) is “slow” in interaction picture Asymptotically ρ(t) is steady in Schrödinger picture Assume instead ρ(t) is slow in Schrödinger picture “Schrödinger picture Bloch Redfield.” I I Correct ∆2 expansion Satisfies sum rule Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 26 Beyond Redfield: Schrödinger picture Bloch Redfield Is BR the best (time-local) theory we can find? Hints it is not: I I Eigenvalues of M vs exact sol’n near ∆ = 0. Sum rule [Salmilehto et al. PRA ’12; Hell et al. PRB ’14]: “For X̂ s.t. [X̂ , Ĥsystem-bath ] = 0, then ∂t hX̂ i should match closed system.” Alternate approach: I I I BR assumes ρ̃(t) is “slow” in interaction picture Asymptotically ρ(t) is steady in Schrödinger picture Assume instead ρ(t) is slow in Schrödinger picture Re[Fab] 0 Here, hX̂ i = ϕ2b Faa + ϕ2a Fbb − 2ϕa ϕb Fab . Fails 2×10 I Exact BR SpBR 0 “Schrödinger picture Bloch Redfield.” I -2 1×10-2 Faa, Fbb I 0.2 Faa Fbb 0 -0.4 -0.2 0 0.2 -∆=2(ωb-ωa) 0.4 Correct ∆2 expansion Satisfies sum rule Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 26 Acknowledgements G ROUP : C OLLABORATORS : F UNDING : Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 27 Summary Vibrational configuration 0.05 1 1.9 0.1 2 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 2.4 2.6 ω [eV] 2.8 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, Kirton, De Liberato, JK arXiv:1506.08974] Modelling incoherent emission into non-degenerate modes 0 0.05 0.1 Re[Fab] Fab 2000 Mode b 1500 J( ν) Mode a ν 2×10-2 1×10-2 Exact BR SpBR 0 1000 Faa, Fbb Time Bath 500 0 -0.4 -0.2 0 0.2 ∆=2(ωa-ωb) 0.4 0.2 Faa Fbb 0 -0.4 -0.2 0 0.2 -∆=2(ωb-ωa) 0.4 [Eastham, Kirton, Cammack, Lovett, JK arXiv:1508.XXXX] Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 28 Extra Slides Jonathan Keeling Modelling matter-light coupling Telluride, July 2015 29