Pairing Phases of Polaritons, and photon condensates

Transcription

1 Pairing Phases of Polaritons, and photon condensates Jonathan Keeling University of St Andrews 6 YEARS ICSCE7, Hakone, April 14 Jonathan Keeling Pairing phases & photons ICSCE7, April 14 1 / 1

2 Outline 1 Pairing phases of polaritons Pairing phases and Feshbach for polaritons Phase diagram: Critical detunings Signatures Phase diagram: Critical temperatures Photon condensation Modelling organic molecules: Vibrational modes Strong coupling? Jonathan Keeling Pairing phases & photons ICSCE7, April 14 / 1

3 Pairing phases of atoms Fermions From Randeria, Nat. Phys. 1 BEC-BCS crossover BEC-BEC transition Ĥ =... + ˆψ m ˆψ a1 ˆψ a + h.c. If ˆψm, If ˆψ a1, ˆψ a. High density metastability. [Eagles, Leggett, Keldysh, Nozières, Randeria,... ] Jonathan Keeling Pairing phases & photons ICSCE7, April 14 3 / 1

4 Pairing phases of atoms Fermions Bosons T (B B ) From Randeria, Nat. Phys. 1 BEC-BCS crossover [Eagles, Leggett, Keldysh, Nozières, Randeria,... ] BEC-BEC transition Ĥ =... + ˆψ m ˆψ a1 ˆψ a + h.c. If ˆψm, If ˆψ a1, ˆψ a. High density metastability. [Nozières, St James, Timmermanns, Mueller, Thouless, Radzihovsky, Stoof, Sachdev... ] Jonathan Keeling Pairing phases & photons ICSCE7, April 14 3 / 1

5 Pairing phases of atoms Fermions Bosons T (B B ) From Randeria, Nat. Phys. 1 BEC-BCS crossover [Eagles, Leggett, Keldysh, Nozières, Randeria,... ] BEC-BEC transition Ĥ =... + ˆψ m ˆψ a1 ˆψ a + h.c. If ˆψm, If ˆψ a1, ˆψ a. High density metastability. [Nozières, St James, Timmermanns, Mueller, Thouless, Radzihovsky, Stoof, Sachdev... ] Jonathan Keeling Pairing phases & photons ICSCE7, April 14 3 / 1

6 Polariton Feshbach Hybridisation of bound states: Biexciton: opposite spins (two-species): ω X [ E b Hybridisation with photons: 1 (ωc + ωx ) ] 1 Ω r + δ Control δ change ν, m, Interaction... [Ivanov, Haug, Keldysh 98], [Wouters 7], [Caursotto et al. 1], [Deveaud-Pledran et al. 13] Jonathan Keeling Pairing phases & photons ICSCE7, April 14 4 / 1

7 Polariton Feshbach Hybridisation of bound states: Biexciton: opposite spins (two-species): ω X [ E b Hybridisation with photons: 1 (ωc + ωx ) ] 1 Ω r + δ Control δ change ν, m, Interaction... [Ivanov, Haug, Keldysh 98], [Wouters 7], [Caursotto et al. 1], [Deveaud-Pledran et al. 13] Jonathan Keeling Pairing phases & photons ICSCE7, April 14 4 / 1

8 Polariton Feshbach Hybridisation of bound states: Biexciton: opposite spins (two-species): ω X [ E b Hybridisation with photons: 1 (ωc + ωx ) ] 1 Ω r + δ E X ω ω XX LP ω E b ν > E ω X [mev] 6 4 UP ω UP Control δ change ν, m, Interaction... C C ω X X ω δ (Ω R +δ ) 1/ 4 LP 4 k [µm 1 ] LP ω [Ivanov, Haug, Keldysh 98], [Wouters 7], [Caursotto et al. 1], [Deveaud-Pledran et al. 13] Jonathan Keeling Pairing phases & photons ICSCE7, April 14 4 / 1

9 Polariton Feshbach Hybridisation of bound states: Biexciton: opposite spins (two-species): ω X [ E b Hybridisation with photons: 1 (ωc + ωx ) ] 1 Ω r + δ E ω = X ω LP ω XX E b ν = X [mev] ω E 6 4 UP C X LP 4 k [µm 1 ] 4 ω UP ω C X ω LP ω δ (Ω R +δ ) 1/ Control δ change ν, m, Interaction... [Ivanov, Haug, Keldysh 98], [Wouters 7], [Caursotto et al. 1], [Deveaud-Pledran et al. 13] Jonathan Keeling Pairing phases & photons ICSCE7, April 14 4 / 1

10 Polariton Feshbach X [mev] ω E 6 4 Hybridisation of bound states: Biexciton: opposite spins (two-species): ω X [ E b Hybridisation with photons: 1 (ωc + ωx ) ] 1 Ω r + δ UP C X LP 4 k [µm 1 ] 4 ω = X ω LP ω XX ω UP ω C X ω LP ω E E b ν = δ (Ω R +δ ) 1/ [mev] ν Control δ change ν, m, Interaction... Bound state δ [mev] [Ivanov, Haug, Keldysh 98], [Wouters 7], [Caursotto et al. 1], [Deveaud-Pledran et al. 13] Jonathan Keeling Pairing phases & photons ICSCE7, April 14 4 / 1 E b

11 Phase diagram (ground state, T = ) δ [mev] N.6.4. µ [mev] PS 1x1 11 n [cm ] δ < : standard BEC. Small δ : 1st order transition Larkin-Pikin mechanism Large phase-separation region Jonathan Keeling Pairing phases & photons ICSCE7, April 14 5 / 1

12 Phase diagram (ground state, T = ) δ [mev] N.6.4. µ [mev] PS 1x1 11 n [cm ] δ < : standard BEC. Small δ : 1st order transition Larkin-Pikin mechanism Large phase-separation region Jonathan Keeling Pairing phases & photons ICSCE7, April 14 5 / 1

13 Phase diagram (ground state, T = ) δ [mev] N.6.4. µ [mev] PS 1x1 11 n [cm ] δ < : standard BEC. Small δ : 1st order transition Larkin-Pikin mechanism Large phase-separation region Jonathan Keeling Pairing phases & photons ICSCE7, April 14 5 / 1

14 Consequences and Signatures Phase separation Phase coherence (1) : standard g σ = ψ σ(r, t)ψ σ (, ). (D, QLRO) Novel half vortices, ψ = e im θ, ψ = e im θ, has (m, m ) = (1/, 1/). Previous half-vortex (m, m ) = (1, ) [Lagoudakis et al. Science 9] Jonathan Keeling Pairing phases & photons ICSCE7, April 14 6 / 1

15 Consequences and Signatures Phase separation Phase coherence (1) : standard g σ = ψ σ(r, t)ψ σ (, ). (D, QLRO) Novel half vortices, ψ = e im θ, ψ = e im θ, has (m, m ) = (1/, 1/). Previous half-vortex (m, m ) = (1, ) [Lagoudakis et al. Science 9] Jonathan Keeling Pairing phases & photons ICSCE7, April 14 6 / 1

16 Consequences and Signatures Phase separation Phase coherence (1) : standard g σ = ψ σ(r, t)ψ σ (, ). (D, QLRO) Novel half vortices, ψ = e im θ, ψ = e im θ, has (m, m ) = (1/, 1/). Previous half-vortex (m, m ) = (1, ) [Lagoudakis et al. Science 9] Jonathan Keeling Pairing phases & photons ICSCE7, April 14 6 / 1

17 Consequences and Signatures Phase separation Phase coherence (1) : standard g σ = ψ σ(r, t)ψ σ (, ). (D, QLRO) Novel half vortices, ψ = e im θ, ψ = e im θ, has (m, m ) = (1/, 1/). Previous half-vortex (m, m ) = (1, ) [Lagoudakis et al. Science 9] Jonathan Keeling Pairing phases & photons ICSCE7, April 14 6 / 1

18 Consequences and Signatures Phase separation Phase coherence (1) : standard g σ = ψ σ(r, t)ψ σ (, ). (D, QLRO) APD : g (1) σ = but Adjustable delay φ APD A g (1) m = ψ (r, t)ψ (r, t)ψ (, )ψ (, ) APD θ APD B Novel half vortices, ψ = e im θ, ψ = e im θ, has (m, m ) = (1/, 1/). Previous half-vortex (m, m ) = (1, ) [Lagoudakis et al. Science 9] Jonathan Keeling Pairing phases & photons ICSCE7, April 14 6 / 1

19 Consequences and Signatures Phase separation Phase coherence (1) : standard g σ = ψ σ(r, t)ψ σ (, ). (D, QLRO) APD : g (1) σ = but Adjustable delay φ APD A g (1) m = ψ (r, t)ψ (r, t)ψ (, )ψ (, ) APD θ APD B Novel half vortices, ψ = e im θ, ψ = e im θ, has (m, m ) = (1/, 1/). Previous half-vortex (m, m ) = (1, ) [Lagoudakis et al. Science 9] Jonathan Keeling Pairing phases & photons ICSCE7, April 14 6 / 1

20 Phase diagram, T > N T=.58 K µ [mev] N PS x1 11 n [cm ] δ [mev] Jonathan Keeling Pairing phases & photons ICSCE7, April 14 7 / 1

21 Phase diagram, T > N T=.58 K N PS δ [mev] 3 δ=1 mev N PS µ [mev] 11 n [cm ] x1 T [K] 1 N µ [mev] 11 x1 n [cm ] Jonathan Keeling Pairing phases & photons ICSCE7, April 14 7 / 1

22 Phase diagram, T > T [K] 1 3 δ=8.3 mev N δ=1 mev N PS N PS T=.58 K N x1 µ [mev] n [cm ] N PS δ [mev] T [K] 1 N µ [mev] 11 x1 n [cm ] Jonathan Keeling Pairing phases & photons ICSCE7, April 14 7 / 1

23 Phase diagram, T > T [K] T [K] δ=6.5 mev N δ=8.3 mev N δ=1 mev N PS N PS N PS T=.58 K N x1 µ [mev] n [cm ] N PS [mev] δ T [K] 1 N µ [mev] 11 x1 n [cm ] Jonathan Keeling Pairing phases & photons ICSCE7, April 14 7 / 1

24 Evolution of triple point Exciton fraction = 1 [ ] δ 1 +. δ + Ω GaAs, need low T/high exciton fraction ZnO, easy to attain Jonathan Keeling Pairing phases & photons ICSCE7, April 14 8 / 1

25 Evolution of triple point δ [mev] GaAs ZnO T [K] Exciton fraction = 1 [ T [K] ] δ. δ + Ω excitonic fraction [%] GaAs, need low T/high exciton fraction ZnO, easy to attain Jonathan Keeling Pairing phases & photons ICSCE7, April 14 8 / 1

26 Evolution of triple point δ [mev] GaAs ZnO T [K] Exciton fraction = 1 [ T [K] ] δ. δ + Ω excitonic fraction [%] GaAs, need low T/high exciton fraction ZnO, easy to attain Jonathan Keeling Pairing phases & photons ICSCE7, April 14 8 / 1

27 Evolution of triple point δ [mev] GaAs ZnO T [K] Exciton fraction = 1 [ T [K] ] δ. δ + Ω excitonic fraction [%] GaAs, need low T/high exciton fraction ZnO, easy to attain Jonathan Keeling Pairing phases & photons ICSCE7, April 14 8 / 1

28 Outline 1 Pairing phases of polaritons Pairing phases and Feshbach for polaritons Phase diagram: Critical detunings Signatures Phase diagram: Critical temperatures Photon condensation Modelling organic molecules: Vibrational modes Strong coupling? Jonathan Keeling Pairing phases & photons ICSCE7, April 14 9 / 1

29 Photon BEC experiments Dye filled microcavity No strong coupling [Klaers et al, Nature, 1] Jonathan Keeling Pairing phases & photons ICSCE7, April 14 1 / 1

30 Photon BEC experiments Dye filled microcavity No strong coupling [Klaers et al, Nature, 1] Jonathan Keeling Pairing phases & photons ICSCE7, April 14 1 / 1

31 Photon BEC experiments Dye filled microcavity No strong coupling [Klaers et al, Nature, 1] Jonathan Keeling Pairing phases & photons ICSCE7, April 14 1 / 1

32 Relation to dye laser No single cavity mode Condensate mode is not maximum gain Gain/Absorption in balance Thermalised many-mode system Jonathan Keeling Pairing phases & photons ICSCE7, April / 1

33 Relation to dye laser 4 Level Dye Laser Energy Cavity nuclear coordinate No single cavity mode Condensate mode is not maximum gain Gain/Absorption in balance Thermalised many-mode system Jonathan Keeling Pairing phases & photons ICSCE7, April / 1

34 Relation to dye laser 4 Level Dye Laser Energy Cavity nuclear coordinate But: No single cavity mode Condensate mode is not maximum gain Gain/Absorption in balance Thermalised many-mode system Jonathan Keeling Pairing phases & photons ICSCE7, April / 1

35 Relation to dye laser 4 Level Dye Laser Energy Cavity nuclear coordinate But: No single cavity mode Condensate mode is not maximum gain Gain/Absorption in balance Thermalised many-mode system Jonathan Keeling Pairing phases & photons ICSCE7, April / 1

36 Modelling [ ɛ σz α + g ( ψ m σ + α + H.c. )] H sys = m ω m ψ mψ m + α D harmonic cavity ω m = ω cutoff + mω H.O. Degeneracies g m = m + 1 Local vibrational mode Phonon frequency Ω Huang-Rhys parameter S phonon coupling Jonathan Keeling Pairing phases & photons ICSCE7, April 14 1 / 1

37 Modelling H sys = m ω m ψ mψ m + α [ ɛ σz α + g ( ψ m σ + α + H.c. )] α { +Ω b αb α + }) Sσα (b z α + b α D harmonic cavity ω m = ω cutoff + mω H.O. Degeneracies g m = m + 1 Local vibrational mode Phonon frequency Ω Huang-Rhys parameter S phonon coupling Jonathan Keeling Pairing phases & photons ICSCE7, April 14 1 / 1

38 Modelling Rate equation ρ = i[h, ρ] L[ψ m] m α [ Γ(δm = ω m ɛ) m,α κ [ Γ L[σ+ α ] + Γ ] L[σ α ] L[σ α + ψ m ] + Γ( δ m = ɛ ω m ) L[σα ψ m] ] Γ( δ) Γ(δ) 1 1 δ [THz] [Marthaler et al PRL 11, Kirton & JK PRL 13] Kennard-Stepanov Γ(+δ) Γ( δ)e βδ Expt: ω < ɛ Γ at large δ Jonathan Keeling Pairing phases & photons ICSCE7, April / 1

39 Modelling Rate equation ρ = i[h, ρ] L[ψ m] m α [ Γ(δm = ω m ɛ) m,α κ [ Γ L[σ+ α ] + Γ ] L[σ α ] L[σ α + ψ m ] + Γ( δ m = ɛ ω m ) L[σα ψ m] ] Γ( δ) Γ(δ) 1 1 δ [THz] [Marthaler et al PRL 11, Kirton & JK PRL 13] Kennard-Stepanov Γ(+δ) Γ( δ)e βδ Expt: ω < ɛ Γ at large δ Jonathan Keeling Pairing phases & photons ICSCE7, April / 1

40 Distribution g m n m Rate equation include spontaneous emission Bose-Einstein distribution without losses [Kirton & JK PRL 13] Jonathan Keeling Pairing phases & photons ICSCE7, April / 1

41 Distribution g m n m Rate equation include spontaneous emission Bose-Einstein distribution without losses Low loss: Thermal [Kirton & JK PRL 13] Jonathan Keeling Pairing phases & photons ICSCE7, April / 1

42 Distribution g m n m Rate equation include spontaneous emission Bose-Einstein distribution without losses Low loss: Thermal [Kirton & JK PRL 13] High loss Laser Jonathan Keeling Pairing phases & photons ICSCE7, April / 1

43 Outline 1 Pairing phases of polaritons Pairing phases and Feshbach for polaritons Phase diagram: Critical detunings Signatures Phase diagram: Critical temperatures Photon condensation Modelling organic molecules: Vibrational modes Strong coupling? Jonathan Keeling Pairing phases & photons ICSCE7, April / 1

44 Strong coupling limit Energy H = ωψ ψ+ α [ ɛ ( ) σz α + g ψσ α + + ψ σα { +Ω b αb α + ) }] S (b α + b α σα z Photon l S l nuclear coordinate Phonon frequency Ω Huang-Rhys parameter S phonon coupling Polaron formation (dressing by vibrational modes) Vibrational replicas and BEC Ultra-strong phonon coupling Jonathan Keeling Pairing phases & photons ICSCE7, April / 1

45 Strong coupling limit Energy H = ωψ ψ+ α [ ɛ ( ) σz α + g ψσ α + + ψ σα { +Ω b αb α + ) }] S (b α + b α σα z Photon l S l nuclear coordinate Strong coupling and vibrational modes Phonon frequency Ω Huang-Rhys parameter S phonon coupling Polaron formation (dressing by vibrational modes) Vibrational replicas and BEC Ultra-strong phonon coupling Jonathan Keeling Pairing phases & photons ICSCE7, April / 1

46 Strong coupling limit Energy H = ωψ ψ+ α [ ɛ ( ) σz α + g ψσ α + + ψ σα { +Ω b αb α + ) }] S (b α + b α σα z Photon l S l nuclear coordinate Strong coupling and vibrational modes Phonon frequency Ω Huang-Rhys parameter S phonon coupling Polaron formation (dressing by vibrational modes) Vibrational replicas and BEC Ultra-strong phonon coupling Jonathan Keeling Pairing phases & photons ICSCE7, April / 1

47 Strong coupling limit Energy H = ωψ ψ+ α [ ɛ ( ) σz α + g ψσ α + + ψ σα { +Ω b αb α + ) }] S (b α + b α σα z Photon l S l nuclear coordinate Strong coupling and vibrational modes Phonon frequency Ω Huang-Rhys parameter S phonon coupling Polaron formation (dressing by vibrational modes) Vibrational replicas and BEC Ultra-strong phonon coupling First attempt equilibrium [Cwik et al. EPL 14] Jonathan Keeling Pairing phases & photons ICSCE7, April / 1

48 Organic materials in microcavities Strong coupling with organic materials Polariton lasing [Lidzey, Nature 98] Ωr '.1eV [Kena Cohen and Forrest, Nat. Photon 1] Thermalisation, condensation interactions T = 3K [Daskalakis et al. Nat. Photon 14; Plumhoff et al. ibid.] Ultrastrong coupling regime Ωr '.6eV! [Canaguier-Durand Ang. Chem. 13,... ] Jonathan Keeling Pairing phases & photons ICSCE7, April / 1

49 Phase diagram H = ωψ ψ+ α [ ( ) { ɛsα z + g ψs α + + ψ Sα +Ω b αb α + ) }] S (b α + b α Sα z S suppresses condensation reduces overlap Reentrant behaviour Min µ at T. [Cwik et al. EPL 14] Jonathan Keeling Pairing phases & photons ICSCE7, April / 1

50 Phase diagram H = ωψ ψ+ α [ ( ) { ɛsα z + g ψs α + + ψ Sα +Ω b αb α + ) }] S (b α + b α Sα z T g=. S=, =4, Ω=.1 S= -x*y Coherent field ψ µ-ω c S suppresses condensation reduces overlap Reentrant behaviour Min µ at T. [Cwik et al. EPL 14] Jonathan Keeling Pairing phases & photons ICSCE7, April / 1

51 [Cwik et Jonathan al. EPL Keeling 14] Pairing phases & photons ICSCE7, April / 1 Critical coupling with increasing S 4 ε - ω=-4 3 Re-orient phase diagram g vs µ, T Colors Jump of ψ g c Ν µ-ω T

52 [Cwik et Jonathan al. EPL Keeling 14] Pairing phases & photons ICSCE7, April / 1 Critical coupling with increasing S 4 ε - ω=-4 3 Re-orient phase diagram g vs µ, T Colors Jump of ψ g c Ν µ-ω T g c Ν µ-ω -3 S= =4, Ω=1 - T.1 7 S= µ-ω -3 g c Ν - T.1 7 S= µ-ω -3 g c Ν - T Jump in ψ

53 Acknowledgements GROUP: COLLABORATORS: Francesca Marchetti, UAM FUNDING: Jonathan Keeling Pairing phases & photons ICSCE7, April 14 / 1

54 T Adjustable delay φ θ APD APD A B Summary Polaritons pairing phase feasible for ZnO, signatures in coherence and vortices δ [mev] N.6.4. µ [mev] PS 1x1 11 n [cm ] T [K] 3 1 δ=1 mev N µ [mev] N PS x1 11 n [cm ] [Marchetti and Keeling, arxiv:138.13] δ [mev] GaAs ZnO T [K] T [K] Photon condensation and thermalisation; vibrational modes [Kirton and Keeling, PRL 13] excitonic fraction [%] APD APD Vibrational modes and strong coupling.6 g=. S=, =4, Ω=.1 -x*y.5 S= µ-ω c Coherent field ψ g c Ν 7 S= 6 =4, Ω= µ-ω -3 - T g c Ν. 7 S= µ-ω T g c Ν. 7 S= µ-ω T Jump in ψ [Cwik, Reja, Littlewood, Keeling EPL 14] Jonathan Keeling Pairing phases & photons ICSCE7, April 14 1 / 1

55 Jonathan Keeling Pairing phases & photons ICSCE7, April 14 / 39

56 3 Pairing phases model Excitons and photons Polaritons Exciton spin 4 Calculation details Variational wavefunction Variational MFT WIDBG result 5 More phase diagrams 6 Photon phase diagram 7 Organic polaritons Polarons Condensation of phonon replicas? Anticrossing vs ρ Jonathan Keeling Pairing phases & photons ICSCE7, April 14 3 / 39

57 Exciton-photon model Microscopic model coupled exciton-photon system H = k + σ=±1 σ=±,±1 ( ) k µ ˆX m ˆX kσ kσ + (δ + k ) µ X m C Ω R (Ĉ kσ ˆX kσ + ˆX kσĉkσ)] + ˆX ( σ R + r ) ˆX τ (R r Interaction U XX has exchange structure σ=±1 d rd R σ,σ,τ,τ =±,±1 ) ˆXτ ( R r U XX Ĉ kσĉkσ σ τ τσ (r) ) ˆXσ ( R + r For large Ω R, neglect σ = ± Interaction supports bound states in U+1, 1, 1,+1 XX channel bipolariton NB, bipolariton, bound polaritons, but larger exciton fraction. Jonathan Keeling Pairing phases & photons ICSCE7, April 14 4 / 39 )

58 Exciton-photon model Microscopic model coupled exciton-photon system H = k + σ=±1 σ=±,±1 ( ) k µ ˆX m ˆX kσ kσ + (δ + k ) µ X m C Ω R (Ĉ kσ ˆX kσ + ˆX kσĉkσ)] + ˆX ( σ R + r ) ˆX τ (R r Interaction U XX has exchange structure σ=±1 d rd R σ,σ,τ,τ =±,±1 ) ˆXτ ( R r U XX Ĉ kσĉkσ σ τ τσ (r) ) ˆXσ ( R + r For large Ω R, neglect σ = ± Interaction supports bound states in U+1, 1, 1,+1 XX channel bipolariton NB, bipolariton, bound polaritons, but larger exciton fraction. Jonathan Keeling Pairing phases & photons ICSCE7, April 14 4 / 39 )

59 Exciton-photon model Microscopic model coupled exciton-photon system H = k + σ=±1 σ=±,±1 ( ) k µ ˆX m ˆX kσ kσ + (δ + k ) µ X m C Ω R (Ĉ kσ ˆX kσ + ˆX kσĉkσ)] + ˆX ( σ R + r ) ˆX τ (R r Interaction U XX has exchange structure σ=±1 d rd R σ,σ,τ,τ =±,±1 ) ˆXτ ( R r U XX Ĉ kσĉkσ σ τ τσ (r) ) ˆXσ ( R + r For large Ω R, neglect σ = ± Interaction supports bound states in U+1, 1, 1,+1 XX channel bipolariton NB, bipolariton, bound polaritons, but larger exciton fraction. Jonathan Keeling Pairing phases & photons ICSCE7, April 14 4 / 39 )

60 Exciton-photon model Microscopic model coupled exciton-photon system H = k + σ=±1 σ=±,±1 ( ) k µ ˆX m ˆX kσ kσ + (δ + k ) µ X m C Ω R (Ĉ kσ ˆX kσ + ˆX kσĉkσ)] + ˆX ( σ R + r ) ˆX τ (R r Interaction U XX has exchange structure σ=±1 d rd R σ,σ,τ,τ =±,±1 ) ˆXτ ( R r U XX Ĉ kσĉkσ σ τ τσ (r) ) ˆXσ ( R + r For large Ω R, neglect σ = ± Interaction supports bound states in U+1, 1, 1,+1 XX channel bipolariton NB, bipolariton, bound polaritons, but larger exciton fraction. Jonathan Keeling Pairing phases & photons ICSCE7, April 14 4 / 39 )

61 Exciton-photon model Microscopic model coupled exciton-photon system H = k + σ=±1 σ=±,±1 ( ) k µ ˆX m ˆX kσ kσ + (δ + k ) µ X m C Ω R (Ĉ kσ ˆX kσ + ˆX kσĉkσ)] + ˆX ( σ R + r ) ˆX τ (R r Interaction U XX has exchange structure σ=±1 d rd R σ,σ,τ,τ =±,±1 ) ˆXτ ( R r U XX Ĉ kσĉkσ σ τ τσ (r) ) ˆXσ ( R + r For large Ω R, neglect σ = ± Interaction supports bound states in U+1, 1, 1,+1 XX channel bipolariton NB, bipolariton, bound polaritons, but larger exciton fraction. Jonathan Keeling Pairing phases & photons ICSCE7, April 14 4 / 39 )

62 Exciton-photon model Microscopic model coupled exciton-photon system H = k + σ=±1 σ=±,±1 ( ) k µ ˆX m ˆX kσ kσ + (δ + k ) µ X m C Ω R (Ĉ kσ ˆX kσ + ˆX kσĉkσ)] + ˆX ( σ R + r ) ˆX τ (R r Interaction U XX has exchange structure σ=±1 d rd R σ,σ,τ,τ =±,±1 ) ˆXτ ( R r U XX Ĉ kσĉkσ σ τ τσ (r) ) ˆXσ ( R + r For large Ω R, neglect σ = ± Interaction supports bound states in U+1, 1, 1,+1 XX channel bipolariton NB, bipolariton, bound polaritons, but larger exciton fraction. Jonathan Keeling Pairing phases & photons ICSCE7, April 14 5 / 39 )

63 Exciton-photon model Microscopic model coupled exciton-photon system H = k + σ=±1 σ=±,±1 ( ) k µ ˆX m ˆX kσ kσ + (δ + k ) µ X m C Ω R (Ĉ kσ ˆX kσ + ˆX kσĉkσ)] + ˆX ( σ R + r ) ˆX τ (R r Interaction U XX has exchange structure σ=±1 d rd R σ,σ,τ,τ =±,±1 ) ˆXτ ( R r U XX Ĉ kσĉkσ σ τ τσ (r) ) ˆXσ ( R + r For large Ω R, neglect σ = ± Interaction supports bound states in U+1, 1, 1,+1 XX channel bipolariton NB, bipolariton, bound polaritons, but larger exciton fraction. Jonathan Keeling Pairing phases & photons ICSCE7, April 14 5 / 39 )

64 Exciton-photon model Microscopic model coupled exciton-photon system H = k + σ=±1 σ=±,±1 ( ) k µ ˆX m ˆX kσ kσ + (δ + k ) µ X m C Ω R (Ĉ kσ ˆX kσ + ˆX kσĉkσ)] + ˆX ( σ R + r ) ˆX τ (R r Interaction U XX has exchange structure σ=±1 d rd R σ,σ,τ,τ =±,±1 ) ˆXτ ( R r U XX Ĉ kσĉkσ σ τ τσ (r) ) ˆXσ ( R + r For large Ω R, neglect σ = ± Interaction supports bound states in U+1, 1, 1,+1 XX channel bipolariton NB, bipolariton, bound polaritons, but larger exciton fraction. Jonathan Keeling Pairing phases & photons ICSCE7, April 14 5 / 39 )

65 Exciton-photon model Microscopic model coupled exciton-photon system H = k + σ=±1 σ=±,±1 ( ) k µ ˆX m ˆX kσ kσ + (δ + k ) µ X m C Ω R (Ĉ kσ ˆX kσ + ˆX kσĉkσ)] + ˆX ( σ R + r ) ˆX τ (R r Interaction U XX has exchange structure σ=±1 d rd R σ,σ,τ,τ =±,±1 ) ˆXτ ( R r U XX Ĉ kσĉkσ σ τ τσ (r) ) ˆXσ ( R + r For large Ω R, neglect σ = ± Interaction supports bound states in U+1, 1, 1,+1 XX channel bipolariton NB, bipolariton, bound polaritons, but larger exciton fraction. Jonathan Keeling Pairing phases & photons ICSCE7, April 14 5 / 39 )

66 Exciton-photon model Microscopic model coupled exciton-photon system H = k + σ=±1 σ=±,±1 ( ) k µ ˆX m ˆX kσ kσ + (δ + k ) µ X m C Ω R (Ĉ kσ ˆX kσ + ˆX kσĉkσ)] + ˆX ( σ R + r ) ˆX τ (R r Interaction U XX has exchange structure σ=±1 d rd R σ,σ,τ,τ =±,±1 ) ˆXτ ( R r U XX Ĉ kσĉkσ σ τ τσ (r) ) ˆXσ ( R + r For large Ω R, neglect σ = ± Interaction supports bound states in U+1, 1, 1,+1 XX channel bipolariton NB, bipolariton, bound polaritons, but larger exciton fraction. Jonathan Keeling Pairing phases & photons ICSCE7, April 14 5 / 39 )

67 Polariton model H = ( ) k m µ k σ=, + d R σ=,,m ( ) ˆψ ˆψ k σk σk + + ν µ ˆψ m ˆψ mk mk m U σσ ˆψ σ ˆψ σ ˆψ σ ˆψ σ + U ˆψ ˆψ ˆψ ˆψ + g ( ˆψ ˆψ ˆψ ) m + h.c. E Polariton dispersion m, detuning ν, interactions depend on δ Resonance width, dispersion derived from dressed exciton T matrix. E ω X [mev] 6 4 UP C X LP 4 4 k [µm 1 ] X ω ω XX LP ω ω UP ω C ω X ω LP E b ν > δ (Ω R +δ ) 1/ Jonathan Keeling Pairing phases & photons ICSCE7, April 14 6 / 39

68 Polariton model H = ( ) k m µ k σ=, + d R σ=,,m ( ) ˆψ ˆψ k σk σk + + ν µ ˆψ m ˆψ mk mk m U σσ ˆψ σ ˆψ σ ˆψ σ ˆψ σ + U ˆψ ˆψ ˆψ ˆψ + g ( ˆψ ˆψ ˆψ ) m + h.c. E Polariton dispersion m, detuning ν, interactions depend on δ Resonance width, dispersion derived from dressed exciton T matrix. E ω X [mev] 6 4 UP C X LP 4 4 k [µm 1 ] X ω ω XX LP ω ω UP ω C ω X ω LP E b ν > δ (Ω R +δ ) 1/ Jonathan Keeling Pairing phases & photons ICSCE7, April 14 6 / 39

69 Polariton model H = ( ) k m µ k σ=, + d R σ=,,m ( ) ˆψ ˆψ k σk σk + + ν µ ˆψ m ˆψ mk mk m U σσ ˆψ σ ˆψ σ ˆψ σ ˆψ σ + U ˆψ ˆψ ˆψ ˆψ + g ( ˆψ ˆψ ˆψ ) m + h.c. E Polariton dispersion m, detuning ν, interactions depend on δ Resonance width, dispersion derived from dressed exciton T matrix. X [mev] ω E 6 4 UP C X LP 4 k [µm 1 ] 4 X ω LP ω = ω XX ω UP ω C X ω LP ω E b ν = δ (Ω R +δ ) 1/ Jonathan Keeling Pairing phases & photons ICSCE7, April 14 6 / 39

70 Polariton model H = ( ) k m µ k σ=, + d R σ=,,m ( ) ˆψ ˆψ k σk σk + + ν µ ˆψ m ˆψ mk mk m U σσ ˆψ σ ˆψ σ ˆψ σ ˆψ σ + U ˆψ ˆψ ˆψ ˆψ + g ( ˆψ ˆψ ˆψ ) m + h.c. E Polariton dispersion m, detuning ν, interactions depend on δ Resonance width, dispersion derived from dressed exciton T matrix. X [mev] ω E 6 4 UP C X LP 4 k [µm 1 ] 4 X ω LP ω = ω XX ω UP ω C X ω LP ω E b ν = δ (Ω R +δ ) 1/ Jonathan Keeling Pairing phases & photons ICSCE7, April 14 6 / 39

71 Polariton model H = ( ) k m µ k σ=, + d R σ=,,m ( ) ˆψ ˆψ k σk σk + + ν µ ˆψ m ˆψ mk mk m U σσ ˆψ σ ˆψ σ ˆψ σ ˆψσ + U ˆψ ˆψ ˆψ ˆψ + g ( ˆψ ˆψ ˆψ ) m + h.c. ν > Biexciton in continuum m LP /m X E b ~m C /m X δ [mev] ν [mev] ν < Bound biexciton. (Excitonic limit) Jonathan Keeling Pairing phases & photons ICSCE7, April 14 7 / 39

72 Exciton and polariton spin degrees of freedom Photon: two circular polarisation modes Exciton: bound state of electron & hole J = 1 ± 1/ hole (p-orbital), J = 1/ electron Spin orbit splits hole bands, 4 states. Quantum well fixes k z of hole states. Exciton spin states J z = +, +1, 1, Optically active states J z = ±1 Jonathan Keeling Pairing phases & photons ICSCE7, April 14 8 / 39

73 Exciton and polariton spin degrees of freedom Photon: two circular polarisation modes Exciton: bound state of electron & hole E c v J = 1 ± 1/ hole (p-orbital), J = 1/ electron Spin orbit splits hole bands, 4 states. Quantum well fixes k z of hole states. k Exciton spin states J z = +, +1, 1, Optically active states J z = ±1 Jonathan Keeling Pairing phases & photons ICSCE7, April 14 8 / 39

74 Exciton and polariton spin degrees of freedom Photon: two circular polarisation modes Exciton: bound state of electron & hole E c v, J=3/ HH v, J=3/ LH v, J=1/ J = 1 ± 1/ hole (p-orbital), J = 1/ electron Spin orbit splits hole bands, 4 states. Quantum well fixes k z of hole states. k Exciton spin states J z = +, +1, 1, Optically active states J z = ±1 Jonathan Keeling Pairing phases & photons ICSCE7, April 14 8 / 39

75 Exciton and polariton spin degrees of freedom Photon: two circular polarisation modes Exciton: bound state of electron & hole z Cavity E c v, J=3/ HH v, J=3/ LH v, J=1/ J = 1 ± 1/ hole (p-orbital), J = 1/ electron Spin orbit splits hole bands, 4 states. Quantum well fixes k z of hole states. k Exciton spin states J z = +, +1, 1, Optically active states J z = ±1 Jonathan Keeling Pairing phases & photons ICSCE7, April 14 8 / 39

76 Exciton and polariton spin degrees of freedom Photon: two circular polarisation modes Exciton: bound state of electron & hole z Cavity E c v, J=3/ HH v, J=3/ LH v, J=1/ J = 1 ± 1/ hole (p-orbital), J = 1/ electron Spin orbit splits hole bands, 4 states. Quantum well fixes k z of hole states. k Exciton spin states J z = +, +1, 1, Optically active states J z = ±1 Jonathan Keeling Pairing phases & photons ICSCE7, April 14 8 / 39

77 Exciton and polariton spin degrees of freedom Photon: two circular polarisation modes Exciton: bound state of electron & hole z Cavity E c v, J=3/ HH v, J=3/ LH v, J=1/ J = 1 ± 1/ hole (p-orbital), J = 1/ electron Spin orbit splits hole bands, 4 states. Quantum well fixes k z of hole states. k Exciton spin states J z = +, +1, 1, Optically active states J z = ±1 Jonathan Keeling Pairing phases & photons ICSCE7, April 14 8 / 39

78 Beyond mean-field Fluctuation effects? Polariton fluctuations irrelevant: mu 1 4. Exciton fluctuations important: m m U 1. Next order theory: [Nozières & St James, J. Phys 8] Ψ exp ψ σ ˆψ k=,σ + tanh(θ kγ )ˆb ˆb kγ kγ. σ=,,m k,γ=a,b,m ( ) ˆψ where ˆb km = ˆψ km and k ˆψ = 1 ( 1 1 k 1 1 Variational functional E[ψ, ψ m, θ kγ ] ) ( ˆb ka ˆb kb Can show minimum θ kγ has form tanh(θ kγ ) = β γ + k /m γ Finite only if αγ < β γ Variational function E(ψ, ψ m, α γ, β γ ) ) α γ, Jonathan Keeling Pairing phases & photons ICSCE7, April 14 9 / 39

79 Beyond mean-field Fluctuation effects? Polariton fluctuations irrelevant: mu 1 4. Exciton fluctuations important: m m U 1. Next order theory: [Nozières & St James, J. Phys 8] Ψ exp ψ σ ˆψ k=,σ + tanh(θ kγ )ˆb ˆb kγ kγ. σ=,,m k,γ=a,b,m ( ) ˆψ where ˆb km = ˆψ km and k ˆψ = 1 ( 1 1 k 1 1 Variational functional E[ψ, ψ m, θ kγ ] ) ( ˆb ka ˆb kb Can show minimum θ kγ has form tanh(θ kγ ) = β γ + k /m γ Finite only if αγ < β γ Variational function E(ψ, ψ m, α γ, β γ ) ) α γ, Jonathan Keeling Pairing phases & photons ICSCE7, April 14 9 / 39

80 Beyond mean-field Fluctuation effects? Polariton fluctuations irrelevant: mu 1 4. Exciton fluctuations important: m m U 1. Next order theory: [Nozières & St James, J. Phys 8] Ψ exp ψ σ ˆψ k=,σ + tanh(θ kγ )ˆb ˆb kγ kγ. σ=,,m k,γ=a,b,m ( ) ˆψ where ˆb km = ˆψ km and k ˆψ = 1 ( 1 1 k 1 1 Variational functional E[ψ, ψ m, θ kγ ] ) ( ˆb ka ˆb kb Can show minimum θ kγ has form tanh(θ kγ ) = β γ + k /m γ Finite only if αγ < β γ Variational function E(ψ, ψ m, α γ, β γ ) ) α γ, Jonathan Keeling Pairing phases & photons ICSCE7, April 14 9 / 39

81 Beyond mean-field Fluctuation effects? Polariton fluctuations irrelevant: mu 1 4. Exciton fluctuations important: m m U 1. Next order theory: [Nozières & St James, J. Phys 8] Ψ exp ψ σ ˆψ k=,σ + tanh(θ kγ )ˆb ˆb kγ kγ. σ=,,m k,γ=a,b,m ( ) ˆψ where ˆb km = ˆψ km and k ˆψ = 1 ( 1 1 k 1 1 Variational functional E[ψ, ψ m, θ kγ ] ) ( ˆb ka ˆb kb Can show minimum θ kγ has form tanh(θ kγ ) = β γ + k /m γ Finite only if αγ < β γ Variational function E(ψ, ψ m, α γ, β γ ) ) α γ, Jonathan Keeling Pairing phases & photons ICSCE7, April 14 9 / 39

82 Finite T calculation Finite T minimize free energy Use Feynman-Jensen inequality: [ ] F = k B T ln Tre Ĥ/k BT F MF + Ĥ ĤMF MF Where... MF calculated using ρ = e (F MF ĤMF)/k B T Ansatz ĤMF Variational F(ψ, ψ m, α γ, β γ ). Ĥ MF = γ { + 1 Aψ γ (α γ + β γ ) (ˆb γ + ˆb ) γ k (ˆb kγ ˆb kγ ) ( ɛ kγ + β γ α γ ) ( ) } α γ ˆbkγ. ɛ kγ + β γ ˆb kγ Jonathan Keeling Pairing phases & photons ICSCE7, April 14 3 / 39

83 Finite T calculation Finite T minimize free energy Use Feynman-Jensen inequality: [ ] F = k B T ln Tre Ĥ/k BT F MF + Ĥ ĤMF MF Where... MF calculated using ρ = e (F MF ĤMF)/k B T Ansatz ĤMF Variational F(ψ, ψ m, α γ, β γ ). Ĥ MF = γ { + 1 Aψ γ (α γ + β γ ) (ˆb γ + ˆb ) γ k (ˆb kγ ˆb kγ ) ( ɛ kγ + β γ α γ ) ( ) } α γ ˆbkγ. ɛ kγ + β γ ˆb kγ Jonathan Keeling Pairing phases & photons ICSCE7, April 14 3 / 39

84 Finite T calculation Finite T minimize free energy Use Feynman-Jensen inequality: [ ] F = k B T ln Tre Ĥ/k BT F MF + Ĥ ĤMF MF Where... MF calculated using ρ = e (F MF ĤMF)/k B T Ansatz ĤMF Variational F(ψ, ψ m, α γ, β γ ). Ĥ MF = γ { + 1 Aψ γ (α γ + β γ ) (ˆb γ + ˆb ) γ k (ˆb kγ ˆb kγ ) ( ɛ kγ + β γ α γ ) ( ) } α γ ˆbkγ. ɛ kγ + β γ ˆb kγ Jonathan Keeling Pairing phases & photons ICSCE7, April 14 3 / 39

85 Variational MFT for WIDBG Test validity. WIDBG Ĥ = k VMFT for WIDBG: k m ψ k ψ k + U d rψ ψ ψψ Ĥ MF = Aψ(α + β)(ˆb + ˆb ) + 1 ) (ˆb ( ) ( ) ɛ k ˆb k + β α ˆbk k. α ɛ k + β k ˆb k Compare to D EOS, ρ(µ) = Tf (µ/t ) CF Hartree-Fock-Popov-Bogoluibov method, include Uρ in Σ Jonathan Keeling Pairing phases & photons ICSCE7, April / 39

86 Variational MFT for WIDBG Test validity. WIDBG Ĥ = k VMFT for WIDBG: k m ψ k ψ k + U d rψ ψ ψψ Ĥ MF = Aψ(α + β)(ˆb + ˆb ) + 1 ) (ˆb ( ) ( ) ɛ k ˆb k + β α ˆbk k. α ɛ k + β k ˆb k Compare to D EOS, ρ(µ) = Tf (µ/t ) CF Hartree-Fock-Popov-Bogoluibov method, include Uρ in Σ Jonathan Keeling Pairing phases & photons ICSCE7, April / 39

87 Variational MFT for WIDBG Test validity. WIDBG Ĥ = k k m ψ k ψ k + U d rψ ψ ψψ VMFT for WIDBG: Ĥ MF = Aψ(α + β)(ˆb + ˆb ) + 1 ) (ˆb ( ) ( 3 ) ɛ k ˆb k + β α ˆbk k. α ɛ k + β 1 k ˆb k Compare to D EOS, ρ(µ) = Tf (µ/t ) Density ρ/t VMFT QMC [PRA ] mu=.1, ε c =1 [a.u.] CF Hartree-Fock-Popov-Bogoluibov method, include Uρ in Σ µ/t Temperature, T [a.u.] Jonathan Keeling Pairing phases & photons ICSCE7, April / 39

88 Variational MFT for WIDBG Test validity. WIDBG Ĥ = k k m ψ k ψ k + U d rψ ψ ψψ VMFT for WIDBG: Ĥ MF = Aψ(α + β)(ˆb + ˆb ) + 1 ) (ˆb ( ) ( 3 ) ɛ k ˆb k + β α ˆbk k. α ɛ k + β 1 k ˆb k Compare to D EOS, ρ(µ) = Tf (µ/t ) Density ρ/t VMFT QMC [PRA ] mu=.1, ε c =1 [a.u.] CF Hartree-Fock-Popov-Bogoluibov method, include Uρ in Σ µ/t Temperature, T [a.u.] Jonathan Keeling Pairing phases & photons ICSCE7, April / 39

89 Phase diagram, finite temperature δ [mev] N T=.58 K µ [mev] N PS x1 11 n [cm ] Jonathan Keeling Pairing phases & photons ICSCE7, April 14 3 / 39

90 Phase diagram, finite temperature δ [mev] δ [mev] N N T=.58 K T=1.16 K µ [mev] N N PS PS n [cm ] x1 11 Jonathan Keeling Pairing phases & photons ICSCE7, April 14 3 / 39

91 Phase diagram, finite temperature δ [mev] δ [mev] δ [mev] N N N T=.58 K T=1.16 K T =.3 K µ [mev] N N N PS PS PS x1 11 n [cm ] Jonathan Keeling Pairing phases & photons ICSCE7, April 14 3 / 39

92 Phase diagram, finite temperature δ [mev] δ [mev] δ [mev] N N N T=.58 K T=1.16 K T =.3 K µ [mev] N N N PS PS PS x1 11 n [cm ] δ [mev] 5 N PS n [cm ] Jonathan Keeling Pairing phases & photons ICSCE7, April 14 3 / 39

93 Phase diagram, vs temperature N T=.58 K N PS δ [mev] 3 δ=1 mev N PS µ [mev] 11 n [cm ] x1 T [K] 1 N µ [mev] 11 x1 n [cm ] Jonathan Keeling Pairing phases & photons ICSCE7, April / 39

94 Phase diagram, vs temperature T [K] 1 3 δ=8.3 mev N δ=1 mev N PS N PS T=.58 K N x1 µ [mev] n [cm ] N PS δ [mev] T [K] 1 N µ [mev] 11 x1 n [cm ] Jonathan Keeling Pairing phases & photons ICSCE7, April / 39

95 Phase diagram, vs temperature T [K] T [K] δ=6.5 mev N δ=8.3 mev N δ=1 mev N PS N PS N PS T=.58 K N x1 µ [mev] n [cm ] N PS [mev] δ T [K] 1 N µ [mev] 11 x1 n [cm ] Jonathan Keeling Pairing phases & photons ICSCE7, April / 39

96 Threshold condition 6 κ κ κ κ =1 MHz κ =.5 GHz κ =5 GHz Increasing loss Compare threshold: Pump rate (Laser) Critical density (condensate) Thermal at low κ/high temperature High loss, κ competes with Γ(±δ ) Low temperature, Γ(±δ ) shrinks [Kirton & JK PRL 13] Jonathan Keeling Pairing phases & photons ICSCE7, April / 39

97 Threshold condition 6 κ κ κ κ =1 MHz κ =.5 GHz κ =5 GHz Increasing loss Compare threshold: Pump rate (Laser) Critical density (condensate) Thermal at low κ/high temperature High loss, κ competes with Γ(±δ ) Low temperature, Γ(±δ ) shrinks [Kirton & JK PRL 13] Jonathan Keeling Pairing phases & photons ICSCE7, April / 39

98 Threshold condition 6 κ κ κ κ =1 MHz κ =.5 GHz κ =5 GHz Increasing loss Compare threshold: Pump rate (Laser) Critical density (condensate) Thermal at low κ/high temperature High loss, κ competes with Γ(±δ ) Low temperature, Γ(±δ ) shrinks [Kirton & JK PRL 13] Γ( δ) Γ(δ) 1 1 δ [THz] Jonathan Keeling Pairing phases & photons ICSCE7, April / 39

99 Threshold condition 6 κ κ κ κ =1 MHz κ =.5 GHz κ =5 GHz Increasing loss Compare threshold: Pump rate (Laser) Critical density (condensate) Thermal at low κ/high temperature High loss, κ competes with Γ(±δ ) Low temperature, Γ(±δ ) shrinks [Kirton & JK PRL 13] Γ( δ) Γ(δ) 1 1 δ [THz] Jonathan Keeling Pairing phases & photons ICSCE7, April / 39

100 Explanation: Polaron formation Unitary transform H α H α = e Kα H α e Kα K = SS z α(b α b α ) Coupling moves to S ± [ ] H α = const. + ɛsα z + Ωb αb α + g ψs α + S(b e α b α ) + H.c. Optimal phonon displacements, S Reduced g eff g exp( S/) For ψ, competition Variational MFT ψ α exp( ηk α ζb α), S α Jonathan Keeling Pairing phases & photons ICSCE7, April / 39

101 Explanation: Polaron formation Unitary transform H α H α = e Kα H α e Kα K = SS z α(b α b α ) Coupling moves to S ± [ ] H α = const. + ɛsα z + Ωb αb α + g ψs α + S(b e α b α ) + H.c. Optimal phonon displacements, S Reduced g eff g exp( S/) For ψ, competition Variational MFT ψ α exp( ηk α ζb α), S α Jonathan Keeling Pairing phases & photons ICSCE7, April / 39

102 Explanation: Polaron formation Unitary transform H α H α = e Kα H α e Kα K = SS z α(b α b α ) Coupling moves to S ± [ ] H α = const. + ɛsα z + Ωb αb α + g ψs α + S(b e α b α ) + H.c. Optimal phonon displacements, S Reduced g eff g exp( S/) For ψ, competition Variational MFT ψ α exp( ηk α ζb α), S α Jonathan Keeling Pairing phases & photons ICSCE7, April / 39

103 Explanation: Polaron formation Unitary transform H α H α = e Kα H α e Kα K = SS z α(b α b α ) Coupling moves to S ± [ ] H α = const. + ɛsα z + Ωb αb α + g ψs α + S(b e α b α ) + H.c. Optimal phonon displacements, S Reduced g eff g exp( S/) For ψ, competition Variational MFT ψ α exp( ηk α ζb α), S α Jonathan Keeling Pairing phases & photons ICSCE7, April / 39

104 Explanation: Polaron formation Unitary transform H α H α = e Kα H α e Kα K = SS z α(b α b α ) Coupling moves to S ± [ ] H α = const. + ɛsα z + Ωb αb α + g ψs α + S(b e α b α ) + H.c. Optimal phonon displacements, S Reduced g eff g exp( S/) For ψ, competition Variational MFT ψ α exp( ηk α ζb α), S α Jonathan Keeling Pairing phases & photons ICSCE7, April / 39

105 Collective polaron formation Compares well at S 1 Coherent bosonic state g Ν 3 Feedback: Large/small g eff λ = ψ Variational free energy F = (ω c µ)λ + N { Ω Effective LS energy in field: [Cwik et al. EPL 14] ( ɛ µ ξ = [ ζ S (a) Exact diagonalization 4 g Ν µ-ω - T T c µ-ω c 6 5 (b) Ansatz S=6, =4, Ω= ] [ ( )]} η( η) ξ T ln cosh 4 T + Ω ) S(1 η)ζ + g λ e Sη Jonathan Keeling Pairing phases & photons ICSCE7, April / 39 λ

106 Collective polaron formation Compares well at S 1 Coherent bosonic state g Ν 3 Feedback: Large/small g eff λ = ψ Variational free energy F = (ω c µ)λ + N { Ω Effective LS energy in field: [Cwik et al. EPL 14] ( ɛ µ ξ = [ ζ S (a) Exact diagonalization 4 g Ν µ-ω - T T c µ-ω c 6 5 (b) Ansatz S=6, =4, Ω= ] [ ( )]} η( η) ξ T ln cosh 4 T + Ω ) S(1 η)ζ + g λ e Sη Jonathan Keeling Pairing phases & photons ICSCE7, April / 39 λ

107 Collective polaron formation Compares well at S 1 Coherent bosonic state g Ν 3 Feedback: Large/small g eff λ = ψ Variational free energy F = (ω c µ)λ + N { Ω Effective LS energy in field: [Cwik et al. EPL 14] ( ɛ µ ξ = [ ζ S (a) Exact diagonalization 4 g Ν µ-ω - T T c µ-ω c 6 5 (b) Ansatz S=6, =4, Ω= ] [ ( )]} η( η) ξ T ln cosh 4 T + Ω ) S(1 η)ζ + g λ e Sη Jonathan Keeling Pairing phases & photons ICSCE7, April / 39 λ

108 Polariton spectrum: photon weight Energy -4.4 T=.4, S=, =4, Ω=.1, g= µ.1. Density ρ Photon weight, Z n Saturating LS: g eff g (1 ρ) What is nature of polariton mode? D(t) = i ψ (t)ψ(), D(ω) = n Z n ω ω n [Cwik et al. EPL 14] Jonathan Keeling Pairing phases & photons ICSCE7, April / 39

109 Polariton spectrum: photon weight Energy -4.4 T=.4, S=, =4, Ω=.1, g= µ.1. Density ρ Photon weight, Z n Saturating LS: g eff g (1 ρ) What is nature of polariton mode? D(t) = i ψ (t)ψ(), D(ω) = n Z n ω ω n [Cwik et al. EPL 14] Jonathan Keeling Pairing phases & photons ICSCE7, April / 39

110 Polariton spectrum: photon weight Energy -4.4 T=.4, S=, =4, Ω=.1, g= µ.1. Density ρ Photon weight, Z n Saturating LS: g eff g (1 ρ) What is nature of polariton mode? D(t) = i ψ (t)ψ(), D(ω) = n Z n ω ω n [Cwik et al. EPL 14] Jonathan Keeling Pairing phases & photons ICSCE7, April / 39

111 Polariton spectrum: what condensed Repeat weight for n-phonon channel Eigenvector that is macroscopically occupied Optimal T Ω [Cwik et al. EPL 14] Jonathan Keeling Pairing phases & photons ICSCE7, April / 39

112 Polariton spectrum: what condensed Repeat weight for n-phonon channel Eigenvector that is macroscopically occupied Sideband spectral weight S=, =4, Ω=.1, g= T=. Optimal T Ω Absorbed phonons: q-p [Cwik et al. EPL 14] Jonathan Keeling Pairing phases & photons ICSCE7, April / 39

113 Polariton spectrum: what condensed Repeat weight for n-phonon channel Eigenvector that is macroscopically occupied Sideband spectral weight S=, =4, Ω=.1, g= T=. T=.5 T=.15 Optimal T Ω Absorbed phonons: q-p [Cwik et al. EPL 14] Jonathan Keeling Pairing phases & photons ICSCE7, April / 39

114 Polariton spectrum: what condensed Repeat weight for n-phonon channel Eigenvector that is macroscopically occupied Sideband spectral weight S=, =4, Ω=.1, g= T=. T=.5 T=.15 T=. T=.3 T=.4 T=.45 Optimal T Ω Absorbed phonons: q-p [Cwik et al. EPL 14] Jonathan Keeling Pairing phases & photons ICSCE7, April / 39

115 Polariton spectrum: what condensed Repeat weight for n-phonon channel Eigenvector that is macroscopically occupied Sideband spectral weight S=, =4, Ω=.1, g= T=. T=.5 T=.15 T=. T=.3 T=.4 T= Absorbed phonons: q-p T Optimal T Ω g=. S=, =4, Ω=.1 -x*y µ-ω c Coherent field ψ [Cwik et al. EPL 14] Jonathan Keeling Pairing phases & photons ICSCE7, April / 39

116 Polariton spectrum coupled oscillators Jonathan Keeling Pairing phases & photons ICSCE7, April / 39

117 Polariton spectrum coupled oscillators 1 Photon Exciton UP Energy -1 - LP Coupling, g Jonathan Keeling Pairing phases & photons ICSCE7, April / 39

118 Polariton spectrum coupled oscillators 1 Photon Exciton Exciton-Ω UP Energy -1 - LP Coupling, g Jonathan Keeling Pairing phases & photons ICSCE7, April / 39

119 Polariton spectrum coupled oscillators 1 Photon Exciton-nΩ UP Energy Coupling, g Jonathan Keeling Pairing phases & photons ICSCE7, April / 39

120 Polariton spectrum coupled oscillators 1 Photon Exciton-nΩ UP Energy Coupling, g Jonathan Keeling Pairing phases & photons ICSCE7, April / 39