600 Pairing Phases of Polaritons, and photon condensates Jonathan Keeling
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600 Pairing Phases of Polaritons, and photon condensates Jonathan Keeling
Pairing Phases of Polaritons, and photon condensates Jonathan Keeling University of St Andrews 600 YEARS ICSCE7, Hakone, April 2014 Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 1 / 21 Outline 1 Pairing phases of polaritons Pairing phases and Feshbach for polaritons Phase diagram: Critical detunings Signatures Phase diagram: Critical temperatures 2 Photon condensation Modelling organic molecules: Vibrational modes Strong coupling? Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 2 / 21 Pairing phases of atoms Fermions BEC-BEC transition † Ĥ = . . . + ψ̂m ψ̂a1 ψ̂a2 + h.c. I From Randeria, Nat. Phys. ’10 I If hψ̂m i = 6 0, MSF If hψ̂a1 i = 6 0, hψ̂a2 i = 6 0. AMSF High density → metastability. BEC-BCS crossover [Eagles, Leggett, Keldysh, Nozières, Randeria, . . . ] Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 3 / 21 Pairing phases of atoms Bosons Fermions T MSF AMSF (B−B0 ) BEC-BEC transition † Ĥ = . . . + ψ̂m ψ̂a1 ψ̂a2 + h.c. I From Randeria, Nat. Phys. ’10 [Eagles, Leggett, Keldysh, Nozières, Jonathan Keeling If hψ̂m i = 6 0, MSF If hψ̂a1 i = 6 0, hψ̂a2 i = 6 0. AMSF High density → metastability. BEC-BCS crossover Randeria, . . . ] I [Nozières, St James, Timmermanns, Mueller, Thouless, Radzihovsky, Stoof, Sachdev . . . ] Pairing phases & photons ICSCE7, April 2014 3 / 21 Pairing phases of atoms Bosons Fermions T MSF AMSF (B−B0 ) BEC-BEC transition † Ĥ = . . . + ψ̂m ψ̂a1 ψ̂a2 + h.c. I From Randeria, Nat. Phys. ’10 [Eagles, Leggett, Keldysh, Nozières, Jonathan Keeling If hψ̂m i = 6 0, MSF If hψ̂a1 i = 6 0, hψ̂a2 i = 6 0. AMSF High density → metastability. BEC-BCS crossover Randeria, . . . ] I [Nozières, St James, Timmermanns, Mueller, Thouless, Radzihovsky, Stoof, Sachdev . . . ] Pairing phases & photons ICSCE7, April 2014 3 / 21 Polariton Feshbach Hybridisation of bound states: I I Biexciton: opposite spins (two-species): 2ω0X − Eb h i p Hybridisation with photons: 2 21 (ω0C + ω0X ) − 12 Ω2r + δ 2 Control δ change ν, m, Interaction . . . [Ivanov, Haug, Keldysh ’98], [Wouters ’07], [Caursotto et al. ’10], [Deveaud-Pledran et al. ’13] Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 4 / 21 Polariton Feshbach Hybridisation of bound states: I I Biexciton: opposite spins (two-species): 2ω0X − Eb h i p Hybridisation with photons: 2 21 (ω0C + ω0X ) − 12 Ω2r + δ 2 Control δ change ν, m, Interaction . . . [Ivanov, Haug, Keldysh ’98], [Wouters ’07], [Caursotto et al. ’10], [Deveaud-Pledran et al. ’13] Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 4 / 21 Polariton Feshbach Hybridisation of bound states: I I Biexciton: opposite spins (two-species): 2ω0X − Eb h i p Hybridisation with photons: 2 21 (ω0C + ω0X ) − 12 Ω2r + δ 2 E X 2ω0 ωXX |Eb| LP 2ω0 X E−ω0 [meV] 6 Control δ change ν, m, Interaction . . . UP 4 UP 2 C ω0 C ω0 0 X ω0 −2 ν>0 −4 LP −2 X δ (ΩR2+δ2)1/2 LP ω0 2 0 k [µm−1] 4 [Ivanov, Haug, Keldysh ’98], [Wouters ’07], [Caursotto et al. ’10], [Deveaud-Pledran et al. ’13] Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 4 / 21 Polariton Feshbach Hybridisation of bound states: I I Biexciton: opposite spins (two-species): 2ω0X − Eb h i p Hybridisation with photons: 2 21 (ω0C + ω0X ) − 12 Ω2r + δ 2 E X 2ω0 LP 2ω0 = ωXX X ν=0 Control δ change ν, m, Interaction . . . UP 6 E−ω0 [meV] |Eb| UP ω0 C 4 C ω0 2 X X 0 ω0 LP ω0 LP −2 −4 −2 0 k [µm−1] 2 δ (ΩR2+δ2)1/2 4 [Ivanov, Haug, Keldysh ’98], [Wouters ’07], [Caursotto et al. ’10], [Deveaud-Pledran et al. ’13] Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 4 / 21 Polariton Feshbach Hybridisation of bound states: I I Biexciton: opposite spins (two-species): 2ω0X − Eb h i p Hybridisation with photons: 2 21 (ω0C + ω0X ) − 12 Ω2r + δ 2 E Control δ change ν, m, Interaction . . . X X ν=0 UP 6 E−ω0 [meV] |Eb| ν [meV] 2ω0 LP 2ω0 = ωXX UP ω0 C 4 C ω0 2 X X 0 ω0 LP ω0 LP −2 −4 −2 0 −1 k [µm ] 2 δ (ΩR2+δ2)1/2 10 8 6 4 2 0 −2 Bound state |Eb | 0 10 20 30 40 50 δ [meV] 4 [Ivanov, Haug, Keldysh ’98], [Wouters ’07], [Caursotto et al. ’10], [Deveaud-Pledran et al. ’13] Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 4 / 21 MSF SF 12 10 8 6 4 2 0 M δ [meV] Phase diagram (ground state, T = 0) AMSF PS N −0.6 AMSF −0.4 −0.2 µ [meV] 0 0 11 1x10 n [cm−2] δ < 0: “standard” BEC. Small |δ|: 1st order transition I I Larkin-Pikin mechanism Large phase-separation region Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 5 / 21 MSF SF 12 10 8 6 4 2 0 M δ [meV] Phase diagram (ground state, T = 0) AMSF PS N −0.6 AMSF −0.4 −0.2 µ [meV] 0 0 11 1x10 n [cm−2] δ < 0: “standard” BEC. Small |δ|: 1st order transition I I Larkin-Pikin mechanism Large phase-separation region Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 5 / 21 MSF SF 12 10 8 6 4 2 0 M δ [meV] Phase diagram (ground state, T = 0) AMSF PS N −0.6 AMSF −0.4 −0.2 µ [meV] 0 0 11 1x10 n [cm−2] δ < 0: “standard” BEC. Small |δ|: 1st order transition I I Larkin-Pikin mechanism Large phase-separation region Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 5 / 21 Consequences and Signatures Phase separation Phase coherence I (1) AMSF: standard gσ = hψσ† (r , t)ψσ (0, 0)i. (2D, QLRO) Novel half vortices, ψ↑ = eim↑θ , ψ↓ = eim↓ θ , MSF has (m↑ , m↓ ) = (1/2, 1/2) . Previous half-vortex (m↑ , m↓ ) = (1, 0) [Lagoudakis et al. Science ’09] Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 6 / 21 Consequences and Signatures Phase separation Phase coherence I (1) AMSF: standard gσ = hψσ† (r , t)ψσ (0, 0)i. (2D, QLRO) Novel half vortices, ψ↑ = eim↑θ , ψ↓ = eim↓ θ , MSF has (m↑ , m↓ ) = (1/2, 1/2) . Previous half-vortex (m↑ , m↓ ) = (1, 0) [Lagoudakis et al. Science ’09] Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 6 / 21 Consequences and Signatures Phase separation Phase coherence I (1) AMSF: standard gσ = hψσ† (r , t)ψσ (0, 0)i. (2D, QLRO) Novel half vortices, ψ↑ = eim↑θ , ψ↓ = eim↓ θ , MSF has (m↑ , m↓ ) = (1/2, 1/2) . Previous half-vortex (m↑ , m↓ ) = (1, 0) [Lagoudakis et al. Science ’09] Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 6 / 21 Consequences and Signatures Phase separation Phase coherence I (1) AMSF: standard gσ = hψσ† (r , t)ψσ (0, 0)i. (2D, QLRO) Novel half vortices, ψ↑ = eim↑θ , ψ↓ = eim↓ θ , MSF has (m↑ , m↓ ) = (1/2, 1/2) . Previous half-vortex (m↑ , m↓ ) = (1, 0) [Lagoudakis et al. Science ’09] Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 6 / 21 Consequences and Signatures Phase separation Phase coherence (1) AMSF: standard gσ = hψσ† (r , t)ψσ (0, 0)i. (2D, QLRO) I MSF: gσ = 0 but APD I APD φ (1) A Adjustable delay (1) B APD gm = hψ↑† (r, t)ψ↓† (r, t)ψ↓ (0, 0)ψ↑ (0, 0)i = 6 0 APD θ Novel half vortices, ψ↑ = eim↑θ , ψ↓ = eim↓ θ , MSF has (m↑ , m↓ ) = (1/2, 1/2) . Previous half-vortex (m↑ , m↓ ) = (1, 0) [Lagoudakis et al. Science ’09] Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 6 / 21 Consequences and Signatures Phase separation Phase coherence (1) AMSF: standard gσ = hψσ† (r , t)ψσ (0, 0)i. (2D, QLRO) I MSF: gσ = 0 but APD I APD φ (1) A Adjustable delay (1) B APD gm = hψ↑† (r, t)ψ↓† (r, t)ψ↓ (0, 0)ψ↑ (0, 0)i = 6 0 APD θ Novel half vortices, ψ↑ = eim↑θ , ψ↓ = eim↓ θ , MSF has (m↑ , m↓ ) = (1/2, 1/2) . Previous half-vortex (m↑ , m↓ ) = (1, 0) [Lagoudakis et al. Science ’09] Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 6 / 21 20 T= 0.58 K MSF 15 AMSF AMSF N N −0.8 −0.6 −0.4 −0.2 µ [meV] Jonathan Keeling Pairing phases & photons 5 PS 0 0 10 δ [meV] MSF Phase diagram, T > 0 −2 n [cm ] 11 0 2x10 ICSCE7, April 2014 7 / 21 20 T= 0.58 K MSF 15 AMSF AMSF N N T [K] 3 δ=10 meV 2 −0.8 −0.6 −0.4 −0.2 µ [meV] PS N 5 PS 0 0 10 δ [meV] MSF Phase diagram, T > 0 −2 n [cm ] 11 0 2x10 N AMSF 1 AMSF MSF MSF 0 −0.5 −0.4 µ [meV] Jonathan Keeling −0.3 0 11 −2 n [cm ] 2x10 Pairing phases & photons ICSCE7, April 2014 7 / 21 Phase diagram, T > 0 N 2 MSF 15 MSF AMSF N AMSF AMSF N −0.8 −0.6 −0.4 −0.2 µ [meV] PS N 0 0 10 5 PS MSF 0 3 δ=10 meV 2 20 T= 0.58 K AMSF N 1 T [K] PS δ [meV] T [K] δ=8.3 meV MSF 0 −2 n [cm ] 11 0 2x10 N AMSF 1 AMSF MSF MSF 0 −0.5 −0.4 µ [meV] Jonathan Keeling −0.3 0 11 −2 n [cm ] 2x10 Pairing phases & photons ICSCE7, April 2014 7 / 21 Phase diagram, T > 0 2 δ=6.5 meV N PS T [K] δ=8.3 meV N 2 MSF 15 MSF AMSF N AMSF AMSF N −0.8 −0.6 −0.4 −0.2 µ [meV] PS N 0 0 10 5 PS MSF 0 3 δ=10 meV 2 20 T= 0.58 K AMSF N 1 T [K] PS δ [meV] 0 AMSF MSF MSF 0.5 MSF N 1 AMSF T [K] 1.5 −2 n [cm ] 11 0 2x10 N AMSF 1 AMSF MSF MSF 0 −0.5 −0.4 µ [meV] Jonathan Keeling −0.3 0 11 −2 n [cm ] 2x10 Pairing phases & photons ICSCE7, April 2014 7 / 21 Evolution of triple point Exciton fraction = 1 δ 1+ √ . 2 δ 2 + Ω2 GaAs, need low T/high exciton fraction ZnO, easy to attain MSF Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 8 / 21 Evolution of triple point GaAs ZnO δ [meV] 40 95 30 90 20 85 10 80 0 0 5 Exciton fraction = 10 T [K] 15 0 5 10 T [K] excitonic fraction [%] 100 50 15 1 δ 1+ √ . 2 δ 2 + Ω2 GaAs, need low T/high exciton fraction ZnO, easy to attain MSF Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 8 / 21 Evolution of triple point GaAs ZnO δ [meV] 40 95 30 90 20 85 10 80 0 0 5 Exciton fraction = 10 T [K] 15 0 5 10 T [K] excitonic fraction [%] 100 50 15 1 δ 1+ √ . 2 δ 2 + Ω2 GaAs, need low T/high exciton fraction ZnO, easy to attain MSF Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 8 / 21 Evolution of triple point GaAs ZnO δ [meV] 40 95 30 90 20 85 10 80 0 0 5 Exciton fraction = 10 T [K] 15 0 5 10 T [K] excitonic fraction [%] 100 50 15 1 δ 1+ √ . 2 δ 2 + Ω2 GaAs, need low T/high exciton fraction ZnO, easy to attain MSF Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 8 / 21 Outline 1 Pairing phases of polaritons Pairing phases and Feshbach for polaritons Phase diagram: Critical detunings Signatures Phase diagram: Critical temperatures 2 Photon condensation Modelling organic molecules: Vibrational modes Strong coupling? Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 9 / 21 Photon BEC experiments Dye filled microcavity No strong coupling [Klaers et al, Nature, 2010] Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 10 / 21 Photon BEC experiments Dye filled microcavity No strong coupling [Klaers et al, Nature, 2010] Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 10 / 21 Photon BEC experiments Dye filled microcavity No strong coupling [Klaers et al, Nature, 2010] Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 10 / 21 Relation to dye laser No single cavity mode I I Condensate mode is not maximum gain Gain/Absorption in balance Thermalised many-mode system Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 11 / 21 Relation to dye laser Energy 4 Level Dye Laser ⇑ Cavity nuclear coordinate ⇓ No single cavity mode I I Condensate mode is not maximum gain Gain/Absorption in balance Thermalised many-mode system Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 11 / 21 Relation to dye laser Energy 4 Level Dye Laser ⇑ Cavity nuclear coordinate ⇓ But: No single cavity mode I I Condensate mode is not maximum gain Gain/Absorption in balance Thermalised many-mode system Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 11 / 21 Relation to dye laser Energy 4 Level Dye Laser ⇑ Cavity nuclear coordinate ⇓ But: No single cavity mode I I Condensate mode is not maximum gain Gain/Absorption in balance Thermalised many-mode system Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 11 / 21 Modelling Hsys = X † ωm ψ m ψm + m Xh α 2 i σαz + g ψm σα+ + H.c. 2D harmonic cavity ωm = ωcutoff + mωH.O. Degeneracies gm = m + 1 Local vibrational mode I I Phonon frequency Ω Huang-Rhys parameter S — phonon coupling Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 12 / 21 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 cavity ωm = ωcutoff + mωH.O. Degeneracies gm = m + 1 Local vibrational mode I I Phonon frequency Ω Huang-Rhys parameter S — phonon coupling Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 12 / 21 Modelling Rate 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 Pairing phases & photons ICSCE7, April 2014 13 / 21 Modelling Rate 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 Pairing phases & photons ICSCE7, April 2014 13 / 21 Distribution gm nm Rate equation — include spontaneous emission Bose-Einstein distribution without losses [Kirton & JK PRL ’13] Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 14 / 21 Distribution gm nm Rate equation — include spontaneous emission Bose-Einstein distribution without losses Low loss: Thermal [Kirton & JK PRL ’13] Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 14 / 21 Distribution gm nm Rate equation — include spontaneous emission Bose-Einstein distribution without losses Low loss: Thermal High loss → Laser [Kirton & JK PRL ’13] Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 14 / 21 Outline 1 Pairing phases of polaritons Pairing phases and Feshbach for polaritons Phase diagram: Critical detunings Signatures Phase diagram: Critical temperatures 2 Photon condensation Modelling organic molecules: Vibrational modes Strong coupling? Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 15 / 21 Energy Strong coupling limit ⇑ Xh σαz + g ψσα+ + ψ † σα− H = ωψ † ψ+ 2 α n oi √ +Ω bα† bα + S bα† + bα σαz Photon nuclear coordinate ⇓ Phonon frequency Ω l Huang-Rhys parameter S — phonon coupling S ×l Polaron formation (dressing by vibrational modes) Vibrational replicas and BEC Ultra-strong phonon coupling Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 16 / 21 Energy Strong coupling limit ⇑ Xh σαz + g ψσα+ + ψ † σα− H = ωψ † ψ+ 2 α n oi √ +Ω bα† bα + S bα† + bα σαz Photon nuclear coordinate ⇓ Phonon frequency Ω l Huang-Rhys parameter S — phonon coupling Strong coupling and vibrational modes S ×l Polaron formation (dressing by vibrational modes) Vibrational replicas and BEC Ultra-strong phonon coupling Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 16 / 21 Energy Strong coupling limit ⇑ Xh σαz + g ψσα+ + ψ † σα− H = ωψ † ψ+ 2 α n oi √ +Ω bα† bα + S bα† + bα σαz Photon nuclear coordinate ⇓ Phonon frequency Ω l Huang-Rhys parameter S — phonon coupling Strong coupling and vibrational modes S ×l Polaron formation (dressing by vibrational modes) Vibrational replicas and BEC Ultra-strong phonon coupling Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 16 / 21 Energy Strong coupling limit ⇑ Xh σαz + g ψσα+ + ψ † σα− H = ωψ † ψ+ 2 α n oi √ +Ω bα† bα + S bα† + bα σαz Photon nuclear coordinate ⇓ Phonon frequency Ω l Huang-Rhys parameter S — phonon coupling Strong coupling and vibrational modes S ×l 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 2014 16 / 21 Organic materials in microcavities Strong coupling with organic materials Polariton lasing [Lidzey, Nature ’98] Ωr ' 0.1eV [Kena Cohen and Forrest, Nat. Photon 2010] Thermalisation, condensation interactions T = 300K [Daskalakis et al. Nat. Photon 2014; Plumhoff et al. ibid.] Ultrastrong coupling regime Ωr ' 0.6eV! [Canaguier-Durand Ang. Chem. ’13, . . . ] Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 17 / 21 Phase diagram n oi Xh √ Sαz + g ψSα+ + ψ † Sα− +Ω bα† bα + S bα† + bα Sαz H = ωψ † ψ+ α S suppresses condensation — reduces overlap Reentrant behaviour — Min µ at T ∼ 0.2 [Cwik et al. EPL ’14] Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 18 / 21 Phase diagram n oi Xh √ Sαz + g ψSα+ + ψ † Sα− +Ω bα† bα + S bα† + bα Sαz H = ωψ † ψ+ α g=2. S=2, ∆=4, Ω=0.1 0.5 1 -x*y S=0 0.8 T 0.4 0.6 0.3 0.4 0.2 0.2 0.1 0 Coherent field 〈ψ〉 0.6 0 -6 -5 -4 -3 µ-ωc -2 -1 0 S suppresses condensation — reduces overlap Reentrant behaviour — Min µ at T ∼ 0.2 [Cwik et al. EPL ’14] Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 18 / 21 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 [Cwik et al. EPL ’14] Pairing phases & photons -4 -3 µ-ω -2 -1 0 0 1 0.6 0.8 0.2 0.4 T ICSCE7, April 2014 19 / 21 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 -4 µ-ω-3 -2 0 Jonathan Keeling [Cwik et al. EPL ’14] 0.1 T 7 6 5 4 3 2 1 0.2 -50 S=6 0.4 gc√ Ν S=3 gc√ Ν gc√ Ν S=0 ∆=4, Ω=1 7 6 5 4 3 2 1 0.2 -50 -4 µ-ω-3 -2 0 0.1 T Pairing phases & photons 0.3 0.2 0.1 0.2 -4 µ-ω-3 -2 0 0.1 T ICSCE7, April 2014 Jump in 〈 ψ 〉 0.5 7 6 5 4 3 2 1 0 -5 0 19 / 21 Acknowledgements G ROUP : C OLLABORATORS : Francesca Marchetti, UAM F UNDING : Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 20 / 21 Summary −0.4 −0.2 µ [meV] 0 0 MSF MSF 0 11 1x10 n [cm−2] AMSF −0.5 −0.3 0 −0.4 µ [meV] 95 30 90 20 85 10 11 −2 n [cm ] 2x10 80 APD APD φ A Adjustable delay B APD T [K] AMSF GaAs ZnO 40 AMSF 1 N −0.6 PS N N excitonic fraction [%] SF PS 2 100 50 3 δ=10 meV MSF AMSF δ [meV] 12 10 8 6 4 2 0 M δ [meV] Polaritons pairing phase feasible for ZnO, signatures in coherence and vortices APD θ 0 0 5 10 T [K] 15 0 5 10 T [K] 15 [Marchetti and Keeling, arXiv:1308.1032] Photon condensation and thermalisation; vibrational modes [Kirton and Keeling, PRL ’13] Vibrational modes and strong coupling 0.2 0.1 0 0 -5 -4 -3 µ-ωc -2 -1 0 7 6 5 4 3 2 1 0.2 -50 S=3 -4 µ-ω-3 -2 0 0.1 T 7 6 5 4 3 2 1 0.2 -50 S=6 0.4 -4 µ-ω-3 -2 0 0.1 T 0.3 0.2 0.1 0.2 -4 µ-ω-3 -2 0 0.1 T Jump in 〈 ψ 〉 0.4 S=0 ∆=4, Ω=1 gc√ Ν T 0.6 0.2 7 6 5 4 3 2 1 0 -5 gc√ Ν 0.8 0.3 -6 0.5 1 -x*y S=0 0.4 gc√ Ν g=2. S=2, ∆=4, Ω=0.1 0.5 Coherent field 〈ψ〉 0.6 0 [Cwik, Reja, Littlewood, Keeling EPL ’14] Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 21 / 21 Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 22 / 39 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 2014 23 / 39 Exciton-photon model Microscopic model — coupled exciton-photon system 2 2 X k k † † H= δ+ − µ X̂kσ X̂kσ + − µ Ĉkσ Ĉkσ 2mX 2mC σ=±1 σ=±2,±1 k # ZZ X X † † + ΩR Ĉkσ X̂kσ + X̂kσ Ĉkσ + d 2 rd 2 R UσXX 0 τ 0 τ σ (r) X X σ,σ 0 ,τ,τ 0 =±2,±1 σ=±1 × X̂σ†0 R + r 2 X̂τ†0 R − r 2 X̂τ R − r r X̂σ R + 2 2 Interaction U XX has exchange structure For large ΩR , neglect σ = ±2 XX Interaction supports bound states in U+1,−1,−1,+1 channel — bipolariton NB, bipolariton, bound polaritons, but larger exciton fraction. Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 24 / 39 Exciton-photon model Microscopic model — coupled exciton-photon system 2 2 X k k † † H= δ+ − µ X̂kσ X̂kσ + − µ Ĉkσ Ĉkσ 2mX 2mC σ=±1 σ=±2,±1 k # ZZ X X † † + ΩR Ĉkσ X̂kσ + X̂kσ Ĉkσ + d 2 rd 2 R UσXX 0 τ 0 τ σ (r) X X σ,σ 0 ,τ,τ 0 =±2,±1 σ=±1 × X̂σ†0 R + r 2 X̂τ†0 R − r 2 X̂τ R − r r X̂σ R + 2 2 Interaction U XX has exchange structure For large ΩR , neglect σ = ±2 XX Interaction supports bound states in U+1,−1,−1,+1 channel — bipolariton NB, bipolariton, bound polaritons, but larger exciton fraction. Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 24 / 39 Exciton-photon model Microscopic model — coupled exciton-photon system 2 2 X k k † † H= δ+ − µ X̂kσ X̂kσ + − µ Ĉkσ Ĉkσ 2mX 2mC σ=±1 σ=±2,±1 k # ZZ X X † † + ΩR Ĉkσ X̂kσ + X̂kσ Ĉkσ + d 2 rd 2 R UσXX 0 τ 0 τ σ (r) X X σ,σ 0 ,τ,τ 0 =±2,±1 σ=±1 × X̂σ†0 R + r 2 X̂τ†0 R − r 2 X̂τ R − r r X̂σ R + 2 2 Interaction U XX has exchange structure For large ΩR , neglect σ = ±2 XX Interaction supports bound states in U+1,−1,−1,+1 channel — bipolariton NB, bipolariton, bound polaritons, but larger exciton fraction. Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 24 / 39 Exciton-photon model Microscopic model — coupled exciton-photon system 2 2 X k k † † H= δ+ − µ X̂kσ X̂kσ + − µ Ĉkσ Ĉkσ 2mX 2mC σ=±1 σ=±2,±1 k # ZZ X X † † + ΩR Ĉkσ X̂kσ + X̂kσ Ĉkσ + d 2 rd 2 R UσXX 0 τ 0 τ σ (r) X X σ,σ 0 ,τ,τ 0 =±2,±1 σ=±1 × X̂σ†0 R + r 2 X̂τ†0 R − r 2 X̂τ R − r r X̂σ R + 2 2 Interaction U XX has exchange structure For large ΩR , neglect σ = ±2 XX Interaction supports bound states in U+1,−1,−1,+1 channel — bipolariton NB, bipolariton, bound polaritons, but larger exciton fraction. Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 24 / 39 Exciton-photon model Microscopic model — coupled exciton-photon system 2 2 X k k † † H= δ+ − µ X̂kσ X̂kσ + − µ Ĉkσ Ĉkσ 2mX 2mC σ=±1 σ=±2,±1 k # ZZ X X † † + ΩR Ĉkσ X̂kσ + X̂kσ Ĉkσ + d 2 rd 2 R UσXX 0 τ 0 τ σ (r) X X σ,σ 0 ,τ,τ 0 =±2,±1 σ=±1 × X̂σ†0 R + r 2 X̂τ†0 R − r 2 X̂τ R − r r X̂σ R + 2 2 Interaction U XX has exchange structure For large ΩR , neglect σ = ±2 XX Interaction supports bound states in U+1,−1,−1,+1 channel — bipolariton NB, bipolariton, bound polaritons, but larger exciton fraction. Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 24 / 39 Exciton-photon model Microscopic model — coupled exciton-photon system 2 2 X k k † † H= δ+ − µ X̂kσ X̂kσ + − µ Ĉkσ Ĉkσ 2mX 2mC σ=±1 σ=±2,±1 k # ZZ X X † † + ΩR Ĉkσ X̂kσ + X̂kσ Ĉkσ + d 2 rd 2 R UσXX 0 τ 0 τ σ (r) X X σ,σ 0 ,τ,τ 0 =±2,±1 σ=±1 × X̂σ†0 R + r 2 X̂τ†0 R − r 2 X̂τ R − r r X̂σ R + 2 2 Interaction U XX has exchange structure For large ΩR , neglect σ = ±2 XX Interaction supports bound states in U+1,−1,−1,+1 channel — bipolariton NB, bipolariton, bound polaritons, but larger exciton fraction. Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 25 / 39 Exciton-photon model Microscopic model — coupled exciton-photon system 2 2 X k k † † H= δ+ − µ X̂kσ X̂kσ + − µ Ĉkσ Ĉkσ 2mX 2mC σ=±1 σ=±2,±1 k # ZZ X X † † + ΩR Ĉkσ X̂kσ + X̂kσ Ĉkσ + d 2 rd 2 R UσXX 0 τ 0 τ σ (r) X X σ,σ 0 ,τ,τ 0 =±2,±1 σ=±1 × X̂σ†0 R + r 2 X̂τ†0 R − r 2 X̂τ R − r r X̂σ R + 2 2 Interaction U XX has exchange structure For large ΩR , neglect σ = ±2 XX Interaction supports bound states in U+1,−1,−1,+1 channel — bipolariton NB, bipolariton, bound polaritons, but larger exciton fraction. Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 25 / 39 Exciton-photon model Microscopic model — coupled exciton-photon system 2 2 X k k † † H= δ+ − µ X̂kσ X̂kσ + − µ Ĉkσ Ĉkσ 2mX 2mC σ=±1 σ=±2,±1 k # ZZ X X † † + ΩR Ĉkσ X̂kσ + X̂kσ Ĉkσ + d 2 rd 2 R UσXX 0 τ 0 τ σ (r) X X σ,σ 0 ,τ,τ 0 =±2,±1 σ=±1 × X̂σ†0 R + r 2 X̂τ†0 R − r 2 X̂τ R − r r X̂σ R + 2 2 Interaction U XX has exchange structure For large ΩR , neglect σ = ±2 XX Interaction supports bound states in U+1,−1,−1,+1 channel — bipolariton NB, bipolariton, bound polaritons, but larger exciton fraction. Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 25 / 39 Exciton-photon model Microscopic model — coupled exciton-photon system 2 2 X k k † † H= δ+ − µ X̂kσ X̂kσ + − µ Ĉkσ Ĉkσ 2mX 2mC σ=±1 σ=±2,±1 k # ZZ X X † † + ΩR Ĉkσ X̂kσ + X̂kσ Ĉkσ + d 2 rd 2 R UσXX 0 τ 0 τ σ (r) X X σ,σ 0 ,τ,τ 0 =±2,±1 σ=±1 × X̂σ†0 R + r 2 X̂τ†0 R − r 2 X̂τ R − r r X̂σ R + 2 2 Interaction U XX has exchange structure For large ΩR , neglect σ = ±2 XX Interaction supports bound states in U+1,−1,−1,+1 channel — bipolariton NB, bipolariton, bound polaritons, but larger exciton fraction. Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 25 / 39 Exciton-photon model Microscopic model — coupled exciton-photon system 2 2 X k k † † H= δ+ − µ X̂kσ X̂kσ + − µ Ĉkσ Ĉkσ 2mX 2mC σ=±1 σ=±2,±1 k # ZZ X X † † + ΩR Ĉkσ X̂kσ + X̂kσ Ĉkσ + d 2 rd 2 R UσXX 0 τ 0 τ σ (r) X X σ,σ 0 ,τ,τ 0 =±2,±1 σ=±1 × X̂σ†0 R + r 2 X̂τ†0 R − r 2 X̂τ R − r r X̂σ R + 2 2 Interaction U XX has exchange structure For large ΩR , neglect σ = ±2 XX Interaction supports bound states in U+1,−1,−1,+1 channel — bipolariton NB, bipolariton, bound polaritons, but larger exciton fraction. Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 25 / 39 Polariton model H= X k 2 X k2 k † † − µ ψ̂σk ψ̂σk + + ν − 2µ ψ̂mk ψ̂mk 2m 2mm σ=↑,↓ Z X + d 2R σ=↑,↓,m Uσσ † † g ψ̂ ψ̂ ψ̂ ψ̂ + U↑↓ ψ̂↓† ψ̂↑† ψ̂↑ ψ̂↓ + ψ̂↑† ψ̂↓† ψ̂m + h.c. 2 σ σ σ σ 2 E X 2ω0 ωXX Polariton dispersion m, detuning ν, interactions depend on δ X E−ω0 [meV] 6 Resonance width, dispersion derived from dressed exciton T matrix. Jonathan Keeling |Eb| ν>0 LP 2ω0 UP 4 UP 2 C ω0 C ω0 0 X ω0 −2 −4 Pairing phases & photons LP −2 X δ (ΩR2+δ2)1/2 LP ω0 2 0 k [µm−1] 4 ICSCE7, April 2014 26 / 39 Polariton model H= X k 2 X k2 k † † − µ ψ̂σk ψ̂σk + + ν − 2µ ψ̂mk ψ̂mk 2m 2mm σ=↑,↓ Z X + d 2R σ=↑,↓,m Uσσ † † g ψ̂ ψ̂ ψ̂ ψ̂ + U↑↓ ψ̂↓† ψ̂↑† ψ̂↑ ψ̂↓ + ψ̂↑† ψ̂↓† ψ̂m + h.c. 2 σ σ σ σ 2 E X 2ω0 ωXX Polariton dispersion m, detuning ν, interactions depend on δ X E−ω0 [meV] 6 Resonance width, dispersion derived from dressed exciton T matrix. Jonathan Keeling |Eb| ν>0 LP 2ω0 UP 4 UP 2 C ω0 C ω0 0 X ω0 −2 −4 Pairing phases & photons LP −2 X δ (ΩR2+δ2)1/2 LP ω0 2 0 k [µm−1] 4 ICSCE7, April 2014 26 / 39 Polariton model H= X k 2 X k2 k † † − µ ψ̂σk ψ̂σk + + ν − 2µ ψ̂mk ψ̂mk 2m 2mm σ=↑,↓ Z X + d 2R σ=↑,↓,m Uσσ † † g ψ̂ ψ̂ ψ̂ ψ̂ + U↑↓ ψ̂↓† ψ̂↑† ψ̂↑ ψ̂↓ + ψ̂↑† ψ̂↓† ψ̂m + h.c. 2 σ σ σ σ 2 E X 2ω0 LP 2ω0 = ωXX Polariton dispersion m, detuning ν, interactions depend on δ X E−ω0 [meV] UP ω0 C 4 C ω0 2 X X 0 ω0 LP ω0 LP −2 −4 Jonathan Keeling ν=0 UP 6 Resonance width, dispersion derived from dressed exciton T matrix. |Eb| Pairing phases & photons −2 0 k [µm−1] 2 δ (ΩR2+δ2)1/2 4 ICSCE7, April 2014 26 / 39 Polariton model H= X k 2 X k2 k † † − µ ψ̂σk ψ̂σk + + ν − 2µ ψ̂mk ψ̂mk 2m 2mm σ=↑,↓ Z X + d 2R σ=↑,↓,m Uσσ † † g ψ̂ ψ̂ ψ̂ ψ̂ + U↑↓ ψ̂↓† ψ̂↑† ψ̂↑ ψ̂↓ + ψ̂↑† ψ̂↓† ψ̂m + h.c. 2 σ σ σ σ 2 E X 2ω0 LP 2ω0 = ωXX Polariton dispersion m, detuning ν, interactions depend on δ X E−ω0 [meV] UP ω0 C 4 C ω0 2 X X 0 ω0 LP ω0 LP −2 −4 Jonathan Keeling ν=0 UP 6 Resonance width, dispersion derived from dressed exciton T matrix. |Eb| Pairing phases & photons −2 0 k [µm−1] 2 δ (ΩR2+δ2)1/2 4 ICSCE7, April 2014 26 / 39 Polariton model σ=↑,↓ k Z X + d 2R σ=↑,↓,m Uσσ † † g ψ̂ ψ̂ ψ̂ ψ̂ + U↑↓ ψ̂↓† ψ̂↑† ψ̂↑ ψ̂↓ + ψ̂↑† ψ̂↓† ψ̂m + h.c. 2 σ σ σ σ 2 ν>0 Biexciton in continuum mLP/mX 10 10 Jonathan Keeling -2 10 10 -1 -3 -4 |Eb| ~mC/mX 0 10 20 30 δ [meV] 40 Pairing phases & photons 10 8 6 4 2 0 -2 ν [meV] H= 2 X k2 k † † − µ ψ̂σk + ν − 2µ ψ̂mk ψ̂σk + ψ̂mk 2m 2mm X ν<0 Bound biexciton. (Excitonic limit) 50 ICSCE7, April 2014 27 / 39 Exciton and polariton spin degrees of freedom Photon: two circular polarisation modes Exciton: bound state of electron & hole I J = 1 ± 1/2 hole (p-orbital), J = 1/2 electron I Spin orbit splits hole bands, 4 × 2 states. I Quantum well fixes kz of hole 2 × 2 states. Exciton spin states Jz = +2, +1, −1, −2 Optically active states Jz = ±1 Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 28 / 39 Exciton and polariton spin degrees of freedom Photon: two circular polarisation modes Exciton: bound state of electron & hole E c v I J = 1 ± 1/2 hole (p-orbital), J = 1/2 electron I Spin orbit splits hole bands, 4 × 2 states. I Quantum well fixes kz of hole 2 × 2 states. k Exciton spin states Jz = +2, +1, −1, −2 Optically active states Jz = ±1 Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 28 / 39 Exciton and polariton spin degrees of freedom Photon: two circular polarisation modes E Exciton: bound state of electron & hole c v, J=3/2 HH v, J=3/2 LH v, J=1/2 I J = 1 ± 1/2 hole (p-orbital), J = 1/2 electron I Spin orbit splits hole bands, 4 × 2 states. I Quantum well fixes kz of hole 2 × 2 states. k Exciton spin states Jz = +2, +1, −1, −2 Optically active states Jz = ±1 Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 28 / 39 Exciton and polariton spin degrees of freedom z Photon: two circular polarisation modes Exciton: bound state of electron & hole E Cavity c v, J=3/2 HH v, J=3/2 LH v, J=1/2 I J = 1 ± 1/2 hole (p-orbital), J = 1/2 electron I Spin orbit splits hole bands, 4 × 2 states. I Quantum well fixes kz of hole 2 × 2 states. k Exciton spin states Jz = +2, +1, −1, −2 Optically active states Jz = ±1 Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 28 / 39 Exciton and polariton spin degrees of freedom z Photon: two circular polarisation modes Exciton: bound state of electron & hole E Cavity c v, J=3/2 HH v, J=3/2 LH v, J=1/2 I J = 1 ± 1/2 hole (p-orbital), J = 1/2 electron I Spin orbit splits hole bands, 4 × 2 states. I Quantum well fixes kz of hole 2 × 2 states. k Exciton spin states Jz = +2, +1, −1, −2 Optically active states Jz = ±1 Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 28 / 39 Exciton and polariton spin degrees of freedom z Photon: two circular polarisation modes Exciton: bound state of electron & hole E Cavity c v, J=3/2 HH v, J=3/2 LH v, J=1/2 I J = 1 ± 1/2 hole (p-orbital), J = 1/2 electron I Spin orbit splits hole bands, 4 × 2 states. I Quantum well fixes kz of hole 2 × 2 states. k Exciton spin states Jz = +2, +1, −1, −2 Optically active states Jz = ±1 Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 28 / 39 Beyond mean-field Fluctuation effects? I I Polariton fluctuations irrelevant: mU ∼ 10−4 . Exciton fluctuations important: mm U ∼ 1. Next order theory: [Nozières & St James, J. Phys ’82] X X † † . |Ψi ∝ exp − ψσ ψ̂k=0,σ + tanh(θkγ )b̂kγ b̂−kγ σ=↑,↓,m where † b̂km = † ψ̂km and k,γ=a,b,m † ψ̂k↑ † ψ̂−k↓ ! 1 =√ 2 1 1 −1 1 † b̂ka † b̂−kb ! , Variational functional E[ψ0 , ψm , θkγ ] Can show minimum θk γ has form tanh(2θkγ ) = I I αγ βγ + k 2 /2mγ Finite only if |αγ | < βγ Variational function E(ψ0 , ψm , αγ , βγ ) Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 29 / 39 Beyond mean-field Fluctuation effects? I I Polariton fluctuations irrelevant: mU ∼ 10−4 . Exciton fluctuations important: mm U ∼ 1. Next order theory: [Nozières & St James, J. Phys ’82] X X † † . |Ψi ∝ exp − ψσ ψ̂k=0,σ + tanh(θkγ )b̂kγ b̂−kγ σ=↑,↓,m where † b̂km = † ψ̂km and k,γ=a,b,m † ψ̂k↑ † ψ̂−k↓ ! 1 =√ 2 1 1 −1 1 † b̂ka † b̂−kb ! , Variational functional E[ψ0 , ψm , θkγ ] Can show minimum θk γ has form tanh(2θkγ ) = I I αγ βγ + k 2 /2mγ Finite only if |αγ | < βγ Variational function E(ψ0 , ψm , αγ , βγ ) Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 29 / 39 Beyond mean-field Fluctuation effects? I I Polariton fluctuations irrelevant: mU ∼ 10−4 . Exciton fluctuations important: mm U ∼ 1. Next order theory: [Nozières & St James, J. Phys ’82] X X † † . |Ψi ∝ exp − ψσ ψ̂k=0,σ + tanh(θkγ )b̂kγ b̂−kγ σ=↑,↓,m where † b̂km = † ψ̂km and k,γ=a,b,m † ψ̂k↑ † ψ̂−k↓ ! 1 =√ 2 1 1 −1 1 † b̂ka † b̂−kb ! , Variational functional E[ψ0 , ψm , θkγ ] Can show minimum θk γ has form tanh(2θkγ ) = I I αγ βγ + k 2 /2mγ Finite only if |αγ | < βγ Variational function E(ψ0 , ψm , αγ , βγ ) Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 29 / 39 Beyond mean-field Fluctuation effects? I I Polariton fluctuations irrelevant: mU ∼ 10−4 . Exciton fluctuations important: mm U ∼ 1. Next order theory: [Nozières & St James, J. Phys ’82] X X † † . |Ψi ∝ exp − ψσ ψ̂k=0,σ + tanh(θkγ )b̂kγ b̂−kγ σ=↑,↓,m where † b̂km = † ψ̂km and k,γ=a,b,m † ψ̂k↑ † ψ̂−k↓ ! 1 =√ 2 1 1 −1 1 † b̂ka † b̂−kb ! , Variational functional E[ψ0 , ψm , θkγ ] Can show minimum θk γ has form tanh(2θkγ ) = I I αγ βγ + k 2 /2mγ Finite only if |αγ | < βγ Variational function E(ψ0 , ψm , αγ , βγ ) Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 29 / 39 Finite T calculation Finite T — minimize free energy Use Feynman-Jensen inequality: h i F = −kB T ln Tre−Ĥ/kB T ≤ FMF + hĤ − ĤMF iMF Where h. . .iMF calculated using ρ = e(FMF −ĤMF )/kB T Ansatz ĤMF → Variational F (ψ0 , ψm , αγ , βγ ). ( ĤMF = X − √ † Aψγ (αγ + βγ ) b̂0γ + b̂0γ γ 1 X † + b̂kγ 2 k Jonathan Keeling b̂−kγ kγ + βγ αγ Pairing phases & photons αγ kγ + βγ b̂kγ † b̂−kγ !) ICSCE7, April 2014 . 30 / 39 Finite T calculation Finite T — minimize free energy Use Feynman-Jensen inequality: h i F = −kB T ln Tre−Ĥ/kB T ≤ FMF + hĤ − ĤMF iMF Where h. . .iMF calculated using ρ = e(FMF −ĤMF )/kB T Ansatz ĤMF → Variational F (ψ0 , ψm , αγ , βγ ). ( ĤMF = X − √ † Aψγ (αγ + βγ ) b̂0γ + b̂0γ γ 1 X † + b̂kγ 2 k Jonathan Keeling b̂−kγ kγ + βγ αγ Pairing phases & photons αγ kγ + βγ b̂kγ † b̂−kγ !) ICSCE7, April 2014 . 30 / 39 Finite T calculation Finite T — minimize free energy Use Feynman-Jensen inequality: h i F = −kB T ln Tre−Ĥ/kB T ≤ FMF + hĤ − ĤMF iMF Where h. . .iMF calculated using ρ = e(FMF −ĤMF )/kB T Ansatz ĤMF → Variational F (ψ0 , ψm , αγ , βγ ). ( ĤMF = X − √ † Aψγ (αγ + βγ ) b̂0γ + b̂0γ γ 1 X † + b̂kγ 2 k Jonathan Keeling b̂−kγ kγ + βγ αγ Pairing phases & photons αγ kγ + βγ b̂kγ † b̂−kγ !) ICSCE7, April 2014 . 30 / 39 Variational MFT for WIDBG Z X k2 † U ψ ψ + d 2 r ψ † ψ † ψψ Test validity. WIDBG Ĥ = 2m k k 2 k VMFT for WIDBG: √ ĤMF = − Aψ(α + β)(b̂0† + b̂0 ) ! + β 1 X † b̂k α k . + b̂k b̂−k † α k + β 2 b̂−k k Compare to 2D EOS, ρ(µ) = Tf (µ/T ) CF Hartree-Fock-Popov-Bogoluibov method, include Uρ in Σ Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 31 / 39 Variational MFT for WIDBG Z X k2 † U ψ ψ + d 2 r ψ † ψ † ψψ Test validity. WIDBG Ĥ = 2m k k 2 k VMFT for WIDBG: √ ĤMF = − Aψ(α + β)(b̂0† + b̂0 ) ! + β 1 X † b̂k α k . + b̂k b̂−k † α k + β 2 b̂−k k Compare to 2D EOS, ρ(µ) = Tf (µ/T ) CF Hartree-Fock-Popov-Bogoluibov method, include Uρ in Σ Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 31 / 39 Variational MFT for WIDBG Z X k2 † U ψ ψ + d 2 r ψ † ψ † ψψ Test validity. WIDBG Ĥ = 2m k k 2 k 6 mU=0.01, εc=100 [a.u.] Density ρ/T √ 4 ĤMF = − Aψ(α + β)(b̂0† + b̂0 ) !3 + β 2 1 X † b̂ α k k . + b̂k b̂−k † α k + β 2 b̂−k 1 2 VMFT QMC [PRA 66 043608] 5 k 1.25 0 Temperature, T [a.u.] VMFT for WIDBG: 0.5 -0.02 0 0.02 µ/T 0.04 0.06 Compare to 2D EOS, ρ(µ) = Tf (µ/T ) CF Hartree-Fock-Popov-Bogoluibov method, include Uρ in Σ Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 31 / 39 Variational MFT for WIDBG Z X k2 † U ψ ψ + d 2 r ψ † ψ † ψψ Test validity. WIDBG Ĥ = 2m k k 2 k 6 mU=0.01, εc=100 [a.u.] Density ρ/T √ 4 ĤMF = − Aψ(α + β)(b̂0† + b̂0 ) !3 + β 2 1 X † b̂ α k k . + b̂k b̂−k † α k + β 2 b̂−k 1 2 VMFT QMC [PRA 66 043608] 5 k 1.25 0 Temperature, T [a.u.] VMFT for WIDBG: 0.5 -0.02 0 0.02 µ/T 0.04 0.06 Compare to 2D EOS, ρ(µ) = Tf (µ/T ) CF Hartree-Fock-Popov-Bogoluibov method, include Uρ in Σ Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 31 / 39 δ [meV] 20 15 MSF Phase diagram, finite temperature T= 0.58 K MSF AMSF 10 5 0 −0.8 −0.6 −0.4 −0.2 µ [meV] Jonathan Keeling AMSF N N PS 0 0 −2 n [cm ] 11 2x10 Pairing phases & photons ICSCE7, April 2014 32 / 39 Phase diagram, finite temperature δ [meV] MSF 20 15 MSF AMSF 10 MSF 0 20 15 PS T=1.16 K AMSF 10 5 AMSF N N 5 δ [meV] T= 0.58 K MSF N N 0 −0.8 −0.6 −0.4 −0.2 µ [meV] Jonathan Keeling AMSF PS 0 0 −2 n [cm ] 11 2x10 Pairing phases & photons ICSCE7, April 2014 32 / 39 Phase diagram, finite temperature δ [meV] MSF 20 15 10 MSF 15 PS T=1.16 K AMSF MSF N 10 5 AMSF N N 0 20 δ [meV] MSF AMSF 5 N AMSF PS MSF 0 20 δ [meV] T= 0.58 K T =2.32 K MSF 15 5 N AMSF 10 N 0 −0.8 −0.6 −0.4 −0.2 µ [meV] Jonathan Keeling AMSF PS 0 0 −2 n [cm ] 11 2x10 Pairing phases & photons ICSCE7, April 2014 32 / 39 Phase diagram, finite temperature MSF 15 10 MSF 15 PS T=1.16 K AMSF MSF N 10 5 AMSF N N 0 20 δ [meV] MSF AMSF 5 N AMSF PS MSF 0 20 δ [meV] T= 0.58 K T =2.32 K MSF 15 AMSF 10 5 N N 0 −0.8 −0.6 −0.4 −0.2 µ [meV] Jonathan Keeling AMSF 5 N 0 PS AMSF 7 PS 0 0 δ [meV] δ [meV] 20 8 9 10 10 10 10 10 −2 n [cm ] −2 n [cm ] 11 10 11 2x10 Pairing phases & photons ICSCE7, April 2014 32 / 39 20 T= 0.58 K MSF 15 AMSF AMSF N N T [K] 3 δ=10 meV 2 −0.8 −0.6 −0.4 −0.2 µ [meV] PS N 5 PS 0 0 10 δ [meV] MSF Phase diagram, vs temperature −2 n [cm ] 11 0 2x10 N AMSF 1 AMSF MSF MSF 0 −0.5 −0.4 µ [meV] Jonathan Keeling −0.3 0 11 −2 n [cm ] 2x10 Pairing phases & photons ICSCE7, April 2014 33 / 39 Phase diagram, vs temperature N 2 MSF 15 MSF AMSF N AMSF AMSF N −0.8 −0.6 −0.4 −0.2 µ [meV] PS N 0 0 10 5 PS MSF 0 3 δ=10 meV 2 20 T= 0.58 K AMSF N 1 T [K] PS δ [meV] T [K] δ=8.3 meV MSF 0 −2 n [cm ] 11 0 2x10 N AMSF 1 AMSF MSF MSF 0 −0.5 −0.4 µ [meV] Jonathan Keeling −0.3 0 11 −2 n [cm ] 2x10 Pairing phases & photons ICSCE7, April 2014 33 / 39 Phase diagram, vs temperature 2 δ=6.5 meV N PS T [K] δ=8.3 meV N 2 MSF 15 MSF AMSF N AMSF AMSF N −0.8 −0.6 −0.4 −0.2 µ [meV] PS N 0 0 10 5 PS MSF 0 3 δ=10 meV 2 20 T= 0.58 K AMSF N 1 T [K] PS δ [meV] 0 AMSF MSF MSF 0.5 MSF N 1 AMSF T [K] 1.5 −2 n [cm ] 11 0 2x10 N AMSF 1 AMSF MSF MSF 0 −0.5 −0.4 µ [meV] Jonathan Keeling −0.3 0 11 −2 n [cm ] 2x10 Pairing phases & photons ICSCE7, April 2014 33 / 39 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 Thermal at low κ/high temperature High loss, κ competes with Γ(±δ0 ) Low temperature, Γ(±δ0 ) shrinks [Kirton & JK PRL ’13] Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 34 / 39 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 Thermal at low κ/high temperature High loss, κ competes with Γ(±δ0 ) Low temperature, Γ(±δ0 ) shrinks [Kirton & JK PRL ’13] Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 34 / 39 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 1 0.8 Thermal at low κ/high temperature 0.6 Γ(δ) Γ(−δ) High loss, κ competes with Γ(±δ0 ) Low temperature, Γ(±δ0 ) shrinks 0.4 0.2 0 −200 −100 0 100 δ [THz] 200 [Kirton & JK PRL ’13] Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 34 / 39 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 1 0.8 Thermal at low κ/high temperature 0.6 Γ(δ) Γ(−δ) High loss, κ competes with Γ(±δ0 ) Low temperature, Γ(±δ0 ) shrinks 0.4 0.2 0 −200 −100 0 100 δ [THz] 200 [Kirton & JK PRL ’13] Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 34 / 39 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 Pairing phases & photons ICSCE7, April 2014 35 / 39 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 Pairing phases & photons ICSCE7, April 2014 35 / 39 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 Pairing phases & photons ICSCE7, April 2014 35 / 39 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 Pairing phases & photons ICSCE7, April 2014 35 / 39 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 Pairing phases & photons ICSCE7, April 2014 35 / 39 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 Pairing phases & photons ICSCE7, April 2014 36 / 39 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 Pairing phases & photons ICSCE7, April 2014 36 / 39 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 Pairing phases & photons ICSCE7, April 2014 36 / 39 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? D(t) = −ihψ † (t)ψ(0)i, D(ω) = X n Zn ω − ωn [Cwik et al. EPL ’14] Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 37 / 39 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? D(t) = −ihψ † (t)ψ(0)i, D(ω) = X n Zn ω − ωn [Cwik et al. EPL ’14] Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 37 / 39 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? D(t) = −ihψ † (t)ψ(0)i, D(ω) = X n Zn ω − ωn [Cwik et al. EPL ’14] Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 37 / 39 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 Pairing phases & photons ICSCE7, April 2014 38 / 39 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 Pairing phases & photons ICSCE7, April 2014 38 / 39 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 Pairing phases & photons ICSCE7, April 2014 38 / 39 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 Pairing phases & photons ICSCE7, April 2014 38 / 39 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 Pairing phases & photons ICSCE7, April 2014 38 / 39 Polariton spectrum — coupled oscillators Jonathan Keeling Pairing phases & photons ICSCE7, April 2014 39 / 39 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 Pairing phases & photons 1.4 1.6 1.8 2 ICSCE7, April 2014 39 / 39 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 Pairing phases & photons 1.4 1.6 1.8 2 ICSCE7, April 2014 39 / 39 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 Pairing phases & photons 1.4 1.6 1.8 2 ICSCE7, April 2014 39 / 39 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 Pairing phases & photons 1.4 1.6 1.8 2 ICSCE7, April 2014 39 / 39