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Soton

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  1. 1. Non-equilibrium phases of coupled matter-lightsystemsJonathan KeelingUniversity ofSt Andrews600YEARSSouthhampton, May 2013Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 1 / 36
  2. 2. Coupling many atoms to lightOld question: What happens to radiation when many atoms interact“collectively” with light.Superradiance — dynamical and steady state.Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 2 / 36
  3. 3. Coupling many atoms to lightOld question: What happens to radiation when many atoms interact“collectively” with light.Superradiance — dynamical and steady state.New relevanceSuperconducting qubitsQuantum dots & NV centresUltra-cold atomsκPumpκCavityPumpRydberg atoms/polaritonsMicrocavity PolaritonsJonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 2 / 36
  4. 4. Dicke effect: Enhanced emissionHint =k,igk ψ†k S−i e−ik·ri + H.c.Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 3 / 36
  5. 5. Dicke effect: Enhanced emissionHint =k,igk ψ†k S−i e−ik·ri + H.c.If |ri − rj| λ, use i Si → SCollective decay:dρdt= −Γ2S+S−ρ − S−ρS++ ρS+S−Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 3 / 36
  6. 6. Dicke effect: Enhanced emissionHint =k,igk ψ†k S−i e−ik·ri + H.c.If |ri − rj| λ, use i Si → SCollective decay:dρdt= −Γ2S+S−ρ − S−ρS++ ρS+S−If Sz = |S| = N/2 initially:I ∝ −Γd Szdt=ΓN24sech2 ΓN2t-N/20N/2tD〈Sz〉tD0ΓN2/2I=-Γd〈Sz〉/dtJonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 3 / 36
  7. 7. Dicke effect: Enhanced emissionHint =k,igk ψ†k S−i e−ik·ri + H.c.If |ri − rj| λ, use i Si → SCollective decay:dρdt= −Γ2S+S−ρ − S−ρS++ ρS+S−If Sz = |S| = N/2 initially:I ∝ −Γd Szdt=ΓN24sech2 ΓN2t-N/20N/2tD〈Sz〉tD0ΓN2/2I=-Γd〈Sz〉/dtProblem: dipole interactions dephase. [Friedberg et al, Phys. Lett. 1972]Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 3 / 36
  8. 8. Collective radiation with a cavity: DynamicsHint =iψ†S−i + ψS+iSingle cavity mode: oscillations[Bonifacio and Preparata PRA ’70]Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 4 / 36
  9. 9. Collective radiation with a cavity: DynamicsHint =iψ†S−i + ψS+i020040060080010001200140016000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8|ψ(t)|2TimeT=2ln(√N__)/√N__1/√N__Single cavity mode: oscillationsIf Sz = |S| = N/2 initially[Bonifacio and Preparata PRA ’70]Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 4 / 36
  10. 10. Dicke model: Equilibrium superradiance transitionH = ωψ†ψ + ω0Sz+ g ψ†S−+ ψS+.Coherent state: |Ψ → eλψ†+ηS+|ΩSmall g, min at λ, η = 0[Hepp, Lieb, Ann. Phys. ’73]Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 5 / 36
  11. 11. Dicke model: Equilibrium superradiance transitionH = ωψ†ψ + ω0Sz+ g ψ†S−+ ψS+.Coherent state: |Ψ → eλψ†+ηS+|ΩSmall g, min at λ, η = 0[Hepp, Lieb, Ann. Phys. ’73]Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 5 / 36
  12. 12. Dicke model: Equilibrium superradiance transitionH = ωψ†ψ + ω0Sz+ g ψ†S−+ ψS+.Coherent state: |Ψ → eλψ†+ηS+|ΩSmall g, min at λ, η = 0Spontaneous polarisation if: Ng2 > ωω0[Hepp, Lieb, Ann. Phys. ’73]Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 5 / 36
  13. 13. Dicke model: Equilibrium superradiance transitionH = ωψ†ψ + ω0Sz+ g ψ†S−+ ψS+.Coherent state: |Ψ → eλψ†+ηS+|ΩSmall g, min at λ, η = 0Spontaneous polarisation if: Ng2 > ωω000ωg-√N⇓ SR[Hepp, Lieb, Ann. Phys. ’73]Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 5 / 36
  14. 14. No go theorem and transitionSpontaneous polarisation if: Ng2 > ωω0[Rzazewski et al PRL ’75]Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 6 / 36
  15. 15. No go theorem and transitionSpontaneous polarisation if: Ng2 > ωω0No go theorem:. Minimal coupling (p − eA)2/2m−iemA · pi ⇔ g(ψ†S−+ ψS+),iA22m⇔ Nζ(ψ + ψ†)2[Rzazewski et al PRL ’75]Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 6 / 36
  16. 16. No go theorem and transitionSpontaneous polarisation if: Ng2 > ωω0No go theorem:. Minimal coupling (p − eA)2/2m−iemA · pi ⇔ g(ψ†S−+ ψS+),iA22m⇔ Nζ(ψ + ψ†)2For large N, ω → ω + 2Nζ. (RWA)[Rzazewski et al PRL ’75]Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 6 / 36
  17. 17. No go theorem and transitionSpontaneous polarisation if: Ng2 > ωω0No go theorem:. Minimal coupling (p − eA)2/2m−iemA · pi ⇔ g(ψ†S−+ ψS+),iA22m⇔ Nζ(ψ + ψ†)2For large N, ω → ω + 2Nζ. (RWA)Need Ng2 > ω0(ω + 2Nζ).[Rzazewski et al PRL ’75]Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 6 / 36
  18. 18. No go theorem and transitionSpontaneous polarisation if: Ng2 > ωω0No go theorem:. Minimal coupling (p − eA)2/2m−iemA · pi ⇔ g(ψ†S−+ ψS+),iA22m⇔ Nζ(ψ + ψ†)2For large N, ω → ω + 2Nζ. (RWA)Need Ng2 > ω0(ω + 2Nζ).But Thomas-Reiche-Kuhn sum rule states: g2/ω0 < 2ζ. No transition[Rzazewski et al PRL ’75]Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 6 / 36
  19. 19. Dicke phase transition: ways outProblem: g2/ω0 < 2ζ for intrinsic parameters. Solutions:InterpretationFerroelectric transition in D · r gauge.[JK JPCM ’07, Vukics & Domokos PRA 2012 ]Circuit QED [Nataf and Ciuti, Nat. Comm. ’10; Viehmann et al. PRL ’11]Grand canonical ensemble:If H → H − µ(Sz+ ψ†ψ), need only:g2N > (ω − µ)(ω0 − µ)Incoherent pumping — polaritoncondensation.Dissociate g, ω0,e.g. Raman scheme: ω0 ω.[Dimer et al. PRA ’07; Baumann et al. Nature’10. Also, Black et al. PRL ’03 ]Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 7 / 36
  20. 20. Dicke phase transition: ways outProblem: g2/ω0 < 2ζ for intrinsic parameters. Solutions:InterpretationFerroelectric transition in D · r gauge.[JK JPCM ’07, Vukics & Domokos PRA 2012 ]Circuit QED [Nataf and Ciuti, Nat. Comm. ’10; Viehmann et al. PRL ’11]Grand canonical ensemble:If H → H − µ(Sz+ ψ†ψ), need only:g2N > (ω − µ)(ω0 − µ)Incoherent pumping — polaritoncondensation.Dissociate g, ω0,e.g. Raman scheme: ω0 ω.[Dimer et al. PRA ’07; Baumann et al. Nature’10. Also, Black et al. PRL ’03 ]Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 7 / 36
  21. 21. Dicke phase transition: ways outProblem: g2/ω0 < 2ζ for intrinsic parameters. Solutions:InterpretationFerroelectric transition in D · r gauge.[JK JPCM ’07, Vukics & Domokos PRA 2012 ]Circuit QED [Nataf and Ciuti, Nat. Comm. ’10; Viehmann et al. PRL ’11]Grand canonical ensemble:If H → H − µ(Sz+ ψ†ψ), need only:g2N > (ω − µ)(ω0 − µ)Incoherent pumping — polaritoncondensation.Dissociate g, ω0,e.g. Raman scheme: ω0 ω.[Dimer et al. PRA ’07; Baumann et al. Nature’10. Also, Black et al. PRL ’03 ]Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 7 / 36
  22. 22. Dicke phase transition: ways outProblem: g2/ω0 < 2ζ for intrinsic parameters. Solutions:InterpretationFerroelectric transition in D · r gauge.[JK JPCM ’07, Vukics & Domokos PRA 2012 ]Circuit QED [Nataf and Ciuti, Nat. Comm. ’10; Viehmann et al. PRL ’11]Grand canonical ensemble:If H → H − µ(Sz+ ψ†ψ), need only:g2N > (ω − µ)(ω0 − µ)Incoherent pumping — polaritoncondensation.Dissociate g, ω0,e.g. Raman scheme: ω0 ω.[Dimer et al. PRA ’07; Baumann et al. Nature’10. Also, Black et al. PRL ’03 ]Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 7 / 36
  23. 23. Dicke phase transition: ways outProblem: g2/ω0 < 2ζ for intrinsic parameters. Solutions:InterpretationFerroelectric transition in D · r gauge.[JK JPCM ’07, Vukics & Domokos PRA 2012 ]Circuit QED [Nataf and Ciuti, Nat. Comm. ’10; Viehmann et al. PRL ’11]Grand canonical ensemble:If H → H − µ(Sz+ ψ†ψ), need only:g2N > (ω − µ)(ω0 − µ)Incoherent pumping — polaritoncondensation.Dissociate g, ω0,e.g. Raman scheme: ω0 ω.[Dimer et al. PRA ’07; Baumann et al. Nature’10. Also, Black et al. PRL ’03 ]κPumpκCavityPumpJonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 7 / 36
  24. 24. Outline1 Introduction: Dicke model and superradiance2 Dynamics of generalized Dicke modelSummary of experiment and classical dynamcsFixed points and dynamical phasesTimescales and consequences for experimentPersistent oscillating phases3 Jaynes Cummings Hubbard modelJCHM vv DickeCoherently driven arrayDisorderJonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 8 / 36
  25. 25. AcknowledgementsGROUP:COLLABORATORS: Simons, Bhaseen, Schmidt, Blatter, T¨ureci, Kr¨ugerEXPERIMENT: Houck, Wallraff, Fink, MylnekFUNDING:Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 9 / 36
  26. 26. Dynamics of generalized Dicke model1 Introduction: Dicke model and superradiance2 Dynamics of generalized Dicke modelSummary of experiment and classical dynamcsFixed points and dynamical phasesTimescales and consequences for experimentPersistent oscillating phases3 Jaynes Cummings Hubbard modelJCHM vv DickeCoherently driven arrayDisorderJonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 10 / 36
  27. 27. Reminder of cold-atom extended Dicke modelκPumpκ2 Level SystemxzΩgψ02 Level system, | ⇓ , | ⇑ :⇓: Ψ(x, z) = 1⇑: Ψ(x, z) =σ,σ =±eik(σx+σ z)H = ωψ†ψ + ω0Sz+ g(ψ + ψ†)(S−+ S+)+USzψ†ψ.∂t ρ = −i[H, ρ]−κ(ψ†ψρ − 2ψρψ†+ ρψ†ψ)[Baumann et al Nature ’10 ]Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 11 / 36
  28. 28. Reminder of cold-atom extended Dicke modelκPumpκ2 Level SystemxzΩgψ02 Level system, | ⇓ , | ⇑ :⇓: Ψ(x, z) = 1⇑: Ψ(x, z) =σ,σ =±eik(σx+σ z)H = ωψ†ψ + ω0Sz+ g(ψ + ψ†)(S−+ S+)+USzψ†ψ.∂t ρ = −i[H, ρ]−κ(ψ†ψρ − 2ψρψ†+ ρψ†ψ)[Baumann et al Nature ’10 ]Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 11 / 36
  29. 29. Reminder of cold-atom extended Dicke modelκPumpκ2 Level SystemxzΩgψ02 Level system, | ⇓ , | ⇑ :⇓: Ψ(x, z) = 1⇑: Ψ(x, z) =σ,σ =±eik(σx+σ z)Feedback: U ∝g20ωc − ωaH = ωψ†ψ + ω0Sz+ g(ψ + ψ†)(S−+ S+)+USzψ†ψ.∂t ρ = −i[H, ρ]−κ(ψ†ψρ − 2ψρψ†+ ρψ†ψ)[Baumann et al Nature ’10 ]Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 11 / 36
  30. 30. Reminder of cold-atom extended Dicke modelκPumpκ2 Level SystemxzΩgψ02 Level system, | ⇓ , | ⇑ :⇓: Ψ(x, z) = 1⇑: Ψ(x, z) =σ,σ =±eik(σx+σ z)Feedback: U ∝g20ωc − ωaH = ωψ†ψ + ω0Sz+ g(ψ + ψ†)(S−+ S+)+USzψ†ψ.∂t ρ = −i[H, ρ]−κ(ψ†ψρ − 2ψρψ†+ ρψ†ψ)ω0 ∼ kHz ω, κ, g√N ∼ MHz. [Baumann et al Nature ’10 ]Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 11 / 36
  31. 31. Reminder of cold-atom extended Dicke modelκPumpκ2 Level SystemxzΩgψ02 Level system, | ⇓ , | ⇑ :⇓: Ψ(x, z) = 1⇑: Ψ(x, z) =σ,σ =±eik(σx+σ z)Feedback: U ∝g20ωc − ωaH = ωψ†ψ + ω0Sz+ g(ψ + ψ†)(S−+ S+)+USzψ†ψ.∂t ρ = −i[H, ρ]−κ(ψ†ψρ − 2ψρψ†+ ρψ†ψ)ω0 ∼ kHz ω, κ, g√N ∼ MHz. [Baumann et al Nature ’10 ]Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 11 / 36
  32. 32. Classical dynamics of the extended Dicke modelOpen dynamical system:H = ωψ†ψ + ω0Sz+ g(ψ + ψ†)(S−+ S+)+USzψ†ψ.∂t ρ = −i[H, ρ]−κ(ψ†ψρ − 2ψρψ†+ ρψ†ψ)Neglects quantum fluctuations — restore via Wigner distributedinitial conditions.Linearisation about fixed point:Recover Retarded Green’s function (spectrum)Cannot recover occupationsJonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 12 / 36
  33. 33. Classical dynamics of the extended Dicke modelOpen dynamical system:H = ωψ†ψ + ω0Sz+ g(ψ + ψ†)(S−+ S+)+USzψ†ψ.∂t ρ = −i[H, ρ]−κ(ψ†ψρ − 2ψρψ†+ ρψ†ψ)Classical EOM(|S| = N/2 1)˙S−= −i(ω0+U|ψ|2)S−+ 2ig(ψ + ψ∗)Sz˙Sz= ig(ψ + ψ∗)(S−− S+)˙ψ = − [κ + i(ω+USz)] ψ − ig(S−+ S+)Neglects quantum fluctuations — restore via Wigner distributedinitial conditions.Linearisation about fixed point:Recover Retarded Green’s function (spectrum)Cannot recover occupationsJonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 12 / 36
  34. 34. Classical dynamics of the extended Dicke modelOpen dynamical system:H = ωψ†ψ + ω0Sz+ g(ψ + ψ†)(S−+ S+)+USzψ†ψ.∂t ρ = −i[H, ρ]−κ(ψ†ψρ − 2ψρψ†+ ρψ†ψ)Classical EOM(|S| = N/2 1)˙S−= −i(ω0+U|ψ|2)S−+ 2ig(ψ + ψ∗)Sz˙Sz= ig(ψ + ψ∗)(S−− S+)˙ψ = − [κ + i(ω+USz)] ψ − ig(S−+ S+)Neglects quantum fluctuations — restore via Wigner distributedinitial conditions.Linearisation about fixed point:Recover Retarded Green’s function (spectrum)Cannot recover occupationsJonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 12 / 36
  35. 35. Classical dynamics of the extended Dicke modelOpen dynamical system:H = ωψ†ψ + ω0Sz+ g(ψ + ψ†)(S−+ S+)+USzψ†ψ.∂t ρ = −i[H, ρ]−κ(ψ†ψρ − 2ψρψ†+ ρψ†ψ)Classical EOM(|S| = N/2 1)˙S−= −i(ω0+U|ψ|2)S−+ 2ig(ψ + ψ∗)Sz˙Sz= ig(ψ + ψ∗)(S−− S+)˙ψ = − [κ + i(ω+USz)] ψ − ig(S−+ S+)Neglects quantum fluctuations — restore via Wigner distributedinitial conditions.Linearisation about fixed point:Recover Retarded Green’s function (spectrum)Cannot recover occupationsJonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 12 / 36
  36. 36. Fixed points (steady states)0 = i(ω0+U|ψ|2)S−+ 2ig(ψ + ψ∗)Sz0 = ig(ψ + ψ∗)(S−− S+)0 = − [κ + i(ω+USz)] ψ − ig(S−+ S+)ψ = 0, S = (0, 0, ±N/2)always a solution.If g > gc, ψ = 0 tooA Sy= − [S−] = 0B ψ = [ψ] = 0Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 13 / 36
  37. 37. Fixed points (steady states)0 = i(ω0+U|ψ|2)S−+ 2ig(ψ + ψ∗)Sz0 = ig(ψ + ψ∗)(S−− S+)0 = − [κ + i(ω+USz)] ψ − ig(S−+ S+)ψ = 0, S = (0, 0, ±N/2)always a solution.If g > gc, ψ = 0 tooA Sy= − [S−] = 0B ψ = [ψ] = 0xSySzSSmall g: ⇑, ⇓ only.(ω = 30MHz, UN = −40MHz)Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 13 / 36
  38. 38. Fixed points (steady states)0 = i(ω0+U|ψ|2)S−+ 2ig(ψ + ψ∗)Sz0 = ig(ψ + ψ∗)(S−− S+)0 = − [κ + i(ω+USz)] ψ − ig(S−+ S+)ψ = 0, S = (0, 0, ±N/2)always a solution.If g > gc, ψ = 0 tooA Sy= − [S−] = 0B ψ = [ψ] = 0xSySzSSmall g: ⇑, ⇓ only. Larger g: SR too.(ω = 30MHz, UN = −40MHz)Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 13 / 36
  39. 39. Steady state phase diagram0 = i(ω0+U|ψ|2)S−+ 2ig(ψ + ψ∗)Sz0 = ig(ψ + ψ∗)(S−− S+)0 = − [κ + i(ω+USz)] ψ − ig(S−+ S+) 00ωg-√NUN=0, κ=0⇓ SRSee also Domokos and Ritsch PRL ’02, Domokos et al. PRL ’10Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 14 / 36
  40. 40. Steady state phase diagram0 = i(ω0+U|ψ|2)S−+ 2ig(ψ + ψ∗)Sz0 = ig(ψ + ψ∗)(S−− S+)0 = − [κ + i(ω+USz)] ψ − ig(S−+ S+) 00ωg-√NUN=0, κ=0⇓ SR-40-20020400 0.5 1 1.5ω(MHz)g√N (MHz)⇓⇑SRSRUN=0⇓ SR(A): Sy = 0See also Domokos and Ritsch PRL ’02, Domokos et al. PRL ’10Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 14 / 36
  41. 41. Steady state phase diagram0 = i(ω0+U|ψ|2)S−+ 2ig(ψ + ψ∗)Sz0 = ig(ψ + ψ∗)(S−− S+)0 = − [κ + i(ω+USz)] ψ − ig(S−+ S+) 00ωg-√NUN=0, κ=0⇓ SR-40-20020400 0.5 1 1.5ω(MHz)g√N (MHz)⇓⇑SRASRA⇓+⇑ SRBUN=-20⇓ SR(A): Sy = 0⇓ + ⇑ SR(B): ψ = 0See also Domokos and Ritsch PRL ’02, Domokos et al. PRL ’10Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 14 / 36
  42. 42. Steady state phase diagram0 = i(ω0+U|ψ|2)S−+ 2ig(ψ + ψ∗)Sz0 = ig(ψ + ψ∗)(S−− S+)0 = − [κ + i(ω+USz)] ψ − ig(S−+ S+) 00ωg-√NUN=0, κ=0⇓ SR-40-20020400 0.5 1 1.5ω(MHz)g√N (MHz)⇓⇑⇓+⇑SRASRASRBUN=-40⇓ SR(A): Sy = 0⇓ + ⇑ SR(B): ψ = 0See also Domokos and Ritsch PRL ’02, Domokos et al. PRL ’10Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 14 / 36
  43. 43. Steady state phase diagram0 = i(ω0+U|ψ|2)S−+ 2ig(ψ + ψ∗)Sz0 = ig(ψ + ψ∗)(S−− S+)0 = − [κ + i(ω+USz)] ψ − ig(S−+ S+) 00ωg-√NUN=0, κ=0⇓ SR-40-20020400 0.5 1 1.5ω(MHz)g√N (MHz)⇓⇑SRASRA⇓+⇑ SRBSRB+⇑SRB+⇑SRA+⇑SRA+⇓SRB+⇓+⇑UN=-40⇓ SR(A): Sy = 0⇓ + ⇑ SR(B): ψ = 0See also Domokos and Ritsch PRL ’02, Domokos et al. PRL ’10Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 14 / 36
  44. 44. Comparison to experiment-40-30-20-1000 0.5 1 1.5 2 2.5(ωp-ωc)(2πMHz)g2N (MHz)2UN = −10MHzAdapted from: [Bhaseen et al. PRA ’12][Baumann et al Nature ’10 ]ω = ωc − ωp +52UN, UN = −g204(ωa − ωc)Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 15 / 36
  45. 45. Dynamics of generalized Dicke model1 Introduction: Dicke model and superradiance2 Dynamics of generalized Dicke modelSummary of experiment and classical dynamcsFixed points and dynamical phasesTimescales and consequences for experimentPersistent oscillating phases3 Jaynes Cummings Hubbard modelJCHM vv DickeCoherently driven arrayDisorderJonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 16 / 36
  46. 46. Dynamics: Evolution from normal stateGray: S = (√N,√N, −N/2)Black: Wigner distribution of S, ψ-40-20020400 0.5 1 1.5ω(MHz)g√N (MHz)(i)(ii)(iii)⇓⇑⇓+⇑SRASRASRBUN=-40Oscillations: ∼ 0.1msDecay: 20ms, 0.1ms, 20ms(i) SR(A)0 20 40 60 80t (ms)04080|ψ|20 1 20100(ii) SR(B)0 0.1 0.2 0.3 0.4t (ms)0100200|ψ|2(iii) SR(A)0 100 200t (ms)04080120|ψ|2150 1514050Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 17 / 36
  47. 47. Asymptotic state: Evolution from normal state(Near to experimental UN = −13MHz).All stable attractors:-40-20020400 0.5 1 1.5ω(MHz)g√N (MHz)⇑⇓ SRASRA⇓+⇑ SRBUN=-10Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 18 / 36
  48. 48. Asymptotic state: Evolution from normal state(Near to experimental UN = −13MHz).All stable attractors:-40-20020400 0.5 1 1.5ω(MHz)g√N (MHz)⇑⇓ SRASRA⇓+⇑ SRBUN=-10Starting from ⇓-40-20020400 0.5 1 1.5ω(MHz)g√N (MHz)⇑⇓SRASRBSRA10-1100101102103|ψ|2Asymptotic stateJonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 18 / 36
  49. 49. Timescales for dynamics: Consequences forexperiment-40-20020400 0.5 1 1.5ω(MHz)g√N (MHz)⇑⇓SRASRBSRA10-1100101102103|ψ|2Asymptotic stateJonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 19 / 36
  50. 50. Timescales for dynamics: Consequences forexperiment-40-20020400 0.5 1 1.5ω(MHz)g√N (MHz)⇑⇓SRASRBSRA10-1100101102103|ψ|2Asymptotic state-40-2002040600.0 0.5 1.0 1.5 2.0 2.5ω(MHz)g2N (MHz2)10-110010110210310ms sweepJonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 19 / 36
  51. 51. Timescales for dynamics: Consequences forexperiment-40-20020400 0.5 1 1.5ω(MHz)g√N (MHz)⇑⇓SRASRBSRA10-1100101102103|ψ|2Asymptotic state-40-2002040600.0 0.5 1.0 1.5 2.0 2.5ω(MHz)g2N (MHz2)10-110010110210310ms sweep-40-2002040600.0 0.5 1.0 1.5 2.0 2.5ω(MHz)g2N (MHz2)10-1100101102103200ms sweepJonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 19 / 36
  52. 52. Timescales for dynamics: What are they?-40-20020400 0.5 1 1.5ω(MHz)g√N (MHz)⇑⇓SRASRBSRA10-1100101102103|ψ|2Asymptotic stateGrowth Most unstable eigenvaluesnear S = (0, 0, −N/2)Decay Slowest stable eigenvaluesnear final state-40-20020400 0.5 1 1.5ω(MHz)g√N (MHz)No unstabledirectionsTwo unstable directionsOne unstable direction10µs100µs1ms10ms100ms1s10sInitial growth timeJonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 20 / 36
  53. 53. Timescales for dynamics: What are they?-40-20020400 0.5 1 1.5ω(MHz)g√N (MHz)⇑⇓SRASRBSRA10-1100101102103|ψ|2Asymptotic stateGrowth Most unstable eigenvaluesnear S = (0, 0, −N/2)Decay Slowest stable eigenvaluesnear final state-40-20020400 0.5 1 1.5ω(MHz)g√N (MHz)No unstabledirectionsTwo unstable directionsOne unstable direction10µs100µs1ms10ms100ms1s10sInitial growth time-40-20020400 0.5 1 1.5ω(MHz)g√N (MHz)⇑⇓SRASRBSRA10µs100µs1ms10ms100ms1s10sAsymptotic decay timeJonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 20 / 36
  54. 54. Timescales for dynamics: Why so slow and varied?Suppose co- and counter-rotating terms differ2 Level SystemΩ∆Ωψba ba∆ψg0g0H = . . . + g(ψ†S−+ ψS+) + g (ψ†S++ ψS−) + . . .SR(A) near phase boundary at small δg → Critical slowing downSR(A), SR(B) continuously connectJonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 21 / 36
  55. 55. Timescales for dynamics: Why so slow and varied?Suppose co- and counter-rotating terms differ2 Level SystemΩ∆Ωψba ba∆ψg0g0H = . . . + g(ψ†S−+ ψS+) + g (ψ†S++ ψS−) + . . .-40-20020400 0.5 1 1.5ω(MHz)g√N (MHz)⇑⇓ SRASRA⇓+⇑ SRBUN=-10SR(A) near phase boundary at small δg → Critical slowing downSR(A), SR(B) continuously connectJonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 21 / 36
  56. 56. Timescales for dynamics: Why so slow and varied?Suppose co- and counter-rotating terms differ2 Level SystemΩ∆Ωψba ba∆ψg0g0H = . . . + g(ψ†S−+ ψS+) + g (ψ†S++ ψS−) + . . .δg = g − g, 2¯g = g + g-40-20020400 0.5 1 1.5ω(MHz)g√N (MHz)⇑⇓ SRASRA⇓+⇑ SRBUN=-10-40-2002040-0.01 -0.005 0 0.005 0.01ω(MHz)δg/g-g-√N=1SR(A) near phase boundary at small δg → Critical slowing downSR(A), SR(B) continuously connectJonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 21 / 36
  57. 57. Timescales for dynamics: Why so slow and varied?Suppose co- and counter-rotating terms differ2 Level SystemΩ∆Ωψba ba∆ψg0g0H = . . . + g(ψ†S−+ ψS+) + g (ψ†S++ ψS−) + . . .δg = g − g, 2¯g = g + g-40-20020400 0.5 1 1.5ω(MHz)g√N (MHz)⇑⇓ SRASRA⇓+⇑ SRBUN=-10-40-2002040-0.01 -0.005 0 0.005 0.01ω(MHz)δg/g-g-√N=1SR(A) near phase boundary at small δg → Critical slowing downSR(A), SR(B) continuously connectJonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 21 / 36
  58. 58. Dynamics of generalized Dicke model1 Introduction: Dicke model and superradiance2 Dynamics of generalized Dicke modelSummary of experiment and classical dynamcsFixed points and dynamical phasesTimescales and consequences for experimentPersistent oscillating phases3 Jaynes Cummings Hubbard modelJCHM vv DickeCoherently driven arrayDisorderJonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 22 / 36
  59. 59. Regions without fixed pointsChanging U:2 Level SystemΩgψ0U ∝g20ωc − ωa-40-20020400 0.5 1 1.5ω(MHz)g√N (MHz)⇓⇑⇓+⇑SRASRASRBUN=-40Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 23 / 36
  60. 60. Regions without fixed pointsChanging U:2 Level SystemΩψg0U ∝g20ωc − ωa-40-20020400 0.5 1 1.5ω(MHz)g√N (MHz)⇓⇑⇓+⇑SRASRASRBUN=-40Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 23 / 36
  61. 61. Regions without fixed pointsChanging U:2 Level SystemΩψg0U ∝g20ωc − ωa-40-20020400 0.5 1 1.5ω(MHz)g√N (MHz)⇓⇑SRSRUN=0Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 23 / 36
  62. 62. Regions without fixed pointsChanging U:2 Level SystemΩψg0U ∝g20ωc − ωa-40-20020400 0.5 1 1.5ω(MHz)g√N (MHz)⇓⇑SRASRAUN=20Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 23 / 36
  63. 63. Regions without fixed pointsChanging U:2 Level SystemΩψg0U ∝g20ωc − ωa-40-20020400 0.5 1 1.5ω(MHz)g√N (MHz)⇓⇑SRASRAUN=40Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 23 / 36
  64. 64. Regions without fixed pointsChanging U:2 Level SystemΩψg0U ∝g20ωc − ωa-40-20020400 0.5 1 1.5ω(MHz)g√N (MHz)⇓⇑SRASRAPersistent OscillationsUN=400 2 4 6 8 10 12 14 16 18t (ms)020040060080010001200|ψ|20 5 10 1504008001200Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 23 / 36
  65. 65. Persistent (optomechanical) oscillations-40-20020400 0.5 1 1.5ω(MHz)g√N (MHz)⇓⇑SRASRAPersistent OscillationsUN=400 2 4 6 8 10 12 14 16 18t (ms)020040060080010001200|ψ|20 5 10 150400800120002004006008001000120018.00 18.02 18.04 18.06 18.08-0.4-0.200.20.4|ψ|2Sx,Sy,Szt(ms)|ψ|2SxSySzJonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 24 / 36
  66. 66. Jaynes Cummings Hubbard model1 Introduction: Dicke model and superradiance2 Dynamics of generalized Dicke modelSummary of experiment and classical dynamcsFixed points and dynamical phasesTimescales and consequences for experimentPersistent oscillating phases3 Jaynes Cummings Hubbard modelJCHM vv DickeCoherently driven arrayDisorderJonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 25 / 36
  67. 67. Equilibrium: Dicke model with chemical potentialH − µN = (ω − µ)ψ†ψ + (ω0 − µ)Sz+ g ψ†S−+ ψS+-5-4-3-2-1012-4 -3 -2 -1 0 1 2(µ-ω)/g(ω0 - ω)/gSRTransition at:g2N > (ω − µ)|ω0 − µ|Reduce critical gUnstable if µ > ωInverted if µ > ω0[Eastham and Littlewood, PRB ’01]Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 26 / 36
  68. 68. Equilibrium: Dicke model with chemical potentialH − µN = (ω − µ)ψ†ψ + (ω0 − µ)Sz+ g ψ†S−+ ψS+-5-4-3-2-1012-4 -3 -2 -1 0 1 2(µ-ω)/g(ω0 - ω)/gunstableSRTransition at:g2N > (ω − µ)|ω0 − µ|Reduce critical gUnstable if µ > ωInverted if µ > ω0[Eastham and Littlewood, PRB ’01]Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 26 / 36
  69. 69. Equilibrium: Dicke model with chemical potentialH − µN = (ω − µ)ψ†ψ + (ω0 − µ)Sz+ g ψ†S−+ ψS+-5-4-3-2-1012-4 -3 -2 -1 0 1 2(µ-ω)/g(ω0 - ω)/g⇑⇓unstableSRTransition at:g2N > (ω − µ)|ω0 − µ|Reduce critical gUnstable if µ > ωInverted if µ > ω0[Eastham and Littlewood, PRB ’01]Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 26 / 36
  70. 70. Jaynes-Cummings Hubbard modelH = −Jzijψ†i ψj +i∆2σzi + g(ψ†i σ−i + H.c.)Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 27 / 36
  71. 71. Jaynes-Cummings Hubbard modelH = −Jzijψ†i ψj +i∆2σzi + g(ψ†i σ−i + H.c.)-2-100.001 0.01 0.1 1µ/gJ/gUnstableNormal∆/g=1Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 27 / 36
  72. 72. Jaynes-Cummings Hubbard modelH = −Jzijψ†i ψj +i∆2σzi + g(ψ†i σ−i + H.c.)-2-100.001 0.01 0.1 1µ/gJ/gUnstableNormal∆/g=1-10-8-6-4-200 1 2 3 4 5 6 7 8 9 10µ/gJ/gUnstableNormal∆/g=-6Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 27 / 36
  73. 73. Dicke vs JCHMJCHM-10-8-6-4-200 1 2 3 4 5 6 7 8 9 10µ/gJ/gUnstableNormal∆/g=-6Dicke-5-4-3-2-1012-4 -3 -2 -1 0 1 2(µ-ω)/g(ω0 - ω)/g⇑⇓unstableSRk = 0 mode of JCHM ↔ Dicke photon mode⇑ ↔ n = 1 Mott lobeJonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 28 / 36
  74. 74. Dicke vs JCHMJCHM-10-8-6-4-200 1 2 3 4 5 6 7 8 9 10µ/gJ/gUnstableNormal∆/g=-6Dicke-5-4-3-2-1012-4 -3 -2 -1 0 1 2(µ-ω)/g(ω0 - ω)/g⇑⇓unstableSRk = 0 mode of JCHM ↔ Dicke photon mode⇑ ↔ n = 1 Mott lobeJonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 28 / 36
  75. 75. Dicke vs JCHMJCHM-10-8-6-4-200 1 2 3 4 5 6 7 8 9 10µ/gJ/gUnstableNormal∆/g=-6Dicke-5-4-3-2-1012-4 -3 -2 -1 0 1 2(µ-ω)/g(ω0 - ω)/g⇑⇓unstableSRk = 0 mode of JCHM ↔ Dicke photon mode⇑ ↔ n = 1 Mott lobeJonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 28 / 36
  76. 76. Dicke vs JCHMJCHM-10-8-6-4-200 1 2 3 4 5 6 7 8 9 10µ/gJ/gUnstableNormal∆/g=-6 EkUPPhotonLP2LS∆JCHM∆DickeDicke-5-4-3-2-1012-4 -3 -2 -1 0 1 2(µ-ω)/g(ω0 - ω)/g⇑⇓unstableSRk = 0 mode of JCHM ↔ Dicke photon mode⇑ ↔ n = 1 Mott lobeJonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 28 / 36
  77. 77. Jaynes Cummings Hubbard model1 Introduction: Dicke model and superradiance2 Dynamics of generalized Dicke modelSummary of experiment and classical dynamcsFixed points and dynamical phasesTimescales and consequences for experimentPersistent oscillating phases3 Jaynes Cummings Hubbard modelJCHM vv DickeCoherently driven arrayDisorderJonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 29 / 36
  78. 78. Coherently pumped JCHMH = −Jzijψ†i ψj +i∆2σzi + g(ψ†i σ−i + H.c.)+f(ψieiωLt+ H.c.)∂t ρ = −i[H, ρ]−κ2Lψ[ρ] −γ2Lσ− [ρ]Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 30 / 36
  79. 79. Coherently pumped single cavity [Bishop et al. Nat. Phys ’09]g H =∆2σz+ g(ψ†σ−+ H.c.)+f(ψeiωpumpt+ H.c.)∂t ρ = −i[H, ρ]−κ2Lψ[ρ] −γ2Lσ− [ρ]Anti-resonance in | ψ |.Effective 2LS:|Empty , |1 polaritonIncreasingPumpingMollow triplet fluorescence[Lang et al. PRL ’11]Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 31 / 36
  80. 80. Coherently pumped single cavity [Bishop et al. Nat. Phys ’09]g H =∆2σz+ g(ψ†σ−+ H.c.)+f(ψeiωpumpt+ H.c.)∂t ρ = −i[H, ρ]−κ2Lψ[ρ] −γ2Lσ− [ρ]Anti-resonance in | ψ |.Effective 2LS:|Empty , |1 polaritonIncreasingPumpingMollow triplet fluorescence[Lang et al. PRL ’11]Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 31 / 36
  81. 81. Coherently pumped single cavity [Bishop et al. Nat. Phys ’09]g H =∆2σz+ g(ψ†σ−+ H.c.)+f(ψeiωpumpt+ H.c.)∂t ρ = −i[H, ρ]−κ2Lψ[ρ] −γ2Lσ− [ρ]Anti-resonance in | ψ |.Effective 2LS:|Empty , |1 polaritonIncreasingPumpingMollow triplet fluorescence[Lang et al. PRL ’11]Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 31 / 36
  82. 82. Coherently pumped dimer & arrayChose detuning a la Dicke modelωpumpωpumpLPUP2gCavityQubitSingle cavity2JLPLPUPEkPhotonQubitArray2gBistability at intermediate JMore/less localised statesConnects to Dicke limit[Nissen et al. PRL ’12]Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 32 / 36
  83. 83. Coherently pumped dimer & arrayChose detuning a la Dicke modelωpumpωpumpLPUP2gCavityQubitSingle cavity2JLPLPUPEkPhotonQubitArray2gEvolution of anti-resonance vs J.00.10.2-1.06 -1.04 -1.02 -1|<a>|ωpump/gBistability at intermediate JMore/less localised statesConnects to Dicke limit[Nissen et al. PRL ’12]Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 32 / 36
  84. 84. Coherently pumped dimer & arrayChose detuning a la Dicke modelωpumpωpumpLPUP2gCavityQubitSingle cavity2JLPLPUPEkPhotonQubitArray2gEvolution of anti-resonance vs J.00.10.2-1.06 -1.04 -1.02 -1|<a>|ωpump/gBistability at intermediate JMore/less localised statesConnects to Dicke limit[Nissen et al. PRL ’12]Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 32 / 36
  85. 85. Coherently pumped dimer & arrayChose detuning a la Dicke modelωpumpωpumpLPUP2gCavityQubitSingle cavity2JLPLPUPEkPhotonQubitArray2gEvolution of anti-resonance vs J.00.10.2-1.06 -1.04 -1.02 -1|<a>|ωpump/gBistability at intermediate JMore/less localised statesConnects to Dicke limit[Nissen et al. PRL ’12]Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 32 / 36
  86. 86. Coherently pumped dimer & arrayChose detuning a la Dicke modelωpumpωpumpLPUP2gCavityQubitSingle cavity2JLPLPUPEkPhotonQubitArray2gEvolution of anti-resonance vs J.00.10.2-1.06 -1.04 -1.02 -1|<a>|ωpump/gBistability at intermediate JMore/less localised statesConnects to Dicke limit[Nissen et al. PRL ’12]Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 32 / 36
  87. 87. Photon blockade picture J gPolariton basisNonlinearity | 2 − 2 1| ∝ g.H =i2τzi + ˜fτxiDecouple hopping:τ+i τ−j → ψτ+ + ψ∗τ−Bistability forJ > Jc =4˜f22˜f2 + (˜κ/2)233/200.10.2-1.06 -1.04 -1.02 -1|<a>|ωpump/g[Nissen et al. PRL ’12]Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 33 / 36
  88. 88. Photon blockade picture J gPolariton basisNonlinearity | 2 − 2 1| ∝ g.H =i2τzi + ˜fτxi −˜Jzijτ+i τ−jDecouple hopping:τ+i τ−j → ψτ+ + ψ∗τ−Bistability forJ > Jc =4˜f22˜f2 + (˜κ/2)233/200.10.2-1.06 -1.04 -1.02 -1|<a>|ωpump/g[Nissen et al. PRL ’12]Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 33 / 36
  89. 89. Photon blockade picture J gPolariton basisNonlinearity | 2 − 2 1| ∝ g.H =i2τzi + ˜fτxi −˜Jzijτ+i τ−jDecouple hopping:τ+i τ−j → ψτ+ + ψ∗τ−Bistability forJ > Jc =4˜f22˜f2 + (˜κ/2)233/200.10.2-1.06 -1.04 -1.02 -1|<a>|ωpump/g[Nissen et al. PRL ’12]Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 33 / 36
  90. 90. Photon blockade picture J gPolariton basisNonlinearity | 2 − 2 1| ∝ g.H =i2τzi + ˜fτxi −˜Jzijτ+i τ−jDecouple hopping:τ+i τ−j → ψτ+ + ψ∗τ−Bistability forJ > Jc =4˜f22˜f2 + (˜κ/2)233/200.10.2-1.06 -1.04 -1.02 -1|<a>|ωpump/g[Nissen et al. PRL ’12]Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 33 / 36
  91. 91. Coherently pumped array: correlations & fluorescence00.51g2(t=0)0.10.20.30.001 0.01 0.1 1 10|〈a〉|Hopping zJ/gCorrelationsg2 : 0 → 1 crossover.Small J: Mollow tripletLarge J: Off resonancefluorescencePump at collectiveresonanceMismatch if J = 0.Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 34 / 36
  92. 92. Coherently pumped array: correlations & fluorescence00.51g2(t=0)0.10.20.30.001 0.01 0.1 1 10|〈a〉|Hopping zJ/gCorrelationsg2 : 0 → 1 crossover.Small J: Mollow tripletLarge J: Off resonancefluorescencePump at collectiveresonanceMismatch if J = 0.Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 34 / 36
  93. 93. Coherently pumped array: correlations & fluorescence00.51g2(t=0)0.10.20.30.001 0.01 0.1 1 10|〈a〉|Hopping zJ/gCorrelationsg2 : 0 → 1 crossover.FluorescenceSmall J: Mollow tripletLarge J: Off resonancefluorescencePump at collectiveresonanceMismatch if J = 0.Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 34 / 36
  94. 94. Coherently pumped array: correlations & fluorescence00.51g2(t=0)0.10.20.30.001 0.01 0.1 1 10|〈a〉|Hopping zJ/gCorrelationsg2 : 0 → 1 crossover.FluorescenceSmall J: Mollow tripletLarge J: Off resonancefluorescencePump at collectiveresonanceMismatch if J = 0.Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 34 / 36
  95. 95. Coherently pumped array: correlations & fluorescence00.51g2(t=0)0.10.20.30.001 0.01 0.1 1 10|〈a〉|Hopping zJ/gCorrelationsg2 : 0 → 1 crossover.FluorescenceSmall J: Mollow tripletLarge J: Off resonancefluorescencePump at collectiveresonanceMismatch if J = 0.Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 34 / 36
  96. 96. Coherently pumped array: correlations & fluorescence00.51g2(t=0)0.10.20.30.001 0.01 0.1 1 10|〈a〉|Hopping zJ/gCorrelationsg2 : 0 → 1 crossover.FluorescenceSmall J: Mollow tripletLarge J: Off resonancefluorescencePump at collectiveresonanceMismatch if J = 0.Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 34 / 36
  97. 97. Coherent pumped array – disorderEffect of disorder, ∆ → ∆iDistribution of ψ – Washes out bistable jumpBistability near resonance — phase of ψ depends on ∆iComplex ψ distributionSuperfluid phases in driven system?-1.02 -1.01 -1 -0.99 -0.98 -0.97 -0.96 -0.95Pump frequency00.10.20.3ψ020406080100||[Kulaitis et al. PRA, ’13]Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 35 / 36
  98. 98. Coherent pumped array – disorderEffect of disorder, ∆ → ∆iDistribution of ψ – Washes out bistable jumpBistability near resonance — phase of ψ depends on ∆iComplex ψ distributionSuperfluid phases in driven system?-1.02 -1.01 -1 -0.99 -0.98 -0.97 -0.96 -0.95Pump frequency00.10.20.3ψ020406080100||[Kulaitis et al. PRA, ’13]Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 35 / 36
  99. 99. Coherent pumped array – disorderEffect of disorder, ∆ → ∆iDistribution of ψ – Washes out bistable jumpBistability near resonance — phase of ψ depends on ∆iComplex ψ distributionSuperfluid phases in driven system?-1.02 -1.01 -1 -0.99 -0.98 -0.97 -0.96 -0.95Pump frequency00.10.20.3ψ020406080100||-0.200.2(a) ωp=-0.988 (b) ωp=-0.987 (c) ωp=-0.986-0.200.2(d) ωp=-0.985 (e) ωp=-0.982 (f) ωp=-0.978-0.2 0 0.2-0.200.2020406080100(g) ωp=-0.975-0.2 0 0.2(h) ωp=-0.971-0.2 0 0.2(i) ωp=-0.968Re( )Im()[Kulaitis et al. PRA, ’13]Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 35 / 36
  100. 100. Coherent pumped array – disorderEffect of disorder, ∆ → ∆iDistribution of ψ – Washes out bistable jumpBistability near resonance — phase of ψ depends on ∆iComplex ψ distributionSuperfluid phases in driven system?-1.02 -1.01 -1 -0.99 -0.98 -0.97 -0.96 -0.95Pump frequency00.10.20.3ψ020406080100||-0.200.2(a) ωp=-0.988 (b) ωp=-0.987 (c) ωp=-0.986-0.200.2(d) ωp=-0.985 (e) ωp=-0.982 (f) ωp=-0.978-0.2 0 0.2-0.200.2020406080100(g) ωp=-0.975-0.2 0 0.2(h) ωp=-0.971-0.2 0 0.2(i) ωp=-0.968Re( )Im()[Kulaitis et al. PRA, ’13]Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 35 / 36
  101. 101. SummaryWide variety of dynamical phases-40-20020400 0.5 1 1.5ω(MHz)g√N (MHz)⇓⇑SRASRAUN=40-40-20020400 0.5 1 1.5ω(MHz)g√N (MHz)⇓⇑SRSRUN=0-40-20020400 0.5 1 1.5ω(MHz)g√N (MHz)⇓⇑⇓+⇑SRASRASRBUN=-40-40-2002040-0.01 -0.005 0 0.005 0.01ω(MHz)δg/g-g-√N=1Slow dynamics for U < 0 & Persistent oscillations for U > 00 100 200t (ms)04080120|ψ|2150 1514050-40-20020400 0.5 1 1.5ω(MHz)g√N (MHz)No unstabledirectionsTwo unstable directionsOne unstable direction10µs100µs1ms10ms100ms1s10sInitial growth time-40-20020400 0.5 1 1.5ω(MHz)g√N (MHz)⇑⇓SRASRBSRA10µs100µs1ms10ms100ms1s10sAsymptotic decay time0 2 4 6 8 10 12 14 16 18t (ms)020040060080010001200|ψ|20 5 10 150400800120002004006008001000120018.00 18.02 18.04 18.06 18.08-0.4-0.200.20.4|ψ|2Sx,Sy,Szt(ms)|ψ|2SxSySzJK et al. PRL ’10, Bhaseen et al. PRA ’12Dicke model and JCHM: connection at J → ∞EkUPPhotonLP2LS∆JCHM∆Dicke-10-8-6-4-200 1 2 3 4 5 6 7 8 9 10µ/gJ/gUnstableNormal∆/g=-6-5-4-3-2-1012-4 -3 -2 -1 0 1 2(µ-ω)/g(ω0 - ω)/g⇑⇓unstableSRCoherently pumped coupled cavity arrayωpumpωpumpLPUP2gCavityQubitSingle cavity2JLPLPUPEkPhotonQubitArray2g00.10.2-1.06 -1.04 -1.02 -1|<a>|ωpump/g00.51g2(t=0)0.10.20.30.001 0.01 0.1 1 10|〈a〉|Hopping zJ/g-1.02 -1.01 -1 -0.99 -0.98 -0.97 -0.96 -0.95Pump frequency00.10.20.3ψ020406080100||-0.200.2(a) ωp=-0.988 (b) ωp=-0.987 (c) ωp=-0.986-0.200.2(d) ωp=-0.985 (e) ωp=-0.982 (f) ωp=-0.978-0.2 0 0.2-0.200.2020406080100(g) ωp=-0.975-0.2 0 0.2(h) ωp=-0.971-0.2 0 0.2(i) ωp=-0.968Re( )Im()Nissen et al. PRL ’12, Kulaitis et al. PRA ’13Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 36 / 36
  102. 102. Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 37 / 44
  103. 103. 4 Ferroelectric transition5 Dicke vs JCHM6 Pumping without symmetry breaking7 Collective dephasingJonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 38 / 44
  104. 104. Ferroelectric transitionAtoms in Coulomb gaugeH = ωk a†k ak +i[pi − eA(ri)]2+ VcoulJonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 39 / 44
  105. 105. Ferroelectric transitionAtoms in Coulomb gaugeH = ωk a†k ak +i[pi − eA(ri)]2+ VcoulTwo-level systems — dipole-dipole couplingH = ω0Sz+ ωψ†ψ + g(S++ S−)(ψ + ψ†) + Nζ(ψ + ψ†)2−η(S+− S−)2(nb g2, ζ, η ∝ 1/V).Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 39 / 44
  106. 106. Ferroelectric transitionAtoms in Coulomb gaugeH = ωk a†k ak +i[pi − eA(ri)]2+ VcoulTwo-level systems — dipole-dipole couplingH = ω0Sz+ ωψ†ψ + g(S++ S−)(ψ + ψ†) + Nζ(ψ + ψ†)2−η(S+− S−)2(nb g2, ζ, η ∝ 1/V).Ferroelectric polarisation if ω0 < 2ηNJonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 39 / 44
  107. 107. Ferroelectric transitionAtoms in Coulomb gaugeH = ωk a†k ak +i[pi − eA(ri)]2+ VcoulTwo-level systems — dipole-dipole couplingH = ω0Sz+ ωψ†ψ + g(S++ S−)(ψ + ψ†) + Nζ(ψ + ψ†)2−η(S+− S−)2(nb g2, ζ, η ∝ 1/V).Ferroelectric polarisation if ω0 < 2ηNGauge transform to dipole gauge D · rH = ω0Sz+ ωψ†ψ + ¯g(S+− S−)(ψ − ψ†)“Dicke” transition at ω0 < N¯g2/ω ≡ 2ηNBut, ψ describes electric displacementJonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 39 / 44
  108. 108. Equilibrium: Dicke model with chemical potentialH − µN = (ω − µ)ψ†ψ + (ω0 − µ)Sz+ g ψ†S−+ ψS+-5-4-3-2-1012-4 -3 -2 -1 0 1 2(µ-ω)/g(ω0 - ω)/gSRTransition at:g2N > (ω − µ)|ω0 − µ|Reduce critical gUnstable if µ > ωInverted if µ > ω0[Eastham and Littlewood, PRB ’01]Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 40 / 44
  109. 109. Equilibrium: Dicke model with chemical potentialH − µN = (ω − µ)ψ†ψ + (ω0 − µ)Sz+ g ψ†S−+ ψS+-5-4-3-2-1012-4 -3 -2 -1 0 1 2(µ-ω)/g(ω0 - ω)/gunstableSRTransition at:g2N > (ω − µ)|ω0 − µ|Reduce critical gUnstable if µ > ωInverted if µ > ω0[Eastham and Littlewood, PRB ’01]Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 40 / 44
  110. 110. Equilibrium: Dicke model with chemical potentialH − µN = (ω − µ)ψ†ψ + (ω0 − µ)Sz+ g ψ†S−+ ψS+-5-4-3-2-1012-4 -3 -2 -1 0 1 2(µ-ω)/g(ω0 - ω)/g⇑⇓unstableSRTransition at:g2N > (ω − µ)|ω0 − µ|Reduce critical gUnstable if µ > ωInverted if µ > ω0[Eastham and Littlewood, PRB ’01]Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 40 / 44
  111. 111. Jaynes-Cummings Hubbard modelH = −Jzijψ†i ψj +i∆2σzi + g(ψ†i σ−i + H.c.)Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 41 / 44
  112. 112. Jaynes-Cummings Hubbard modelH = −Jzijψ†i ψj +i∆2σzi + g(ψ†i σ−i + H.c.)-2-100.001 0.01 0.1 1µ/gJ/gUnstableNormal∆/g=1Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 41 / 44
  113. 113. Jaynes-Cummings Hubbard modelH = −Jzijψ†i ψj +i∆2σzi + g(ψ†i σ−i + H.c.)-2-100.001 0.01 0.1 1µ/gJ/gUnstableNormal∆/g=1-10-8-6-4-200 1 2 3 4 5 6 7 8 9 10µ/gJ/gUnstableNormal∆/g=-6Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 41 / 44
  114. 114. Dicke vs JCHMJCHM-10-8-6-4-200 1 2 3 4 5 6 7 8 9 10µ/gJ/gUnstableNormal∆/g=-6Dicke-5-4-3-2-1012-4 -3 -2 -1 0 1 2(µ-ω)/g(ω0 - ω)/g⇑⇓unstableSRk = 0 mode of JCHM ↔ Dicke photon mode⇑ ↔ n = 1 Mott lobeJonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 42 / 44
  115. 115. Dicke vs JCHMJCHM-10-8-6-4-200 1 2 3 4 5 6 7 8 9 10µ/gJ/gUnstableNormal∆/g=-6Dicke-5-4-3-2-1012-4 -3 -2 -1 0 1 2(µ-ω)/g(ω0 - ω)/g⇑⇓unstableSRk = 0 mode of JCHM ↔ Dicke photon mode⇑ ↔ n = 1 Mott lobeJonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 42 / 44
  116. 116. Dicke vs JCHMJCHM-10-8-6-4-200 1 2 3 4 5 6 7 8 9 10µ/gJ/gUnstableNormal∆/g=-6Dicke-5-4-3-2-1012-4 -3 -2 -1 0 1 2(µ-ω)/g(ω0 - ω)/g⇑⇓unstableSRk = 0 mode of JCHM ↔ Dicke photon mode⇑ ↔ n = 1 Mott lobeJonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 42 / 44
  117. 117. Dicke vs JCHMJCHM-10-8-6-4-200 1 2 3 4 5 6 7 8 9 10µ/gJ/gUnstableNormal∆/g=-6 EkUPPhotonLP2LS∆JCHM∆DickeDicke-5-4-3-2-1012-4 -3 -2 -1 0 1 2(µ-ω)/g(ω0 - ω)/g⇑⇓unstableSRk = 0 mode of JCHM ↔ Dicke photon mode⇑ ↔ n = 1 Mott lobeJonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 42 / 44
  118. 118. Raman pumpingHow to pump without breaking symmetryCounter-rotating terms — Raman pumpingAtom proposal [Dimer et al. PRA ’07]Atom experiment [Baumann et al. Nature ’10]Qubit — allowed transitions ∆n = 1Qubit dephasing much bigger than atomJK, T¨ureci, Houck in progressJonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 43 / 44
  119. 119. Raman pumpingHow to pump without breaking symmetryCounter-rotating terms — Raman pumpingAtom proposal [Dimer et al. PRA ’07]Atom experiment [Baumann et al. Nature ’10]Qubit — allowed transitions ∆n = 1Qubit dephasing much bigger than atomJK, T¨ureci, Houck in progressJonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 43 / 44
  120. 120. Raman pumpingHow to pump without breaking symmetryCounter-rotating terms — Raman pumpingAtom proposal [Dimer et al. PRA ’07]Atom experiment [Baumann et al. Nature ’10]Qubit — allowed transitions ∆n = 1Qubit dephasing much bigger than atomTunable-coupling-qubit000110110220gg01ΩΩabPumpCavityJK, T¨ureci, Houck in progressJonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 43 / 44
  121. 121. Raman pumpingHow to pump without breaking symmetryCounter-rotating terms — Raman pumpingAtom proposal [Dimer et al. PRA ’07]Atom experiment [Baumann et al. Nature ’10]Qubit — allowed transitions ∆n = 1Qubit dephasing much bigger than atomTunable-coupling-qubit000110110220gg01ΩΩabPumpCavity0 0.5 1g001234Ωa=Ωb=Ω⇓SR?JK, T¨ureci, Houck in progressJonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 43 / 44
  122. 122. Raman pumpingHow to pump without breaking symmetryCounter-rotating terms — Raman pumpingAtom proposal [Dimer et al. PRA ’07]Atom experiment [Baumann et al. Nature ’10]Qubit — allowed transitions ∆n = 1Qubit dephasing much bigger than atomTunable-coupling-qubit000110110220gg01ΩΩabPumpCavity0 0.5 1g001234Ωa=Ωb=Ω⇓SR?JK, T¨ureci, Houck in progressJonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 43 / 44
  123. 123. Collective dephasingReal environment is not Markovian[Carmichael & Walls JPA ’73] Requirements for correct equilibrium[Ciuti & Carusotto PRA ’09] Dicke SR and emissionCannot assume fixed κ, γPhase transition → soft modesStrong coupling → varying decayJonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 44 / 44
  124. 124. Collective dephasingReal environment is not Markovian[Carmichael & Walls JPA ’73] Requirements for correct equilibrium[Ciuti & Carusotto PRA ’09] Dicke SR and emissionCannot assume fixed κ, γPhase transition → soft modesStrong coupling → varying decayJonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 44 / 44
  125. 125. Collective dephasingReal environment is not Markovian[Carmichael & Walls JPA ’73] Requirements for correct equilibrium[Ciuti & Carusotto PRA ’09] Dicke SR and emissionCannot assume fixed κ, γPhase transition → soft modesStrong coupling → varying decayJonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 44 / 44
  126. 126. Collective dephasingReal environment is not Markovian[Carmichael & Walls JPA ’73] Requirements for correct equilibrium[Ciuti & Carusotto PRA ’09] Dicke SR and emissionCannot assume fixed κ, γPhase transition → soft modesStrong coupling → varying decayJonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 44 / 44
  127. 127. Collective dephasingReal environment is not Markovian[Carmichael & Walls JPA ’73] Requirements for correct equilibrium[Ciuti & Carusotto PRA ’09] Dicke SR and emissionCannot assume fixed κ, γPhase transition → soft modesStrong coupling → varying decayDicke model linewidth:H = ωψ†ψ+Ni=1i2σzi +g σ+i ψ + h.c.+iσziqγq b†q + bq +qβqb†iqbq.[Nissen, Fink et al. arXiv:1302.0665]Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 44 / 44
  128. 128. Collective dephasingReal environment is not Markovian[Carmichael & Walls JPA ’73] Requirements for correct equilibrium[Ciuti & Carusotto PRA ’09] Dicke SR and emissionCannot assume fixed κ, γPhase transition → soft modesStrong coupling → varying decayDicke model linewidth:H = ωψ†ψ+Ni=1i2σzi +g σ+i ψ + h.c.+iσziqγq b†q + bq +qβqb†iqbq.0.0080.010.0120.0141 2 3 4 5linewidth/gnumber of qubits, Nexperimenttheory〈a〉2(a.u.)frequency (a.u.)123[Nissen, Fink et al. arXiv:1302.0665]Jonathan Keeling Non-eqbm. phases of matter-light systems Southhampton, May 2013 44 / 44

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