Pairing Phases of Polaritons

Transcription

1 Pairing Phases of Polaritons Jonathan Keeling University of St Andrews 6 YEARS St Petersburg, March 14 Jonathan Keeling Pairing Phases of Polaritons St Petersburg, March 14 1 / 11

2 Outline 1 Introduction Pairing phases of atoms Feshbach for polaritons Pairing phases for polaritons Phase diagram: Critical detunings Signatures Phase diagram: Critical temperatures Jonathan Keeling Pairing Phases of Polaritons St Petersburg, March 14 / 11

3 Tuning pair interactions Different F z states (F=J + I), tune by E(r ) = const + (µ a µ b )B z Energy "closed" µb r "open" Jonathan Keeling Pairing Phases of Polaritons St Petersburg, March 14 3 / 11

4 Tuning pair interactions Different F z states (F=J + I), tune by E(r ) = const + (µ a µ b )B z Energy "closed" µb r "open" Energy "closed" µb r "open" Jonathan Keeling Pairing Phases of Polaritons St Petersburg, March 14 3 / 11

5 Tuning pair interactions Different F z states (F=J + I), tune by E(r ) = const + (µ a µ b )B z Energy "closed" µb r "open" Energy "closed" µb r "open" Energy "closed" µb r "open" Jonathan Keeling Pairing Phases of Polaritons St Petersburg, March 14 3 / 11

6 Tuning pair interactions Different F z states (F=J + I), tune by E(r ) = const + (µ a µ b )B z Energy "closed" µb r "open" Energy "closed" µb r "open" Energy "closed" µb r "open" E Open channel (atoms) Closed channel Bound state (B-B ) Jonathan Keeling Pairing Phases of Polaritons St Petersburg, March 14 3 / 11

7 Tuning pair interactions Different F z states (F=J + I), tune by E(r ) = const + (µ a µ b )B z Energy "closed" µb r "open" Energy "closed" µb r "open" Energy "closed" µb r "open" E = Repulsive E = Attractive E Open channel (atoms) Closed channel Bound state (B-B ) Jonathan Keeling Pairing Phases of Polaritons St Petersburg, March 14 3 / 11

8 Pairing phases of atoms Fermions From Randeria, at. Phys. 1 o phase transition, BEC-BCS crossover BEC of atoms or pairs Ĥ =... + ˆψ m ˆψ a1 ˆψa + h.c. If ˆψ m, If ˆψ a1, ˆψ a. High density metastability. [Eagles, Leggett, Keldysh, ozières, Randeria,... ] Jonathan Keeling Pairing Phases of Polaritons St Petersburg, March 14 4 / 11

9 Pairing phases of atoms Fermions Bosons T (B B ) From Randeria, at. Phys. 1 o phase transition, BEC-BCS crossover [Eagles, Leggett, Keldysh, ozières, Randeria,... ] BEC of atoms or pairs Ĥ =... + ˆψ m ˆψ a1 ˆψa + h.c. If ˆψ m, If ˆψ a1, ˆψ a. High density metastability. [ozières, St James, Timmermanns, Mueller, Thouless, Radzihovsky, Stoof, Sachdev... ] Jonathan Keeling Pairing Phases of Polaritons St Petersburg, March 14 4 / 11

10 Pairing phases of atoms Fermions Bosons T (B B ) From Randeria, at. Phys. 1 o phase transition, BEC-BCS crossover [Eagles, Leggett, Keldysh, ozières, Randeria,... ] BEC of atoms or pairs Ĥ =... + ˆψ m ˆψ a1 ˆψa + h.c. If ˆψ m, If ˆψ a1, ˆψ a. High density metastability. [ozières, St James, Timmermanns, Mueller, Thouless, Radzihovsky, Stoof, Sachdev... ] Jonathan Keeling Pairing Phases of Polaritons St Petersburg, March 14 4 / 11

11 Polariton Feshbach Hybridisation of bound states: Biexciton: opposite spins (two-species): ω X [ E b Hybridisation with photons: 1 (ωc + ωx ) ] 1 Ω r + δ Control δ change ν, m, Interaction... [Ivanov, Haug, Keldysh 98], [Wouters 7], [Caursotto et al. 1], [Deveaud-Pledran et al. 13] Jonathan Keeling Pairing Phases of Polaritons St Petersburg, March 14 5 / 11

12 Polariton Feshbach Hybridisation of bound states: Biexciton: opposite spins (two-species): ω X [ E b Hybridisation with photons: 1 (ωc + ωx ) ] 1 Ω r + δ Control δ change ν, m, Interaction... [Ivanov, Haug, Keldysh 98], [Wouters 7], [Caursotto et al. 1], [Deveaud-Pledran et al. 13] Jonathan Keeling Pairing Phases of Polaritons St Petersburg, March 14 5 / 11

13 Polariton Feshbach Hybridisation of bound states: Biexciton: opposite spins (two-species): ω X [ E b Hybridisation with photons: 1 (ωc + ωx ) ] 1 Ω r + δ E X ω ω XX LP ω E b ν > E ω X [mev] 6 4 UP ω UP Control δ change ν, m, Interaction... C C ω X X ω δ (Ω R +δ ) 1/ 4 LP 4 k [µm 1 ] LP ω [Ivanov, Haug, Keldysh 98], [Wouters 7], [Caursotto et al. 1], [Deveaud-Pledran et al. 13] Jonathan Keeling Pairing Phases of Polaritons St Petersburg, March 14 5 / 11

14 Polariton Feshbach Hybridisation of bound states: Biexciton: opposite spins (two-species): ω X [ E b Hybridisation with photons: 1 (ωc + ωx ) ] 1 Ω r + δ E ω = X ω LP ω XX E b ν = X [mev] ω E 6 4 UP C X LP 4 k [µm 1 ] 4 ω UP ω C X ω LP ω δ (Ω R +δ ) 1/ Control δ change ν, m, Interaction... [Ivanov, Haug, Keldysh 98], [Wouters 7], [Caursotto et al. 1], [Deveaud-Pledran et al. 13] Jonathan Keeling Pairing Phases of Polaritons St Petersburg, March 14 5 / 11

15 Polariton Feshbach X [mev] ω E 6 4 Hybridisation of bound states: Biexciton: opposite spins (two-species): ω X [ E b Hybridisation with photons: 1 (ωc + ωx ) ] 1 Ω r + δ UP C X LP 4 k [µm 1 ] 4 ω = X ω LP ω XX ω UP ω C X ω LP ω E E b ν = δ (Ω R +δ ) 1/ [mev] ν Control δ change ν, m, Interaction... Bound state δ [mev] [Ivanov, Haug, Keldysh 98], [Wouters 7], [Caursotto et al. 1], [Deveaud-Pledran et al. 13] Jonathan Keeling Pairing Phases of Polaritons St Petersburg, March 14 5 / 11 E b

16 Phase diagram (ground state, T = ) δ [mev] µ [mev] PS 1x1 11 n [cm ] δ <, no biexciton physics standard BEC. Small δ : Fluctuations drive 1st order Larkin-Pikin mechanism, coupling gψ ψ m Large phase-separation region aive resonance δ r = 3.84,replaced by critical end-point at δ > δ r Large δ: - transition always occurs as U mm renormalises energy. Jonathan Keeling Pairing Phases of Polaritons St Petersburg, March 14 6 / 11

17 Phase diagram (ground state, T = ) δ [mev] µ [mev] PS 1x1 11 n [cm ] δ <, no biexciton physics standard BEC. Small δ : Fluctuations drive 1st order Larkin-Pikin mechanism, coupling gψ ψ m Large phase-separation region aive resonance δ r = 3.84,replaced by critical end-point at δ > δ r Large δ: - transition always occurs as U mm renormalises energy. Jonathan Keeling Pairing Phases of Polaritons St Petersburg, March 14 6 / 11

18 Phase diagram (ground state, T = ) δ [mev] µ [mev] PS 1x1 11 n [cm ] δ <, no biexciton physics standard BEC. Small δ : Fluctuations drive 1st order Larkin-Pikin mechanism, coupling gψ ψ m Large phase-separation region aive resonance δ r = 3.84,replaced by critical end-point at δ > δ r Large δ: - transition always occurs as U mm renormalises energy. Jonathan Keeling Pairing Phases of Polaritons St Petersburg, March 14 6 / 11

19 Phase diagram (ground state, T = ) δ [mev] µ [mev] PS 1x1 11 n [cm ] δ <, no biexciton physics standard BEC. Small δ : Fluctuations drive 1st order Larkin-Pikin mechanism, coupling gψ ψ m Large phase-separation region aive resonance δ r = 3.84,replaced by critical end-point at δ > δ r Large δ: - transition always occurs as U mm renormalises energy. Jonathan Keeling Pairing Phases of Polaritons St Petersburg, March 14 6 / 11

20 Phase diagram (ground state, T = ) δ [mev] µ [mev] PS 1x1 11 n [cm ] δ <, no biexciton physics standard BEC. Small δ : Fluctuations drive 1st order Larkin-Pikin mechanism, coupling gψ ψ m Large phase-separation region aive resonance δ r = 3.84,replaced by critical end-point at δ > δ r Large δ: - transition always occurs as U mm renormalises energy. Jonathan Keeling Pairing Phases of Polaritons St Petersburg, March 14 6 / 11

21 Consequences and Signatures 1st-order near resonance phase separation Direct access to polariton phase coherence : standard g σ (1) = ψ σ(r, t)ψ σ (, ). (D, QLRO) Vortex structure novel half vortices, ψ = e im θ, ψ = e im θ, has (m, m ) = (1/, 1/). Previous half-vortex (m, m ) = (1, ) [Lagoudakis et al. Science 9] Jonathan Keeling Pairing Phases of Polaritons St Petersburg, March 14 7 / 11

22 Consequences and Signatures 1st-order near resonance phase separation Direct access to polariton phase coherence : standard g σ (1) = ψ σ(r, t)ψ σ (, ). (D, QLRO) Vortex structure novel half vortices, ψ = e im θ, ψ = e im θ, has (m, m ) = (1/, 1/). Previous half-vortex (m, m ) = (1, ) [Lagoudakis et al. Science 9] Jonathan Keeling Pairing Phases of Polaritons St Petersburg, March 14 7 / 11

23 Consequences and Signatures 1st-order near resonance phase separation Direct access to polariton phase coherence : standard g σ (1) = ψ σ(r, t)ψ σ (, ). (D, QLRO) Vortex structure novel half vortices, ψ = e im θ, ψ = e im θ, has (m, m ) = (1/, 1/). Previous half-vortex (m, m ) = (1, ) [Lagoudakis et al. Science 9] Jonathan Keeling Pairing Phases of Polaritons St Petersburg, March 14 7 / 11

24 Consequences and Signatures 1st-order near resonance phase separation Direct access to polariton phase coherence : standard g σ (1) = ψ σ(r, t)ψ σ (, ). (D, QLRO) Vortex structure novel half vortices, ψ = e im θ, ψ = e im θ, has (m, m ) = (1/, 1/). Previous half-vortex (m, m ) = (1, ) [Lagoudakis et al. Science 9] Jonathan Keeling Pairing Phases of Polaritons St Petersburg, March 14 7 / 11

25 Consequences and Signatures 1st-order near resonance phase separation Direct access to polariton phase coherence : standard g σ (1) = ψ σ(r, t)ψ σ (, ). (D, QLRO) : no atomic coherence, but: APD φ APD A g (1) m = ψ (r, t)ψ (r, t)ψ (, )ψ (, ) Adjustable delay APD B not g see time/space labels θ APD Vortex structure novel half vortices, ψ = e im θ, ψ = e im θ, has (m, m ) = (1/, 1/). Previous half-vortex (m, m ) = (1, ) [Lagoudakis et al. Science 9] Jonathan Keeling Pairing Phases of Polaritons St Petersburg, March 14 7 / 11

26 Consequences and Signatures 1st-order near resonance phase separation Direct access to polariton phase coherence : standard g σ (1) = ψ σ(r, t)ψ σ (, ). (D, QLRO) : no atomic coherence, but: APD φ APD A g (1) m = ψ (r, t)ψ (r, t)ψ (, )ψ (, ) Adjustable delay APD B not g see time/space labels θ APD Vortex structure novel half vortices, ψ = e im θ, ψ = e im θ, has (m, m ) = (1/, 1/). Previous half-vortex (m, m ) = (1, ) [Lagoudakis et al. Science 9] Jonathan Keeling Pairing Phases of Polaritons St Petersburg, March 14 7 / 11

27 Phase diagram, T > T=.58 K µ [mev] PS x1 11 n [cm ] δ [mev] Jonathan Keeling Pairing Phases of Polaritons St Petersburg, March 14 8 / 11

28 Phase diagram, T > T=.58 K PS δ [mev] 3 δ=1 mev PS µ [mev] 11 n [cm ] x1 T [K] µ [mev] 11 x1 n [cm ] Jonathan Keeling Pairing Phases of Polaritons St Petersburg, March 14 8 / 11

29 Phase diagram, T > T [K] 1 3 δ=8.3 mev δ=1 mev PS PS T=.58 K x1 µ [mev] n [cm ] PS δ [mev] T [K] µ [mev] 11 x1 n [cm ] Jonathan Keeling Pairing Phases of Polaritons St Petersburg, March 14 8 / 11

30 Phase diagram, T > T [K] T [K] δ=6.5 mev δ=8.3 mev δ=1 mev PS PS PS T=.58 K x1 µ [mev] n [cm ] PS [mev] δ T [K] µ [mev] 11 x1 n [cm ] Jonathan Keeling Pairing Phases of Polaritons St Petersburg, March 14 8 / 11

31 Evolution of triple point Required T, δ for : Triple point Excitonic fraction c = 1 [ ] δ 1 +. δ Ω: Pure exciton δ + Ω GaAs, need low T/high exciton fraction ZnO, easy to attain Jonathan Keeling Pairing Phases of Polaritons St Petersburg, March 14 9 / 11

32 Evolution of triple point Required T, δ for : Triple point Excitonic fraction c = 1 [ ] δ 1 +. δ Ω: Pure exciton δ + Ω δ [mev] GaAs ZnO T [K] GaAs, need low T/high exciton fraction ZnO, easy to attain T [K] excitonic fraction [%] Jonathan Keeling Pairing Phases of Polaritons St Petersburg, March 14 9 / 11

33 Evolution of triple point Required T, δ for : Triple point Excitonic fraction c = 1 [ ] δ 1 +. δ Ω: Pure exciton δ + Ω δ [mev] GaAs ZnO T [K] GaAs, need low T/high exciton fraction ZnO, easy to attain T [K] excitonic fraction [%] Jonathan Keeling Pairing Phases of Polaritons St Petersburg, March 14 9 / 11

34 Evolution of triple point Required T, δ for : Triple point Excitonic fraction c = 1 [ ] δ 1 +. δ Ω: Pure exciton δ + Ω δ [mev] GaAs ZnO T [K] GaAs, need low T/high exciton fraction ZnO, easy to attain T [K] excitonic fraction [%] Jonathan Keeling Pairing Phases of Polaritons St Petersburg, March 14 9 / 11

35 Acknowledgements GROUP: COLLABORATORS: Francesca Marchetti, UAM FUDIG: Jonathan Keeling Pairing Phases of Polaritons St Petersburg, March 14 1 / 11

36 Adjustable delay φ θ APD APD A B ν Summary Polariton biexciton detuning Feshbach resonance E ω X [mev] 6 4 LP UP C X 4 4 k [µm 1 ] [mev] Bound state δ [mev] Polaritons can show pairing phase δ [mev] µ [mev] PS 1x1 11 n [cm ] T [K] 3 1 δ=1 mev E b µ [mev] PS x1 11 n [cm ] easily attainable for ZnO; for GaAs δ [mev] GaAs ZnO T [K] T [K] excitonic fraction [%] Possible signatures in coherence and vortices APD APD [Marchetti and Keeling, arxiv:138.13] Jonathan Keeling Pairing Phases of Polaritons St Petersburg, March / 11

37 Jonathan Keeling Pairing Phases of Polaritons St Petersburg, March 14 1 /

38 3 Model Exciton spin 4 Calculation details Variational wavefunction Variational MFT 5 More phase diagrams Jonathan Keeling Pairing Phases of Polaritons St Petersburg, March /

39 Exciton-photon model Microscopic model coupled exciton-photon system H = k + σ=±1 σ=±,±1 ( ) k µ ˆX m ˆX kσ kσ + (δ + k ) µ X m C Ω R (Ĉ kσ ˆX kσ + ˆX kσĉkσ)] + ˆX ( σ R + r ) ˆX τ (R r Interaction U XX has exchange structure σ=±1 d rd R σ,σ,τ,τ =±,±1 ) ˆXτ ( R r U XX Ĉ kσĉkσ σ τ τσ (r) ) ˆXσ ( R + r For large Ω R, neglect σ = ± Interaction supports bound states in U+1, 1, 1,+1 XX channel bipolariton B, bipolariton, bound polaritons, but larger exciton fraction. Jonathan Keeling Pairing Phases of Polaritons St Petersburg, March / )

40 Exciton-photon model Microscopic model coupled exciton-photon system H = k + σ=±1 σ=±,±1 ( ) k µ ˆX m ˆX kσ kσ + (δ + k ) µ X m C Ω R (Ĉ kσ ˆX kσ + ˆX kσĉkσ)] + ˆX ( σ R + r ) ˆX τ (R r Interaction U XX has exchange structure σ=±1 d rd R σ,σ,τ,τ =±,±1 ) ˆXτ ( R r U XX Ĉ kσĉkσ σ τ τσ (r) ) ˆXσ ( R + r For large Ω R, neglect σ = ± Interaction supports bound states in U+1, 1, 1,+1 XX channel bipolariton B, bipolariton, bound polaritons, but larger exciton fraction. Jonathan Keeling Pairing Phases of Polaritons St Petersburg, March / )

41 Exciton-photon model Microscopic model coupled exciton-photon system H = k + σ=±1 σ=±,±1 ( ) k µ ˆX m ˆX kσ kσ + (δ + k ) µ X m C Ω R (Ĉ kσ ˆX kσ + ˆX kσĉkσ)] + ˆX ( σ R + r ) ˆX τ (R r Interaction U XX has exchange structure σ=±1 d rd R σ,σ,τ,τ =±,±1 ) ˆXτ ( R r U XX Ĉ kσĉkσ σ τ τσ (r) ) ˆXσ ( R + r For large Ω R, neglect σ = ± Interaction supports bound states in U+1, 1, 1,+1 XX channel bipolariton B, bipolariton, bound polaritons, but larger exciton fraction. Jonathan Keeling Pairing Phases of Polaritons St Petersburg, March / )

42 Exciton-photon model Microscopic model coupled exciton-photon system H = k + σ=±1 σ=±,±1 ( ) k µ ˆX m ˆX kσ kσ + (δ + k ) µ X m C Ω R (Ĉ kσ ˆX kσ + ˆX kσĉkσ)] + ˆX ( σ R + r ) ˆX τ (R r Interaction U XX has exchange structure σ=±1 d rd R σ,σ,τ,τ =±,±1 ) ˆXτ ( R r U XX Ĉ kσĉkσ σ τ τσ (r) ) ˆXσ ( R + r For large Ω R, neglect σ = ± Interaction supports bound states in U+1, 1, 1,+1 XX channel bipolariton B, bipolariton, bound polaritons, but larger exciton fraction. Jonathan Keeling Pairing Phases of Polaritons St Petersburg, March / )

43 Exciton-photon model Microscopic model coupled exciton-photon system H = k + σ=±1 σ=±,±1 ( ) k µ ˆX m ˆX kσ kσ + (δ + k ) µ X m C Ω R (Ĉ kσ ˆX kσ + ˆX kσĉkσ)] + ˆX ( σ R + r ) ˆX τ (R r Interaction U XX has exchange structure σ=±1 d rd R σ,σ,τ,τ =±,±1 ) ˆXτ ( R r U XX Ĉ kσĉkσ σ τ τσ (r) ) ˆXσ ( R + r For large Ω R, neglect σ = ± Interaction supports bound states in U+1, 1, 1,+1 XX channel bipolariton B, bipolariton, bound polaritons, but larger exciton fraction. Jonathan Keeling Pairing Phases of Polaritons St Petersburg, March / )

44 Polariton model H = ( ) k m µ k σ=, + d R σ=,,m ( ) ˆψ ˆψ k σk σk + + ν µ ˆψ m ˆψ mk mk m U σσ ˆψ σ ˆψ σ ˆψ σ ˆψ σ + U ˆψ ˆψ ˆψ ˆψ + g ( ˆψ ˆψ ˆψ ) m + h.c. E Polariton dispersion m, detuning ν, interactions depend on δ Resonance width, dispersion derived from dressed exciton T matrix. E ω X [mev] 6 4 UP C X LP 4 4 k [µm 1 ] X ω ω XX LP ω ω UP ω C ω X ω LP E b ν > δ (Ω R +δ ) 1/ Jonathan Keeling Pairing Phases of Polaritons St Petersburg, March /

45 Polariton model H = ( ) k m µ k σ=, + d R σ=,,m ( ) ˆψ ˆψ k σk σk + + ν µ ˆψ m ˆψ mk mk m U σσ ˆψ σ ˆψ σ ˆψ σ ˆψ σ + U ˆψ ˆψ ˆψ ˆψ + g ( ˆψ ˆψ ˆψ ) m + h.c. E Polariton dispersion m, detuning ν, interactions depend on δ Resonance width, dispersion derived from dressed exciton T matrix. E ω X [mev] 6 4 UP C X LP 4 4 k [µm 1 ] X ω ω XX LP ω ω UP ω C ω X ω LP E b ν > δ (Ω R +δ ) 1/ Jonathan Keeling Pairing Phases of Polaritons St Petersburg, March /

46 Polariton model H = ( ) k m µ k σ=, + d R σ=,,m ( ) ˆψ ˆψ k σk σk + + ν µ ˆψ m ˆψ mk mk m U σσ ˆψ σ ˆψ σ ˆψ σ ˆψ σ + U ˆψ ˆψ ˆψ ˆψ + g ( ˆψ ˆψ ˆψ ) m + h.c. E Polariton dispersion m, detuning ν, interactions depend on δ Resonance width, dispersion derived from dressed exciton T matrix. X [mev] ω E 6 4 UP C X LP 4 k [µm 1 ] 4 X ω LP ω = ω XX ω UP ω C X ω LP ω E b ν = δ (Ω R +δ ) 1/ Jonathan Keeling Pairing Phases of Polaritons St Petersburg, March /

47 Polariton model H = ( ) k m µ k σ=, + d R σ=,,m ( ) ˆψ ˆψ k σk σk + + ν µ ˆψ m ˆψ mk mk m U σσ ˆψ σ ˆψ σ ˆψ σ ˆψ σ + U ˆψ ˆψ ˆψ ˆψ + g ( ˆψ ˆψ ˆψ ) m + h.c. E Polariton dispersion m, detuning ν, interactions depend on δ Resonance width, dispersion derived from dressed exciton T matrix. X [mev] ω E 6 4 UP C X LP 4 k [µm 1 ] 4 X ω LP ω = ω XX ω UP ω C X ω LP ω E b ν = δ (Ω R +δ ) 1/ Jonathan Keeling Pairing Phases of Polaritons St Petersburg, March /

48 Polariton model H = ( ) k m µ k σ=, + d R σ=,,m ( ) ˆψ ˆψ k σk σk + + ν µ ˆψ m ˆψ mk mk m U σσ ˆψ σ ˆψ σ ˆψ σ ˆψσ + U ˆψ ˆψ ˆψ ˆψ + g ( ˆψ ˆψ ˆψ ) m + h.c. ν > Biexciton in continuum m LP /m X E b ~m C /m X δ [mev] ν [mev] ν < Bound biexciton. (Excitonic limit) Jonathan Keeling Pairing Phases of Polaritons St Petersburg, March /

49 Exciton and polariton spin degrees of freedom Photon: two circular polarisation modes Exciton: bound state of electron & hole J = 1 ± 1/ hole (p-orbital), J = 1/ electron Spin orbit splits hole bands, 4 states. Quantum well fixes k z of hole states. Exciton spin states J z = +, +1, 1, Optically active states J z = ±1 Jonathan Keeling Pairing Phases of Polaritons St Petersburg, March /

50 Exciton and polariton spin degrees of freedom Photon: two circular polarisation modes Exciton: bound state of electron & hole E c v J = 1 ± 1/ hole (p-orbital), J = 1/ electron Spin orbit splits hole bands, 4 states. Quantum well fixes k z of hole states. k Exciton spin states J z = +, +1, 1, Optically active states J z = ±1 Jonathan Keeling Pairing Phases of Polaritons St Petersburg, March /

51 Exciton and polariton spin degrees of freedom Photon: two circular polarisation modes Exciton: bound state of electron & hole E c v, J=3/ HH v, J=3/ LH v, J=1/ J = 1 ± 1/ hole (p-orbital), J = 1/ electron Spin orbit splits hole bands, 4 states. Quantum well fixes k z of hole states. k Exciton spin states J z = +, +1, 1, Optically active states J z = ±1 Jonathan Keeling Pairing Phases of Polaritons St Petersburg, March /

52 Exciton and polariton spin degrees of freedom Photon: two circular polarisation modes Exciton: bound state of electron & hole z Cavity E c v, J=3/ HH v, J=3/ LH v, J=1/ J = 1 ± 1/ hole (p-orbital), J = 1/ electron Spin orbit splits hole bands, 4 states. Quantum well fixes k z of hole states. k Exciton spin states J z = +, +1, 1, Optically active states J z = ±1 Jonathan Keeling Pairing Phases of Polaritons St Petersburg, March /

53 Exciton and polariton spin degrees of freedom Photon: two circular polarisation modes Exciton: bound state of electron & hole z Cavity E c v, J=3/ HH v, J=3/ LH v, J=1/ J = 1 ± 1/ hole (p-orbital), J = 1/ electron Spin orbit splits hole bands, 4 states. Quantum well fixes k z of hole states. k Exciton spin states J z = +, +1, 1, Optically active states J z = ±1 Jonathan Keeling Pairing Phases of Polaritons St Petersburg, March /

54 Exciton and polariton spin degrees of freedom Photon: two circular polarisation modes Exciton: bound state of electron & hole z Cavity E c v, J=3/ HH v, J=3/ LH v, J=1/ J = 1 ± 1/ hole (p-orbital), J = 1/ electron Spin orbit splits hole bands, 4 states. Quantum well fixes k z of hole states. k Exciton spin states J z = +, +1, 1, Optically active states J z = ±1 Jonathan Keeling Pairing Phases of Polaritons St Petersburg, March /

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

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

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

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

59 Finite T calculation Finite T minimize free energy Use Feynman-Jensen inequality: [ ] F = k B T ln Tre Ĥ/k BT F MF + Ĥ ĤMF MF Where... MF calculated using ρ = e (F MF ĤMF)/k B T Ansatz ĤMF Variational F(ψ, ψ m, α γ, β γ ). Ĥ MF = γ { + 1 Aψ γ (α γ + β γ ) (ˆb γ + ˆb ) γ k (ˆb kγ ˆb kγ ) ( ɛ kγ + β γ α γ ) ( ) } α γ ˆbkγ. ɛ kγ + β γ ˆb kγ Jonathan Keeling Pairing Phases of Polaritons St Petersburg, March /

60 Finite T calculation Finite T minimize free energy Use Feynman-Jensen inequality: [ ] F = k B T ln Tre Ĥ/k BT F MF + Ĥ ĤMF MF Where... MF calculated using ρ = e (F MF ĤMF)/k B T Ansatz ĤMF Variational F(ψ, ψ m, α γ, β γ ). Ĥ MF = γ { + 1 Aψ γ (α γ + β γ ) (ˆb γ + ˆb ) γ k (ˆb kγ ˆb kγ ) ( ɛ kγ + β γ α γ ) ( ) } α γ ˆbkγ. ɛ kγ + β γ ˆb kγ Jonathan Keeling Pairing Phases of Polaritons St Petersburg, March /

61 Finite T calculation Finite T minimize free energy Use Feynman-Jensen inequality: [ ] F = k B T ln Tre Ĥ/k BT F MF + Ĥ ĤMF MF Where... MF calculated using ρ = e (F MF ĤMF)/k B T Ansatz ĤMF Variational F(ψ, ψ m, α γ, β γ ). Ĥ MF = γ { + 1 Aψ γ (α γ + β γ ) (ˆb γ + ˆb ) γ k (ˆb kγ ˆb kγ ) ( ɛ kγ + β γ α γ ) ( ) } α γ ˆbkγ. ɛ kγ + β γ ˆb kγ Jonathan Keeling Pairing Phases of Polaritons St Petersburg, March /

62 Variational MFT for WIDBG Test validity. WIDBG Ĥ = k VMFT for WIDBG: k m ψ k ψ k + U d rψ ψ ψψ Ĥ MF = Aψ(α + β)(ˆb + ˆb ) + 1 ) (ˆb ( ) ( ) ɛ k ˆb k + β α ˆbk k. α ɛ k + β k ˆb k Compare to D EOS, ρ(µ) = Tf (µ/t ) CF Hartree-Fock-Popov-Bogoluibov method, include Uρ in Σ Jonathan Keeling Pairing Phases of Polaritons St Petersburg, March 14 /

63 Variational MFT for WIDBG Test validity. WIDBG Ĥ = k VMFT for WIDBG: k m ψ k ψ k + U d rψ ψ ψψ Ĥ MF = Aψ(α + β)(ˆb + ˆb ) + 1 ) (ˆb ( ) ( ) ɛ k ˆb k + β α ˆbk k. α ɛ k + β k ˆb k Compare to D EOS, ρ(µ) = Tf (µ/t ) CF Hartree-Fock-Popov-Bogoluibov method, include Uρ in Σ Jonathan Keeling Pairing Phases of Polaritons St Petersburg, March 14 /

64 Variational MFT for WIDBG Test validity. WIDBG Ĥ = k k m ψ k ψ k + U d rψ ψ ψψ VMFT for WIDBG: Ĥ MF = Aψ(α + β)(ˆb + ˆb ) + 1 ) (ˆb ( ) ( 3 ) ɛ k ˆb k + β α ˆbk k. α ɛ k + β 1 k ˆb k Compare to D EOS, ρ(µ) = Tf (µ/t ) Density ρ/t VMFT QMC [PRA ] mu=.1, ε c =1 [a.u.] CF Hartree-Fock-Popov-Bogoluibov method, include Uρ in Σ µ/t Temperature, T [a.u.] Jonathan Keeling Pairing Phases of Polaritons St Petersburg, March 14 /

65 Variational MFT for WIDBG Test validity. WIDBG Ĥ = k k m ψ k ψ k + U d rψ ψ ψψ VMFT for WIDBG: Ĥ MF = Aψ(α + β)(ˆb + ˆb ) + 1 ) (ˆb ( ) ( 3 ) ɛ k ˆb k + β α ˆbk k. α ɛ k + β 1 k ˆb k Compare to D EOS, ρ(µ) = Tf (µ/t ) Density ρ/t VMFT QMC [PRA ] mu=.1, ε c =1 [a.u.] CF Hartree-Fock-Popov-Bogoluibov method, include Uρ in Σ µ/t Temperature, T [a.u.] Jonathan Keeling Pairing Phases of Polaritons St Petersburg, March 14 /

66 Phase diagram, finite temperature δ [mev] T=.58 K µ [mev] PS x1 11 n [cm ] Jonathan Keeling Pairing Phases of Polaritons St Petersburg, March 14 1 /

67 Phase diagram, finite temperature δ [mev] δ [mev] T=.58 K T=1.16 K µ [mev] PS PS n [cm ] x1 11 Jonathan Keeling Pairing Phases of Polaritons St Petersburg, March 14 1 /

68 Phase diagram, finite temperature δ [mev] δ [mev] δ [mev] T=.58 K T=1.16 K T =.3 K µ [mev] PS PS PS x1 11 n [cm ] Jonathan Keeling Pairing Phases of Polaritons St Petersburg, March 14 1 /

69 Phase diagram, finite temperature δ [mev] δ [mev] δ [mev] T=.58 K T=1.16 K T =.3 K µ [mev] PS PS PS x1 11 n [cm ] δ [mev] 5 PS n [cm ] Jonathan Keeling Pairing Phases of Polaritons St Petersburg, March 14 1 /

70 Phase diagram, vs temperature T=.58 K PS δ [mev] 3 δ=1 mev PS µ [mev] 11 n [cm ] x1 T [K] µ [mev] 11 x1 n [cm ] Jonathan Keeling Pairing Phases of Polaritons St Petersburg, March 14 /

71 Phase diagram, vs temperature T [K] 1 3 δ=8.3 mev δ=1 mev PS PS T=.58 K x1 µ [mev] n [cm ] PS δ [mev] T [K] µ [mev] 11 x1 n [cm ] Jonathan Keeling Pairing Phases of Polaritons St Petersburg, March 14 /

72 Phase diagram, vs temperature T [K] T [K] δ=6.5 mev δ=8.3 mev δ=1 mev PS PS PS T=.58 K x1 µ [mev] n [cm ] PS [mev] δ T [K] µ [mev] 11 x1 n [cm ] Jonathan Keeling Pairing Phases of Polaritons St Petersburg, March 14 /