Pairing Phases of Polaritons

Transcription

1 Pairing Phases of Polaritons Jonathan Keeling 1 University of St Andrews 6 YEARS Pisa, January 14 Jonathan Keeling Pairing Phases of Polaritons Pisa, January 14 1 / 34

2 Bose-Einstein condensation: macroscopic occupation Atoms. 1 7 K Polaritons. K [Anderson et al. Science 95] [Kasprzak et al. Nature, 6] Jonathan Keeling Pairing Phases of Polaritons Pisa, January 14 / 34

3 Microcavity polaritons "Cavity" Quantum Wells Jonathan Keeling Pairing Phases of Polaritons Pisa, January 14 3 / 34

4 Microcavity polaritons "Cavity" Quantum Wells Cavity photons: ω k = ω + c k ω + k /m m 1 4 m e Energy n=4 n=3 n= n=1 Bulk Momentum Jonathan Keeling Pairing Phases of Polaritons Pisa, January 14 3 / 34

5 Microcavity polaritons "Cavity" Quantum Wells Cavity photons: ω k = ω + c k ω + k /m m 1 4 m e Energy Photon Exciton In plane momentum Jonathan Keeling Pairing Phases of Polaritons Pisa, January 14 3 / 34

6 Microcavity polaritons "Cavity" Quantum Wells Cavity photons: ω k = ω + c k ω + k /m m 1 4 m e Energy Photon Exciton In plane momentum Jonathan Keeling Pairing Phases of Polaritons Pisa, January 14 3 / 34

7 Microcavity polaritons θ "Cavity" Quantum Wells Cavity Cavity photons: ω k = ω + c k ω + k /m m 1 4 m e Energy Photon Exciton In plane momentum Jonathan Keeling Pairing Phases of Polaritons Pisa, January 14 3 / 34

8 Polariton experiments: occupation and coherence [Kasprzak, et al. Nature, 6] Jonathan Keeling Pairing Phases of Polaritons Pisa, January 14 4 / 34

9 Polariton experiments: occupation and coherence Sample Coherence map: CCD Beam Splitter Tunable Delay + = Retroreflector [Kasprzak, et al. Nature, 6] Jonathan Keeling Pairing Phases of Polaritons Pisa, January 14 4 / 34

10 (Some) other polariton condensation experiments Quantised vortices [Lagoudakis et al. Nat. Phys. 8. Science 9, PRL 1; Sanvitto et al. Nat. Phys. 1; Roumpos et al. Nat. Phys. 1 ] Josephson oscillations [Lagoudakis et al. PRL 1] = + Pattern formation/hydrodynamics [Amo et al. Science 11, Nature 9; Wertz et al. Nat. Phys 1] Jonathan Keeling Pairing Phases of Polaritons Pisa, January 14 5 / 34

11 What can you do beyond BEC (atoms) Jonathan Keeling Pairing Phases of Polaritons Pisa, January 14 6 / 34

12 What can you do beyond BEC (atoms) Optical lattices Feshbach resonance strong interactions Image: Bloch group Energy "closed" µb r "open" a/a bg (B-B )/ Degenerate fermions BEC BCS crossover Images: Jin Group, Zweierlein group Jonathan Keeling Pairing Phases of Polaritons Pisa, January 14 6 / 34

13 What can you do beyond BEC (atoms) Coupled matter-light systems κ κ z x Pump Spinor condensates, spin-orbit, gauges P3/ 66MHz F =3 F =.5GHz F =1 F = 7THz P1/ F = F =1 1.6eV=39THz=78nm S1/ F= 7GHz F=1 Quenches, thermalisation, dynamics. Image: Weiss group//stamper Kurn group Jonathan Keeling Pairing Phases of Polaritons Pisa, January 14 7 / 34

14 What of this can polaritons do? Engineered potentials; lattices Bloch etched micropillars Yamamoto metal lattices Snoke stress traps Krizhanovskii Surface Acoustic Waves TE-TM interaction Spin-Orbit Theory: Malpuech, Rubo Feshbach resonances Theory: Wouters, Carusotto Deveaud, Biexciton resonance Jonathan Keeling Pairing Phases of Polaritons Pisa, January 14 8 / 34

15 What of this can polaritons do? Engineered potentials; lattices Bloch etched micropillars Yamamoto metal lattices Snoke stress traps Krizhanovskii Surface Acoustic Waves TE-TM interaction Spin-Orbit Theory: Malpuech, Rubo Feshbach resonances Theory: Wouters, Carusotto Deveaud, Biexciton resonance This talk: Spin structure, pairing phases of bosons Jonathan Keeling Pairing Phases of Polaritons Pisa, January 14 8 / 34

16 Outline 1 Introduction Pairing phases of atoms: review 3 Modelling polariton pairing Exciton spin structure Microscopic and effective Hamiltonian 4 Ground state phase diagram Origin of multicritical behaviour Signatures of phases 5 Finite temperature phase diagram Variational MFT Required temperatures, detuning Candidate material systems Jonathan Keeling Pairing Phases of Polaritons Pisa, January 14 9 / 34

17 Acknowledgements GROUP: COLLABORATORS: Francesca Marchetti, UAM FUNDING: Jonathan Keeling Pairing Phases of Polaritons Pisa, January 14 1 / 34

18 Outline 1 Introduction Pairing phases of atoms: review 3 Modelling polariton pairing Exciton spin structure Microscopic and effective Hamiltonian 4 Ground state phase diagram Origin of multicritical behaviour Signatures of phases 5 Finite temperature phase diagram Variational MFT Required temperatures, detuning Candidate material systems Jonathan Keeling Pairing Phases of Polaritons Pisa, January / 34

19 Tuning atomic interactions Open/closed channel different F z states (F=J + I) Magnetic field, E(r ) = const + (µ a µ b )B z Hybridisation by hyperfine coupling Energy "closed" µb r "open" Jonathan Keeling Pairing Phases of Polaritons Pisa, January 14 1 / 34

20 Tuning atomic interactions Open/closed channel different F z states (F=J + I) Magnetic field, E(r ) = const + (µ a µ b )B z Hybridisation by hyperfine coupling Energy "closed" µb Energy "closed" µb r r "open" "open" Jonathan Keeling Pairing Phases of Polaritons Pisa, January 14 1 / 34

21 Tuning atomic interactions Open/closed channel different F z states (F=J + I) Magnetic field, E(r ) = const + (µ a µ b )B z Hybridisation by hyperfine coupling Energy "closed" µb Energy "closed" µb Energy "closed" µb r r r "open" "open" "open" Jonathan Keeling Pairing Phases of Polaritons Pisa, January 14 1 / 34

22 Tuning atomic interactions Open/closed channel different F z states (F=J + I) Magnetic field, E(r ) = const + (µ a µ b )B z Hybridisation by hyperfine coupling Energy "closed" µb Energy "closed" µb Energy "closed" µb r r r "open" "open" "open" E Closed channel (atoms) Open channel Bound state (B-B ) Jonathan Keeling Pairing Phases of Polaritons Pisa, January 14 1 / 34

23 Feshbach resonance B < B Energy "closed" µb r "open" True bound state exists. Open channel a > (repulsive) Resonance a Open channel a < (attractive) Hybridisation with continuum: Fano Feshbach resonance Jonathan Keeling Pairing Phases of Polaritons Pisa, January / 34

24 Feshbach resonance B < B Energy "closed" µb r "open" True bound state exists. Open channel a > (repulsive) B = B Energy "closed" µb r "open" Resonance a Open channel a < (attractive) Hybridisation with continuum: Fano Feshbach resonance Jonathan Keeling Pairing Phases of Polaritons Pisa, January / 34

25 Feshbach resonance B < B Energy "closed" µb r "open" True bound state exists. Open channel a > (repulsive) B = B Energy "closed" µb r "open" Resonance a B > B Energy "closed" µb r "open" Open channel a < (attractive) Hybridisation with continuum: Fano Feshbach resonance Jonathan Keeling Pairing Phases of Polaritons Pisa, January / 34

26 Pairing phases of Fermions Attractive interaction of open channel: BCS condensate. True bound state. Molecule BEC Always pair coherence: No phase transition (for s-wave). BEC-BCS crossover [Eagles, Leggett, Keldysh, Nozières, Randeria,... ] From Randeria, Nat. Phys. News & Views 1 Jonathan Keeling Pairing Phases of Polaritons Pisa, January / 34

27 Pairing phases of Fermions Attractive interaction of open channel: BCS condensate. True bound state. Molecule BEC Always pair coherence: No phase transition (for s-wave). BEC-BCS crossover [Eagles, Leggett, Keldysh, Nozières, Randeria,... ] From Randeria, Nat. Phys. News & Views 1 Jonathan Keeling Pairing Phases of Polaritons Pisa, January / 34

28 Pairing phases of Fermions Attractive interaction of open channel: BCS condensate. True bound state. Molecule BEC Always pair coherence: No phase transition (for s-wave). BEC-BCS crossover [Eagles, Leggett, Keldysh, Nozières, Randeria,... ] From Randeria, Nat. Phys. News & Views 1 Jonathan Keeling Pairing Phases of Polaritons Pisa, January / 34

29 Pairing phases of Fermions Attractive interaction of open channel: BCS condensate. True bound state. Molecule BEC Always pair coherence: No phase transition (for s-wave). BEC-BCS crossover [Eagles, Leggett, Keldysh, Nozières, Randeria,... ] From Randeria, Nat. Phys. News & Views 1 Jonathan Keeling Pairing Phases of Polaritons Pisa, January / 34

30 Pairing phases of Bosons Pair (molecule) coherence,, vs atom coherence ASF, vs both A. Homonuclear/heteronuclear cases distinct, symmetry U(1) Z vs U(1) U(1). T A (B B ) Heteronuclear: Ĥ =... + ˆψ m ˆψ a1 ˆψ a + h.c. If ˆψm, free atomic phase. If ˆψ a1, ˆψ a no free phase. A Homonuclear: Ĥ =... + ˆψ m ˆψ a ˆψa + h.c. If ˆψ m, free atomic sign Z. [Radzihovsky et al. PRL 4, Romans et al. PRL 4, building on Nozières & St James, Timmermanns, Mueller, Thouless... ] Jonathan Keeling Pairing Phases of Polaritons Pisa, January / 34

31 Pairing phases of Bosons Pair (molecule) coherence,, vs atom coherence ASF, vs both A. Homonuclear/heteronuclear cases distinct, symmetry U(1) Z vs U(1) U(1). T A (B B ) Heteronuclear: Ĥ =... + ˆψ m ˆψ a1 ˆψ a + h.c. If ˆψm, free atomic phase. If ˆψ a1, ˆψ a no free phase. A Homonuclear: Ĥ =... + ˆψ m ˆψ a ˆψa + h.c. If ˆψ m, free atomic sign Z. [Radzihovsky et al. PRL 4, Romans et al. PRL 4, building on Nozières & St James, Timmermanns, Mueller, Thouless... ] Jonathan Keeling Pairing Phases of Polaritons Pisa, January / 34

32 Pairing phases of Bosons Pair (molecule) coherence,, vs atom coherence ASF, vs both A. Homonuclear/heteronuclear cases distinct, symmetry U(1) Z vs U(1) U(1). T A (B B ) Heteronuclear: Ĥ =... + ˆψ m ˆψ a1 ˆψ a + h.c. If ˆψm, free atomic phase. If ˆψ a1, ˆψ a no free phase. A Homonuclear: Ĥ =... + ˆψ m ˆψ a ˆψa + h.c. If ˆψ m, free atomic sign Z. [Radzihovsky et al. PRL 4, Romans et al. PRL 4, building on Nozières & St James, Timmermanns, Mueller, Thouless... ] Jonathan Keeling Pairing Phases of Polaritons Pisa, January / 34

33 Pairing phases of Bosons Pair (molecule) coherence,, vs atom coherence ASF, vs both A. Homonuclear/heteronuclear cases distinct, symmetry U(1) Z vs U(1) U(1). T A (B B ) Heteronuclear: Ĥ =... + ˆψ m ˆψ a1 ˆψ a + h.c. If ˆψm, free atomic phase. If ˆψ a1, ˆψ a no free phase. A Homonuclear: Ĥ =... + ˆψ m ˆψ a ˆψa + h.c. If ˆψ m, free atomic sign Z. [Radzihovsky et al. PRL 4, Romans et al. PRL 4, building on Nozières & St James, Timmermanns, Mueller, Thouless... ] Jonathan Keeling Pairing Phases of Polaritons Pisa, January / 34

34 Problems with cold atoms Signatures? Momentum distribution indirect Half vortices T Cold atoms (meta)stability issues Enhanced scattering at resonance First order transitions high densities Vibrationally hot molecules A (B B ) Jonathan Keeling Pairing Phases of Polaritons Pisa, January / 34

35 Problems with cold atoms Signatures? Momentum distribution indirect Half vortices T Cold atoms (meta)stability issues Enhanced scattering at resonance First order transitions high densities Vibrationally hot molecules A (B B ) Jonathan Keeling Pairing Phases of Polaritons Pisa, January / 34

36 Problems with cold atoms Signatures? Momentum distribution indirect Half vortices T Cold atoms (meta)stability issues Enhanced scattering at resonance First order transitions high densities Vibrationally hot molecules A (B B ) Jonathan Keeling Pairing Phases of Polaritons Pisa, January / 34

37 Outline 1 Introduction Pairing phases of atoms: review 3 Modelling polariton pairing Exciton spin structure Microscopic and effective Hamiltonian 4 Ground state phase diagram Origin of multicritical behaviour Signatures of phases 5 Finite temperature phase diagram Variational MFT Required temperatures, detuning Candidate material systems Jonathan Keeling Pairing Phases of Polaritons Pisa, January / 34

38 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 Pisa, January / 34

39 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 Pisa, January / 34

40 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 Pisa, January / 34

41 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 Pisa, January / 34

42 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 Pisa, January / 34

43 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 Pisa, January / 34

44 Exciton-photon model Microscopic model coupled exciton-photon system H = k + σ=±1 σ=±,±1 ( ) k µ ˆX m ˆX kσ kσ + (δ + k ) µ X m C Ω R (Ĉ kσ ˆX kσ + ˆX kσĉkσ)] + ˆX ( σ R + r ) ˆX τ (R r Interaction U XX has exchange structure σ=±1 d rd R σ,σ,τ,τ =±,±1 ) ˆXτ ( R r U XX Ĉ kσĉkσ σ τ τσ (r) ) ˆXσ ( R + r For large Ω R, neglect σ = ± Interaction supports bound states in U+1, 1, 1,+1 XX channel bipolariton NB, bipolariton, bound polaritons, but larger exciton fraction. Jonathan Keeling Pairing Phases of Polaritons Pisa, January / 34 )

45 Exciton-photon model Microscopic model coupled exciton-photon system H = k + σ=±1 σ=±,±1 ( ) k µ ˆX m ˆX kσ kσ + (δ + k ) µ X m C Ω R (Ĉ kσ ˆX kσ + ˆX kσĉkσ)] + ˆX ( σ R + r ) ˆX τ (R r Interaction U XX has exchange structure σ=±1 d rd R σ,σ,τ,τ =±,±1 ) ˆXτ ( R r U XX Ĉ kσĉkσ σ τ τσ (r) ) ˆXσ ( R + r For large Ω R, neglect σ = ± Interaction supports bound states in U+1, 1, 1,+1 XX channel bipolariton NB, bipolariton, bound polaritons, but larger exciton fraction. Jonathan Keeling Pairing Phases of Polaritons Pisa, January / 34 )

46 Exciton-photon model Microscopic model coupled exciton-photon system H = k + σ=±1 σ=±,±1 ( ) k µ ˆX m ˆX kσ kσ + (δ + k ) µ X m C Ω R (Ĉ kσ ˆX kσ + ˆX kσĉkσ)] + ˆX ( σ R + r ) ˆX τ (R r Interaction U XX has exchange structure σ=±1 d rd R σ,σ,τ,τ =±,±1 ) ˆXτ ( R r U XX Ĉ kσĉkσ σ τ τσ (r) ) ˆXσ ( R + r For large Ω R, neglect σ = ± Interaction supports bound states in U+1, 1, 1,+1 XX channel bipolariton NB, bipolariton, bound polaritons, but larger exciton fraction. Jonathan Keeling Pairing Phases of Polaritons Pisa, January / 34 )

47 Exciton-photon model Microscopic model coupled exciton-photon system H = k + σ=±1 σ=±,±1 ( ) k µ ˆX m ˆX kσ kσ + (δ + k ) µ X m C Ω R (Ĉ kσ ˆX kσ + ˆX kσĉkσ)] + ˆX ( σ R + r ) ˆX τ (R r Interaction U XX has exchange structure σ=±1 d rd R σ,σ,τ,τ =±,±1 ) ˆXτ ( R r U XX Ĉ kσĉkσ σ τ τσ (r) ) ˆXσ ( R + r For large Ω R, neglect σ = ± Interaction supports bound states in U+1, 1, 1,+1 XX channel bipolariton NB, bipolariton, bound polaritons, but larger exciton fraction. Jonathan Keeling Pairing Phases of Polaritons Pisa, January / 34 )

48 Exciton-photon model Microscopic model coupled exciton-photon system H = k + σ=±1 σ=±,±1 ( ) k µ ˆX m ˆX kσ kσ + (δ + k ) µ X m C Ω R (Ĉ kσ ˆX kσ + ˆX kσĉkσ)] + ˆX ( σ R + r ) ˆX τ (R r Interaction U XX has exchange structure σ=±1 d rd R σ,σ,τ,τ =±,±1 ) ˆXτ ( R r U XX Ĉ kσĉkσ σ τ τσ (r) ) ˆXσ ( R + r For large Ω R, neglect σ = ± Interaction supports bound states in U+1, 1, 1,+1 XX channel bipolariton NB, bipolariton, bound polaritons, but larger exciton fraction. Jonathan Keeling Pairing Phases of Polaritons Pisa, January / 34 )

49 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 Pisa, January 14 / 34

50 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 Pisa, January 14 / 34

51 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 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 Pisa, January 14 / 34

52 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 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 Pisa, January 14 / 34

53 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 Pisa, January 14 1 / 34

54 Outline 1 Introduction Pairing phases of atoms: review 3 Modelling polariton pairing Exciton spin structure Microscopic and effective Hamiltonian 4 Ground state phase diagram Origin of multicritical behaviour Signatures of phases 5 Finite temperature phase diagram Variational MFT Required temperatures, detuning Candidate material systems Jonathan Keeling Pairing Phases of Polaritons Pisa, January 14 / 34

55 Phase diagram (ground state, T = ) δ [mev] N.6.4. µ [mev] A PS 1x1 11 n [cm ] A Parameters for GaAs Ω R = 4.4meV, E b = mev. Dashed: Mean-field. Solid: Next order fluctuations Two condensed phases:, ˆψ = ˆψ =, ˆψ m. A σ : ˆψ σ. Small (or ve) δ: ν g, no biexciton physics standard WIDBG. Jonathan Keeling Pairing Phases of Polaritons Pisa, January 14 3 / 34

56 Phase diagram (ground state, T = ) δ [mev] N.6.4. µ [mev] A PS 1x1 11 n [cm ] A Parameters for GaAs Ω R = 4.4meV, E b = mev. Dashed: Mean-field. Solid: Next order fluctuations Two condensed phases:, ˆψ = ˆψ =, ˆψ m. A σ : ˆψ σ. Small (or ve) δ: ν g, no biexciton physics standard WIDBG. Jonathan Keeling Pairing Phases of Polaritons Pisa, January 14 3 / 34

57 Phase diagram (ground state, T = ) δ [mev] N.6.4. µ [mev] A PS 1x1 11 n [cm ] A Parameters for GaAs Ω R = 4.4meV, E b = mev. Dashed: Mean-field. Solid: Next order fluctuations Two condensed phases:, ˆψ = ˆψ =, ˆψ m. A σ : ˆψ σ. Small (or ve) δ: ν g, no biexciton physics standard WIDBG. Jonathan Keeling Pairing Phases of Polaritons Pisa, January 14 3 / 34

58 Mean field ansatz Hamiltonian: Weakly Interacting Bose Gas First ansatz: mean-field theory [Zhou et al. PRA 8] Ψ exp ψ σ ˆψk=,σ. σ=,,m No magnetic field, take ψ = ψ = ψ (real): E = Ψ H Ψ = µψ + ( U + U + U ) ψ 4 + (ν µ)ψ m + U mm ψ4 m + gψ ψ m Minimise E(ψ, ψ m ): first order transitions exist. Jonathan Keeling Pairing Phases of Polaritons Pisa, January 14 4 / 34

59 Mean field ansatz Hamiltonian: Weakly Interacting Bose Gas First ansatz: mean-field theory [Zhou et al. PRA 8] Ψ exp ψ σ ˆψk=,σ. σ=,,m No magnetic field, take ψ = ψ = ψ (real): E = Ψ H Ψ = µψ + ( U + U + U ) ψ 4 + (ν µ)ψ m + U mm ψ4 m + gψ ψ m Minimise E(ψ, ψ m ): first order transitions exist. Jonathan Keeling Pairing Phases of Polaritons Pisa, January 14 4 / 34

60 Mean field ansatz Hamiltonian: Weakly Interacting Bose Gas First ansatz: mean-field theory [Zhou et al. PRA 8] Ψ exp ψ σ ˆψk=,σ. σ=,,m No magnetic field, take ψ = ψ = ψ (real): E = Ψ H Ψ = µψ + ( U + U + U ) ψ 4 + (ν µ)ψ m + U mm ψ4 m + gψ ψ m Minimise E(ψ, ψ m ): first order transitions exist. Jonathan Keeling Pairing Phases of Polaritons Pisa, January 14 4 / 34

61 Phase boundaries (ground state, T = ) δ [mev] N.6.4. µ [mev] A PS 1x1 11 n [cm ] A Small δ: Fluctuations drive N A 1st order (Larkin-Pikin mechanism, coupling gψ ψ m ) Large phase-separation region high densities. First-order BEC transition! Naive resonance δ r = 3.84, tricritical point at (δ, µ) = (δ r, ) replaced by critical end-point at δ > δ r Large δ: -A transition always occurs as U mm renormalises energy. Jonathan Keeling Pairing Phases of Polaritons Pisa, January 14 5 / 34

62 Phase boundaries (ground state, T = ) δ [mev] N.6.4. µ [mev] A PS 1x1 11 n [cm ] A Small δ: Fluctuations drive N A 1st order (Larkin-Pikin mechanism, coupling gψ ψ m ) Large phase-separation region high densities. First-order BEC transition! Naive resonance δ r = 3.84, tricritical point at (δ, µ) = (δ r, ) replaced by critical end-point at δ > δ r Large δ: -A transition always occurs as U mm renormalises energy. Jonathan Keeling Pairing Phases of Polaritons Pisa, January 14 5 / 34

63 Phase boundaries (ground state, T = ) δ [mev] N.6.4. µ [mev] A PS 1x1 11 n [cm ] A Small δ: Fluctuations drive N A 1st order (Larkin-Pikin mechanism, coupling gψ ψ m ) Large phase-separation region high densities. First-order BEC transition! Naive resonance δ r = 3.84, tricritical point at (δ, µ) = (δ r, ) replaced by critical end-point at δ > δ r Large δ: -A transition always occurs as U mm renormalises energy. Jonathan Keeling Pairing Phases of Polaritons Pisa, January 14 5 / 34

64 Phase boundaries (ground state, T = ) δ [mev] N.6.4. µ [mev] A PS 1x1 11 n [cm ] A Small δ: Fluctuations drive N A 1st order (Larkin-Pikin mechanism, coupling gψ ψ m ) Large phase-separation region high densities. First-order BEC transition! Naive resonance δ r = 3.84, tricritical point at (δ, µ) = (δ r, ) replaced by critical end-point at δ > δ r Large δ: -A transition always occurs as U mm renormalises energy. Jonathan Keeling Pairing Phases of Polaritons Pisa, January 14 5 / 34

65 Beyond mean-field Fluctuation effects? Polariton fluctuations irrelevant: mu 1 4. Exciton fluctuations important: m m U 1. Next order theory: [Nozières & St James, J. Phys 8] Ψ exp ψ σ ˆψ k=,σ + tanh(θ kγ )ˆb ˆb kγ kγ. σ=,,m k,γ=a,b,m ( ) ˆψ where ˆb km = ˆψ km and k ˆψ = 1 ( 1 1 k 1 1 Variational functional E[ψ, ψ m, θ kγ ] ) ( ˆb ka ˆb kb Can show minimum θ kγ has form tanh(θ kγ ) = β γ + k /m γ Finite only if αγ < β γ Variational function E(ψ, ψ m, α γ, β γ ) ) α γ, Jonathan Keeling Pairing Phases of Polaritons Pisa, January 14 6 / 34

66 Beyond mean-field Fluctuation effects? Polariton fluctuations irrelevant: mu 1 4. Exciton fluctuations important: m m U 1. Next order theory: [Nozières & St James, J. Phys 8] Ψ exp ψ σ ˆψ k=,σ + tanh(θ kγ )ˆb ˆb kγ kγ. σ=,,m k,γ=a,b,m ( ) ˆψ where ˆb km = ˆψ km and k ˆψ = 1 ( 1 1 k 1 1 Variational functional E[ψ, ψ m, θ kγ ] ) ( ˆb ka ˆb kb Can show minimum θ kγ has form tanh(θ kγ ) = β γ + k /m γ Finite only if αγ < β γ Variational function E(ψ, ψ m, α γ, β γ ) ) α γ, Jonathan Keeling Pairing Phases of Polaritons Pisa, January 14 6 / 34

67 Beyond mean-field Fluctuation effects? Polariton fluctuations irrelevant: mu 1 4. Exciton fluctuations important: m m U 1. Next order theory: [Nozières & St James, J. Phys 8] Ψ exp ψ σ ˆψ k=,σ + tanh(θ kγ )ˆb ˆb kγ kγ. σ=,,m k,γ=a,b,m ( ) ˆψ where ˆb km = ˆψ km and k ˆψ = 1 ( 1 1 k 1 1 Variational functional E[ψ, ψ m, θ kγ ] ) ( ˆb ka ˆb kb Can show minimum θ kγ has form tanh(θ kγ ) = β γ + k /m γ Finite only if αγ < β γ Variational function E(ψ, ψ m, α γ, β γ ) ) α γ, Jonathan Keeling Pairing Phases of Polaritons Pisa, January 14 6 / 34

68 Beyond mean-field Fluctuation effects? Polariton fluctuations irrelevant: mu 1 4. Exciton fluctuations important: m m U 1. Next order theory: [Nozières & St James, J. Phys 8] Ψ exp ψ σ ˆψ k=,σ + tanh(θ kγ )ˆb ˆb kγ kγ. σ=,,m k,γ=a,b,m ( ) ˆψ where ˆb km = ˆψ km and k ˆψ = 1 ( 1 1 k 1 1 Variational functional E[ψ, ψ m, θ kγ ] ) ( ˆb ka ˆb kb Can show minimum θ kγ has form tanh(θ kγ ) = β γ + k /m γ Finite only if αγ < β γ Variational function E(ψ, ψ m, α γ, β γ ) ) α γ, Jonathan Keeling Pairing Phases of Polaritons Pisa, January 14 6 / 34

69 Outline 1 Introduction Pairing phases of atoms: review 3 Modelling polariton pairing Exciton spin structure Microscopic and effective Hamiltonian 4 Ground state phase diagram Origin of multicritical behaviour Signatures of phases 5 Finite temperature phase diagram Variational MFT Required temperatures, detuning Candidate material systems Jonathan Keeling Pairing Phases of Polaritons Pisa, January 14 7 / 34

70 Detecting vs A Phase separation Direct access to polariton phase coherence A: standard g σ (1) = ψ σ(r, t)ψ σ (, ). (D, QLRO) Vortex structure half vortices, ψ = e im θ, ψ = e im θ, has (m, m ) = (1/, 1/). Polariton half-vortex (m, m ) = (1, ) [Lagoudakis et al. Science 9] Jonathan Keeling Pairing Phases of Polaritons Pisa, January 14 8 / 34

71 Detecting vs A Phase separation Direct access to polariton phase coherence A: standard g σ (1) = ψ σ(r, t)ψ σ (, ). (D, QLRO) Vortex structure half vortices, ψ = e im θ, ψ = e im θ, has (m, m ) = (1/, 1/). Polariton half-vortex (m, m ) = (1, ) [Lagoudakis et al. Science 9] Jonathan Keeling Pairing Phases of Polaritons Pisa, January 14 8 / 34

72 Detecting vs A Phase separation Direct access to polariton phase coherence A: standard g σ (1) = ψ σ(r, t)ψ σ (, ). (D, QLRO) Vortex structure half vortices, ψ = e im θ, ψ = e im θ, has (m, m ) = (1/, 1/). Polariton half-vortex (m, m ) = (1, ) [Lagoudakis et al. Science 9] Jonathan Keeling Pairing Phases of Polaritons Pisa, January 14 8 / 34

73 Detecting vs A Phase separation Direct access to polariton phase coherence A: standard g σ (1) = ψ σ(r, t)ψ σ (, ). (D, QLRO) Vortex structure half vortices, ψ = e im θ, ψ = e im θ, has (m, m ) = (1/, 1/). Polariton half-vortex (m, m ) = (1, ) [Lagoudakis et al. Science 9] Jonathan Keeling Pairing Phases of Polaritons Pisa, January 14 8 / 34

74 Detecting vs A Phase separation Direct access to polariton phase coherence A: 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 half vortices, ψ = e im θ, ψ = e im θ, has (m, m ) = (1/, 1/). Polariton half-vortex (m, m ) = (1, ) [Lagoudakis et al. Science 9] Jonathan Keeling Pairing Phases of Polaritons Pisa, January 14 8 / 34

75 Detecting vs A Phase separation Direct access to polariton phase coherence A: 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 half vortices, ψ = e im θ, ψ = e im θ, has (m, m ) = (1/, 1/). Polariton half-vortex (m, m ) = (1, ) [Lagoudakis et al. Science 9] Jonathan Keeling Pairing Phases of Polaritons Pisa, January 14 8 / 34

76 Outline 1 Introduction Pairing phases of atoms: review 3 Modelling polariton pairing Exciton spin structure Microscopic and effective Hamiltonian 4 Ground state phase diagram Origin of multicritical behaviour Signatures of phases 5 Finite temperature phase diagram Variational MFT Required temperatures, detuning Candidate material systems Jonathan Keeling Pairing Phases of Polaritons Pisa, January 14 9 / 34

77 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 Pisa, January 14 3 / 34

78 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 Pisa, January 14 3 / 34

79 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 Pisa, January 14 3 / 34

80 Phase diagram, finite temperature δ [mev] N T=.58 K A µ [mev] N PS A x1 11 n [cm ] Jonathan Keeling Pairing Phases of Polaritons Pisa, January / 34

81 Phase diagram, finite temperature δ [mev] δ [mev] N N T=.58 K A T=1.16 K A µ [mev] N N A PS A PS n [cm ] x1 11 Jonathan Keeling Pairing Phases of Polaritons Pisa, January / 34

82 Phase diagram, finite temperature δ [mev] δ [mev] δ [mev] N N N T=.58 K A T=1.16 K A T =.3 K A µ [mev] N N N PS PS PS A A A x1 11 n [cm ] Jonathan Keeling Pairing Phases of Polaritons Pisa, January / 34

83 Phase diagram, finite temperature δ [mev] δ [mev] δ [mev] N N N T=.58 K A T=1.16 K A T =.3 K A µ [mev] N N N PS PS PS A A A x1 11 n [cm ] δ [mev] 5 N PS A n [cm ] Jonathan Keeling Pairing Phases of Polaritons Pisa, January / 34

84 Phase diagram, vs temperature N T=.58 K A N PS A δ [mev] 3 δ=1 mev N PS µ [mev] 11 n [cm ] x1 T [K] 1 N A µ [mev] A 11 x1 n [cm ] Jonathan Keeling Pairing Phases of Polaritons Pisa, January 14 3 / 34

85 Phase diagram, vs temperature T [K] 1 3 δ=8.3 mev N δ=1 mev A N PS A N PS T=.58 K A N x1 µ [mev] n [cm ] N A PS δ [mev] T [K] 1 N A µ [mev] A 11 x1 n [cm ] Jonathan Keeling Pairing Phases of Polaritons Pisa, January 14 3 / 34

86 Phase diagram, vs temperature T [K] T [K] δ=6.5 mev N δ=8.3 mev N δ=1 mev N PS A A A N PS A N PS T=.58 K A N x1 µ [mev] n [cm ] N A PS [mev] δ T [K] 1 N A µ [mev] A 11 x1 n [cm ] Jonathan Keeling Pairing Phases of Polaritons Pisa, January 14 3 / 34

87 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 Pisa, January / 34

88 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 Pisa, January / 34

89 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 Pisa, January / 34

90 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 Pisa, January / 34

91 Adjustable delay φ θ APD APD A B Summary Polariton biexciton detuning Feshbach resonance E ω X [mev] 6 4 LP UP C X 4 4 k [µm 1 ] m LP /m X E b ~m C /m X δ [mev] Polaritons can show pairing phase δ [mev] N.6.4. µ [mev] A PS 1x1 11 n [cm ] A T [K] 3 1 δ=1 mev N ν [mev] A µ [mev] N PS A x1 11 n [cm ] easily attainable for ZnO; possible for GaA 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 Pisa, January / 34

92 Jonathan Keeling Pairing Phases of Polaritons Pisa, January / 38

93 Jonathan Keeling Pairing Phases of Polaritons Pisa, January / 38

94 Differences & new opportunities Atoms Metastable, isolated Polaritons Finite lifetime, in semiconductor Fixed mass 1 3 m Tunable mass 1 4 m Couples to light Can be 3D Is part light Always D or less. Spin: S z = S, S + 1,..., S Light polarisations S z = ±1 Jonathan Keeling Pairing Phases of Polaritons Pisa, January / 38

95 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 Pisa, January / 38

96 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 Pisa, January / 38

97 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 Pisa, January / 38

98 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 Pisa, January / 38