Nonequilibrium quantum condensates: from microscopic theory to macroscopic phenomenology

Transcript

1 Nonequilibrium quantum condensates: from microscopic theory to macroscopic phenomenology J. M. J. Keeling N. G. Berloff, M. O. Borgh, P. B. Littlewood, F. M. Marchetti, M. H. Szymanska. Royal Holloway Condensed Matter Seminar, October 2009 Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 1 / 36

2 Acknowledgements People: Funding: Pembroke College Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 2 / 36

3 Microcavity Polaritons Cavity Quantum Wells Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 3 / 36

4 Microcavity Polaritons Energy Photon UP LP Exciton [Pekar, JETP(1958)] [Hopfield, Phys. Rev.(1958)] Cavity Quantum Wells Momentum Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 3 / 36

5 Microcavity Polaritons Energy Photon UP LP Exciton [Pekar, JETP(1958)] [Hopfield, Phys. Rev.(1958)] Cavity Quantum Wells Momentum Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 3 / 36

6 Microcavity Polaritons Energy Photon UP LP Exciton [Pekar, JETP(1958)] [Hopfield, Phys. Rev.(1958)] Cavity Quantum Wells Momentum Cavity photons: ω k = ω0 2 + c2 k 2 ω 0 + k 2 /2m m 10 4 m e Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 3 / 36

7 Microcavity Polaritons Energy Photon UP LP Exciton [Pekar, JETP(1958)] [Hopfield, Phys. Rev.(1958)] Cavity Quantum Wells Momentum Cavity photons: ω k = ω0 2 + c2 k 2 ω 0 + k 2 /2m m 10 4 m e Energy [mev] UP Photon Exciton -10 LP Angle [degrees] Wavevector [10 7 cm -1 ] Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 3 / 36

8 Non-equilibrium: flux and baths θ Photon Exciton Energy In plane momentum Cavity Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 4 / 36

9 Polariton experiments: Momentum/Energy distribution [Kasprzak, et al., Nature, 2006] Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 5 / 36

10 Polariton experiments: Coherence Basic idea: Sample Beam Splitter CCD Tunable Delay Retroreflector Coherence map: + = [Kasprzak, et al., Nature, 2006] Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 6 / 36

11 Other polariton condensation experiments Stress traps for polaritons [Balili et al Science (2007)] Temporal coherence and line narrowing [Love et al Phys. Rev. Lett (2008)] Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 7 / 36

12 Other polariton condensation experiments Quantised vortices in disorder potential [Lagoudakis et al Nature Phys. 4, 706 (2008)] Changes to excitation spectrum [Utsunomiya et al Nature Phys (2008)] Soliton propagation [Amo et al Nature (2009)] Driven superfluidity [Amo et al Nature Phys. (2009) Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 8 / 36

13 Overview 1 Introduction to microcavity polaritons 2 Model and review of equilibrium results Disorder-localised exciton model Equilibrium mean field theory 3 Microscopic non-equilibrium model Model and mean-field theory Fluctuations Stability of normal state lasing vs condensation Condensed spectrum 4 Macroscopic phenomenology Gross Pitaevskii equation in an harmonic trap Internal Josephson effect and spatial variation Spin degree of freedom Spin and spatial degrees of freedom 5 Conclusions Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 9 / 36

14 Excitons in a disorderd Quantum well Cavity Quantum Wells Exciton states in disorder: [ ] 2 R + V (R) Φ α (R) = ε α Φ α (R) 2m X V ( R) smoothed by exciton Bohr radius [PRL (2006); PRB (2007)] Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 10 / 36

15 Excitons in a disorderd Quantum well Cavity Quantum Wells Exciton states in disorder: [ ] 2 R + V (R) Φ α (R) = ε α Φ α (R) 2m X V ( R) smoothed by exciton Bohr radius Want: Energies ε α Oscillator strengths: g α,p ψ 1s (0)Φ α,p [PRL (2006); PRB (2007)] Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 10 / 36

16 Excitons in a disorderd Quantum well Cavity Quantum Wells Exciton states in disorder: [ ] 2 R + V (R) Φ α (R) = ε α Φ α (R) 2m X V ( R) smoothed by exciton Bohr radius Want: Energies ε α Oscillator strengths: g α,p ψ 1s (0)Φ α,p g αp=0 2 [arb. units] g 2 (ε,0) DoS(ε) ε-e x [mev] [PRL (2006); PRB (2007)] Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 10 / 36

17 Polariton system model Polariton model Disorder-localised excitons Treat disorder sites as two-level (exciton/no-exciton) Propagating (2D) photons Energy Cavity Quantum Wells Exciton photon coupling g. Position Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 11 / 36

18 Polariton system model Polariton model Disorder-localised excitons Treat disorder sites as two-level (exciton/no-exciton) Propagating (2D) photons Exciton photon coupling g. Energy Cavity Quantum Wells H sys = k ω k ψ k ψ k + α Position [ɛ α (b αb α a αa ) α + 1 ] g α,k ψ k b αa α + H.c. Area System Excitons Cavity mode Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 11 / 36

19 Equilibrium: Mean-field theory H sys = k ω k ψ k ψ k + α [ɛ α (b αb α a αa ) α + g ] α,k ψ k b αa α + H.c. A g αp=0 2 [arb. units] g 2 (ε,0) DoS(ε) ε-e x [mev] Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 12 / 36

20 Equilibrium: Mean-field theory H sys = k ω k ψ k ψ k + α [ɛ α (b αb α a αa ) α + g ] α,k ψ k b αa α + H.c. A Mean-field theory: Self-consistent polarisation and field [ ] i t + ω 0 2 ψ = 1 g α P α 2m A α g αp=0 2 [arb. units] g 2 (ε,0) DoS(ε) ε-e x [mev] Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 12 / 36

21 Equilibrium: Mean-field theory H sys = k ω k ψ k ψ k + α [ɛ α (b αb α a αa ) α + g ] α,k ψ k b αa α + H.c. A Mean-field theory: Self-consistent polarisation and field ] [ µ + ω 0 2 ψ = 1 g α a 2m αb α A α g αp=0 2 [arb. units] g 2 (ε,0) DoS(ε) ε-e x [mev] Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 12 / 36

22 Equilibrium: Mean-field theory H sys = k ω k ψ k ψ k + α [ɛ α (b αb α a αa ) α + g ] α,k ψ k b αa α + H.c. A Mean-field theory: Self-consistent polarisation and field ] [ µ + ω 0 2 ψ = 1 g α ψ g α tanh (βe α ) 2m A 2E α α ( ) E 2 ɛα µ 2 α = + g 2 2 g αp=0 2 [arb. units] g 2 (ε,0) DoS(ε) αψ ε-e x [mev] Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 12 / 36

23 Equilibrium: Mean-field theory H sys = k ω k ψ k ψ k + α [ɛ α (b αb α a αa ) α + g ] α,k ψ k b αa α + H.c. A Mean-field theory: Self-consistent polarisation and field ] [ µ + ω 0 2 ψ = 1 g α ψ g α 2m A Density ρ = ψ A α α ( E 2 ɛα µ α = 2 tanh (βe α ) 2E α ) 2 + gαψ 2 2 [ 1 2 ɛ ] α µ tanh(βe α ) 4E α g αp=0 2 [arb. units] g 2 (ε,0) DoS(ε) ε-e x [mev] Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 12 / 36

24 Equilibrium: Mean-field theory H sys = k ω k ψ k ψ k + α [ɛ α (b αb α a αa ) α + g ] α,k ψ k b αa α + H.c. A Mean-field theory: Self-consistent polarisation and field ] [ µ + ω 0 2 ψ = 1 g α ψ g α 2m A α ( E 2 ɛα µ α = 2 tanh (βe α ) 2E α ) 2 + gαψ 2 2 g αp=0 2 [arb. units] 40 g 2 (ε,0) DoS(ε) ε-e x [mev] non-condensed Density ρ = ψ A α [ 1 2 ɛ ] α µ tanh(βe α ) 4E α k B T condensed n [cm -2 ] Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 12 / 36

25 Overview 1 Introduction to microcavity polaritons 2 Model and review of equilibrium results Disorder-localised exciton model Equilibrium mean field theory 3 Microscopic non-equilibrium model Model and mean-field theory Fluctuations Stability of normal state lasing vs condensation Condensed spectrum 4 Macroscopic phenomenology Gross Pitaevskii equation in an harmonic trap Internal Josephson effect and spatial variation Spin degree of freedom Spin and spatial degrees of freedom 5 Conclusions Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 13 / 36

26 Non-equilibrium system Photon Exciton Ph Ex Energy In plane momentum Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 14 / 36

27 Non-equilibrium system Photon Exciton Ph Ex Energy In plane momentum Lifetime Thermalisation Atoms 10s 10ms Excitons a 50ns 0.2ns Polaritons 5ps 0.5ps Magnons b 1µs(??) 100ns(?) a Coupled quantum wells. [Hammack et al PRB (2007)] b Yttrium Iron Garnett. [Demokritov et al, Nature (2007)] Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 14 / 36

28 Non-equilibrium system Photon Exciton Ph Ex Energy In plane momentum Lifetime Thermalisation Linewidth Temperature Atoms 10s 10ms mev 10 8 K 10 9 mev Excitons a 50ns 0.2ns mev 1K 0.1meV Polaritons 5ps 0.5ps 0.5meV 20K 2meV Magnons b 1µs(??) 100ns(?) mev 300K 30meV a Coupled quantum wells. [Hammack et al PRB (2007)] b Yttrium Iron Garnett. [Demokritov et al, Nature (2007)] Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 14 / 36

29 Non-equilibrium model: baths Pumping Bath γ Excitons System Cavity mode κ Bulk photon modes H = H sys + H sys,bath + H bath Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 15 / 36

30 Non-equilibrium model: baths Pumping Bath γ Excitons System Cavity mode κ Bulk photon modes H = H sys + H sys,bath + H bath Schematically: pump γ, decay κ H sys,bath κψk Ψ p + ) γ (a αa β + b αb β + H.c. p, k α,β Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 15 / 36

31 Non-equilibrium model: baths Pumping Bath γ Excitons System Cavity mode κ Bulk photon modes H = H sys + H sys,bath + H bath Schematically: pump γ, decay κ H sys,bath κψk Ψ p + ) γ (a αa β + b αb β + H.c. p, k α,β Bath correlations, Ψ Ψ, A A, B B fixed: Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 15 / 36

32 Non-equilibrium model: baths Pumping Bath γ Excitons System Cavity mode κ Bulk photon modes H = H sys + H sys,bath + H bath Schematically: pump γ, decay κ H sys,bath κψk Ψ p + ) γ (a αa β + b αb β + H.c. p, k α,β Bath correlations, Ψ Ψ, A A, B B fixed: Ψ bath is empty. Pumping bath thermal, µ B, T : na ε nb x µ /2 0 +µ B B/2 ε x Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 15 / 36

33 Non-equilibrium theory; mean-field Look for mean-field solution, ψ(r, t) = ψ 0 e iµ S t. Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 16 / 36

34 Non-equilibrium theory; mean-field Look for mean-field solution, ψ(r, t) = ψ 0 e iµ S t. Gap equation: (µ s ω 0 + iκ) ψ 0 = χ(ψ 0, µ s )ψ 0 Susceptibility: χ(ψ 0, µ s ) = g 2 γ excitons dν (F a + F b )ν + (F b F a )(iγ + ɛ α 1 2 µ S) 2π [(ν E α ) 2 + γ 2 ][(ν + E α ) 2 + γ 2 ] Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 16 / 36

35 Non-equilibrium theory; mean-field Look for mean-field solution, ψ(r, t) = ψ 0 e iµ S t. Gap equation: (µ s ω 0 + iκ) ψ 0 = χ(ψ 0, µ s )ψ 0 Susceptibility: E 2 α = ɛ 2 α + g 2 ψ 0 2 χ(ψ 0, µ s ) = g 2 γ excitons dν (F a + F b )ν + (F b F a )(iγ + ɛ α 1 2 µ S) 2π [(ν E α ) 2 + γ 2 ][(ν + E α ) 2 + γ 2 ] Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 16 / 36

36 Non-equilibrium theory; mean-field Look for mean-field solution, ψ(r, t) = ψ 0 e iµ S t. Gap equation: (µ s ω 0 + iκ) ψ 0 = χ(ψ 0, µ s )ψ 0 Susceptibility: Eα 2 = ɛ 2 α + g 2 ψ 0 2, F a,b (ν) = F [ν 1 2 (µ s µ B )] χ(ψ 0, µ s ) = g 2 γ dν (F a + F b )ν + (F b F a )(iγ + ɛ α 1 2 µ S) 2π [(ν E α ) 2 + γ 2 ][(ν + E α ) 2 + γ 2 ] excitons Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 16 / 36

37 Non-equilibrium theory; mean-field Look for mean-field solution, ψ(r, t) = ψ 0 e iµ S t. Gap equation: (µ s ω 0 + iκ) ψ 0 = χ(ψ 0, µ s )ψ 0 Susceptibility: Eα 2 = ɛ 2 α + g 2 ψ 0 2, F a,b (ν) = F [ν 1 2 (µ s µ B )] χ(ψ 0, µ s ) = g 2 γ dν (F a + F b )ν + (F b F a )(iγ + ɛ α 1 2 µ S) 2π [(ν E α ) 2 + γ 2 ][(ν + E α ) 2 + γ 2 ] excitons Bath Temperature Increasing γ Density Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 16 / 36

38 Limits of gap equation Gap equation: µ s ω 0 + iκ = g 2 γ excitons dν (F a + F b )ν + (F b F a )(iγ + ɛ α 1 2 µ S) 2π [(ν E α ) 2 + γ 2 ][(ν + E α ) 2 + γ 2 ] Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 17 / 36

39 Limits of gap equation Gap equation: µ s ω 0 + iκ = g 2 γ excitons dν (F a + F b )ν + (F b F a )(iγ + ɛ α 1 2 µ S) 2π [(ν E α ) 2 + γ 2 ][(ν + E α ) 2 + γ 2 ] Laser Limit Imaginary part: Gain vs Loss. Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 17 / 36

40 Limits of gap equation Gap equation: µ s ω 0 + iκ = g 2 γ excitons dν (F a + F b )ν + (F b F a )(iγ + ɛ α 1 2 µ S) 2π [(ν E α ) 2 + γ 2 ][(ν + E α ) 2 + γ 2 ] Laser Limit Imaginary part: Gain vs Loss. If F b (ω) F a (ω) 2N 0 κ = g 2 γ excitons N 0 2(E 2 α + γ 2 ) Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 17 / 36

41 Limits of gap equation Gap equation: µ s ω 0 + iκ = g 2 γ excitons dν (F a + F b )ν + (F b F a )(iγ + ɛ α 1 2 µ S) 2π [(ν E α ) 2 + γ 2 ][(ν + E α ) 2 + γ 2 ] Laser Limit Imaginary part: Gain vs Loss. If F b (ω) F a (ω) 2N 0 κ = g 2 γ excitons N 0 2(Eα 2 + γ 2 ) = g 2 2γ Inversion Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 17 / 36

42 Limits of gap equation Gap equation: µ s ω 0 + iκ = g 2 γ excitons dν (F a + F b )ν + (F b F a )(iγ + ɛ α 1 2 µ S) 2π [(ν E α ) 2 + γ 2 ][(ν + E α ) 2 + γ 2 ] Laser Limit Imaginary part: Gain vs Loss. If F b (ω) F a (ω) 2N 0 κ = g 2 γ excitons N 0 2(Eα 2 + γ 2 ) = g 2 2γ Inversion Equilibrium limit: finite T set by pumping, need κ γ. Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 17 / 36

43 Limits of gap equation Gap equation: µ s ω 0 + iκ = g 2 γ excitons dν (F a + F b )ν + (F b F a )(iγ + ɛ α 1 2 µ S) 2π [(ν E α ) 2 + γ 2 ][(ν + E α ) 2 + γ 2 ] Laser Limit Imaginary part: Gain vs Loss. If F b (ω) F a (ω) 2N 0 κ = g 2 γ excitons N 0 2(Eα 2 + γ 2 ) = g 2 2γ Inversion Equilibrium limit: finite T set by pumping, need κ γ. Require: F a (ω) = F b (ω) so µ S = µ B Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 17 / 36

44 Limits of gap equation Gap equation: µ s ω 0 + iκ = g 2 γ excitons dν (F a + F b )ν + (F b F a )(iγ + ɛ α 1 2 µ S) 2π [(ν E α ) 2 + γ 2 ][(ν + E α ) 2 + γ 2 ] Laser Limit Imaginary part: Gain vs Loss. If F b (ω) F a (ω) 2N 0 κ = g 2 γ excitons N 0 2(Eα 2 + γ 2 ) = g 2 2γ Inversion Equilibrium limit: finite T set by pumping, need κ γ. Require: F a (ω) = F b (ω) so µ S = µ B Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 17 / 36

45 Limits of gap equation Gap equation: µ s ω 0 + iκ = g 2 γ excitons dν (F a + F b )ν + (F b F a )(iγ + ɛ α 1 2 µ S) 2π [(ν E α ) 2 + γ 2 ][(ν + E α ) 2 + γ 2 ] Laser Limit Imaginary part: Gain vs Loss. If F b (ω) F a (ω) 2N 0 κ = g 2 γ excitons N 0 2(Eα 2 + γ 2 ) = g 2 2γ Inversion Equilibrium limit: finite T set by pumping, need κ γ. Require: F a (ω) = F b (ω) so µ S = µ B ω 0 µ s = g 2 ( ) βe 2E tanh 2 Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 17 / 36

46 Overview 1 Introduction to microcavity polaritons 2 Model and review of equilibrium results Disorder-localised exciton model Equilibrium mean field theory 3 Microscopic non-equilibrium model Model and mean-field theory Fluctuations Stability of normal state lasing vs condensation Condensed spectrum 4 Macroscopic phenomenology Gross Pitaevskii equation in an harmonic trap Internal Josephson effect and spatial variation Spin degree of freedom Spin and spatial degrees of freedom 5 Conclusions Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 18 / 36

47 Fluctuations Stability, Luminescence, Absorption Keldysh approach: [ D R,A = iθ[±(t t )] ψ, ψ ] [ D K = i ψ, ψ ] + Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 19 / 36

48 Fluctuations Stability, Luminescence, Absorption Keldysh approach: [ D R D A = i ψ, ψ ] [ D K = i ψ, ψ ] + Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 19 / 36

49 Fluctuations Stability, Luminescence, Absorption Keldysh approach: [ D R D A = i ψ, ψ ] [ D K = i ψ, ψ ] = (2n(ω) + 1)(D R D A ) + Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 19 / 36

50 Fluctuations Stability, Luminescence, Absorption [ D R D A = i ψ, ψ ] Keldysh approach: [ D K = i ψ, ψ ] = (2n(ω) + 1)(D R D A ) + [ 1 D (ω)] R = A(ω) + ib(ω), [ D 1 (ω) ] K = ic(ω), Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 19 / 36

51 Fluctuations Stability, Luminescence, Absorption [ D R D A = i ψ, ψ ] Keldysh approach: [ D K = i ψ, ψ ] = (2n(ω) + 1)(D R D A ) + [ 1 D (ω)] R = A(ω) + ib(ω), [ D 1 (ω) ] K = ic(ω), D K = 1 [D R ] 1 [D 1 ] K 1 [D A ] 1 D R D A 1 = [D R ] 1 1 [D A ] 1 Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 19 / 36

52 Fluctuations Stability, Luminescence, Absorption [ D R D A = i ψ, ψ ] Keldysh approach: [ D K = i ψ, ψ ] = (2n(ω) + 1)(D R D A ) + [ 1 D (ω)] R = A(ω) + ib(ω), [ D 1 (ω) ] K = ic(ω), D K = D R D A = ic(ω) B(ω) 2 + A(ω) 2 2B(ω) B(ω) 2 + A(ω) 2 Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 19 / 36

53 Fluctuations Stability, Luminescence, Absorption Keldysh approach: [ D R D A = i ψ, ψ ] [ D K = i ψ, ψ ] = (2n(ω) + 1)(D R D A ) + [ 1 D (ω)] R = A(ω) + ib(ω), A(ω) [ D 1 (ω) ] 1 K = ic(ω), D K = D R D A = ic(ω) B(ω) 2 + A(ω) 2 2B(ω) B(ω) 2 + A(ω) 2 Intensity (a.u.) Density of states, 2 Im[D R ] Energy (units of g) Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 19 / 36

54 Fluctuations Stability, Luminescence, Absorption Keldysh approach: [ D R (ω)] 1 = A(ω) + ib(ω), [ D 1 (ω) ] K = ic(ω), [ D R D A = i ψ, ψ ] [ D K = i ψ, ψ ] = (2n(ω) + 1)(D R D A ) A(ω) B(ω) 1 0 D K = D R D A = ic(ω) B(ω) 2 + A(ω) 2 2B(ω) B(ω) 2 + A(ω) 2 Intensity (a.u.) Density of states, 2 Im[D R ] Energy (units of g) Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 19 / 36

55 Fluctuations Stability, Luminescence, Absorption Keldysh approach: [ D R (ω)] 1 = A(ω) + ib(ω), [ D 1 (ω) ] K = ic(ω), [ D R D A = i ψ, ψ ] [ D K = i ψ, ψ ] = (2n(ω) + 1)(D R D A ) A(ω) B(ω) 1 C(ω) 0 D K = D R D A = ic(ω) B(ω) 2 + A(ω) 2 2B(ω) B(ω) 2 + A(ω) 2 Intensity (a.u.) Density of states, 2 Im[D R ] Occupation, n(ω) Energy (units of g) Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 19 / 36

56 Fluctuations Stability, Luminescence, Absorption Keldysh approach: [ D R (ω)] 1 = A(ω) + ib(ω), [ D 1 (ω) ] K = ic(ω), D K = D R D A = [ D R D A = i ψ, ψ ] [ D K = i ψ, ψ ] = (2n(ω) + 1)(D R D A ) + ic(ω) B(ω) 2 + A(ω) 2 2B(ω) B(ω) 2 + A(ω) 2 [ D R (ω)] 1 = (ω ω k )+iα(ω µ eff ) Intensity (a.u.) A(ω) B(ω) 1 C(ω) Density of states, 2 Im[D R ] Occupation, n(ω) Energy (units of g) Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 19 / 36

57 Linewidth, inverse Green s function and gap equation A(ω) B(ω) 1 C(ω) 0 Intensity (a.u.) Density of states, 2 Im[D R ] Occupation, n(ω) Energy (units of g) Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 20 / 36

58 Linewidth, inverse Green s function and gap equation Intensity (a.u.) A(ω) B(ω) 1 C(ω) Density of states, 2 Im[D R ] Occupation, n(ω) Energy (units of g) Energy of zero (units of g) Zero of Re Zero of Im Bath chemical potential, µ B /g Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 20 / 36

59 [D R ] 1 for a laser Maxwell-Bloch equations: t ψ = iω k ψ κψ + gp t P = 2iɛP ΓP + gψn t N = Γ(N 0 N) 2g(ψ P + P ψ) [ 1 D (ω)] R g 2 N 0 = ω ωk +iκ+ ω 2ɛ + iγ Energy of zero Zero of Re Zero of Im -(2γ/g) 2 (2γκ/g 2 ) System inversion, N Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 21 / 36

60 Fluctuations above transition When condensed [ 1 Det D R (ω, k)] = ω 2 c 2 k 2 Poles: ω = c k frequency Sound mode momentum [Szymańska et al., PRL 06; PRB 07] Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 22 / 36

61 Fluctuations above transition When condensed [ ] 1 Det D R (ω, k) = (ω+ix) 2 +x 2 c 2 k 2 Poles: ω = ix ± c 2 k 2 x 2 frequency Real Imaginary momentum [Szymańska et al., PRL 06; PRB 07] Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 22 / 36

62 Fluctuations above transition When condensed [ ] 1 Det D R (ω, k) = (ω+ix) 2 +x 2 c 2 k 2 Poles: ω = ix ± c 2 k 2 x 2 frequency Real Imaginary momentum Correlations (in 2D): ψ (r, t)ψ(0, r ) ψ 0 2 exp [ D φφ (t, r, r )] [Szymańska et al., PRL 06; PRB 07] Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 22 / 36

63 Fluctuations above transition When condensed [ ] 1 Det D R (ω, k) = (ω+ix) 2 +x 2 c 2 k 2 Poles: ω = ix ± c 2 k 2 x 2 frequency Real Imaginary momentum Correlations (in 2D): ψ (r, t)ψ(0, r ) ψ 0 2 exp [ D φφ (t, r, r )] [ { ] ψ (r, t)ψ(0, 0 ψ 0 2 ln(r/ξ) r, t 0 exp η 1 2 ln(c2 t/xξ 2 ) r, t [Szymańska et al., PRL 06; PRB 07] Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 22 / 36

64 Overview 1 Introduction to microcavity polaritons 2 Model and review of equilibrium results Disorder-localised exciton model Equilibrium mean field theory 3 Microscopic non-equilibrium model Model and mean-field theory Fluctuations Stability of normal state lasing vs condensation Condensed spectrum 4 Macroscopic phenomenology Gross Pitaevskii equation in an harmonic trap Internal Josephson effect and spatial variation Spin degree of freedom Spin and spatial degrees of freedom 5 Conclusions Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 23 / 36

65 Limits of gap equation Gap equation: (µ s ω 0 + iκ) ψ = χ(ψ, µ s )ψ Local density limit: Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 24 / 36

66 Limits of gap equation Gap equation: (µ s ω 0 + iκ) ψ = χ(ψ, µ s )ψ Local density limit: Gross-Pitaevskii equation ( i t + iκ [V (r) 2 2 ]) ψ(r) = χ(ψ(r, t))ψ(r, t) 2m Nonlinear, complex susceptibility χ(ψ(r, t)) Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 24 / 36

67 Limits of gap equation Gap equation: (µ s ω 0 + iκ) ψ = χ(ψ, µ s )ψ Local density limit: Gross-Pitaevskii equation ( i t + iκ [V (r) 2 2 ]) ψ(r) = χ(ψ(r, t))ψ(r, t) 2m Nonlinear, complex susceptibility χ[e(ψ(r, t))], E 2 α = ɛ 2 α + g 2 ψ 0 2 i t ψ nlin = U ψ 2 ψ Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 24 / 36

68 Limits of gap equation Gap equation: (µ s ω 0 + iκ) ψ = χ(ψ, µ s )ψ Local density limit: Gross-Pitaevskii equation ( i t + iκ [V (r) 2 2 ]) ψ(r) = χ(ψ(r, t))ψ(r, t) 2m Nonlinear, complex susceptibility χ[e(ψ(r, t))], E 2 α = ɛ 2 α + g 2 ψ 0 2 i t ψ nlin = U ψ 2 ψ i t ψ loss = iκψ i t ψ gain = iγ eff (µ B )ψ Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 24 / 36

69 Limits of gap equation Gap equation: (µ s ω 0 + iκ) ψ = χ(ψ, µ s )ψ Local density limit: Gross-Pitaevskii equation ( i t + iκ [V (r) 2 2 ]) ψ(r) = χ(ψ(r, t))ψ(r, t) 2m Nonlinear, complex susceptibility χ[e(ψ(r, t))], E 2 α = ɛ 2 α + g 2 ψ 0 2 i t ψ loss = iκψ i t ψ nlin = U ψ 2 ψ i t ψ gain = iγ eff (µ B )ψ iγ ψ 2 ψ Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 24 / 36

70 Limits of gap equation Gap equation: (µ s ω 0 + iκ) ψ = χ(ψ, µ s )ψ Local density limit: Gross-Pitaevskii equation ( i t + iκ [V (r) 2 2 ]) ψ(r) = χ(ψ(r, t))ψ(r, t) 2m Nonlinear, complex susceptibility χ[e(ψ(r, t))], E 2 α = ɛ 2 α + g 2 ψ 0 2 i t ψ loss = iκψ i t ψ nlin = U ψ 2 ψ i t ψ gain = iγ eff (µ B )ψ iγ ψ 2 ψ i t ψ = [ 2 2 2m + V (r) + U ψ 2 + i ( γ eff (µ B ) κ Γ ψ 2) ] ψ Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 24 / 36

71 Gross-Pitaevskii equation: Harmonic trap i t ψ = [ 2 2 2m + mω2 2 r 2 + U ψ 2 + i ( γ eff κ Γ ψ 2) ] ψ [Keeling & Berloff, PRL, 08] Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 25 / 36

72 Gross-Pitaevskii equation: Harmonic trap i t ψ = [ 2 2 2m + mω2 2 r 2 + U ψ 2 + i ( γ eff κ Γ ψ 2) ] ψ Density Radius [Keeling & Berloff, PRL, 08] Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 25 / 36

73 Stability of Thomas-Fermi solution 1 2 tρ+ (ρv) = 1 (γ net Γρ)ρ 3γ net 2Γ Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 26 / 36

74 Stability of Thomas-Fermi solution High m modes: δρ n,m e imθ r m tρ+ (ρv) = 1 (γ net Γρ)ρ Unstable growth 3γ net 2Γ Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 26 / 36

75 Stability of Thomas-Fermi solution High m modes: δρ n,m e imθ r m tρ+ (ρv) = 1 (γ netθ(r 0 r) Γρ)ρ Stabilised 3γ net 2Γ Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 26 / 36

76 Stability of Thomas-Fermi solution High m modes: δρ n,m e imθ r m tρ+ (ρv) = 1 (γ netθ(r 0 r) Γρ)ρ???? 3γ net 2Γ Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 26 / 36

77 Time evolution: t=0 t=2 t=22 t=30 t=35 t=40 t=45 t=56 [Keeling & Berloff, PRL, 08] Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 27 / 36

78 Why vortices Density profile Thomas-Fermi in flattened trap Density Cross Section [ρ(v Ω r)] = (γ net Θ(r 0 r) Γρ) ρ, µ = m 2 v Ω r 2 + m 2 r 2 (ω 2 Ω 2 ) + Uρ 2 2 ρ 2m ρ v = Ω r, Ω = ω, ρ = γ net Γ Θ(r 0 r) = µ U Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 28 / 36

79 Why vortices Density profile Thomas-Fermi in flattened trap Density Cross Section Rotating solution: i t ψ = (µ 2ΩL z )ψ [ρ(v Ω r)] = (γ net Θ(r 0 r) Γρ) ρ, µ = m 2 v Ω r 2 + m 2 r 2 (ω 2 Ω 2 ) + Uρ 2 2 ρ 2m ρ v = Ω r, Ω = ω, ρ = γ net Γ Θ(r 0 r) = µ U Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 28 / 36

80 Why vortices Density profile Thomas-Fermi in flattened trap Density Cross Section Rotating solution: i t ψ = (µ 2ΩL z )ψ [ρ(v Ω r)] = (γ net Θ(r 0 r) Γρ) ρ, µ = m 2 v Ω r 2 + m 2 r 2 (ω 2 Ω 2 ) + Uρ 2 2 ρ 2m ρ v = Ω r, Ω = ω, ρ = γ net Γ Θ(r 0 r) = µ U Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 28 / 36

81 Why vortices Density profile Thomas-Fermi in flattened trap Density Cross Section Rotating solution: i t ψ = (µ 2ΩL z )ψ [ρ(v Ω r)] = (γ net Θ(r 0 r) Γρ) ρ, µ = m 2 v Ω r 2 + m 2 r 2 (ω 2 Ω 2 ) + Uρ 2 2 ρ 2m ρ v = Ω r, Ω = ω, ρ = γ net Γ Θ(r 0 r) = µ U Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 28 / 36

82 Why vortices Density profile Thomas-Fermi in flattened trap Density Cross Section Rotating solution: i t ψ = (µ 2ΩL z )ψ [ρ(v Ω r)] = (γ net Θ(r 0 r) Γρ) ρ, µ = m 2 v Ω r 2 + m 2 r 2 (ω 2 Ω 2 ) + Uρ 2 2 ρ 2m ρ v = Ω r, Ω = ω, ρ = γ net Γ Θ(r 0 r) = µ U Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 28 / 36

83 Why vortices Density profile Thomas-Fermi in flattened trap Density Cross Section Rotating solution: i t ψ = (µ 2ΩL z )ψ [ρ(v Ω r)] = (γ net Θ(r 0 r) Γρ) ρ, µ = m 2 v Ω r 2 + m 2 r 2 (ω 2 Ω 2 ) + Uρ 2 2 ρ 2m ρ v = Ω r, Ω = ω, ρ = γ net Γ Θ(r 0 r) = µ U Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 28 / 36

84 Why vortices Density profile Thomas-Fermi in flattened trap Density Cross Section Rotating solution: i t ψ = (µ 2ΩL z )ψ [ρ(v Ω r)] = (γ net Θ(r 0 r) Γρ) ρ, µ = m 2 v Ω r 2 + m 2 r 2 (ω 2 Ω 2 ) + Uρ 2 2 ρ 2m ρ v = Ω r, Ω = ω, ρ = γ net Γ Θ(r 0 r) = µ U Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 28 / 36

85 Why vortices Density profile Thomas-Fermi in flattened trap Density Cross Section Rotating solution: i t ψ = (µ 2ΩL z )ψ [ρ(v Ω r)] = (γ net Θ(r 0 r) Γρ) ρ, µ = m 2 v Ω r 2 + m 2 r 2 (ω 2 Ω 2 ) + Uρ 2 2 ρ 2m ρ v = Ω r, Ω = ω, ρ = γ net Γ Θ(r 0 r) = µ U Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 28 / 36

86 Why vortices Density profile Thomas-Fermi in flattened trap Density Cross Section Rotating solution: i t ψ = (µ 2ΩL z )ψ [ρ(v Ω r)] = (γ net Θ(r 0 r) Γρ) ρ, µ = m 2 v Ω r 2 + m 2 r 2 (ω 2 Ω 2 ) + Uρ 2 2 ρ 2m ρ v = Ω r, Ω = ω, ρ = γ net Γ Θ(r 0 r) = µ U Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 28 / 36

87 Why vortices Density profile Thomas-Fermi in flattened trap Density Cross Section Rotating solution: i t ψ = (µ 2ΩL z )ψ [ρ(v Ω r)] = (γ net Θ(r 0 r) Γρ) ρ, µ = m 2 v Ω r 2 + m 2 r 2 (ω 2 Ω 2 ) + Uρ 2 2 ρ 2m ρ v = Ω r, Ω = ω, ρ = γ net Γ Θ(r 0 r) = µ U Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 28 / 36

88 Why vortices: chemical potential vs size µ Thomas-Fermi : µ = f (r 0 ) stable 1 Vortex : µ = Uγ net Γ unstable r 0 Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 29 / 36

89 Overview 1 Introduction to microcavity polaritons 2 Model and review of equilibrium results Disorder-localised exciton model Equilibrium mean field theory 3 Microscopic non-equilibrium model Model and mean-field theory Fluctuations Stability of normal state lasing vs condensation Condensed spectrum 4 Macroscopic phenomenology Gross Pitaevskii equation in an harmonic trap Internal Josephson effect and spatial variation Spin degree of freedom Spin and spatial degrees of freedom 5 Conclusions Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 30 / 36

90 Polariton spin degree of freedom Results so far do not involve polariton spin: Left- and Right-circular polarised polaritons states. For weakly-interacting dilute Bose gas model: Tendency to biexciton formation: U 1 Magnetic field: Quantum well interface, break rotation symmetry: J 1 Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 31 / 36

91 Polariton spin degree of freedom Results so far do not involve polariton spin: Left- and Right-circular polarised polaritons states. For weakly-interacting dilute Bose gas model: H = Ψ L 2 2m + Ψ R 2 2m + U 0 ( Ψ L 2 + Ψ R 2) 2 2 Tendency to biexciton formation: U 1 Magnetic field: Quantum well interface, break rotation symmetry: J 1 Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 31 / 36

92 Polariton spin degree of freedom Results so far do not involve polariton spin: Left- and Right-circular polarised polaritons states. For weakly-interacting dilute Bose gas model: H = Ψ L 2 2m + Ψ R 2 2m + U 0 ( Ψ L 2 + Ψ R 2) 2 2 2U 1 Ψ L 2 Ψ R 2 + ( Ψ L 2 Ψ R 2) ) + J 1 (Ψ 2 L Ψ R + H.c. Tendency to biexciton formation: U 1 Magnetic field: Quantum well interface, break rotation symmetry: J 1 Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 31 / 36

93 Non-equilibrium spinor system Spinor Gross-Pitaevskii equation: i t ψ L = [ 2 2 2m + V (r) U 0 ψ L 2 + (U 0 2U 1 ) ψ R 2 + i ( γ eff (µ B ) κ Γ ψ L 2)] ψ L + J 1 ψ R J 1 interconversion. How does this interact with currents. Two-mode case (neglect spatial variation): [Wouters PRB 08] To recap results write ψ L,R = ρ L,R e i(φ±θ/2), ρ L,R = R ± z. Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 32 / 36

94 Non-equilibrium spinor system Spinor Gross-Pitaevskii equation: i t ψ L = [ 2 2 2m + V (r) U 0 ψ L 2 + (U 0 2U 1 ) ψ R 2 + i ( γ eff (µ B ) κ Γ ψ L 2)] ψ L + J 1 ψ R J 1 interconversion. How does this interact with currents. Two-mode case (neglect spatial variation): [Wouters PRB 08] To recap results write ψ L,R = ρ L,R e i(φ±θ/2), ρ L,R = R ± z. Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 32 / 36

95 Non-equilibrium spinor system Spinor Gross-Pitaevskii equation: i t ψ L = [ 2 2 2m + V (r) U 0 ψ L 2 + (U 0 2U 1 ) ψ R 2 + i ( γ eff (µ B ) κ Γ ψ L 2)] ψ L + J 1 ψ R J 1 interconversion. How does this interact with currents. Two-mode case (neglect spatial variation): [Wouters PRB 08] To recap results write ψ L,R = ρ L,R e i(φ±θ/2), ρ L,R = R ± z. Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 32 / 36

96 Non-equilibrium spinor system Spinor Gross-Pitaevskii equation: i t ψ L = [ 2 2 2m + V (r) U 0 ψ L 2 + (U 0 2U 1 ) ψ R 2 + i ( γ eff (µ B ) κ Γ ψ L 2)] ψ L + J 1 ψ R J 1 interconversion. How does this interact with currents. Two-mode case (neglect spatial variation): [Wouters PRB 08] To recap results write ψ L,R = ρ L,R e i(φ±θ/2), ρ L,R = R ± z. Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 32 / 36

97 Non-equilibrium spinor system Spinor Gross-Pitaevskii equation: i t ψ L = [ 2 2 2m + V (r) U 0 ψ L 2 + (U 0 2U 1 ) ψ R 2 + i ( γ eff (µ B ) κ Γ ψ L 2)] ψ L + J 1 ψ R J 1 interconversion. How does this interact with currents. Two-mode case (neglect spatial variation): [Wouters PRB 08] To recap results write ψ L,R = ρ L,R e i(φ±θ/2), ρ L,R = R ± z. ( Ṙ = 2σ R γ net Γ R2 z 2) θ = 4U 1 z + 2 J 1z cos(θ) R 2 z 2 ż = 2(γ net 2ΓR)z 2J 1 R 2 z 2 sin(θ) Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 32 / 36

98 Two-mode system summary ( Ṙ = 2σ R γ net Γ R2 z 2) θ = 4U 1 z + 2 J 1z cos(θ) R 2 z 2 ż = 2(γ net 2ΓR)z 2J 1 R 2 z 2 sin(θ). Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 33 / 36

99 Two-mode system summary ( Ṙ = 2σ R γ net Γ R2 z 2) θ = 4U 1 z + 2 J 1z cos(θ) R 2 z 2 ż = 2(γ net 2ΓR)z 2J 1 R 2 z 2 sin(θ) Josephson regime J U 1 R. Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 33 / 36

100 Two-mode system summary ( Ṙ = 2σ R γ net Γ R2 z 2) θ = 4U 1 z + 2 J 1z cos(θ) R 2 z 2 ż = 2(γ net 2ΓR)z 2J 1 R 2 z 2 sin(θ) Josephson regime J U 1 R & z R. Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 33 / 36

101 Two-mode system summary ( Ṙ = 2σ R γ net Γ R2 z 2) θ = 4U 1 z + 2 J 1z cos(θ) R 2 z 2 ż = 2(γ net 2ΓR)z 2J 1 R 2 z 2 sin(θ) Josephson regime J U 1 R & z R. R = γ net /Γ θ = 4U 1 z, ż = 2γ net z 2J 1 R 0 sin(θ) Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 33 / 36

102 Two-mode system summary ( Ṙ = 2σ R γ net Γ R2 z 2) θ = 4U 1 z + 2 J 1z cos(θ) R 2 z 2 ż = 2(γ net 2ΓR)z 2J 1 R 2 z 2 sin(θ) Josephson regime J U 1 R & z R. R = γ net /Γ θ = 4U 1 z, ż = 2γ net z 2J 1 R 0 sin(θ) Damped, driven pendulum γ net 8U J 1 1 θ + 2γ net θ = 8U 1 J 1 R 0 sin(θ) 2γ net Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 33 / 36

103 Two-mode system summary ( Ṙ = 2σ R γ net Γ R2 z 2) θ = 4U 1 z + 2 J 1z cos(θ) R 2 z 2 ż = 2(γ net 2ΓR)z 2J 1 R 2 z 2 sin(θ) γ net 8U J 1 1 Josephson regime J U 1 R & z R. R = γ net /Γ θ = 4U 1 z, ż = 2γ net z 2J 1 R 0 sin(θ) Damped, driven pendulum γ net Bistable θ + 2γ net θ = 8U 1 J 1 R 0 sin(θ) 2γ net Stable limit cycle Stable fixed point 1.5 γ net Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 33 / 36

104 Trapped spinor system Consider: V (r) = mω 2 r 2 /2 γ net γ net Θ(r 0 r) Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 34 / 36

105 Trapped spinor system Consider: V (r) = mω 2 r 2 /2 γ net γ net Θ(r 0 r) Track µ L,R = t ln ψ L,R vs. Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 34 / 36

106 Trapped spinor system Consider: V (r) = mω 2 r 2 /2 γ net γ net Θ(r 0 r) Track µ L,R = t ln ψ L,R vs. 22 Simple case J 1 = γ net ; r 0 < r TF µ µ R L Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 34 / 36

107 Trapped spinor system Consider: V (r) = mω 2 r 2 /2 γ net γ net Θ(r 0 r) Track µ L,R = t ln ψ L,R vs. 22 Simple case J 1 = γ net ; r 0 < r TF. 20 Marginal case J 1 = γ net ; r 0 r TF. 18 θ = 4U 1 z = 0 16 causes L(R) to grow (shrink) µ µ R L Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 34 / 36 µ µ R L

108 Trapped spinor system Consider: V (r) = mω 2 r 2 /2 γ net γ net Θ(r 0 r) Track µ L,R = t ln ψ L,R vs. 22 Simple case J 1 = γ net ; r 0 < r TF. 20 Marginal case J 1 = γ net ; r 0 r TF. 18 µ R θ = 4U 1 z = 0 16 causes L(R) to grow (shrink) µ L µ µ R L Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 34 / 36

109 Trapped spinor system Consider: V (r) = mω 2 r 2 /2 γ net γ net Θ(r 0 r) Track µ L,R = t ln ψ L,R vs. 22 Simple case J 1 = γ net ; r 0 < r TF. 20 Marginal case J 1 = γ net ; r 0 r TF. 18 θ = 4U 1 z = 0 16 causes L(R) to grow (shrink) µ µ R L Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 34 / 36 µ µ R L

110 Trapped spinor system phase portraits Simple case not so simple µ µ R L Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 35 / 36

111 Trapped spinor system phase portraits Simple case not so simple µ µ R L Examine phase portrait t θ vs θ Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 35 / 36

112 Trapped spinor system phase portraits Simple case not so simple µ µ R L Examine phase portrait t θ vs θ Retrograde motion; limit cycles with winding 0,1,2; chaotic behaviour (large J 1, ) t θ=µ L µ R = Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 35 / 36

113 Trapped spinor system phase portraits Simple case not so simple µ µ R L Examine phase portrait t θ vs θ Retrograde motion; limit cycles with winding 0,1,2; chaotic behaviour (large J 1, ) t θ=µ L µ R t θ=µ L µ R = = 3.25 Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 35 / 36

114 Conclusions Effects of pumping on mean-field theory Pumping Bath γ Excitons System Cavity mode κ Bulk photon modes Bath Temperature Increasing γ Density 1 Instability of normal state Translating: condensation lasing Energy of zero (units of g) Zero of Re Zero of Im Bath chemical potential, µ B /g Energy of zero Zero of Re Zero of Im -(2γ/g) 2 (2γκ/g 2 ) System inversion, N Modification to Thomas-Fermi profile Spontaneous rotating vortex lattice t=22 t=35 t=56 Spinor model. Steady states & fluctuations. Temperature neither polarised normal γ net Stable limit cycle Bistable Stable fixed point t θ=µ L µ R r l+r Magnetic field, l 1.5 γnet Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 36 / 36

115 Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 37 / 55

116 Extra slides 6 Equilibrium results 7 Mean-field Keldysh theory 8 Condensate lineshape 9 More on vortices Instability of Thomas-Fermi Stability of lattice Observation 10 Spinor problem Two level systems; phase diagram Two model model,dispersion 11 Superfluidity Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 38 / 55

117 Fluctuation corrections to phase boundary Fluctuation corrections to density ρ ρ 0 + q δψ δψ In 2D system: modified critical condition: ρ s = ρ ρ normal = # 2m 2 k B T Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 39 / 55

118 Fluctuation corrections to phase boundary Fluctuation corrections to density ρ ρ 0 + q δψ δψ In 2D system: modified critical condition: ρ s = ρ ρ normal = # 2m non-condensed condensed 2 k B T k B T n [cm -2 ] Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 39 / 55

119 Fluctuation corrections to phase boundary Fluctuation corrections to density ρ ρ 0 + q δψ δψ In 2D system: modified critical condition: ρ s = ρ ρ normal = # 2m non-condensed condensed 2 k B T k B T n [cm -2 ] Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 39 / 55

120 Fluctuation corrections to phase boundary Fluctuation corrections to density ρ ρ 0 + q δψ δψ In 2D system: modified critical condition: ρ s = ρ ρ normal = # 2m 2 k B T Energy k k 40 non-condensed 30 condensed k B T n [cm -2 ] Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 39 / 55

121 Fluctuation corrections to phase boundary Fluctuation corrections to density ρ ρ 0 + q δψ δψ In 2D system: modified critical condition: ρ s = ρ ρ normal = # 2m 2 k B T Energy k k 40 non-condensed 30 condensed k B T n [cm -2 ] Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 39 / 55

122 Fluctuation corrections to phase boundary Fluctuation corrections to density ρ ρ 0 + q δψ δψ In 2D system: modified critical condition: ρ s = ρ ρ normal = # 2m 2 k B T Energy k k 40 non-condensed condensed k B T n [cm -2 ] Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 39 / 55

123 Fluctuation corrections to phase boundary Fluctuation corrections to density ρ ρ 0 + q δψ δψ In 2D system: modified critical condition: ρ s = ρ ρ normal = # 2m 2 k B T Energy k k 40 non-condensed 30 condensed k B T Second BCS crossover at na 2 B n [cm -2 ] Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 39 / 55

124 Blueshfit and experimental phase boundary Blueshift: Energy [mev] mean field threshold δ=5.3mev, k B T eff =17K n [cm 2 ] Clean limit: δe LP Ry X a 2 X n + Ω Ra 2 X n Here: Ω R a 2 X Ω Rξ 2 [PRB ] Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 40 / 55

125 Blueshfit and experimental phase boundary Blueshift: Energy [mev] mean field threshold Phase diagram: k B T δ=5.3mev, k B T eff =17K n [cm 2 ] non-condensed condensed n [cm -2 ] Clean limit: δe LP Ry X a 2 X n + Ω Ra 2 X n Here: Ω R a 2 X Ω Rξ 2 [PRB ] CdTe: T [K] n [cm -2 ] δ=5.5mev δ=5.3mev T cryo =5K δ=6.9mev, T cryo =15K δ=7.2mev, T cryo =5K 0 20 δ=7.6mev, T cryo =12K 0 20 δ=10.5mev, T cryo =5K 0 20 δ=12.5mev, T cryo 5K n [cm -2 ] Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 40 / 55

126 Zero temperature phase diagram ω 0 iκ = g 2 γ α dν (F a + F b )ν + (F b F a )( ɛ α + iγ) 2π [(ν E α ) 2 + γ 2 ][(ν + E α ) 2 + γ 2 ]. Pump bath coupling γ/g κ/g= µ B /g Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 41 / 55

127 Zero temperature phase diagram ω 0 iκ = g 2 γ α dν (F a + F b )ν + (F b F a )( ɛ α + iγ) 2π [(ν E α ) 2 + γ 2 ][(ν + E α ) 2 + γ 2 ]. Pump bath coupling γ/g κ/g= µ B /g Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 41 / 55

128 Zero temperature phase diagram ω 0 iκ = g 2 γ α dν (F a + F b )ν + (F b F a )( ɛ α + iγ) 2π [(ν E α ) 2 + γ 2 ][(ν + E α ) 2 + γ 2 ]. 1 Pump bath coupling γ/g µ B /g=-0.05 κ/g=0.05 µ B /g= κ/g µ B /g κ/g Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 41 / 55

129 Finite size effects: Single mode vs many mode ψ (r, t)ψ(0, r ) ψ 0 2 exp [ D φφ (t, r, r ) ] Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 42 / 55

130 Finite size effects: Single mode vs many mode ψ (r, t)ψ(0, r ) ψ 0 2 exp [ D φφ (t, r, r ) ] D φφ (t, r, r ) from sum of phase modes. Study ct r limit: n max D φφ (t, r, r) n dω 2π ϕ n (r) 2 (1 e iωt ) (ω + ix) 2 + x 2 (n ) 2 2 Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 42 / 55

131 Finite size effects: Single mode vs many mode ψ (r, t)ψ(0, r ) ψ 0 2 exp [ D φφ (t, r, r ) ] D φφ (t, r, r ) from sum of phase modes. Study ct r limit: n max D φφ (t, r, r) n dω 2π ϕ n (r) 2 (1 e iωt ) (ω + ix) 2 + x 2 (n ) 2 2 x/t E max E max Energy D φφ 1 + ln(e max t/x) Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 42 / 55

132 Finite size effects: Single mode vs many mode ψ (r, t)ψ(0, r ) ψ 0 2 exp [ D φφ (t, r, r ) ] D φφ (t, r, r ) from sum of phase modes. Study ct r limit: n max D φφ (t, r, r) n dω 2π ϕ n (r) 2 (1 e iωt ) (ω + ix) 2 + x 2 (n ) 2 2 E max x/t E max Energy D φφ 1 + ln(e max t/x) x/t Emax E max Energy D φφ ( πc 2x ) ( t ) 2x Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 42 / 55

133 Relating finite-size spectrum to self phase modulation Single mode spectrum: D φφ (t, r, r) dω ϕ n (r) 2 (1 e iωt ) 2π (ω + ix) 2 + x 2 (n ) 2 2 Connection to Kubo Formula [Eastham and Whittaker, arxiv: ] Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 43 / 55

134 Relating finite-size spectrum to self phase modulation Single mode spectrum: D φφ (t, r, r) dω ϕ n (r) 2 (1 e iωt ) 2π (ω + ix) 2 + x 2 2 Connection to Kubo Formula [Eastham and Whittaker, arxiv: ] Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 43 / 55

135 Relating finite-size spectrum to self phase modulation Single mode spectrum: D φφ (t, r, r) dω ϕ n (r) 2 (1 e iωt ) 2π (ω + ix) 2 + x 2 2 Connection to Kubo Formula [Eastham and Whittaker, arxiv: ] Imψ t φ = UδN φ Reψ Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 43 / 55

136 Relating finite-size spectrum to self phase modulation Single mode spectrum: D φφ (t, r, r) dω ϕ n (r) 2 (1 e iωt ) 2π (ω + ix) 2 + x 2 2 Connection to Kubo Formula [Eastham and Whittaker, arxiv: ] Imψ t φ = UδN t δn = ΓδN + F (t), F (t)f (t ) = Cδ(t t ) φ Reψ Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 43 / 55

137 Relating finite-size spectrum to self phase modulation Single mode spectrum: D φφ (t, r, r) dω ϕ n (r) 2 (1 e iωt ) 2π (ω + ix) 2 + x 2 2 Connection to Kubo Formula [Eastham and Whittaker, arxiv: ] Imψ φ Reψ t φ = UδN t δn = ΓδN + F (t), F (t)f (t ) = Cδ(t t ) dω D φφ (t) = 2π (1 eiωt ) φ ω φ ω Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 43 / 55

138 Relating finite-size spectrum to self phase modulation Single mode spectrum: D φφ (t, r, r) dω ϕ n (r) 2 (1 e iωt ) 2π (ω + ix) 2 + x 2 2 Connection to Kubo Formula [Eastham and Whittaker, arxiv: ] Imψ φ Reψ t φ = UδN t δn = ΓδN + F (t), F (t)f (t ) = Cδ(t t ) dω D φφ (t) = 2π (1 CU 2 eiωt ) ω(ω + iγ) 2 Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 43 / 55

139 Relating finite-size spectrum to self phase modulation Single mode spectrum: D φφ (t, r, r) dω ϕ n (r) 2 (1 e iωt ) 2π (ω + ix) 2 + x 2 2 Connection to Kubo Formula [Eastham and Whittaker, arxiv: ] Imψ φ Reψ t φ = UδN t δn = ΓδN + F (t), F (t)f (t ) = Cδ(t t ) dω D φφ (t) = 2π (1 CU 2 eiωt ) ω(ω + iγ) 2 = CU2 [ 2Γ 3 Γt 1 + e Γt] Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 43 / 55

140 Instability of Thomas-Fermi: details ψ = ρe iφ v = m φ 1 2 tρ + (ρv) = (γ net Γρ)ρ t v + (Uρ + mω2 2 r 2 + m 2 v 2 ) = 0 Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 44 / 55

141 Instability of Thomas-Fermi: details ψ = ρe iφ v = m φ 1 2 tρ + (ρv) = (γ net Γρ)ρ t v + (Uρ + mω2 2 r 2 + m 2 v 2 ) = 0 Consider ρ ρ + δρ, v v + δv. Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 44 / 55

142 Instability of Thomas-Fermi: details ψ = ρe iφ v = m φ If γ net, Γ 0, can find normal modes in 2D trap: 1 2 tρ + (ρv) = (γ net Γρ)ρ t v + (Uρ + mω2 2 r 2 + m 2 v 2 ) = 0 δρ n,m (r, θ, t) = e imθ h n,m (r)e iωn,mt ω n,m = ω2 m(1 + 2n) + 2n(n + 1) Consider ρ ρ + δρ, v v + δv. Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 44 / 55

143 Instability of Thomas-Fermi: details ψ = ρe iφ v = m φ If γ net, Γ 0, can find normal modes in 2D trap: 1 2 tρ + (ρv) = (γ net Γρ)ρ t v + (Uρ + mω2 2 r 2 + m 2 v 2 ) = 0 δρ n,m (r, θ, t) = e imθ h n,m (r)e iωn,mt ω n,m = ω2 m(1 + 2n) + 2n(n + 1) Consider ρ ρ + δρ, v v + δv. Add weak pumping/decay: Instability [ ] m(1 + 2n) + 2n(n + 1) m 2 ω n,n ω n,m + iγ net 2m(1 + 2n) + 4n(n + 1) + m 2 Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 44 / 55

144 Why vortices Density profile Thomas-Fermi in flattened trap Density Cross Section [ρ(v Ω r)] = (γ net Θ(r 0 r) Γρ) ρ, µ = m 2 v Ω r 2 + m 2 r 2 (ω 2 Ω 2 ) + Uρ 2 2 ρ 2m ρ v = Ω r, Ω = ω, ρ = γ net Γ Θ(r 0 r) = µ U Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 45 / 55

145 Why vortices Density profile Thomas-Fermi in flattened trap Density Cross Section Rotating solution: i t ψ = (µ 2ΩL z )ψ [ρ(v Ω r)] = (γ net Θ(r 0 r) Γρ) ρ, µ = m 2 v Ω r 2 + m 2 r 2 (ω 2 Ω 2 ) + Uρ 2 2 ρ 2m ρ v = Ω r, Ω = ω, ρ = γ net Γ Θ(r 0 r) = µ U Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 45 / 55

146 Why vortices Density profile Thomas-Fermi in flattened trap Density Cross Section Rotating solution: i t ψ = (µ 2ΩL z )ψ [ρ(v Ω r)] = (γ net Θ(r 0 r) Γρ) ρ, µ = m 2 v Ω r 2 + m 2 r 2 (ω 2 Ω 2 ) + Uρ 2 2 ρ 2m ρ v = Ω r, Ω = ω, ρ = γ net Γ Θ(r 0 r) = µ U Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 45 / 55

147 Why vortices Density profile Thomas-Fermi in flattened trap Density Cross Section Rotating solution: i t ψ = (µ 2ΩL z )ψ [ρ(v Ω r)] = (γ net Θ(r 0 r) Γρ) ρ, µ = m 2 v Ω r 2 + m 2 r 2 (ω 2 Ω 2 ) + Uρ 2 2 ρ 2m ρ v = Ω r, Ω = ω, ρ = γ net Γ Θ(r 0 r) = µ U Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 45 / 55

148 Why vortices Density profile Thomas-Fermi in flattened trap Density Cross Section Rotating solution: i t ψ = (µ 2ΩL z )ψ [ρ(v Ω r)] = (γ net Θ(r 0 r) Γρ) ρ, µ = m 2 v Ω r 2 + m 2 r 2 (ω 2 Ω 2 ) + Uρ 2 2 ρ 2m ρ v = Ω r, Ω = ω, ρ = γ net Γ Θ(r 0 r) = µ U Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 45 / 55

149 Why vortices Density profile Thomas-Fermi in flattened trap Density Cross Section Rotating solution: i t ψ = (µ 2ΩL z )ψ [ρ(v Ω r)] = (γ net Θ(r 0 r) Γρ) ρ, µ = m 2 v Ω r 2 + m 2 r 2 (ω 2 Ω 2 ) + Uρ 2 2 ρ 2m ρ v = Ω r, Ω = ω, ρ = γ net Γ Θ(r 0 r) = µ U Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 45 / 55

150 Why vortices Density profile Thomas-Fermi in flattened trap Density Cross Section Rotating solution: i t ψ = (µ 2ΩL z )ψ [ρ(v Ω r)] = (γ net Θ(r 0 r) Γρ) ρ, µ = m 2 v Ω r 2 + m 2 r 2 (ω 2 Ω 2 ) + Uρ 2 2 ρ 2m ρ v = Ω r, Ω = ω, ρ = γ net Γ Θ(r 0 r) = µ U Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 45 / 55

151 Why vortices Density profile Thomas-Fermi in flattened trap Density Cross Section Rotating solution: i t ψ = (µ 2ΩL z )ψ [ρ(v Ω r)] = (γ net Θ(r 0 r) Γρ) ρ, µ = m 2 v Ω r 2 + m 2 r 2 (ω 2 Ω 2 ) + Uρ 2 2 ρ 2m ρ v = Ω r, Ω = ω, ρ = γ net Γ Θ(r 0 r) = µ U Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 45 / 55

152 Why vortices Density profile Thomas-Fermi in flattened trap Density Cross Section Rotating solution: i t ψ = (µ 2ΩL z )ψ [ρ(v Ω r)] = (γ net Θ(r 0 r) Γρ) ρ, µ = m 2 v Ω r 2 + m 2 r 2 (ω 2 Ω 2 ) + Uρ 2 2 ρ 2m ρ v = Ω r, Ω = ω, ρ = γ net Γ Θ(r 0 r) = µ U Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 45 / 55

153 Why vortices Density profile Thomas-Fermi in flattened trap Density Cross Section Rotating solution: i t ψ = (µ 2ΩL z )ψ [ρ(v Ω r)] = (γ net Θ(r 0 r) Γρ) ρ, µ = m 2 v Ω r 2 + m 2 r 2 (ω 2 Ω 2 ) + Uρ 2 2 ρ 2m ρ v = Ω r, Ω = ω, ρ = γ net Γ Θ(r 0 r) = µ U Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 45 / 55

154 Why vortices: chemical potential vs size µ Thomas-Fermi : µ = f (r 0 ) stable 1 Vortex : µ = Uγ net Γ unstable r 0 Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 46 / 55

155 Observing vortices: fringe pattern Sample Beam Splitter CCD Tunable Delay Retroreflector + = Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 47 / 55

156 Observing vortices: fringe pattern Sample t=35 t=40 t=45 Beam Splitter CCD Tunable Delay Retroreflector t=35 t=40 t=45 + = Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 47 / 55

157 Spin in terms of twofour-level systems To include spin, replace 2 level system with 4 levels: 0, L, R, LR Bi-exciton binding E XX U 1 Mean-field: find polarisation given ψ L, ψ R. E XX has weak effect on T c [Marchetti et al PRB, 08] Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 48 / 55

158 Spin in terms of twofour-level systems To include spin, replace 2 level system with 4 levels: 0, L, R, LR ĥ α = 0 gψ L gψ R 0 gψl ε α µ 0 gψ L gψr 0 ε α + µ gψ R 0 gψl gψr 2(ε α µ) E XX Bi-exciton binding E XX U 1 Mean-field: find polarisation given ψ L, ψ R. E XX has weak effect on T c [Marchetti et al PRB, 08] Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 48 / 55

159 Spin in terms of twofour-level systems To include spin, replace 2 level system with 4 levels: 0, L, R, LR ĥ α = 0 gψ L gψ R 0 gψl ε α µ 0 gψ L gψr 0 ε α + µ gψ R 0 gψl gψr 2(ε α µ) E XX Bi-exciton binding E XX U 1 Mean-field: find polarisation given ψ L, ψ R. E XX has weak effect on T c [Marchetti et al PRB, 08] Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 48 / 55

160 Spin in terms of twofour-level systems To include spin, replace 2 level system with 4 levels: 0, L, R, LR ĥ α = 0 gψ L gψ R 0 gψl ε α µ 0 gψ L gψr 0 ε α + µ gψ R 0 gψl gψr 2(ε α µ) E XX Bi-exciton binding E XX U 1 Mean-field: find polarisation given ψ L, ψ R. E XX has weak effect on T c [Marchetti et al PRB, 08] Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 48 / 55

161 Spin in terms of twofour-level systems To include spin, replace 2 level system with 4 levels: 0, L, R, LR ĥ α = 0 gψ L gψ R 0 gψl ε α µ 0 gψ L gψr 0 ε α + µ gψ R 0 gψl gψr 2(ε α µ) E XX 30 Bi-exciton binding E XX U 1 Mean-field: find polarisation given ψ L, ψ R. 5 E XX has weak effect on T c 0 0 [Marchetti et al PRB, 08] T[K] No binding E XX = 4 mev E XX = 8 mev n [cm -2 ] Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 48 / 55

162 Equilibrium phase diagrams J 1 = J 2 = 0. For U 1 = 0.5, Ψ L,R decouple neither T/T deg HFB approx Exact EOS 0.1 r l+r l /k B T deg Circular Eliptical transitions. [Rubo et al Phys. Lett. A 06; Keeling, Phys. Rev. B 08] Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 49 / 55

163 Equilibrium phase diagrams J 1 = J 2 = 0. For U 1 = 0.5, Ψ L,R decouple neither J 1 = 0, J 2 0. Phase locking J 2 cos(2(θ L θ R )). T/T deg HFB approx Exact EOS 0.1 r l+r l /k B T deg Circular Eliptical transitions. [Rubo et al Phys. Lett. A 06; Keeling, Phys. Rev. B 08] Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 49 / 55

164 Equilibrium phase diagrams J 1 = J 2 = 0. For U 1 = 0.5, Ψ L,R decouple. T/T deg HFB approx Exact EOS neither 0.1 r l+r l /k B T deg Circular Eliptical transitions. J 1 = 0, J 2 0. Phase locking J 2 cos(2(θ L θ R )). Temperature r neither l+r Magnetic field, polarised normal Separate Ising/XY transitions. l [Rubo et al Phys. Lett. A 06; Keeling, Phys. Rev. B 08] Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 49 / 55

165 Equilibrium phase diagrams J 1 = J 2 = 0. For U 1 = 0.5, Ψ L,R decouple. T/T deg HFB approx Exact EOS neither 0.1 r l+r l /k B T deg Circular Eliptical transitions. J 1 = 0, J 2 0. Phase locking J 2 cos(2(θ L θ R )). Temperature r neither l+r Magnetic field, polarised normal Separate Ising/XY transitions. J 1 0: Eqbm state locked. l [Rubo et al Phys. Lett. A 06; Keeling, Phys. Rev. B 08] Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 49 / 55

166 Mathematical outline 2D Single component equation of state: n(µ, T ) = Tf (x = µ/t ) For two components: n 0 = T [ f ( µ + Ω T ) + f At critical point for one component: [ ( n 0 = T f c + f x c + 2Ω T ( )] µ Ω T )] Hence: T = n 0 f c + f ( x c + 2Ω T ) Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 50 / 55

167 Mathematical outline 2D Single component equation of state: n(µ, T ) = Tf (x = µ/t ) For two components: n 0 = T [ f ( µ + Ω T ) + f At critical point for one component: [ ( n 0 = T f c + f x c + 2Ω T ( )] µ Ω T )] Hence: T = n 0 f c + f ( x c + 2Ω T ) Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 50 / 55

168 Mathematical outline 2D Single component equation of state: n(µ, T ) = Tf (x = µ/t ) For two components: n 0 = T [ f ( µ + Ω T ) + f At critical point for one component: [ ( n 0 = T f c + f x c + 2Ω T ( )] µ Ω T )] Hence: T = n 0 f c + f ( x c + 2Ω T ) Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 50 / 55

169 Mathematical outline 2D Single component equation of state: n(µ, T ) = Tf (x = µ/t ) For two components: n 0 = T [ f ( µ + Ω T ) + f At critical point for one component: [ ( n 0 = T f c + f x c + 2Ω T ( )] µ Ω T )] Hence: T = n 0 f c + f ( x c + 2Ω T ) Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 50 / 55

170 [ Graphical implementation of T = n 0 / f c + f ( x c + 2Ω T )] 4 3 Bogoliubov µ/g Monte Carlo HFP f x=µ/t Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 51 / 55

171 [ Graphical implementation of T = n 0 / f c + f ( x c + 2Ω T )] 4 3 Bogoliubov µ/g Monte Carlo HFP f Ω/T = x-x c Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 51 / 55

172 [ Graphical implementation of T = n 0 / f c + f ( x c + 2Ω T )] n 0 /T Bogoliubov µ/g Monte Carlo HFP Ω/T Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 51 / 55

173 [ Graphical implementation of T = n 0 / f c + f ( x c + 2Ω T )] 0.4 Bogoliubov µ/g Monte Carlo HFP T/n Ω/T Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 51 / 55

174 [ Graphical implementation of T = n 0 / f c + f ( x c + 2Ω T )] 0.4 Bogoliubov µ/g Monte Carlo HFP T/n Ω/n 0 Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 51 / 55

175 [ Graphical implementation of T = n 0 / f c + f ( x c + 2Ω T )] 0.4 Bogoliubov µ/g Monte Carlo HFP T/n Ω/n 0 Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 51 / 55

176 Two-mode model bistability γ net Stable limit cycle Bistable c Stable fixed point 1.5 γ net Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 52 / 55

177 Two-mode model bistability γ net Stable limit cycle 4 =6.6 =6.8 =7.0 Bistable c Stable fixed point γ net z dθ/dt =7.2 =7.4 =7.6 4 =7.8 0 =8.0 = π 2π 0 π 2π 0 π 2π θ Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 52 / 55

178 Two-mode model bistability γ net Stable limit cycle 4 =6.6 =6.8 =7.0 ω Bistable ω= c Stable fixed point 1.5 γ net =7.8 c : ω [ln( c )] 1 0 π 2π z dθ/dt =7.2 =7.4 =7.6 =8.0 0 π 2π 0 π 2π θ =8.2 Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 52 / 55

179 Spatial freedom: Homgeneous case < c θ + 2γ net θ = 8U 1 J 1 R 0 sin(θ) 2γ net Steady state condition: ( 8U 1 J 1 R 0 sin(θ) = 2γ net ψ L,R e iµt ψl,r 0 + u 1e ik r+( iω κ)t + v1 e ik r+(iω κ)t) Define Ω 2 p = 8U 1 J 1 R 0 cos(θ). At k = 0 Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 53 / 55

180 Spatial freedom: Homgeneous case < c θ + 2γ net θ = 8U 1 J 1 R 0 sin(θ) 2γ net Steady state condition: ( 8U 1 J 1 R 0 sin(θ) = 2γ net ψ L,R e iµt ψl,r 0 + u 1e ik r+( iω κ)t + v1 e ik r+(iω κ)t) Define Ω 2 p = 8U 1 J 1 R 0 cos(θ). At k = 0 Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 53 / 55

181 Spatial freedom: Homgeneous case < c θ + 2γ net θ = 8U 1 J 1 R 0 sin(θ) 2γ net Steady state condition: ( 8U 1 J 1 R 0 sin(θ) = 2γ net ψ L,R e iµt ψl,r 0 + u 1e ik r+( iω κ)t + v1 e ik r+(iω κ)t) Define Ω 2 p = 8U 1 J 1 R 0 cos(θ). At k = 0 Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 53 / 55

182 Spatial freedom: Homgeneous case < c θ + 2γ net θ = 8U 1 J 1 R 0 sin(θ) 2γ net Steady state condition: ( 8U 1 J 1 R 0 sin(θ) = 2γ net ψ L,R e iµt ψl,r 0 + u 1e ik r+( iω κ)t + v1 e ik r+(iω κ)t) Define Ω 2 p = 8U 1 J 1 R 0 cos(θ). At k = 0 ω iκ = 0, 2iγ net iγ net ± i γ net Ω 2 p ω π/4 π/2 3π/4 π 0 π/4 π/2 3π/4 π θ θ Stability requires Ω 2 p > 0. If Ω 2 p < γ net overdamped. Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 53 / 55

183 Spatial variation Varieties of behaviour possible as θ(r), not θ needed to define state. Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 54 / 55

184 Spatial variation Varieties of behaviour possible as θ(r), not θ needed to define state. Plot J 1 sin(θ) vs r. Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 54 / 55

185 Spatial variation Varieties of behaviour possible as θ(r), not θ needed to define state. Plot J 1 sin(θ) vs r. J 1 = 0.5; r 0 > r TF ; = 6 Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 54 / 55

186 Spatial variation Varieties of behaviour possible as θ(r), not θ needed to define state. Plot J 1 sin(θ) vs r. J 1 = 0.5; r 0 > r TF ; = 6 J 1 = 1; r 0 > r TF ; = 6 Counter-rotating. Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 54 / 55