Nonequilibrium quantum condensates: from microscopic theory to macroscopic phenomenology

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

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

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

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

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

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] 40 30 20 10 0 UP Photon Exciton -10 LP 0 10 20 30 40 Angle [degrees] 10-3 10-2 10-1 Wavevector [10 7 cm -1 ] 40 30 20 10 0-10 Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 3 / 36

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

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

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

Other polariton condensation experiments Stress traps for polaritons [Balili et al Science 316 1007 (2007)] Temporal coherence and line narrowing [Love et al Phys. Rev. Lett. 101 067404 (2008)] Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 7 / 36

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. 4 700 (2008)] Soliton propagation [Amo et al Nature 457 291 (2009)] Driven superfluidity [Amo et al Nature Phys. (2009) Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 8 / 36

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

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 96 066405 (2006); PRB 76 115326 (2007)] Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 10 / 36

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 96 066405 (2006); PRB 76 115326 (2007)] Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 10 / 36

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) -3-2 -1 0 1 2 ε-e x [mev] [PRL 96 066405 (2006); PRB 76 115326 (2007)] Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 10 / 36

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

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

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) -3-2 -1 0 1 2 ε-e x [mev] Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 12 / 36

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) -3-2 -1 0 1 2 ε-e x [mev] Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 12 / 36

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) -3-2 -1 0 1 2 ε-e x [mev] Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 12 / 36

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 2-3 -2-1 0 1 2 ε-e x [mev] Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 12 / 36

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 2 + 1 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) -3-2 -1 0 1 2 ε-e x [mev] Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 12 / 36

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) -3-2 -1 0 1 2 ε-e x [mev] non-condensed Density ρ = ψ 2 + 1 A α [ 1 2 ɛ ] α µ tanh(βe α ) 4E α k B T condensed 30 20 10 0 9 9 0 1 10 2 10 n [cm -2 ] Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 12 / 36

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

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 76 193308 (2007)] b Yttrium Iron Garnett. [Demokritov et al, Nature 443 430 (2007)] Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 14 / 36

Non-equilibrium system Photon Exciton Ph Ex Energy In plane momentum Lifetime Thermalisation Linewidth Temperature Atoms 10s 10ms 2.5 10 13 mev 10 8 K 10 9 mev Excitons a 50ns 0.2ns 5 10 5 mev 1K 0.1meV Polaritons 5ps 0.5ps 0.5meV 20K 2meV Magnons b 1µs(??) 100ns(?) 2.5 10 6 mev 300K 30meV a Coupled quantum wells. [Hammack et al PRB 76 193308 (2007)] b Yttrium Iron Garnett. [Demokritov et al, Nature 443 430 (2007)] Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 14 / 36

Non-equilibrium model: baths Pumping Bath 00 11 00 11 00 11 00 11 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

Non-equilibrium model: baths Pumping Bath 00 11 00 11 00 11 00 11 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

Non-equilibrium model: baths Pumping Bath 00 11 00 11 00 11 00 11 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

Non-equilibrium model: baths Pumping Bath 00 11 00 11 00 11 00 11 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 000000000000000000 111111111111111111 0000000000000000000 1111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 1 x 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 0000000000000000000000 1111111111111111111111 0000000000000000000000 1111111111111111111111 0000000000000000000000 1111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 µ /2 0 +µ B B/2 ε x Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 15 / 36

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

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

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

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

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 0.4 0.2 Increasing γ 0 0 0.1 0.2 0.3 0.4 0.5 Density Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 16 / 36

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

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

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

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

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

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

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

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

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

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

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

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

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

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 -1.5-1 = A(ω) + ib(ω), A(ω) [ D 1 (ω) ] 1 K = ic(ω), -0.5 0 0.5 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.) 3 2 1 Density of states, 2 Im[D R ] 0-1.5-1 -0.5 0 0.5 1 Energy (units of g) Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 19 / 36

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 ) + -1.5-1 -0.5 0 0.5 1 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.) 3 2 1 Density of states, 2 Im[D R ] 0-1.5-1 -0.5 0 0.5 1 Energy (units of g) Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 19 / 36

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 ) + -1.5-1 -0.5 0 0.5 1 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.) 3 2 1 Density of states, 2 Im[D R ] Occupation, n(ω) 0-1.5-1 -0.5 0 0.5 1 Energy (units of g) Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 19 / 36

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.) -1.5-1 -0.5 0 0.5 1 A(ω) B(ω) 1 C(ω) 0 3 2 1 Density of states, 2 Im[D R ] Occupation, n(ω) 0-1.5-1 -0.5 0 0.5 1 Energy (units of g) Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 19 / 36

Linewidth, inverse Green s function and gap equation -1.5-1 -0.5 0 0.5 1 A(ω) B(ω) 1 C(ω) 0 Intensity (a.u.) 3 2 1 Density of states, 2 Im[D R ] Occupation, n(ω) 0-1.5-1 -0.5 0 0.5 1 Energy (units of g) Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 20 / 36

Linewidth, inverse Green s function and gap equation Intensity (a.u.) -1.5-1 -0.5 0 0.5 1 A(ω) B(ω) 1 C(ω) 0 3 2 1 Density of states, 2 Im[D R ] Occupation, n(ω) 0-1.5-1 -0.5 0 0.5 1 Energy (units of g) Energy of zero (units of g) 1 0-1 -2-3 Zero of Re Zero of Im -0.6-0.5-0.4-0.3 Bath chemical potential, µ B /g Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 20 / 36

[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

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

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

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

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

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

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

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

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

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

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

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

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 30 25 Density 20 15 10 5 0 0 2 4 6 8 Radius [Keeling & Berloff, PRL, 08] Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 25 / 36

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

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

Stability of Thomas-Fermi solution High m modes: δρ n,m e imθ r m... 1 2 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

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

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

Why vortices 25 20 Density profile Thomas-Fermi in flattened trap Density 15 10 5 0-15 -10-5 0 5 10 15 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

Why vortices 25 20 Density profile Thomas-Fermi in flattened trap Density 15 10 5 0-15 -10-5 0 5 10 15 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

Why vortices: chemical potential vs size µ 35 30 25 20 15 10 Thomas-Fermi : µ = f (r 0 ) stable 1 Vortex : µ = Uγ net Γ unstable 1 1 1 2 2 1 3 3 3 4 6 17 28 44 53 75 5 0 0 2 4 6 8 10 r 0 Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 29 / 36

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

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

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

Non-equilibrium spinor system Spinor Gross-Pitaevskii equation: i t ψ L = [ 2 2 2m + V (r) + 2 + 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

Non-equilibrium spinor system Spinor Gross-Pitaevskii equation: i t ψ L = [ 2 2 2m + V (r) + 2 + 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

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

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

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

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

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

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

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

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

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 18 16 14 12 10 0 1 2 3 4 5 6 µ µ R L Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 34 / 36

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) 14 30 12 25 20 15 10 5 0 1 2 3 4 5 6 7 8 9 10 µ µ R L 10 0 1 2 3 4 5 6 Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 34 / 36 µ µ R L

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) 14 30 12 µ L 25 10 0 1 2 3 20 µ 4 5 6 15 10 5 0 1 2 3 4 5 6 7 8 9 10 µ R L Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 34 / 36

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) 14 30 12 25 20 15 10 5 0 1 2 3 4 5 6 7 8 9 10 µ µ R L 10 0 1 2 3 4 5 6 Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 34 / 36 µ µ R L

Trapped spinor system phase portraits Simple case not so simple 22 20 18 16 14 12 10 0 1 2 3 4 5 6 µ µ R L Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 35 / 36

Trapped spinor system phase portraits Simple case not so simple 22 20 18 16 14 12 10 0 1 2 3 4 5 6 µ µ R L Examine phase portrait t θ vs θ Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 35 / 36

Trapped spinor system phase portraits Simple case not so simple 22 20 18 16 14 12 10 0 1 2 3 4 5 6 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 30 25 20 15 10 5 0 5 3 2 1 0 1 2 3 = 3.20 3 2 1 0 1 1 1.5 2 Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 35 / 36

Trapped spinor system phase portraits Simple case not so simple 22 20 18 16 14 12 10 0 1 2 3 4 5 6 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 30 25 20 15 10 5 0 5 3 2 1 0 1 2 3 8 7 6 5 4 3 2 1 0 1 3 2 1 0 1 2 3 = 3.20 3 2 1 0 1 1 1.5 2 = 3.25 Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 35 / 36

00 11 00 11 00 11 00 11 Conclusions Effects of pumping on mean-field theory Pumping Bath γ Excitons System Cavity mode κ Bulk photon modes Bath Temperature 0.4 0.2 Increasing γ 0 0 0.1 0.4 0.5 0.2 0.3 Density 1 Instability of normal state Translating: condensation lasing Energy of zero (units of g) 0-1 -2-3 Zero of Re Zero of Im -0.6-0.5-0.4-0.3 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 30 25 20 15 10 5 0 r l+r Magnetic field, l 1.5 γnet 5 3 2 1 0 1 2 3 Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 36 / 36

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

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

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

Fluctuation corrections to phase boundary Fluctuation corrections to density ρ ρ 0 + q δψ δψ In 2D system: modified critical condition: 40 30 ρ s = ρ ρ normal = # 2m non-condensed condensed 2 k B T k B T 20 10 0 0 1 10 9 2 10 9 n [cm -2 ] Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 39 / 55

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 20 10 0 0 1 10 9 2 10 9 n [cm -2 ] Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 39 / 55

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 30 100 condensed k B T 20 10 10 1 10 7 10 8 10 9 10 10 0 0 1 10 9 2 10 9 n [cm -2 ] Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 39 / 55

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 20 100 10 Second BCS crossover at na 2 B 1 10 1 10 7 10 8 10 9 10 10 0 0 1 10 9 2 10 9 n [cm -2 ] Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 39 / 55

Blueshfit and experimental phase boundary Blueshift: Energy [mev] 10 1 0.1 0.01 0.001 mean field threshold δ=5.3mev, k B T eff =17K 10 7 10 8 10 9 10 10 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 77 235313] Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 40 / 55

Blueshfit and experimental phase boundary Blueshift: Energy [mev] 10 1 0.1 0.01 0.001 mean field threshold Phase diagram: k B T 40 30 20 10 0 δ=5.3mev, k B T eff =17K 10 7 10 8 10 9 10 10 n [cm 2 ] 100 10 non-condensed condensed 1 10 7 10 8 10 9 10 10 0 1 10 9 2 10 9 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 77 235313] CdTe: T [K] 20 0 20 0 20 0 0 1 10 10 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 0 1 10 10 2 10 10 n [cm -2 ] Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 40 / 55

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 0.8 0.6 0.4 0.2 κ/g=0.05 0-0.5 0 0.5 µ B /g Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 41 / 55

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 0.8 0.6 0.4 0.2 µ B /g=-0.05 κ/g=0.05 µ B /g=0.05 0 0 0.2 0.4 0.6 κ/g -0.5 0 0.5 µ B /g 0 0.2 0.4 0.6 κ/g Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 41 / 55

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

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

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

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

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:0811.4333] Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 43 / 55

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:0811.4333] Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 43 / 55

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:0811.4333] Imψ t φ = UδN φ Reψ Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 43 / 55

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:0811.4333] 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

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:0811.4333] 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

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:0811.4333] 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

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:0811.4333] 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

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

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

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

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

Why vortices 25 20 Density profile Thomas-Fermi in flattened trap Density 15 10 5 0-15 -10-5 0 5 10 15 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

Why vortices 25 20 Density profile Thomas-Fermi in flattened trap Density 15 10 5 0-15 -10-5 0 5 10 15 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

Why vortices: chemical potential vs size µ 35 30 25 20 15 10 Thomas-Fermi : µ = f (r 0 ) stable 1 Vortex : µ = Uγ net Γ unstable 1 1 1 2 2 1 3 3 3 4 6 17 28 44 53 75 5 0 0 2 4 6 8 10 r 0 Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 46 / 55

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

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

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

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

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] 25 20 15 10 No binding E XX = 4 mev E XX = 8 mev 10 5 10-4 10 0 10 4 10 8 5 10 9 1 10 10 2 10 10 2 10 10 n [cm -2 ] Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 48 / 55

Equilibrium phase diagrams J 1 = J 2 = 0. For U 1 = 0.5, Ψ L,R decouple. 0.6 0.5 0.4 neither T/T deg 0.3 0.2 HFB approx Exact EOS 0.1 r l+r l 0-0.04-0.03-0.02-0.01 0 0.01 0.02 0.03 0.04 /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

Equilibrium phase diagrams J 1 = J 2 = 0. For U 1 = 0.5, Ψ L,R decouple. 0.6 0.5 neither J 1 = 0, J 2 0. Phase locking J 2 cos(2(θ L θ R )). T/T deg 0.4 0.3 0.2 HFB approx Exact EOS 0.1 r l+r l 0-0.04-0.03-0.02-0.01 0 0.01 0.02 0.03 0.04 /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

Equilibrium phase diagrams J 1 = J 2 = 0. For U 1 = 0.5, Ψ L,R decouple. T/T deg 0.6 0.5 0.4 0.3 0.2 HFB approx Exact EOS neither 0.1 r l+r l 0-0.04-0.03-0.02-0.01 0 0.01 0.02 0.03 0.04 /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

Equilibrium phase diagrams J 1 = J 2 = 0. For U 1 = 0.5, Ψ L,R decouple. T/T deg 0.6 0.5 0.4 0.3 0.2 HFB approx Exact EOS neither 0.1 r l+r l 0-0.04-0.03-0.02-0.01 0 0.01 0.02 0.03 0.04 /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

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

[ Graphical implementation of T = n 0 / f c + f ( x c + 2Ω T )] 4 3 Bogoliubov µ/g Monte Carlo HFP f 2 1 0-0.02 0 0.02 0.04 0.06 x=µ/t Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 51 / 55

[ Graphical implementation of T = n 0 / f c + f ( x c + 2Ω T )] 4 3 Bogoliubov µ/g Monte Carlo HFP f 2 1 0-0.04-0.02 0 0.02 0.04 2Ω/T = x-x c Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 51 / 55

[ Graphical implementation of T = n 0 / f c + f ( x c + 2Ω T )] n 0 /T 6 5 4 3 2 1 0 Bogoliubov µ/g Monte Carlo HFP -0.04-0.02 0 0.02 0.04 2Ω/T Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 51 / 55

[ Graphical implementation of T = n 0 / f c + f ( x c + 2Ω T )] 0.4 Bogoliubov µ/g Monte Carlo HFP T/n 0 0.2 0-0.04-0.02 0 0.02 0.04 2Ω/T Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 51 / 55

[ Graphical implementation of T = n 0 / f c + f ( x c + 2Ω T )] 0.4 Bogoliubov µ/g Monte Carlo HFP T/n 0 0.2 0-0.02-0.01 0 0.01 0.02 2Ω/n 0 Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 51 / 55

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

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

Two-mode model bistability γ net Stable limit cycle 4 =6.6 =6.8 =7.0 ω 0-10 -20-30 Bistable ω= c Stable fixed point 1.5 γ net 0 2 4 6 8 10 12 14 4 =7.8 c : ω [ln( c )] 1 0 π 2π z dθ/dt+ 0-4 4 0-4 0-4 =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

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

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 5 0-5 0 π/4 π/2 3π/4 π 0 π/4 π/2 3π/4 π θ θ 15 10 5 0-5 Stability requires Ω 2 p > 0. If Ω 2 p < γ net overdamped. Jonathan Keeling Nonequilibrium quantum condensates Royal Holloway CM Seminar 53 / 55

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

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

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

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