SUPA University of St Andrews 600 YEARS RETUNE, Heidelberg, June - PowerPoint PPT Presentation

Stability of Thomas-Fermi solution − ∇ 2 2 m + m ω 2 � γ eff − κ − Γ | ψ | 2 �� r 2 + U | ψ | 2 + i � i ∂ t ψ = ψ 2 30 Density 25 Unstable growth 20 15 10 5 3 γ net 0 2 Γ 0 2 4 6 8 Radius Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 9 / 22

Stability of Thomas-Fermi solution − ∇ 2 2 m + m ω 2 � γ eff − κ − Γ | ψ | 2 �� r 2 + U | ψ | 2 + i � i ∂ t ψ = ψ 2 30 Density 25 Unstable growth 20 15 10 5 3 γ net 0 2 Γ 0 2 4 6 8 Radius High m modes: δρ n , m ≃ e im θ r m . . . 1 2 ∂ t ρ + ∇· ( ρ v ) = ( γ net − Γ ρ ) ρ Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 9 / 22

Stability of Thomas-Fermi solution − ∇ 2 2 m + m ω 2 � γ eff − κ − Γ | ψ | 2 �� r 2 + U | ψ | 2 + i � i ∂ t ψ = ψ 2 30 Density 25 Stabilised 20 15 10 5 3 γ net 0 2 Γ 0 2 4 6 8 Radius High m modes: δρ n , m ≃ e im θ r m . . . 1 2 ∂ t ρ + ∇· ( ρ v ) = ( γ net Θ( r 0 − r ) − Γ ρ ) ρ Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 9 / 22

Stability of Thomas-Fermi solution − ∇ 2 2 m + m ω 2 � γ eff − κ − Γ | ψ | 2 �� r 2 + U | ψ | 2 + i � i ∂ t ψ = ψ 2 30 Density 25 ???? 20 15 10 5 3 γ net 0 2 Γ 0 2 4 6 8 Radius High m modes: δρ n , m ≃ e im θ r m . . . 1 2 ∂ t ρ + ∇· ( ρ v ) = ( γ net Θ( r 0 − r ) − Γ ρ ) ρ Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 9 / 22

Time evolution: t=0 t=2 t=22 t=30 t=35 t=40 t=45 t=56 [Keeling & Berloff PRL ’08] Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 10 / 22

Time evolution: t=0 t=2 t=22 t=30 t=35 t=40 t=45 t=56 i ∂ t ψ = ( µ − 2 Ω L z ) ψ Ω = ω , cancels trap. Why? [Keeling & Berloff PRL ’08] Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 10 / 22

Observability of vortex lattices Not seen experimentally (yet?) Observation: Fast rotation Stability: Disorder, ellipticity Relaxation, thermalisation? [Borgh et al. PRB ’12 in press] Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 11 / 22

Observability of vortex lattices Not seen experimentally (yet?) Observation: Fast rotation 30 25 20 Spectrum: ω 15 10 5 0 −5 0 5 k x Stability: Disorder, ellipticity Relaxation, thermalisation? [Borgh et al. PRB ’12 in press] Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 11 / 22

Observability of vortex lattices Not seen experimentally (yet?) Observation: Fast rotation 30 25 20 Spectrum: ω Interference: 15 10 5 0 −5 0 5 k x Stability: Disorder, ellipticity Relaxation, thermalisation? [Borgh et al. PRB ’12 in press] Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 11 / 22

Observability of vortex lattices Not seen experimentally (yet?) Observation: Fast rotation 30 25 20 Spectrum: ω Interference: 15 10 5 0 −5 0 5 k x Stability: Disorder, ellipticity Relaxation, thermalisation? [Borgh et al. PRB ’12 in press] Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 11 / 22

Observability of vortex lattices Not seen experimentally (yet?) Observation: Fast rotation 30 25 20 Spectrum: ω Interference: 15 10 5 0 −5 0 5 k x Stability: Disorder, ellipticity Relaxation, thermalisation? [Borgh et al. PRB ’12 in press] Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 11 / 22

Superfluidity Introduction to polariton condensation 1 Approaches to modelling Pattern formation 2 Non-equilibrium pattern formation Spontaneous vortex lattices Superfluidity 3 Non-equilibrium condensate spectrum Experiments and aspects of superfluidity Current-current response function Coherence 4 Experiments Power law decay of coherence Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 12 / 22

Fluctuations above transition When condensed Sound mode � − 1 � = ω 2 − ξ 2 frequency D R ( ω, k ) Det k momentum With ξ k ≃ ck Poles: ω ∗ = ξ k Generic structure of Green’s function: � ω + i γ net − ǫ k − µ � i γ net − µ [ D R ] − 1 = − i γ net − µ − ω − i γ net − ǫ k − µ Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 13 / 22

Fluctuations above transition When condensed Real Imaginary � − 1 � frequency D R ( ω, k ) = ( ω + i γ net ) 2 + γ 2 net − ξ 2 Det k momentum With ξ k ≃ ck Poles: � ω ∗ = − i γ net ± ξ 2 k − γ 2 net Generic structure of Green’s function: � ω + i γ net − ǫ k − µ � i γ net − µ [ D R ] − 1 = − i γ net − µ − ω − i γ net − ǫ k − µ Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 13 / 22

Fluctuations above transition When condensed Real Imaginary � − 1 � frequency D R ( ω, k ) = ( ω + i γ net ) 2 + γ 2 net − ξ 2 Det k momentum With ξ k ≃ ck Poles: � ω ∗ = − i γ net ± ξ 2 k − γ 2 net Generic structure of Green’s function: � ω + i γ net − ǫ k − µ � i γ net − µ [ D R ] − 1 = − i γ net − µ − ω − i γ net − ǫ k − µ Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 13 / 22

Aspects of superfluidity Quantised Landau Metastable Two-fluid Local Solitary vortices critical persistent hydrody- thermal waves velocity flow namics equilib- rium ✧ ✧ ✧ ✧ ✧ ✧ Superfluid 4 He/cold atom Bose-Einstein condensate ✧ ✪ ✪ ✧ ✧ ✪ Non-interacting Bose-Einstein condensate ✪ ✧ ✪ ✧ ✧ ✧ Classical irrotational fluid ✧ ✪ ✪ ? ? ? Incoherently pumped polariton condensates Lagoudakis et al. Nat. Phys. ’08. Utsunomiya et al. Nat. Phys. ’08. Amo et al. Nature ’09; Nat. Phys. ’09 Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 14 / 22

Aspects of superfluidity Quantised Landau Metastable Two-fluid Local Solitary vortices critical persistent hydrody- thermal waves velocity flow namics equilib- rium ✧ ✧ ✧ ✧ ✧ ✧ Superfluid 4 He/cold atom Bose-Einstein condensate ✧ ✪ ✪ ✧ ✧ ✪ Non-interacting Bose-Einstein condensate ✪ ✧ ✪ ✧ ✧ ✧ Classical irrotational fluid ✧ ✪ ✪ ? ? ? Incoherently pumped polariton condensates Lagoudakis et al. Nat. Phys. ’08. Utsunomiya et al. Nat. Phys. ’08. Amo et al. Nature ’09; Nat. Phys. ’09 Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 14 / 22

����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ������ ������ ������ ������ Superfluid density Two-fluid hydrodynamics Experimentally, rotation: 1 ρ / ρ total ρ normal ρ superfluid To calculate, 0 transverse/longitudinal: 0 1 T/T c ρ s , ρ n distinguished by slow rotation Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 15 / 22

����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ������ ������ ������ ������ Superfluid density Two-fluid hydrodynamics Experimentally, rotation: 1 ρ / ρ total ρ normal ρ superfluid To calculate, 0 transverse/longitudinal: 0 1 T/T c ρ s , ρ n distinguished by slow rotation Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 15 / 22

Superfluid density Two-fluid hydrodynamics Experimentally, rotation: 1 ρ / ρ total ρ normal ρ superfluid To calculate, 0 transverse/longitudinal: 0 1 ����������� ����������� ����������� ����������� T/T c ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ������ ������ ρ s , ρ n distinguished by slow ������ ������ rotation Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 15 / 22

����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ������ ������ ������ ������ Superfluid density Currrent: J = ρ v = ψ † i ∇ ψ = | ψ | 2 ∇ φ 2 k i + q i J i ( q ) = ψ † ψ k k + q 2 m Response function: � H → H − f ( q ) · J i ( q ) J i ( q ) = χ ij ( q ) f j ( q ) q Vertex corrections essential for superfluid part. Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 16 / 22

����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ������ ������ ������ ������ Superfluid density Currrent: J = ρ v = ψ † i ∇ ψ = | ψ | 2 ∇ φ 2 k i + q i J i ( q ) = ψ † ψ k k + q 2 m Response function: � H → H − f ( q ) · J i ( q ) J i ( q ) = χ ij ( q ) f j ( q ) q Vertex corrections essential for superfluid part. Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 16 / 22

Superfluid density ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� Currrent: ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ������ ������ ������ ������ J = ρ v = ψ † i ∇ ψ = | ψ | 2 ∇ φ 2 k i + q i J i ( q ) = ψ † ψ k k + q 2 m Response function: � H → H − f ( q ) · J i ( q ) J i ( q ) = χ ij ( q ) f j ( q ) q χ ij ( ω = 0 , q → 0 ) = � [ J i ( q ) , J j ( − q )] � = ρ S q i q j q 2 + ρ N m δ ij m Vertex corrections essential for superfluid part. Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 16 / 22

Superfluid density ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� Currrent: ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ������ ������ ������ ������ J = ρ v = ψ † i ∇ ψ = | ψ | 2 ∇ φ 2 k i + q i J i ( q ) = ψ † ψ k k + q 2 m Response function: � H → H − f ( q ) · J i ( q ) J i ( q ) = χ ij ( q ) f j ( q ) q χ ij ( ω = 0 , q → 0 ) = � [ J i ( q ) , J j ( − q )] � = ρ S q i q j q 2 + ρ N m δ ij m Vertex corrections essential for superfluid part. Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 16 / 22

Non-equilibrium superfluid response Superfluid response exists because: � i ψ 0 q j = i ψ 0 q i � 1 2 m ( 1 , − 1 ) D R ( q , ω = 0 ) − 1 2 m D R ( ω = 0 ) ∝ 1 /ǫ q despite pumping/decay — superfluid response exists. Normal density: � d ω � � σ z D K σ z ( D R + D A ) � d d k ǫ k ρ N = 2 π Tr Is affected by pump/decay: Does not vanish at T → 0. [JK PRL ’11] Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 17 / 22

Non-equilibrium superfluid response Superfluid response exists because: � i ψ 0 q j = i ψ 0 q i � 1 2 m ( 1 , − 1 ) D R ( q , ω = 0 ) − 1 2 m D R ( ω = 0 ) ∝ 1 /ǫ q despite pumping/decay — superfluid response exists. Normal density: � d ω � � σ z D K σ z ( D R + D A ) � d d k ǫ k ρ N = 2 π Tr Is affected by pump/decay: Does not vanish at T → 0. [JK PRL ’11] Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 17 / 22

Non-equilibrium superfluid response Superfluid response exists because: � i ψ 0 q j = i ψ 0 q i � 1 2 m ( 1 , − 1 ) D R ( q , ω = 0 ) − 1 2 m D R ( ω = 0 ) ∝ 1 /ǫ q despite pumping/decay — superfluid response exists. Normal density: � d ω � � σ z D K σ z ( D R + D A ) � d d k ǫ k ρ N = 2 π Tr Is affected by pump/decay: Does not vanish at T → 0. [JK PRL ’11] Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 17 / 22

Non-equilibrium superfluid response Superfluid response exists because: � i ψ 0 q j = i ψ 0 q i � 1 2 m ( 1 , − 1 ) D R ( q , ω = 0 ) − 1 2 m D R ( ω = 0 ) ∝ 1 /ǫ q despite pumping/decay — superfluid response exists. Normal density: � d ω � � σ z D K σ z ( D R + D A ) � d d k ǫ k ρ N = 2 π Tr Is affected by pump/decay: 3 γ net / µ = 0.0 Does not vanish at T → 0. γ net / µ = 0.1 γ net / µ = 0.5 2 ρ N / m µ 1 0 0 1 2 3 4 5 T/ µ [JK PRL ’11] Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 17 / 22

Coherence: Introduction to polariton condensation 1 Approaches to modelling Pattern formation 2 Non-equilibrium pattern formation Spontaneous vortex lattices 3 Superfluidity Non-equilibrium condensate spectrum Experiments and aspects of superfluidity Current-current response function Coherence 4 Experiments Power law decay of coherence Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 18 / 22

Correlations in a 2D Gas = Correlations: + � � g 1 ( r , r ′ , t ) = ψ † ( r , t ) ψ ( r ′ , 0 ) D < = D K − D R + D A Generally, get: � � � ln ( r / r 0 ) t ≃ 0 � � ≃ | ψ 0 | 2 exp ψ † ( r , t ) ψ ( 0 , 0 ) − a p 2 ln ( c 2 t /γ net r 2 1 0 ) r ≃ 0 [Szyma´ nska et al. PRL ’06; PRB ’07] [Wouters and Savona PRB ’09] Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 19 / 22

Correlations in a 2D Gas = Correlations: (in 2D) + � � g 1 ( r , r ′ , t ) = ψ † ( r , t ) ψ ( r ′ , 0 ) ≃ | ψ 0 | 2 exp � � − D < φφ ( r , r ′ , t ) D < = D K − D R + D A Generally, get: � � � ln ( r / r 0 ) t ≃ 0 � � ≃ | ψ 0 | 2 exp ψ † ( r , t ) ψ ( 0 , 0 ) − a p 1 2 ln ( c 2 t /γ net r 2 0 ) r ≃ 0 [Szyma´ nska et al. PRL ’06; PRB ’07] [Wouters and Savona PRB ’09] Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 19 / 22

Correlations in a 2D Gas = Correlations: (in 2D) + � � g 1 ( r , r ′ , t ) = ψ † ( r , t ) ψ ( r ′ , 0 ) ≃ | ψ 0 | 2 exp � � − D < φφ ( r , r ′ , t ) D < = D K − D R + D A Generally, get: � � � ln ( r / r 0 ) t ≃ 0 � � ≃ | ψ 0 | 2 exp ψ † ( r , t ) ψ ( 0 , 0 ) − a p 1 2 ln ( c 2 t /γ net r 2 0 ) r ≃ 0 [Szyma´ nska et al. PRL ’06; PRB ’07] [Wouters and Savona PRB ’09] Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 19 / 22

Experimental observation of power-law decay G. Rompos, Y. Yamamoto et al. submitted Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 20 / 22

Experimental observation of power-law decay � r � − a p g 1 ( r , − r ) ∝ r 0 G. Rompos, Y. Yamamoto et al. submitted Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 20 / 22

Exponent in a non-equilibrium 2D gas � � 2 r �� � � = | ψ 0 | 2 exp � � ψ † ( r , 0 ) ψ ( − r , 0 ) − D < lim φφ ( r , − r ) ∝ exp − a p ln r 0 r →∞ Experimentally, a P ≃ 1 . 2 In equilibrium a p = mk B T < 1 4 (BKT transition) 2 π � 2 n s Non-equilibrium theory depends on thermalisation. ◮ Thermalised (yet diffusive modes) a p = mk B T 2 π � 2 n s ◮ Non-thermalised, a P ∝ Pumping noise . n s Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 21 / 22

Exponent in a non-equilibrium 2D gas � � 2 r �� � � = | ψ 0 | 2 exp � � ψ † ( r , 0 ) ψ ( − r , 0 ) − D < lim φφ ( r , − r ) ∝ exp − a p ln r 0 r →∞ Experimentally, a P ≃ 1 . 2 In equilibrium a p = mk B T < 1 4 (BKT transition) 2 π � 2 n s Non-equilibrium theory depends on thermalisation. ◮ Thermalised (yet diffusive modes) a p = mk B T 2 π � 2 n s ◮ Non-thermalised, a P ∝ Pumping noise . n s Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 21 / 22

Exponent in a non-equilibrium 2D gas � � 2 r �� � � = | ψ 0 | 2 exp � � ψ † ( r , 0 ) ψ ( − r , 0 ) − D < lim φφ ( r , − r ) ∝ exp − a p ln r 0 r →∞ Experimentally, a P ≃ 1 . 2 In equilibrium a p = mk B T < 1 4 (BKT transition) 2 π � 2 n s Non-equilibrium theory depends on thermalisation. ◮ Thermalised (yet diffusive modes) a p = mk B T 2 π � 2 n s ◮ Non-thermalised, a P ∝ Pumping noise . n s Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 21 / 22

Exponent in a non-equilibrium 2D gas � � 2 r �� � � = | ψ 0 | 2 exp � � ψ † ( r , 0 ) ψ ( − r , 0 ) − D < lim φφ ( r , − r ) ∝ exp − a p ln r 0 r →∞ Experimentally, a P ≃ 1 . 2 In equilibrium a p = mk B T < 1 4 (BKT transition) 2 π � 2 n s Non-equilibrium theory depends on thermalisation. ◮ Thermalised (yet diffusive modes) a p = mk B T 2 π � 2 n s ◮ Non-thermalised, a P ∝ Pumping noise . n s Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 21 / 22

Exponent in a non-equilibrium 2D gas � � 2 r �� � � = | ψ 0 | 2 exp � � ψ † ( r , 0 ) ψ ( − r , 0 ) − D < lim φφ ( r , − r ) ∝ exp − a p ln r 0 r →∞ Experimentally, a P ≃ 1 . 2 In equilibrium a p = mk B T < 1 4 (BKT transition) 2 π � 2 n s Non-equilibrium theory depends on thermalisation. ◮ Thermalised (yet diffusive modes) a p = mk B T 2 π � 2 n s ◮ Non-thermalised, a P ∝ Pumping noise . n s Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 21 / 22

Conclusion Instability of Thomas-Fermi and spontaneous rotation t=22 t=35 t=56 Survival of superfluid response 3 Real γ net / µ = 0.0 Imaginary γ net / µ = 0.1 frequency γ net / µ = 0.5 2 ρ N / m µ momentum 1 0 0 1 2 3 4 5 T/ µ Power law decay of correlations Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 22 / 22

Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 23 / 35

Extra slides Condensation vs Lasing 5 GPE stability 6 Detecting vortex lattice 7 Calculating superfluid density 8 Measuring superfluid density 9 Finite size coherence and Schawlow-Townes 10 Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 24 / 35

Simple Laser: Maxwell Bloch equations α + g α, k � ǫ α S z H = ω 0 ψ † ψ + ψ S + √ α + H.c. N γ N 0 A γ α Maxwell-Bloch eqns: P = − i � S − � , N = 2 � S z � g κ ∂ t ψ = − i ω 0 ψ − κψ + � α g α P α ∂ t P α = − 2 i ǫ α P α − 2 γ P + g α ψ N α ∂ t N α = 2 γ ( N 0 − N α ) − 2 g α ( ψ ∗ P α + P ∗ α ψ ) Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 25 / 35

Simple Laser: Maxwell Bloch equations α + g α, k � ǫ α S z H = ω 0 ψ † ψ + ψ S + √ α + H.c. N γ N 0 A γ α Maxwell-Bloch eqns: P = − i � S − � , N = 2 � S z � g κ ∂ t ψ = − i ω 0 ψ − κψ + � α g α P α ∂ t P α = − 2 i ǫ α P α − 2 γ P + g α ψ N α ∂ t N α = 2 γ ( N 0 − N α ) − 2 g α ( ψ ∗ P α + P ∗ α ψ ) 0.2 6 0 Absorption Strong coupling. κ, γ < g √ n 4 -0.2 N 0 2 Inversion causes collapse -0.4 before lasing -0.6 0 -1 0 1 ω /g Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 25 / 35

Simple Laser: Maxwell Bloch equations α + g α, k � ǫ α S z H = ω 0 ψ † ψ + ψ S + √ α + H.c. N γ N 0 A γ α Maxwell-Bloch eqns: P = − i � S − � , N = 2 � S z � g κ ∂ t ψ = − i ω 0 ψ − κψ + � α g α P α ∂ t P α = − 2 i ǫ α P α − 2 γ P + g α ψ N α ∂ t N α = 2 γ ( N 0 − N α ) − 2 g α ( ψ ∗ P α + P ∗ α ψ ) 0.2 6 0 Absorption Strong coupling. κ, γ < g √ n 4 -0.2 N 0 2 Inversion causes collapse -0.4 before lasing -0.6 0 -1 0 1 ω /g Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 25 / 35

Simple Laser: Maxwell Bloch equations α + g α, k � ǫ α S z H = ω 0 ψ † ψ + ψ S + √ α + H.c. N γ N 0 A γ α Maxwell-Bloch eqns: P = − i � S − � , N = 2 � S z � g κ ∂ t ψ = − i ω 0 ψ − κψ + � α g α P α ∂ t P α = − 2 i ǫ α P α − 2 γ P + g α ψ N α ∂ t N α = 2 γ ( N 0 − N α ) − 2 g α ( ψ ∗ P α + P ∗ α ψ ) 0.2 6 0 Absorption Strong coupling. κ, γ < g √ n 4 -0.2 N 0 2 Inversion causes collapse -0.4 before lasing -0.6 0 -1 0 1 ω /g Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 25 / 35

Maxwell-Bloch Equations: Retarded Green’s function 0.2 6 0 Absorption 4 -0.2 N 0 2 -0.4 -0.6 0 -1 0 1 ω /g Introduce D R ( ω ) : Response to perturbation Absorption = − 2 ℑ [ D R ( ω )] Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 26 / 35

Maxwell-Bloch Equations: Retarded Green’s function 0.2 6 0 Absorption 4 -0.2 N 0 2 -0.4 -0.6 0 -1 0 1 ω /g ∂ t ψ = − i ω 0 ψ − κψ + � α g α P α Introduce D R ( ω ) : ∂ t P α = − 2 i ǫ α P α − 2 γ P + g α ψ N α Response to perturbation ∂ t N α = 2 γ ( N 0 − N α ) − 2 g α ( ψ ∗ P α + P ∗ α ψ ) Absorption = − 2 ℑ [ D R ( ω )] g 2 N 0 � − 1 � D R ( ω ) = ω − ω k + i κ + ω − 2 ǫ + i 2 γ Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 26 / 35

Maxwell-Bloch Equations: Retarded Green’s function 0.2 6 0 Absorption 4 -0.2 N 0 2 -0.4 -0.6 0 -1 0 1 ω /g ∂ t ψ = − i ω 0 ψ − κψ + � α g α P α Introduce D R ( ω ) : ∂ t P α = − 2 i ǫ α P α − 2 γ P + g α ψ N α Response to perturbation ∂ t N α = 2 γ ( N 0 − N α ) − 2 g α ( ψ ∗ P α + P ∗ α ψ ) 2 B ( ω ) Absorption = − 2 ℑ [ D R ( ω )] = A ( ω ) 2 + B ( ω ) 2 g 2 N 0 � − 1 � D R ( ω ) = ω − ω k + i κ + ω − 2 ǫ + i 2 γ = A ( ω ) + iB ( ω ) Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 26 / 35

Maxwell-Bloch Equations: Retarded Green’s function (a) 0.2 6 1 0 Absorption 4 -0.2 N 0 0 2 -0.4 -0.6 0 -1 A( ω ) -1 0 1 -1 0 1 ω /g ω /g ∂ t ψ = − i ω 0 ψ − κψ + � α g α P α Introduce D R ( ω ) : ∂ t P α = − 2 i ǫ α P α − 2 γ P + g α ψ N α Response to perturbation ∂ t N α = 2 γ ( N 0 − N α ) − 2 g α ( ψ ∗ P α + P ∗ α ψ ) 2 B ( ω ) Absorption = − 2 ℑ [ D R ( ω )] = A ( ω ) 2 + B ( ω ) 2 g 2 N 0 � − 1 � D R ( ω ) = ω − ω k + i κ + ω − 2 ǫ + i 2 γ = A ( ω ) + iB ( ω ) Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 26 / 35

Maxwell-Bloch Equations: Retarded Green’s function (a) 0.2 6 1 0 Absorption 4 -0.2 N 0 0 2 -0.4 A( ω ) -0.6 0 -1 B( ω ) -1 0 1 -1 0 1 ω /g ω /g ∂ t ψ = − i ω 0 ψ − κψ + � α g α P α Introduce D R ( ω ) : ∂ t P α = − 2 i ǫ α P α − 2 γ P + g α ψ N α Response to perturbation ∂ t N α = 2 γ ( N 0 − N α ) − 2 g α ( ψ ∗ P α + P ∗ α ψ ) 2 B ( ω ) Absorption = − 2 ℑ [ D R ( ω )] = A ( ω ) 2 + B ( ω ) 2 g 2 N 0 � − 1 � D R ( ω ) = ω − ω k + i κ + ω − 2 ǫ + i 2 γ = A ( ω ) + iB ( ω ) Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 26 / 35

Maxwell-Bloch Equations: Retarded Green’s function (a) 0.2 (b) 6 1 1 0 Absorption 4 -0.2 N 0 0 0 2 -0.4 A( ω ) A( ω ) -0.6 0 -1 -1 B( ω ) B( ω ) -1 0 1 -1 0 1 -1 0 1 ω /g ω /g ω /g ∂ t ψ = − i ω 0 ψ − κψ + � α g α P α Introduce D R ( ω ) : ∂ t P α = − 2 i ǫ α P α − 2 γ P + g α ψ N α Response to perturbation ∂ t N α = 2 γ ( N 0 − N α ) − 2 g α ( ψ ∗ P α + P ∗ α ψ ) 2 B ( ω ) Absorption = − 2 ℑ [ D R ( ω )] = A ( ω ) 2 + B ( ω ) 2 g 2 N 0 � − 1 � D R ( ω ) = ω − ω k + i κ + ω − 2 ǫ + i 2 γ = A ( ω ) + iB ( ω ) Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 26 / 35

Evolution of poles with Inversion (a) 0.2 (b) 1 6 1 0 Absorption 4 -0.2 N 0 0 0 2 -0.4 A( ω ) A( ω ) -0.6 0 -1 -1 B( ω ) B( ω ) -1 0 1 -1 0 1 -1 0 1 ω /g ω /g ω /g 2 (a) (b) 1 ω /g 0 -1 Zero of Re Zero of Im -2 -(2 γ /g) 2 2 γκ /g 2 Inversion, N 0 Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 27 / 35

Evolution of poles with Inversion (a) 0.2 (b) 1 6 1 0 Absorption 4 -0.2 N 0 0 0 2 -0.4 A( ω ) A( ω ) -0.6 0 -1 -1 B( ω ) B( ω ) -1 0 1 -1 0 1 -1 0 1 ω /g ω /g ω /g Laser: Equilibrium: 2 (a) (b) 2 UP 1 ω /g 0 LP 0 ω /g µ -2 -1 Zero of Re Zero of Im non-condensed condensed -4 -2 -(2 γ /g) 2 2 γκ /g 2 -2 -1.5 -1 -0.5 0 Inversion, N 0 µ /g Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 27 / 35

Luminescence spectrum and Green’s functions − 2 ℑ [ D R ( ω )] = DoS ( ω ) � − 1 � D R ( ω ) = A ( ω ) + iB ( ω ) 2 B ( ω ) DoS ( ω ) = B ( ω ) 2 + A ( ω ) 2 Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 28 / 35

Luminescence spectrum and Green’s functions − 2 ℑ [ D R ( ω )] = DoS ( ω ) iD K ( ω ) = ( 2 n ( ω ) + 1 ) DoS ( ω ) � − 1 � D R ( ω ) = A ( ω ) + iB ( ω ) 2 B ( ω ) DoS ( ω ) = B ( ω ) 2 + A ( ω ) 2 Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 28 / 35

Luminescence spectrum and Green’s functions − 2 ℑ [ D R ( ω )] = DoS ( ω ) iD K ( ω ) = ( 2 n ( ω ) + 1 ) DoS ( ω ) � − 1 � D R ( ω ) = A ( ω ) + iB ( ω ) 2 B ( ω ) DoS ( ω ) = B ( ω ) 2 + A ( ω ) 2 Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 28 / 35

Luminescence spectrum and Green’s functions − 2 ℑ [ D R ( ω )] = DoS ( ω ) iD K ( ω ) = ( 2 n ( ω ) + 1 ) DoS ( ω ) � − 1 -1.5 -1 -0.5 0 0.5 1 � D R ( ω ) = A ( ω ) + iB ( ω ) A( ω ) 1 2 B ( ω ) 0 DoS ( ω ) = B ( ω ) 2 + A ( ω ) 2 3 R ] Density of states, 2 Im[ D Intensity (a.u.) 2 1 0 -1.5 -1 -0.5 0 0.5 1 Energy (units of g) Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 28 / 35

Luminescence spectrum and Green’s functions − 2 ℑ [ D R ( ω )] = DoS ( ω ) iD K ( ω ) = ( 2 n ( ω ) + 1 ) DoS ( ω ) � − 1 -1.5 -1 -0.5 0 0.5 1 � D R ( ω ) = A ( ω ) + iB ( ω ) A( ω ) B( ω ) 1 2 B ( ω ) 0 DoS ( ω ) = B ( ω ) 2 + A ( ω ) 2 3 R ] Density of states, 2 Im[ D Intensity (a.u.) 2 1 0 -1.5 -1 -0.5 0 0.5 1 Energy (units of g) Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 28 / 35

Luminescence spectrum and Green’s functions − 2 ℑ [ D R ( ω )] = DoS ( ω ) iD K ( ω ) = ( 2 n ( ω ) + 1 ) DoS ( ω ) � − 1 -1.5 -1 -0.5 0 0.5 1 � D R ( ω ) = A ( ω ) + iB ( ω ) A( ω ) B( ω ) 1 2 B ( ω ) 0 DoS ( ω ) = B ( ω ) 2 + A ( ω ) 2 3 R ] Density of states, 2 Im[ D C ( ω ) Intensity (a.u.) iD K ( ω ) = 2 B ( ω ) 2 + A ( ω ) 2 1 0 -1.5 -1 -0.5 0 0.5 1 Energy (units of g) Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 28 / 35

Luminescence spectrum and Green’s functions − 2 ℑ [ D R ( ω )] = DoS ( ω ) iD K ( ω ) = ( 2 n ( ω ) + 1 ) DoS ( ω ) � − 1 -1.5 -1 -0.5 0 0.5 1 � D R ( ω ) = A ( ω ) + iB ( ω ) A( ω ) B( ω ) 1 C( ω ) 2 B ( ω ) 0 DoS ( ω ) = B ( ω ) 2 + A ( ω ) 2 3 R ] Density of states, 2 Im[ D C ( ω ) Intensity (a.u.) iD K ( ω ) = Occupation, n( ω ) 2 B ( ω ) 2 + A ( ω ) 2 1 0 -1.5 -1 -0.5 0 0.5 1 Energy (units of g) Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 28 / 35

Stability and evolution with pumping -1.5 -1 -0.5 0 0.5 1 A( ω ) B( ω ) 1 C( ω ) 0 3 R ] Density of states, 2 Im[ D Intensity (a.u.) Occupation, n( ω ) 2 1 0 -1.5 -1 -0.5 0 0.5 1 Energy (units of g) Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 29 / 35

Stability and evolution with pumping -1.5 -1 -0.5 0 0.5 1 A( ω ) B( ω ) 1 C( ω ) 0 3 R ] Density of states, 2 Im[ D Intensity (a.u.) Occupation, n( ω ) 2 1 0 -1.5 -1 -0.5 0 0.5 1 Energy (units of g) � − 1 � D R ( ω ) = ( ω − ξ k )+ i α ( ω − µ eff ) Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 29 / 35

Stability and evolution with pumping -1.5 -1 -0.5 0 0.5 1 A( ω ) B( ω ) 1 C( ω ) 0 3 R ] Density of states, 2 Im[ D Intensity (a.u.) Occupation, n( ω ) 2 1 0 -1.5 -1 -0.5 0 0.5 1 Energy (units of g) � − 1 � D R ( ω ) = ( ω − ξ k )+ i α ( ω − µ eff ) � − 1 � D R ( ω ∗ = 0 → ℑ ( ω ∗ ) ∝ µ eff − ξ k k ) Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 29 / 35

Stability and evolution with pumping -1.5 -1 -0.5 0 0.5 1 A( ω ) B( ω ) 1 C( ω ) 0 1 3 R ] Density of states, 2 Im[ D Energy of zero (units of g) Intensity (a.u.) Occupation, n( ω ) 0 2 1 -1 0 -1.5 -1 -0.5 0 0.5 1 -2 Energy (units of g) Zero of Re Zero of Im � − 1 � D R ( ω ) -3 = ( ω − ξ k )+ i α ( ω − µ eff ) -0.6 -0.5 -0.4 -0.3 Bath occupation, µ B /g � − 1 � D R ( ω ∗ = 0 → ℑ ( ω ∗ ) ∝ µ eff − ξ k k ) Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 29 / 35

Strong coupling and lasing — low temperature phenomenon Eqbm. polariton Non-eqbm. polariton Laser 1 ξ µ eff 0 ω /g -1 non- non- non- -2 condensed condensed condensed condensed condensed condensed -2 -1 0 -2 -1 0 -1 0 1 µ B /g µ /g Inversion, N 0 A( ω ) Laser: Uniformly invert TLS B( ω ) 1 Non-equilibrium polaritons: Cold bath 0 If T B ≫ γ → Laser limit -1.5 -1 -0.5 0 0.5 1 Energy (units of g) Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 30 / 35

Strong coupling and lasing — low temperature phenomenon Eqbm. polariton Non-eqbm. polariton Laser 1 ξ µ eff 0 ω /g -1 non- non- non- -2 condensed condensed condensed condensed condensed condensed -2 -1 0 -2 -1 0 -1 0 1 µ B /g µ /g Inversion, N 0 A( ω ) Laser: Uniformly invert TLS B( ω ) 1 Non-equilibrium polaritons: Cold bath 0 If T B ≫ γ → Laser limit -1.5 -1 -0.5 0 0.5 1 Energy (units of g) Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 30 / 35

Strong coupling and lasing — low temperature phenomenon Eqbm. polariton Non-eqbm. polariton Laser 1 ξ µ eff 0 ω /g -1 non- non- non- -2 condensed condensed condensed condensed condensed condensed -2 -1 0 -2 -1 0 -1 0 1 µ B /g µ /g Inversion, N 0 A( ω ) Laser: Uniformly invert TLS B( ω ) 1 Non-equilibrium polaritons: Cold bath 0 If T B ≫ γ → Laser limit -1.5 -1 -0.5 0 0.5 1 Energy (units of g) Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 30 / 35

Instability of Thomas-Fermi: details 1 2 ∂ t ρ + ∇ · ( ρ v ) = ( γ net − Γ ρ ) ρ ∂ t v + ∇ ( U ρ + m ω 2 r 2 + m 2 | v | 2 ) = 0 2 3 γ net 2 Γ Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 31 / 35

Instability of Thomas-Fermi: details 1 2 ∂ t ρ + ∇ · ( ρ v ) = ( γ net − Γ ρ ) ρ ∂ t v + ∇ ( U ρ + m ω 2 r 2 + m 2 | v | 2 ) = 0 2 3 γ net 2 Γ Normal modes for γ net , Γ → 0: δρ n , m ( r , θ, t ) = e im θ h n , m ( r ) e i ω n , m t � ω n , m = ω 2 m ( 1 + 2 n ) + 2 n ( n + 1 ) Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 31 / 35

Instability of Thomas-Fermi: details 1 2 ∂ t ρ + ∇ · ( ρ v ) = ( γ net − Γ ρ ) ρ ∂ t v + ∇ ( U ρ + m ω 2 r 2 + m 2 | v | 2 ) = 0 2 3 γ net 2 Γ Normal modes for γ net , Γ → 0: δρ n , m ( r , θ, t ) = e im θ h n , m ( r ) e i ω n , m t � ω n , m = ω 2 m ( 1 + 2 n ) + 2 n ( n + 1 ) Add weak pumping/decay: � m ( 1 + 2 n ) + 2 n ( n + 1 ) − m 2 � ω n , n → ω n , m + i γ net 2 m ( 1 + 2 n ) + 4 n ( n + 1 ) + m 2 Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 31 / 35

Instability of Thomas-Fermi: details 1 2 ∂ t ρ + ∇ · ( ρ v ) = ( γ net − Γ ρ ) ρ Unstable growth ∂ t v + ∇ ( U ρ + m ω 2 r 2 + m 2 | v | 2 ) = 0 2 3 γ net 2 Γ Normal modes for γ net , Γ → 0: δρ n , m ( r , θ, t ) = e im θ h n , m ( r ) e i ω n , m t � ω n , m = ω 2 m ( 1 + 2 n ) + 2 n ( n + 1 ) Add weak pumping/decay: � m ( 1 + 2 n ) + 2 n ( n + 1 ) − m 2 � ω n , n → ω n , m + i γ net 2 m ( 1 + 2 n ) + 4 n ( n + 1 ) + m 2 Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 31 / 35

Instability of Thomas-Fermi: details 1 2 ∂ t ρ + ∇ · ( ρ v ) = ( γ net − Γ ρ ) ρ Stabilised ∂ t v + ∇ ( U ρ + m ω 2 r 2 + m 2 | v | 2 ) = 0 2 3 γ net 2 Γ Normal modes for γ net , Γ → 0: δρ n , m ( r , θ, t ) = e im θ h n , m ( r ) e i ω n , m t � ω n , m = ω 2 m ( 1 + 2 n ) + 2 n ( n + 1 ) Add weak pumping/decay: � m ( 1 + 2 n ) + 2 n ( n + 1 ) − m 2 � ω n , n → ω n , m + i γ net 2 m ( 1 + 2 n ) + 4 n ( n + 1 ) + m 2 Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 31 / 35

Detecting vortex lattices 30 25 20 ω 15 10 Snapshot Spectrum: 5 0 −5 0 5 k x Defocussed homodyne intereference: Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 32 / 35

Calculating superfluid response function Using Keldysh generating functional d 2 Z [ f , θ ] χ ij ( q ) = − i � df i ( q ) d θ j ( − q ) , Z [ f , θ ] = D ψ exp ( iS [ f , θ ]) 2 f , θ couple as force/response current. � ¯ � � 2 k i + q i � ψ cl � θ i f i + θ i � ¯ � S [ f , θ ] = S + ψ cl ψ q k + q f i − θ i − θ i ψ q 2 m q k k , q Saddle point + fluctuations: Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 33 / 35

Calculating superfluid response function Using Keldysh generating functional d 2 Z [ f , θ ] χ ij ( q ) = − i � df i ( q ) d θ j ( − q ) , Z [ f , θ ] = D ψ exp ( iS [ f , θ ]) 2 f , θ couple as force/response current. � ¯ � � 2 k i + q i � ψ cl � θ i f i + θ i � ¯ � S [ f , θ ] = S + ψ cl ψ q k + q f i − θ i − θ i ψ q 2 m q k k , q Saddle point + fluctuations: Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 33 / 35

Calculating superfluid response function Using Keldysh generating functional d 2 Z [ f , θ ] χ ij ( q ) = − i � df i ( q ) d θ j ( − q ) , Z [ f , θ ] = D ψ exp ( iS [ f , θ ]) 2 f , θ couple as force/response current. � ¯ � � 2 k i + q i � ψ cl � θ i f i + θ i � ¯ � S [ f , θ ] = S + ψ cl ψ q k + q f i − θ i − θ i ψ q 2 m q k k , q Saddle point + fluctuations: + + + + . . . + Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 33 / 35

Calculating superfluid response function Using Keldysh generating functional d 2 Z [ f , θ ] χ ij ( q ) = − i � df i ( q ) d θ j ( − q ) , Z [ f , θ ] = D ψ exp ( iS [ f , θ ]) 2 f , θ couple as force/response current. � ¯ � � 2 k i + q i � ψ cl � θ i f i + θ i � ¯ � S [ f , θ ] = S + ψ cl ψ q k + q f i − θ i − θ i ψ q 2 m q k k , q Saddle point + fluctuations: Only one diagram for χ N + + + + . . . + Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 33 / 35

Measuring superfluid density 1. Effect rotating frame Polariton polarization: ( ψ � , ψ � ) (a) ℓ 2 r 2 e 2 i φ � � µ c H = λ r 2 e − 2 i φ − ℓ 2 Jonathan Keeling Condensation, superfluidity and lasing RETUNE, June 2012 34 / 35