Critical Dynamics in Driven-Dissipative Bose-Einstein Condensation - PowerPoint PPT Presentation

Critical Dynamics in Driven-Dissipative Bose-Einstein Condensation auber 1 and Sebastian Diehl 2 Uwe C. T¨ Weigang Liu 1 1 Department of Physics, Virginia Tech, Blacksburg, Virginia, USA 2 Institute of Theoretical Physics, TU Dresden, Germany Renormalization Methods in Statistical Physics and Lattice Field Theories Montpellier, 28 August 2015 Ref.: Phys. Rev. X 4 , 021010 (2014); arXiv:1312.5182

Outline Experimental Motivation Langevin Description of Critical Dynamics Driven-Dissipative Bose–Einstein Condensation Relationship with Equilibrium Critical Dynamics Scaling Laws and Critical Exponents Onsager–Machlup Functional Janssen–De Dominicis Response Functional One-Loop Renormalization Group Analysis Two-Loop Renormalization Group Analysis Conserved Dynamics Variant Critical Aging and Outlook

Experimental Motivation Pumped semiconductor quantum wells in optical cavities: driven Bose–Einstein condensation of exciton-polaritons J. Kasprzak et al. , Nature 443 , 409 (2006); K.G. Lagoudakis et al. , Nature Physics 4 , 706 (2008) Theoretical approach: ◮ nonlinear Langevin dynamics, mapped to path integral ◮ perturbatively analyze ultraviolet divergences ( d ≥ d c = 4) ◮ scale ( µ ) dependence, flow equations for running couplings ◮ emerging symmetry : ξ → ∞ induces scale invariance ◮ critical RG fixed point → scale invariance , infrared scaling laws ◮ loop expansion in ϵ = d c − d ≪ 1 → critical exponents

Langevin Description of Critical Dynamics Critical slowing-down as correlated regions grow ( τ ∝ T − T c ): → relaxation time t c ( τ ) ∼ ξ ( τ ) z ∼ | τ | − z ν , dynamic exponent z coarse-grained description: ◮ fast modes → random noise ◮ mesoscopic Langevin equation for slow variables S α ( x , t ) Example: purely relaxational critical dynamics (“model A”): ∂ S α ( x , t ) δ H [ S ] δ S α ( x , t ) + ζ α ( x , t ) , ⟨ ζ α ( x , t ) ⟩ = 0 , = − D ∂ t ⟨ ζ α ( x , t ) ζ β ( x ′ , t ′ ) ⟩ = 2 D k B T δ ( x − x ′ ) δ ( t − t ′ ) δ αβ Einstein relation guarantees that P [ S , t ] → e − H [ S ] / k B T as t → ∞ non-conserved order parameter: D = const . conserved order parameter: relaxes diffusively , D → − D ∇ 2 Generally: mode couplings to additional conserved , slow fields → various dynamic universality classes

Driven-Dissipative Bose–Einstein Condensation Noisy Gross–Pitaevskii equation for complex bosonic field ψ : [ i ∂ψ ( x , t ) − ( A − iD ) ∇ 2 − µ + i χ = ∂ t + ( λ − i κ ) | ψ ( x , t ) | 2 ] ψ ( x , t ) + ζ ( x , t ) A = 1 / 2 m eff ; D diffusivity (dissipative); µ chemical potential; χ ∼ pump rate - loss; λ, κ > 0: two-body interaction / loss noise correlators: ( γ = 4 D k B T in equilibrium) ⟨ ζ ( x , t ) ⟩ = 0 = ⟨ ζ ( x , t ) ζ ( x ′ , t ′ ) ⟩ ⟨ ζ ∗ ( x , t ) ζ ( x ′ , t ′ ) ⟩ = γ δ ( x − x ′ ) δ ( t − t ′ ) r = − χ D , r ′ = − µ D , u ′ = 6 κ D , r K = A D , r U = λ κ , ζ → − i ζ → time-dependent complex Ginzburg–Landau equation [ ∂ψ ( x , t ) r + ir ′ − (1 + ir K ) ∇ 2 = − D ∂ t 6 (1 + ir U ) | ψ ( x , t ) | 2 ] + u ′ ψ ( x , t ) + ζ ( x , t )

Relationship with Equilibrium Critical Dynamics ”Model A” relaxational kinetics for non-conserved order parameter: δ ¯ ∂ψ ( x , t ) H [ ψ ] = − D δψ ∗ ( x , t ) + ζ ( x , t ) ∂ t with non-Hermitean “Hamiltonian” ∫ [( r + ir ′ ) | ψ ( x , t ) | 2 + (1 + ir K ) |∇ ψ ( x , t ) | 2 ¯ d d x H [ ψ ] = 12 (1 + ir U ) | ψ ( x , t ) | 4 ] + u ′ ◮ (1) r ′ = r K = r U = 0: equilibrium model A for non-conserved two-component order parameter, GL-Hamiltonian H [ ψ ] ◮ (2) r ′ = r U r , r K = r U ̸ = 0: S 1 / 2 = Re / Im ψ , ¯ H = (1 + ir K ) H δ H [ ⃗ ∑ δ H [ ⃗ ∂ S α ( x , t ) S ] S ] = − D δ S α ( x , t ) + Dr K ϵ αβ δ S β ( x , t ) + η α ( x , t ) ∂ t β ⟨ η α ( x , t ) ⟩ = 0 , ⟨ η α ( x , t ) η β ( x ′ , t ′ ) ⟩ = γ 2 δ αβ δ ( x − x ′ ) δ ( t − t ′ ) → effective equilibrium dynamics with detailed balance (FDT)

Scaling Laws and Critical Exponents (Bi-)critical point τ, τ ′ = r K τ → 0: correlation length ξ ( τ ) ∼ | τ | − ν universal scaling for dynamic response and correlation functions: ( ) 1 ω χ ( q , ω, τ ) ∝ | q | 2 − η (1 + ia | q | η − η c ) ˆ χ | q | z (1 + ia | q | η − η c ) , | q | ξ ( ω | q | z , | q | ξ, a | q | η − η c ) 1 | q | 2+ z − η ′ ˆ C ( q , ω, τ ) ∝ C five independent critical exponents (three in equilibrium: ν, η, z ) Non-perturbative (numerical) renormalization group study: d = 3: ν ≈ 0 . 716, η = η ′ ≈ 0 . 039, z ≈ 2 . 121, η c ≈ − 0 . 223 L.M. Sieberer, S.D. Huber, E. Altman, S. Diehl, Phys. Rev. Lett. 88 , 045702 (2013); Phys. Rev. B 89 , 134310 (2014) Thermalization : one-loop → scenario (2); two-loop → model A (1) Critical exponents in ϵ = 4 − d expansion: 10 + O ( ϵ 2 ) , η = ϵ 2 ν = 1 2 + ϵ 50 + O ( ϵ 3 ) z = 2 + c η , c = 6 ln 4 3 − 1 + O ( ϵ ) as for equilibrium model A; in addition, novel critical exponent : ( ) η c = c ′ η , c ′ = − + O ( ϵ ) , but FDT → η ′ = η 4 ln 4 3 − 1 ϵ = 1: ν ≈ 0 . 625, η = η ′ ≈ 0 . 02, z ≈ 2 . 01452, η c ≈ − 0 . 0030146

Onsager–Machlup Functional Coupled Langevin equations for mesoscopic stochastic variables: ∂ S α ( x , t ) = F α [ S ]( x , t ) + ζ α ( x , t ) , ⟨ ζ α ( x , t ) ⟩ = 0 , ∂ t ⟨ ζ α ( x , t ) ζ β ( x ′ , t ′ ) ⟩ = 2 L α δ ( x − x ′ ) δ ( t − t ′ ) δ αβ ◮ systematic forces F α [ S ], stochastic forces (noise) ζ α ◮ noise correlator L α : can be operator, functional of S α Assume Gaussian stochastic process → probability distribution: [ ∫ ∫ t f ]] ∑ [ − 1 ζ α ( x , t ) ( L α ) − 1 ζ α ( x , t ) d d x W [ ζ ] ∝ exp dt 4 0 α switch variables ζ α → S α : W [ ζ ] D [ ζ ] = P [ S ] D [ S ] ∝ e −G [ S ] D [ S ], with Onsager-Machlup functional providing field theory action: ∫ ∫ ∑ [ ] G [ S ] = 1 ( ∂ t S α − F α [ S ]) ( L α ) − 1 ( ∂ t S α − F α [ S ]) d d x dt 4 α ◮ functional determinant = 1 with forward (Itˆ o) discrectization ∫ ◮ normalization: D [ ζ ] W [ ζ ] = 1 → “partition function” = 1 ◮ problems: ( L α ) − 1 , high non-linearities F α [ S ] ( L α ) − 1 F α [ S ]

Janssen–De Dominicis Response Functional ∫ Average over noise “histories” : ⟨ A [ S ] ⟩ ζ ∝ D [ ζ ] A [ S ( ζ )] W [ ζ ]: ∫ ( ) D [ S ] ∏ ∏ ∂ t S α ( x , t ) − F α [ S ]( x , t ) − ζ α ( x , t ) use 1 = ( x , t ) δ α ∫ ∫ [ ∫ ∫ dt ∑ ] S α ( ∂ t S α − F α [ S ] − ζ α ) D [ i � α � d d x = S ] D [ S ] exp − [ ] ∫ ∫ ∫ ∫ ∑ S α ( ∂ t S α − F α [ S ]) D [ i � d d x � ⟨ A [ S ] ⟩ ζ ∝ S ] D [ S ] exp − dt α ∫ ( ∫ ∫ [ 1 ]) ∑ 4 ζ α ( L α ) − 1 ζ α − � S α ζ α d d x × A [ S ] D [ ζ ] exp − dt α perform Gaussian integral over noise ζ α : ∫ ∫ S ] e −A [ � S , S ] , D [ i � ⟨ A [ S ] ⟩ ζ = D [ S ] A [ S ] P [ S ] , P [ S ] ∝ with Janssen–De Dominicis response functional ∫ ∫ t f [ S α ] ∑ S α ( ∂ t S α − F α [ S ]) − � A [ � d d x � S α L α � S , S ] = dt 0 α ∫ ∫ D [ S ] e −A [ � S , S ] = 1; integrate out � S α → Onsager–Machlup D [ i � S ]

One-Loop Renormalization Group Analysis Causality : propagator → directed line, noise → two-point vertex k two-point vertex function with k γ = 4 DT , r ′ = r U r , u = u ′ T : q q q + u [ ∫ ] 1 r (1+ ir U )+(1+ ir K ) q 2 + 2 Γ ˜ ψψ ∗ ( q , ω ) = − i ω + D 3 u (1+ ir U ) r + k 2 k → fluctuation-induced shift of critical point: τ = r − r c , τ ′ = r U τ q q q q 2 2 2 q q q 2 u u q−k k k 2 2 2 q q + k k + q k q + q 2 u 2 2 u 2 u 2 q k k k q q q 2 2 2 2 ( ) 1 + 2 r KR ∆ R +∆ R 2 u R ∆ = r U − r K : β ∆ = ∆ R ⇒ ∆ R → 0 1+ r 2 3 KR [ ] ∆ R 2 − ϵ + 5 ⇒ u R → u ∗ β u = u R 3 u R − KR ) u R 3(1+ r 2 → thermalization ; to O ( ϵ ): ν − 1 = 2 − 2 5 ϵ , η = η c = η ′ = 0, z = 2

Two-Loop Renormalization Group Analysis q u q two-loop Feynman graphs for two-point k+k -q k vertex functions Γ ˜ ψ ∗ ( q , ω ), Γ ˜ ψψ ∗ ( q , ω ) k ψ ˜ + k k special case r ′ = 0: q k+k -q T. Risler, J. Prost, and F. J¨ ulicher, Phys. Rev. E 72 , 016130 (2005) q u k k k k k k k+k−q k q+k−k k u u u u k k + + + q q q q k k u u k k q k q k u u k k q q RG beta function β r K : 9 Β r K 2 u R ⇒ r KR → 0, hence to O ( ϵ 2 ): 12 10 η = η ′ = ϵ 2 50 + O ( ϵ 3 ) 8 ( ) z = 2 + ϵ 2 6 ln 4 3 − 1 6 50 ( ) 4 η c = − ϵ 2 4 ln 4 3 − 1 50 2 r KR subleading scaling exponent 0.5 1.0 1.5 2.0 2.5 3.0