Critical Dynamics in Driven-Dissipative Bose-Einstein Condensation

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1 Critical Dynamics in Driven-Dissipative Bose-Einstein Condensation Uwe C. Täber 1 and Sebastian Diehl Weigang Li 1 1 Department of Physics, Virginia Tech, Blacsbrg, Virginia, USA Institte of Theoretical Physics, TU Dresden, Germany Renormalization Methods in Statistical Physics and Lattice Field Theories Montpellier, 8 Agst 015 Ref.: Phys. Rev. X 4, (014); arxiv:

2 Otline Experimental Motivation Langevin Description of Critical Dynamics Driven-Dissipative Bose Einstein Condensation Relationship with Eilibrim Critical Dynamics Scaling Laws and Critical Exponents Onsager Machlp Fnctional Janssen De Dominicis Response Fnctional One-Loop Renormalization Grop Analysis Two-Loop Renormalization Grop Analysis Conserved Dynamics Variant Critical Aging and Otloo

Experimental Motivation Pmped semicondctor antm wells in optical cavities: driven Bose Einstein condensation of exciton-polaritons J. Kasprza et al., Natre 443, 409 (006); K.G. Lagodais et al.

3 Experimental Motivation Pmped semicondctor antm wells in optical cavities: driven Bose Einstein condensation of exciton-polaritons J. Kasprza et al., Natre 443, 409 (006); K.G. Lagodais et al., Natre Physics 4, 706 (008) Theoretical approach: I nonlinear Langevin dynamics, mapped to path integral I pertrbatively analyze ltraviolet divergences (d dc = 4) I scale (µ) dependence, flow eations for rnning coplings I emerging symmetry: ξ indces scale invariance I critical RG fixed point scale invariance, infrared scaling laws I loop expansion in ϵ = dc d 1 critical exponents

4 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 zν, dynamic exponent z coarse-grained description: fast modes random noise mesoscopic Langevin eation for slow variables S α (x, t) Example: prely relaxational critical dynamics ( model A ): S α (x, t) t = D δh[s] δs α (x, t) + ζα (x, t), ζ α (x, t) = 0, ζ α (x, t) ζ β (x, t ) = D B T δ(x x ) δ(t t ) δ αβ Einstein relation garantees that P[S, t] e H[S]/ BT as t non-conserved order parameter: D = const. conserved order parameter: relaxes diffsively, D D Generally: mode coplings to additional conserved, slow fields varios dynamic niversality classes

5 Driven-Dissipative Bose Einstein Condensation Noisy Gross Pitaevsii eation for complex bosonic field ψ: ψ(x, t) [ i = (A id) µ + iχ t + (λ iκ) ψ(x, t) ] ψ(x, t) + ζ(x, t) A = 1/m eff ; D diffsivity (dissipative); µ chemical potential; χ pmp rate - loss; λ, κ > 0: two-body interaction / loss noise correlators: (γ = 4D B T in eilibrim) 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, = 6κ D, r K = A D, r U = λ κ, ζ iζ time-dependent complex Ginzbrg Landa eation ψ(x, t) t = D [ r + ir (1 + ir K ) + 6 (1 + ir U) ψ(x, t) ] ψ(x, t) + ζ(x, t)

6 Relationship with Eilibrim Critical Dynamics Model A relaxational inetics for non-conserved order parameter: ψ(x, t) t = D δ H[ψ] δψ + ζ(x, t) (x, t) with non-hermitean Hamiltonian [ (r H[ψ] = d d x + ir ) ψ(x, t) + (1 + ir K ) ψ(x, t) + 1 (1 + ir U) ψ(x, t) 4] (1) r = r K = r U = 0: eilibrim model A for non-conserved two-component order parameter, GL-Hamiltonian H[ψ] () r = r U r, r K = r U 0: S 1/ = Re/Imψ, H = (1 + ir K ) H S α (x, t) t = D δh[ S] δs α (x, t) + Dr K β ϵ αβ δh[ S] δs β (x, t) + η α(x, t) η α (x, t) = 0, η α (x, t) η β (x, t ) = γ δ αβ δ(x x ) δ(t t ) effective eilibrim dynamics with detailed balance (FDT)

7 Scaling Laws and Critical Exponents (Bi-)critical point τ, τ = r K τ 0: correlation length ξ(τ) τ ν niversal scaling for dynamic response and correlation fnctions: 1 ( ω ) χ(, ω, τ) η (1 + ia η η ˆχ c ) z (1 + ia η η, ξ c ) 1 ( ω ) C(, ω, τ) Ĉ z, ξ, a η η c +z η five independent critical exponents (three in eilibrim: ν, η, z) Non-pertrbative (nmerical) renormalization grop stdy: d = 3: ν 0.716, η = η 0.039, z.11, η c 0.3 L.M. Sieberer, S.D. Hber, E. Altman, S. Diehl, Phys. Rev. Lett. 88, (013); Phys. Rev. B 89, (014) Thermalization: one-loop scenario (); two-loop model A (1) Critical exponents in ϵ = 4 d expansion: ν = 1 + ϵ 10 + O(ϵ ), η = ϵ 50 + O(ϵ3 ) z = + cη, c = 6 ln O(ϵ) as for eilibrim model A; in addition, novel critical exponent: η c = c η, c = ( 4 ln 4 3 1) + O(ϵ), bt FDT η = η ϵ = 1: ν 0.65, η = η 0.0, z.0145, η c

8 Onsager Machlp Fnctional Copled Langevin eations 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 ) = 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, fnctional of S α Assme Gassian stochastic process probability distribtion: [ W[ζ] exp 1 tf d d x dt ζ α (x, t) [ (L α ) 1 ζ α (x, t) ]] 4 0 α switch variables ζ α S α : W[ζ] D[ζ] = P[S]D[S] e G[S] D[S], with Onsager-Machlp fnctional providing field theory action: G[S] = 1 d d x dt ( t S α F α [S]) [ (L α ) 1 ( t S α F α [S]) ] 4 α fnctional determinant = 1 with forward (Itô) discrectization normalization: D[ζ]W [ζ] = 1 partition fnction = 1 problems: (L α ) 1, high non-linearities F α [S] (L α ) 1 F α [S]

9 Janssen De Dominicis Response Fnctional Average over noise histories : A[S] ζ D[ζ] A[S(ζ)] W [ζ]: se 1 = D[S] α (x,t) δ( t S α (x, t) F α [S](x, t) ζ α (x, t) ) = D[i S] D[S] exp [ d d x dt S α α ( t S α F α [S] ζ α ) ] [ A[S] ζ D[i S] D[S] exp d d x dt ] S α ( t S α F α [S]) α ( A[S] D[ζ] exp d d x dt α perform Gassian integral over noise ζ α : A[S] ζ = D[S] A[S] P[S], P[S] [ ]) 1 4 ζα (L α ) 1 ζ α S α ζ α D[i S] e A[ S,S], with Janssen De Dominicis response fnctional tf A[ S, S] = d d x dt [ Sα ( t S α F α [S]) S ] α L α Sα 0 α D[i S] D[S] e A[ S,S] = 1; integrate ot S α Onsager Machlp

10 One-Loop Renormalization Grop Analysis Casality: propagator directed line, noise two-point vertex two-point vertex fnction with γ = 4DT, r = r U r, = T : [ Γ ψψ (, ω) = iω+d r(1+ir U )+(1+ir K ) + 3 (1+ir U) + ] 1 r + flctation-indced shift of critical point: τ = r r c, τ = r U τ ( ) = r U r K : β = R 1 + r KR R + R R3 1+rKR R 0 [ ] β = R ϵ R R 3(1+rKR ) R R thermalization; to O(ϵ): ν 1 = 5 ϵ, η = η c = η = 0, z =

11 Two-Loop Renormalization Grop Analysis two-loop Feynman graphs for two-point vertex fnctions Γ ψ ψ (, ω), Γ ψψ (, ω) special case r = 0: T. Risler, J. Prost, and F. Jülicher, Phys. Rev. E 7, (005) RG beta fnction β rk : r KR 0, hence to O(ϵ ): + 9ΒrK R η = η = ϵ 50 + O(ϵ3 ) ( z = + ϵ 50 6 ln 4 3 ( 1) η c = ϵ 50 4 ln 4 3 1) sbleading scaling exponent rkr

12 Conserved Dynamics Variant Complex model B variant for conserved order parameter: t ψ(x, t) = D δ H[ψ] δψ + ζ(x, t), ζ(x, t) = 0 (x, t) ζ (x, t) ζ(x, t ) = 4T D δ(x x ) δ(t t ) non-linear vertex to all orders: Γ ψψ ( = 0, ω) = iω, Γ ψ ψ (, ω = 0) =0 = DT exact scaling relations: η = η, z = 4 η dynamic scaling laws: 1 ( χ(, ω, τ) η (1 + ia η η ˆχ c ) C(, ω, τ) 1 6 η Ĉ ( ω 4 η, ξ, a η ηc ) ω 4 η (1 + ia η η c ), ξ to two-loop order: η c = η + O(ϵ 3 ) = ϵ 50 + O(ϵ3 ) non-eilibrim drive indces no independent critical exponent )

13 Critical Aging and Otloo Qench from random initial conditions onto critical point: time translation invariance broen t c : system always remembers disordered initial state critical aging scaling in limit t /t 0: ( t ) θ χ(, t, t z +η (, τ) t 1 + ia η η ˆχ z ( 1 + ia ) ) η ηc t, ξ c model A: θ = γ 0 /z, γ 0 = 5 ϵ + O(ϵ ) as in eilibrim inetics model B: θ = 0 exactly Crrent and ftre projects: critical ench of driven complex Model A: two-loop analysis coarsening, critical aging in sitable spherical model limit? thermalization and emerging Model E dynamics?

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