Anomalous scaling at non-thermal fixed points of Gross-Pitaevskii and KPZ turbulence

Transcription

1 Anomalous scaling at non-thermal fixed points of Gross-Pitaevskii and KPZ turbulence Thomas Gasenzer Steven Mathey Jan M. Pawlowski ITP - Heidelberg 24 September 2014

2 Non-thermal fixed points Non-thermal fixed point are far from equilibrium quasi stationary states of matter. X Scale invariance ɛ(k) k d+η Depending on the initial conditions, the system may take an algebraically long time on the way to thermalisation.

3 Introduction GPE KPZ DDGPE to KPZ Conclusions Plan Ultra-cold Bose gases, Gross-Pitaevskii Equation (GPE) Interface dynamics, Kardar-Parisi-Zhang (KPZ) z/ξ 1.5 ξ/lc 50 0 x/ξ t2 α = 2.5 α=3 α=4 α=6 α = 10 t y/ξ t3 0.5 FIG. 5: Vortex tangle structures emerging in the gas for the t2 t1 Burn paper by CrazzHky, used under CC BYα/ = Cropped, two different extremes of initial 2.5, at time t3 0 NIST/JILA/CU-Boulder; Nowak etconditions, al., arxiv: v t = 2424 τ (left) and α = 10, at time t = 606 τ (right). ξ/ld 1

4 Driven-Dissipative GPE Classical field equation for the average Bose wave-function φ(x, t): i t φ(x, t) = [ ( ) ] 1 2m iν 2 + µ + g φ(x, t) 2 φ(x, t) + ζ(x, t) With complex parameters µ = µ 1 + iµ 2 g = g 1 ig 2 Single particle pump 2 particle losses and stochastic driving ζ(x, t) = 0 ζ(x, t)ζ(x, t ) = γ δ(t t )δ(x x ) Sieberer et al., PRL 110, (2013); Sieberer et al., Phys. Rev. B 89, (2014); Täuber et al., Phys. Rev. X, (2014)

5 Driven-Dissipative GPE Classical field equation for the average Bose wave-function φ(x, t): i t φ(x, t) = [ ( ) ] 1 2m iν 2 + µ + g φ(x, t) 2 φ(x, t) + ζ(x, t) With complex parameters µ = µ 1 + iµ 2 g = g 1 ig 2 Single particle pump 2 particle losses and stochastic driving ζ(x, t) = 0 ζ(x, t)ζ(x, t ) = γ δ(t t )δ(x x ) Sieberer et al., PRL 110, (2013); Sieberer et al., Phys. Rev. B 89, (2014); Täuber et al., Phys. Rev. X, (2014)

6 Driven-Dissipative GPE Classical field equation for the average Bose wave-function φ(x, t): i t φ(x, t) = [ ( ) ] 1 2m iν 2 + µ + g φ(x, t) 2 φ(x, t) + ζ(x, t) With complex parameters µ = µ 1 + iµ 2 g = g 1 ig 2 Single particle pump 2 particle losses and stochastic driving ζ(x, t) = 0 ζ(x, t)ζ(x, t ) = γ δ(t t )δ(x x ) Sieberer et al., PRL 110, (2013); Sieberer et al., Phys. Rev. B 89, (2014); Täuber et al., Phys. Rev. X, (2014)

7 Scaling in ultra-cold Bose gases We focus on the kinetic energy density, ɛ kin = 1 2m φ 2 = ɛ(k) k Scaling from dimensional analysis with an anomalous correction ɛ(k) = ɛ kin k d (kξ) η d η num 1 small small Nowak et al., Phys. Rev. B 84, (R) (2011)

8 Scaling in ultra-cold Bose gases We focus on the kinetic energy density, ɛ kin = 1 2m φ 2 = ɛ(k) k Scaling from dimensional analysis with an anomalous correction ɛ(k) = ɛ kin k d (kξ) η d η num 1 small small Nowak et al., Phys. Rev. B 84, (R) (2011)

9 Scaling in ultra-cold Bose gases We focus on the kinetic energy density, ɛ kin = 1 2m φ 2 = ɛ(k) k Scaling from dimensional analysis with an anomalous correction ɛ(k) = ɛ kin k d (kξ) η d η num 1 small small Nowak et al., Phys. Rev. B 84, (R) (2011)

10 Scaling in ultra-cold Bose gases We focus on the kinetic energy density, ɛ kin = 1 2m φ 2 = ɛ(k) k Scaling from dimensional analysis with an anomalous correction ɛ(k) ɛ kin k d+η kξ d η num 1 small small Nowak et al., Phys. Rev. B 84, (R) (2011)

11 Kardar Parisi Zhang equation A model for interface growth, t θ(x, t) = ν 2 θ(x, t) + λ 2 [ θ(x, t)]2 + η(x, t) with diffusion, perpendicular expansion and stochastic driving, η(x, t) = 0 η(x, t)η(x, t ) = D δ(t t )δ(x x ) θ(x,t) x

12 Kardar Parisi Zhang equation A model for interface growth, t θ(x, t) = ν 2 θ(x, t) + λ 2 [ θ(x, t)]2 + η(x, t) with diffusion, perpendicular expansion and stochastic driving, η(x, t) = 0 η(x, t)η(x, t ) = D δ(t t )δ(x x ) θ(x,t) x

13 Kardar Parisi Zhang equation A model for interface growth, t θ(x, t) = ν 2 θ(x, t) + λ 2 [ θ(x, t)]2 + η(x, t) with diffusion, perpendicular expansion and stochastic driving, η(x, t) = 0 η(x, t)η(x, t ) = D δ(t t )δ(x x ) θ(x,t) x

14 Kardar Parisi Zhang equation A model for interface growth, t θ(x, t) = ν 2 θ(x, t) + λ 2 [ θ(x, t)]2 + η(x, t) with diffusion, perpendicular expansion and stochastic driving, η(x, t) = 0 η(x, t)η(x, t ) = D δ(t t )δ(x x ) θ(x,t) x

15 Scaling in interface growth The stationary state has scaling correlation functions, with exponents given by: θ(t + τ, x + r)θ(t, x) c = r 2χ g (τ/r z ) d χ 1/ ? z = 2 χ 3/ ? Kloss et al., Phys. Rev. E 86, (2012) and references therein

16 From DDGPE to KPZ Density and phase decomposition φ(x, t) = n(x, t) e iθ(x,t) t θ 1 2m ( θ)2 ν 2 θ = U, t n 1 m (n θ) = S, with sources of phase and density fluctuations, U = U[θ, n] + Re(ζeiθ ) n S = S[θ, n] + 2 n Im(ζe iθ ). Altman et al., arxiv: v2 [cond-mat.stat-mech]

17 From DDGPE to KPZ Density and phase decomposition φ(x, t) = n(x, t) e iθ(x,t) t θ 1 2m ( θ)2 ν 2 θ = U, KPZ equation t n 1 m (n θ) = S, with sources of phase and density fluctuations, U = U[θ, n] + Re(ζeiθ ) n S = S[θ, n] + 2 n Im(ζe iθ ). Altman et al., arxiv: v2 [cond-mat.stat-mech]

18 ɛ kin = 1 2m φ 2 = n 2m Comparing scaling exponents ɛ kin (k) k d+η η = z χ k k2 θ(k, t) 2 ɛ kin (k) k z d χ GPE simulations KPZ literature X

19 ɛ kin = 1 2m φ 2 = n 2m Comparing scaling exponents ɛ kin (k) k d+η η = z χ k k2 θ(k, t) 2 ɛ kin (k) k z d χ GPE simulations d η num 1 small small KPZ literature X d η Nowak et al., Phys. Rev. B 84, (R) (2011)

20 volution in d = 3 Introduction GPE FIG. 3: Single-particle KPZ mode DDGPE occupation to KPZ numbers as Conclusions functions of the radial momentum k = [ d ensity falls below e: g = i=1, 4sin2 (k i /2)] 1/2, k = 2πn/N s, n = (n 1,..., n d ), n i = N s /2,..., N s /2, for the four = 103 (top right) different Comparing times ofscaling the run inexponents d = 2 dimensions shown in Fig. 1. right) Note the double-logarithmic scale. An early development of ɛ kin = 1 2m φ 2 a scaling n(k) k 4 ɛ kin is (k) followed k d+η by a bimodal scaling with n(k) k η = z χ garithmic scale. = n 2m k 2 at larger wave numbers. k2 θ(k, t) 2 ɛ kin (k) k z d χ During the inidually spread to n(k) antivortex d = 2pairs n 10 observed. Durphase two dis- n i (k) GPE simulations q (k) KPZ literature n c (k) ɛ(k) = k 2 n(k) d to be in excelction k X 4 10 ν of = Ref. µ 2 [6]. = 0 4 k 3 g k = ζ = 0 Nowak et al., Phys. Rev. B 84, (R) (2011) UV = d = 2 exed the exponent q. (2), corrobo- 4]. Note that in V = 2 is identi- Rayleigh-Jeans momentum dishown again, for Occupation number n(k) initial occupation k ξ Radial momentum k

21 Comparing scaling exponents FIG. 3: Single-particle mode occupation numbers as functions of the radial momentum k = [ d 2πn/N s, n = (n 1,..., n d), n i = N s/2,..., N s/2, for the four different times of the run in = n d = 2 dimensions shown in Fig. 1. Note the double-logarithmic scale. An early development of 2m a scaling n(k) k 4 is followed by a bimodal scaling with n(k) k 2 at larger wave numbers. Occupation number n(k) ɛ kin = 1 2m φ 2 ɛ kin (k) k d+η i=1 4sin2 (k i/2)] 1/2, k = η = z χ k k2 θ(k, t) 2 ɛ kin (k) k z d χ initial occupation d = 2 d = 3 GPE simulations k ξ Radial momentum k n(k) n q(k) n c(k) n i(k) k 4 k 3 k 4.66 Occupation number n(k) X initial occupation KPZ literature k ξ 3 Radial momentum k n(k) n q(k) n c(k) n i(k) k 5 k 4 k 5.66 Nowak et al., Phys. Rev. B 84, FIG (R) 4: Single-particle (2011) occupation numbers of different FIG. fractions of the system, at time t = 10 5 of the d = 2 run shown in Fig. 2, at the time t = 1640 of the situation in 5: Same as in Fig. 4, for the run in d = 3 dimensions its 5.66

22 ɛ cin = 1 2m φ 2 = n 2m Compressible excitations ɛ c (k) k d+η+1 η = z χ 1 k k2 θ(k, t) 2 ɛ c (k) k z d χ GPE simulations d η num 0 small small KPZ literature X d η Nowak et al., Phys. Rev. A 85, (2012)

23 ɛ cin = 1 2m φ 2 = n 2m Compressible excitations ɛ c (k) k d+η+1 η = z χ 1 k k2 θ(k, t) 2 ɛ c (k) k z d χ Occupation number n(k) d = 2 d = 3 GPE simulations Occupation number n(k) 10 6 X k ξ 1 3 Radial momentum k n(k) n q(k) n c(k) n i(k) k 3 k 2 1 KPZ literature k ξ 3 Radial momentum k n(k) n q(k) n c(k) n i(k) k 4 k 2 Nowak et al., Phys. Rev. A 85, (2012)

24 ɛ cin = 1 2m φ 2 = n 2m Compressible excitations ɛ c (k) k d+η+1 η = z χ 1 k k2 θ(k, t) 2 ɛ c (k) k z d χ Occupation number n(k) d = 2 d = 3 GPE simulations Occupation number n(k) 10 6 X k ξ 1 3 Radial momentum k n(k) n q(k) n c(k) n i(k) k 3 k 2 k KPZ literature k ξ 3 Radial momentum k n(k) n q(k) n c(k) n i(k) k 4 k 2 k 3.6 Nowak et al., Phys. Rev. A 85, (2012)

25 Conclusions Interface dynamics described by KPZ equation does not capture vortex dynamics. It does captures the rest. We have made an estimation of anomalous scaling exponents of the ultra-cold Bose gas at a non-thermal fixed point. Gasenzer, SM, Pawlowski, arxiv: [cond-mat.quant-gas]