Theory of fractional Lévy diffusion of cold atoms in optical lattices

Transcription

1 Theory of fractional Lévy diffusion of cold atoms in optical lattices, Erez Aghion, David Kessler Bar-Ilan Univ. PRL, (2012) PRX, (2014)

2 Fractional Calculus, Leibniz (1695) L Hospital: Can the meaning of derivatives with integral order be generalized to derivatives with nonintegral order? d 1/2 /dx 1/2? Leibniz: It will lead to a paradox, from which one day useful consequences will be drawn. d α exp(λx) dx α = λ α exp(λx) Leibnitz. d α x β Γ(1 + β) = dxα Γ(β α + 1) xβ α Euler.

3 Diffusion of atoms in optical lattice (2012) (2012) Sagi et al: Diffusion of 87 Rb in optical lattice β P (x,t) t β = K ν ν P (x, t) u β P (k, u) u β 1 = K µ k µ P (k, u). P (x, t) fitted to Lévy s distribution. Our goal: derive equations describing the atomic cloud. Metzler Klafter Physics Reports (2000).

4 Main themes Lévy statistics. Semiclassical theory of Sisyphus cooling. Area under Brownian and Bessel excursions. Scaling Green-Kubo relation. Open problems. Relation with experiment.

5 Anomalous diffusion of 87 Rb atoms

6 Lévy Central Limit theorem (1930) Sum of independent identically distributed random variables N i=1 χ i. Gaussian statistics if the variance of the random variable χ is finite. If the variance diverges, Lévy central limit theorem holds. N i=1 χ i/n 1/ν is Lévy distributed. q(χ) χ (1+ν) and 0 < ν < 2. Fourier Transform of Lévy distribution L ν (x) is exp( k ν ).

7 How does it look like (Mandelbrot)? (1960) y y x x Physics: Lévy flights are unphysical since x 2 = (causality?)

8 Lévy Walks (Shlesinger, West, Klafter 1987) Particle travels with constant speed between collision events. Waiting times are power law distributed x 2 < t 2

9 Stochastic theory- Summary P (x,t) t = K ν ν P (x, t) Solution in Fourier space exp( K ν t k ν ) (Lévy statistics). x 2 = so diffusion constant is infinite. What is the meaning of this?

10 Strange friction force Basic mechanism: Castin, Dalibard, Cohen-Tannoudji (1991) Connection to anomalous diffusion: Marksteiner, Ellinger, Zoller (1996).

11 Sisyphus Cooling Castin, Dalibard, Cohen-Tannoudji Atoms with degenerate ground state. Two counter propagating lasers produce optical lattice. E(z) = E 0 e iqz ( ê x + ê y e 2iqz)

12 Castin, Dalibard, Cohen-Tannoudji (1991) Marksteiner, Ellinger, Zoller (1996).

13 Momentum Dynamics, Dimensionless Representation W (p, t) = t [D 2 p 2 ] p F (p) W (p, t) Damping force F (p) = p 1 + p 2. The cooling allows unique control of dynamics. D = ce R /U 0 Damping not effective when p >> 1 where F (p) 1/p.

14 Why F (p) 1/p? Friction force F (p) Γδp. Energy conservation (P +δp)2 2M P 2 2M = U 0. Hence δp U 0 (P /M) Damping force inversely proportional to P F (p) c U 0Γs 0 (P /M) with s 0 1 saturation parameter.

15 Velocity distribution is fat tailed Equilibrium Density: W eq = N (1 + p 2 ) 1/(2D) Power-Law Tail Divergent moments. Experiments verify this behavior (Renzoni, Walther). For D > 1/3 energy diverges! p 2 /2m =? If D > 1: no equilibrium distribution. D > 1/5 p 4 diverges (D = 1/5 will become important). Kessler, EB PRL 105, (2010).

16 Diverging energy? Walther For 1/3 < D the average kinetic energy p 2 /2m =, which is unphysical.

17 Semi-classical dynamics in phase space dp dt = F (p) + 2Dξ(t), dx dt = p. Our goal: find P (x, t) for initial conditions centered on the origin.

18 Waiting times τ and jump displacements χ τ(1) τ(2) τ(3) τ(4) 5 χ(3) 0 p -5 χ(2) χ(1) χ(4) t χ random area under velocity excursion = jump size

19 Standard Approach Einstein x 2 = 2D 1 t and D 1 = χ2 2 τ. Green-Kubo D 1 = 0 v(t + τ)v(t) dτ. But that gives D 1 = if D = ce R /U 0 > 1/5. What then?

20 Lévy walks The τ s and χ s are correlated. Problem of sum of large number of random variables x = N χ(i) + χ i=1 t = N τ(i) + τ PDF of τ is g(τ) τ 3/2 1/(2D). Hence when D > 1 τ =. PDF of χ is q(χ) χ 4/3 1/(3D). Hence when D > 1/5 χ 2 =. i=1 Neglect correlation expect x Lévy distributed.

21 τ and χ are correlated. χ τ 3/ Jump Displacement ( χ ) Simulation τ 3/ Jump Duration (τ) ψ(χ, τ) = g(τ)p(χ τ) the joint probability density of χ and τ. Scaling implies p(χ τ) τ 3/2 B(χ/τ 3/2 ).

22 Plan Find ψ(χ, τ) the joint probability density of χ and τ. With Fourier Laplace transform of ψ(χ, τ) find P (k, u) = Ψ M(k, u) 1 ψ(k, u) where Ψ M (k, u) is the contribution from the last step. Invert to find P (x, t). Take home: find ψ(χ, τ) get P (x, t).

23 Correlations are important ψ(χ, τ) the joint probability density of χ and τ ψ(χ, τ) = g(τ)p(χ τ). p(χ τ) conditional probability density Simple scaling argument p(χ τ) τ 3/2 B(χ/τ 3/2 ). χ = τ 0 p(t)dt τ t 1/2 dt τ 3/2. It follow x 2 ct 3.

24 Bessel excursions D = D = 2/3 D = 2/ p(t) t Attractive force seems to be repelling? Surviving trajectories sample the large p outskirts.

25 Area Under the Brownian and Bessel Excursion X ( ) 0 0 T Brownian paths constrained that they start at the origin and end there for the first time after time τ. Majumdar, Comtet, Darling, Louchard.

26 Path integrals for Brownian excursions (Majumdar) Let x(τ) be a Brownian excursion in (0, T ). A = t 0 x(τ)dτ Area under excursion P (A, T ) x(t )=ɛ x(0)=ɛ Dx(τ)e 1 2 T 0 dτ(dx/dτ)2 π T τ=0θ[x(τ)]δ ( T 0 x(τ)dτ A ) P (u, T ) x(t )=ɛ x(0)=ɛ Dx(τ)e T 0 dτ [ 1 2 (dx/dτ)2 +ux(τ)] π T τ=0 θ[x(τ)] Problem of QM particle in triangular potential V (x) = ux for x > 0.

27 Distribution of χ with fixed τ p(χ τ) = [ d k ] [Γ 2 k ( Γ(1 + α) 4D 1/3 ) α+1 τ 2πχ (χ) 2/3 ( ) ( ν sin π 2 + 3ν ) ( 43 2F ν2, 56 + ν2 ; 13, 23 ; 4Dλ3 k τ 3 27χ 2 ( ) ( D1/3 λ k τ 7 (χ) Γ 2/3 3 + ν sin π 4 + 3ν ) ( 76 2F ν2, 53 + ν2 ; 23, 43 ; 4Dλ3 k τ 3 ) 27χ 2 ( D 1/3 ) 2 λ k τ Γ (3 + ν) sin (πν) 2 F 2 (2 + ν2, 32 + ν2 ; 43, 53 ; 4Dλ3 k τ 3 ) ] ) χ 2/3 27χ 2 Barkai, Aghion, Kessler PRX (2014)

28 (2Dτ) 3/2 p( χ τ ) Airy Distribution D =, τ = 10 4 D = 0.4, τ = 10 4 D = 0.4, τ = 10 5 D = 0.4, τ = χ / (2Dτ) 3/2

29 Lévy distribution, weakly correlated phase When 1/5 < D < 1 Lévy statistics describes the center part of the packet. D < 1/5, deep optical lattices, Gaussian diffusion. 1/5 < D we get χ 2 =. Correlations are important only in the tails of P (x, t), for x t 3/2. We find β = 1, ν = (1 + D)/(3D) and K ν β P (x, t) t β = K ν ν P (x, t) P (x, t) [ ] 1 (K ν t) L x 1/ν ν,0 ( Kν t 1/ν).

30 Lévy distribution for P (x, t) t 1/ν P(x,t) t = 10 4 t = 10 5 t = 10 6 t = 10 7 Lévy x / t 1/ν t (1+3ν)/2 P(x,t) t = 10 4 t = 10 5 t = x / t 3/2 The cutoff gives superdiffusion x 2 t η with 1 < η = 4 3ν/2 < 3.

31 Diffusion constant K ν anomalous diffusion coefficient, units [cm ν /sec]. Cooling force F (p) = αp/[1 + (p/p c ) 2 ]. Reminder: ν = (1 + D)/(3D), and 2/3 < ν < 2. K ν = π(3ν 1) ν 1 Γ ( ) 3ν 1 2 Γ ( ) 3ν ν 1 [Γ(ν)] 2 sin ( ) πν 2 ( pc m ) ν (α) ν+1. K ν is found from average jump duration τ and x defined through q(χ) (x ) ν / χ 1+ν K ν = π(x ) ν τ Γ(1 + ν) sin πν 2. We see that correlation are not important.

32 Obhukov-Richardson diffusion: the correlated phase When D > 1 average flight time τ =. Lévy index ν approaches 2/3 as D 1, x scales like t 3/2. Here P (x, t) t 3/2 h(x/t 3/2 ). Indeed when D >> 1, damping negligible, we have free diffusion P (x, t) [ ] 3 4πDt exp 3x2. 3 4Dt 3 Obhukov (1956) Richardson (1926) model of tracer particle in turbulence. Here x 2 t 3.

33 Comparison with experiment Renzoni measured equilibrium distribution of momentum, semiclassical theory works well. So do simulations. Our work shows transitions from Gaussian D < 1/5 to Lévy 1/5 < D < 1 to Obukhov-Richardson scaling D > 1. Experiment shows that depth of optical potential controls the Lévy exponent. Experiments: fit to Lévy distribution, a new exponent was introduced, to describe full width at half maximum. In experiments no x 2 t 3, at most ballistic.

34 Green Kubo Relation Green-Kubo relation between diffusion constant and velocity correlation function. x 2 = 2D 1 t D 1 = 0 dτ v(t + τ)v(t). In our case D 1. What then? Dechant, Lutz, Kessler Barkai PRX (2014)

35 Scaling Green Kubo relation For non stationary processes, exhibiting aging, ( ) τ v(t + τ)v(t) = Ct η 2 φ. t Then x 2 (t) = 2D η t η with D η = C η 0 ds φ(s) (1 + s) η. However this relation is valid for a process starting at t = 0. For [x(t 0 + t) x(t 0 )] 2 = 2D η,s t η for t << t 0. Is D η,s = D η?

36 Persistence of initial conditions D η,s 2.0 D η D η D 1 = D 1,s /D

37 The last jump... the meander x = N χ(i) + χ i=1

38 Summary Strange friction force is responsible for non-boltzmann Gibbs equilibrium state for cold atoms. As long as the heat bath (=laser) is coupled to the system. Usual transport theory, Green-Kubo, Gaussian central limit theorem and the diffusion equation are replaced. Rich dynamical phase diagram, Normal, Lévy, Richardson. Many unsolved problems remain. Persistent initial condition leave their mark on the diffusivity D η. All this without heavy-tailed waiting times and without disorder.

39 Refs. and Thanks D. Kessler, E. Barkai Infinite covariant density for diffusion in logarithmic potentials and optical lattices Phys. Rev. Lett. 105, (2010). A. Dechant, E. Lutz, D. Kessler, E. Barkai Fluctuations of time averages for Langevin dynamics in a binding force field Phys. Rev. Lett. 107, (2011). D. A. Kessler, and E. Barkai Theory of fractional-lévy kinetics for cold atoms diffusing in optical lattices Phys. Rev. Lett (2012). E. Barkai, E. Aghion, and D. Kessler From the area under the Bessel excursion to anomalous diffusion of cold atoms Physical Review X 4, (2014). A. Dechant, E. Lutz, D. Kessler, E. Barkai Scaling Green-Kubo relation and application to three aging systems. Physical Review X 4, (2014). A. Dechant, D. A. Kessler and E. Barkai Deviations from Boltzmann-Gibbs equilibrium in confined optical lattices arxiv: [cond-mat.stat-mech]

40 Level diagram

41 Polarization Optical Lattice

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44 Momentum distribution, Renzoni prl (2006)

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49 x 2 (t) = 2D ν t ν D 1 = dτ C(τ) 0 Green-Kubo D ν = ds (s + 1) ν φ(s) 0 Scaling Green-Kubo

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51 Second moment: Kinetic energy 1/3 < D < 1 W eq useless for calculating p 2. p 2 = 0 p2 W (p, t)dt grows with time! p 2 is determined by the scaling function F(z) p 2 = 2t γ z 2 F(z)dz. 0 Here 0 < γ = 3D 1 2D < 1 anomalous subdiffusion. For D 1, p 2 t, as in normal diffusion force fields.

52 Sagi (abstract): The shape of the the distribution is found to be well fitted by a Levy distribution, but with a characteristic exponent that differs from the temporal one. Add fig. 5 in Sagi et al.