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Theory of fractional Lévy diffusion of coldatoms in optical lattices

Eli Barkai, Erez Aghion, David Kessler

Bar-Ilan Univ.

PRL, 108 230602 (2012)PRX, 4 011022 (2014)

Eli Barkai

Fractional Calculus, Leibniz (1695)

• L’Hospital: Can the meaning of derivatives with integralorder be generalized to derivatives with nonintegralorder?

• d1/2/dx1/2 ?

• Leibniz: It will lead to a paradox, from which one dayuseful consequences will be drawn.

dα exp(λx)

dxα= λ

αexp(λx) Leibnitz.

dαxβ

dxα=

Γ(1 + β)

Γ(β − α+ 1)xβ−α Euler.

Eli Barkai

Diffusion of atoms in optical lattice (2012)

• (2012) Sagi et al: Diffusion of 87Rb in optical lattice

∂βP (x,t)

∂tβ= Kν∇νP (x, t)

uβP (k, u)− uβ−1 = −Kµ|k|µP (k, u).

• P (x, t) fitted to Lévy’s distribution.

• Our goal: derive equations describing the atomic cloud.

Metzler Klafter Physics Reports (2000).

Eli Barkai

Main themes

• Lévy statistics.

• Semiclassical theory of Sisyphus cooling.

• Area under Brownian and Bessel excursions.

• Scaling Green-Kubo relation.

• Open problems. Relation with experiment.

Eli Barkai

Anomalous diffusion of 87Rb atoms

Eli Barkai

Lévy Central Limit theorem (1930)

• Sum of independent identically distributed randomvariables

∑Ni=1χi.

• Gaussian statistics if the variance of the random variableχ is finite.

• If the variance diverges, Lévy central limit theoremholds.

•∑Ni=1χi/N

1/ν is Lévy distributed.

• q(χ) ∼ χ−(1+ν) and 0 < ν < 2.

• Fourier Transform of Lévy distribution Lν(x) isexp(−|k|ν).

Eli Barkai

How does it look like (Mandelbrot)? (1960)

x

y

x

y

Physics: Lévy flights are unphysical since 〈x2〉 =∞ (causality?)

Eli Barkai

Lévy Walks (Shlesinger, West, Klafter 1987)

- 200 - 100 0 100 200 300 400- 300

- 200

- 100

0

100

• Particle travels with constant speed between collision events.

• Waiting times are power law distributed 〈x2〉 < t2

Eli Barkai

Stochastic theory- Summary

∂P (x,t)∂t = Kν∇νP (x, t)

• Solution in Fourier space exp(−Kνt|k|ν) (Lévy statistics).

• 〈x2〉 =∞ so diffusion constant is infinite.

• What is the meaning of this?

Eli Barkai

Strange friction force

• Basic mechanism: Castin, Dalibard, Cohen-Tannoudji (1991)

• Connection to anomalous diffusion: Marksteiner, Ellinger, Zoller (1996).

Eli Barkai

Sisyphus Cooling Castin, Dalibard, Cohen-Tannoudji

• Atoms with degenerate ground state.

• Two counter propagating lasers produce optical lattice.

• E(z) = E0eiqz(ex + eye

−2iqz)

Eli Barkai

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

Eli Barkai

Momentum Dynamics, Dimensionless Representation

∂

∂tW (p, t) =

[D∂2

∂p2− ∂

∂pF (p)

]W (p, t)

Damping forceF (p) = − p

1 + p2.

The cooling allows unique control of dynamics.

D = cER/U0

Damping not effective when p >> 1 where F (p) ∼ −1/p.

Eli Barkai

Why F (p) ∝ −1/p?

• Friction force F (p) ' −Γδp.

• Energy conservation (P+δp)2

2M − P 2

2M = U0.

• Hence δp ' U0(P/M)

• Damping force inversely proportional to P

F (p) ∼ −cU0Γs0

(P/M)

with s0 � 1 saturation parameter.

Eli Barkai

Velocity distribution is fat tailed

• Equilibrium Density:

Weq = N (1 + p2)−1/(2D)

• Power-Law Tail ⇒ Divergent moments.

• Experiments verify this behavior (Renzoni, Walther).

• For D > 1/3 energy diverges! 〈p2/2m〉 =∞?

• If D > 1: no equilibrium distribution.

• D > 1/5 〈p4〉 diverges (D = 1/5 will become important).

Kessler, EB PRL 105, 120602 (2010).

Eli Barkai

Diverging energy? Walther

• For 1/3 < D the average kinetic energy 〈p2〉/2m =∞, which is unphysical.

Eli Barkai

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.

Eli Barkai

Waiting times τ and jump displacements χ

0 200 400 600 800t

-20

-15

-10

-5

0

5

10

15p

τ(1)τ(2)

τ(3) τ(4)

χ(4)χ(4)χ(4)

χ(3)

χ(2)

χ(1)

χ random area under velocity excursion = jump size

Eli Barkai

Standard Approach

Einstein 〈x2〉 = 2D1t and

D1 =〈χ2〉2〈τ〉.

Green-Kubo

D1 =

∫ ∞0

〈v(t+ τ)v(t)〉dτ.

But that gives D1 =∞ if D = cER/U0 > 1/5.

What then?

Eli Barkai

Lévy walks

The τ ’s and χ’s are correlated.Problem of sum of large number of random variables

x =

N∑i=1

χ(i) + χ∗

t =

N∑i=1

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 〈χ2〉 =∞.

Neglect correlation expect x Lévy distributed.

Eli Barkai

τ and χ are correlated. χ ∼ τ3/2.

104 105 106 107

Jump Duration (τ)105

106

107

108

109

1010

Jum

p D

ispl

acem

ent (

|χ| )

Simulationτ3/2

ψ(χ, τ) = g(τ)p(χ|τ) the joint probability density of χ and τ .Scaling implies p(χ|τ) ∼ τ−3/2B(χ/τ3/2).

Eli Barkai

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) =ΨM(k, u)

1− ψ(k, u)

where ΨM(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).

Eli Barkai

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

p(χ|τ) ∼ τ−3/2B(χ/τ3/2).

Simple scaling argument

χ =

∫ τ

0

p(t)dt ∝∫ τ

t1/2dt ∼ τ3/2.

It follow 〈x2〉 ≤ ct3.

Eli Barkai

Bessel excursions

0 20000 40000 60000 800000

100

200

300

400

p(t)

D = ∞

0 20000 40000 60000 80000t

0

100

200

300

400

D = 2/3

0 20000 40000 60000 800000

100

200

300

400

D = 2/5

Attractive force seems to be repelling?

Surviving trajectories sample the large p outskirts.

Eli Barkai

Area Under the Brownian and Bessel Excursion

784 Majumdar and Comtet

00

(X

)

T

Fig. 2. A Brownian excursion over the time interval 0 ! ! ! T starting at x(0) = " andending at x(T )= " and staying positive in between.

Note that we have suppressed the " dependence of P(A,T ) for brev-ity. The normalization of the distribution,

! !0 P(A,T )dA = 1, is ensured

by the following definition of the partition function ZE of the Brownianexcursion

ZE =" x(T )="

x(0)="Dx(! ) e" 1

2! T

0 d! (dx/d! )2T#

!=0

# [x(! )] . (5)

All paths inside the path integrals in Eqs. (4) and (5) propagate from theirinitial value x(0)= " at ! =0 to their final value x(0)= " at ! =T .

We first evaluate the partition function ZE . This is easy since one canidentify the quantity inside the exponential in Eq. (5) as the action corre-sponding to a single particle quantum Hamiltonian, H0 #" 1

2d2

dx2 +V0(x),where the potential V0(x)=0 for x >0 and V0(x)=! for x ! 0. The infi-nite potential for x ! 0 ensures that the path never crosses zero and thustakes care of the indicator function

$T!=0 # [x(! )] in Eq. (5). Using the

standard bra–ket notation, the partition function ZE is then simply thepropagator

ZE = $"|e"H0T |"%. (6)

It is easy to see that the Hamiltonian H0 has continuous eigenvalues E0 =k2/2 labeled by k " 0 and the corresponding eigenfunctions that vanish at

Brownian paths constrained that they start at the origin and endthere for the first time after time τ.

Majumdar, Comtet, Darling, Louchard.

Eli Barkai

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 )=ε

x(0)=ε

Dx(τ)e−12

∫ T0 dτ(dx/dτ)2

πTτ=0θ[x(τ)]δ

(∫ T

0

x(τ)dτ −A)

P (u, T ) ∝∫ x(T )=ε

x(0)=ε

Dx(τ)e−∫ T

0 dτ[12(dx/dτ)2+ux(τ)]πTτ=0θ[x(τ)]

Problem of QM particle in triangular potential V (x) = ux for x > 0.

Eli Barkai

Distribution of χ with fixed τ

p(χ|τ) =

−Γ(1 + α)

2πχ

(4D1/3τ

(χ)2/3

)α+1

∑k

[dk]2

[Γ

(5

3+ ν

)sin

(π

2 + 3ν

3

)2F2

(4

3+ν

2,5

6+ν

2;1

3,2

3;−4Dλ3

kτ3

27χ2

)

−D1/3λkτ

(χ)2/3Γ

(7

3+ ν

)sin

(π

4 + 3ν

3

)2F2

(7

6+ν

2,5

3+ν

2;2

3,4

3;−4Dλ3

kτ3

27χ2

)

+1

2

(D1/3λkτ

χ2/3

)2

Γ (3 + ν) sin (πν) 2F2

(2 +

ν

2,3

2+ν

2;4

3,5

3;−4Dλ3

kτ3

27χ2

)]Barkai, Aghion, Kessler PRX (2014)

Eli Barkai

0 0.5 1 1.5

|χ| / (2Dτ)3/2

0

0.5

1

1.5

2

2.5

3

(2D

τ)3/

2p(

|χ| |

τ )

Airy DistributionD = ∞, τ = 104

D = 0.4, τ = 104

D = 0.4, τ = 105

D = 0.4, τ = 106

Eli Barkai

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 〈χ2〉 =∞.

• Correlations are important only in the tails of P (x, t), for x ' t3/2.

• We find β = 1, ν = (1 +D)/(3D) and Kν

∂βP (x, t)

∂tβ= Kν∇ν

P (x, t)

P (x, t) ∼1

(Kνt)1/νLν,0

[x(

Kνt1/ν)] .

Eli Barkai

Lévy distribution for P (x, t)

0 25 50 75|x| / t1/ν

0

0.01

0.02

0.03

0.04

t1/ν P(

x,t)

t = 104

t = 105

t = 106

t = 107

Lévy

10-4 10-3 10-2 10-1 100

|x| / t3/2

100

103

106

t(1+3

ν)/2

P(x

,t)

t = 104

t = 105

t = 106

The cutoff gives superdiffusion 〈x2〉 ∼ tη with 1 < η = 4− 3ν/2 < 3.

Eli Barkai

Diffusion constant

• Kν anomalous diffusion coefficient, units [cmν/sec].

• Cooling force F (p) = −αp/[1 + (p/pc)2].

• Reminder: ν = (1 +D)/(3D), and 2/3 < ν < 2.

Kν =

√π(3ν − 1)ν−1Γ

(3ν−1

2

)Γ(

3ν−22

)32ν−1[Γ(ν)]2 sin

(πν2

) (pcm

)ν(α)

−ν+1.

• Kν is found from average jump duration 〈τ〉 and x∗ definedthrough q(χ) ∼ (x∗)ν/|χ|1+ν

Kν =π(x∗)ν

〈τ〉Γ(1 + ν) sin πν2

.

• We see that correlation are not important.

Eli Barkai

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 t3/2.

• Here P (x, t) ∼ t−3/2h(x/t3/2).

• Indeed when D >> 1, damping negligible, we have free diffusion

P (x, t) ∼√

3

4πDt3exp

[−

3x2

4Dt3

].

• Obhukov (1956) Richardson (1926) model of tracer particle in turbulence.

• Here 〈x2〉 ∼ t3.

Eli Barkai

Comparison with experiment

• Renzoni measured equilibrium distribution of momentum, semiclassical theoryworks well. So do simulations.

• Our work shows transitions from Gaussian D < 1/5 to Lévy 1/5 < D < 1 toObukhov-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, todescribe full width at half maximum.

• In experiments no x2 ∼ t3, at most ballistic.

Eli Barkai

Green Kubo Relation

• Green-Kubo relation between diffusion constant and velocity correlationfunction.

〈x2〉 = 2D1t

D1 =

∫ ∞0

dτ〈v(t+ τ)v(t)〉.

• In our case D1 →∞.

• What then?

Dechant, Lutz, Kessler Barkai PRX (2014)

Eli Barkai

Scaling Green Kubo relation

• For non stationary processes, exhibiting aging,

〈v(t+ τ)v(t)〉 = Ctη−2φ

(τ

t

).

• Then 〈x2(t)〉 = 2Dηtη with

Dη =Cη

∫ ∞0

dsφ(s)

(1 + s)η.

• However this relation is valid for a process starting at t = 0.

• For 〈[x(t0 + t)− x(t0)]2〉 = 2Dη,st

η for t << t0. Is Dη,s = Dη?

Eli Barkai

Persistence of initial conditions

0 1 2 3 4 5 6 7 80 . 0

0 . 5

1 . 0

1 . 5

2 . 0

2 . 5

3 . 0

3 . 5

D 1 = D 1 , sD η

D η

1 / D

D η, s

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The last jump.... the meander

x =

N∑i=1

χ(i) + χ∗

Eli Barkai

Summary

• Strange friction force is responsible for non-Boltzmann Gibbsequilibrium state for cold atoms. As long as the heat bath(=laser) is coupled to the system.

• Usual transport theory, Green-Kubo, Gaussian central limittheorem 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 diffusivityDη.

• –All this without heavy-tailed waiting times and without disorder.

Eli Barkai

Refs. and Thanks

• D. Kessler, E. Barkai Infinite covariant density for diffusion in logarithmicpotentials and optical lattices Phys. Rev. Lett. 105, 120602 (2010).

• A. Dechant, E. Lutz, D. Kessler, E. Barkai Fluctuations of time averages forLangevin dynamics in a binding force field Phys. Rev. Lett. 107, 240603(2011).

• D. A. Kessler, and E. Barkai Theory of fractional-Lévy kinetics for cold atomsdiffusing in optical lattices Phys. Rev. Lett. 108 230602 (2012).

• E. Barkai, E. Aghion, and D. Kessler From the area under the Bessel excursionto anomalous diffusion of cold atoms Physical Review X 4, 021036 (2014).

• A. Dechant, E. Lutz, D. Kessler, E. Barkai Scaling Green-Kubo relation andapplication to three aging systems. Physical Review X 4, 011022 (2014).

• A. Dechant, D. A. Kessler and E. Barkai Deviations from Boltzmann-Gibbsequilibrium in confined optical lattices arXiv:1412.5402 [cond-mat.stat-mech]

Eli Barkai

Level diagram

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Polarization Optical Lattice

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Eli Barkai

Momentum distribution, Renzoni prl (2006)

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Eli Barkai

𝑫𝝂 = 𝒅𝒔 (𝒔 + 𝟏)−𝝂 𝝓(𝒔)∞

𝟎

Scaling Green-Kubo

𝑫𝟏 = 𝒅𝝉 𝑪(𝝉)∞

𝟎

Green-Kubo 𝒙𝟐(𝒕) = 𝟐𝑫𝝂 𝒕

𝝂

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Eli Barkai

Second moment: Kinetic energy 1/3 < D < 1

Weq useless for calculating 〈p2〉.

〈p2〉 =∫∞

0p2W (p, t)dt grows with time!

〈p2〉 is determined by the scaling function F(z)

〈p2〉 = 2tγ∫ ∞

0

z2F(z)dz.

Here 0 < γ = 3D−12D < 1 anomalous subdiffusion.

For D → 1, 〈p2〉 ∼ t, as in normal diffusion

force fields.

Eli Barkai

Sagi (abstract): The shape of the the distribution is found to be wellfitted by a Levy distribution, but with a characteristic exponent thatdiffers from the temporal one.

Add fig. 5 in Sagi et al.

Eli Barkai