Theory of fractional Lévy diffusion of coldatoms in optical lattices

Eli Barkai

Bar-Ilan Univ.

Kessler, EB PRL, 108 230602 (2012)EB, Aghion, Kessler PRX, 4 011022 (2014)Dechant, Kessler, EB PRL, 115, 173006 (2015) .Aghion, Kessler, EB PRL, 118, 173006 (2017) .

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.

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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 = −Kν|k|νP (k, u).

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

• Our goal: derive equations describing the atomic cloud.

Metzler Klafter Physics Reports (2000).

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Main themes

• Lévy statistics.

• Semiclassical theory of Sisyphus cooling.

• Area under Brownian and Bessel excursions.

• Scaling Green-Kubo relation.

• Open problems. Relation with experiment.

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Anomalous diffusion of 87Rb atoms

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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).

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

x

y

x

y

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

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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

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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?

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Strange friction force

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

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

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Sisyphus Cooling Castin, Dalibard, Cohen-Tannoudji

• Atoms with degenerate ground state.

• Two counter propagating lasers produce optical lattice.

• E(z) = E0eiqz(ex + eye

−2iqz)

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

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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.

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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.

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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 (2010). Hirschberg, Mukamel, Schutz PRE (2011).

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Diverging energy? Walther

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

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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.

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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

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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?

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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.

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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).

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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).

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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.

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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.

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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.

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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.

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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)

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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

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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)] .

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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.

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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.

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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.

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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.

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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)

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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?

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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∗

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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.

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Refs. and Thanks

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

• 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 Phys. Rev. Lett. 115, 173006 (2015).

• E. Aghion, D. A. Kessler, and E. Barkai Large-fluctuations for spatial diffusionof cold atoms Phys. Rev. Lett. 118, 260601 (2017).

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Level diagram

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Polarization Optical Lattice

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Momentum distribution, Renzoni prl (2006)

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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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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.

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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.

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