From localization to coherence: A tunable Bose-Einstein condensate in disordered potentials

Benjamin DeisslerLENS and Dipartimento di Fisica, Università di Firenze

June 03, 2010

Introduction

Superfluids in porous media

GrapheneLight propagation in random media

Disorder is ubiquitous in nature. Disorder, even if weak, tends to inhibit transport and can destroy superfluidity.

Still under investigation, despite several decades of research; also important for applications (e.g. wave propagation in engineered materials)

Ultracold atoms: ideal model system

Granular and thin-film superconductors

Reviews:Aspect & Inguscio: Phys. Today, August 2009Sanchez-Palencia & Lewenstein:

Nature Phys. 6, 87-95 (2010)

Adding interactions – schematic phase diagram

localization through disorder

localization through interactions

cf. Roux et al., PRA 78, 023628 (2008)Deng et al., PRA 78, 013625 (2008)

Bosons with repulsive interactions

Our approach to disorder & localization

• A binary incommensurate lattice in 1D: quasi-disorder is easier to realize than random disorder, but shows the same phenomenology (“quasi-crystal”)

• An ultracold Bose gas of 39K atoms: precise tuning of the interaction to zero

• Fine tuning of the interactions permits the study of the competition between disorder and interactions

• Investigation of momentum distribution: observation of localization and phase coherence properties

• Investigation of transport properties

Realization of the Aubry-André model

The first lattice sets the tunneling energy JThe second lattice controls the site energydistribution D

S. Aubry and G. André, Ann. Israel Phys. Soc. 3, 133 (1980); G. Harper, Proc. Phys. Soc. A 68, 674 (1965)

J

4J

J

4J

2D

J

4J

2D

quasiperiodic potential:localization transition at finite D = 2J

4.4 lattice sites

Experimental scheme

348 350 352

-1

0

1

G. Roati et al., Phys. Rev. Lett. 99, 010403 (2007)

Probing the momentum distribution – non-interacting

experiment theory Density distribution afterballistic expansion of the initialstationary state

Measure• Width of the central peak • exponent of generalized exponential

Scaling behavior with D/J

G. Roati et al., Nature 453, 896 (2008)

Adding interactions…

Anderson ground-state

Anderson glass

Extended BEC

Fragmented BEC

Quasiperiodic lattice: energy spectrum

4J+2Δ

cf. M. Modugno: NJP 11, 033023 (2009)

Energy spectrum: Appearance of “mini-bands”

lowest “mini-band” corresponds to lowest lying energy eigenstates

width of lowest energies 0.17Dmean separation of energies 0.05D

Momentum distribution – observables

2. Fourier transform :average local shape of the wavefunctionFit to sum of two generalized exponential functions

exponent

3. Correlations:Wiener-Khinchin theorem

gives us spatially averaged correlation function

fit to same function, get spatially averaged correlation g(4.4 lattice sites)

1. Momentum distributionwidth of central peak

Probing the delocalization

momentum width

exponent

correlations

0.05D

Probing the phase coherence

Increase in correlations and decrease in the spread of phase number of phases in the system decreases

0.05D 0.17D

Comparison experiment - theory

Experiment Theory

0.05D

independent exponentially localized states

formation of fragments

single extended state

B. Deissler et al., Nature Physics 6, 354 (2010)

1 1010

100

Gau

ssia

n w

idth

(m

)

D/J1 10

10

100

Gau

ssia

n w

idth

(m

)

D/J

Expansion in a lattice

Prepare interacting system in optical trap + lattice, then release from trap and change interactions

radial confinement ≈ 50 Hz

many theoretical predictions:Shepelyansky: PRL 70, 1787 (1993)Shapiro: PRL 99, 060602 (2007)Pikovsky & Shepelyansky: PRL 100, 094101 (2008)Flach et al.: PRL 102, 024101 (2009)Larcher et al.: PRA 80, 053606 (2009)

initial size

Expansion in a lattice

Characterize expansion by exponent a:a = 1: ballistic expansiona = 0.5: diffusiona < 0.5: sub-diffusion

fit curves to

1 10 100 1000 10000

10

15

20

25

30

35

40 2a0

230a0

500a0

690a0

800a0

1130a0

Gau

ssia

n w

idth

(m

)

time (ms)

Expansion in a lattice

Expansion mechanisms:resonances between states (interaction energy enables coupling of states within localization volume)

but: not only mechanism for our system radial modes become excited over 10sreduce interaction energy, but enable coupling between states

(cf. Aleiner, Altshuler & Shlyapnikov: arXiv:0910.4534) combination of radial modes and interactions enable delocalization

3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.00.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

s1 = 7, a = 800a

0

s1 = 5, a = 800a

0

expo

nent

a

D/J0 200 400 600 800 1000 1200

0.0

0.2

0.4

0.6

0.8 s

1 = 7, D/J = 4.9

s1 = 5, D/J = 5.7

expo

nent

a

a (a0)

Conclusion and Outlook

What’s next?• Measure of phase coherence for different length scales• What happens for attractive interactions?• Strongly correlated regime 1D, 2D, 3D systems• Random disorder• Fermions in disordered potentials

…and much more

• control of both disorder strength and interactions• observe crossover from Anderson glass to coherent, extended state by

probing momentum distribution• interaction needed for delocalization proportional to the disorder strength• observe sub-diffusive expansion in quasi-periodic lattice with non-linearity

B. Deissler et al., Nature Physics 6, 354 (2010)

The Team

Massimo InguscioGiovanni Modugno

Experiment:Ben DeisslerMatteo ZaccantiGiacomo RoatiEleonora LucioniLuca TanziChiara D’ErricoMarco Fattori

Theory:Michele Modugno

Counting localized states

one localized state

two localized states

three localized states

many localized states

controlled by playing with harmonic confinement and loading time

reaching the Anderson-localized ground state is very difficult, since Jeff 0

G. Roati et al., Nature 453, 896 (2008)

Adiabaticity?

Preparation of system not always adiabatic in localized regime, populate several states where theory expects just one

see non-adiabaticity as transfer of energy into radial direction

0.05D

Theory density profiles

Eint

AG

fBEC

BEC