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

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

Benjamin Deissler

LENS
and
Dipartimento

di

Fisica
,
Università

di

Firenze


June 03, 2010

Introduction

Superfluids in
porous media

Graphene

Light 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 2009

Sanchez
-
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
39
K 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 J

The second lattice controls the site energy

distribution
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

2
D

J

4J

2
D

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 after

ballistic expansion
of the initial

stationary 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

4
J
+2
Δ

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
0.17
D

mean separation of energies
0.05
D


Momentum distribution


observables

2. Fourier transform


:

average local shape of the
wavefunction

Fit 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 distribution

width of central peak

Probing the delocalization

momentum width

exponent

correlations

0.05
D

Probing the phase coherence

Increase in correlations and decrease in the spread of
phase


number of phases in the system decreases

0.05
D

0.17
D

Comparison experiment
-

theory

Experiment

Theory

0.05
D

independent exponentially
localized states

formation of
fragments

single extended
state

B. Deissler
et al
.,
Nature
Physics

6
, 354 (2010)

1
10
10
100
Gaussian width (

m)
D
/J
1
10
10
100
Gaussian width (

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 expansion


a
= 0.5: diffusion


a
< 0.5: sub
-
diffusion

fit curves to

1
10
100
1000
10000
10
15
20
25
30
35
40
2a
0
230a
0
500a
0
690a
0
800a
0
1130a
0
Gaussian width (

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 10s

reduce 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
7.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
s
1
= 7, a = 800a
0
s
1
= 5, a = 800a
0


exponent
a
D
/J
0
200
400
600
800
1000
1200
0.0
0.2
0.4
0.6
0.8
s
1
= 7,
D
/J = 4.9
s
1
= 5,
D
/J = 5.7


exponent
a
a (a
0
)
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 Inguscio

Giovanni Modugno


Experiment:

Ben Deissler

Matteo Zaccanti

Giacomo Roati

Eleonora Lucioni

Luca Tanzi

Chiara D’Errico

Marco 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
J
eff


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

0.05
D

Theory density profiles

E
int

cutoff for evaluating
different regimes

AG

fBEC

BEC