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
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