Transport of an Interacting Bose Gas
in 1D Disordered Lattices
Chiara D’Errico
CNR-INO, LENS and Dipartimento di Fisica, Università di Firenze
15° International Conference on
Transport in Interacting Disordered Systems,
Sant Feliu , September 2013
Disorder in quantum systems
There is a growing interest in determining exactly how disorder
affects the properties of quantum systems.
Superconducting thin
films
Superfluids
in porous
media
Graphene
Biological systems
Light propagation
in random media
Anderson localization
•
•
•
•
Non-interacting particles hopping in a the lattice
With random on-site energy
A critical value of disorder is able to localize the particle wavefunction
The eigenstates are spatially localized with exponentially decreasing tails.
Disorder and quantum gases
also Shlyapnikov, Burnett, Roth, Sanchez-Palencia, Giamarchi,
Natterman, Garcia-Garcia ….
Urbana
Hannover
Rice U.
Paris
Florence
L. Sanchez-Palencia and M. Lewenstein, Nat. Phys. 6, 87 (2010);
G. Modugno, Rep. Prog. Phys. 73, 102401 (2010).
Interplay between disorder and interaction
Many-body problem to investigate the interplay between disorder &
interaction
Theoretical interest on the investigation of 1D bosons at T=0, which is a
simple prototype of disordered interacting systems
Giamarchi & Schultz, PRB 37 325 (1988)
Fisher et al PRB 40, 546 (1989), …
Rapsch, Schollwoeck, Zwerger
EPL 46 559 (1999), …
A 1D quasiperiodic lattice
1D system in a quasiperiodic potential
4J
2D
d /(  1)
In the tight binding limit: Aubry-Andrè or Harper model
Hˆ   J  bˆi bˆ j  D  cos( 2i ) nˆi
i, j
i
Metal-insulator transition at D=2J
S. Aubry and G. André, Ann. Israel Phys. Soc. 3, 133 (1980).
L. Fallani et al., PRL 98, 130404 (2007).
M. Modugno, New J. Phys. 11, 033023 (2009).

k2
k1
A 1D quasiperiodic lattice
Energy correlation function
gE(x) (arb. units)
g E ( x)   E ( x) E ( x  x) dx '
d /(  1)
-20
-10
0
10
Position (lattice sites)
20
A 1D quasiperiodic lattice
Energy (units of J)
4
2
Miniband structure
0
-2
-4
0
10
0
100
200
300
400
500
| (x)|
2
Eigenstate #
-10
10
Short, uniform localization length:
  d / logD / 2 J 
-20
10
420
440
460
Position (lattice sites)
480
Interplay between disorder and interaction
1D system in a quasiperiodic potential
4J
2D
d /
In the tight binding limit: Aubry-Andrè or Harper model
Hˆ   J  bˆi bˆ j  D  cos( 2i ) nˆi+ U å nˆi (nˆi -1)
i, j
i
i
Metal-insulator transition at D=2J
2p 2
U=
a ò | f (x) |4 d 3 x
m
S. Aubry and G. André, Ann. Israel Phys. Soc. 3, 133 (1980).
L. Fallani et al., PRL 98, 130404 (2007).
M. Modugno, New J. Phys. 11, 033023 (2009).
Tuned on the Feshbach
resonance
Interplay between disorder and interaction
Potassium-39
scattering length (a0)
400
Eint 
2  2
m
a   4 dx
0
BEC
340
350
360
370
380
390
magnetic field (G)
G. Roati, et al. Phys. Rev. Lett. 99, 010403 (2007).
400
410
Interplay between disorder and interaction
Disorder
Anderson localization
Glass?
???
Superfluid
Mott
insulator
Interaction
Anomalous diffusion with disorder, noise and interactions
2
Position
Anomalous diffusion with disorder, noise and interactions
Disorder
60
D/J=4
(m)
40
D/J=2.5
20
0
2
3
4
D/J
time
D/J=0
5
6
7 8 9
Interaction
Disorder
Anomalous diffusion with disorder, noise and interactions
time
Interaction
Anomalous diffusion with disorder, noise and interactions
 (t )  t
40
35
non interacting
30
Eint= 0.8 J
  0.5
Eint= 1.2 J
width (m)
25
normal diffusion
20
15
0.1
1.0
10.0
time (s)
E. Lucioni et al. , Phys. Rev. Lett. 106, 230403 (2011).
E. Lucioni et al. , Phys. Rev. E 87, 042922 (2013).
Eint=Un(x,t)
Anomalous diffusion with disorder, noise and interactions
log
Levy flights
1
=

tic
s
i
ll
a
b
superdiffusive
ive
s
u
f
f
di
5
.
=0
Brownian motion
subdiffusive
log(t)
Many classes of linear disordered
systems
Localized interacting systems?
J-P. Bouchaud and A .Georges, Phys. Rep. 195, 127 (1990)
D. L. Shepelyansky, Phys. Rev. Lett. 70, 1787 (1993)
S. Flach, et al, Phys. Rev. Lett. 102, 024101 (2009)
Coherent hopping between localized states

H   Hint

 U  n j (n j 1)
Instantaneous diffusion coefficient:
Width-dependent diffusion coefficient:
D  
2


i H int f
E
2

1
2
D  n( x, t )     
Standard Diffusion Equation with
Gaussian solution:
n( x, t ) 1   n( x, t ) 

D

t
2 x 
x 
Subdiffusive behaviour, i.e. decreasing
diffusion coefficient:
 2
 D   (t )  t   t 1/(2  )
t
D(t )   (t ) 21/ 
E. Lucioni et al. , Phys. Rev. E 87, 042922 (2013).
Nonlinear diffusion equation
What about the evolution of the distribution n(x,t)?
D  n( x, t )     
n (arb. units)
Experiment
Gaussian fit
t = 0.1s
0
20
40
t = 10s
60
x(m)
Nonlinear Diffusion Equation:
n( x, t )  
n( x, t ) 
  D0 n( x, t ) a

t
x 
x 
0
20
40
60
80
x(m)
1/ a

x2 
n( x, t )  1  2 
 w (t ) 
w(t )  t
1
2 a
B. Tuck, Journal of Physics D: Applied Physics 9, 1559 (1976)
Nonlinear diffusion equation
What about the evolution of the distribution n(x,t)?
n (arb. units)
Experiment
Gaussian fit
fit with solution of NDE
0
20
t = 0.1s
b = 0.06
0.03
40
60
x(m)
t = 10s
b = 0.57
0
20
40
0.06
60
80
x(m)
Solution of NDE:
1/ a

x2 
n( x, t )  1  2 
 w (t ) 
 b(t ) x
n( x, t )  B(b, w)1  2
w (t )

2
w(t )  t
1
2 a
1/ b ( t )



b(t )  a(1  e t / )
E. Lucioni et al. , Phys. Rev. E 87, 042922 (2013).
Noise- and interaction-assisted transport
Can we learn something abouth the complex properties of the energy
transport in biological systems with our ultracold atom system?
 Disorder
 Noise
 Interactions ?
Collaboration with F. Caruso and
M. Plenio, Ulm University
Chin et al., New J. Phys. 12 065002 (2010)
Noise-assisted diffusion
Our noise: sine modulation of the secondary lattice with a random frequency
Vdis  D cos(2 x) (1  A cos(it ))
PSD (dB/Hz)
Frequencies are
changed randomly with
time step Td
-40
-50
-60
100
200
300
frequency (Hz)
  const   (t )  Dt
normal diffusion
400
Noise-assisted diffusion
50
 0.5
40
(m)
increasing noise amplitude
 2 (t )  Dt
30
20
1
10
t (s)
Also observed in atomic ionization (Walther), kicked rotor (Raizen) and photonic lattices
(Segev&Fishman):
M. Arndt et al, Phys. Rev. Lett. 67, 2435 (1991); D. A. Steck, et al, Phys. Rev. E 62, 3461 (2000).
Noise-assisted diffusion



H '  DA cos(it ) cos(2 x)
Normal diffusion:
General expectation:
 2
D
t
D   2
Our perturbative result for qp lattices:
(works for both experiment and DNLSE)
C. D’Errico et al., New J. Phys.15, 045007 (2013).
 2  Dt

i H' f
E
2
 const
A2 J (  d ) 2
D
3 1  e / d
Noise-assisted diffusion
0.4
 4.5 d
2
D (m /ms)
0.3
0.2
0.1
 0.7 d
0.0
0.0
0.2
0.4
2
A
C. D’Errico et al., New J. Phys.15, 045007 (2013).
0.6
0.8
1.0
A2 J (  d ) 2
D
3 1  e / d
Noise-assisted diffusion
0.4
1
A=1
2
2
D (m /ms)
2
0.3
Experiment
Perturbative model
A<Ac
D /A (m /ms)
 4.5 d
0.2
0.1
0.1
 0.7 d
0.0
0.0
0.6
0.2
0.4
2
0.6
0.8
1.0
1
/d
2
3
4
A
C. D’Errico et al., New J. Phys.15, 045007 (2013).
A2 J (  d ) 2
D
3 1  e / d
Noise + interactions?
Anderson localization
noise alone
interactions alone
noise + interactions
Noise and interaction: generalized diffusion equation
DNLSE
Experiment
50
20
a
c
(iii)
40
(ii)
10
 (m)
 (m)
30
(i)
20
2
15
b
0.1
d


0.1
0.0
0.1
0.0
t (s)
1
noise alone
interactions alone
noise + interactions
10
0.01
0.1
t (s)
1
 2
 Dnoise  Dint (t )
t
10
Experimental scheme and parameters for 1D system
Strong 2D lattice (s=30) with weak 3D harmonic trapping
+ weaker 1D q.p. lattice (s=10)
Optical lattices create an array of
quasi one-dimensional systems:
nr=50 kHz; J/h=100 Hz
| (k ) |2
D=0, U=J
Inhomogeneous filling factor
(3D Thomas-Fermi):
nmean ~ 2
-2k1
k
0
2k1
D=0, U=J
Transport in 1D system
t*=0
System at
equilibrium
t=0
trap minimum
is shifted
t=t*
all fields are
switched off
A. Polkovnikov et al. Phys. Rev. A 71, 063613 (2005);
applied on Bose gases by DeMarco, Naegerl, Schneble.
t*≠0
D
k
TOF image (16.6 ms)
Transport in the weakly interacting regime: clean system
0.5
Experiment
no damping
low damped fit
high damped fit
0.4
p0 (h/1)
Without disorder:
D/J=0
0.3
0.2
0.1
0.0
0
1
2
t (ms)
A. Smerzi et al., Phys. Rev. Lett. 89, 170402 (2002)
E. Altman et al., Phys. Rev. Lett. 95, 020402 (2005)
L. Fallani et al., Phys. Rev. Lett. 93, 140406 (2004)
J. Mun et al., Phys. Rev. Lett. 99, 150604 (2007)
I. Danshita, ArXiv:1303.1616
3
4
Dynamical instability driven by
quantum and thermal fluctuations.
Transport in the weakly interacting regime: clean system
0.5
Without disorder:
D/J=0
0.4
pC
p0 (h/1)
0.3
0.2
0.1
At p=pc we observe a sudden
increase of the damping and
of the width
0.0
0.4
p0- pth (h/)
0.3
0.2
-2
0.1
0
p (h/1)
2
0.0
0
1
2
t (ms)
3
4
Transport in the weakly interacting regime:clean system
Without disorder:
D/J=0
0.5
Experiment
piecewise fit
quantum phase slips model
pc (h/)
0.4
0.3
0.2
0.1
0.0
0
2
4
6
8
10 20 22
U/J
J. Mun et al., Phys. Rev. Lett. 99, 150604 (2007).
L. Tanzi et al., ArXiv:1307.4060, accepted by PRL
Also in 1D the onset of the
Mott regime can be detected
from a vanishing of pc, as in 3D
Transport in the weakly interacting regime:clean system
0.4
Without disorder:
D/J=0
Experiment
quantum phase slip model
thermal phase slip model
pc(h/)
0.3
The observed dependences of
pc and  on U suggest a quantum
activation of phase slip
0.2
0.1
0.0
0
2
4
6
8
10
12
500
 (Hz)
U/J
E. Altman et al., PRL 95,020402 (2005)
A Polkovnikov et al., PRA 71 063613 (2005)
I. Danshita and A Polkovnikov, PRA 85, 023638 (2012)
I. Danshita, PRL 111, 025303 (2013)
L. Tanzi et al., ArXiv:1307.4060, accepted by PRL
50
0
2
4
U/J
6
Transport in the weakly interacting regime: with disorder
D=0
D = 3.6 J
D = 10 J
pC
0.3
p0 (h/1)
pC
0.2
The damping rate is
enhanced and the
critical momentum is
reduced by disorder
0.1
0.0
0
1
2
t (ms)
Fixed interaction
energy: U/J=1.26
3
Transport in the weakly interacting regime: with disorder
D=0
D = 3.6 J
D = 10 J
pC
0.3
p0 (h/1)
pC
0.2
0.1
0.52
0.30
0.0
0.50
1
2
0.48
t (ms)
0.20
pc(h/)
Fixed interaction
energy: U/J=1.26
3
0.25
DC
0.15
0.46
0.44
0.10
0.42
0.05
0.40
0.00
0
L. Tanzi et al., ArXiv:1307.4060, accepted by PRL
2
4
6
D/J
8
10
12
0.38
p (h/)
0
Transport in the weakly interacting regime: with disorder
10
Insulator
8
A = 1.3 ± 0.4
 = 0.83 ± 0.22
D/J
6
4
Fluid
2
0
0
(Dc  2) / J  A(nU / J )
2
4
6
8
nU/J
P. Lugan, et al., Phys. Rev. Lett. 98, 170403 (2007);
L. Fontanesi, et al., Phys. Rev. A 81, 053603 (2010).
Conclusions & Outlook
 We have studied the diffusion of a localized disordered system,
assisted by interaction and noise
 We have studied the momentum-dependent transport for a weakly
interacting disordered Bose gas on the BG – SF transition
 Study a strongly correlated, disordered Bose gas in 1D: correlations,
excitations, compressibility, and transport
 Investigation of a quantum quench on a strongly correlated system
and effect of the disorder on the thermalization of a closed system
 Exploration of the role of temperature on the many-body fluidinsulator transition at large T
I. L. Aleiner, B. L. Altshuler, G. V. Shlyapnikov, Nat. Phys. 6, 900 (2010)
The Team
Team
Massimo Inguscio
Giovanni Modugno
Eleonora Lucioni
Luca Tanzi
Lorenzo Gori
Avinash Kumar
Saptarishi Chaudhuri
C.D.
For Noise-assisted transport: collaboration with
F. Caruso
B. Deissler (Ulm University)
M. Moratti
M. B. Plenio (Ulm University)
Thank you for the attention