Disorder physics Disorder physics
with Bosewith Bose--Einstein condensatesEinstein condensates
Giovanni Modugno
LENS and Dipartimento di Fisica, Università di Firenze
New quantum states of matter in and out of equilibrium
GGI , May 2012
Biological systems
Disorder is everywhere, but it is hardly controllable.
Superconductors Photonic mediaGraphene
Disorder and quantum gases
Ultracold quantum gases can help answering to open questions.
Current experimental studies
Anderson localization in 1D-3D (Palaiseau, Florence, Urbana)
Aspect, Inguscio, Phys. Today 62, 30 (2009); Sanchez-Palencia, Lewenstein, Nat. Phys. 6, 87 (2010)Modugno, Rep. Progr. Phys. 73, 102401 (2010)
Transport in disordered potentials (Rice, Florence)
Current experimental studies
BKT physics and disorder in 2D (Palaiseau, NIST-Maryland)
Strongly correlated lattice phases (Firenze, Urbana, Stony Brook)
Current studies in Florence
Quantum phases
Bosons in 1D lattices with tunable disorder and interactions (and noise).
Transport
The phase diagram of disordered lattice bosons
Dis
ord
er
Anderson localization
Bose glass
Interaction
Superfluid
Mott insulator
The phase diagram of disordered lattice bosons
Rapsch, Schollwoeck, Zwerger
EPL 46 559 (1999)
Theory: Giamarchi and Schultz , PRB 37 325 (1988)
Fisher et al PRB 40, 546 (1989).
Experiment: Fallani et al., PRL 98, 130404 (2007);
Pasienski et al., Nat. Phys. 6, 677 (2010).
4J
~2∆
β/d
A quasi-periodic lattice
1
2
k
k=β
S. Aubry and G. André, Ann. Israel Phys. Soc. 3, 133 (1980); Fallani et al., PRL 98, 130404 (2007); M. Modugno, New J. Phys. 11, 033023 (2009)
i
i
ji
ji
nibbJH ˆ)2cos(ˆˆˆ
,
∑∑ ∆+−= + πβ
Harper or Aubry-Andrè model:
Metal-insulator transition at ∆=2J
A quasi-periodic lattice
Rapidly oscillating correlation of the eigen-energies:
short localization length
gE(x
) (a
rb.
un
its)
)1/( −βd
( )Jd 2/log/ ∆≈ξ
0 100 200 300 400 500
-4
-2
0
2
4
Energ
y (
un
its o
f J)
Eigenstate #
… and energy gaps
∆≈ 15.0
-20 -10 0 10 20
Position (lattice sites)
)1/( −βd
Tunable interactions
Potassium-39
∑∑∑ −+∆+−= +
i
iii
i
ji
ji
nnaUnibbJH )1ˆ(ˆ)(ˆ)2cos(ˆˆˆ
,
πβ
Quasi-periodic Bose-Hubbard model:
G. Roati, et al. Phys. Rev. Lett. 99, 010403 (2007).
∫= xdxam
U34
2
|)(|2
ϕπh
Weak interaction regime
Optical traps
BEC
Feshbach and gravity compensation coils
Quasiperiodic lattice
Weak radial trapping (νr=50Hz): a 3D system with 1D disorder
Strong interaction regime
Lattices
BEC
Feshbach and gravity compensation coils
Quasiperiodic lattice
Strong 2D lattices (νr=50 kHz): many quasi-1D systems with 1D disorder
Momentum distribution
Dis
ord
er
Interaction
G. Roati et al., Nature 453, 895 (2008); B. Deissler et al, Nat. Phys. 6, 354 (2010).
Weak interactions: momentum distribution
Dis
ord
er
Interaction
G. Roati et al., Nature 453, 895 (2008); B. Deissler et al, Nat. Phys. 6, 354 (2010).
Phase diagram from momentum distribution
Bose Glass
Mott Insulator
Bose Glass
∆∆∆∆=2J
Superfluid MI+SF
Mott Insulator
+Bose Glass
D‘Errico et al, in preparation.
Superfluid
Mott insulator
Weak interaction cartoon
Anderson ground-state
Bose Glass (Anderson Glass)
Bose Glass (Fragmented BEC)
)(2
int EgfHi 12
2
2
1 >≈ ∫ dxE
Uϕϕ
δ
Aleiner, Altshuler, Shlyapnikov, Nat. Phys. 6, 900 (2010).
Fermi golden rule
Phase diagram from momentum distribution
Bose Glass
Mott Insulator
Bose Glass
∆∆∆∆=2J
Superfluid
MI+SF
Mott Insulator
+Bose Glass
D‘Errico et al, in preparation. Theory by T. Giamarchi, M. Modugno.
Correlation function
2|)(|)()()( kFxxxxdxg Ψ=′+ΨΨ′= ∫+
B. Deissler et al. New J. Phys. 13, 023020 (2011)
Global and local lengths
2.5
0.5
1.0
1.5
Local le
ngth
(lattic
e s
ites)
2
4
Co
rre
latio
n le
ng
th
(la
ttic
e s
ite
s)
ξ
Global Local
0.1 1
1.0
1.5
2.0
2.5
exponent
Eint
/J0.1 1
0.5
1.0
exp
on
en
t
U/J
B. Deissler et al. New J. Phys. 13, 023020 (2011)
Phase diagram from momentum distribution
Bose Glass
Mott Insulator
Bose GlassAnderson Glass
∆∆∆∆=2J
Superfluid
MI+SF
Mott Insulator
+Bose Glass
D‘Errico et al, in preparation. Theory by T. Giamarchi, M. Modugno.
Phase diagram from momentum distribution
Bose Glass
Mott Insulator
Bose GlassAnderson Glass
∆∆∆∆=2JSuperfluid
MI+SF
Mott Insulator
+Bose Glass
D‘Errico et al, in preparation. Theory by T. Giamarchi, M. Modugno.
Strong interaction cartoon
4J
4J
Bose glass
Bose glass
Strongly interacting Bose glass: insulating but gapless
Diagnostics:
� momentum distribution/correlation function� impulsive transport� excitation spectrum
-20
-10
0
10P
ositio
n a
fter
kic
k (
µm
)
Impulsive transport
0.5ms kick free expansionprepare in equilibrium
-20
-10
0
10P
ositio
n a
fter
kic
k (
µm
)
0.1 1 10-60
-50
-40
-30
-20
∆/J=0
∆/J=15Positio
n a
fter
kic
k (
U/JA. Polkovnikov et al. Phys. Rev. A 71, 063613 (2005); applied on Bose gases by DeMarco, Naegerl,
Schneble
0.1 1 10-60
-50
-40
-30
-20
∆/J=0
Positio
n a
fter
kic
k (
U/J
100
120
MI
SF
Mom
entm
wid
th (
arb
. un
its)
Excitation spectrum
free expansionprepare in equilibriummain lattice modulation
(10%, 500ms)
0 1 2 3 4 5 6
40
60
80
Mom
entm
wid
th (
arb
. un
its)
Modulation frequency (kHz)
U ~ 4.2 kHz
Outlook
Comparison with theory, several issues: finite temperature, trap, averaging
DMRG data, Roux et al., PRA 78, 023628 (2008)
Anomalous transport in disorder
An open problem since decades, little data with nonlinearities:
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)
Klages, Radons, Sokolv, Anomalous transport (Wiley, 2010)
Interaction-assisted transport
30
40
50
wid
th (
µm
)in
cre
asin
g in
tera
ctio
n s
trength
Dtt ≠2)(σ
0.1 1.0 10.0
20
wid
th
time (s)
incre
asin
g in
tera
ctio
n s
trength
Lucioni et al. Phys. Rev. Lett. 106, 230403 (2011)
Diffusion as hopping between localized states
Γ≈=∂
∂ 22
ξσ
Dt
Γ
σ
Instantaneous diffusion∂t
2
2
int 1
σδ∝≈Γ
E
fHi
Several experts in theory: Shepeliansky, Fishman, Flach, Pikovsky, M.Modugno …
Intuitive description of the coupling: Aleiner, Altshuler, Shlyapnikov, Nat. Phys. 6, 900 (2010).
t∝2σwith density-dependent rate
Subdiffusion exponent
experiment
numerical simulations (DNLSE)
Various regimes of sub-diffusion, depending on the interaction energy:very weak interaction, self-trapping.
0.8
1.0
den
sity
(ar
b.
un
its)
Spatial profiles
0.8
1.0
den
sity
(ar
b.
un
its)
space-dependent diffusion constant
0 50 100 150
0.2
0.4
0.6
den
sity
(ar
b.
un
its)
position (µm)
0 50 100 150
0.2
0.4
0.6
den
sity
(ar
b.
un
its)
position (µm)
∂
∂
∂
∂=
∂
∂
x
nnD
xt
n a0
B. Tuck, Jour. Phys. D 9, 1559 (1976)
Nonlinear diffusion equation
a
xxn
/1
2
2
1)(
−≈
σ
Noise-assisted transport
))cos(1()2cos( tAxV idis ωπβ +∆=
0 100 200P
ow
er
Frequency (Hz)
Non stationary situation: no fluctuation-dissipation relation holds
Frequencies are picked
randomly from a given interval, with time step Td
Noise-assisted transport
30
40
50
wid
th (
µm
)in
cre
asin
g n
ois
e a
mplitu
de
Dtt =)(2σ
1 10
20
time (s)
incre
asin
g n
ois
e a
mplitu
de
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).
Diffusion as hopping between localized states
Γ≈=∂
∂ 22
ξσ
Dt
2
1fHi
Γ
σ
Noise:
ftVi2
)('
Interaction:
2
2
int 1
σδ∝≈Γ
E
fHi
Theory: Ovchinnikov, Ott, Shepeliansky, Bouchaud&Georges, … and many
others.
constE
ftVi=≈Γ
δ
2)('
Dt=2σt∝2σ
Normal diffusionSub-diffusion
An extended perturbative model
C. D’Errico et al., arXiv:1204.1313
E. Ott, T. M. Antonsen and J. D. Hanson, Phys. Rev. Lett. 53, 2187 (1984);
J.P. Bouchaud, D. Toutati and D. Sornette, Phys. Rev. Lett. 68, 1787 (1992).
40
50
60m
)
40
50
60m
)
ασσ /11
2−+=
∂
∂intnoise DD
t
noisenoise + interaction
Noise and interaction
1 1020
30σ (
µm
)
time (s)
1 1020
30σ (
µm
)
time (s)
interaction
Anomalies: self-trapping and super-diffusion
initial int. energybandwidth
initial kin.+ pot. energy
noise +
interaction
superdiffusion!
noise
interaction
superdiffusion!
Outlook
Towards improved spatial resolution, noise-induced heating, stationary noise (another atomic species):
Noise-assisted trasport in natural disordered media
Many-body localization transition in disorderd Bose and Fermi systems
Out-of-equilibrium quantum phase transitions
Future directions in Florence
Impurity and other disorder types: effect of disorder correlations and distributions
Strongly interacting1D bosons with weak disorder
Interactions and Anderson localization in higher dimensions
Strongly correlated phases and frustration in higher dimensionsStrongly correlated phases and frustration in higher dimensions
Fermi gases; dipolar systems; …
39K
87Rb
Impurities and thermal baths
Bragg spectr.
Engineered disorder
Experiment:Chiara D’Errico Eleonora LucioniLuca Tanzi Lorenzo GoriBenjamin DeisslerG.M. Massimo Inguscio
Theory:Filippo Caruso (Ulm-LENS) Martin Plenio (Ulm)Marco Moratti Michele Modugno (Bilbao)
The team
Marco Moratti Michele Modugno (Bilbao)Marco Larcher (Trento) Franco Dalfovo (Trento)
acknowledgments:B. Altshuler, S. Flach, T. Giamarchi, G. Shlyapnikov, M. Mueller, …C. Fort, L. Fallani, M. Fattori, F. Minardi, G. Roati, A. Smerzi, …