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).

nmean ~ 3

Phase diagram from momentum distribution

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.

Strong interaction cartoon

BEC

4J

4J

Bose glass

4J

Bose glass

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)

Expansion measurements

Dis

ord

er

Interaction

time

latt

ice d

irection

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, …