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

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

Biological systems

There is a growing interest in determining exactly how disorder affects the properties of quantum systems.

Superfluids in porous

media

Superconducting thin films

Light propagation in random media

Graphene

Disorder in quantum systems

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

Hannover

Florence

Paris

Urbana

Rice U.

L. Sanchez-Palencia and M. Lewenstein, Nat. Phys. 6, 87 (2010); G. Modugno, Rep. Prog. Phys. 73, 102401 (2010).

also Shlyapnikov, Burnett, Roth, Sanchez-Palencia, Giamarchi, Natterman, Garcia-Garcia ….

Giamarchi & Schultz, PRB 37 325 (1988)Fisher et al PRB 40, 546 (1989), …

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

Rapsch, Schollwoeck, Zwerger EPL 46 559 (1999), …

Interplay between disorder and interaction

4J

2

)1/( d

In the tight binding limit: Aubry-Andrè or Harper model

Metal-insulator transition at =2J

ii

jiji

nibbJH ˆ)2cos(ˆˆˆ,

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).

1D system in a quasiperiodic potential

A 1D quasiperiodic lattice

1

2

k

k


-20 -10 0 10 20

g E(x

) (a

rb. u

nits

)

Position (lattice sites)

')()()( dxxxExExgE

Energy correlation function

)1/( d

420 440 460 48010-20

10-10

100

| (

x)|2

Position (lattice sites)

0 100 200 300 400 500

-4

-2

0

2

4

Ene

rgy

(uni

ts o

f J)

Eigenstate #

Jd 2/log/

Short, uniform localization length:

Miniband structure


4J

2

/d

In the tight binding limit: Aubry-Andrè or Harper model

Metal-insulator transition at =2J

ii

jiji

nibbJH ˆ)2cos(ˆˆˆ,

Tuned on the Feshbach resonance

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).


1D system in a quasiperiodic potential

340 350 360 370 380 390 400 410

0

400

scat

terin

g le

ngth

(a0)

magnetic field (G)

G. Roati, et al. Phys. Rev. Lett. 99, 010403 (2007).

Potassium-39

dxam

E 42

int

2

BEC


Interaction

Dis

ord

er

Superfluid

Anderson localization

Glass?

???

Mott insulator


Anomalous diffusion with disorder, noise and interactions

2

Position

Interaction

Dis

ord

er

/J=0

/J=2.5

/J=4

time

2 3 4 5 6 7 8 90

20

40

60

(

m)

/J


Interaction

Dis

ord

er

time


0.1 1.0 10.0

15

20

25

30

35

40

non interacting

Eint

= 0.8 J

Eint

= 1.2 J

normal diffusion

wid

th (m

)

time (s)

E. Lucioni et al. , Phys. Rev. Lett. 106, 230403 (2011).E. Lucioni et al. , Phys. Rev. E 87, 042922 (2013).

5.0)( tt


Eint=Un(x,t)

superdiffusive

subdiffusive

=0.5 diffusive

log

log(t)

=1 ballis

tic

Levy flights

Many classes of linear disordered systems

Brownian motion

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)

Localized interacting systems?


)2/(12

)(

tttDt

)1(int jj nnUHH

),( txnD

Coherent hopping between localized states

2D 2

2

int 1

E

fHi

Instantaneous diffusion coefficient:

Standard Diffusion Equation with Gaussian solution:

Width-dependent diffusion coefficient:

E. Lucioni et al. , Phys. Rev. E 87, 042922 (2013).

Subdiffusive behaviour, i.e. decreasing diffusion coefficient:

/12)()( ttD

x

txnD

xt

txn ),(

2

1),(

0 20 40 60 80

t = 10s

x(m)0 20 40 60

t = 0.1s

n (a

rb. u

nits

)

x(m)

Experiment Gaussian fit

What about the evolution of the distribution n(x,t)?

Nonlinear diffusion equation

x

txntxnD

xt

txn a ),(),(

),(0

Nonlinear Diffusion Equation:

B. Tuck, Journal of Physics D: Applied Physics 9, 1559 (1976)

a

a

ttwtw

xtxn

2

1/1

2

2

)()(

1),(

),( txnD

0 20 40 60 80

t = 10sb = 0.57 0.06

x(m)0 20 40 60

t = 0.1sb = 0.06 0.03 n

(arb

. uni

ts)

x(m)

Experiment Gaussian fit fit with solution of NDE

What about the evolution of the distribution n(x,t)?

Nonlinear diffusion equation

a

a

ttwtw

xtxn

2

1/1

2

2

)()(

1),(

E. Lucioni et al. , Phys. Rev. E 87, 042922 (2013).

)1()()(

)(1),(),( /

)(/1

2

2t

tb

eatbtw

xtbwbBtxn

Solution of NDE:

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 ?

Chin et al., New J. Phys. 12 065002 (2010)

Collaboration with F. Caruso and M. Plenio, Ulm University

Noise-assisted diffusion

))cos(1()2cos( tAxV idis

Dttconst )(

Our noise: sine modulation of the secondary lattice with a random frequency

Frequencies are changed randomly with time step Td

normal diffusion

100 200 300 400

-60

-50

-40

PS

D (d

B/H

z)

frequency (Hz)

1 10

20

30

40

50

(m

)

t (s)


0.5

increasing noise amplitude

Dtt )(2

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).


Dt

2

C. D’Errico et al., New J. Phys.15, 045007 (2013).

Dt2

de

dJAD

/

22

1

)(

3

2D

Normal diffusion:

General expectation:

Our perturbative result for qp lattices:(works for both experiment and DNLSE)

)2cos()cos(' xtAH i

constE

fHi

2'


0.0 0.2 0.4 0.6 0.8 1.00.0

0.1

0.2

0.3

0.4

0.7 d

4.5 d

A2

D (m

2 /ms)

de

dJAD

/

22

1

)(

3



1 2 3 4

0.1

1

Experiment Perturbative model A<A

c

A=1

0.6

D /A

2 (m

2 /ms)

/d0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.1

0.2

0.3

0.4

0.7 d

4.5 d

A2

D (m

2 /ms)


de

dJAD

/

22

1

)(

3

Noise + interactions?

Anderson localization interactions alonenoise alone noise + interactions

Noise and interaction: generalized diffusion equation

)(2

tDDt intnoise

10

2

(

m)

c (iii)

(ii)

(i)

20

0.01 0.1 1 10

0.0

0.1d

t (s)

20

30

40

50

(

m)

15

0.1 1 10

0.0

0.1 b

a

t (s)

Experiment DNLSE

noise alone interactions alonenoise + interactions

2|)(| k

k

2k1

-2k 1

0

=0, U=J

r=50 kHz; J/h=100 Hz

Experimental scheme and parameters for 1D system

Strong 2D lattice (s=30) with weak 3D harmonic trapping + weaker 1D q.p. lattice (s=10)

Inhomogeneous filling factor (3D Thomas-Fermi):nmean ~ 2

=0, U=J

Optical lattices create an array of quasi one-dimensional systems:

t=0trap minimum

is shifted

t=t*all fields are switched off

TOF image (16.6 ms)

System at equilibrium

t*=0

t*≠0

k

Transport in 1D system

A. Polkovnikov et al. Phys. Rev. A 71, 063613 (2005); applied on Bose gases by DeMarco, Naegerl, Schneble.

0 1 2 3 40.0

0.1

0.2

0.3

0.4

0.5

Experiment no damping low damped fit high damped fit

p 0 (

h/ 1

)

t (ms)

Transport in the weakly interacting regime: clean system

Dynamical instability driven by quantum and thermal fluctuations.

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

Without disorder: /J=0


Transport in the weakly interacting regime: clean system



0.0

0.1

0.2

0.3

0.4

0.5

0 1 2 3 4

0.0

0.1

0.2

0.3

0.4

p 0 (h

/1)

p 0- p th

(h/

)

t (ms)

-2 0 2p (h/

1)

pC

At p=pc we observe a sudden increase of the damping and of the width

0 2 4 6 8 10 20 220.0

0.1

0.2

0.3

0.4

0.5

p c (h

/)

Experiment piecewise fit quantum phase slips model

U/J

Transport in the weakly interacting regime:clean system



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



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

0 2 4 6 8 10 120.0

0.1

0.2

0.3

0.4Experiment quantum phase slip model thermal phase slip model

p c(h/

)

U/J

0 2 4 6

50

500

(H

z)

U/J

The observed dependences of pc and on U suggest a quantum activation of phase slip

0 1 2 3

0.0

0.1

0.2

0.3 = 0 = 3.6 J = 10 J

p 0

(h/

1)

t (ms)

Fixed interaction energy: U/J=1.26Fixed interaction energy: U/J=1.26

pC

pC

Transport in the weakly interacting regime: with disorder

The damping rate is enhanced and the critical momentum is reduced by disorder

pC

pC


Fixed interaction energy: U/J=1.26Fixed interaction energy: U/J=1.26

0 1 2 3

0.0

0.1

0.2

0.3 = 0 = 3.6 J = 10 J

p 0

(h/

1)

t (ms)

0 2 4 6 8 10 12

0.00

0.05

0.10

0.15

0.20

0.25

0.30

/J

0.38

0.40

0.42

0.44

0.46

0.48

0.50

0.52

p (

h/

)

p c(h/

) CC

L. Tanzi et al., ArXiv:1307.4060, accepted by PRL

P. Lugan, et al., Phys. Rev. Lett. 98, 170403 (2007);L. Fontanesi, et al., Phys. Rev. A 81, 053603 (2010).

0 2 4 6 80

2

4

6

8

10

/J

nU/J

Insulator

Fluid

A = 1.3 ± 0.4 = 0.83 ± 0.22


)/(/)2( JnUAJc

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 fluid-insulator transition at large T I. L. Aleiner, B. L. Altshuler, G. V. Shlyapnikov, Nat. Phys. 6, 900 (2010)

Massimo Inguscio

TeamThe Team

Eleonora LucioniLuca TanziLorenzo GoriAvinash KumarSaptarishi ChaudhuriC.D.

Giovanni Modugno

For Noise-assisted transport: collaboration withF. Caruso B. Deissler (Ulm University) M. Moratti M. B. Plenio (Ulm University)

Thank you for the attention

Documents

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