aspects of anderson localization with cold atoms...cooling of atomic cloud in optical dipole trap,...

aspects of Anderson Localization with Cold Atoms

Tobias Micklitz collaborations: A. Altland, C. Müller (theory)

A. Aspect, V. Josse (experiment)

Motivation

tunable system parameters:full control of the trapping potentials (lattice geometry, effective dimensionality), random potentials (disorder), interactions (by Feshbach resonances or density), type of quantum statistics (ultracold bosons or fermions) closed system (no coupling to bath, decoherence)

focus here: single-particle physics...

observation of many-body localization?

weak-localization echorecent examples:

(dynamical) localization transition

Outlineintroduction Anderson localization (AL)

quantum quench experiment

forward scattering peak as a signature of strong AL

echo-spectroscopy at the onset of AL

Anderson Localization (AL)

single-particle wave-function in a (macroscopic) disordered

quantum system can be exponentially localized

introduction

“Absence of diffusion in certain lattices” (Anderson,1958)

H = tX

hiji

c†jci +X

i

✏ic†i ci ✏i 2 [✏0 ��, ✏0 + �] (random)1d example:

classically: random walk(Markovian process)hr2i = Dt

(system has memory)hr2i !t!1

const.

quantum mechanically: interference

quantum vs. classical diffusion

“localization of high-energy/short wave-length waves by random potentials”:different from classical trapping in a random potential...

introduction

“all states are (Anderson) localized by disorder in d<3 ”in an infinite system at T=0:

“gang of four” (1979) Abrahams, Anderson, Licciardello, Ramakrishnan

�(g) =d ln g

d lnL

depends only on conductance itself!

g(bL) = fb(g(L))

role of dimensionality:

...a quantum interference phenomenon!

“What is the probability to propagate from A to B ?”

A B

microscopic origin: the onset of localizationintroduction

Semiclassical limit

A� A�A⇤�0

classical paths

different classical paths with same actionPA!B '

X

�

��2

+X

��0

⇣ ⌘S�/~� 1

(classical diffusion) (quantum interference)

Q: What are the relevant classical paths giving rise to quantum-interference corrections?

A: self-intersecting paths, which exactly depends on the symmetries of the system...

self-intersecting paths

A B

microscopic origin: the onset of localization

PA!B 'X

�

��2

+X

��0A�A⇤

�0different classical paths with

same action

⇣ ⌘symmetries/class:

unitary (U)

orthogonal (O)

symplectic (Sp)

introduction

role of symmetries:perturbative RG d=2+ : ✏

(O)

(U)

(Sp)

�(t) = ✏� t� 34⇣(3)t4 +O(t5)

�(t) = ✏� 12t2 � 3

8t4 +O(t6)

�(t) = ✏ + t� 34⇣(3)t4 +O(t5)

strong localization(non-perturbative)

...

onset of localization

introduction

�(g)

ln gd = 2

d = 1

d = 3

critical region


strong localization

tunable interactionsBloch (2015)Billy et

coherent backscatteringJosse/Aspect (2012)

some recent cold atom experiments

introduction

1d exponential localization of atomic matter wave

cold atoms released into 1d wave-guide in presence of disorder

(direct observation of wave-function)

initial state

| (z)|2 ⇠ e�2|z|/⇠loc

hz2i(t)

density profiles

introduction

�(g)

ln gd = 2

d = 1

d = 3

critical region


strong localization

tunable interactionsBloch (2015)


Billy et


introduction

3d AL-transition in cold atom realization of kicked rotor

see also Lemaríe et al. (2010)

(dynamical) localization in momentum-space

ˆH =

p2

2

+ K cos x [1 + ✏ cos(!2t) cos(!3t)]X

n

�(t� n)

quasi-periodic kicked rotor

✏ = 0 equivalent to 1d disordered system ✏ 6= 0 equivalent to 3d disordered system

hp2i(t)diffusive

critical t2/3

localized

critical exponents

⇠ ⇠loc

⇠ 1/D

⇠ ⇠ |K �Kc|�⌫

⌫ = 1.60 (theory)(experiment)⌫ = 1.4± 0.3

Moore et al. (1995)

diffusive

localized

Chabé et al. (2008)

introduction

�(g)

ln gd = 2

d = 1

d = 3

critical region


strong localization



Billy et


introduction

many-body localization of interacting atoms in quasi-random potential?

I(t) =N

e

(t)�No

(t)N

e

(t) + No

(t)6= 0?

t!1

imbalance:

as

t=0: “charge density wave”

3 independently tunable parameters

schematic phase diagram

does localized phase withstand interactions in a closed system? does an initial local perturbation (here density) decay to zero?

introduction

�(g)

ln gd = 2

d = 1

d = 3

critical region


strong localization



Billy et


in this talk: cold atom “quench experiment”

quantum quench experiment: controlled observation of strong Anderson localization control parameter: time

flow as time increases

�(g)

ln gd = 2

d = 1


strong localization

introduction

coherent backscatteringJosse/Aspect 2012

forward scattering peak Karpiuk, et al. (2012)TM, Müller, Altland, (2014)

Lee, Grémaud, Miniatura, (2014)Ghosh et al. (2014)

echo-spectroscopyTM, Müller, Altland, (2015)

http://arxiv.org/find/cond-mat/1/au:+Ghosh_S/0/1/0/all/0/1


http://arxiv.org/find/cond-mat/1/au:+Lee_K/0/1/0/all/0/1


http://arxiv.org/find/cond-mat/1/au:+Gremaud_B/0/1/0/all/0/1


http://arxiv.org/find/cond-mat/1/au:+Miniatura_C/0/1/0/all/0/1


cold atom quench experiment

Quantum Quench Experiment

preparation of initial state:cooling of atomic cloud in optical dipole trap, BEC of atoms in F =2, mF =-2 ground sublevelsuppress interatomic interactions by releasing atomic cloud and letting it expand freeze motion of atoms by switching on harmonic potential for well chosen amount of time almost give atoms finite momentum without changing spread by applying additional magnetic gradient during 12 ms

1.

2.

3.

4.

Quantum Quench Experimentcold atom quench experiment

initial state (k-space)

experiment:release atoms from optical trap and suspend against gravity by magnetic levitationswitch on anisotropic laser speckle disordered potential (2d: elongated along one axis)

let atoms scatter for a time t switch off the disorder and monitor momentum distribution at time t (time of flight imaging)

“quantum-quench”

low densities... “it’s all single-particle physics”

study momentum-correlations in timeCkikf (t) = |hkf |e�iHt|kii|2observable:

quantum quench experiment: controlled observation of Anderson localization

observable: momentum -distribution

control parameter: time flow as t increases

strong AL:t & tH ln g

�(g)

t⌧ tHprecursor:


no momentum-correlationst

... ... ...

isotropic redistribution of atoms over energy-shell

(Markovian process)

Classical vs. quantum diffusion

quantum interference(system has memory)

momentum-correlations

classical:


Momentum-CorrelationsPrecursor Effects 1

peak in back-scattering direction


“coherent back-scattering”

leading contribution (orthogonal class)

P (r! r, t) P (k! �k, t)

space-correlations momentum-correlations

Coherent Backscattering peakExperiment 2012 (orthogonal class)


Momentum-Correlationscold atom quench experiment

leading contribution (unitary class)

peak in forward-scattering direction“coherent forward-scattering”

P (r! r, t)

P (k! k, t)

space-correlations momentum-correlations

Precursor Effects 2

forward scattering peak

simulation + phenomenology(2d, orthogonal class)

time perturbation theory + self-consistent theory for localization

t & tHfwd-peak builds up on long time-scales ( ), i.e. pronounced in strongly localized regime

a signature of strong localization...

[T. Karpiuk, N. Cherroret, K.L. Lee, B. Grémaud, C.A. Müller, C. Miniatura, PRL 109, 190601 (2012)]

S[Q] = ⇡⌫

Zdx str

✓D

4(@

x

Q)2 � i⌘Q⇤◆


H[~S] =Z

dx⇣J(@

x

~S)2 � ~S · ~h⌘

analogy:symmetry-breaking

⌘ ⇠ tH/t ~S 2

“sphere x hyperboloid dressed with Grasmann

variables”

Q(x) 2

t⌧ tHprecursor:

t & tHstrong AL:(perturbative regime,

Goldstone-modes = diffusion)(non-perturbative regime)

�(g)

ln g

fwd-peak: field theory

Fwd peak = ???

C(q, ⌘) = �0,q?htr (P�+ Q(q)P+�Q(�q))iS h...iS =ZDQeS[Q](. . . )q = ki � kf

+ ...Fwd peak =

/p

t/tH

solution strategyS[Q] =

Zdx

⇣↵ str(Q2) + V (Q)

⌘V [Q] = �i⇡⌫⌘ str (Q⇤)

↵ str(Q2) = ⇡⌫str✓

D

4(@

x

Q)2◆quasi 1d geometry:

C(q, ⌘) /Z 1

1

Z 1

�1

d�1d�

�1 � �

⇥ q

1(�1, �) + �q1 (�1, �)

⇤ 0(�1, �)

(�Q + V (�1, �)) 0 = 0(2�Q + 2V (�

1

, �)� iq⇠loc

) q1

= (�1

� �) 0

�1, �radial-symmetric V[Q]: few relevant variablesFunctional integral “Schrödinger equation”7!

elliptic coordinates 7! “3d Coulomb-problem”� = (r � r1)/2�1 = (r + r1)/2

r0 = (1, 0, 0)t

r1(⇢, z) =p

(z � 2)2 + ⇢2

r(⇢, z) =p

z2 + ⇢2

H0 = �r21r

2

�0 �

2

r

�1r1

⌘ �i⌘tH/2

3d-Laplace + 1/r-potential

�0 ⌘ @2z + ⇢�1@⇢⇢@⇢


fwd-peak from Coulomb-problem

Green’s function for 3d non-relativistic Coulomb-problem:�0 �

2

r

�G0(r, r0) = �(r� r0)

C(0, ⌘) = 32⇡⇠hr0|G01rG0

1rG0|r0i

= 8⇡⇠@2G0(r0, r0)

C(0, !) /Z

drr

�0(r)�1(r)⇣@2r +

r

⌘�0(r, t) = 0

using 3d-variables:

⇣@2r +

r

⌘�1(r, t) =

1r�0(r, t)


analytical resultsG0(r, r0) =

(@u � @v)p

uK1(2p

u)p

vI1(2p

v)2⇡|r� r0| ,

u = r + r0 + |r� r0|v = r + r0 � |r� r0|

0 5 10 15 20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t/tH

C fs(

t)/C !

0 5 100

0.5

1

!k!

C !(i

!k)/C !

−10 0 100

0.5

1

q!

C !(q

)/C !

time-evolution of forward peak

L. Hostler (1964):

for details see: T. M., C. A. Müller, A. Altland, Phys. Rev. Lett. 112, 110602 (2014)

saturates to value which is twice the isotropic background

Cfs(t)C1

=

8<

:1p2⇡

⇣ ⇣t

2tH

⌘ 12

+ 18

⇣t

2tH

⌘3/2+ . . .

⌘, t⌧ tH ,

1� 2 tH

t + 3�

tH

t

�2 + . . . , t� tH ,



(@u � @v)p

uK1(2p

u)p

vI1(2p

v)2⇡|r� r0| ,

u = r + r0 + |r� r0|v = r + r0 � |r� r0|

0 5 10 15 20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t/tH

C fs(

t)/C !

0 5 100

0.5

1

!k!

C !(i

!k)/C !

−10 0 100

0.5

1

q!

C !(q

)/C !

L. Hostler (1964):

momentum-space structure


sharp on 1/⇠ ⌧ 1/l



(@u � @v)p

uK1(2p

u)p

vI1(2p

v)2⇡|r� r0| ,

u = r + r0 + |r� r0|v = r + r0 � |r� r0|

0 5 10 15 20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t/tH

C fs(

t)/C !

0 5 100

0.5

1

!k!

C !(i

!k)/C !

−10 0 100

0.5

1

q!

C !(q

)/C !

L. Hostler (1964):

dependence on initial state



fwd-peak and level statisticsCfs(t) is the form factor, i.e. Fourier-transform of level-level correlation function K2(!) =

1⌫20

h⌫(✏)⌫(✏ + !)i � 1


eigenstate-representation

✏± = ✏± !/2

saturation value as t� tH

in momentum-space is GUE independent of L/⇠loc

h⌫(✏+)⌫(✏�)i =X

↵�

h�(✏+ � E↵)�(✏� � E�)i

level-level correlations

Cfs

Cbgn=

h↵(ki)↵⇤(ki)↵(ki)↵⇤(ki)ih↵(ki)↵⇤(k)↵(k)↵(ki)i

= 2

Cfs(t) /Z

d✏

Zd! ei!t

X

↵�

h|↵(ki)|2|�(ki)|2�(✏+ � E↵)�(✏� � E�)i


↵(k) = h↵|ki

wave-function statistics

Cfs(t)C1

=

8<

:1p2⇡

⇣ ⇣t

2tH

⌘ 12

+ 18

⇣t

2tH

⌘3/2+ . . .

⌘, t⌧ tH ,

1� 2 tH

t + 3�

tH

t

�2 + . . . , t� tH ,

fwd-peak and level statistics

K2(!) / �(!)self-correlations Mott scale (2 resonant levels)

...� 4 ln!

Cfs(t) is the form factor, i.e. Fourier-transform of level-level correlation function K2(!) =

1⌫20

h⌫(✏)⌫(✏ + !)i � 1

K2(!) / !�3/2

KL(!) = �⇠loc

LK(4!/�⇠),

K(z) = Re8piz

⇣K1(

piz)I0(

piz)�K0(

piz)I1(

piz)

⌘

spectral correlations of a finite-size Anderson insulator



experiments: still challenging...

fwd-peak: long-time asymptotics

H =

0

@ ✏ + �✏ �⇠e� |x�x

0|⇠loc

�⇠e� |x�x

0|⇠loc ✏� �✏

1

A“two resonant levels”

long-time asymptotic general d

K(!) / �✓

⇠loc

L

◆d

lnd (!/�⇠)

Cfs(t)C1

= 1� �1tHt

lnd�1(�2t/tH)

[S. Ghosh, N. Cherroret, B. Grémaud, C. Miniatura, D. Delande, Phys. Rev. A 90, 063602 (2014)]

d=2

[K. L. Lee, B. Grémaud, C. Miniatura, Phys. Rev. A 90, 043605 (2014)]

d=1









http://arxiv.org/find/cond-mat/1/au:+Cherroret_N/0/1/0/all/0/1

http://arxiv.org/find/cond-mat/1/au:+Cherroret_N/0/1/0/all/0/1





http://arxiv.org/find/cond-mat/1/au:+Delande_D/0/1/0/all/0/1




echo spectroscopy

tunability of cold atomspossible to manipulate system on short time scales! (compared to elastic scattering time)

full control of system paramaters opens new ways to study localization phenomena

release atoms from trap, suspend against gravity by magnetic levitation

in the quench experiment:

field pulse weak enough to not change the path of atom

atoms pick up a coordinate-dependent phase

��k = �k · r�(t1)

t1

a proposal to study onset of localization

possible to change magnetic field on short time scales

[TM, C. A. Müller, A. Altland, PRB. 91, 064203 (2015)]

Coherent backscattering echo

t > 2t1�

�

t < 2t1 t = 2t1

a single pulse at t = t1(orthogonal class)

t/⌧e

q/|�p|�k0

echo-structure in momentum-

space

echo spectroscopy

Two-mode echoes

t/⌧e

q/|�p|

echo-structure in momentum-

space

k0

two pulses att = t1, t2

(orthogonal class)(unitary class)

forwardscattering echoes

⌧ = 2(t2 � t1)

⌧ = (2t2 � t1)

⌧ = 2(t2 � t1)

(orthogonal class)

backscattering echo

⌧ = t2 + t1

echo spectroscopy

Echo-spectroscopy at the onset of AL

echo-signals in forward- and backward-scattering directions appear at moments which are in well-defined relations to applied pulses

a systematic way to test elementary processes driving AL

echo spectroscopy

[TM, C. A. Müller, A. Altland, PRB. 91, 064203 (2015)]

recent experiment: A. Aspect, V. Josse[K. Müller, J. Richard, V.V. Volchkov, V. Denechaud, P. Bouyer, A. Aspect, V. Josse, PRL (2015)]

coherent backscattering echo

echo spectroscopy

Summarycold atom quantum quench experiment...

waiting for new experiments...

time-resolved portrait of a strong localization phenomenon, wi th the perspect ive of observation using current device technology

0 5 10 15 20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t/tH

C fs(

t)/C !

0 5 100

0.5

1

!k!

C !(i

!k)/C !

−10 0 100

0.5

1

q!

C !(q

)/C !direct observation of level-level correlation

function of finite size Anderson insulator

full analytical description by mapping to 3d Coulomb problem

forward peak

echo-spectroscopysystematic way to study processes driving AL

aspects of anderson localization with cold atoms...cooling of atomic cloud in optical dipole trap,...

Documents