3D Anderson Localization
of Noninteracting Cold Atoms
Bart van Tiggelen
Université Joseph Fourier – Grenoble 1 / CNRS
Warsaw may 2011
HKUST april 2010
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My precious collaborators
• Sergey Skipetrov, Anna Minguzzi (Grenoble)
• Afifa Yedjour (PhD) (Grenoble and Oran-Algeria)
HKUST april 2010
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50 years of Anderson localization
Localization [..] very few believed it at the time, and even fewer saw its
importance, among those who failed was certainly its author.
It has yet to receive adequate mathematical treatment,
and one has to resort to the indignity of numerical simulations
to settle even the simplest questions about it.
P.W. Anderson, Nobel lecture, 1977
…..and now we have (numerical) experiments !
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Physics Today, August 2009
50 years of Anderson Localization
http://www.andersonlocalization.com/
c
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Diffusion of Waves
Diffusion = random walk of waves
 t  (r, t )  D   (r, t )  S  (t )  (r  rS )
2
r (t ) 
2
 (r, t )r
 (r, t )
2
D 1
 6D t
3
v 
*
diffusion constant
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Small diffusion constant ≠ localization
Trapped Rb85
Temperature T = 0,0001 K (v=15 cm/s)
1010 atomes
ℓ
ℓ
Labeyrie, Miniatura, Kaiser (2006, Nice)
photon
  0.8 µm
5 mm
Random walk of photons
D
vE   0.9 m / sec
2
1
3
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vE  ........ 0.00003 c0
  0.3 mm k  4000
6
Dimension < 3
V(r)
r
« Trivial »
Localization
(most mathematical
proofs)
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Dimension < 3
V(r)
« Tunnel /percolation
assisted »
localization
(Anderson model)
r
« Trivial »
Localization
(most mathematical
proofs)
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Dimension < 3
« genuine »
Localization
E > Vmax
(Classical waves,
cold atoms ??)
V(r)
« Tunnel /percolation
assisted »
localization
(Anderson model)
r
« Trivial »
Localization
(most mathematical
proofs)
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Dimension = 3
« metal »
V(r)
Mobility
edge
r
«insulator»
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Mott minimum conductivity
•Thouless criterion and scaling theory
•Quantum Hall effect
MIT and role of interactions
dense point spectrum
Chaos theory (DMPK equation)
Multifractal eigenfunctions
Full statistics of conductance and transmission
Random laser
•Transverse localization
• Anderson tight bindingHKUST
modelapril
&2010
Kicked rotor
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Mesoscopic Wave Transport
1
G (E , r, r')  r
E
p
r'  G (E ,k ,k ')
2
One particle Green function
 V (r )  i 0
2m
G ( E , k , k ' )  G ( E , k ' , k )
reciprocity
 kk '
G ( E , k , k ' )  G ( E , k ) kk ' :
2
E  V  
 k
2
 (E , k )
2m
Dyson Green function
(E, k )  (E )  G (E, k ) 
Self energy
2m / 
2
k=1/2ℓ :
strong scattering
2

i 
2
k
(
E
)


k


2

(
E
)


Mean free path
A( E , k )  
1

Im G ( E , k )
 ( E )   A( E , k )
k
Spectral function
HKUST april 2010
Average
LDOS

dE A ( E , k )  1
12
Mesoscopic Wave Transport
 kk '
G ( E , k , k ' )  G ( E , k ) kk ' :
2
E  V  
 k
2
 (E , k )
2m
Dyson Green function
(E, k )  (E )  G (E, k ) 
2m / 
2
2

i 
2
k
(
E
)


k


2( E ) 

OK for white noise fluctuations:
V
 0 ;  V ( r ) V ( r ' )  4  U  ( r  r ' )
 
  
 2m
2
Mean free path:
2
 1

 U

3
1  2m 
E   2  U
4  
2
is 2010
strongly scattering (localized)
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Mesoscopic Wave Transport
G ( E , r1 , r1 ' ) G * ( E ' , r2 , r2 ' )  G ( E 

2
,k 
q
, k '
2
q
)G * ( E 

2
2
,k 
q'
, k '
2
q'
)
2
Two particle Green function
 E kk ' (  , q ) qq '
Momentum conservation
 E kk ' ( t , r )
Wigner function
(looks like phase space distribution)
 (r , t )
2


kk '
(r , t ) S (k ' )
J (r , t ) 
kk '

kk '
k
m
 kk ' ( r , t ) S ( k ' )
Proba current density
Proba density
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Diffusion approximation
  0, q  0
reciprocity
 E kk' (  , q ) 
2
 ( E , k, q ) ( E , k ' , q )
 (E )
 i  D ( E ) q
2
Proba of
quantum diffusion
normalization
 ( E , k , q )  A ( E , k )  ik  q F ( E , k )
D(E ) 

1
3m

 (E )


d r n (r , t )  N
d k'
( 2 )
Kubo Greenwood formula
k
3
n (r , t ) 
2
k F (E, k )
  (k ' )
3
2



dE A ( E , k ' ) PE ( r , t )
3

d k
 N
3
HKUST
( 2  ) april 2010

dE
2
1

dr  1
15
Diffusion approximation
 E kk' (  , q ) 
2
 ( E , k, q ) ( E , k ' , q )
 (E )
 i  D ( E ) q
k+q/2
x
k-q/2
D(E ) 
1
3
v 
  0, q  0
x

E+hΩ/2
x
x
E-hΩ/2
x
x
k’+q/2
HKUST april 2010
Boltzmann
approximation
-k’-q/2
k  D ( E )   / 3m
3m
2
near mobility edge kℓ=1
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Diffusion approximation
 E k 'k (  , q )  
E
k  k ' q k ' k  q
2
k+q/2
 E kk' (  , q ) 
« ladder »
+
« most-crossed »
  0, q  0
k-q/2
k+q/2
k-q/2
(  , k  k ' ) ???
2
E+hΩ/2
x
x
x
x
x
x
x
E-hΩ/2
E+hΩ/2
x
x
k’+q/2
E-hΩ/2
k’-q/2
x
k’+q/2
x
k’-q/2
x
1
HKUST april 2010
 i   D ( E )( k  k ' )
2
17
Diffusion approximation
1
D(E )

1
DB

  0, q  0
2m
 ( E ) 
2

q
1
D ( E )q
2
Diffuse return Green function
Diverges in 3D: q < 1/ℓ or 1/ℓ*?
Infinite medium with white noise

1

D  DB 1 
2



k


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



Critical exponent =1
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Inhibition of transport of Q1D BEC in random potential
n(x,t)
V(x)
expansion
V
1

Time after trap extinction
Palaiseau
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2010
group, Firenze group
PRL oct 2005
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Localization of noninteracting cold atoms in 3D white noise
Skipetrov, Minguzzi, BAvT, Shapiro PRL, 2008
 (r )
2
expansion stage (t=0)
Trap stage
 (r, t )  e i t
  V (r )
g
Localization
with x < ℓ
Random potential
2 2

 k 

 (k , t  0)     
2m 

Localization
with x > ℓ
2
n(r, t )   (r, t )  ??
t 
Diffusive regime
D( )
µ
band edge
mobility edge
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kℓ ~1
chemical potential
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Density profile of atoms at large times
Skipetrov, Minguzzi, BAvT, Shapiro PRL, 2008
n(r, t )  nloc (r)  nAD (r, t )
localized
x ( ) 
1

( c   )
 n loc ( r ) 
1
r
3 1 / 
anomalous diffusion
D (  )  (    c )   n AD ( r , t ) 
1
s
r
3 2 / s
Selfconsistent theory with white noise
(ν=s=1)
3 % localized
nloc(r)
45 % localized
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t
1/ s
Cold atoms in a 3D speckle potential
Kuhn, Miniatura, Delande etal NJP 2007
Yedjour, BavT, EPJD 2010
nonGaussian!
V (r ) 
 V ( r ) V ( r ' )  U sinc (  r /  )
2
U

 7 mm / s
m


 600 µm / s
2
Mott minimum
3m
2
2m
2
 E 
U  µ  h  220 Hz
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    0.22
3 µm
Self-consistent Born Approximation
1
G (E, p) 
(E, p) 
E  p / 2m  U  (E , p)
2

p'
S (p  p ' )
E  p' / 2m  U  (E , p')
2
E  E
p in units of  / m 
U  E
2
p in units of  / m 
HKUST april 2010
Mean
free path?
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D Drude ( E ) 
1 
3 m

2
2
p 2 Im G ( E , p )
p

 Im G ( E , p )

1 
 " k "
3 m
p
D Boltz ( E ) 
D Drude ( E )
1  cos 
 D Drude ( E )
Selfconsistent theory of localization
D ( E )  D Boltz ( E )
K (E )  
6


2
2
p
Im
G
(
E
,
p
)


p


2

 1  K (E ) 
2
Im G ( E , p ' )
p p'
HKUST april 2010
p p'
(p  p ' )
Im  ( E , 1 2 p  p ' ) G ( E , p )
2
2
24
2
U  E
2
<V>
D/DB
D/DB {1-K}
0
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Is 3D cold atom localization « trivial »?

  k
U
Kuhn, Miniatura, etal
FBA (2007):
kℓ=0.95 (1-<cos ϑ>)
kℓ=1.12
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Cold atoms in 3D speckle
« metal »
V(r)
Mobility
edge
r
«insulator»
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U=Eξ2
Energy
distribution
N  (E ) 
3

d k
( 2 )
3
A( E , k )   (k )
Fraction of localized
atoms
f loc (  ) 

Ec
0
2
*
dE N  ( E )
HKUST april 2010
* 45 % in white noise (Skipetrov etal 2008)
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Anderson Localization is still a
major theme in condensed matter physics,
full of surprises
New experiments (in high dimensions and
with « new » matter waves) exist and are underway.
Need of accurate description of self-energy
Thank you for your attention
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