Physique des Lasers, Atomes et Molecules

Université des Sciences et Technologies de Lille, Lille, France
Laboratoire de Physique des Lasers, Atomes et Molécules
Groupe de Physique des Atomes Refroidis par Laser
Équipe Chaos Quantique
Transition d'Anderson avec un système
d'atomes froids quantiquement chaotique
Jean-Claude Garreau
ENS-Lyon
– 28/9/2009
1

Université des Sciences et Technologies de Lille, Lille, France
Laboratoire de Physique des Lasers, Atomes et Molécules
Groupe de Physique des Atomes Refroidis par Laser
Julien Chabé, thésitif (→6/2008)
Hans Lignier (→9/2009), Jean-François Clément, post-docs
Pascal Szriftgiser, Véronique Zehnlé, J. C. G.
Gabriel Lemarié, thésitif (→9/2009)
Benoît Grémaud, Dominique Delande
2

Anderson model
The Anderson model
Anderson model: how disorder affects the quantum dynamics of electrons in a crystal?
• Perfect crystal → Delocalized wavefunctions (Bloch states)
• An electron can be found anywhere in the crystal → diffusion = conduction
P. W. Anderson, Absence of Diffusion in Certain Random Lattices, Phys. Rev. 109, 1492 (1958)
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Anderson model
Anderson localization
• “Anderson” crystal
m
m
+∑ Vrum+
r
r
T u
=
Eu
m
W
−
2
≤ T
m
≤
W
2
→ Localized wavefunctions
• An electron stays “close” to its initial position = insulator behavior
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Anderson localization
1D Anderson localization
E(q)
q
dψ
( x)
dx
2
≈ −t
ψ ( x)
2
ψ ( x)
2
≈
e
− x / L
5/38

“Intuitive” (naïve?) introduction to the Anderson transition
Pedestrian’s approach to the Anderson transition
ψ in
ψ out
2
ψ =
out
tψ in
2
r
t
= 1−
t
J. T. Edwards and D. J. Thouless, Numerical studies of localization in disordered systems,
J. Phys. C: Solid State Phys. 5, 807 (1972)
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Anderson transition
out
2
ψ =
L
L
t ψ
in
2
R
=
1−t
L
t
L
=
( )
L
1+
r −1
7/38

Anderson transition
If scaling:
R =
lnR = α lnL
d
d
ln
ln
α
L
R
L
=α
L →∞
α > 0
α < 0
Insulator
Conductor
r 1
R ≈ rL
ln R ≈ lnr
+ lnL
R
=
( )
L
1+
r −1
lnR
L
R ≈ r
≈ Llnr
= e
α ≈1> 0
α ≈ Lln r > 0
ln
L lnr
Insulator
Insulator
8/38

Anderson transition
r
1
R ≈ rL
/L 2 1
~ rL
−
R
~
r
L
L
−2
α ~ −1
α = 0
α ≈ Lln r −2
> 0
Conductor
Insulator
9/38

Experimental observations
Experimental observations of Anderson
localization and transition
• Not easy to control decoherence
• Bulk quantities x wavefunctions
• Strong interactions
• Localization has been observed with sound waves, microwaves…
…and with a Bose-Einstein condensate, but only in 1D
J. Billy et al., Direct observation of Anderson localization of matter-waves in a controlled
disorder, Nature 453, 891 (2008)
• Transition observed with light
M. Störzer et al., Observation of the Critical Regime Near
Anderson Localization of Light , PRL 96, 063904 (2006)
…and sound waves
H. Hu et al., Localization of ultrasound in a three-dimensional elastic
network, Nature Physics 4, 945 (2008)
…experiments plagued by strong absorption
10
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Anderson transition with matter waves
Why with cold atoms?
• Control of decoherence
• Direct access to wavefunctions
• Negligible particle-particle interactions
11
/38

The optical potential
The optical potential
p
after
=
p
before
+ 2hk
L
12
/38

Kicked rotor
The kicked rotor
2
P
H = + K cos x∑δ
( t − n)
2
n
13
/38

Kicked rotor
Why is dynamics quantum?
λ / 2
~10-1000 µK
3 µK
λ ≈ λ /
dB
3
14
/38

Kicked rotor
lnP(p)
classical
quantum
〈P 2 (t)〉
Momentum distribution is exponentially
“Dynamical” localization
localized for t > t L
/38
F. L. Moore et al., Observation of Dynamical Localization in Atomic Momentum Transfer: A New Testing Ground for
Quantum Chaos, PRL 73, 2974(1994)
15
/38

Controlling decoherence
Controlling decoherence
V (x)
V
( x)
~
I
Δ
sin
( 2k
x)
L
Intensity
Detuning
Γ
sp
I
~ Δ
2
Coherent effects
Decoherence
V I / Δ
~ ~
2
Γ I / Δ
sp
~
Δ
Δ 10 GHz ~ 2000Γ
~ 1 spontaneously-emitted photon ~ 500 kicks
16
/38

Measuring momentum distributions
Stimulated Raman transitions
Δ R
>>Γ
ω
ω + 9.2 GHz + δ
F = 4
F = 3
δ
Measuring the population of the F = 3 hyperfine sublevel is equivalent to
measure to population of the p = p 0
momentum class population!
17
/38

Measuring momentum distributions
3.3 µ K
18
/38

Anderson localization x dynamical localization
How is all this connected to Anderson?
Unitary transformation
T m v +∑ m
V rvm+
r
=
r
T
m
⎛ ε
m
= tan
⎜
⎝
−
2
2
m
⎞
⎟
⎠
Ev
m
Fishman, Prenge, Grempel, PRL 49, 509 (1982); ibid. PRA 29, 1639 (1984)
19
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Anderson localization x dynamical localization
A “3D” Kicked rotor?
2
P
H = + V ( x,
x t x t )∑ 1(
),
2(
) δ ( t − n)
2
x1
and x 2
are treated as new independent variables, with conjugates P and
1 2
ψ
⎛ ∂x
−it
⎜
⎝ ∂t
∂
∂x
∂x
+
∂t
2
P
H = + ω P + P + V x x + t x + t ∑
1 1
ω2
2
( ,
10
ω1
,
20
ω2
) δ ( t − n)
2
∂
∂x
1 2
1 2
→ e
Apply again the Fishman-Prenge-Grempel transformation to obtain a model
⎞
⎟
n
⎠ψ
“substantially equivalent”
to an “anisotropic” 3D Anderson model
n
P
G. Casati et al., Anderson transition in a one-dimensional system with three incommensurate
frequencies, PRL 62, 345 (1989)
20
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Anderson localization x dynamical localization
2
P
H = + K cos x( 1+
ε cos( ω t) cos( t)
)∑ 1
ω2
δ ( t − n)
2
1/L D
n
ω 1
and
ω 2
are irrational numbers:
Localisation
Quasiperiodic kicked rotor
Diffusion
ε
21
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Experiment
22/38

Momentum distributions
Experimental momentum distributions
Linear scale
Log scale
ψ ( p)
2
ψ ( p)
2
K = 5.0
Π 0
K = 5.0
K = 9.0
K = 9.0
p
p
G. Lemarié et al., PRA (2009) to appear
23
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Critical point
Experimental determination of the critical point
1/3
Λ ≡ t Π ( 0
t)
Localized Critical Diffusive
Π ( 0
t)
→ cte
Π
0
( t)
→
t
−1/3
Π
0
( t)
→
t
−1/
2
Λ
1/3
→ t
−1/ 6
Λ → cte Λ → t
α ≡
d ln Λ
d ln t
α =1/ 3
α = 0
α = −1/
6
24/38

Critical point
2
p
H = + K cos( 2k
)( +
)∑ Lx
1 ε cosω1t
cosω2t
δ ( t − n)
2
n
25
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Critical point
Localized
Critical
Diffusive
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Kinetic energy
Average kinetic energy evolution
P
= Π
2 −2
0
linear
log-log
〈 P 2 (t) 〉
t
t
G. Lemarié et al., PRA (2009) to appear
27
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Critical exponent
Extracting a critical exponent
Small samples : no singular behavior
Ferromagnetic-paramagnetic transition
28
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Critical exponent
Finite time scaling
Scaling hypothesis
(
1/3
)
−2
2 −2/
3
Π t = P t
0
f ( K,
t)
→ F(
g(
K)
t)
log( g(
K)
t)
→
logt
+ log( g)
=
logt
+ ξ ( K)
log(〈P 2 〉 t -2/3 )
ξ(K)
log(t)
E. Abrahams et al., Scaling Theory of Localization: Absence of Quantum Diffusion in Two Dimensions,
PRL 42, 673 (1979)
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Critical exponent
J. Chabé et al., PRL 101, 255702 (2008); G. Lemarié et al., PRA (2009), to appear
30
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Critical exponent
logξ
ξ ~
K
−
K c
−ν
K c
log K
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Critical exponent
diffusive
localized
KR (num.)
K c
= 6.4
v = 1 .59 ± 0.01
K c
v
= 6 .9 ±
= 1 .6 ±
0.2
0.2
Anderson (num)
K c
= 6.4
v = 1 .6 ± 0.01
J. Chabé et al., PRL 101, 255702 (2008); G. Lemarié et al., PRA (2009), to appear
32
/38

Critical wavefunction: work in progress!
Momentum distribution in the localized regime
|ψ | 2
p
=
P
2hk
p
G. Lemarié et al. - unpublished results
33
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Critical wavefunction: work in progress!
Diffusive regime
|ψ | 2
1/ 2
~ t
2
~ t
−1/
p
|ψ | 2 t 1/2
−1/ 2
pt
G. Lemarié et al. - unpublished results
34
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Critical wavefunction: work in progress!
Critical regime
“Self-consistent” theory:
ψ ( p,
t)
2
=
3
2
Ai
( ) )
3/ 2 −1/3
3a
t p
(
3/ 2
3a
t) 1/ 3
|ψ | 2
p
|ψ | 2 t 1/3
−1/3
pt
G. Lemarié et al. - unpublished results
35
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Critical wavefunction: work in progress!
|ψ| 2 t 1/3
Exponential
Airy
Gaussian
χ 2 = 4.5
χ 2 = 1.1
χ 2 = 8.8
p
|ψ| 2 t 1/3
p
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36
/38

Critical wavefunction (numerical)
Critical wavefunction at 10 4 kicks…
multifractality?
G. Lemarié, Ph. Thesis (2009); unpublished results
37
/38

Conclusion
Prospects for future work
• Can one observe multifractality experimentally?
• 4D, 5D… ?
• 2D… ?
More generally
• What is the effect of interactions on the transition?
→ Use a BEC (and Feshbach resonances) to study interactions
→ Use a FDG to study long-range interactions
Still more generally
• Simulation of condensed matter systems by dynamical, cold atom systems?
→ For example: Harper model :
J. Wang and J. Gong, Proposal of a cold-atom realization of quantum maps with
Hofstadter's butterfly spectrum, PRA 77, 031405(R) (2008)
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