ultracold atoms in controlled disorder...
TRANSCRIPT
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Ultracold Atoms in Controlled Disorder I:Ultracold Atoms in Controlled Disorder I:
Anderson Localization in Dimension d=1Anderson Localization in Dimension d=1
Laurent Sanchez-PalenciaLaboratoire Charles Fabry - UMR8501
Institut d'Optique, CNRS, Univ. Paris-sud 11
2 av. Augustin Fresnel, Palaiseau, France
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AcknowledgementsAcknowledgements
« Theory of Ultracold Quantum Gases » team➢M. Piraud, S. Lellouch, G. Boeris, G. Carleo, LSP➢ former members : L. Pezzé, T.-L. Dao, B. Hambrecht, P. Lugan
Collaboration and stimulating discussions➢A. Aspect's experimental group, P. Bouyer, V. Josse, T. Bourdel et al.➢B. van Tiggelen, T. Giamarchi, M. Lewenstein, G.V. Shlyapnikov
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Disordered Quantum GasesDisordered Quantum Gases
Urbana Champain
RiceLENS
Institut d'Optique
Singapore
Grenoble
New York
Geneva
Orsay
Lille
LSP & Lewenstein, Nat. Phys. 6, 87 (2010)special issue in Phys. Today 2009Modugno, Rep. Prog. Phys. 73, 102401 (2010)Fallani et al., Adv. At. Mol. Opt. Phys. 56, 119 (2008)Shapiro, J. Phys. A: Math. Theor. 45, 143001 (2012)
Barcelona
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OutlookOutlook
Lecture #1
1. Basics of quantum transport theory in disordered media
2. Ultracold atoms and speckle potentialsSpecific features of a controlled disorder
3. Anderson localization in 1D speckle potentials3.1 Effective mobility edges3.2 Expansion of Bose-Einstein condensates (exp. vs theory)
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Basics of Transport Theory in Disordered MediaBasics of Transport Theory in Disordered Media
Homogeneous (underlying) medium|k⟩ states
characterized by (k) and v=∂(k)/∂ℏk
Anderson localization : microscopic approach
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Basics of Transport Theory in Disordered MediaBasics of Transport Theory in Disordered Media
Homogeneous (underlying) medium|k⟩ states
characterized by (k) and v=∂(k)/∂ℏk
DisorderV=0 (sets the zero of energy)
characterized by Cn(r1, ..., rn)=V(r1) ... V(rn)
most importantly C2(r)=VR
2c2({zj/j}) and its
Fourier transform C2(k) (disorder power spectrum)
If c2 is isotropic, j,l=j/l (anisotropy factors)
+
y
x
Anderson localization : microscopic approach
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Basics of Transport Theory in Disordered MediaBasics of Transport Theory in Disordered Media
Single scattering
V
kk'
(con
ti nu
um)
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Basics of Transport Theory in Disordered MediaBasics of Transport Theory in Disordered Media
Single scattering
V
kk'
(con
ti nu
um)
finite life time s of |k⟩ states (Fermi golden rule)
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Basics of Transport Theory in Disordered MediaBasics of Transport Theory in Disordered Media
scattering mean free pathls(k) = |v| s(k)
Single scattering
V
kk'
(con
ti nu
um)
In general C(q) decreases with |q| (maximum coupling kk, ie forward scattering) ➔
s : characteristic time of phase loss
For white noise, C(q)=cst and the scattering is isotropic ➔ s describes
phase and direction loss
finite life time s of |k⟩ states (Fermi golden rule)
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Basics of Transport Theory in Disordered MediaBasics of Transport Theory in Disordered Media
Multiple scattering
V
k k'
V
k''
V
k'''
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Basics of Transport Theory in Disordered MediaBasics of Transport Theory in Disordered Media
transport mean free pathDB(E) = (1/d) |v| lB(E)
Multiple scattering
V
k k'
V
k''
V
k'''
In general C(q) decreases with |q| (maximum coupling kk, ie forward scattering) ➔ l
B(E) > l
s(E)
For white noise, C(q)=cst and the scattering is isotropic ➔ s describes
phase and direction loss, and lB(E) = l
s(E)
deflection of of |k⟩ states : random walk ➔ normal spatial diffusion, DB(E)
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Basics of Transport Theory in Disordered MediaBasics of Transport Theory in Disordered Media
Interference effects
V
k k'
V
k''
V
k'''
V V
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Basics of Transport Theory in Disordered MediaBasics of Transport Theory in Disordered Media
Interference effects
V
k k'
V
k''
V
k'''
V V
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Basics of Transport Theory in Disordered MediaBasics of Transport Theory in Disordered Media
localization length(E) = Lloc(E)-1 = lB
-1 Fd(k lB)
Interference effects
V
k k'
V
k''
V
k'''
weak localization ➔ D*(E) < DB(E) : diffusion
strong (Anderson) localization ➔ D*(E) -iLloc(E)2 exponential localization
V V
decreases with- d (return prob.)- k lB (interf. param.)
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Basics of Transport Theory in Disordered MediaBasics of Transport Theory in Disordered Media
Anderson localization : scaling theory
dimensionless conductance : g=0 (insulator) ; g= ∞ (metal)
RG based on the fundamental « on-parameter scaling » hypothesis : d ln(g) / d ln(L)=(g)
P.W. Anderson, Phys. Rev. 109, 1492 (1958)Edwards and Thouless, J. Phys. C 5, 807 (1972)Abrahams et al., Phys. Rev. Lett. 42, 673 (1979)
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Basics of Transport Theory in Disordered MediaBasics of Transport Theory in Disordered Media
Anderson localization : scaling theory
dimensionless conductance : g=0 (insulator) ; g= ∞ (metal)
RG based on the fundamental « on-parameter scaling » hypothesis : d ln(g) / d ln(L)=(g)
P.W. Anderson, Phys. Rev. 109, 1492 (1958)Edwards and Thouless, J. Phys. C 5, 807 (1972)Abrahams et al., Phys. Rev. Lett. 42, 673 (1979)
1D : no diffusion & all states localized2D : all states localized3D : mobility edge, kl*~1
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OutlookOutlook
Lecture #1
1. Basics of quantum transport theory in disordered media
2. Ultracold atoms and speckle potentialsSpecific features of a controlled disorder
3. Anderson localization in 1D speckle potentials3.1 Effective mobility edges3.2 Expansion of Bose-Einstein condensates (exp. vs theory)
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High-precision measurementscomplementary to condensed-matter toolsvariety of diagnostic tools (direct imaging, velocity distribution, oscillations, Bragg spectroscopy, ....)
Flexibilitycontrol of parameters (dimensionality, shape, interactions, bosons/fermions/mixtures, ...)
Model systemsparameters known ab-initio ⇒ direct comparison with theorytowards quantum simulators ⇒ artificial, controlled systems
to study fundamental questions
Condensed Matter with Ultracold AtomsCondensed Matter with Ultracold Atoms
specific ingredients of ultracold atoms ...
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ℏa t ℏL
VR
R
Controlled Optical DisorderControlled Optical Disorder
J. W. Goodman, Speckle Phenomena in Optics (2007)Clément et al., New J. Phys. 8, 165 (2006)
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Experimental control
C2(r) = V(r)V(0) – V 2
VR I
L / ∆
R: correlation length
(depends only on the intensity profile of the ground-glass plate)C
2(r) can be tailored almost at will
ℏa t ℏL
VR
R
Controlled Optical DisorderControlled Optical Disorder
Speckles: an original class of disorder
E(r) is a Gaussian random process
V(r) |E(r)|2 - |E|2 is not Gaussian is not symmetric
Cn(r
1, ..., r
n) = V(r
1) ... V(r
n) all
determined by CE(r) = E(r)*E(0)
J. W. Goodman, Speckle Phenomena in Optics (2007)Clément et al., New J. Phys. 8, 165 (2006)
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VR
R
Controlled Optical DisorderControlled Optical Disorder
ℏa t ℏL
J. W. Goodman, Speckle Phenomena in Optics (2007)Clément et al., New J. Phys. 8, 165 (2006)
Experimental control
C2(r) = V(r)V(0) – V 2
VR I
L / ∆
R: correlation length
(depends only on the intensity profile of the ground-glass plate)C
2(r) can be tailored almost at will
Speckles: an original class of disorder
E(r) is a Gaussian random process
V(r) |E(r)|2 - |E|2 is not Gaussian is not symmetric
Cn(r
1, ..., r
n) = V(r
1) ... V(r
n) all
determined by CE(r) = E(r)*E(0)
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OutlookOutlook
Lecture #1
1. Basics of quantum transport theory in disordered media
2. Ultracold atoms and speckle potentialsSpecific features of a controlled disorder
3. Anderson localization in 1D speckle potentials3.1 Effective mobility edges3.2 Expansion of Bose-Einstein condensates (exp. vs theory)
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ln(|(z)|/|(0)|) ~ -|z|/Lloc
Anderson Localization in 1D Speckle PotentialsAnderson Localization in 1D Speckle Potentials
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24V V
q = 0,-2k
+k +k
-k
+k-k
+k-k
q = +2k,0,-2k
Anderson Localization in 1D Speckle PotentialsAnderson Localization in 1D Speckle Potentials
ln(|(z)|/|(0)|) ~ -|z|/Lloc
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25V V
q = 0,-2k
+k +k
-k
+k-k
+k-k
q = +2k,0,-2k
C2(2k)=V(+2k)V(-2k)≠0
coherent superposition
Quantity of interest:
Anderson Localization in 1D Speckle PotentialsAnderson Localization in 1D Speckle Potentials
ln(|(z)|/|(0)|) ~ -|z|/Lloc
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z2kc
Potential cut-off:CE(k)=0 for k>kc
C2(2k)=0 for k>kckc=(2/L)sin R
1
finitegroundglass plate
laser
Anderson Localization in 1D Speckle PotentialsAnderson Localization in 1D Speckle Potentials
J. W. Goodman, Speckle Phenomena in Optics (2007)Clément et al., New J. Phys. 8, 165 (2006)
C(2p)
pℏ/σR
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Anderson Localization in 1D Speckle PotentialsAnderson Localization in 1D Speckle Potentials
Perturbation theory in the phase formalism
Lifshits, Gredeskul & Pastur, Introduction to the Theory of Disordered Systems (1988)
(z)
k-1∂/∂z
r(z)
kE is defined byE = ℏ2kE
2/2m = pE2/2m
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Anderson Localization in 1D Speckle PotentialsAnderson Localization in 1D Speckle Potentials
Perturbation theory in the phase formalism
(z) = kz + (1)(z)+ (2)(z)+ ...
Include (z) into equation for r(z)
Lifshits, Gredeskul & Pastur, Introduction to the Theory of Disordered Systems (1988)
(z)
k-1∂/∂z
r(z)
kE is defined byE = ℏ2kE
2/2m = pE2/2m
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Anderson Localization in 1D Speckle PotentialsAnderson Localization in 1D Speckle Potentials
determined by thenpoint correlation function
Perturbation theory
LSP et al., Phys. Rev. Lett 98, 210401 (2007)P. Lugan et al., Phys. Rev. A 80, 023605 (2009)E. Gurevich and O. Kenneth, Phys. Rev. A 79, 063617 (2009)
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Anderson Localization in 1D Speckle PotentialsAnderson Localization in 1D Speckle Potentials
→ order 2 Perturbation theory
kc
determined by thenpoint correlation function
Order 2cut-off at kc defines an effective mobility edge
no difference between VR > 0 and VR < 0
LSP et al., Phys. Rev. Lett 98, 210401 (2007)P. Lugan et al., Phys. Rev. A 80, 023605 (2009)E. Gurevich and O. Kenneth, Phys. Rev. A 79, 063617 (2009)
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Anderson Localization in 1D Speckle PotentialsAnderson Localization in 1D Speckle Potentials
→ order 3
kc
Order 3same cut-off at kc
opposite correction value of (3) for VR > 0 and VR < 0
this term can be relevant for experiments, see M. Piraud et al., Phys. Rev. A 85, 063611 (2012)
LSP et al., Phys. Rev. Lett 98, 210401 (2007)P. Lugan et al., Phys. Rev. A 80, 023605 (2009)E. Gurevich and O. Kenneth, Phys. Rev. A 79, 063617 (2009)
Perturbation theory
determined by thenpoint correlation function
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2kc
Anderson Localization in 1D Speckle PotentialsAnderson Localization in 1D Speckle Potentials
→ order 4
kc
Order 4new cut-off at 2kc defines a second effective mobility edge
no difference between VR > 0 and VR < 0
complicated behavior of (4)
Perturbation theory
determined by thenpoint correlation function
P. Lugan et al., Phys. Rev. A 80, 023605 (2009)E. Gurevich and O. Kenneth, Phys. Rev. A 79, 063617 (2009)
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Anderson Localization in 1D Speckle PotentialsAnderson Localization in 1D Speckle Potentials
Orders 2 to 4excellent agreement with exact numerical calculations (by D. Delande [P. Lugan et al., Phys. Rev. A 80, 023605 (2009)])
Perturbation theory
determined by thenpoint correlation function
P. Lugan et al., Phys. Rev. A 80, 023605 (2009)E. Gurevich and O. Kenneth, Phys. Rev. A 79, 063617 (2009)
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Anderson Localization in 1D Speckle PotentialsAnderson Localization in 1D Speckle Potentials
Perturbation theory
P. Lugan et al., Phys. Rev. A 80, 023605 (2009)E. Gurevich and O. Kenneth, Phys. Rev. A 79, 063617 (2009)
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OutlookOutlook
Lecture #1
1. Basics of quantum transport theory in disordered media
2. Ultracold atoms and speckle potentialsSpecific features of a controlled disorder
3. Anderson localization in 1D speckle potentials3.1 Effective mobility edges3.2 Expansion of Bose-Einstein condensates (exp. vs theory)
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Anderson Localization of an Expanding Anderson Localization of an Expanding Matterwave in a 1D Speckle PotentialMatterwave in a 1D Speckle Potential
LSP et al., Phys. Rev. Lett. 98, 210401 (2007)M. Piraud et al., Phys. Rev. A 83, 031603(R) (2011)J. Billy et al., Nature 453, 891 (2008)see also G. Roati et al., Nature 453, 895 (2008)
Superposition of matterwaves with a broadenergy distribution
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Bose-Einstein condensate in a harmonic trap
Thomas-Fermi regime :
t=0 Vtrap ≠ 0
VR = 0
g ≠ 0
Anderson Localization of an Expanding Anderson Localization of an Expanding Matterwave in a 1D Speckle PotentialMatterwave in a 1D Speckle Potential
LSP et al., Phys. Rev. Lett. 98, 210401 (2007)M. Piraud et al., Phys. Rev. A 83, 031603(R) (2011)
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0<t<tiVtrap = 0
VR = 0
g ≠ 0
Interaction-driven expansion of the BEC
Scaling theory of expanding BEC (in TF regime)
Momentum distribution :
Yu. Kagan, E. L. Surkov, and G.V. Shlyapnikov, Phys. Rev. A 54, R1753 (1996)Y. Castin and R. Dum, Phys. Rev. Lett. 77, 5315 (1996)
Anderson Localization of an Expanding Anderson Localization of an Expanding Matterwave in a 1D Speckle PotentialMatterwave in a 1D Speckle Potential
LSP et al., Phys. Rev. Lett. 98, 210401 (2007)M. Piraud et al., Phys. Rev. A 83, 031603(R) (2011)
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t=tiVtrap = 0
VR ≠ 0
g = 0
Interactions off and disorder on
« p » is no longer a good quantum number but « E » is !
R
VR
Anderson Localization of an Expanding Anderson Localization of an Expanding Matterwave in a 1D Speckle PotentialMatterwave in a 1D Speckle Potential
LSP et al., Phys. Rev. Lett. 98, 210401 (2007)M. Piraud et al., Phys. Rev. A 83, 031603(R) (2011)
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t=tiVtrap = 0
VR ≠ 0
g = 0
Interactions off and disorder on
« p » is no longer a good quantum number but « E » is !
R
VR
spectral function
FT of the correlation
function of the disorder
fixed value of p
E
2''(p)
A(p,E)
Anderson Localization of an Expanding Anderson Localization of an Expanding Matterwave in a 1D Speckle PotentialMatterwave in a 1D Speckle Potential
LSP et al., Phys. Rev. Lett. 98, 210401 (2007)M. Piraud et al., Phys. Rev. A 83, 031603(R) (2011)
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ti<t Vtrap = 0
VR ≠ 0
g = 0
Localization of the wave packet
V.L. Berezinskii, Sov. Phys. JETP 38, 620 (1974)A.A. Gogolin et al., Sov. Phys. JETP 42, 168 (1976)
short distance : ln(P) ~ -2(E)|z|long distance : ln(P) ~ -0.5(E)|z|
Anderson Localization of an Expanding Anderson Localization of an Expanding Matterwave in a 1D Speckle PotentialMatterwave in a 1D Speckle Potential
LSP et al., Phys. Rev. Lett. 98, 210401 (2007)M. Piraud et al., Phys. Rev. A 83, 031603(R) (2011)
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ti<t Vtrap = 0
VR ≠ 0
g = 0
Localization of the wave packet
FT of the correlation function of the disorder
V.L. Berezinskii, Sov. Phys. JETP 38, 620 (1974)A.A. Gogolin et al., Sov. Phys. JETP 42, 168 (1976)
Anderson Localization of an Expanding Anderson Localization of an Expanding Matterwave in a 1D Speckle PotentialMatterwave in a 1D Speckle Potential
LSP et al., Phys. Rev. Lett. 98, 210401 (2007)M. Piraud et al., Phys. Rev. A 83, 031603(R) (2011)
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M. Piraud et al., Phys. Rev. A 83, 031603(R) (2011)
versus numerics
Anderson Localization of an Expanding Anderson Localization of an Expanding Matterwave in a 1D Speckle PotentialMatterwave in a 1D Speckle Potential
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E
γ(E)
E
versus numerics
Anderson Localization of an Expanding Anderson Localization of an Expanding Matterwave in a 1D Speckle PotentialMatterwave in a 1D Speckle Potential
M. Piraud et al., Phys. Rev. A 83, 031603(R) (2011)
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E
D(pE)/pE
γ(E)
2µ
versus numerics
Anderson Localization of an Expanding Anderson Localization of an Expanding Matterwave in a 1D Speckle PotentialMatterwave in a 1D Speckle Potential
M. Piraud et al., Phys. Rev. A 83, 031603(R) (2011)
(«on-shell» approximation)
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E
D(pE)/pE
γeff
γ(E)
2µ
versus numerics
Anderson Localization of an Expanding Anderson Localization of an Expanding Matterwave in a 1D Speckle PotentialMatterwave in a 1D Speckle Potential
M. Piraud et al., Phys. Rev. A 83, 031603(R) (2011)
(«on-shell» approximation)
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E
D(pE)/pE
γ(E)
0.5∗γeff
2µ
versus numerics
γeff
Anderson Localization of an Expanding Anderson Localization of an Expanding Matterwave in a 1D Speckle PotentialMatterwave in a 1D Speckle Potential
M. Piraud et al., Phys. Rev. A 83, 031603(R) (2011)
(«on-shell» approximation)
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0.5∗γeff
∼2∗γeff
E
γ(E)
D(pE)/pE
2µ
versus numerics
γeff
Anderson Localization of an Expanding Anderson Localization of an Expanding Matterwave in a 1D Speckle PotentialMatterwave in a 1D Speckle Potential
M. Piraud et al., Phys. Rev. A 83, 031603(R) (2011)
(«on-shell» approximation)
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0.5∗γeff
∼2∗γeff spectral broadening
E
D(pE)/pE
γ(E)
2µ
versus numerics
Anderson Localization of an Expanding Anderson Localization of an Expanding Matterwave in a 1D Speckle PotentialMatterwave in a 1D Speckle Potential
M. Piraud et al., Phys. Rev. A 83, 031603(R) (2011)
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J. Billy et al., Nature 453, 891 (2008)see also G. Roati et al., Nature 453, 895 (2008)
BEC (t=0)
Exponential fit
Semilog plot
BEC parameters : N=1.7 104 atoms, (in ≅ 1.5R)Weak disorder : VR/µin =0.12 << 1
Anderson Localization of an Expanding Anderson Localization of an Expanding Matterwave in a 1D Speckle PotentialMatterwave in a 1D Speckle Potential
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LSP et al., Phys. Rev. Lett. 98, 210401 (2007)J. Billy et al., Nature 453, 891 (2008)see also G. Roati et al., Nature 453, 895 (2008)
Fair agreement withtheoretical prediction
Anderson Localization of an Expanding Anderson Localization of an Expanding Matterwave in a 1D Speckle PotentialMatterwave in a 1D Speckle Potential
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FT of the correlation function of the disorder
pEℏ/σR
effective mobility edge
γ ≠ 0 γ ≃ 0
C(2pE)
z2kc
laser
LSP et al., Phys. Rev. Lett. 98, 210401 (2007)P. Lugan et al., Phys. Rev. A 80, 023605 (2009)M. Piraud et al., Phys. Rev. A 83, 031603(R) (2011)
Correlation function of a 1D speckle potential
Anderson Localization of an Expanding Anderson Localization of an Expanding Matterwave in a 1D Speckle PotentialMatterwave in a 1D Speckle Potential
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0.5∗γeff
∼2∗γeff spectral broadening
E
D(pE)/pE
γ(E)
ℏ2/2mσR2
For ξin > σR
2µ=ℏ2/2mξin
2
versus numerics
M. Piraud et al., Phys. Rev. A 83, 031603(R) (2011)
Anderson Localization of an Expanding Anderson Localization of an Expanding Matterwave in a 1D Speckle PotentialMatterwave in a 1D Speckle Potential
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E
D(pE)/pE
γ(E)
For ξin < σR
ℏ2/2mσR2 2µ=ℏ2/2mξin
2
versus numerics
M. Piraud et al., Phys. Rev. A 83, 031603(R) (2011)
Anderson Localization of an Expanding Anderson Localization of an Expanding Matterwave in a 1D Speckle PotentialMatterwave in a 1D Speckle Potential
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E
D(pE)/pE
γ(E)
For ξin < σRalgebraic localizationn(z) ∝ 1/|z|2
versus numerics
M. Piraud et al., Phys. Rev. A 83, 031603(R) (2011)
Anderson Localization of an Expanding Anderson Localization of an Expanding Matterwave in a 1D Speckle PotentialMatterwave in a 1D Speckle Potential
ℏ2/2mσR2 2µ=ℏ2/2mξin
2
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For ξin < σRalgebraic localizationn(z) ∝ 1/|z|2
versus numerics
E
D(pE)/pE
γ(E)
Anderson Localization of an Expanding Anderson Localization of an Expanding Matterwave in a 1D Speckle PotentialMatterwave in a 1D Speckle Potential
M. Piraud et al., Phys. Rev. A 83, 031603(R) (2011)
ℏ2/2mσR2 2µ=ℏ2/2mξin
2
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J. Billy et al., Nature 453, 891 (2008)
kmax / kc = 1.16 +/-0.14
n(z) ~ 1/|z|
≅ 2.010.03
BEC parameters : N=1.7 105 atoms, (in ≅ 0.8R)Weak disorder : VR/µin =0.15 << 1
Anderson Localization of an Expanding Anderson Localization of an Expanding Matterwave in a 1D Speckle PotentialMatterwave in a 1D Speckle Potential
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Anderson Localization in a 1D Speckle Anderson Localization in a 1D Speckle Potential : Potential : ConclusionsConclusions
Model of 1D Anderson localizationFor ξin > σR,
short distance : ln[n(z)] ~ -2eff observed
large distance : ln[n(z)] ~ -0.5eff possible signatures (?)
+ deviations from exponential decay
For ξin < σR,
central role of special finite-range correlations in 1D speckle potentials effective mobility edgealgebraic localization, n(z) ~ 1/|z|2 observednot in contradiction with theorems [P. Lugan et al., PRA (2009)] !
Perspectivestailored correlations [M. Piraud et al., Phys. Rev. A 85, 063611 (2012)]
Extension to many-body localization
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Anderson Localization in a 1D Speckle Anderson Localization in a 1D Speckle Potential : Potential : ConclusionsConclusions
Model of 1D Anderson localizationFor ξin > σR,
short distance : ln[n(z)] ~ -2eff observed
large distance : ln[n(z)] ~ -0.5eff possible signatures (?)
+ deviations from exponential decay
For ξin < σR,
central role of special finite-range correlations in 1D speckle potentials effective mobility edgealgebraic localization, n(z) ~ 1/|z|2 observednot in contradiction with theorems [P. Lugan et al., PRA (2009)] !
Perspectivestailored correlations [M. Piraud et al., Phys. Rev. A 85, 063611 (2012)]
Extension to many-body localization
J. Billy et al., Nature 453, 891 (2008)
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Anderson Localization in a 1D Speckle Anderson Localization in a 1D Speckle Potential : Potential : ConclusionsConclusions
Model of 1D Anderson localizationFor ξin > σR,
short distance : ln[n(z)] ~ -2eff observed
large distance : ln[n(z)] ~ -0.5eff possible signatures (?)
+ deviations from exponential decay
For ξin < σR,
central role of special finite-range correlations in 1D speckle potentials effective mobility edgealgebraic localization, n(z) ~ 1/|z|2 observednot in contradiction with theorems [P. Lugan et al., PRA (2009)] !
Perspectivestailored correlations [M. Piraud et al., Phys. Rev. A 85, 063611 (2012)]
Extension to many-body localizationJ. Billy et al., Nature 453, 891 (2008) Courtesy of R. Hulet, unpublished (2009)
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Anderson Localization in a 1D Speckle Anderson Localization in a 1D Speckle Potential : Potential : ConclusionsConclusions
Model of 1D Anderson localizationFor ξin > σR,
short distance : ln[n(z)] ~ -2eff observed
large distance : ln[n(z)] ~ -0.5eff possible signatures (?)
+ deviations from exponential decay
For ξin < σR,
central role of special finite-range correlations in 1D speckle potentials effective mobility edgealgebraic localization, n(z) ~ 1/|z|2 observednot in contradiction with theorems [P. Lugan et al., PRA (2009)] !
Perspectivestailored correlations [M. Piraud et al., Phys. Rev. A 85, 063611 (2012)]
Extension to many-body localization
J. Billy et al., Nature 453, 891 (2008)
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Anderson Localization in a 1D Speckle Anderson Localization in a 1D Speckle Potential : Potential : ConclusionsConclusions
Model of 1D Anderson localizationFor ξin > σR,
short distance : ln[n(z)] ~ -2eff observed
large distance : ln[n(z)] ~ -0.5eff possible signatures (?)
+ deviations from exponential decay
For ξin < σR,
central role of special finite-range correlations in 1D speckle potentials effective mobility edgealgebraic localization, n(z) ~ 1/|z|2 observednot in contradiction with theorems [P. Lugan et al., PRA (2009)] !
Perspectivestailored correlations [M. Piraud et al., Phys. Rev. A 85, 063611 (2012)]
extension to many-body localization
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Bibliography (I)Bibliography (I)
Anderson localization in 1D disorderLifshits, Gredeskul & Pastur, Introduction to the Theory of Disordered Systems (1988)V.L. Berezinskii, Sov. Phys. JETP 38, 620 (1974)A.A. Gogolin et al., Sov. Phys. JETP 42, 168 (1976)
Speckle potentials in ultracold atom systemsJ. W. Goodman, Speckle Phenomena in Optics (2007)D. Clément et al., New J. Phys. 8, 165 (2006)B. Shapiro, J. Phys. A: Math. Theor. 45, 143001 (2012)
Anderson localization in 1D speckle potentialsLSP et al., Phys. Rev. Lett. 98, 210401 (2007)J. Billy et al., Nature 453, 891 (2008)P. Lugan et al., Phys. Rev. A 80, 023605 (2009)E. Gurevich and O. Kenneth, Phys. Rev. A 79, 063617 (2009)M. Piraud et al., Phys. Rev. A 83, 031603(R) (2011)
Localization in bichromatic lattices (not discussed here)S. Aubry and G. André, Ann. Israel Phys. Soc. 3, 133 (1980)G. Roati et al., Nature 453, 895 (2008)G. Modugno, Rep. Prog. Phys. 73, 102401 (2010)