Mesoscopic Physics in Complex Media, 01012 (2010)
DOI:10.1051/iesc/2010mpcm01012
© Owned by the authors, published by EDP Sciences, 2010
Transport properties of coherent matter waves :
superfluidity & Anderson localization
Patricio Leboeuf
Laboratoire de Physique Théorique et Modèles Statistiques
Université Paris 11 & CNRS, Orsay
In collaboration with:
M. Albert, S. Moulieras
T. Paul, N. Pavloff, K. Richter, P. Schlagheck
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Article published online by EDP Sciences and available at http://www.iesc-proceedings.org or http://dx.doi.org/10.1051/iesc/2010mpcm01012
Transport properties of coherent matter waves :
superfluidity & Anderson localization
Patricio Leboeuf
Laboratoire de Physique Théorique et Modèles Statistiques
Université Paris 11 & CNRS, Orsay
In collaboration with:
M. Albert, S. Moulieras
T. Paul, N. Pavloff, K. Richter, P. Schlagheck
Cargèse, July 2010
Transport of a Bose Einstein Condensate
Multipole Oscillations
(sudden offset of the trap)
« Stir »
« Atom Lasers »
Atoms chip
Guerin, Riou, Gaebler, Josse, Bouyer, Aspect,
PRL 97 (2006)
Equations of motion, 1D regime:
regime
a sc m ω ⊥ / h << na
sc
<< 1
Order parameter:
i h ∂ tψ = −
h2
2m
Adiabatic approximation:
a sc : s - wave scattering length ( > 0)
ψ (x, t )e−iμ t / h
∂ ψ + [U ( x − Vt ) + g ψ
2
x
n ( x , t ) = ∫ d 2 r⊥ Ψ
2
2
− μ ]ψ
= ψ (x , t )
g = 2 h ω ⊥ a sc
r
r
Ψ (r , t ) = ψ ( x , t ) φ (r⊥ , n )e − i μ t / h
r
V ⊥ (r ⊥
)= 1 ω
2
2
⊥
r
2
⊥
U ( x ),
μ
U (x → ∞ )= 0
2
Longitudinal
density
Interaction
parameter
Longitudinal
potential
Chemical
potential
Stationary transmission modes
no
ψ (x, t ) =ψ ( X = x −Vt)
⎧μ = gno
⎪
⎨c = μ / m
⎪ξ = h / (mc)
⎩
Chemical potential
Speed of sound
Healing length
ψ ( X ) = n( X ) eiS ( X )
⎧ (v − V )n = J ∞
⎪
2
dA
d
⎨ U
=
⎪
dX
dX
⎩
V
A =
n / no
v = (h / m )∂ X S
⎡ h 2 ⎛ dA ⎞ 2
⎤
⎜
⎟ + W ( A )⎥
⎢
⎢⎣ 2 m ⎝ dX ⎠
⎥⎦
Boundary conditions
ψ ( X → −∞ ) =
no
ξ
Free modes
0.5
20
0
−10
+
Ecl
n(x)/n2
0
Amax
A1
A ( X ) = 1 + δρ ( X )
T=
20
0
10
20
0
x / ξ 10
20
1
Transmission coefficient
2
10
0.5
0
−10
Amin A0=1
0
1
n(x)/n2
W(A) / m c
2
n(x)/n2
1
δρ <<
1
1 + mEcl+ /( 2h 2κ 2 )
2
V
−1
2
c
0.5
0
−10
κ=
m
V 2 − c2
h
Superfluidity
• Superfluidity is a phase of matter in which unusual physical effects are observed
- No viscosity (frictionless flow)
- quantized vortices that persist forever
- zero thermodynamic entropy
- infinite thermal conductivity (no temperature gradient can exist)
- fountain effect
• superfluidity and superconductivity are low temperature quantum mechanical effects
• The potential applications of superfluids seem more limited compared to those
of superconductors
Landau criterion
Assume we have a SF moving at velocity v. In order to create an excitation of the fluid (« heat »), the
fluid must have enough energy. Suppose we create an excitation of momentum p and energy ε(p)
(dispersion relation). The fluid looses momentum (and energy), with
p=Mδv
where δv the velocity variation and M the fluid mass. The energy variation of the fluid is
δE=(1/2) M 2 v δv = M v δv = v p
The excitations are then possible if δEr ε(p) => v p r ε(p)
⎛ ε (p )⎞
⎟⎟
v ≥ v = min ⎜⎜
⎝ p ⎠
L
L
c
Interactions!
Linear analysis of the stability of the uniform flow under a small perturbation
4
He
Superfluidity
−
h
∂ 2xψ + [U ( x ) + g ψ
2m
2
V <c
]ψ = μ ψ
ε ( p ) = p m 2c 2 + p 2 / 4 / m
•
No dissipation
•
No drag force
•
Perfect transmission
• Normal fluid fraction
L
c=
gn / m
Speed of sound
In measurements,
v c ≤ v cL
Flow past an impurity
U ( x ) = λδ ( x )
2.5
Stationary quasi-ideal
flow
2
1
2
Non-stationary
flow
0
Leboeuf.-Pavloff
PRA 2001
vv/c
/c∞
∞
1.5
1
0.5
0
V ≥c
V <c
(
⎧F ∝ V 2 1 − c 2 / V 2
⎪⎪
2
2
⎨F ∝ V − c / V
⎪ F ∝ cst
⎪⎩
(
)
)
2
M n ∝ λ2 / c 3
2
2
1
1
0
0
−1
−0.5
3D
2D
1D
0
ξλ
AstrakharchikPitaevskii 2004
Pavloff 2002
Stationary superfluid
flow
0.5
1
Experimental tests
P. Engels, C. Atherton, Phys. Rev. Lett. 99, 160405 (2007)
Dipolar oscillation + defect
D. Dries, S. E. Pollack, J. M. Hitchcock and R. G. Hulet, arXiv:1004.1891
Interacting bosons in a random potential: Two contrasting phenomena
Superfluidity
Anderson localization
E
U(x)
V <c
L
•
•
•
No dissipation
No drag force
Perfect transmission
• Normal fluid fraction
Interaction
• Exponential decay
•
T ∝ exp(-L/Lloc)
•
Large L, no transmission
Disorder
Flow through a random potential
h 2 Ni
U (x ) =
δ ( x − xn )
∑
mb n =1
A series of uncorrelated δ peaks
Correlated Gaussian potential
1
W(A)
Speckle potential
20
2
Ecl
0
Ecl
A1
A0=1
1
Uncorrelated δ peaks
2
U = h ni /(mb)
(U
<< μ + mV 2 / 2,
2
Ecl
3
A
Ecl
1
0
U ( x )U (0 ) − U
X/ξ
2
(
5
10
)
= h / m σδ (x )
2
σ = ni / b 2 , ni = N i / L )
2
Summary of Results
L/ξ
rs
de
*
An
L
time−dependent
on
loc
ali
1500
1000
Lloc
superfluid
500
0
Lloc < L
za
tio
n
2000
Ohmic
L
loc > L
0
0.5
1
2
4
6
8
V/c
10
12
14
Phys. Rev. Lett. 98, 210602 (2007)
loc
ali
za
tio
n
2000
mn
δn ( X ) ≅ − 2 0
h κ
m
κ=
h
Mn / M =
c2 −V 2
λ2 niξ
2
L/ξ
1000
0
∫
∞
−∞
superfluid
Ohmic
0
0.5
1
2
4
6
dyU ( y ) exp [− 2κ X − y ]
Effective wave vector
(1 − V / c )−3 / 2 ,
de
Lloc
500
n ( X ) = n0 + δn ( X )
*
An
SF Regime:
Subsonic motion
L
time−dependent
rs
on
1500
(κ >> ni )
Density locally perturbed in the region of the potential,
Æperfect transmission
Æno dissipation nor drag exerted on the potential
8
V/c
10
12
14
2000
loc
ali
za
tio
n
Lloc (κ ) < L
L
W(A)
20
2
cl
E
0
1
cl
E
1
cl
E
A
t = j /(κ 2b 2 )
2
cl
E
*
1000
P(λ , j )dλ
A1
A0=1
3
λ j = mEcl( j ) /(2h 2κ 2 )
L/ξ
time−dependent
Lloc
superfluid
500
0
on
1500
An
de
rs
Non-perturbative supersonic flow
Ohmic
0
0.5
1
2
4
6
8
V/c
10
12
14
1
0
X/ξ
5
10
Diffusion process:
(DMPK Equation)
(Dorokov-Mello-Pereyra-Kumar)
ln T = −
κ=
L
Lloc (κ )
m
V 2 − c2 → k
h
κ 2 /σ
Lloc (κ ) =
Cˆ (2κ )
Lloc (κ ) → Lloc (k )
κ=
m
V 2 − c2
h
Antsygina-Pastur-Slyusarev
formula
Speckle potential
Uncorrelated delta peaks
10
3
0
(a)
< ln T >
8
6
(d)
4
2
0
−2
2
0
0.5
1
L / Lloc
1.5
2
(a)
1
(c)
(b)
(c)
0
(b)
P(T)
P(T)
−1
0.2
0.4
T
0.6
0.8
1
0
0
0.2
L / Lloc (κ ) = 0.1, 0.5, 1, 2
P(T ) =
T
0.6
0.8
L / Lloc (κ ) = 0.31, 0.52, 0.68
Lloc
⎤
⎡ L
exp ⎢− loc (1 − T )⎥
L
⎦
⎣ L
⎡ L
Lloc
P(ln T ) =
exp ⎢− loc
4πL
⎢⎣ 4 L
0.4
⎛ L
⎞
⎜⎜
+ ln T ⎟⎟
⎝ Lloc
⎠
L / Lloc (κ ) << 1
2
⎤
⎥
⎥⎦
L / Lloc (κ ) >> 1
1
Crossover to the time-dependent regime
W(A)
2
0
Ecl
1
cl
0
5
0
0.5
1
2
4
6
8
V/c
P (λmax , t ) = 0
10
2000
1
Ps (t)
L /ξ
1500
*
0.5
1000
500
0
0
5
10
t = L / Lloc
Ps (t * ) = 1 / 2
0
15
2
4
6
V/c
8
10
12
⎡ ⎛ V2 ⎞ ⎤
⎟ − 1⎥
L (κ ) ≈ Lloc (κ )⎢ln⎜⎜
2 ⎟
⎣ ⎝ 16c ⎠ ⎦
*
n
so
Ohmic
E
X/ξ
er
superfluid
1
0
An
d
Lloc
2
cl
E
3
A
*
1000
500
A1
A0=1
L
time−dependent
1
Ecl
loc
1500
L/ξ
20
ali
za
t
ion
2000
10
12
14
Crossover from superfluid to time-dependent regime:
A problem of extreme value statistics
Slowly varying disordered potential :
ξ / lc << 1
n(x)
8
n
lc
v(x)=c(x)
Umax
U(x)
<U>
L
100
1
L/lc
Dissipative
Superfluid
0
0
0.2
0.4
0.6
v/c
0.8
1
0
Albert, Paul, Pavloff, Leboeuf PRA(R) 2010
Experiments with Dipole Oscillations
Disordered potential
Theory: M. Albert, T. Paul, N. Pavloff, P. Leboeuf, Phys. Rev. Lett. 100, 250405 (2008)
Experiment: D. Dries, S. E. Pollack, J. M. Hitchcock and R. G. Hulet, arXiv:1004.1891.
« Half » way between a periodic and a random system
• C : tunneling rate between adjacent waveguides
• εκ : on-site energy
• γ : strength of the nonlinearity of the medium
(>0, self defocusing medium)
• Paraxial approximation
Superfluid Motion of Light
Non-linear equation, similar to the mean-field description
of cold atoms => does superfluid motion of light exists?
v cL ∝
Localized impurity
(attractive or repulsive)
γ C A
2
≈ 2 × 10 − 2
Critical speed
(or altern. twisted array of waveguides)
1
v/vcL
1.5
Dissipative
Motion
1
d
c
Superfluid
Motion
a
0.5
b
0
-0.5
-0.25
0
U0/μ
0.25
0.5
0
P. Leboeuf, S. Moulieras
to appear
Concluding remarks
• We provide analytical and numerical evidence of AL in the presence of
interaction for # types of disorder. Generalization of DMPK diffusion
equation to include interactions. No localization as L goes to infinity.
• Renormalization of the localization length due to interactions
• The flow is SF at small velocities either for an obstacle or a disordered potential
• Existence of a time-dependent regime, at intermediate velocities
or large sample lengths
• Estimates of crossovers to an unstable flow, either from the sub or supersonic
regimes
• Dipole oscillations. Experiments on BEC show SF as well as the presence of
unstable flow in presence of disorder
• Extension of the concept of superfluidity to motion motion of light in NL media
Some Related References
# Bose-Einstein beams: coherent propagation through a guide
P. Leboeuf, N. Pavloff
Phys. Rev. A 64 (2001) 033602
# Breakdown of superfluidity of an atom laser past an obstacle
N. Pavloff
Phys. Rev. A 66 (2002) 013610
# Solitonic transmission of Bose-Einstein matter waves
P. Leboeuf, N. Pavloff, S. Sinha
Phys. Rev. A 68 (2003) 063608
# Nonlinear transport of Bose-Einstein condensates through waveguides with disorder
T. Paul, P. Leboeuf, N. Pavloff, K. Richter, P. Schlagheck
Phys. Rev. A 72, 063621 (2005)
# Superfluidity versus Anderson localization in a dilute Bose gas
T. Paul, P. Schlagheck, P. Leboeuf, N. Pavloff
Phys. Rev. Lett. 98, 210602 (2007)
# Dipole oscillations of a BEC in the presence of defects and disorder
M. Albert, T. Paul, N. Pavloff, P. Leboeuf
Phys. Rev. Lett. 100, 250405 (2008)
# Anderson localization of a weakly interacting one-dimensional Bose gas
T. Paul, M. Albert, P. Schlagheck, P. Leboeuf, N. Pavloff
Phys. Rev. A 80, 033615 (2009)
# Localization by bichromatic potentials versus Anderson localization
M. Albert, P. Leboeuf
Phys. Rev. A 81, 013614 (2010)
# Breakdown of superfluidity of a matter wave in a random environment
M. Albert, T. Paul, N. Pavloff, P. Leboeuf
Phys. Rev. A 82, 011602 (2010)
# Superfluid motion of light
P. Leboeuf, S. Moulieras
to appear