Upload
vuongdiep
View
213
Download
0
Embed Size (px)
Citation preview
Coworkers
Michael Hartung(Regensburg)
Thomas Wellens(Freiburg)
Cord Muller(Singapore)
Klaus Richter(Regensburg)
Timo Hartmann(Regensburg)
Cyril Petitjean(Grenoble)
Juan-Diego Urbina(Regensburg)
Outline
Weak localization in 2D disorder and chaotic billiards
Coherent backscattering of condensates from 2D disorder:numerical approach and diagrammatic theory
Weak localization of condensates in 2D chaotic billiards:preliminary numerical and semiclassical results
Conclusion
Weak localization in two-dimensional disorder
Constructive interference between reflected paths and theirtime-reversed counterparts
������
������
������
������
������
������
������
������
������
������
���
���
������
������
���
���
������
������
���
���
������
������
������
������
������
������
������
������
������
������
���
���
��������
���
���
������
������
������
������
���
���
���
���
������
������
���
���
������
��������
����
������
������
������
������
������
������
������
������
������
���
���
������
������
������
������
������
������
������
������
���
���
���
���
������
������
���
������
���
���
���
θ
-0.4π -0.2π 0 0.2π 0.4π0
1
2
3
4
back
scat
tere
dcu
rren
t
angle θ
→ coherent backscattering
Weak localization in two-dimensional disorder
Constructive interference between reflected paths and theirtime-reversed counterparts
������
������
������
������
������
������
������
������
������
������
���
���
������
������
���
���
������
������
���
���
������
������
������
������
������
������
������
������
������
������
���
���
��������
���
���
������
������
������
������
���
���
���
���
������
������
���
���
������
��������
����
������
������
������
������
������
������
������
������
������
���
���
������
������
������
������
������
������
������
������
���
���
���
���
������
������
���
������
���
���
���
θ
→ enhanced coherent backscattering of laser light fromdisordered mediaM. P. Van Albada and A. Lagendijk, PRL 55, 2692 (1985)
P.-E. Wolf and G. Maret, PRL 55, 2696 (1985)
Weak localization in two-dimensional disorder
Constructive interference between reflected paths and theirtime-reversed counterparts
������
������
������
������
������
������
������
������
������
������
���
���
������
������
���
���
������
������
���
���
������
������
������
������
������
������
������
������
������
������
���
���
��������
���
���
������
������
������
������
���
���
���
���
������
������
���
���
������
��������
����
������
������
������
������
������
������
������
������
������
���
���
������
������
������
������
������
������
������
������
���
���
���
���
������
������
���
������
���
���
���
θ
→ enhanced coherent backscattering of laser light fromdisordered mediaM. P. Van Albada and A. Lagendijk, PRL 55, 2692 (1985)
P.-E. Wolf and G. Maret, PRL 55, 2696 (1985)
→ magnetoresistance in disordered 2D metalsB. L. Altshuler et al., PRB 22, 5142 (1980)
A. G. Aronov and Yu. V. Sharvin, Rev. Mod. Phys. 59, 755 (1987); . . .
Weak localization in two-dimensional chaotic billiards
Constructive interference between retro-reflectedtrajectories within the same transverse channel of the lead
Weak localization in two-dimensional chaotic billiards
Constructive interference between retro-reflectedtrajectories within the same transverse channel of the lead
0magnetic field
0
0.5
reflection probability (5 open channels)
dephasing in the presence of a magnetic field
Weak localization in two-dimensional chaotic billiards
Constructive interference between retro-reflectedtrajectories within the same transverse channel of the lead
dephasing in the presence of a magnetic field
→ magnetoresistance in ballistic nanostructuresA. M. Chang et al., PRL 73, 2111 (1994)
Transport of condensates through 2D disorder
Possible experimental realization:
BEC
optical lattice+ speckle field
−→ measure angle-resolved flux of backscattered atoms
Transport of condensates through 2D disorder
Gross-Pitaevskii description of the condensate in the2D confinement: g
i~∂
∂tψ(~r, t) =
(
− ~2
2m
∂2
∂~r2+ V (~r) + g|ψ(~r, t)|2
)
ψ(~r, t)
with g =√
8πas
a⊥
~2
m≡ g(x): effective 2D interaction strength
BEC
Transport of condensates through 2D disorder
−→ integrate Gross-Pitaevskii equation in the presence of a−→ source term (atom laser)
i~∂
∂tψ(~r, t) =
(
− ~2
2m
∂2
∂~r2+ V (~r) + g|ψ(~r, t)|2
)
ψ(~r, t)
+S0δ(x− x0) exp(−iµt/~)
BEC reservoirµchem. pot.
scattering potential
coupling (source)
T. Paul, K. Richter, and P.S., PRL 94, 020404 (2005)
Transport of condensates through 2D disorder
Definition of the spatial 2D geometry:
source
periodic boundary conditions
periodic boundary conditions
absorbing boundaryab
sorb
ing
boun
dary
disorder(Gaussian)
Transport of condensates through 2D disorder
Definition of the spatial 2D geometry:
source
periodic boundary conditions
periodic boundary conditions
absorbing boundaryab
sorb
ing
boun
dary
disorder(Gaussian)
h(x)
x
g(x)
Transport of condensates through 2D disorder
Definition of the spatial 2D geometry:
source
periodic boundary conditions
periodic boundary conditions
absorbing boundaryab
sorb
ing
boun
dary
disorder(Gaussian)
BEC
Transport of condensates through 2D disorder
Stationary scattering state of the condensate:
decomposition of reflected wave into transverse eigenmodes→ angle-resolved backscattered current
Transport of condensates through 2D disorder
Stationary scattering state of the condensate:
decomposition of reflected wave into transverse eigenmodes→ angle-resolved backscattered current
Coherent backscattering of the condensate
-0.4π -0.2π 0 0.2π 0.4π0
1
2
3
4 g=0
curr
ent[
arb.
units
]
angle θ [rad]
Coherent backscattering of the condensate
-0.4π -0.2π 0 0.2π 0.4π0
1
2
3
4 g=0g=0.0025
curr
ent[
arb.
units
]
angle θ [rad]
Coherent backscattering of the condensate
-0.4π -0.2π 0 0.2π 0.4π0
1
2
3
4 g=0g=0.005
curr
ent[
arb.
units
]
angle θ [rad]
Coherent backscattering of the condensate
-0.4π -0.2π 0 0.2π 0.4π0
1
2
3
4 g=0g=0.0075
curr
ent[
arb.
units
]
angle θ [rad]
Coherent backscattering of the condensate
-0.4π -0.2π 0 0.2π 0.4π0
1
2
3
4 g=0g=0.01
curr
ent[
arb.
units
]
angle θ [rad]
−→ inverted cone in presence of finite interaction:−→ crossover from constructive to destructive interference
Analytical theory of nonlinear coherent backscattering
T. Wellens and B. Gremaud, PRL 100, 033902 (2008)
Nonlinear Lippmann-Schwinger equation:
ψ(~r) = ψ0eikx +
∫
d2r′ G0(~r − ~r′)[
Vdis(~r′) + g|ψ(~r′)|2
]
ψ(~r′)
with G0(~r)kr≫1= −m
~2
1√2πkr
ei(kr+π/4) :
Green function of free motion in two dimensions
Analytical theory of nonlinear coherent backscattering
T. Wellens and B. Gremaud, PRL 100, 033902 (2008)
Nonlinear Lippmann-Schwinger equation:
ψ(~r) = ψ0eikx +
∫
d2r′ G0(~r − ~r′)[
Vdis(~r′) + g|ψ(~r′)|2
]
ψ(~r′)
−→ diagrams:ψ
ψ
ψ∗
V g
Analytical theory of nonlinear coherent backscattering
T. Wellens and B. Gremaud, PRL 100, 033902 (2008)
Nonlinear Lippmann-Schwinger equation:
ψ(~r) = ψ0eikx +
∫
d2r′ G0(~r − ~r′)[
Vdis(~r′) + g|ψ(~r′)|2
]
ψ(~r′)
−→ diagrams:ψ
ψ
ψ∗
V g
Regime of weak localization (k × ℓtransport ≫ 1):
→ main contributions to disorder average of densitiesfrom ladder (diffuson) and crossed (Cooperon) diagrams
Analytical theory of nonlinear coherent backscattering
T. Wellens and B. Gremaud, PRL 100, 033902 (2008)
Ladder diagrams:
Analytical theory of nonlinear coherent backscattering
T. Wellens and B. Gremaud, PRL 100, 033902 (2008)
Ladder diagrams:
→ no net effect of nonlinearity→ on 〈|ψ(~r)|2〉disorder av.
Analytical theory of nonlinear coherent backscattering
T. Wellens and B. Gremaud, PRL 100, 033902 (2008)
Crossed diagrams:
Analytical theory of nonlinear coherent backscattering
T. Wellens and B. Gremaud, PRL 100, 033902 (2008)
Crossed diagrams:
Analytical theory of nonlinear coherent backscattering
T. Wellens and B. Gremaud, PRL 100, 033902 (2008)
Crossed diagrams:
Analytical theory of nonlinear coherent backscattering
T. Wellens and B. Gremaud, PRL 100, 033902 (2008)
Crossed diagrams:
→ nontrivial contribution to the→ backscattering peak height:
γC =
∫
d2r
Wℓse−x/ℓsRe
{
C(~r) − e−x/ℓs}
with ℓs = scattering mean-free path,
C(~r) = e−x/ℓs +
∫
d2r′e−|~r−~r′|/ℓs
2πℓs|~r − ~r′|C(~r′)
×(
1 + 2ikℓsmg
~2k2〈|ψ(~r′)|2〉
)
00
crossedcontrib.
ladder contrib.
angle
curr
ent
γC
−→ effective phase shift between the reflected paths
Comparison of the total peak height
0 0.01 0.020
numerical result
analytical prediction
back
scat
tere
dcu
rren
tatθ
=0
peak
dip
nonlinearity g
Coherent backscattering of the condensate
-0.4π -0.2π 0 0.2π 0.4π0
1
2
3
4 g=0g=0.01
curr
ent[
arb.
units
]
angle θ [rad]
Coherent backscattering of the condensate
-0.4π -0.2π 0 0.2π 0.4π0
1
2
3
4 g=0g=0.01g=0.015
curr
ent[
arb.
units
]
angle θ [rad]
Coherent backscattering of the condensate
-0.4π -0.2π 0 0.2π 0.4π0
1
2
3
4 g=0g=0.01g=0.02
curr
ent[
arb.
units
]
angle θ [rad]
Coherent backscattering of the condensate
-0.4π -0.2π 0 0.2π 0.4π0
1
2
3
4 g=0g=0.01g=0.03
curr
ent[
arb.
units
]
angle θ [rad]
−→ permanently time-dependent scattering at g = 0.03
−→ loss of coherence−→ T. Ernst, T. Paul, and P.S., PRA 81, 013631 (2010)
Transport of condensates through 2D billiards
−→ two waveguides connected to a chaotic scattering region−→ (hard walls, flat potential background)
−→ inject condensate in the transverse ground mode−→ of the incident waveguide
Transport of condensates through 2D billiards
−→ introduce effective “magnetic field” within the billiard−→ (via laser beams with finite orbital angular momentum)
G. Juzeli unaset al., PRA 71, 053614 (2005)
Y.-J. Lin et al., PRL 102, 130401 (2009)
Transport of condensates through 2D billiards
−→ introduce effective “magnetic field” within the billiard−→ (via laser beams with finite orbital angular momentum)
G. Juzeli unaset al., PRA 71, 053614 (2005)
Y.-J. Lin et al., PRL 102, 130401 (2009)
−→ dephasing of the coherent backscattering contribution
Weak localization in 2D billiards
Probability for retroreflection into the incident channel:(energy average and variation of obstacle position)
-0.001 0 0.0010.05
0.1
0.15 g=0
magnetic field
Weak localization in 2D billiards
Probability for retroreflection into the incident channel:(energy average and variation of obstacle position)
-0.001 0 0.0010.05
0.1
0.15 g=0g=0.01
magnetic field
Weak localization in 2D billiards
Probability for retroreflection into the incident channel:(energy average and variation of obstacle position)
-0.001 0 0.0010.05
0.1
0.15 g=0g=0.02
magnetic field
Weak localization in 2D billiards
Probability for retroreflection into the incident channel:(energy average and variation of obstacle position)
-0.001 0 0.0010.05
0.1
0.15 g=0g=0.03
magnetic field
Weak localization in 2D billiards
Probability for retroreflection into the incident channel:(energy average and variation of obstacle position)
-0.001 0 0.0010.05
0.1
0.15 g=0g=0.04
magnetic field
Weak localization in 2D billiards
Probability for retroreflection into the incident channel:(energy average and variation of obstacle position)
-0.001 0 0.0010.05
0.1
0.15 g=0g=0.05
magnetic field
Weak localization in 2D billiards
Probability for retroreflection into the incident channel:(energy average and variation of obstacle position)
-0.001 0 0.0010.05
0.1
0.15 g=0g=0.05
magnetic field
−→ signature for weak antilocalization
Semiclassical theory of nonlinear transport
ψ(~r) =
∫
d2r′ G0(~r − ~r′)[
S(~r′) + g|ψ(~r′)|2ψ(~r′)]
with
G0(~r − ~r′) = (linear) Green function within the billiard
S(~r′) = S0δ(x− x0)χn(y′): source amplitude in channel n
χn(y) = transverse eigenmode of channel n in the incident lead
Semiclassical theory of nonlinear transport
ψ(~r) =
∫
d2r′ G0(~r − ~r′)[
S(~r′) + g|ψ(~r′)|2ψ(~r′)]
with
G0(~r − ~r′) = (linear) Green function within the billiard
S(~r′) = S0δ(x− x0)χn(y′): source amplitude in channel n
χn(y) = transverse eigenmode of channel n in the incident lead
Probability for retroreflection:
Rnn =
∣
∣
∣
∣
∫
dyχn(y)ψ(x0, y)
∣
∣
∣
∣
2
Semiclassical theory of nonlinear transport
−→ semiclassical expansion of G0 in terms of trajectories
−→ identify ladder and crossed contributions−→ (diagonal approximation)
−→ apply sum rules within the billiard
−→ solve transport equation for the nonlinear crossed intensity
Semiclassical theory of nonlinear transport
−→ semiclassical expansion of G0 in terms of trajectories
−→ identify ladder and crossed contributions−→ (diagonal approximation)
−→ apply sum rules within the billiard
−→ solve transport equation for the nonlinear crossed intensity
⇒ Semiclassical prediction for the retroreflection probability:
Rnn =1
2Nc
(
1 +C0
1 + (2tDgρC0/~)2
)
with C0(B) =tDtH
1
1 + (B/B0)2
tD = dwell time, Nc = number of open channels in the lead,tH = Heisenberg time, ρ = mean density within the billiard
Weak localization in 2D billiards
Fit of horizontal scale, vertical scale, and dwell time:
-0.001 0 0.0010.05
0.1
0.15
g=0g=0.01g=0.02g=0.03g=0.04
magnetic field
T. Hartmann et al., in preparation
Conclusion
Transport of Bose-Einsteincondensates through 2D disorder:
weak nonlinearity reverts the peak ofcoherent backscattering
Quantitative understanding in terms ofanalytical diagrammatic theory
Transport through chaotic billiards:signature for weak antilocalization
00
curr
ent
angle
g = 0g > 0
0magnetic field
retr
oref
lect
ion
g=0g>0
M. Hartung, T. Wellens, C. A. Muller, K. Richter, P.S.,
PRL 101, 020603 (2008)
T. Hartmann, T. Wellens, C. Petitjean, J.-D. Urbina, K. Richter, P.S.,
in preparation