coherent backscattering of bec in 2d disorder and chaotic...

Coherent backscattering of BEC in 2D disorderand chaotic billiards

Peter Schlagheck

15.12.2010

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

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θ

-0.4π -0.2π 0 0.2π 0.4π0

1

2

3

4

back

scat

tere

dcu

rren

t

angle θ

→ coherent backscattering



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→ 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)



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→ 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



0magnetic field

0

0.5

reflection probability (5 open channels)

dephasing in the presence of a magnetic field



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


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


−→ 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)


Definition of the spatial 2D geometry:

source

periodic boundary conditions


absorbing boundaryab

sorb

ing

boun

dary

disorder(Gaussian)



source




sorb

ing

boun

dary

disorder(Gaussian)

h(x)

x

g(x)



source




sorb

ing

boun

dary

disorder(Gaussian)

BEC


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]


-0.4π -0.2π 0 0.2π 0.4π0

1

2

3

4 g=0g=0.0025

curr

ent[

arb.

units

]

angle θ [rad]


-0.4π -0.2π 0 0.2π 0.4π0

1

2

3

4 g=0g=0.005

curr

ent[

arb.

units

]

angle θ [rad]


-0.4π -0.2π 0 0.2π 0.4π0

1

2

3

4 g=0g=0.0075

curr

ent[

arb.

units

]

angle θ [rad]


-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




ψ(~r) = ψ0eikx +

∫

d2r′ G0(~r − ~r′)[

Vdis(~r′) + g|ψ(~r′)|2

]

ψ(~r′)

−→ diagrams:ψ

ψ

ψ∗

V g




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



Ladder diagrams:



Ladder diagrams:

→ no net effect of nonlinearity→ on 〈|ψ(~r)|2〉disorder av.



Crossed diagrams:



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


-0.4π -0.2π 0 0.2π 0.4π0

1

2

3

4 g=0g=0.01

curr

ent[

arb.

units

]

angle θ [rad]


-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]


-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]


-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


−→ 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)


−→ 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



-0.001 0 0.0010.05

0.1

0.15 g=0g=0.01

magnetic field



-0.001 0 0.0010.05

0.1

0.15 g=0g=0.02

magnetic field



-0.001 0 0.0010.05

0.1

0.15 g=0g=0.03

magnetic field



-0.001 0 0.0010.05

0.1

0.15 g=0g=0.04

magnetic field



-0.001 0 0.0010.05

0.1

0.15 g=0g=0.05

magnetic field



-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


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 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 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


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

coherent backscattering of bec in 2d disorder and chaotic...

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