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Aharonov-Bohm oscillations of bosonic matter-wave beams in the presence of disorder and interaction

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Aharonov-Bohm oscillations of bosonic matter-wave beams in

the presence of disorder and interaction

R. Chr´etien, J. Dujardin, C. Petitjean, and P. Schlagheck

D´epartement de Physique, University of Liege, 4000 Li`ege, Belgium

Abstract

We study the one-dimensional (1D) transport properties of an ultracold gas of Bose-Einstein condensed atoms through Aharonov-Bohm (AB) rings. Our system consists of a Bose-Einstein condensate (BEC) that is outcoupled from a mag-netic trap into a 1D waveguide which is made of two semi-infinite leads that join a ring geometry exposed to a synthetic magnetic flux φ. We specifically investigate the effects both of a disorder potential and of a small atom-atom contact interac-tion strength on the AB oscillainterac-tions. The main numerical tools that we use for this purpose are a mean-field Gross-Pitaevskii (GP) description and the truncated Wigner (tW) method. We find that a correlated disorder suppress the AB oscillations leaving thereby place to weaker amplitude, half period oscil-lations on transmission, namely the Aronov-Al’tshuler-Spivak (AAS) oscillations. The competition between disorder and in-teraction leads to a flip of the transmission at the AB flux φ = π. This flip could be a possible preliminary signature of an inversion of the coherent backscattering (CBS) peak. Our study paves the way to an analytical description of the inversion of that peak.

Aharonov-Bohm rings

• Toro¨ıdal optical dipole trap

[A. Ramanathan et al. PRL 106, 130401 (2011)] [L. Amico et al. PRL 95, 063201 (2005)] Sheet beam Laser beam V latt x y

Intersection of two red-detuned beams

• Connection to two waveguides

Theoretical description

• Ring geometry connected to two semi-infinite

homogeneous leads

Perfect condensation of the reservoir (T = 0 K)

with chemical potential µ

• Discretisation of a 1D Bose-Hubbard system

[J. Dujardin et al. Phys. Rev. A 91, 033614 (2015)]

φ

δ αS

S

Interaction and disorder

κ(t) α α′ R L L −1 0 n n + 1 • Hamiltonian ˆ H = ˆHL + ˆHLR + ˆHR + ˆHS where ˆ HL = X α∈L h Eδˆa†αˆaα − Eδ 2 

ˆa†α−1ˆaα + ˆa†α+1ˆaα i ˆ

HLR = −Eδ

2



ˆa†−1ˆa0 + ˆa†0ˆa−1 + ˆa†nˆan+1 + ˆa†n+1ˆan ˆ HR = h X α∈R (Eδ + Vα) ˆa†αˆaα − Eδ 2 

ˆa†α−1ˆaα + ˆa†α+1ˆaα + gˆa†αˆa†αˆaαˆaαi

ˆ

HS = κ(t)ˆa†αSˆb + κ∗(t)ˆb†ˆaαS + µˆb†ˆb with :

• ˆaα (ˆb) and ˆa†α (ˆb†) the annihilation and

creation operators at site α (of the source),

• Eδ ∝ 1/δ2 the on-site energy,

• Vα the disorder potential at site α,

• g the interaction strength,

• N → ∞ the number of Bose-Einstein

condensed atoms within the source,

• κ(t) → 0 the coupling strength, which tends to

zero such that N |κ(t)|2 remains finite.

Aharonov-Bohm effect

Potentials act on charged particles even if all

fields vanish

[Y. Aharonov and D. Bohm, PR 115, 485-491 (1959)] [Y. Imry and R.A. Webb, Scient. Am. 260, 56 (1989)]

Electron gun Plate Screen Magnet Shield

• Interference pattern shifted due to the presence

of vector potential A with depashing

∆ϕ = k∆l + e ~ I A · dl | {z } φ = k∆l + 2π φ φ0 | {z } Φ

with φ0 = h/2e the magnetic flux quantum

Same effect within a two-arm ring

φ

1

2

Oscillations in transport properties due to

interferences of partial waves crossing each arm of the ring

Transmission periodic w.r.t. the AB flux φ

T = |t1 + t2|2 = |t1|2 + |t2|2 + 2|t1| · |t2| cos ∆ϕ

with period φ0

Aharonov-Bohm oscillations

Periodic transmission oscillation with φ.

T ra n sm is si on 0 Φ π/2 π 3π/2 2π 0 0.5 1 µ/Eδ = 1, g = 0 Total density Total density No interaction

• Clear signature of the AB effect

Incoherent transmission when g 6= 0

T ra n sm is si on Φ π/2 π 3π/2 2π 0 0 0.5 1 Total Coh. Incoh. With interaction µ/Eδ = 1, g/µ = 0.1 GP

Resonant transmission peaks move with g and

disappear if g is strong enough

T ra n sm is si on Φ π/2 π 3π/2 2π 0 0 0.5 1 g/µ = 0 g/µ = 0.05 g/µ = 0.1 g/µ = 0.15 µ/Eδ = 1 T ra n sm is si o n g 0 0 0.075 0.15 0.1 0.2 Total Incoh. With interaction

More incoherent particles created as g ↑

Higher order interferences

Presence of higher harmonics of weak intensity

• Diagrammatic approach of the problem

... . ... . . .... . .... . . .. . . ... .... ...... . .... . . . .... . . .... . . . . ... . . .. . . .. . . ... ... . . . .. . . . . . . . . . . . . . . . . (a) (b) (c) φ φ φ

[Ihn T., Semiconductor nanostructures, Oxford (2010).]

The reflection probability is given by

R = r0 + r1eiΦ + r1e−iΦ + . . . 2

= |r0|2 + |r1|2 + . . . (1)

+ 4|r0| · |r1| cos Λ cos Φ + . . . (2)

+ 2|r1|2 cos (2Φ) + . . . (3)

with Λ the disorder-dependent phase accumulated after one turn with φ = 0.

(1) no Φ-dependence, classical contributions

(2) Φ-periodicity, AB contribution, damped to zero

when averaged over the disorder

(3) Φ/2-periodicity, AAS contribution, robust to

averages over the disorder

AAS oscillations

Averages over the disorder suppress

Aharonov-Bohm oscillations T ra n sm is si on (V = 0) T ra n sm is si on (V 6= 0) Φ π/2 π 3π/2 2π0.35 0.45 0.55 0 0 0.2 0.4 0.6 0.8 1 µ/Eδ = 0.75, g/µ = 0 Disorder No disorder

• Appearance of Φ/2 periodic oscillations :

Altshuler-Aronov-Spivak oscillations

What happens if we set a weak interaction ?

T ra n sm is si on Φ Φ π/2 π/2 π 3π/2 2π 0 π 3π/2 2π 0 µ/Eδ = 0.75, V = Eδ/2 0.4 0.5 0.6 0.7 0.8 g/µ = 0 g/µ = 0.003 g/µ = 0.03 0.48 0.51 0.54

The oscillations amplitude is reduced

The minimum at Φ = π becomes a maximum !

Towards coherent backscattering

Same origin for CBS and AAS

[E. Akkermans et al., PRL 56, 1471 (1986)]

Constructive wave interference between

reflected classical paths and their time-reversed counterparts angle t 0 1 2 3 4 -0.4p0 -0.2p 00 0.2p 0.4p 1 2 3 4 −0.4π −0.2π 0.2π 0.4π angle θ b ac ks ca tt er ed cu rr en t

Recent verification with BEC

[F. Jendrzejewski, et al., PRL 109, 195302 (2012)]

Inversion in the presence of nonlinearity (2D)

[M. Hartung, et al., PRL 101, 020603 (2008)]

Analytical calculations with our 1D model

more feasible

Computational resources have been provided by the Consortium des Equipements de Calcul Intensif (C´ECI), funded by the Fonds de la Recherche Scientifique de Belgique (F.R.S.-FNRS) under Grant No. 2.5020.11

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