Transport of Bose-Einstein Condensates through Aharonov-Bohm rings

Transport of Bose-Einstein Condensates through

Aharonov-Bohm rings

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 mag-netic flux φ. We specifically investigate the effects of a small atom-atom contact interaction strength on the AB oscilla-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. The latter allows for the de-scription of incoherent transport and corresponds to a classical sampling of the evolution of the quantum bosonic many-body state through effective GP trajectories. We find that reso-nant transmission peaks move with an increasing interaction strength and can be suppressed for sufficiently strong interac-tion. We also observe that the coherent transmission blockade due to destructive interference at the AB flux φ = π is very robust with respect to the interaction strength.

Aharonov-Bohm rings

• Toro¨ıdal optical dipole trap

Sheet beam Laguerre-Gaussian beam LG1₀ [1] Vlatt x y [2]

• 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)

• Discretisation of 1D space • Bose-Hubbard system [3] φ δ α_S S Interaction κ(t) α α′ J_αα′ • Hamiltonian ˆ H = X α∈Z E_δˆa†_αˆa_α + X α′

J_αα′ˆa†_αˆa_α + g_αˆa†_αˆa†_αˆa_αˆa_α

! + κ(t)ˆa†_α Sˆb + κ ∗_(t)ˆa α_Sˆb† + µˆb†ˆb with : • Jαα′ = − E_δ 2 (

1 if site α next to site α′

0 otherwise ,

• ˆaα and ˆa†_α the annihilation and creation

operators at site α,

• ˆb and ˆb† the annihilation and creation operators

of the source with chemical potential µ,

• 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,

• gα =

(

g if site α within the ring

0 otherwise .

References

[1] A. Ramanathan et al. Phys. Rev. Lett. 106, 130401 (2011). [2] L. Amico et al. Phys. Rev. Lett. 95, 063201 (2005).

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

[4] Y. Aharonov and D. Bohm, Phys. Rev. 115, 485-491 (1959). [5] Y. Imry and R.A. Webb, Scient. Am. 260, 56 (1989).

[6] R.A. Webb et al. Phys. Rev. Lett. 54, 2696 (1985).

Aharonov-Bohm effect

• Potentials act on charged particles even if all

fields vanish [4] Electron gun Plate Screen Magnet Shield [5]

• Additional phase shift of the electron

wavefunction due to the vector potential A

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

with φ₀ = h/e _{≈ 4.13566727 · 10}−15 Wb the

magnetic flux quantum

• Interference pattern shifted when a shielded

magnet is added

• 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 + t_2|2 = _|t_1|2 + _|t_2|2 + 2_|t_{1| · |}t_{2| cos ∆ϕ}

with period φ₀

Truncated Wigner Method

• Sampling of the initial quantum state by

classical field _{ψα(t₀)_{}α∈{S,0,±1,±2,...}}.

• Time propagation according to classical

(slightly modified) GP equation.

• Sampling the initial quantum state at t₀ [3]

◦ Empty waveguide WG({ψα, ψ_α∗}, t0) = Y α  2 π  e−2|ψα|2 • ψα(t = t0) = 1 2 (Aα + iBα)

• A_α and B_α: real gaussian random variables

Aα = Bα = Aα′Bα = 0

Aα′Aα = Bα′Bα = δα′,α

◦ Source populated with _|ψS0 _|2 = N _{≫ 1 atoms}

WS(ψS, ψ_S∗ , t0) =

 2 π



e−2|ψS−ψ_S0|2

• Relative uncertainty negligible

• ψ_S(t = t0) =

√

N ≈ ψS(t) for |κ| → 0

• Equation of propagation for the classical field

i~∂ψα ∂t = (Eδ − µ) ψα + X α′ J_αα′ψ_α′ + g_{α |ψα|}2₋₁
ψ_α + κ(t)√N δ_α,αS

• Compute mean values of observables by

average over GP trajectories

◦ total density n_α = _|ψα|2 ₋ 1

2

◦ coherent density ncoh_α = ψ_α2

◦ incoherent density nincoh_α = n_{α − n}coh_α

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

Aharonov-Bohm oscillations

• Interferences induced by the AB flux φ [6].

• 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

• Clear signature of the AB effect.

• No atom-atom interaction so far.

Interaction effects

• Incoherent transmission even at φ = π

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

• Inhibition of perfect transmission (see [3])

• Suspension of transmission blockade at φ = π

due to incoherent atoms created within the ring

Total density

Incoherent density φ = π/2

φ = π µ/E_δ = 1, g/µ = 0.1

• 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 = 0.125 µ/E_δ = 1 T ra n sm is si o n g 0 0 0.075 0.15 0.1 0.2 Total Incoh.

• Destructive interferences at φ = π, which is

robust on the variations of g.

• More incoherent particles created within the

ring as g increases.

• Transmission totally incoherent at φ = π, for

all g > 0.

Perspectives

• Disorder within the ring

• Continuous limit δ _{→ 0}

• Finite temperature for the reservoirs

φ

Interaction

• Boson sampling