Fast transport of Bose-Einstein condensates in anharmonic traps Jing Li1Xi Chen2 3and Andreas Ruschhaupt1 1Department of Physics University College Cork Cork Ireland T12 H6T1

2025-04-27 0 0 424.25KB 8 页 10玖币

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Fast transport of Bose-Einstein condensates in anharmonic traps

Jing Li,1, ∗Xi Chen,2, 3 and Andreas Ruschhaupt1

1Department of Physics, University College Cork, Cork, Ireland, T12 H6T1

2Department of Physical Chemistry, University of the Basque Country UPV/EHU, Apartado 644, 48080 Bilbao, Spain

3EHU Quantum Center, University of the Basque Country UPV/EHU, 48940 Leioa, Spain

(Dated: October 18, 2022)

We present a method to transport Bose-Einstein condensates (BECs) in anharmonic traps and in the presence

of atom-atom interactions in short times without residual excitation. Using a combination of a variational ap-

proach and inverse engineering methods, we derive a set of Ermakov-like equations that take into account the

coupling between the center of mass motion and the breathing mode. By an appropriate inverse engineering

strategy of those equations, we then design the trap trajectory to achieve the desired boundary conditions. Nu-

merical examples for cubic or quartic anharmonicities are provided for fast and high-ﬁdelity transport of BECs.

Potential applications are atom interferometry and quantum information processing.

I. INTRODUCTION

The accurate manipulation of ultracold atoms is a key pre-requisite to implement quantum technologies within atomic, molec-

ular and optical science [1]. In particular, the transport of individual atoms and of thermal or Bose-condensed clouds using

moving traps has been demonstrated in many experiments [2–12] for diﬀerent goals in quantum information processing and

metrology. In all quantum technologies, preserving quantum coherence and achieving high ﬁnal ﬁdelities in short times is of

crucial importance. One possibility is called shortcuts to adiabaticity (STA) [13,14] which provides a toolbox to control both

the internal and external degrees of freedom of a quantum system in faster-than adiabatic times.

Various shortcuts to adiabatic transport have been proposed: Lewis-Riesenfeld invariant-based inverse engineering [15–19],

enhanced STA scheme [20–22], the Fourier optimization [23], fast-forward scaling method [24,25], and the counter-diabatic

driving [26] have been theoretically put forward, and experimentally demonstrated for various systems [7,10,11,27]. The

possibility to operate with short times not only reduces the sensitivity to low frequency noise, but also allows for improved

measurement statistics in the total time available for the experiment.

Diﬀerent approaches for transporting particles have been implemented. Neutral atoms have been transported as Bose-Einstein

condensates (BECs) [2], thermal atomic clouds [28], or individually [5], using magnetic or optical traps. The commonly used

traps for ultracold atoms based on electromagnetic ﬁelds are never perfectly harmonic. The weak cubic anharmonicity plays

a role when a BEC is transported perpendicular to the atom chip surface [29]. The quartic anharmonicity is signiﬁcant when

approximating the potential of an optical tweezers for transport [7,16]. Thus cancelling the anharmonic contributions of the

trapping potential is vital for useful control schemes and is already a diﬃcult technical challenge for a static trap [30].

Anharmonicities can have an important impact on the dynamics as observed in atom cooling [31], collective modes [32], or

wave packet dynamics [33]. In most cases, the anharmonic traps are considered as a perturbation of a harmonic one. Perturbation

theory has been used to design shortcut protocols for expansion/compression [34] and transport [35]. Of course the results are

limited by the premises of perturbation theory, i.e., by small anharmonicities. Considering a non-perturbative scenario is thus of

much interest.

In this paper, we propose to inverse engineer rapid and robust transport of an interacting Bose-Einstein condensate (BEC) in

anharmonic traps using a variational approach. The method relies on a variational formulation of the dynamics to derive a set of

coupled Ermakov-like and Newton-like equations, from which the trap trajectory is inferred interpolating between the desired

boundary conditions. In Sec. II, we explain the variational formalism. In Sec. III, we work out the explicit solutions for quartic

and cubic anharmonicities of the conﬁning potential, and illustrate the eﬃciency of the method with various numerical examples.

In section IV, we will discuss the results.

II. MODEL, HAMILTONIAN AND METHOD

For a cigar-shaped trap with strong transverse conﬁnement, e.g. ω⊥>> ω, it is appropriate to consider a 1D dimensionless

formula by freezing the transverse dynamics to the respective ground state and integrating over the transverse variables[36]. The

∗jli@ucc.ie

arXiv:2210.03788v1 [cond-mat.quant-gas] 7 Oct 2022

(a)

(b)

FIG. 1. Cubic (a) and quartic (b) anharmonic potentials (red solid lines) compared to the harmonic counterparts (blue dashed lines). The

ground states are plotted for diﬀerent potentials.

eﬀective atomic interaction is denoted by g=2asω⊥N/ωaho, with asthe interatomic scattering length and aho =√~/(mω). The

resulting dimensionless form of Gross-Pitaevskii equation (GPE) [37] can be written as

i∂ψ(x,t)

∂t="−1

∂2

∂x2+V(x,t)+g|ψ(x,t)|2#ψ(x,t),(1)

where

V(x,t)=1

2[x−x0(t)]2+κ

3! [x−x0(t)]3+λ

4! [x−x0(t)]4,(2)

where ψ(x,t) is the axial wavefunction of the condenstate with normalization condition R+∞

−∞ |ψ(x,t)|2dx =N. The attractive and

repulsive interactions are denoted by g<0 and g>0, respectively. The axial harmonic trap frequency is ω. The potential

center x0(t) is time-dependent for transport. Note that the potential in Eq. (2) consists two types of anharmonicities, one is cubic

(κ≥0) and the other is quartic (λ≥0) anharmonicity, which is shown in Fig. 1.

To apply the variational approach we ﬁrst deﬁne an ansatz for the wave function with a few free parameters and evaluate the

Lagrangian density. The minimization of the total Lagrangian with respect to the free parameters provides equations of motion

for the free parameters [38]. This approach is equivalent to a moment method [39].

We assume a general Gaussian ansatz,

ψ(x,t)=A(t) exp "−(x−xc(t))2

2a(t)2#exp hib(t)(x−xc(t))2+ic(t)(x−xc(t)) +iφ(t)i(3)

where the time-dependent parameters A(t), a(t), b(t), c(t), and φ(t) represent respectively the amplitude, width, chirp, velocity,

and global phase. The wave function center of mass is xc(t). In the following we omit tin those variables for simpliﬁcation. The

normalization condition yields A=qN/(a√π).

The Lagrangian density which corresponds to Eq. (1) reads [38]

L=i

2 ∂ψ

∂tψ∗−∂ψ∗

∂tψ!−1

2

∂ψ

∂x

−g

2|ψ|4−V(x)|ψ|2.(4)

Inserting the ansatz (3) into Eq. (4), we ﬁnd an eﬀective Lagrangian [38] by integrating the Lagrangian density over the whole

coordinate space, L=R+∞

−∞ Ldx. The Euler-Lagrange minimization is performed over Land with respect to the free parameters

and the conditions δL/δξ =0 where ξ=a,b,cor xc. Four coupled equations result for (˙a,˙

b,˙xc,˙c), are given by

˙a=2ab,(5)

b=1

2a4−1

2h1−4κ(x0−xc)+18λ(x0−xc)2i−2b2+gN

2√2πa3,(6)

˙c=(1 +18λa2)(x0−xc)−2κ(x0−xc)2+12λ(x0−xc)3−κa2,(7)

˙xc=c,(8)

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FasttransportofBose-EinsteincondensatesinanharmonictrapsJingLi,1,XiChen,2,3andAndreasRuschhaupt11DepartmentofPhysics,UniversityCollegeCork,Cork,Ireland,T12H6T12DepartmentofPhysicalChemistry,UniversityoftheBasqueCountryUPV/EHU,Apartado644,48080Bilbao,Spain3EHUQuantumCenter,UniversityoftheBasqueCount...

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Fast transport of Bose-Einstein condensates in anharmonic traps Jing Li1Xi Chen2 3and Andreas Ruschhaupt1 1Department of Physics University College Cork Cork Ireland T12 H6T1

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