Bose-Einstein condensates and the thin-shell limit in anisotropic bubble traps Elias J. P. Biral1 Natália S. Móller2 Axel Pelster3 F. Ednilson

2025-05-06 0 0 714.96KB 20 页 10玖币

侵权投诉

Bose-Einstein condensates and the thin-shell limit in

anisotropic bubble traps

Elias J. P. Biral1, Natália S. Móller2, Axel Pelster3, F. Ednilson

A. dos Santos4

santos.ufscar.br

1Instituto de Física de São Carlos, Universidade de São Paulo, 13560-550 São Carlos,

SP, Brazil

2RCQI, Institute of Physics, Slovak Academy of Sciences, Dúbravská Cesta 9,

84511 Bratislava, Slovakia

3Physics Department and Research Center Optimas, Rheinland-Pfälzische Technische

Universität Kaiserslautern-Landau, 67663 Kaiserslautern, Germany

4Department of Physics, Federal University of São Carlos, 13565-905 São Carlos,

SP, Brazil

Abstract. Within the many diﬀerent models, that appeared with the use of cold

atoms to create BECs, the bubble trap shaped potential has been of great interest.

However, the relationship between the physical parameters and the resulting manifold

geometry remains yet to be fully understood for the anisotropic bubble trap physics

in the thin-shell limit. In this paper, we work towards this goal by showing how

the parameters of the system must be manipulated in order to allow for a non-

collapsing thin-shell limit. In such a limit, a dimensional compactiﬁcation takes place,

thus leading to an eﬀective 2D Hamiltonian which relates to up-to-date bubble trap

experiments. At last, the resulting Hamiltonian is perturbatively solved for both

the ground-state wave function and the excitation frequencies in the leading order

of deviations from a spherical bubble trap.

1. Introduction

In the 1990’s, the experimental realization of Bose-Einstein Condensates (BEC) [1, 2] gave rise to

a myriad of both theoretical and experimental studies with contributions ranging from a basic

understanding of the underlying physics of this macroscopic quantum phenomenon to various

applications in particular cases of interest. Among the vast knowledge developed, there is the creation

of bubble trap physics [3–7], which consists of thin-shell traps created using a radiofrequency ﬁeld in

an adiabatic potential based on a quadrupolar magnetic trap.

The idea to work with two-dimensional superﬂuid manifolds soon proved to be appealing to

physicists since the ﬁne tune in the geometry opens new possibilities of physical interest. As a natural

consequence, many experiments appeared in the literature [8–12]. Unfortunately, there are various

technical diﬃculties in creating bubble trap experiments among which is the gravitational sag, i.e.,

the sinking of the BEC atoms into the bottom of the trap. With the current developments, it is

possible to escape this problem working with microgravity either with free-falling experiments on earth-

based laboratories [13,14] or space-based in the International Space Station (ISS) with the Cold Atom

Laboratory (CAL) [15–21]. Up until today, the usual microgravity [22] seems to be the best solution

arXiv:2210.08074v3 [cond-mat.quant-gas] 29 Jan 2024

Bose-Einstein condensates and the thin-shell limit in anisotropic bubble traps 2

for conﬁning atomic gases into shells in order to study its properties, but some new alternatives such

as gravity compensation mechanisms are arising [23,24]. Also, an interesting substitute to the usual

procedure of radio-frequency dressing was proposed for dual-species atomic mixtures [25], which led to

the creation of a BEC on Earth’s gravity [26].

Conﬁnements in three-dimensional shell shaped condensates inspired some theoretical works worth

mentioning here. For instance, in [27], the authors apply analytical methods to investigate the ground

state wave function of a BEC and its collective modes. In [28], both analytical and numerical techniques

are used in order to obtain expansion properties. The interesting paper [29] employs thermodynamic

arguments in order to survey the formation of clusters. The thermodynamics of a BEC on a spherical

shell is analyzed, including the critical temperature, in Refs. [30–32]. The topological hollowing

transition from a full sphere to a thin-shell was studied in [33] and [34], thus ﬁnding some universal

properties. The ground state and collective excitations of a dipolar BEC was considered in [35]. The

general physical relevance of cold atoms on curved manifolds is also addressed in [36]. The contribution

of [37] plays with the idea that the external potential is equivalent to the harmonic trap for a large

radius thin-shell. Universal scaling relations are found for topological superﬂuid transitions in bubble

traps in [38]. And non-Hermitian phase transitions are meticulously described in [39]. Although it

is not the focus of this work, it is also worth citing vortex studies on spherical-like surfaces with no

holes [40–47].

In this paper, we work out an explicit relation between the conﬁnement of the particles in the

thin-shell limit and the geometrical distortion of a bubble trap for a family of conﬁnement potentials.

They are chosen in such a way that the current experiments are included as special cases. All potentials

in this family turn out to exhibit the same angular dependency for the conﬁnement strength. Section 2

describes the mathematical background by deﬁning the concepts in which our theory is developed. In a

second step we calculate in section 3 the harmonic radial frequency in the Gaussian Normal Coordinate

System (GNCS) and expose its resulting angular dependency. Furthermore, the deﬁnition of the thin-

shell limit is discussed in a more rigorous way by elucidating how it depends on the geometrical

distortion of the bubble trap. The general Gross-Pitaevskii Hamiltonian of the system is deduced

using a perturbative approach near the thin-shell limit in section 4. In section 5, special topics of

the spherical shell and the Thomas-Fermi approximation are considered. Finally, in section 6, the

corresponding eﬀective Gross-Pitaevskii equation is perturbatively solved for small distortions from a

spherical bubble trap in order to determine both the ground-state wave function and the oscillation

frequencies.

2. Gaussian Normal Coordinate System

In this section, some preliminary concepts are deﬁned and explained in order to establish the

mathematical background in which our theory is developed. One of such main concepts concerns the

manifold [48, 49] considered here. In this work, we study 2D surfaces embedded into a 3D Euclidean

space. More speciﬁcally, ellipsoidal surfaces [50–52] are considered since they correspond to the bubble

trap potentials in BEC experiments. Therefore, our manifolds are compact, smooth and diﬀerentiable

everywhere. In order to describe the 3D region around the 2D manifold we choose as a suitable

coordinate system, the so-called Gaussian Normal Coordinate System (GNCS) [53,54]. Further features

and particularities on its application can be found in [27].

It is always possible to describe the region around smooth manifolds with the aid of a GNCS.

The main idea is to consider two coordinates x1and x2over the 2D manifold M, also called tangent

coordinates, describing arbitrary points in the manifold. Thus, any point pof this manifold Mis

portrayed by the position vector −→

p(x1, x2). Any point qin such a vicinity of Mcan be represented

by a coordinate x0referred to as the orthogonal coordinate, and a normal unit vector ˆnat the point p

through the following equation

−→

q(x0, x1, x2) = −→

p(x1, x2) + x0ˆn(x1, x2).(1)

Bose-Einstein condensates and the thin-shell limit in anisotropic bubble traps 3

Figure 1. Drawing of the manifold Mas an ellipsoid including the GNCS with

(x0, x1, x2) = (s, ν, ϕ)described with the aid of the prolate spheroidal coordinates

developed in this section.

We deﬁne the geometrical shape of the manifold in question with the prolate spheroidal coordinates

[58,59]. The transformation equations between such a system of coordinates and the Cartesian System

allows us to establish the following family of ellipsoidal surfaces











x=Asin νcos ϕ

y=Asin νsin ϕ

z=A

√1+ϵcos ν

,(ϕ∈[0,2π)

ν∈[0, π).(2)

Here ν=x1and ϕ=x2represent the tangent coordinates, whereas the parameter ϵstands for the

geometrical distortion between an ellipsoid and a sphere according to the equation x2+y2+(1+ϵ)z2=

A2, where Adenotes a quantity analogous to a sphere radius characterizing the overall size of the

ellipsoid.

For each point on the manifold Mit is possible to determine a pair of mutually orthogonal vectors

tangent to the manifold by taking partial derivatives of −→

p(ν, ϕ)







⃗v1(ν, ϕ) = ∂−→

p(ν,ϕ)

∂ν =Acos νcos ϕˆx+Acos νsin ϕˆy−A

√1+ϵsin νˆz,

⃗v2(ν, ϕ) = ∂−→

p(ν,ϕ)

∂ϕ =−Asin νsin ϕˆx+Asin νcos ϕˆy. (3)

In this way the unitary normal vector can be written as

ˆn=⃗v1×⃗v2

|⃗v1×⃗v2|=sin νcos ϕˆx+ sin νsin ϕˆy+√1 + ϵcos νˆz

√1 + ϵcos2ν.(4)

These vectors form an orthogonal basis according to the following properties

ˆn·ˆn= 1, ⃗v1·ˆn= 0, ⃗v2·ˆn= 0, ⃗v1·⃗v2= 0.(5)

Choosing x0=s, the set of coordinates (x0, x1, x2)=(s, ν, ϕ)deﬁnes the GNCS.

The visualization of the coordinates and vectors outlined in this section is exposed in Figure 1.

It shows a drawing of the manifold Mas an ellipsoid including the GNCS. The vector −→

p(ν, ϕ)points

to the point pat the manifold, where s= 0. Also the two tangent vectors ⃗v1(ν, ϕ)and ⃗v2(ν, ϕ), as well

as the unit normal vector ˆn(ν, ϕ)to the manifold at the point pare illustrated.

Now, let us consider the 3D metric tensor Gαβ =∂−→

q(s,ν,ϕ)

∂xα·∂−→

q(s,ν,ϕ)

∂xβ, which has in matrix

Bose-Einstein condensates and the thin-shell limit in anisotropic bubble traps 4

notation according to the properties (5) the typical form within GNCS

Gαβ =







1 0 0

0 cos2ν+sin2ν

1+ϵ! A+s1+ϵ

(1+ϵcos2ν)3/2!2

0 0 sin2ν A+s1

√1+ϵcos2ν!2







.(6)

It is important to realize that in the case of a spherical shell, with ϵ= 0, we recover the result of the

metric tensor for spherical coordinates, where r=A+sdenotes the radial coordinate and νstands for

the polar angle.

3. Thin-shell limit for bubble traps

In this section, we discuss the types of potentials that are relevant to this work. Namely, we consider

a family of 3D potentials which are constant and have their lowest value along the manifold M, and

that have their conﬁnement strength proportional to the geometrical distortion of the ellipsoid. Later

we show how such conditions are consistent with actual experimental potentials.

3.1. Conﬁnement potential in a bubble trap

The initial idea is to introduce a parameter Λ, which controls the strength of the conﬁnement in

the direction perpendicular to the manifold. To this end, we simplify the notation according to

(x0, x1, x2)≡(s, xi)with i= 1,2, so the general potential can be considered as

V(s, xi)=Λ²v(s, xi),(7)

where the limit Λ→ ∞ corresponds to an inﬁnitely tight potential thus deﬁning the thin-shell limit. In

addition, the factor v(s, xi)does not depend on the shell thickness and must be chosen in such a way

that it fulﬁlls the requisites of being constant at the manifold Mand having a vanishing ﬁrst derivative

with respect to sfor s= 0. Notice that due to the dimension of Λ,v(s, xi)does not necessarily have

dimension of energy. In this section, additional considerations on the deﬁnition of the thin-shell limit

will be analyzed by relating it to the geometrical distortion of the shell.

In order to study the vicinity of the manifold M, let us start with a Taylor expansion along the

orthogonal direction

v(s, xi) = K+1

2!s2∂2v(s, xi)

∂s2M

+O(s3),(8)

with K=v|Mbeing a constant at the minimum M, which is characterized by s= 0, and the ﬁrst

derivative in svanishes there. The second derivative of the potential, in the case of the bubble trap,

deﬁnes the harmonic frequency ΛΩ around the vicinity of the shell. Such derivative follows from

∂2v(s, xi)

∂s2M

α,β=1

∂2v(x, y, z)

∂rα∂rβM

nαnβ= ˆn·(∇∇v(x, y, z)|M)·ˆn=mΩ2(x, y, z),(9)

where mdeﬁnes the mass of the particles, r1=x,r2=y,r3=zrepresent the Cartesian coordinates,

and nα,nβdenote the respective components of the normal vector. Observe that Ωdoes not have the

same dimension as the harmonic frequency ΛΩ, due to the prefactor Λ.

In order to take into account the geometry in the usual bubble trap experiments [4,5,8–11,13–21],

let us consider the ﬁnite factor of our potential in a generalized form as

v(x, y, z)≡f(x2+y2+ (1 + ϵ)z2).(10)

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

Bose-Einsteincondensatesandthethin-shelllimitinanisotropicbubbletrapsEliasJ.P.Biral1,NatáliaS.Móller2,AxelPelster3,F.EdnilsonA.dosSantos4santos.ufscar.br1InstitutodeFísicadeSãoCarlos,UniversidadedeSãoPaulo,13560-550SãoCarlos,SP,Brazil2RCQI,InstituteofPhysics,SlovakAcademyofSciences,DúbravskáCesta9,8...

展开>> 收起<<

Bose-Einstein condensates and the thin-shell limit in anisotropic bubble traps Elias J. P. Biral1 Natália S. Móller2 Axel Pelster3 F. Ednilson.pdf

共20页,预览4页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Bose-Einstein condensates and the thin-shell limit in anisotropic bubble traps Elias J. P. Biral1 Natália S. Móller2 Axel Pelster3 F. Ednilson

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: