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

. 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.


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][4][5][6][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][9][10][11][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 earthbased laboratories [13,14] or space-based in the International Space Station (ISS) with the Cold Atom Laboratory (CAL) [15][16][17][18][19][20][21].Up until today, the usual microgravity [22] seems to be the best solution 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][31][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][41][42][43][44][45][46][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 thinshell 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.

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][51][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 x 1 and x 2 over the 2D manifold M , also called tangent coordinates, describing arbitrary points in the manifold.Thus, any point p of this manifold M is portrayed by the position vector − → p (x 1 , x 2 ).Any point q in such a vicinity of M can be represented by a coordinate x 0 referred to as the orthogonal coordinate, and a normal unit vector n at the point p through the following equation Figure 1.Drawing of the manifold M as an ellipsoid including the GNCS with (x 0 , x 1 , x 2 ) = (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 Here ν = x 1 and ϕ = x 2 represent the tangent coordinates, whereas the parameter ϵ stands for the geometrical distortion between an ellipsoid and a sphere according to the equation , where A denotes a quantity analogous to a sphere radius characterizing the overall size of the ellipsoid.
For each point on the manifold M it is possible to determine a pair of mutually orthogonal vectors tangent to the manifold by taking partial derivatives of − → p (ν, ϕ) In this way the unitary normal vector can be written as These vectors form an orthogonal basis according to the following properties Choosing x 0 = s, the set of coordinates (x 0 , x 1 , x 2 ) = (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 M as an ellipsoid including the GNCS.The vector − → p (ν, ϕ) points to the point p at the manifold, where s = 0. Also the two tangent vectors ⃗ v 1 (ν, ϕ) and ⃗ v 2 (ν, ϕ), as well as the unit normal vector n(ν, ϕ) to the manifold at the point p are illustrated.Now, let us consider the 3D metric tensor , which has in matrix notation according to the properties (5) the typical form within GNCS 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 + s denotes the radial coordinate and ν stands for the polar angle.

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.

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 (x 0 , x 1 , x 2 ) ≡ (s, x i ) with i = 1, 2, so the general potential can be considered as where the limit Λ → ∞ corresponds to an infinitely tight potential thus defining the thin-shell limit.In addition, the factor v(s, x i ) 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 M and having a vanishing first derivative with respect to s for s = 0. Notice that due to the dimension of Λ, v(s, x i ) 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 with K = v| M being a constant at the minimum M , which is characterized by s = 0, and the first derivative in s vanishes 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 where m defines the mass of the particles, r 1 = x, r 2 = y, r 3 = z represent the Cartesian coordinates, and n α , n β denote the respective components of the normal vector.Observe that Ω does 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][9][10][11][13][14][15][16][17][18][19][20][21], let us consider the finite factor of our potential in a generalized form as In this case the geometry of the potential is well established to be an ellipsoidal surface, which becomes spherical for ϵ = 0. Thus, our manifold M is characterized by With such a general form, let us explicitly calculate mΩ 2 (s, x i ).The derivatives can be written in vector form as ∇v(x, y, z By working out the second derivatives as well, we read off from (9) that mΩ²(x, y, z which reduces in terms of the GNCS to This gives the final form for the harmonic frequency ΛΩ considering our generalized potential in form of an ellipsoidal surface with the appropriate constraints.Therefore, the confinement strength turns out to be proportional to the geometrical distortion of the ellipsoid through the dependence on the parameter ϵ.Moreover, the dependence on the angle ν establishes that the confinement varies from the poles to the equator of the ellipsoid.Apart from the mentioned conditions, the function f can be quite general, only with the natural assumption that f ′′ (A 2 ) > 0. Now, some careful considerations must be made for the thin-shell limit.For infinitely tight potentials, we know in advance, that the motion of particles along the normal direction is restricted to the ground state of an harmonic oscillator with frequency ΛΩ(x i ) which is described by a Gaussian wave function with width α = ℏ/mΛΩ.In previous works, mainly devoted to the spherical case, some authors defined the thin-shell limit as a rather generic situation, where the radius R of the sphere-shaped trap is much larger than the thickness α of the shell.Since these are not the only length scales in this system, different results can be obtained by either considering R → ∞ or α → 0. For example in [47] the authors considered the situation where R → ∞.In the numerical work [31], the authors made some qualitative comparisons between α and R. In some papers [33][34][35], the ratio R/α is chosen to be as large as possible without more detailed considerations.The objective in this paper is to consider the specific case α → 0, i.e., the case where α is much smaller than any other length scale of the system.This constitutes a more precise definition of a thin-shell limit, which is equivalent to consider Λ → ∞.
As already mentioned, in the thin-shell limit we expect the motion in the normal direction to be confined to the ground state of a harmonic oscillator with frequency ΛΩ.Since for the ellipsoidal case, Ω is space dependent, every particle in the system will experience a site-dependent energy due to its confinement.Such an energy is provided by the ground-state energy of a harmonic oscillator with frequency ΛΩ, i.e., Here we see that for ϵ = 0, this energy diverges for Λ → ∞, thus adding a uniform infinite potential which does not have any consequence for the dynamics of the particles moving in the manifold.However for ϵ ̸ = 0 the situation changes considerably since the particles would be subjected to an infinite spacedependent potential.Let us for example consider the energy difference between the poles and the equator ∆E = E(ν = 0) − E(ν = π/2), for which we find ∆E ∝ Λ for any non-vanishing ϵ, which means that it diverges in the thin-shell limit.This would then induce a complete localization of particles either at the poles or at the equator, depending on the sign of ϵ.Thus, in order to restrict our limit to physically acceptable situations, we must consider only the case of small geometrical distortions where ε is considered to be finite.By taking into account (13), the induced energy difference then becomes which is finite in the thin-shell limit.For arbitrary values of ν, Eq. ( 12) becomes Thus, we recognize that the energy per particle generated due to the compactfication process separates into one physically irrelevant infinite site-independent part and a finite site-dependent part.It means that, in the thin-shell limit, each particle is effectively subjected to a compactification potential given by In experimental terms, the infinitesimal eccentricity ϵ = Λ −1 ε corresponds to the situation, where the difference between the larger axis and the smaller axis of the ellipsoid is of the same order of the Gaussian width α.This means that, in order to avoid that the particles collapse to either the poles or the equator, the ellipsoidal eccentricity must be kept within such bounds.
In the next subsections, let us consider an experimental realization of the general potential (10) as a generic example where the theory developed in here can be realized.

Confinement potential in experiments
In this subsection, we apply the calculations to a particular confinement potential.Note that there are some variations of this potential in the literature that differ in the way the potential is defined, for instance, by including gravity effects or considering frequency anisotropy [4,5].
Let us consider here the more concrete experimental example in [8] as an application of our theory, where Here ℏ△ = ℏω rf − V 0 denotes the rf detuning with respect to the rf transition at the center of the magnetic trap, and Ω RF = gµ B B 1 /2ℏ stands for the Rabi frequency of the magnetic field B 1 cos(ω rf t) with the Landé factor g = 1/2 and the Bohr magneton µ B , whereas ϵ represents the geometrical distortion of the ellipsoid.This potential has a local minimum provided that V trap (x, y, z) = ℏ△ for the state Let us express the potential according to Eq. ( 7), i.e., which defines a function f according to (10), where Λ 2 equals to ω 2 in this particular case.The thinshell limit for such a potential can be obtained by considering ω → ∞, △ → ∞, and Ω RF → ∞, while the radii ℏ△/ω 2 and ℏΩ RF /ω 2 are kept finite.Also ϵω must be kept finite in order to prevent the collapse of the condensate.The expression for the harmonic frequency ΛΩ according to (11) and considering the GNCS with A = 2ℏ△/mω 2 reads In more experimental terms, the range of parameters corresponding to the thin-shell limit occur for α ≪ A. Thus, the thin-shell limit corresponds to the inequality which represents a condition involving all three frequencies ∆, ω, Ω RF of the bubble trap potential (19).Conversely, in order to prevent the collapse of the condensate, we must also have ϵA ∼ α.This leads to the eccentricity which is small due to the inequality (20).Within the thin-shell limit for this particular experimental case Eq. ( 7) with the aid of (8) becomes And the corresponding compactification potential follows from Eq. ( 16) Such expressions confirm that the general theory presented here is well suited to deal with the already created experimental bubble-trap potentials.

Effective Hamiltonian in the thin-shell limit
In this section, we will consider the thin-shell limit for interacting particles.Let us begin with the usual 3D Gross-Pitaevskii Hamiltonian written in Cartesian coordinates, where V stands for the overall potential and g int represents the interaction parameter.The total number of particles is given by Rewriting the Hamiltonian (24) in the GNCS by using the Laplace-Beltrami form of the Laplace operator yields approximately with the abbreviation dx i = dx 1 dx 2 .Here the reduced covariant 2D metric turns out to be which has only the elements corresponding to the tangent coordinates.The limits of the s integral are chosen in such a way that both (ΛΩms -2 )/ℏ ≫ 1 and (ΛΩms + 2 )/ℏ ≫ 1.This assures that the difference between ( 24) and ( 26) decays exponentially for Λ → ∞ since Ψ ∼ e −[ΛΩ(x i )ms 2 ]/2ℏ in the thin-shell limit.
In order to simplify our calculations, we follow [27] and consider the alternative wave function Ψ Ψ(s, where |g 0 (x i , ϵ)| = |g(0, x i , ϵ)|.Note that the normalization condition 25 for Ψ then reduces to This allows us to use the 2D s-independent Jacobian |g 0 | also for the Hamiltonian where we have introduced γ(s, x i , ϵ) = |g(s, x i , ϵ)|/|g 0 (x i , ϵ)|.As already mentioned, in the thin-shell limit we expect Ψ ∼ e −(ΛΩms 2 )/2ℏ , i.e., Ψ depends implicitly on Λ.In order to make such a dependency explicit, while maintaining the normalization of Ψ as well as to keep the interaction term finite in the limit Λ → ∞, we perform the following scale transformations that gives us some extra control over the Taylor expansions to be performed inside the integrals.These rescaled quantities are the normal direction s, the wave function ψ, and the particle interaction g int .
Introducing the abbreviations γ 1 = ∂γ/∂s, γ 2 = ∂ 2 γ/∂s 2 , we obtain together with ( 7) and (31) for the Hamiltonian (30) Now we can expand both v and γ in a power series with respect to Λ −1/2 which yields Note that we have and according to Appendix A we conclude that Note that the presence of the harmonic frequency in ( 11) considering ( 13) leads to mΩ 2 = 4f ′′ (A 2 )A 2 + Λ −1 ε4f ′′ (A 2 )A 2 cos 2 ν = mΩ 2 0 + Λ −1 εmΩ 2 0 cos 2 ν, where Substituting these equations and neglecting exponentially decaying contributions, the Hamiltonian can be written as ), where the respective terms are sorted according to decreasing contributions with with limits for the integrals in u being −∞ to ∞.Since H 2 turns out to be a simple additive constant, our analysis effectively starts with H 1 .
Note that H 1 lacks an interaction term and essentially corresponds to the harmonic oscillator Hamiltonian whose spectrum is given by the linear eigenvalue equation Therefore, at order Λ 0 the degenerate eigenfunctions of 41 are given by where H n denote the Hermite polynomials, ψ n⊥ represent the harmonic oscillator eigenstates, and ξ l stand for arbitrary functions of x i obeying the condition l dx 1 x 2 |g 0 |ξ * l (x i )ξ l (x i ) = N .The dominant contribution to the energy spectrum of system reads Therefore, in the thin-shell limit, where we have Λ → ∞, the system acquires infinitely separated energy bands.This implies that, physically, only the lowest energy band with energy E 1 = E (1) 0 = N ΛℏΩ 0 /2 will be populated with particles, i.e., ξ l = 0 for l ̸ = 0.In particular, the normalization condition for ξ 0 becomes Now, let us deal with the effects of the H 1/2 contribution to the total Hamiltonian.Similar to H 1 , also H 1/2 acts only in the subspace defined by the variable u.This means that it will induce corrections of order Λ −1/2 to ψ n⊥ as well as corrections of order Λ 1/2 to the energies (43).Such corrections of order Λ −1/2 to ψ n⊥ can be neglected in the thin-shell limit, while the contribution with power Λ 1/2 to (43) vanishes since it involves the infinite integral of an odd function of u according to Although the contribution to the energy with power Λ 1/2 vanishes, an application of second-order perturbation theory considering H 1/2 as a perturbation over H 1 shows that contributions with power Λ 0 to (43) do not vanish.In particular the correction Further contributions to (43) due to H 1/2 are of order Λ −1/2 and are negligible in the thin-shell limit.Finally, we must consider the effect of H 0 for the energy spectrum and eigenstates.The first thing to observe is that H 0 also acts over the subspace defined by the coordinates x i , which implies that H 0 is able to break the degeneracy.Therefore, at order Λ 0 the system is effectively separated into disjoint subspaces, each one having wave functions given by ψ n⊥ (u)ξ n (x i ).According to (43), the energy gap between such subspaces is of the order of Λ, which means that in the thin-shell limit Λ → ∞ only the subspace with lowest energy becomes occupied.Consequently, the system wave function in the thin-shell limit must be where ξ(x i ) = ξ 0 (x i ).Substituting ( 47) into H 0 and integrating where possible yields the constant contribution in addition to the effective Hamiltonian The latter simplifies to where the resulting effective interaction strength in 2D turns out to be With this, we conclude that ( 50) is the appropriate Hamiltonian in the thin-shell limit apart from the diverging additive constant

Particular cases
Evaluating the functional derivative of the effective Hamiltonian (50), it is possible to obtain the wave function that minimizes it.To this end, however, we have to take into account the constraint (44) and its corresponding Lagrange multiplier, the chemical potential µ according to which finally leads to Let us consider now some particular cases for this 2D time-independent GPE.

Spherical shell
At first we deal with the particular situation of the spherical shell, i.e. ε = 0, which allows some simplifications.Most importantly, the wave function corresponding to the lowest energy state ξ(x i ) is a constant along the shell, since the BEC particles are uniformly distributed around the bubble.Let us determine the constant value of ξ through the normalization (44), which yields for the ground state of the system Substituting this value for the wave function into the effective Hamiltonian (50), we find with ε = 0 the corresponding energy This is the Λ 0 contribution to the energy for the spherical shell.Thus, the total ground-state energy amounts in this case to 0 + E (0) 0 + C 0 + E S , which is consistent with Ref. [28].

Thomas-Fermi approximation
For situations, where the interaction energy is much larger than the kinetic energy, the so-called Thomas-Fermi approximation can be applied.In case of the effective Hamiltonian 50, this corresponds to the situation where N → ∞ or A → ∞.In order to calculate the wave function ξ(x i ), we take (53) without the kinetic energy term, which gives As expected, ξ(x i ) has an angular dependence along the shell.Inserting it into equation ( 50) and solving the integral, we find the Thomas-Fermi energy which represents the ground state Λ 0 energy contribution without the diverging additive constants.Taking into account (44), it can be rewritten in terms of the total number of particles N as Thus, the total ground-state energy in this case is 0 + E (0) 0 + C 0 + E TF , which is also consistent with Ref. [28].

Perturbative solutions
Although the main result of our work is the step-by-step construction of the theory for BECs in the thinshell limit of a bubble trap, this paper also provides solutions to particular cases of the spherical shell and the Thomas-Fermi regime for the effective Hamiltonian (50), thus revealing some mathematical aspects of the corresponding wave functions.Now, in this chapter, we determine both the ground-state wave function and the excitation frequencies perturbatively by considering the geometrical distortion ε as the smallness parameter.

Perturbative ground state
Let us now perturbatively solve (53), which we rewrite as where we have introduced the smallness parameter λ = εℏΩ 0 /4.In the following, we also have to take into account the normalization (44), which reads explicitly The square of the angular momentum operator L2 has the eigenvalues ℏ 2 l(l + 1) and normalized eigenfunctions given by the spherical harmonics Y m l (ν, ϕ).Since the set {Y m l } represents a complete orthonormal basis over the spherical domain, the wave function ξ can be expanded in such a basis.But, as we are looking for the unique lowest-energy state, which is represented by a ϕ-independent real function ξ(ν), its expansion becomes so that the normalization reduces to In the following we use an important property of the spherical harmonics Y m l , which reads cos(ν . With this we get Taking these properties in (59) into account leads to where we define the effective interaction strength g eff = ḡ2D N/4πA 2 .Furthermore, we introduce the abbreviations which have the specific values where δ ij denotes the Kroenecker delta.Now we apply a generalization of Rayleigh-Schrödinger perturbation theory [55][56][57] in order to solve the nonlinear equation ( 62) perturbatively.To this end we employ the Taylor expansions of both the expansion coefficients c i and the chemical potential µ with respect to the dimensionless smallness parameter λ: Considering the zeroth-order solution, we find c (0) i = δ i,0 and µ (0) = g eff .In order to use the same normalization as in Rayleigh-Schrödinger perturbation theory, we follow Ref. [55] and define the renormalization constant Z(λ) according to so we get The new coefficients a i are then normalized by i a i c (0) i = a 0 = 1.Furthemore, with this Eq.( 62) becomes with the corresponding expansion i + λ 2 a (2) Note that here the coefficients a (l) i depend on |Z| 2 .Therefore, we treat |Z| 2 as another constant and expand the results according to (68) and (69) at the end.In addition, since a 0 = 1, we have which implies that |Z| 2 = 1 + O(λ 2 ).Thus, if one is interested only in first-order perturbation theory, one could consider |Z| 2 equal to 1.
Substituting (71) into (70) and applying (66), we get Considering i = 0 in (73), we obtain a recursion relation for the respective perturbative orders of the chemical potential µ (n) while for i > 0 in (73), we get a recursion relation for the perturbative coefficients a Note that in the linear case, where we have g eff = 0, Eqs.74) and (75) reduce to the usual Rayleigh-Schrödinger recursion formulae, where the normalization constant Z does not appear.Furthermore, we read off from 74) and (75) the first-order perturbative result for i > 0  76) and (80), respectively.Blue dots are obtained by numerically by solving Eq. ( 62).(b) Ratio between the coefficients c 2 and c 0 as a function of the surface eccentricity ϵ according to Eq. ( 60) for 2mN ḡ2D /ℏ 2 = 1.The black solid line corresponds to the first-order perturbative result in Eq. ( 78), while blue dots are obtained from a numerical solution of Eq. ( 62).
Since we have |Z| 2 = 1 + O(λ 2 ), we conclude from (68) which gives for the ground state wave function according to (60) In addition we obtain from (74) the second-order correction for the chemical potential: In order to determine the range of validity for these perturbative results, we performed a comparative analysis, as depicted in Fig. 2. To this end we examined the chemical potential and the wave-function coefficients, as defined in Eq. ( 60), and contrasted them with the results obtained from a numerical solution of Eq. ( 62) using the relaxation method.Thus, we read off that firstorder perturbative results are indistiguishable from the numerical ones for surface eccentricities ϵ < 2ℏ/mΩ 0 A 2 .

Perturbative excitation spectrum
We consider now the dynamics of small excitations over the ground state ξ (0) , which are described by the time-dependent GP equation and the ansatz ξ(t) = e −iµt/ℏ [ξ 0 + δξ(t)].Here δξ(t) = ue −iωt − ū * e iωt represent small elongations out of the ground state, yielding the Bogouliubov equations [61] T U = ℏω σU, where we have introduced the abbreviations According to [62,63], we can apply a generalized version of Rayleigh-Schrödinger perturbation theory to the generalized eigenvalue problem (82).
To this end, we must expand (83) in a power series of λ From ( 76) and (79), we have Let us now express the problem in the basis of the spherical harmonics {Y m l }, i.e., where the expansion coefficients are given by Thus, the operator T is represented in the basis of the spherical harmonics via the matrix elements From (87) we then deduce at zeroth order while ( 61) and (88) yield the corresponding first-order contribution for T : A direct diagonalization of (92) yields for the eigenvalues for each (l, m) with the corresponding eigenvectors The first-order correction to the eigenvalues ℏω With this, we obtain the final expression for the excitation frequencies: In order to assess the validity range of the first-order perturbative result in Eq. ( 99), we compare some eigenvalues as shown in Fig. 3 with the corresponding results obtained through direct numerical diagonalization of Eq. ( 82).An intriguing behavior arises when considering the case of l = 3 and m = 2, which corresponds to Fig. 3-h.For these specific values of l and m, the first-order correction in Eq. (99) vanishes, i.e. changes in the eccentricity of the trap have no effect on the eigenfrequency.Thus, only higher-order corrections could potentially change the eigenfrequency.To date, we have no explanation for this occurrence, which is limited to such specific l and m values, nor have we determined if this result holds true for all eccentricity values.Further investigation is needed to comprehend this phenomenon.In addition, such results are consistent with Refs.[31,33] for ε = 0, when their thin-shell limit is considered.

Summary and conclusions
In this paper, we have shown how the parameters of a bubble trap have to be carefully chosen in order to allow the existence of a thin-shell limit.Under the adequate assumptions, we were able to perform a dimensional compactification which leads to an effective 2D Hamiltonian for interacting BECs.From the experimental point of view, we demonstrated that, in order to avoid the collapse of the condensate, the thin-shell limit must be taken in such way that the spatial distortion caused by the eccentricity of the surface must be kept in the same order of magnitude as the width of the Gaussian distribution perpendicular to the surface.Details on how the experimentally obtained bubble-trap potentials fit into our general theory were also provided.In addition, applications of this theory to stationary systems as well as the excitation frequencies were calculated.

Figure 2 .
Figure 2. (a) Chemical Potential µ as a function of the surface eccentricity ϵ for 2mN ḡ2D /ℏ 2 = 1.Black dashed and solid line represent analytical results corresponding to first-and second-order perturbation theory according to Eqs. (76) and (80), respectively.Blue dots are obtained by numerically by solving Eq. (62).(b) Ratio between the coefficients c 2 and c 0 as a function of the surface eccentricity ϵ according to Eq. (60) for 2mN ḡ2D /ℏ 2 = 1.The black solid line corresponds to the first-order perturbative result in Eq. (78), while blue dots are obtained from a numerical solution of Eq. (62).

Figure 3 .
Figure 3.Comparison of excitation frequencies between the first-order perturbative results in Eq.(99) and the eigenvalues obtained through direct numerical diagonalization of Eq. (82) for varying values of l and m for 2mN ḡ2D /ℏ 2 = 1.The figure demonstrates the validity range of the perturbative approximation and highlights an intriguing case for l = 3 and m = 2 (shown as Fig.3-h), where the first-order correction vanishes and the eccentricity of the trap has no effect on the corresponding eigenfrequency.