Efficient multipole representation for matter-wave optics

Technical optics with matter waves requires a universal description of three-dimensional traps, lenses, and complex matter-wave fields. In analogy to the two-dimensional Zernike expansion in beam optics, we present a three-dimensional multipole expansion for Bose-condensed matter waves and optical devices. We characterize real magnetic chip traps, optical dipole traps, and the complex matter-wave field in terms of spherical harmonics and radial Stringari polynomials. We illustrate this procedure for typical harmonic model potentials as well as real magnetic and optical dipole traps. Eventually, we use the multipole expansion to characterize the aberrations of a ballistically interacting expanding Bose-Einstein condensate in (3+1)-dimensions. In particular, we find deviations from the quadratic phase ansatz in the popular scaling approximation. This universal multipole description of aberrations can be used to optimize matter-wave optics setups, for example in matter-wave interferometers.


I. INTRODUCTION
In 1934, Frits Zernike introduced the orthogonal "Kreisflächenpolynome" to describe the optical path difference between light waves and a spherical reference wavefront 1 .Understanding the phase differences and minimizing the optical aberrations laid the base for the first phase-contrast microscope 2 .This invention was awarded with the Nobel Prize in Physics in 1953.Nowadays, Zernike polynomials are widely used in optical system design as a standard description of imperfections in optical imaging 3 .Typical wavefront errors are known as defocus, astigmatism, coma, spherical aberration, etc. 4 .Balancing aberrations is also relevant for optical imaging with electron microscopes [5][6][7][8] .In contrast to visible light, massive particles, such as electrons, atoms, and even larger molecules 9 , have a much smaller de Broglie wavelength, λ dB = h/p, and therefore a higher resolution.
Nowadays, atom-interferometers with ultracold atoms are used to study fundamental scientific questions like tests of the Einstein equivalence principle [10][11][12] , probing the quantum superposition on macroscopic scales 13 , the search for dark matter candidates 14 and gravitational waves 15,16 .Being very sensitive to accelerations and rotations, atom interferometry could be used for inertial sensing, replacing commercial laser gyroscopes, and satellite navigation in space 17 .Common to all µg-interferometric measurements with Bose-Einstein condensates are long expansion times 18 to reduce mean-field interaction as well as to increase the sensitivity of the interferometer.Hence, it is crucial to understand the actual shape of the condensate's phase as it determines the interference patterns at the end of the interferometer [19][20][21] .
Inspired by Zernike's work, we will adopt his approach to analyze these aberrations in the world of matter waves: First, we introduce a multipole expansion with suitable polynomial basis functions in Sec.II.We consider spherical-, spheroidal-, displaced asymmetric harmonic-and generally asymmetric trapping potentials in Sec.III.In particular, we characterize the magnetic potential from a realistic atom chip model.In Sec.IV, we extend the multipole analysis to Bose-Einstein condensates in the strongly interacting Thomas-Fermi as well as in the low interacting limit.Finally, we investigate the shape of the phase profile for a ballistically expanding condensate in Sec.V and conclude the discussion in Sec.VI.

II. MULTIPOLE EXPANSION WITH STRINGARI POLYNOMIALS
A. Orthogonal function within a sphere Cold atoms can be trapped or guided in either optical dipole or Zeeman potentials 22 .If these potentials are applied for short times (impact approximation, phase imprinting 23 ), one modifies only the phase of the atomic wave packet, thus, the momentum distribution of the condensate.This process is equivalent to a thin lens in optics, but now in (3+1) dimensions 24 .Systematically analyzing the features of different potentials becomes crucial for achieving the ultimate precision in long-time atom interferometry 25 .
The standard Taylor expansion of a potential, in the vicinity of a point r 0 , in three-dimensional Cartesian coordinates r = r 0 + ζ reads This interpolation polynomial is useful for extracting forces from the gradient or trapping frequencies from the eigenvalues of the Hesse matrix K = (∇ ⊗ ∇U) |r 0 .However, the Taylor series are notoriously inefficient approximation schemes.Alternatively, we introduce a multipole expansion in spherical coordinates (r, ϑ , ϕ) in terms of multipole coefficients U nlm = nlm|U and orthonormal basis functions r|nlm = S nlm (r) = S nl (r)Y lm (ϑ , ϕ), consisting of spherical harmonics Y lm (ϑ , ϕ) and radial polynomials S nl (r) (see App. A).Explicitly, they have support within a hard three-dimensional "aperture" of radius R. In terms of a scaled aperture radius 0 ≤ r = r/R ≤ 1 , they read The polynomials in Eq. ( 5) describe the sound wave excitations of a Bose-Einstein condensate in the strong interacting Thomas-Fermi limit for an isotropic threedimensional harmonic oscillator potential, originally described by S. Stringari 26 and P. Öhberg et al. 27 .Thus, these fundamental modes are well adapted for condensates in harmonic as well as in more realistic trapping geometries (see Sec. III C and V).In hindsight, it almost seems natural to recover Jacobi polynomials J (l+1/2,0) n with half-integral indices in the three-dimensional situation, if we compare this to the circle polynomials of F. Zernike in two-dimensional beam optics 28 .The normalization Therefore, the potential expansion coefficients U nlm in Eq. ( 2) are given by the overlap integral within a spherical volume V = 4πR 3 /3.Typically, the characteristic size of the wave packet in the trap determines the aperture radius R. For cold clouds, this size is of the order of the harmonic oscillator length ℓ = h/(Mω) for a non-interacting gas or of the Thomas-Fermi radius r TF = 2µ/(Mω 2 ) for an interacting condensate.For thermal clouds, the radius R ≈ k B T /(Mω 2 ) is proportional to the temperature T .
Instead of a multipole expansion of the potential in Eq. (36), it is also possible to decompose the logarithm of the potential This cumulant expansion 29 is particularly useful for Gaussian functions as the series terminates quickly after the second order.In Sec.III, we analyze both the potentials and their cumulants in terms of these multipole expansions.

B. Spectral powers
For interpreting and visualizing the complex expansion coefficient U nlm or θ nlm , we introduce relative spectral powers, p nl ≥ 0. The latter is independent of the potential's orientation to the reference coordinate system.This also holds for the cumulant expansion in Eq. (8).Let us consider a coordinate transformation r ′ = Rr, where R denotes an orthogonal rotation matrix RR ⊤ = ½.In both frames, the values of the potential agree The coefficients in the new reference frame U ′ nlm are given by where the R|lm are the Wigner D-matrices as the matrix representation of the rotation operator R in the angular momentum basis [30][31][32] .Using the unitarity of the rotation operator, R R † = R † R = ½, as well the orthonormality of the Stringari polynomials (3), we specify rotational invariant measures such as the total power Moreover, we define the marginals P nl , P l , as well as the relative fractional powers p nl (U) by Thus, the relative fractional powers add up to one.

III. MULTIPOLE EXPANSION OF TRAPS
A. Harmonic, isotropic, three-dimensional oscillator A stiffness parameter k characterizes a harmonic, isotropic three-dimensional oscillator potential For a particle with mass M, the angular frequency ω = k/M.An isotropic potential is invariant under rotations.Thus, only the s-waves contribute.A Stringari polynomial S n0 from Eq. ( 5) is of the order r 2n , which limits the radial modes to two monopoles.The expansion coefficients U n00 are given in terms of a dimensional factor [J m For the total power, one finds as well as the fractional powers B. Harmonic, anisotropic, three-dimensional oscillator

Spheroidal potential
For a prolate or oblate spheroidal harmonic oscillator, the potential is characterized by two stiffness constants k ⊥ and k , but it is still rotational symmetric around the z-axis The anisotropy is measured by α 2 = k /k ⊥ .In contrast to the isotropic oscillator, one needs also a quadrupole within the multipole expansion Using the coefficients in Eq. ( 19) we can evaluate the total power and the relative fractional powers p 00 (U) = 7 5 (2 + α 2 ) 2 8 + 4α 2 + 3α 4 , p 10 (U) = 4 15 for the cylindrical symmetric oscillator potential.It is noteworthy that this expansion encompasses degenerate traps with vanishing z-confinement α = 0, as well as isotropic traps with α = 1 from Eq. (17).

Tilted, shifted anisotropic harmonic oscillator potential
We consider a general harmonic oscillator potential with a symmetric stiffness matrix K localized at a position r 0 and gather it into homogeneous potentials U (n) (r) of degree n.
In order to determine the multipole expansion, we transform the position vectors r = xe x + ye y + ze z given in a Cartesian basis {e x , e y , e z } to the spherical basis {e 1 , e 0 , e −1 } which are orthogonal with respect to the standard complex scalar-product e n e * m = δ n,m .It is convenient to introduce also a dual basis {e 1 , e 0 , e −1 } with e n e m = e n e * m = δ n,m .
Now, the position vector reads in the co-and contravariant spherical basis 30 r More generally, one can express the spherical vector components 33 q m by with the spherical tensor T (1) m (ϑ , ϕ).Obviously, the constant U (0) (r 0 ) = r ⊤ 0 Kr 0 /2.The dipole coefficients for U (1) (r, r 0 ) in Eq. ( 22) are then given by the spherical components of the vector r 0

U
(1) The second order contribution U (2) (r) in Eq. ( 22) contains a product of two spherical tensors that can be simplified using the Clebsch-Gordan expansion 33 T (l 1 ) with Clebsch-Gordan coefficients 32 C l 1 l 2 l m 1 m 2 m = l 1 m 1 l 2 m 2 |lm .Hence, we can rewrite U (2) (r) as with multipole coefficients that contain the matrix elements of K in the spherical basis and Clebsch-Gordan coefficients.
For the radial part, one can use the results from the expansion of isotropic oscillator III A in terms of the s-wave Stringari polynomials.The total power in Eq. ( 11) of U (2) evaluates to For an isotropic stiffness K = k½, Eq. ( 34) reduces to the result of the isotropic harmonic oscillator in Eq. ( 16).

C. Magnetic Zeeman potential of an atom chip
Magnetizable atoms can be trapped in a static magnetic field 34 Here µ B denotes the Bohr magneton, g F the Landé factor, m F the magnetic quantum number of the total angular momentum, and B(r) the magnetic induction field.In this work, we analyze the Zeeman potential in Eq. ( 35) obtained by a realworld model of a magnetic chip trap 35 .These have become popular when a miniaturization of the experimental setup is required, for example at drop towers 36,37 and in space 38,39 .
The atom chip and its functionalities are described in detail in Ref. 40 .In App.B, we present a realistic finite-wire model of the microtrap from which we deduce the magnetic induction field (see Eq. (B2)).Here, we consider 87 Rb atoms in the magnetic hyperfine state |F = 2, m F = 2 with g F = 1/2 with the set of currents as presented in Tab.I.The corresponding Zeeman potential, evaluated in the chip coordinate system, is depicted in Fig. 1.
For an efficient representation of the three-dimensional Zeeman potential, we use the multipole expansion in Eq. (2).For a comparison, we extract also the multipoles of the cumulant as discussed in Eq. ( 8).As we represent the potential on discrete lattice points, we use a least-square optimization (see App. C) to calculate the expansion coefficients U nlm and θ nlm respectively.The results of the multipole expansion are summarized in Fig. 2. There, we show the relative fractional angular powers p nl for the harmonic approximation p nl (U ho ) (a), the full Zeeman potential p nl (U Z ) (b) and the cumulant p nl (θ ) (c).In each subfigure, we have used the same number of basis functions with maximal principle and angular momentum quantum numbers n max = 3, l max = 5.Moreover, the multipole expansion is performed at the position of the trap minimum, shifting the position vector in Eq. ( 22) by r = r ′ + r 0 .The latter implies a vanishing dipole component U (1) (r, r 0 ) = 0 for the harmonic approximation.
As discussed in Sec.III B, the anisotropic harmonic oscillator potential exhibits just two monopoles and one quadrupole contributions, depicted in Fig. 2 (a).For a real Z-wire trap on the atom chip, the Zeeman potential exhibits all multipoles: in particular the monopoles p 00 , p 10 , dipoles with p 01 , p 11 , a quadrupole p 02 , as well as the octupole with p 03 , which is de-picted in Fig. 2 (b).From the dipole coefficients, we deduce that in the anharmonic trap, the position of the trap minimum does not coincide with the center-of-mass position of the trap.Thus, the application of a Zeeman lens-potential causes a finite momentum-kick to the atomic density distribution.While the dipoles affect the center of mass motion, the additional octupole causes density distortions in long-time matter-wave optics and interferometry.
Finally, we note that multipoles of higher order l > 3 are decreasing rapidly for our trap configuration.Comparing the direct multipole expansion to the cumulant expansion, one finds that the expansion of the cumulant series converges more slowly due to the logarithmic character of Eq. ( 8), see Fig. 2  (c).

D. Optical dipole potential from Laguerre-Gaussian beams
Besides magnetic trapping, atoms can be also trapped by an optical dipole potential 41,42 which is created by laser light far detuned from the atomic resonance.Here, ∆ = ω L − ω 0 describes the laser detuning and Ω 0 is the Rabi frequency.For red-detuned lasers ω L < ω 0 with respect to the atomic transition ω 0 , the dipole potential is attractive and it is repulsive for blue-detuning ω L > ω 0 43 .For a single Laguerre-Gaussian beam 44 the exponent θ (r) in Eq. ( 36) has the spatial dependence where the Rayleigh range z R = πw 2 0 /λ L , is typically much larger than the extension of the condensate wave packet.The laser wavelength is λ L = 2π/k L and the beam waist is denoted by w(z) = w 0 (1 + z 2 /z 2 R ) 1/2 .The dipole potential Eq. (37) describes an optical waveguide 45 or can act as an optical matterwave lens in the time domain 46 .For the latter, we depict the optical potential for 87 Rb in Fig. 3.
The harmonic approximation of Eq. ( 37), corresponds to the spheroidal trapping potential in Eq. ( 18) with stiffness and the anisotropy α 2 = w 2 0 /(2z R ) depending on the ratio of the minimal waist and the Rayleigh length.
As in the previous subsection, we evaluate the relative powers, see Fig. 4, for the harmonic approximation p nl (U ho ) Eq. ( 18) (a), the dipole potential p nl (U D ) Eq. ( 36) (b) and the cumulant p nl (θ ) in Eq. (37) (c).Due to the Gaussian laser beam, the cumulant expansion of the dipole potential is much more efficient than the direct multipole expansion (compare Fig. 4 (b) and (c)).For large Rayleigh length z R ≫ R, the spatial dependence of the exponent θ (r) is almost Gaussian, which is shown in subfigures 4 (a) and (b).Additional corrections to the harmonic cumulant in higher angular momentum components l > 2 are of the order 10 −9 and smaller.

IV. MULTIPOLE EXPANSION OF BOSE-EINSTEIN CONDENSATES
Besides the external trapping potentials, we are also interested in an efficient representation of a three-dimensional Bose-Einstein condensate.Within the classical field approximation for Bosons, the evolution of the complex matter-wave field is described by the time-dependent Gross-Pitaevskii equation.Here, the contact-interaction strength is denoted by g = 4π h2 a s /M, with the atomic mass M and the s-wave scattering length a s .The atomic density n(r,t) = |Ψ(r,t)| 2 is normalized to the particle number N. Equivalently to Eq. ( 39), one can represent the classical complex Gross-Pitaevskii field in terms of the two real hydrodynamical variables, the density of the condensates n(r,t) and its phase φ (r,t), also known as a Madelung transform 47 .
To represent the three-dimensional matter-wave field, we choose the same multipole expansion as in Eq. (2), for the density as well as for the phase.
In the limit of large repulsive interactions, one can neglect the quantum pressure that arises from the localization energy of the classical field Eq. ( 45) admits algebraic solutions of the form known as the Thomas-Fermi approximation 48 .
In the following, we apply the multipole expansion in Eq. (42) for the condensate density in some of the trapping potentials discussed in the previous Sec.III.Thereby, we discuss the strongly interacting Thomas-Fermi regime as well as the exact numerical solution of the stationary Gross-Pitaevskii equation.
We should note that in principle the Stringari polynomials in Eq. ( 42) can exhibit negative values which are nonphysical when regarding positive-valued atomic densities.In order to avoid this anomaly, we consider coordinates in the interval 0 ≤ r ≤ R and densities n(r) ≥ 0.
For the Thomas-Fermi density in Eq. ( 46), we expect an exact interpolation by the Stringari basis functions, if the potential is of polynomial form.In contrast, setting the interaction strength in Eq. ( 44) to g = 0 and considering a harmonic oscillator potential, one obtains a Gaussian density distribution as the harmonic oscillator ground state 32 The widths of the Gaussian correspond to the harmonic oscillator lengths σ j = h/(Mω j ), j = 1, 2, 3 for the three spatial directions.For the latter, the cumulant expansion is always of a quadratic form.Thus we recover the expansion coefficients for the optical dipole potential of a single Laguerre-Gaussian beam in Sec.III D.

A. Isotropic, three-dimensional density
We consider atomic density distributions in an isotropic harmonic oscillator potential (14).As the symmetry of the external potential determines the symmetry of the density, the condensate is interpolated by monopoles only.The efficiency of the interpolation depends on the actual radial shape n(r) = n(r), which will be determined by either the Thomas-Fermi density (46) or the stationary Gross-Pitaevskii equation (44).For the Gross-Pitaevskii density, we also evaluate the cumulant expansion to investigate the effect of different mean-field interactions.

Thomas-Fermi density
As the Thomas-Fermi density is directly proportional to the trapping potential, the interpolation is obtained by two Stringari polynomials only, as discussed in Sec.III A. Using the Thomas-Fermi approximation in its dimensionless form with dimensionless radial coordinate r ′ = r/r TF and the Thomas-Fermi radius r TF = 2µ TF (N)/(Mω 2 ).The chemi- cal potential scales with the particle number as We find the monopole coefficients,

Gross-Pitaevskii density
We represent the three-dimensional matter-wave field on a discrete, Cartesian Fourier grid.Therefore, we solve the sta-tionary Gross-Pitaevskii equation ( 44) using Fourier spectral methods.To study the behavior of the radial expansion coefficients in different interaction regimes, we choose different particle numbers for the condensate.The n nlm 's in the multipole expansion of density in Eq. ( 42) are obtained using the method of least-squares (see Eq. ( C2)) replacing the target potential with the numerical, discrete Gross-Pitaevskii target density n t (r j ).In contrast to the Thomas-Fermi density the radius R = r TF of the spherical integration volume in Eq. ( 5) is not known a priori.Therefore, we are minimizing the leastsquare error ε(R) with respect to a variable aperture radius R for a fixed number of basis functions.The monopole coefficients p n0 (n) for the isotropic Gross-Pitaevskii density are depicted in Fig. 5 on a semi-logarithmic scale for n max = 14 basis functions.For small particle numbers 5 (a), the coefficients are declining exponentially for n > 6 whereas the magnitude of the tenth coefficient is p 10,0 (n) < 10 −6 .Entering the intermediate and strong interacting regime, the decline in magnitude becomes more irregular, as the Gaussian-like shape of the density is modified towards a polynomial shape.In Fig. 5 (b), the most contributions to the polynomial expansion are within the first five coefficients, and in Fig. 5 (c) within the first three.The latter reflects the transition to the pure quadratic Thomas-Fermi regime.However, we also recognize that the magnitude of the expansion coefficients does not converge to the same level as in the low interacting regime.In contrast to the Thomas-Fermi solution, the density also con-tains high-energetic modes close to the Thomas-Fermi radius, whose interpolation requires a lot of Stringari polynomials.
As we are interested in good interpolation in the region of significant density, we introduce a cutoff at which disregards some of the highly energetic modes.In Fig. 6, we plot the corresponding expansion in terms of the polynomials which matches the Gross-Pitaevskii density quite well.We also use the Stringari polynomials with a reduced number of basis functions, neglecting coefficients smaller than the chosen cutoff.
Besides the multipole coefficients for the density, we also look into the cumulant expansion in Eq. ( 8), for which we expect a faster convergence in the low-interacting limit.The p n0 (θ ) in Fig. 7 (a) confirm that the density distribution is more of Gaussian shape, as the cumulant expansion almost terminates for monopole powers n > 3.For larger particle numbers, subfigures (b) and (c), the cumulant expansion works quite efficiently as the polynomial series converges faster as in Fig. 5. Cross-sections of the cumulant and the interpolation with the Stringari polynomials are shown in Fig. 8.

B. Anisotropic three-dimensional density
As discussed in Ref. 49 , the Thomas-Fermi field in a general harmonic oscillator potential can always be re-scaled to an isotropic s-wave by an affine coordinate transformation.Hence, this simplifies the search for the optimal aperture radius R and adapts the polynomial expansion on the finite interval to the anisotropic extension of the density distribution.The latter becomes necessary for an optimal and efficient interpolation of the Gross-Pitaevskii matter-wave field which reaches beyond the Thomas-Fermi radius.
For this purpose, we evaluate the covariance matrix for the three-dimensional Gross-Pitaevskii field.The positive, semi-definite matrix Σ r admits a Cholesky decomposition of the form The matrices Q and σ are defined by the eigenvalue equation The matrix C and the expectation value r 0 define the required affine coordinate transformation that we use to evaluate a multipole expansion of the form where the Stringari polynomials are evaluated with respect to the new coordinates ζ.For the new multipole coefficients n nlm , Θ nlm we define also the corresponding spectral powers x, z (µm) n FIG. 10.Cross-sections of the scaled Thomas-Fermi density n TF (x, 0, 0), n TF (0, 0, z) (blue ) versus Cartesian coordinates x, z in a spheroidal harmonic oscillator Eq. (18).Interpolation of the density with Stringari polynomials n(r) Eq. ( 42) (red ).Parameters: α = 2, Thomas-Fermi radii x TF = y TF = 14.4 µm, z TF = 7.2 µm, particle number N = 10 5 , chemical potential µ TF /h = 304 Hz.
a. Thomas-Fermi density As a benchmark test, we investigate the Thomas-Fermi density in an anisotropic harmonic oscillator with cylindrical symmetry, which we discussed in Sec.III B 1.For the ratio of angular frequencies, we use α = 2. From analyzing the potential, we know that the multipole expansion in Eq. ( 48) just exhibits monopoles as well as one quadrupole.Applying the coordinate transformation Eq. ( 58), we obtain the angular powers p nl (n TF ) shown in Fig. 9.As the multipole expansion is now performed in the scaled reference frame (59), where the ellipsoid is re-scaled to a sphere, we expect monopoles only, see Eq. (51).Indeed, we find good agreement with the isotropic Thomas-Fermi density as the quadrupoles are p n2 (n TF ) < 10 −9 as displayed in Fig. 9. Using the monopoles n n00 and the quadrupoles within the transformation matrix C, one can reconstruct the original oblate-shaped Thomas-Fermi density as depicted in Fig. 10.
particle numbers varying the effective mean-field interaction in Eq. ( 44).In addition, we compare the multipole expansion of the density with the multipole expansion of the cumulant.In both cases, the expansion coefficients are evaluated in the scaled reference frame defined by Eq. (59).In contrast to the ellipsoidal Thomas-Fermi density, we observe non-negligible quadrupole contributions in the relative angular powers p nl (n) for the low as well as for the high interacting regime, which is presented in Fig. 11.For low particle numbers, subfigure (a), the angular powers p n0 (n), p n2 (n), p n4 (n) are decaying exponentially with respect to the principle number n as we already stated in the isotropic case.Increasing the angular momentum for a fixed value of n, the magnitudes of the p nl (n) decrease by roughly 1.5-2 orders of magnitude.The angu-lar momentum dependence decreases for increasing interactions as shown in the subfigures 11 (b), (c).In particular, the powers p n4 < 10 −6 , are emphasizing the change of the Gross-Pitaevskii density towards the Thomas-Fermi shape.Moreover, the spectrum of the monopoles p n0 (n) in the re-scaled reference frame exhibits the same structure as in the isotropic case (see Fig. 5), reflecting again the high-energetic modes in the Gross-Pitaevskii density that require a large number of Stringari polynomials.
Nevertheless, we can interpolate the ground-state density distributions also in the anisotropic harmonic oscillator as depicted in Fig. 12.In particular, we can neglect modes with l = 4 for the condensate with large particle numbers to obtain a good approximation with the Stringari polynomials.48) for the spheroidal harmonic oscillator Eq. ( 18) with anisotropy α = 2. Relative angular powers p nl (θ ) versus principle number n. Different angular momenta: red l = 0, blue l = 2, green l = 4 with n max = 14, l max = 4.For the reconstruction of the cumulant in Fig. 14 we mark the cutoff p c = 10 −6 (gray ).Parameters as in Fig. 11.
The results for the anisotropic cumulant expansion are presented in Fig. 13 and 14.The monopoles p n0 (θ ) exhibit again the same features as for the isotropic density, while the cumulant expansion works more efficiently describing the lowinteracting regime.The latter is well described by just three multipole coefficients Θ 000 , Θ 020 and Θ 200 , subfigure 13 (a).In contrast to the direct multipole expansion of the density, the cumulant expansion contains significant angular powers p n4 (θ ) which needs to be considered for the polynomial interpolation.

V. RELEASE AND FREE EXPANSION OF A BOSE-EINSTEIN CONDENSATE
Time of flight measurements is one of the standard techniques to image the density distribution of a Bose-Einstein condensate 50 after a ballistic expansion and to extract equilibrium as well as dynamical properties.Here, we investigate the release of the condensate initially trapped in the Zeeman potential of the magnetic chip trap that we characterized in Sec.III C.
Within the Thomas-Fermi approximation, Eq. ( 46), it is well-known [51][52][53] that the time evolution of the density n TF (r,t) as well as the phase φ TF (r,t) is given terms of the three adap- tive scales λ j (t), j = 1, 2, 3, if the condensate is initially trapped in a harmonic trap.The adaptive scales evolve during the ballistic expansion according to the differential equations with initial conditions chosen as λ j (0) = 1 and λ j (0) = 0. Therefore, the conjugate variables n TF (r,t), φ TF (r,t) evolve quadratically in time making the multipole expansions with the introduced Stringari polynomials in Eq. ( 42), ( 43) very efficiently, as only monopoles as well as quadrupole are contributing to Eq. ( 42) and Eq. ( 43).Beyond, the Thomas-Fermi approximation the initial Gross-Pitaevskii matter-wave field in the harmonic trap consists of non-quadratic polynomials as analyzed numerically by our multipole expansion in Sec.IV and described analytically in Ref. 54 .During the ballistic expansion of the condensate the density deviation n 1 (r) leads to additional phase perturbations φ (r,t) = φ TF (r,t) + φ 1 (r,t), to the quadratic Thomas-Fermi phase φ TF in Eq. ( 63).We obtain the total phase in Eq. (66) by solving the differential equation (64) for the adaptive scales λ j (t) and the Gross-Pitaevksii equation in the co-expanding frame of reference with the coordinates ξ j = r j /λ j (t).The transformed field ψ(ξ,t) is related to the original one in Eq. ( 39) by At t = 0, the field ψ(ξ, 0) satisfies the stationary Gross-Pitaevskii equation (44) with N = 10 5 87 Rb atoms.As an external potential U(r) we choose the Zeeman potential U Z as well as the harmonic approximation.For the chosen parameter we find the Thomas-Fermi size r TF = (25.5,8.3, 9.4)µm and maximal density deviations of max δ n = 0.1% within the two different trapping configurations.To analyze the impact on the phase during the ballistic expansion after a flight of flight t 1 = 80 ms, we apply the multipole expansion of the total phase in Eq. (66) with respect to the re-scaled coordinates ξ for these two different initial states.The relative angular powers p nl (φ ) are shown in Fig. 15.In the two subfigures, we compare the phase of the condensate that was initially trapped in the Zeeman potential (a) to the condensate initially in the harmonic approximation (b).From the latter, we state that the next leading orders to the scaling approximation (63) are of the form r 4 with spectral powers p 20 (φ ), p 12 (φ ), and p 04 (φ ).In addition, we find that their values are approximately three orders of magnitude higher than the anharmonic corrections which populate the dipole p n1 (φ ), and the octupole moments p n3 (φ ) in the phase of the condensate.

VI. CONCLUSION AND OUTLOOK
In conclusion, we have introduced a multipole expansion with suitable radial polynomials to characterize differ-ent trapping geometries and the matter-wave field of a threedimensional Bose-Einstein condensate.Besides the optical dipole potential for a single Laguerre-Gaussian beam, we have examined the multipole moments for the Zeeman potential of a realistic atom chip model.For both, we quantified deviations from their harmonic approximation and introduced an expansion of the cumulant which is superior for Gaussian-shaped functions.In the Thomas-Fermi approximation, the shape of the condensate is directly proportional to the external potential.Hence, it is natural to characterize the three-dimensional shapes of density and phase in terms of the same polynomial basis functions.Moreover, we have examined the efficiency of our multipole expansion for the different mean-field interactions in the Gross-Pitaevskii equation.In addition, we studied the phase of an expanding condensate in the same manner.We identified possible aberrations for long-time atom interferometry in the different multipole moments that are caused either by the external potentials or the intrinsic properties of interacting Bose-Einstein condensates.
Our work provides a general and universal framework for an aberration analysis in matter-wave optics with interacting Bose-Einstein condensates.The multipole analysis allows the design for aberrations balanced matter-wave lenses in single or multiple lens setups 37,46 , e.g. with programmable optical dipole potentials using digital micromirror devices 55 .Finally, the multipole expansion of the magnetic field could be used to exploit different trapping geometries and for designing new atom chips 56 .where the unit vectors e 2 , (e 1 ) are pointing from the conductor's end (start) to the observation point r and ρ = |r − r 1 | denotes the distance to the head of the wire element at r 1 .Hence, the total magnetic induction field of M wires in N segments is just the sum of all individual fields 2 − e (i) 1 • w (i) . (B2)

FIG. 1 .FIG. 2 .
FIG. 1. Magnetic Zeeman potential of an atomic chip trap for 87 Rb in the magnetic substate |F = 2, m F = 2 .Two-dimensional contour plots along all three spatial planes at position r ′ = r − r 0 .The principle axis of the trap obtained by the Hesse matrix K Eq. (1) marked as lines.Parameters of the trap are summarized in Tab.I and II.

FIG. 16 .
FIG.16.QUANTUS II atom chip model40,58 .Light-gray wires belong to the base chip structure.Gray wires belong to the science chip structure.Active conductors in Z-trap configuration in blue (science chip) and red (base chip) colors.External Helmholtz coils creating a homogeneous field are not depicted.