Interaction of a Bose-Einstein Condensate and a Superconductor via Eddy Currents

We study center-of-mass oscillations of a dipolar Bose-Einstein condensate in the vicinity of a superconducting surface. We show that the magnetic field of the magnetic dipoles induces eddy currents in the superconductor, which act back on the Bose-Einstein condensate. This leads to a shift of its oscillation frequency and to an anharmonic coupling of the Bose-Einstein condensate with the superconductor. The anharmonicity creates a coupling to one of the collective modes of the condensate that can be resonantly enhanced, if the parameters of the condensate are chosen properly. This provides a new physical mechanism to couple a Bose-Einstein condensate and a superconductor which becomes significant for 52Cr, 168Er or 164Dy condensates in superconducting mircotraps.


Introduction
In the last years hybrid quantum systems have come into focus of research in the context of quantum information processing [1,2,3,4]. They are able to combine the strengths of qubits based on solid state devices, which can be better controlled, and qubits based on atomic systems, which promise longer coherence times. Such a hybrid can consist of a superconductor coupled to a Bose-Einstein condensate (BEC). To achieve a controlled coupling between a superconductor and a BEC it is necessary to understand the interaction between those two systems.
Coupled quantum systems based on BECs and solid state devices have been suggested theoretically [5,6,7]. While the influence of a solid on a BEC is sizable and has been studied well in the past, normally the influence of a BEC on a solid is very weak due to the low density of the BEC. In recent years, BECs have been condensed in superconducting microtraps [8,9,10,11,12] allowing a close approach of a BEC to a superconducting surface [13,14,15,16]. The use of a superconducting microtrap as opposed to a metallic one allows for a significantly longer lifetime of the atomic cloud in close vicinity of the surface [13,17,18,19], and therefore also longer coherence times. It also promises a successful coupling of these two macroscopic quantum phenomena [6,7,20,21,22,23].
In the last decade there has also been intensive research, theoretical as well as experimental, in the field of dipolar BECs. The high magnetic dipole moment of the atoms leads to a long ranged anisotropic interaction between the atoms in a BEC. This interaction is responsible for a number of interesting phenomena observable in dipolar BECs [24]. For some time 52 Cr [25] has been the only experimentally realized dipolar BEC. But most recently also the condensation of 168 Er and 164 Dy [26,27] has been achieved.
Here we will study a dipolar BEC in close vicinity of a superconductor. We suggest that the mutual interaction of a dipolar BEC with a superconductor can become sizeable due to the large magnetic dipole moment of the atoms. Specifically, centerof-mass oscillations of the dipolar condensate within its trap create eddy currents in the superconductor surface. These eddy currents, in turn, shift the oscillation frequency of the condensate. Their anharmonicity creates a coupling of the centerof-mass oscillations of the condensate with one of its collective modes. We show that this anharmonic coupling can be resonantly enhanced, allowing for a sizeable interaction of the BEC with the solid and a new mechanism for coupling a BEC with a superconductor.

Interaction between the superconducting surface and the Bose-Einstein condensate
Consider a weakly interacting dipolar BEC of N atoms at temperature T = 0 confined by an external harmonic trapping potential V T (r). The trapping potential could be generated by the magnetic field of a superconducting atom chip or a laser field, for example. The spins are all aligned alongê z by an external magnetic field (see Fig. 1). The interaction U (r, r ′ ) between two atoms consists of two contributions. One contribution is the isotropic contact interaction U s (r, r ′ ) = g s δ (3) (r − r ′ ), where g s = 4π 2 as M gives the strength of the interaction and is determined by the mass M and the s-wave scattering length a s . The other contribution is the long ranged magnetic dipole-dipole interaction The strength is given by g D = µ 0 m 2 , where m is the magnetic dipole moment. The time evolution of the BEC is given by the time dependent Gross-Pitaevskii equation (GPE) [28,29] For simplicity we will model the superconductor by a superconducting half space, which is a valid approximation when the BEC is sufficiently close to a plane superconductor surface. The presence of a superconducting half space modifies the magnetic field distribution of a nearby magnetic dipole due to the currents induced in the superconductor. As long as the distance between the magnetic dipole and the surface of the superconductor (in our case ∼ 10 µm) is larger than the magnetic penetration depth of the superconductor (∼ 100 nm for Nb) and as long as the oscillation frequency of the dipole motion (in our case ∼ 1 Hz-1 kHz) is smaller than the gap frequency (∼ 100 GHz for Nb), the superconductor acts as a perfect magnetic mirror. This means that at the surface of a superconductor the normal component of a magnetic induction field has to vanish B ·n = 0. The field distribution of the magnetic dipole close to a superconductor can thus be found by introducing a mirror dipole in the superconductor and adding up the field of the dipole and the mirror dipole. The mirror dipole emulates the effect of the induced eddy currents. This way the magnetic interaction between a dipolar BEC and a superconductor can be described by an additional external potential felt by the atoms in the BEC due to the mirror BEC: Here, n (r ′ ) is the density distribution of the mirror BEC. Note, that this potential depends on the number of atoms in the BEC in contrast to other single-particle potentials like the Casimir-Polder force for example. The BEC ground state is the stationary solution of (2), which will be determined numerically below. Before we do that, let us discuss first a useful approximation for the potential of the mirror. With a sufficiently large number of atoms in the BEC the kinetic energy can be neglected, which leads to the Thomas-Fermi (TF) approximation [28,29]. Within this approximation an analytical expression for the density distribution N · |ψ (r)| 2 of a BEC can be given. In an harmonic potential the density distribution has an ellipsoidal shape and n TF (r) = 0 for r / ∈ D TF . Here n 0 is the central density and λ x , λ y and λ z are the semi-axes of the ellipsoid. In the case where only contact interaction is present it is easy to see that n TF (r) is of the form (4). However, this is not so obvious for the case of a dipolar BEC. As has been discussed by Eberlein et al. [30] the BEC density distribution remains of ellipsoidal shape also in the presence of the dipoledipole interaction only the semi axes being modified. They have also shown that the BEC may become unstable if the dipole-dipole interaction becomes too large. The dimensionless parameter ε D = gD 3gs provides a measure for the strength of the dipoledipole interaction compared to the strength of the contact interaction. In the region −1/2 < ε D < 1 the ground state is stable while beyond this region it may or may not be stable depending on the trap geometries. In the following we will only consider values of ε D in the stable region. In the case where no dipole-dipole interaction is present (ε D = 0) the semi axes are given by with the chemical potential µ (0) = g s n (0) 0 and n (0) 0 being the central density of a non-dipolar BEC fixed by the normalization condition´dr n TF = N and given by [28,29] For a dipolar BEC these quantities need to be determined numerically, see for example [30,24,31]. Using n (r) = n TF (r) integral (3) cannot be solved analytically. However, if the distance x d between the BEC and the superconductor is large enough, i.e. λ x , λ y ≪ x d , the problem can be further simplified. Density distribution (4) can be integrated over x and y yielding for |z| ≤ λ z and n 1D (z) = 0 elsewhere. n 1D (z) is the so-called column density [33] and represents an effective one dimensional density distribution of the mirror BEC. Note that n 1D (z) is a good approximation for the 3D mirror BEC even for λ z ≫ x d . Using n 1D (z) the interaction potential (3) is reduced to an one dimensional integral along the axis of the column density of the mirror BEC. The potential generated by the mirror BEC at a position r = (x, y = 0, z) T can be written as Here, we have evaluated U SC only in the plane y = 0, since the column density of the BEC is located in this plane and we are interested in oscillations in x-direction (see next section). The analytical solution of this integral is straight forward. The discussed model is depicted in Fig. 1.

Center-of-mass frequency shift
In this section we will discuss the change of the center-of-mass frequency of the dipolar BEC due to the presence of the superconductor. The external trapping potential V T (r) provides a certain oscillation frequency which is modified by the interaction with the Figure 1. The dipole at z = 0 interacts with the mirror dipoles distributed with n 1D (z) along the z-axis. The interaction sign with the dipoles in the red region is negative and with the dipoles beyond the red region its positive. The BEC has an optimal length when all dipoles are in the red region.
mirror BEC. In particular, we are interested in the frequency of the center-of-mass motion perpendicular to the surface. The frequency shift due to the superconductor is related to the curvature in x-direction generated by U SC . By calculating the curvature we have to take into account that the motion of the BEC also leads to motion of the mirror BEC. When the BEC moves towards the superconductor, so does the mirror BEC. This means that we have to take the derivative with respect to the distance to the superconductor rather than with respect to the distance to the mirror BEC. Using expression (8) as the interaction potential we have to take the derivative with respect to x/2 and the curvature change along the z-axis of the BEC reads For small amplitude oscillations the center-of-mass frequency ω ′ x perpendicular to the surface is determined by [33] ω ′2 here n 1D (z) represents the column density of the BEC and ω x is the frequency of the harmonic trapping potential V T (r). If the frequency change is small we have x and with that the relative frequency shift can be written as Using the semi-axes of a non-dipolar BEC λ a = λ (0) a we find γ ∝ ε D . The change of the semi-axes of the ellipsoidal BEC due to the dipole-dipole interaction only appears as a higher order correction. The integral in expression (10) is best evaluated numerically. Although the atoms in the BEC do not experience an individual frequency shift, we can still consider a single atom at the center of the BEC interacting with the mirror BEC in order to get an analytical order of magnitude estimate for the frequency shift. In this case we have γ = g(z=0; x d ) The frequency shift as function of the number of atoms for different models and interaction strengths ε D = 0.15 and ε D = 0.5. The curves labeled with "non-dipolar TF" represent calculations based on a TF density distribution for a BEC without dipole-dipole interaction. The curves labeled with "dipolar TF" are based on a TF density distribution which takes the dipole-dipole interaction into account. In both cases the frequency shift was calculated using equation (10). The points labeled with "num" are the results of a numerical timeevolution of the GPE using the effective potential (13). The parameters for the curves are λ on the relative position of the interacting dipoles, it can be attractive or repulsive. Considering a single dipole at z = 0, the strongest frequency shift is obtained when the interaction sign is the same with all the dipoles in the mirror BEC (see Fig. 1). Then, all contributions to the interaction integral (8) add up constructively. Depending on the distance x d there is an optimal length of the BEC, which reads for the semi-axis which represents a rule of thumb for the magnitude of the maximal frequency shift.
As an example, for 52 Cr with ε D ≈ 0.15 [32] assuming a distance of x d = 2λ (0) x we find γ max ≈ 10 −3 . A frequency shift of this magnitude is well within experimental resolution. A precision of 10 −5 was demonstrated in an experiment where the Casimir-Polder force was measured via the frequency shift of a BEC [34].
In Fig. 2 we show the frequency shift (10) as a function of the number of atoms N in the BEC. For the calculations we assumed a distance of x d = 14 µm. Experimental findings [13,23] and theoretical analysis [14,15] both suggest that using a superconducting microtrap such distances can be achieved. The frequency shift is presented for two different values of the dipole-dipole interaction parameter using three different models. First we calculated the frequency shift (10) for a non-dipolar BEC, meaning that we used the semi-axes λ (0) a according to (5). So the dipole-dipole interaction is only taken into account in the interaction between the BEC and its mirror. In Fig. 2 this model is labeled by "non-dipolar" in the plot legend. In this case the ratio of the trapping frequencies ν = ω x /ω z is given by the inverse ratio of the semi axes ν = λ (0) z /λ (0) x according to (5). The larger the value of ν the more elongated is the BEC in z-direction. If the semi axes λ Varying the number of atoms this way is equivalent to changing length of the BEC. Experimentally this could be achieved by adjusting the trap frequency ω z according to the number of atoms N , such that relation (12) remains satisfied. Fig. 2 shows that γ has a maximum. This maximum appears at an optimal length of the BEC as has been discussed above. Increasing the number of atoms above the optimal number leads to a smaller frequency shift, because contributions from the edges of the mirror BEC with opposite sign compensate contributions from the central region. In the limit N → ∞ the frequency shift approaches 0. In order to detect the eddy current effect experimentally, we suggest to use the frequencyω x of a long BEC as reference frequency. If the BEC is long enough, the frequency shift due to the eddy current effect is smaller than experimentally detectable. However,ω x would still include possible shifts due to other effects, like for example the Casimir-Polder force, which do not depend on N . Withω x as a reference the frequency shift can be measured as a function of N . Since other surface forces do not have this characteristic dependence on the number of atoms the curve is a fingerprint for the eddy current effect.
We also calculated the frequency shift using a dipolar BEC. In Fig. 2 these results are labeled with "dipolar". Here we have used the same parameters as in the other calculation, meaning that the trap frequencies remain the same as well as the distance to the surface. However, this time we used the correct dipolar semi axis λ z instead of λ (0) z in the density distribution. While the trap ratio ν is still proportional to N , it is no longer given exactly be the ratio λ z /λ x . Also the central density n 0 slightly changes while varying N . Fig. 2 shows that the effect of the modified semi axes is negligible for ε D = 0.15 but somewhat changes the result for ε D = 0.5. However, the main features of the curve are preserved.
The points labeled with "num" in the plot legend of Fig. 2 are the results of numerical calculations. We obtained these results by solving the time dependent GPE (2) numerically in three spatial dimensions using a time-splitting spectral method [35]. We only considered U s (r, r ′ ) in the GPE for the numerical calculations and will discuss the effect of U md (r, r ′ ) below. As potential in the GPE we used the following effective potential where the function f (z; x d ) describes the relative curvature change of the potential in x-direction due to U SC and is defined by The effective potential (13) is a good approximation if f (z; x d ) ≪ 1. We first determine the ground state of the GPE numerically. To excite the center-of-mass oscillation we then shift the potential by a distance x s in x-direction, and calculate the time evolution. In every time step t i we calculate the x-coordinate of the centerof-mass where ψ (r; t i ) is the numerically determined solution of the GPE at that particular time step. After the time evolution is completed we perform a Fourier analysis of the data to obtain the oscillation frequency. The results for the center-of-mass frequency are presented in Fig. 2 and labeled "num". We can see a very nice agreement with the results obtained for γ using (10). The reason that it does not agree with the results for the dipolar BEC is that we did not take into account the modified semi axes when we calculated V eff (r). Since we neglected the dipole-dipole interaction in the GPE, neglecting it in f (z; x d ) is consistent. Again the dipole-dipole interaction is only taken into account between the BEC and its mirror. The numerically obtained results are expected to follow the curves labeled "dipolar" in Fig. 2 if we include the dipole-dipole interaction in the GPE and consider it in f (z; x d ).

Coupling of the center-of-mass motion with the breather mode
Next we want to discuss the aforementioned coupling of a collective mode with the center-of-mass motion due to the eddy current effect. Center-of-mass motions can be excited by a sudden shift x s of the trap minimum. In a harmonic potential the center-of-mass motion does not excite collective shape fluctuations of the BEC and the shape of the BEC will remain constant during motion. However, in the vicinity of the superconductor the effective potential is no longer purely harmonic. The interaction with the mirror BEC generates additional anharmonic terms to the harmonic trapping potential. The excitation of collective modes due to terms like x 3 , x 4 etc. has been discussed previously [36,37,38,39]. The lowest order anharmonic term in (13) is of the form x 2 z 2 . Transforming into the center-of-mass system we have x (t) ∝ sin (ω ′ x t). The anharmonic term thus generates a time dependent change of the trap frequency in z-direction of the form ∆ω ′ z (t) ∝ sin 2 (ω ′ x t). This excites monopole-quadrupole modes of the BEC [40] with frequency 2ω ′ x . We thus expect to see a resonance, if one of the monopole-quadrupole modes happens to have the frequency 2ω ′ x . The best candidate for this is the so-called breather mode.
In Fig. 3 we show the breather mode frequency Ω B of a BEC calculated within the TF approximation [31,41]. The mode frequency is shown as a function of the aspect ratio ω x /ω z of the trapping frequencies. In a spherical trap with ω x = ω y = ω z the mode frequency is Ω B = √ 2ω x . In a uni-axial elongated trap with ω x = ω y > ω z the mode frequency approaches 2ω x (red line). In a tri-axial elongated trap with ω x > ω y > ω z the mode frequency crosses 2ω x (dashed blue line). In the latter case we expect to see a strong resonant coupling of the breather mode and the center-ofmass motion at the crossing point.  To study the excitation of the breather mode, we again numerically solve the time dependent GPE using potential (13). However, this time we analyze the fluctuations of the width of the wave function After calculating the time evolution a Fourier analysis of σ x provides the frequency spectrum. The peaks in the frequency spectrum correspond to collective modes of the BEC. We have identified the peaks in the spectrum by comparing them to the results from TF calculations. In Fig. 4 we present the frequency spectra of ∆σ x = σ x (t) − σ x (0) in a slightly tri-axial trap for different ω x /ω z . The yellow line starting at ≈ 2.3 ω x represents the breather mode frequency. For increasing trap ratio ω x /ω z it approaches another yellow line at ≈ 2ω x . This second line represents the double oscillation frequency of the center-of-mass motion. A resonance can be observed where the two lines meet (red). There is also a third yellow line visible in the spectrum. It belongs to another collective mode of the BEC, which also is excited due to the anharmonicity of the trap. However, this excitation is very weak compared to the resonance which occurs for the breather mode.
In Fig. 5 an enlarged region of the excitation spectrum of ∆σ x near the resonance is shown. While the spectra are plotted on a linear scale the color map on the bottom is plotted on a logarithmic scale. There are two peaks visible in the spectra shown: one corresponding to the breather mode frequency and the other representing twice the center-of-mass frequency (≈ 2.0012 ω x ). One can clearly see the resonance peak at the crossing point just as we have already conjectured.
As shown in Fig. 3 the breather mode for a BEC with ε D > 0 has a different frequency than a BEC with ε D = 0. Since we have neglected U md (r, r ′ ) in the GPE this effect is not included in our numerical results. Fig. 3 shows that the crossing point    of the breather mode and the double oscillation frequency is shifted towards smaller values of ω x /ω z (dot-dashed blue line). Therefore we also expect the resonance peak to appear at a smaller value of ω x /ω z .

Discussion and Conclusion
We have studied a new physical mechanism to couple a BEC and a superconductor. The mechanism rests on the interaction of the magnetic dipole moments of a dipolar BEC with the superconductor. Center-of-mass oscillations of such a dipolar BEC create eddy currents in the superconductor, which act back on the BEC. We have demonstrated that this eddy current effect leads to a frequency change and can resonantly excite a collective mode of the BEC.
The frequency change of the center-of-mass motion of the BEC has a characteristic dependence on the number of atoms in the BEC, which can serve as an experimental fingerprint to distinguish this effect from other effects that may change the oscillation frequency. We have also shown that the resonant excitation of the collective mode of the BEC becomes possible, if the parameters of the external trapping potential are tuned properly. Both effects become significant in 52 Cr, 168 Er or 164 Dy BECs.
In principle there should be similar effects using a thermal cloud instead of a BEC. However, it will be much harder to observe those effects, because the density in a thermal cloud is much smaller. Also such a system would not represent a coupled quantum system.
The eddy current effect we described here requires a trap to be formed a few ten micrometers away from a superconductor surface. For superconducting microtraps it has been pointed out that the Meissner effect of the superconductor modifies the magnetic field distribution in its vicinity in such a way that a magnetic trap cannot be formed anymore, if the trap position is brought too close to the superconducting surface [10]. In Refs. [14] and [15] this effect has been studied theoretically for a finite conductor with rectangular cross-section and the limits for such traps have been discussed. It was shown that a trap distance of less than 10 µm can be achieved for a Niobium conductor. A recent experiment [23] has demonstrated a distance of 14 µm from a superconducting strip.
In our calculations we assumed a superconducting half-space. For this approximation to be appropriate a finite superconducting strip with a rectangular cross-section should fulfil the following requirements: • Thickness: as the eddy currents are flowing at the surface of the superconductor within the magnetic penetration depth λ, the thickness of the superconductor should be 2λ or more, which for Niobium is ∼200 nm. These are typical film thicknesses used in thin film technology. • Width: the width of the strip should be larger than both the width of the BEC and its distance from the superconductor. In the recent experiment of Ref. [23] it was shown that this requirement can be met. • Length: the length should be larger than both the length of the BEC and its distance from the superconductor. Such an axial confinement could be made by a Z-shape trap, for example, as in the work of Ref. [23].
If some of these requirements cannot be met in an experiment, it is clear that the eddy current effect we describe will be reduced by a geometrical factor. This factor depends on the solid angle under which the BEC sees the superconductor surface.
In this work we have only considered small amplitude oscillations. For large amplitudes we expect to see corrections to the results presented here. On the other hand, larger amplitudes should increase the discussed effects since the potential becomes more anharmonic. For example the potential can no longer be assumed to be symmetric in the x-direction.
So far experiments on superconducting microtraps for BECs have only been done with 87 Rb atoms, which have a comparatively small magnetic dipole moment. Our calculations show that it is beneficial to study strongly dipolar BECs in superconducting microtraps. In such a system a mutual coupling of a BEC and a superconductor via eddy currents becomes possible.