Dynamics of a dipolar Bose-Einstein condensate in the vicinity of a superconductor

We study the dynamics of a dipolar Bose-Einstein condensate, like, for example, a ${}^{52}\mathrm{Cr}$ or ${}^{164}\mathrm{Dy}$ condensate, interacting with a superconducting surface. The magnetic dipole moments of the atoms in the Bose-Einstein condensate induce eddy currents in the superconductor. The magnetic field generated by eddy currents modifies the trapping potential such that the center-of-mass oscillation frequency is shifted. We numerically solve the Gross-Pitaevskii equation for this system and compare the results with analytical approximations. We present an approximation that gives excellent agreement with the numerical results. The eddy currents give rise to anharmonic terms, which leads to the excitation of shape fluctuations of the condensate. We discuss how the strength of the excitation of such modes can be increased by exploiting resonances, and we examine the strength of the resonances as a function of the center-of-mass oscillation amplitude of the condensate. Finally, we study different orientations of the magnetic dipoles and discuss favorable conditions for the experimental observation of the eddy-current effect.

Figure 1

Depicted is a schematic sketch of the system under investigation. A BEC is placed at a distance $x_d$ above a superconducting surface. The BEC consists of atoms which carry a magnetic dipole moment. The dipoles are all aligned in the same direction by a magnetic field. The magnetic field generated by the dipoles must satisfy the boundary condition $\mathbf{B}·\widehat{\mathbf{n}}=0$ at the surface of the superconductor, where $\widehat{\mathbf{n}}$ is the normal vector of the superconducting plane. By introducing a magnetic mirror BEC, at a distance $x_d$ below the superconducting surface, this boundary condition can be satisfied. With that, the interaction between BEC and superconductor can be modeled as an interaction between BEC and mirror BEC.

Figure 2

Depicted are the relative positions of the BEC and its mirror. The coordinate system $K^{\prime }$ is comoving with the mirror BEC and has its origin $O^{\prime }$ at the center of the mirror BEC. The coordinate system $K$ is stationary and its origin $O$ coincides with the minimum of the harmonic trap. As becomes clear from this graphic, the transformation between $K^{\prime }$ and $K$ is given by $x=x^{\prime }-2x_d-\langle x\rangle$.

Figure 3

Frequency shift for small-amplitude oscillations. We calculate the frequency shift for two different dipole-dipole interaction strengths, ${\varepsilon }_D={\varepsilon }_D^{\left(m\right)}=0.15$ and ${\varepsilon }_D={\varepsilon }_D^{\left(m\right)}=0.5$. The data points show the results from the numerical solution of the time-dependent GPE (20). The red stars show the results for an oscillation amplitude of $x_s=0.001{\lambda }_x^{\left(0\right)}$ and the blue squares for $x_s=0.1{\lambda }_x^{\left(0\right)}$. The solid black lines show the frequency shift based on the Thomas-Fermi approximation for a three-dimensional BEC for small-amplitude oscillations (23). The dot-dashed green lines show the results based on the column density model [34]. Other parameters: $\kappa ={\omega }_y/{\omega }_x=1$; distance to surface, $x_d=2{\lambda }_x^{\left(0\right)}$; length of the time evolution for the GPE, $t={10}^4{T}_x$, with $T_x=2\pi /{\omega }_x$.

Figure 4

Frequency shift for large-amplitude oscillations. Again the frequency shift is shown for two different dipole-dipole interaction strengths, ${\varepsilon }_D={\varepsilon }_D^{\left(m\right)}=0.15$ and ${\varepsilon }_D={\varepsilon }_D^{\left(m\right)}=0.5$. The data points show the frequency shift based on the numerical solution of the time-dependent GPE (20), where two different oscillation amplitudes are presented, $x_s=0.25{\lambda }_x^{\left(0\right)}$ (red stars) and $x_s=0.5{\lambda }_x^{\left(0\right)}$ (blue squares). The lines show the frequency shift based on the three-dimensional Thomas-Fermi approximation. The solid black line shows the result for small-amplitude oscillations (23). The dashed red line and the dotted blue line show the frequency shift with large-amplitude correction (25). Other parameters: $\kappa ={\omega }_y/{\omega }_x=1$; length of the time evolution for the GPE, $t={10}^4{T}_x$, with $T_x=2\pi /{\omega }_x$.

Figure 5

Frequency spectra for the relative fluctuation of the BEC aspect ratio $\Delta a\left(t\right)=\left[a\right(0)-a(t\left)\right]/a\left(0\right)$ of the BEC. The simulations were performed for various trap aspect ratios $\nu ={\omega }_x/{\omega }_z$, ranging from $\nu =1$ to $\nu =2.4$ in steps of $\Delta \nu =0.05$. The plots in (a) and (b) show the region of the crossing point between the single oscillation frequency ${\omega }_x^{\prime }$ and one of the monopole-quadrupole modes. From (a) to (b) the oscillation amplitude increases by an order of magnitude, from $0.01{\lambda }_x^{\left(0\right)}$ to $0.1{\lambda }_x^{\left(0\right)}$. The resonance peak at the crossing also increases by an order of magnitude. In (c) and (d) the region of the crossing between the double oscillation frequency $2{\omega }_x^{\prime }$ and the breather mode is presented. Again, the amplitude $x_s$ increases from (c) to (d) by an order of magnitude. The resonance peak increases by two orders of magnitude. Parameters: ${\varepsilon }_D={\varepsilon }_D^{\left(m\right)}=0.2$; $\kappa ={\omega }_y/{\omega }_x=0.99$; length of time evolution for (a) and (b) $t=100T_x$ and for (c) and (d) $t=500T_x$, with $T_x=2\pi /{\omega }_x$.

Figure 6

The left setup depicts the configuration where the dipoles are oriented in the $y$ direction. The interaction between dipoles $A$ and $B$ differs from the interaction between $A$ and $C$ only in the distance. The relative orientation of the dipoles is the same; therefore, also the interaction sign is the same. In this configuration, all contributions add up constructively. The right setup depicts the situation where all dipoles are oriented in the $z$ direction. The relative orientation between $A$ and $B$ is different than the relative orientation between $A$ and $C$; therefore, also the interaction sign may change. In this configuration the contributions from the edges partially compensate for the contributions from the center.

Figure 7

The frequency shift as a function of the dipole orientation angle $\phi$ for various trap aspect ratios $\nu$. In the inset the frequency shift is shown as a function of the trap aspect ratio $\nu$ for various orientation angles $\phi$. The curves were calculated with the column density model. Parameters: ${\varepsilon }_D^{\left(m\right)}=0.5$, ${\varepsilon }_D=0$, ${\omega }_y/{\omega }_x=1$, and $x_d=2{\lambda }_x^{\left(0\right)}$.

Figure 8

Frequency shift vs aspect ratio for a BEC with dipoles oriented in the $y$ direction. All curves were calculated with the column density model. We compare the case ${\varepsilon }_D^{\left(m\right)}=0.5$ and ${\varepsilon }_D=0$ (red dashed lines) to the case ${\varepsilon }_D^{\left(m\right)}={\varepsilon }_D=0.5$ (blue solid lines). In the inset, the frequency shift is shown as a function of the magnetic field orientation angle ${\phi }_B$ for two different values of $\nu$. Again, the two above mentioned cases are compared. Note that for ${\varepsilon }_D=0$ we have $\phi ={\phi }_B$, since ${\phi }_{\mathrm{TF}}=0$.

Figure 9

The orientation of the Thomas-Fermi ellipsoid relative to the external polarizing field $\mathbf{B}$. The direction of the $z$ axis of the trap is indicated by ${\widehat{\mathbf{e}}}_{z,\mathrm{trap}}$ and the direction of the BEC is indicated by ${\widehat{\mathbf{e}}}_{z,\mathrm{TF}}$. Due to the dipole-dipole interaction those two are no longer aligned. This modifies the angle between the BEC axis and the magnetic field: $\phi ={\phi }_B-{\phi }_{\mathrm{TF}}$.