Mean-field predictions for a dipolar Bose-Einstein condensate with ${}^{164}\mathrm{Dy}$

Dipolar Bose-Einstein condensates are systems well suited for the investigation of effects caused by the nonlocal and anisotropic dipole-dipole interaction. In this paper we are interested in properties which are directly connected to the realization of a condensate with ${}^{164}\mathrm{Dy}$, such as stability and phase diagrams. Additionally, we study the expansion of dipolar condensates and find signatures of the dipole-dipole interaction in terms of structured states and a deviation of the well-known inversion of the aspect ratio of the cloud during a time of flight. Our analysis is based on the extended Gross-Pitaevskii equation, which we solve numerically exactly on a grid by means of an imaginary- and real-time evolution.

Figure 1

Absorption images in $y$ and $z$ directions $({\left|\psi \right|}^2$ integrated along the $y$ and $z$ axis, respectively) for three different expansions of the condensate at times $t=0,3$, and 6 ms (from left to right). The symbol in the left top corner illustrates the polarization axis, which always points in the $z$ direction. Top: Absorption images in the $y$ direction $(x$ abscissa, $z$ ordinate) $\perp$ polarization axis with ${\omega }_{x,y,z}=2\pi \times (205,195,760)$ Hz, $\epsilon =1.1$, and $\alpha ={60}^{\circ }$. The expansion is dominated by the anisotropy of the momentum distribution. Center: Absorption images in the $y$ direction $(x$ abscissa, $z$ ordinate) $\perp$ polarization axis with ${\omega }_{x,y,z}=2\pi \times (205,195,360)$ Hz, $\epsilon =1.1$, and $\alpha ={45}^{\circ }$. Here the anisotropy in the momentum distribution is lower than in the upper case, which yields an expansion dynamics showing a clear signature of the dipole-dipole interaction. Bottom: Absorption images in the $z$ direction $(x$ abscissa, $y$ ordinate) ‖ polarization axis with ${\omega }_{x,y,z}=2\pi \times (205,195,760)$ Hz, $\epsilon =1.1$, and $\alpha =0^{\circ }$. Here the occurrence of structured states during the TOF is recognizable (see also Fig. 2).

Figure 2

One-dimensional absorption images in the $z$ direction for $\alpha =0$ and ${30}^{\circ }$ after 6 ms of TOF. Note that for $\alpha =0^{\circ }$ the graphs for the quantities $|{\mathrm{\Psi }}_x{|}^2$ and $|{\mathrm{\Psi }}_y{|}^2$ are congruent. Since the visibility $c$ of the structures decreases with an increase of $\alpha$ the largest value of $c=0.023$ is given for $\alpha =0^{\circ }$. This is below the benchmark of $c\approx 0.1$, which is a necessary condition for the experimental observation of such structures. Additionally, we see that the rotation of the trap destroys the rotational symmetry in the $xy$ plane.

Figure 3

(a) Stability diagrams for four values of $\epsilon$ with respect to the angle of rotation $\alpha$ and the aspect ratio of the trap $\lambda ={\omega }_z/{\omega }_{x,y}$ with ${\omega }_z=2\pi \times 760$ Hz, and $N{a}_{\mathrm{dd}}=3.6$ (which corresponds to ${10}^4$ particles). We have plotted the maximum value of $\alpha$ which allows for a stable condensate for a given aspect ratio $\lambda$. A smaller value of $\epsilon$ stabilizes the condensate against the unstable head-to-tail configuration, which corresponds to large angles of rotation and small aspect ratios. (b) Phase diagram for the visibility $c$ for $\epsilon =1.3$. Structured stationary ground states appear at the border of stability. Since we find $c\approx 0.1$ for a broad strip, an experimental observation should be possible.

Figure 4

(a) Stability diagram with respect to the number of particles $N$ and $\epsilon$. Here we have chosen $\alpha ={90}^{\circ },{\omega }_{x,y}=2\pi \times 50$ Hz, and four different values of ${\omega }_z$, where we have plotted the maximum value of $N$ for a given ratio $\epsilon$ which allows for a stable condensate. An increase of $N$ or $\epsilon$ destabilizes the condensate due to the DDI. (b) Stability diagram with respect to the angle of rotation $\alpha$ and $\epsilon$ for ${\omega }_{x,y,z}=2\pi \times (50,50,250)$ Hz and three different values for $N$, where we have plotted the maximum admissible value of $\alpha$. Reducing $\alpha$ stabilizes the condensate and therefore allows for a greater amount of particles in the condensates.

Figure 5

One- and two-dimensional absorption image in the $x$ direction $(y$ abscissa, $z$ ordinate) $\perp$ polarization axis with $\alpha ={90}^{\circ },{\omega }_{x,y,z}=2\pi \times (50,50,250)$ Hz, $\epsilon =1.5$, and $N=700$. The symbol in the left top corner illustrates the polarization axis, which points in the $z$ direction. The density distribution has the structure of a macroscopic quantum self-trapping state.