Expansion dynamics of a dipolar Bose-Einstein condensate

Our recent measurements on the expansion of a chromium dipolar condensate after release from an optical trapping potential are in good agreement with an exact solution of the hydrodynamic equations for dipolar Bose gases. We report here the theoretical method used to interpret the measurement data as well as more details of the experiment and its analysis. The theory reported here is a tool for the investigation of different dynamical situations in time-dependent harmonic traps.

Figure 1

(Color) Inverted parabolic profile of a BEC in the Thomas-Fermi limit without a dipole-dipole interaction. When the trapping potential is symmetric the distribution is also spherically symmetric.

Figure 2

(Color) Saddle shaped dipole potential generated by dipolar atoms of a BEC in a spherical trap. The atomic dipoles which are illustrated as small magnets in the figure are aligned by an external magnetic field $B$. Note the orientation of the saddle potential relative to the magnetic field direction.

Figure 3

(Color) Figure to illustrate the change of the condensate shape under the influence of magnetization in trap (top row) and during time of flight (bottom row). Left column, magnetization in transversal direction; center column, nondipolar atoms; right column, longitudinal polarization. Dashed ellipses represent the nondipolar condensate.

Figure 4

(Color) Experimental cycle for measuring the dipole-dipole interaction. (a) and (c): alignment of the field relative to the chamber just before releasing them. Gravity $\vec{g}$ marks $y$ as the vertical axis. (b) and (d): schematic cycle. The magnetic field during preparation is in both cases along the $y$ axis. To measure with the $z$ polarization (a) and (b), the field has to be turned slowly (within $40\mathrm{ms}$) within the trap before releasing the atoms. After $1\mathrm{ms}$ of expansion, the field is switched to the $x$ direction. For $y$ polarization (c) and (d), the field is kept in $y$ direction until $1\mathrm{ms}$ after release.

Figure 5

(Color) (a) and (b) show the density profiles in the $y$ and $z$ directions of an almost pure condensate after $90\mathrm{ms}$ of expansion, respectively. Data in the gray shaded areas have been used to fit the thermal cloud.

Figure 6

(Color) Aspect ratio of the expanding dipolar condensate. Data points in (a) are averaged with an 11-point moving average. Error bars in the upper graph represent errors from the fits to the density distribution. Upper, red data: Field aligned in a vertical $y$ direction. Lower, black data: Field in a horizontal $z$ direction. The solid lines represent the corresponding theoretical predictions. The blue dotted line is the behavior that one would expect for pure contact interaction without the presence of dipole-dipole interaction among the atoms. (b) illustrates the evolution of the condensate shape as seen by the camera. (c) shows in detail the gray shaded area in (a). At ten milliseconds time of flight, we have performed a series of measurements under the same conditions. The two data points represent the mean values of 42 measurements with $y$ polarization (red circle) and 32 with $z$ polarization (black square). The solid error bars are derived from the statistics of the measured value and represent one standard deviation in both directions. They do not include the systematic error on the single measurements. These errors affect both the measurements with $z$ and $y$ polarization in the same way and do not change the significance of the measured difference in the expansion dynamics with different polarization. The shift of the measured aspect ratios due to such systematic errors is indicated by the dashed-dotted representation of the error bars (displaced laterally for clarity) in (c).

Figure 7

Log-linear plot of the function $f(x,y)$ defined in Eq. (A2) versus $x$ and for different values of $y$ (dashed lines). The chosen values of $y$ correspond to different powers of 2. Due to the symmetry property $f(x,y)=f(y,x)$ of the function $f$ the asymptotic values of $f(x,y)$ for large $x$ correspond to the function $f_a\left(y\right)$ defined in (A8) which is also shown as a function of $x$ (solid line).

Figure 8

(Color) Log-linear three-dimensional plot of the function $f(x,y)$ defined in Eq. (A2) versus $x,y$.