Roton Excitations in an Oblate Dipolar Quantum Gas

We observe signatures of radial and angular roton excitations around a droplet crystallization transition in dipolar Bose-Einstein condensates. In situ measurements are used to characterize the density fluctuations near this transition. The static structure factor is extracted and used to identify the radial and angular roton excitations by their characteristic symmetries. These fluctuations peak as a function of the interaction strength indicating the crystallization transition of the system. We compare our observations to a theoretically calculated excitation spectrum allowing us to connect the crystallization mechanism with the softening of the angular roton modes.

Figure 1

(a) Normalized mode patterns of the density fluctuations close to the transition at $a_s\simeq 77.5a_0$ and their spatial power spectra in the $x\text{−}y$ plane. Shown are the lowest radial ($m=0$) and angular rotons ($m=1$, 2, 3) having $m$ angular nodal lines. (b) Corresponding Bogoliubov excitation energies as a function of the scattering length $a_s$, with the insets indicating the emergence of a blood cell shape in the ground-state density close to the instability at $a_s\simeq 77.1a_0$. For decreasing scattering lengths, first the $m=2$ followed by the $m=1$ and $m=3$ angular roton modes drops below the radial dipole modes at the trap frequency and softens further toward the instability. For information on higher modes (e.g., cyan line) and details, see Refs. [27, 50].

Figure 2

(a) Single-shot images for four different relative scattering lengths, in the BEC regime ($+5.2a_0$), closer to the transition point ($+2.0a_0$), in the transition region ($+0.0a_0$), and for a droplet crystal ($-4.2a_0$). (b) Mean images of the unrotated images showing no clear crystalline structure. (c) Aligned images (see text) indicate the presence of droplets in the mean image. (d) 2D structure factor showing an increasing height of the peaks at finite momentum $|\mathit{k}|$ indicating the approaching transition point. The central area below $k_{\min }/2\pi \simeq 0.11\mu {\mathrm{m}}^{-1}$ (see text) was masked out.

Figure 3

(a) Radial distribution $S(k)$ of the two-dimensional structure factor after integration over the angular coordinate for different relative scattering lengths. A clear peak at finite wave vector rises toward the phase transition. For clarity, the lines for smaller scattering lengths were shifted vertically. (b) The amplitude of this peak is obtained from a Gaussian fit reaching its maximum at the phase transition. (c) Roton momentum shifts toward larger values in the droplet regime where it stays constant. The dash-dotted line on the left indicates the smallest momentum $k_{\min }/2\pi \simeq 0.11\mu {\mathrm{m}}^{-1}$. The green diamond indicates the maximum $S(k)$ and the dashed line the roton momentum at that maximum. The gray area in (b) and (c) indicates the transition region.

Figure 4

(a) Angular distribution $S(\varphi )$ of the two-dimensional static structure factor integrated over the interval $k/2\pi \in [0.2,0.45]\mu {\mathrm{m}}^{-1}$ around the roton momentum $k_{\mathrm{rot}}$ for different relative scattering lengths. (b) Decomposition of $S(\varphi )$ into Fourier components matching the symmetry of the lowest two angular roton modes $m=2$ and $m=3$ in the BEC. For clarity, the lines in (a) were shifted vertically for smaller scattering lengths. The gray area in (b) indicates the transition region. Error bars are obtained by bootstrapping [65].