Scissors Mode of Dipolar Quantum Droplets of Dysprosium Atoms

We report on the observation of the scissors mode of a single dipolar quantum droplet. The existence of this mode is due to the breaking of the rotational symmetry by the dipole-dipole interaction, which is fixed along an external homogeneous magnetic field. By modulating the orientation of this magnetic field, we introduce a new spectroscopic technique for studying dipolar quantum droplets. This provides a precise probe for interactions in the system, allowing us to extract a background scattering length for ${}^{164}\mathrm{Dy}$ of $69(4)a_0$. Our results establish an analogy between quantum droplets and atomic nuclei, where the existence of the scissors mode is also only due to internal interactions. They further open the possibility to explore physics beyond the available theoretical models for strongly dipolar quantum gases.

Figure 1

(a) Experimental method: The quantum droplet is held in a cylindrically symmetric trap (around $z$); the orientation of the field is modulated around its mean value along $y$ for $\mathrm{\Delta }t=20\mathrm{ms}$ at variable frequency. The amplitude ${\theta }_m$ follows ${\theta }_m(f)=12°/\sqrt{f/100\mathrm{Hz}}$. (b) Experimental response measured as a growth in the visible size of the droplet ${\sigma }_x\times {\sigma }_y$ as a function of atom number and modulation frequency. (c) The theory shows the same value extracted from solutions of the equations of motion within the Gaussian ansatz using the best-fit value of $a=68a_0$. It takes into account the finite excitation time as well as departure from the linear response regime. The shot-to-shot fluctuations in atom number are simulated, and finite resolution is also implemented in the theory calculation of ${\sigma }_x\times {\sigma }_y$. Note the different scales for theory and experiments; the $\sim 20\%$ difference in droplet size could be due to imaging miscalibration. The dashed line shows the theoretical scissors mode frequency in linear response theory; see Ref. [17].

Figure 2

Variation of the response with the angle modulation amplitude. Data and theory in Fig. 1 correspond to ${\theta }_0=12°$. (a) Theory: We take $a=68a_0$, as it gives the best fit over the whole range of atom numbers in Fig. 1; this explains the slight horizontal disagreement with experiment. For increasing excitation amplitude, the response increases, and a line splitting is observed due to departure from the linear response regime. The dashed line shows the linear repsonse prediction. (b) Experiment, with atom number 390(100): The same increase in response and line splitting is observed. We surmise that the small peaks at high frequency absent in the experiment are due to the simplifying assumptions of our model.

Figure 3

Long axis oscillation frequency following a 90° field angle quench. Experiments are performed with an atom number of 690 (150). The error bar in blue shows the result of a squared sum of the fit standard error and the fitted decay time. The grey error bars show a 100 Hz interval corresponding to the short observation time. The red lines are theory calculations using the equations of motion for different scattering lengths (indicated on the figure).