Excitation Spectrum of a Trapped Dipolar Supersolid and Its Experimental Evidence

We study the spectrum of elementary excitations of a dipolar Bose gas in a three-dimensional anisotropic trap across the superfluid-supersolid phase transition. Theoretically, we show that, when entering the supersolid phase, two distinct excitation branches appear, respectively associated with dominantly crystal and superfluid excitations. These results confirm infinite-system predictions, showing that finite-size effects play only a small qualitative role, and connect the two branches to the simultaneous occurrence of crystal and superfluid orders. Experimentally, we probe compressional excitations in an Er quantum gas across the phase diagram. While in the Bose-Einstein condensate regime the system exhibits an ordinary quadrupole oscillation, in the supersolid regime we observe a striking two-frequency response of the system, related to the two spontaneously broken symmetries.

Figure 1

Axial excitation spectra of a trapped dipolar quantum gas across the BEC-supersolid-ID phase transition. The trap frequencies are $2\pi \times (260,29.6,171)\mathrm{Hz}$. The upper (lower) row shows calculations for a ${}^{164}\mathrm{Dy}$ (${}^{166}\mathrm{Er}$) quantum gas of $4\times {10}^4$ ($5\times {10}^4$) atoms in the BEC (a),(b), supersolid (c)–(f), and ID (g),(h) regimes, together with the corresponding ground-state density profiles (insets). (a), (c), (e), and (g) correspond to $a_s=(92,91,90,81)a_0$, and (b), (d), (f), and (h) to $a_s=(50.8,50.5,50,48)a_0$, respectively. In (e) and (f), the dashed and dash-dotted lines are guides to the eyes, indicating the two excitation branches. The color map indicates the calculated DSF, and $l_z$ is the harmonic oscillator length along the dipoles’ direction.

Figure 2

Evolution of three different even modes of the system calculated for $5\times {10}^4$ Er atoms at $a_s=49.8a_0$: (a) fourth-, (b) second-, and (c) third-lowest-lying even modes in energy with frequencies (67.4, 40.3, 49.8) Hz, corresponding to crystal, phase, and mixed modes, respectively. Each panel shows $n=|\psi (0,y,0,t){|}^2$ for $t=\pi /2{\omega }_l$ and $t=3\pi /2{\omega }_l$ with $\eta =0.15$ and the corresponding $\delta \phi (0,y,0)$. (d) DSF for the same setting as in (a)–(c), where the modes are colored according to their associated phase (red) or crystal (blue) character via $C$ [37].

Figure 3

(a) Example of a measured mean interference pattern in the renormalized central cut of the density distribution $n(k_y)$ for $t_h=5\mathrm{ms}$ in the supersolid regime at $a_s=49.8a_0$ (filled circles) and in the BEC regime at $a_s=51.7a_0$ (open circles). (b)–(d) PCA results at $a_s=49.8a_0$. (b) Time evolution of the weights of PC1 (filled circles) and PC2 (open circles) together with their sine fit. Error bars denote the standard error of the mean. (c),(d) Evolution of the partially recomposed $n(k_y)$ accounting for the population of PC1 (c) and PC2 (d) only. (e),(f) Calculated time evolution of $n(k_y)$ from excitation of the mode shown in Figs. 2 and 2, respectively, and using $\eta =0.15$.

Figure 4

Comparison between the mode energy obtained from the theory calculations and the energies extracted from the PCs (circles). The gradual color code of the theory lines represents the relative strength of $R_l$ going from strong (red) to no (gray) coupling. Error bars denote one standard deviation from the fit. The background color indicates the BEC, supersolid, and ID regions (see upper labels), identified using a matter-wave interferometric analysis of the experimental data [17].