Fate of the Amplitude Mode in a Trapped Dipolar Supersolid

We theoretically investigate the spectrum of elementary excitations of a trapped dipolar quantum gas across the BEC-supersolid phase transition. Our calculations reveal the existence of distinct Higgs amplitude and Nambu-Goldstone modes that emerge from the softening roton modes of the dipolar BEC at the phase transition point. On the supersolid side of the transition, the energy of the Higgs amplitude mode increases rapidly, leading to a strong coupling to higher-lying modes. Our Letter highlights how the symmetry-breaking nature of the supersolid state translates to finite-size systems.

Figure 1

The excitation frequencies $\omega /2\pi$ of the eight lowest BdG modes as a function of the scattering length $a_s$. Red (blue) triangles indicate that the density variation function of the corresponding mode has odd (even) parity. The background color indicates the ratio between density in between central and side droplets and peak density of the center droplet as a measure of the superfluid fraction [10, 13]. The vertical dashed line indicates the position of the phase transition. On top, we show density profiles for the isolated droplets (left), supersolid (middle), and BEC (right) regime.

Figure 2

The upper panel (a) shows the points of Fig. 1 for symmetric modes, color coded with the Higgs coupling $\mathcal{I}$ to other modes. For clarity, the points of antisymmetric modes have been removed, as the coupling to these modes is identically zero. The red line is a guide to the eye along the path of strongest Higgs character within the modes. The lower panel (b)–(g) shows time density evolution of the mode patterns at the selected points in (a) according to Eq. (3), where (b) corresponds to $f^{\text{Higgs}}$ and defines the coupling to all other modes. (b), (d), (f) show mode patterns at scattering lengths of optimal coupling on the (first), (second), (third) symmetric branch. (c), (e), (g) show mode patterns at lower scattering lengths on the respective branches. One can see that the top row (b), (d), (f) shows the strong amplitude character, whereas the lower row (c), (e), (g) displays the mixing with breathing patterns and decay of the Higgs character within these respective branches. Every time the axis has been rescaled to show two periods $T=2\pi /\omega$.