Moment of inertia and dynamical rotational response of a supersolid dipolar gas

We show that knowledge of the time-dependent response of a trapped gas, subject to a sudden rotation of a confining harmonic potential, allows for the determination of the moment of inertia of dipolar supersolid configurations. While in the presence of one-dimensional arrays of droplets the frequency of the resulting scissors oscillation provides accurate access to the value of the moment of inertia, two-dimensional-like configurations are characterized by low-frequency resonances in the rotating signal, reflecting the presence of significant rigid-body components in the rotational motion. Using the formalism of response-function theory and simulations based on the so-called extended time-dependent Gross-Pitaevskii equation, we point out the crucial role played by the low-frequency components in the determination of the moment of inertia and of its deviations from the irrotational value. We also propose a protocol based on the stationary rotation of the trap, followed by its sudden stop, which might provide a promising alternative to the experimental evaluation of the moment of inertia.

Figure 1

(a)–(d) Examples of integrated ground-state density profiles ${n(x,y)=\int dz\mathrm{\Psi }(x,y,z)|}^2$ in the superfluid and supersolid phases for a gas of ${}^{164}\mathrm{Dy}$ atoms. (a) and (b) refer to a gas of 40 000 atoms in a harmonic trap of frequencies ${\omega }_{x,y,z}=2\pi (20,40,80)\text{Hz}$, and ${\epsilon }_{dd}=1.32$ and 1.434, respectively. (c) and (d) refer to a gas of 60 000 atoms in a harmonic trap of frequencies ${\omega }_{x,y,z}=2\pi (40,80,160)\text{Hz}$, and ${\epsilon }_{dd}=1.32$ and 1.467, respectively. (e) Fraction of nonclassical rotational inertia $f_{\mathrm{NCRI}}$ [Eq. (5); symbols and solid lines] and the ratio ${\beta }^2={\mathrm{\Theta }}_{\mathrm{irr}}/{\mathrm{\Theta }}_{\mathrm{rig}}$ as a function of ${\epsilon }_{dd}$ (symbols and dashed lines) for the zigzag (red circles and solid line) configuration [25] and for the small linear configuration (black squares and solid line) considered in [13, 15]. ${\epsilon }_{dd}^{*}$ is the value of ${\epsilon }_{dd}$ at which the superfluid-supersolid phase transition occurs.

Figure 2

Time dependence (insets) and Fourier transform $S^T\left(\omega \right)/\omega$ (main panel) corresponding to different observables of the zigzag configuration for ${\epsilon }_{dd}=1.467$: (a) Response $\langle xy\rangle$ to a sudden trap rotation. (b) Response $\langle xy\rangle$ to a sudden stop of the trap rotation (c) Response $\langle L_z\rangle$ to a sudden stop of the trap rotation.

The vertical red dashed line is the value ${\omega }_{31}$ [Eq. (3)] given by the irrotational moment of inertia, while the vertical red dash-dotted line is the value ${\omega }_{1,-1}$ [Eq. (2)] given by the true moment of inertia of the system.

Figure 3

Top: Incoherent droplet configuration for the same parameters as in Figs. 1 and 1, except ${\epsilon }_{dd}=1.553$. Bottom: response $\langle xy\rangle$ of the incoherent droplets to a sudden rotation of the trap.