Observation of Anisotropic Superfluid Density in an Artificial Crystal

We experimentally and theoretically investigate the anisotropic speed of sound of an atomic superfluid (SF) Bose-Einstein condensate in a 1D optical lattice. Because the speed of sound derives from the SF density, this implies that the SF density is itself anisotropic. We find that the speed of sound is decreased by the optical lattice, and the SF density is concomitantly reduced. This reduction is accompanied by the appearance of a zero entropy normal fluid in the purely Bose condensed phase. The reduction in SF density—first predicted [A. J. Leggett, Phys. Rev. Lett. 25, 1543 (1970).] in the context of supersolidity—results from the coexistence of superfluidity and density modulations, but is agnostic about the origin of the modulations. We additionally measure the moment of inertia of the system in a scissors mode experiment, demonstrating the existence of rotational flow. As such we shed light on some supersolid properties using imposed, rather than spontaneously formed, density order.

Figure 1

Concept. (a) A BEC is confined in a harmonic trap superimposed with a 1D optical lattice (along ${\mathbf{e}}_x$, green), spatially modulating the condensate density (red). The dashed and dotted lines call out a region of nominally constant mean density and the left and right columns indicate the (b) state of the condensate and (c) SF in the presence of a current. These were computed for a $5E_{\mathrm{r}}$ deep lattice and plot: $i$. density (red), ii. current (green), iii. phase (orange), and iv. local velocity (blue). The red dashed line plots the mean density $\overline{\rho }$.

Figure 2

Bragg spectroscopy. Black and red symbols mark excitations created along ${\mathbf{e}}_x$ and ${\mathbf{e}}_y$, respectively. (a) Transferred population fraction $p$ as a function of frequency difference $\delta \omega$ with wave vector $\delta k/2\pi =0.26\mu {\mathrm{m}}^{-1}$ and lattice depth $U_0=5.7E_{\mathrm{r}}$. The solid curve is a Lorentzian fit giving the resonance frequency marked by the vertical dashed line. (b) Phonon dispersion obtained from Bragg spectra. The bold symbols resulted from (a) and the linear fit (with zero intercept) gives the speed of sound. (c) Anisotropic speed of sound. The bold symbols are derived from (b) and the solid curves are from BdG simulations (no free parameters [21]). (d) SF density obtained from speed of sound measurements (blue markers, error bars mark single-sigma statistical uncertainties). We compare with two models: the red dashed curve plots a homogeneous gas BdG calculation, and the solid black curve plots the result of Eq. (1). The simulations used our calibrated experimental parameters. In (a)–(c) each point has uncertainty as shown on the last point.

Figure 3

Moment of inertia from scissors mode. (a),inset Measured dipole mode frequencies (circles) along with fits (curves) where the bare trap frequency is the only free parameter for each curve. (a) Normalized scissors mode frequency. Blue and green correspond to $U_0=0$ trap frequencies (34,51) and (54,36) Hz, respectively. (b) Moment of inertia in units of $I_{\mathrm{c}}$. In (a) and (b) each point has uncertainty as shown on the first point. Symbols are the data computed as described in the text, and the solid curves are GPE predictions.