Two-Dimensional Supersolid Formation in Dipolar Condensates

Dipolar condensates have recently been coaxed to form the long-sought supersolid phase. While one-dimensional supersolids may be prepared by triggering a roton instability, we find that such a procedure in two dimensions (2D) leads to a loss of both global phase coherence and crystalline order. Unlike in 1D, the 2D roton modes have little in common with the supersolid configuration. We develop a finite-temperature stochastic Gross-Pitaevskii theory that includes beyond-mean-field effects to explore the formation process in 2D and find that evaporative cooling directly into the supersolid phase—hence bypassing the first-order roton instability—can produce a robust supersolid in a circular trap. Importantly, the resulting supersolid is stable at the final nonzero temperature. We then experimentally produce a 2D supersolid in a near-circular trap through such an evaporative procedure. Our work provides insight into the process of supersolid formation in 2D and defines a realistic path to the formation of large two-dimensional supersolid arrays.

Figure 1

(a) Crystal preparation from interaction quench, evolved with the eGPE, for $N\approx 2.1\times {10}^5$ Dy atoms [quench (i)]. Isosurfaces are at 5% max density, with color indicating phase. Insets: $z$ column densities normalized to max value from the entire simulation. (b) Dynamic structure factor for an unmodulated BEC ($a_s=92a_0$) in energy-momentum space, normalized to peak value. The lowest-energy roton modes are indicated, and the ground state with an $m=2$ roton mode added is shown, revealing the localized nature of the rotons. Parameters are otherwise the same as in (a). (c) Crystal preparation from temperature quench (evaporative cooling) evolved with the SeGPE [quench (ii)]. The temperature decreases as the chemical potential and condensate number rise, with scattering length fixed at $a_s=88a_0$. For all subplots $f_{x,y,z}=(33,33,167)\mathrm{Hz}$, and $l_z=\sqrt{\hslash /m{\omega }_z}$.

Figure 2

Supersolid quality. (a) Global phase coherence $C^p$ over time for interaction quenches [quench (i)] into linear chain (blue) and pancake crystal (red) and temperature quenches [quench (ii)] into the pancake crystal (black). Diamonds link to example frames in Figs. 1 and 1. Each curve is averaged over 3–5 runs with an error band marking one standard deviation. Time $t=0$ corresponds to when the crystals first fully mature. (b) Density overlap $C^d$ between the time-dependent and ground state densities. Parameters are the same as Fig. 1, but for linear chain $f_{x,y,z}=(33,110,167)\mathrm{Hz}$ and $N=82\times {10}^3$.

Figure 3

Experimental realization of a 7-droplet hexagon state. (a) Exemplary in situ image of the density profile. (b) Image after 36 ms time-of-flight (TOF) expansion, averaged over 68 trials of the experiment. Hexagonal modulation structure is clearly present in the averaged image. Note the rotation of the hexagon between in situ and TOF images. (c),(d) Corresponding simulations for the same trap, and with $a_s=90a_0$ and $\approx 4.4\times {10}^4$ atoms within the droplets.