3 Some of the obtained energies are shown in Fig.3b. For small values of α the BEC groundstate is a homogenous superfluid. At a critical value of α$_{suso}$ ≈ 30 . 1, one finds a transition to a stable supersolid state. This first-order transition precedes the roton-instability [20] and takes place at a finite roton gap of δ$_{suso}$ ≈ 0 . 66 ¯ h $^{2}$k2 rot 2 M . The ex- istence of several competing states with similar energies but different crystal symmetries (see Fig.3b), may gener- ally complicate the experimental preparation of ordered states. In this respect, the dynamical tunability of the interaction strength ˜ C$_{6}$ via changing the laser intensity can serve a useful tool to steer the BEC evolution. In order to demonstrate this point, we also studied the time evolution, starting from a homogenous BEC. As a specific example, we discuss the BEC dynamics for a simple time-dependence of α , shown in Fig.4a. The calculation starts from a homogenous condensate with small random phase noise and uses a complex-time in- tegration with a small imaginary contribution [22]. The instantaneous increase of α at time t = 0 from α = 0 to 60 induces the roton instability. This sudden param- eter quench, however, causes relaxation towards a short- range ordered, ”glassy” [5] state (Fig.4b), as discussed above. As α is decreased close to the phase transition some of the structures vanish entirely, leading to a mixed phase in which extended superfluid fractions of nearly constant density coexist with density-modulated domains (Fig.4c). The latter increase in time (Fig.4d), and ul- timately merge to form sizable ”crystallites” of regular density modulations (Fig.4e). instability, at which density modulations may develop without energy cost. In ultracold gases, a similar behav- ior was found in tightly confined, pancake-shaped dipolar BECs [13, 14]. While it is the partially attractive nature of the anisotropic interactions that generates the 2 D ro- ton excitations in this case [15], it also leads to collapse in the post-roton-instability phase, thus, precluding forma- tion of supersolid density modulations [16, 17] in higher than one dimension [18]. The present interactions, on the other hand, are entirely repulsive, which may indeed provide for stable regular density modulations. Turning to a discussion of the experimental feasibility, we consider a particular example of coupling to 60 S Ry- dberg states in a $^{87}$Rb condensate, for which Rydberg excitation has recently been demonstrated [23]. Fig.5 shows the corresponding ”phase diagram” for a typical density of 10 $^{14}$cm − $^{3}$, also including a finite s-wave scat- tering length a > 0. The latter only leads to some in- crease of the critical Rabi frequencies for inducing the roton instability. Importantly, the transition to a su- persolid can be realized with Rabi frequencies of a few hundred kHz, and the condition | Δ | ≫ Ω can be well fulfilled deep in the roton-instability regime. Yet, the de- tuning is sufficiently small to avoid excitation of nearby Addressing this question, requires us to go beyond the linear analysis discussed above. We performed numerical simulations of Eq.(2) on a three-dimensional grid of ∼ 10 7 grid points with periodic boundary conditions. Because of the nonlinear character of the Gross Pitaevskii equa- tion a standard imaginary-time integration may generally not converge to the true groundstate of the system. As also found for lattice-confined dipolar bosons [19], there is a large number of stationary states, that correspond to local minima of the total energy. As a consequence, the system generally approaches a glassy state with short- range ordered density modulations, when starting from a homogenous initial state (see below). We, thus, used variational calculations, based on periodically arranged Gaussians with varying width and lattice constant, to provide the proper initial wavefunction for a subsequent imaginary time-evolution according to Eq.(2). FIG. 5: Required laser parameters to induce the roton insta- bility (solid lines) for n = 60 and ρ 0 = 10 $^{14}$cm − $^{3}$. The upper curve also accounts for s-wave scattering with a = 100 a 0 . FIG. 4: Snapshots of the BEC dynamics for a time-varying interaction parameter α ( t ) shown in (a). Panels (b)-(e) show the density along orthogonal slices through the simulation box at times indicated in (a). The upper and right axes in (a) show the actual time and Rabi frequency for a $^{87}$Rb BEC with n = 60, Δ = 50MHz and ρ 0 = 2 · 10 $^{14}$cm − $^{3}$. 0 10 20 30 40 50 60 !  [MHz] 0.2 0.4 0.6 0.8 "  [MHz] a = 0 a = 100 a 0 roton-unstable