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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. 

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). 

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). 

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 

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