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potential energy of the many-body groundstate | G 〉 is separated from the excited states by at least Δ [10], jus- tifying the use of a Born-Oppenheimer (BO) treatment for the atomic dynamics on the potential surface W$_{G}$ . The BO surface W$_{G}$ can be determined from a many- body perturbation expansion in the small parameter Ω / | Δ | ≪ 1, which up to fourth order only involves cou- pling to the singly and doubly excited many-body states | E$_{i}$ 〉 = | e$_{i}$ 〉 ⊗ k ̸ = i | g$_{k}$ 〉 and | E$_{ij}$ 〉 = | e$_{i}$e$_{j}$ 〉 ⊗ k ̸ = i,j | g$_{k}$ 〉 . Ex- plicitly, we obtain 

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Omitting constant terms, the leading-order, N -body BO surface is given by a sum of binary effective potentials U ( r$_{ij}$ ) = ˜ C$_{6}$/ ( r 6 ij + R 6 $_{c}$), exemplarily shown in Fig.1b. 

The peculiar shape of U is easily understood within a simple two-atom picture: For far-distant atomic pairs ( r$_{ij}$ ≫ R$_{c}$ ), a small fraction (Ω / 2Δ) 2 is indepen- dently admixed to each groundstate atom, such that U ( r$_{ij}$ ) is of vdW-type with an effective coefficient ˜ C$_{6}$ = (Ω / 2Δ) $^{4}$C$_{6}$ . At smaller distances, however, the inter- action shift V$_{ee}$ ( r$_{ij}$ ) renders dressing to the doubly ex- cited | E$_{ij}$ 〉 states ineffective, such that the effective po- tential approaches a constant value below a critical dis- tance R$_{c}$ = ( C$_{6}$/ 2¯ h | Δ | ) 1 / $^{6}$. Typically, R$_{c}$ can take on rather large values of a few µ m. 

For simplicity, the above discussion has ignored inter- actions between pairs of groundstate atoms as well as groundstate-Rydberg atom interactions. Both, however, can be included in the calculations of the effective po- tential (see Fig.1c). Although being comparably strong, the range of the corresponding contributions is consider- ably smaller than R$_{c}$ . Upon avoiding photo-excitation of Rydberg-molecular resonances [11] these additional po- tentials may, thus, be described in terms of an s-wave scattering pseudopotential, with a scattering length a . On the other hand, the large value of R$_{c}$ prevents such 

a simplified treatment for U ( r ). However, as long as ˜ C$_{6}$ ≪ ¯ h $^{2}$R 4 $_{c}$/M its effect can be described within a first Born approximation [12] [26]. With these simplifications and in the zero-temperature limit one arrives at the fol- lowing nonlocal, nonlinear Schr¨ odinger equation 

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for the condensate wavefunction Ψ( r ). The contact in- teraction term ∝ g = 4 π ¯ h $^{2}$a/M may be tuned to very small values. For clarity, it will, hence, be omitted in the following and briefly discussed at the end of this work. 

For small ˜ C$_{6}$ the BEC ground state corresponds to a homogeneous superfluid with density ρ$_{0}$ . Its elementary excitations with wave number k and corresponding en- ergy ϵ are calculated from the corresponding Bogoliubov- de Gennes equations, which yield ϵ ( k ) 2 = ¯ h $^{2}$k2 2 M [ ¯ h $^{2}$k2 2 M + 2 ρ$_{0}$ ˜ U ( k )], where ˜ U ( k ) is the Fourier transform of the in- teraction potential U ( r ). Upon appropriate length and energy scaling (see Fig.2), the dispersion is determined by a single dimensionless parameter α = ρ$_{0}$M ˜ C$_{6}$/ ¯ h $^{2}$R$_{c}$ , parametrizing the interaction strength. Asymptotically the dispersion relation ϵ ( k ) has phonon and free particle character, at small and large k , respectively. However, due to the inner potential plateau at r < R$_{c}$ , the momen- tum space potential ˜ U has negative attractive contribu- tions around k = k$_{rot}$ ∼ 2 π/R$_{c}$ , such that the spectrum develops a roton minimum at k = k$_{rot}$ (see Fig.2) for suf- ficiently large α > α$_{rot}$ ≈ 4 . 8. The corresponding roton g a p δ decreases with increasing α and ultimately vanishe s a t α$_{inst}$ ≈ 50 . 1 (see Fig.3a), marking the onset of a roton 

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