2 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 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 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 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 F IG. 3: (a) Roton gap δ as a function of the interaction pa- r a meter α . (b) Energy density ε = ρ$_{0}$ 2 $_{N}$〈 Ψ | − ¯ h $^{2}$∇2 2 M + ˆ H | Ψ 〉 f or different crystal symmetries relative to the energy den- si ty ε hom = π $^{2}$α/ 3 of a homogeneous BEC. Panel (c) pro- v ides an enlarged view around the transition point. T h e e ffective Rydberg state lifetime for excitation of $^{87}$Rb t o n = 60( C 6 = 9 . 7 · 10 $^{20}$a . u . [21] ) with Δ = 50MHz and ρ 0 = 10 $^{14}$cm − 3 is shown in (d). FIG. 2: Dispersion relation ϵ ( k ) for different values of the interaction parameter α . The arrow indicates the roton gap δ . i ¯ h ∂ Ψ ∂t = [ − ¯ h $^{2}$∇ 2 2 M + g | Ψ | 2 + ∫ d r $^{′}$U ( r − r $^{′}$) | Ψ( r $^{′}$) | 2 ] Ψ , (2) W$_{G}$ = 2 ∑ i = j |〈 G | ˆ H L | E$_{i}$ 〉〈 E$_{i}$ | ˆ H L | E$_{ij}$ 〉| 2 ¯ h $^{2}$Δ$^{2}$(2¯ h Δ − V ee ( r $_{ij}$)) + ∑ i |〈 G | ˆ H L | E$_{i}$ 〉| 2 ¯ h Δ = ∑ i = j ¯ h $^{4}$Ω4 8¯ h $^{2}$Δ$^{2}$(2¯ h Δ − V ee ( r $_{ij}$)) + ¯ h $^{2}$Ω$^{2}$N 4¯ h Δ = N ¯ h $^{2}$Ω2 4¯ h Δ + N ( N − 1)¯ h $^{4}$Ω4 16¯ h $^{3}$Δ 3 + ∑ i<j ˜ C 6 r 6 $_{ij}$+ R 6 c (1) k$_{·}$ R$_{c}$ ϵ [ ℏ 2 ◁MR 2$_{c}$ ] δ 0 2 4 6 8 0 10 20 30 ! =3 ! =30 ! =60 0 2 4 6 8 -0.6 -0.4 -0.2 0 10 20 30 40 50 0 3 6 9 homogeneous fcc bcc hcp α$_{suso}$ $^{α}$min α inst ε − ε hom [ ℏ 2 MR 5$_{c}$ ] γ − 1 e ff [s] δ [ ℏ 2 MR 2$_{c}$ ] α (a) (b) (d) (c)