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# Three-dimensional Roton-Excitations and Supersolid formation in Rydberg-excited Bose-Einstein Condensates 

N. Henkel, R. Nath and T. Pohl 

Max Planck Institute for the Physics of Complex Systems, N¨ othnitzer Straße 38, 01187 Dresden, Germany 

We study the behavior of a Bose-Einstein condensate in which atoms are weakly coupled to a highly excited Rydberg state. Since the latter have very strong van der Waals interactions, this coupling induces effective, nonlocal interactions between the dressed groundstate atoms, which, opposed to dipolar interactions, are isotropically repulsive. Yet, one finds partial attraction in momentum space, giving rise to a roton-maxon excitation spectrum and a transition to a super solid state in three-dimensional condensates. A detailed analysis of decoherence and loss mechanisms, suggests that these phenomena are observable with current experimental capabilities. 

Since introduced by Landau in a series of seminal arti- cles [1], the notion of a roton minimum in the dispersion of a quantum liquid has been pivotal to understanding superfluidity in helium. This later led to the prediction of a peculiar solid state upon softening of the roton ex- citation energy [2], simultaneously possessing crystalline and superfluid properties. In such a supersolid [3], the particles that must supply the rigidity to form a crys- tal, at the same time provide for superfluid nonviscous flow. Forty years after its conjecture, this apparent con- tradiction continues to attract theoretical interest and has ushered in an intense search for experimental evi- dence in solid $^{4}$He, whose interpretations are currently under active debate [4, 5]. 

Here, we demonstrate how three-dimensional roton ex- citations can be realized in atomic Bose-Einstein con- densates (BECs), thereby introducing an alternative sys- tem to study supersolidity. The supersolid phase transi- tion is shown to arise from effective interactions, realized through off-resonant optical coupling [6–8] to highly ex- cited Rydberg states. Owing to the strong increase of atomic interactions with their principal quantum num- ber n , resonantly excited Rydberg gases have proved to be an ideal platform to study strong interactions in many-body systems [9] on short µ s time scales. The present approach − based on off-resonant two-photon excitation [see Fig.1a] of Bose-condensed alkaline atoms − permits to utilize the strong Rydberg interactions over much longer times of ∼ 100ms. In particular, we consider coupling to nS Rydberg states with vanishing orbital an- gular momentum, which, as opposed to dipole-dipole in- teractions, gives rise to isotropically repulsive interaction potentials for the groundstate atoms, and, thus, ensures stability of the condensate. 

The system is described as a gas of N atoms with mass M at positions r$_{i}$ , each possessing a ground state | g$_{i}$ 〉 and an excited nS Rydberg state, denoted by | e$_{i}$ 〉 . The two states are optically coupled with a two-photon Rabi frequency Ω and detuning Δ (see Fig.1a). Defin- ing corresponding transition and projection operators 

PACS numbers: 32.80.Ee,03.75.Kk,67.80.K-,32.80.Qk 

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ˆ σ ( i ) αβ = | α$_{i}$ 〉〈 β$_{i}$ | ( α, β = e , g), the resulting N -particle in- teraction can be written as 

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where ˆ H$_{L}$ = ¯ h Ω 2 ∑ i ˆ σ ( i ) eg + ˆ σ ( i ) ge describes the laser cou- pling and V$_{ee}$ ( r$_{ij}$ ) = C$_{6}$/r 6 ij > 0 denotes the van der Waals (vdW) interaction between two Rydberg atoms at a distance r$_{ij}$ = r$_{i}$ − r$_{j}$ . Because of the strong C$_{6}$ ∼ n 11 scaling of the vdW coefficient, such Rydberg-Rydberg atom interactions are orders of magnitude larger than those of groundstate atoms. We are interested in the potential surface W$_{G}$ ( r$_{1}$ , ..., r$_{N}$ ) that asymptotically con- nects to the many-body groundstate | G 〉 = ⊗ k | g$_{k}$ 〉 , un- der the condition of far-off resonant driving Ω / | Δ | ≪ 1 and for Δ < 0. Under the latter condition the N -body 

