Three-Dimensional Roton Excitations and Supersolid Formation in Rydberg-Excited Bose-Einstein Condensates

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 ground state 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 supersolid state in three-dimensional condensates. A detailed analysis of decoherence and loss mechanisms suggests that these phenomena are observable with current experimental capabilities.

Figure 1

(a) Schematics of the considered three-level atom, illustrating the laser coupling between the atomic ground state $|n_{0}S⟩$ and the Rydberg state $|nS⟩$. For ${\Delta }_1\gg {\Omega }_1$, the system reduces to an effective two-level atom, with the states $|g⟩\equiv |n_{0}S⟩$ and $|e⟩\equiv |nS⟩$ coupled with a two-photon Rabi frequency $\Omega$ and detuning $\Delta$. (b) Effective potential resulting from the off-resonant coupling to the strongly interacting Rydberg states for $n=60$ and $\Delta =50\mathrm{MHz}$. Panels (c) and (d) provide an enlarged view of the potential showing the contributions from both ground state-Rydberg atom and ground state-ground state atom interactions (solid line) as well as the sole contribution from the latter (dashed line).

Figure 2

Dispersion relation $ϵ(k)$ for different values of the interaction parameter $\alpha$. The arrow indicates the roton gap $\delta$.

Figure 3

(a) Roton gap $\delta$ as a function of the interaction parameter $\alpha$. (b) Energy density $\epsilon =\frac{{\rho }_0}{2N}⟨\Psi |\frac{-{\hslash }^2{\nabla }^2}{2M}+\widehat{H}|\Psi ⟩$ for different crystal symmetries relative to the energy density ${\epsilon }_{\mathrm{hom}}={\pi }^2\alpha /3$ of a homogeneous BEC. Panel (c) provides an enlarged view around the transition point. The effective Rydberg state lifetime for excitation of ${}^{87}\mathrm{Rb}$ to $n=60$ ($C_6=9.7\times {10}^{20}\mathrm{a}.\mathrm{u}.$ [26]) with $\Delta =50\mathrm{MHz}$ and ${\rho }_0={10}^{14}{\mathrm{cm}}^{-3}$ is shown in (d).

Figure 4

Snapshots of the BEC dynamics for a time-varying interaction parameter $\alpha (t)$ 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 ${}^{87}\mathrm{Rb}$ BEC with $n=60$, $\Delta =50\mathrm{MHz}$, and ${\rho }_0=2\times {10}^{14}{\mathrm{cm}}^{-3}$.

Figure 5

Required laser parameters to induce the roton instability (solid lines) for $n=60$ and ${\rho }_0={10}^{14}{\mathrm{cm}}^{-3}$. The upper curve also accounts for $s$-wave scattering with $a=100a_0$.