Scalable $W$-type entanglement resource in neutral-atom arrays with Rydberg-dressed resonant dipole-dipole interaction

While the Rydberg-blockade regime provides the natural setting for creating $W$-type entanglement with cold neutral atoms, it is demonstrated here that a scalable entanglement resource of this type can even be obtained under completely different physical circumstances. To be more precise, a special instance of twisted $W$ states -- namely, $\pi$-twisted ones -- can be engineered in one-dimensional arrays of cold neutral atoms with Rydberg-dressed resonant dipole-dipole interaction. In particular, it is shown here that this is possible even when a (dressed) Rydberg excitation is coupled to the motional degrees of freedom of atoms in their respective, nearly-harmonic optical-dipole microtraps, which are quantized into dispersionless (zero-dimensional) bosons. For a specially chosen ("sweet-spot") detuning of the off-resonant dressing lasers from the relevant internal atomic transitions, the desired $\pi$-twisted $W$ state of Rydberg-dressed qubits is the ground state of the effective excitation -- boson Hamiltonian of the system in a broad window of the relevant parameters. Being at the same time separated from the other eigenstates by a gap equal to the single-boson energy, this $W$ state can be prepared using a Rabi-type driving protocol. The corresponding preparation times are independent of the system size and several orders of magnitude shorter than the effective lifetimes of the relevant atomic states.