  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.
