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Using the quantum-brachistochrone formalism, we address the problem offinding the fastest possible (time-optimal) deterministic conversion between$W$ and Greenberger-Horne-Zeilinger (GHZ) states in a system of three identicaland equidistant neutral atoms that are acted upon by four external laserpulses. Assuming that all four pulses are close to being resonant with the sameinternal (atomic) transition -- the one between the atomic ground state and ahigh-lying Rydberg state -- each atom can be treated as an effective two-levelsystem ($gr$-type qubit). Starting from an effective system Hamiltonian, whichis valid in the Rydberg-blockade regime and defined on a four-state manifold,we derive the quantum-brachistochrone equations pertaining to the fastestpossible $W$-to-GHZ state conversion. By numerically solving these equations,we determine the time-dependent Rabi frequencies of external laser pulses thatcorrespond to the time-optimal state conversion. In particular, we show thatthe shortest possible $W$-to-GHZ state-conversion time is given by$T_{\textrm{QB}}= 6.8\:\hbar/E$, where $E$ is the total laser-pulse energyused, this last time being significantly shorter than the state-conversiontimes previously found using a dynamical-symmetry-based approach[$T_{\textrm{DS}}=(1.33-1.66)\:T_{\textrm{QB}}$].

Quantum-brachistochrone approach to the conversion from $W$ to Greenberger-Horne-Zeilinger states for Rydberg-atom qubits