  We present a robust protocol for implementing high-fidelity multiqubit
controlled phase gates $(C^kZ)$ on neutral atom qubits coupled to highly
excited Rydberg states. Our approach is based on extending adiabatic rapid
passage to two-photon excitation via a short-lived intermediate excited state
common to alkali-atom Rydberg experiments, accounting for the full impact of
spontaneous decay and differential AC Stark shifts from the complete manifold
of hyperfine excited states. We evaluate and optimize gate performance,
concluding that for Cs and currently available laser frequencies and powers, a
$CCZ$ gate with fidelity $\mathcal{F}>0.995$ for three qubits and $CCCZ$ with
$\mathcal{F}>0.99$ for four qubits is attainable in $\sim 1.8$~$\mu$s via this
protocol. Higher fidelities are accessible with future technologies, and our
results highlight the ultility of neutral atom arrays for the native
implementation of multiqubit unitaries.
