  Neutral atoms are promising for large-scale quantum computing, but accurate
neutral-atom entanglement depends on large Rydberg interactions which strongly
limit the interatomic distances. Via a phase accumulation in detuned Rabi
cycles enabled by a Rydberg interaction of similar magnitude to the Rydberg
Rabi frequency, we study a controlled-phase gate with an arbitrary phase and
extend it to the controlled-NOT gate. The gates need only three steps for
coupling one Rydberg state, depend on easily accessible van der Waals
interaction that naturally arises between distant atoms, and have no rotation
error in the weak interaction regime. Importantly, they can work with very weak
interactions so that well-separated qubits can be entangled. The gates are
sensitive to the irremovable fluctuation of Rydberg interactions, but can still
have a fidelity over 98\% with realistic position fluctuation of qubits
separated over 20~$\mu$m.
