Saturation of the Cramér-Rao bound for atomic resonance frequency measurements with a phased array of hyperbolic secant pulses

Precise estimation of the atomic resonance frequency is fundamental for the characterization and control of quantum systems. The resonance experiment is a standard method for this measurement, wherein the drive field frequency is swept to invert the system population. We analyze the classical and quantum Fisher information for the resonance experiment driven by hyperbolic-secant-shaped $\ensuremath{\pi}$ pulses, setting a fundamental limit on the precision obtainable using the resonance method. We show that measurements using sequences of pulses with alternating phases nearly saturate the quantum Cram\'er-Rao bound, improving the precision of atomic resonance frequency measurements.