Saturation of the Cram\'er-Rao Bound for the Atomic Resonance Frequency with 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 $\pi$-pulses; setting a fundamental limit on the precision obtainable using the resonance method. We show that measurements using sequences of pulses with alternating phases globally saturates the quantum Cram\'er-Rao bound, achieving the theoretical limit of precision for atomic resonance frequency estimation.