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We design, by invariant-based inverse engineering, driving fields that invertthe population of a two-level atom in a given time, robustly with respect todephasing noise and/or systematic frequency shifts. Without imposingconstraints, optimal protocols are insensitive to the perturbations but need aninfinite energy. For a constrained value of the Rabi frequency, a flat $\pi$pulse is the least sensitive protocol to phase noise but not to systematicfrequency shifts, for which we describe and optimize a family of protocols.

Fast and robust population transfer in two-level quantum systems with dephasing noise and/or systematic frequency errors