Application of the small-tip-angle approximation in the toggling frame for the design of analytic robust pulses in quantum control

We apply the small-tip-angle approximation in the toggling frame in order to analytically design robust pulses against resonance offsets for state to state transfer in two-level quantum systems. We show that a broadband or a local robustness up to an arbitrary order can be achieved. We provide different control parametrizations to satisfy experimental constraints and limitations on the amplitude or energy of the pulse. A comparison with numerical optimal solutions is made.