Fig. 2

Hybrid state design space. a The well-known steady state is characterized by a negligible population of all (real-valued and complex) transient eigenstates and its adiabaticity condition is given by Eq. (3): Both population densities are small whenever the change of the flip angle Δα is small compared to the theoretical limit (black volume). The hybrid state described here is characterized by a population of the real-valued eigenstate while simultaneously avoiding a population of the complex-valued eigenstate. b, c In simulation, this occurs exactly in the predicted area of the parameter space (purple volume). In this volume, Eq. (4) is fulfilled, yet Eq. (3) is not, so that the hybrid state can occur. For the illustrative example shown here, we assumed T₁ = T₂, a constant Δα, and Δϕ = 0. The simulation departs from the steady-state with α = ϕ and we depict the population density after 100 repetitions, each associated with a constant increase of α by Δα