Fig. 1

Comparison of spin ensemble states. a In a fully transient state (here visualized at the example of a random pattern of radio frequency pulses), the spin trajectories on the Bloch sphere are, in general, very sensitive to magnetic field inhomogeneities. b The hybrid state is explicitly designed to mitigate these sensitivities, while still allowing the magnetization to visit the entire Bloch sphere. c Fully adiabatic transitions between steady states have the same robustness to magnetic field deviations, however, they trap the magnetization on the steady-state ellipse^{9, 12,13,14}, which diminishes the capability to encode tissue properties such as relaxation times. The steady-state ellipse is described by setting the left-hand side of Eq. (2) to zero. a–c Deviations of B₀, which dictates the Larmor frequency, has strong practical implications, because local magnetic field variations in a sample in nuclear magnetic resonance (or in a volume element in magnetic resonance imaging (MRI), also known as a voxel and here visualized as a cube) give rise to a distribution of different Larmor frequencies. Consequently, the observed signal, given by the integral over all measured frequencies, depends on the unknown distribution of Larmor frequencies²³. Here, ϕ denotes the phase accumulated over one repetition time T_R, and we define ϕ = π as the on-resonance condition. d–f Deviations of the radio-frequency field B₁, which dictates the Rabi frequencies, can lead to strong variations of the spin trajectories in the transient state. Note that in clinical MRI, the Rabi frequency within a voxel is approximately constant