Figure 4

Drops representation of the propagators. Trajectories of the propagators U(t) for standard rectangular pulses of constant amplitude and phase (A) and by TRE pulses (B) as a function of time. In panels (A,B), T ₁₈₀ and T _R are pulse durations corresponding to a 180°-rotation around the x-axis. In the DROPS representation²⁹, operators are depicted by complex spherical functions \(f(\theta ,\varphi )=|f(\theta ,\varphi )|{e}^{i\eta }\), where for given azimuthal and polar angles θ and ϕ, the absolute value \(|f(\theta ,\varphi )|\) is represented by the distance from the origin and the phase angle is color coded (\(\eta =0\): red, \(\eta =\frac{\pi }{2}\): yellow, \(\eta =\pi \): green, \(\eta =\frac{3\pi }{2}\): blue). At t = 0, the propagator is the identity operator 1 (represented by red spheres), while at (A) t = T _360° = 2T _180° and (B) t = 2T _R the propagator is −1 (represented by green spheres). In panels (A,B), the identity operator is created again at t = 4T _180° = T _720° and t = 4T _R , respectively.