On the geometric phases during radio frequency pulses with sine and cosine amplitude and frequency modulation

In this work, we describe the formation of geometric phases during nonadiabatic frequency swept (FS) radio frequency (RF) pulses with sine amplitude modulation and cosine frequency modulation functions. The geometric phases during the FS pulse were analyzed using a Schrödinger equation formalism, and the unified analytical expression for the geometric phase was derived. We present the solutions for sub-geometric phase components incorporated in spinor wavefunctions for the RF Hamiltonian of spin ½ nuclei. We demonstrate that the geometric phases during sine/cosine RF pulses are opposite in signs for different initial conditions of the spinor and that geometric phases can accumulate in correspondence to different magnetization trajectories. The derived formalism could be extended for the evaluation of the geometric phases during a wide class of amplitude- and frequency-modulated pulses used in MRI and in high-resolution NMR.