Magnetic resonance in an elliptic magnetic field

The behaviour of a particle with a spin 1/2 and a dipole magnetic moment in a time-varying magnetic field in the form $(h_0 cn(\omega t,k), h_0 sn(\omega t,k), H_0 dn(\omega t,k))$, where $\omega$ is the driving field frequency, $t$ is the time, $h_0$ and $H_0$ are the field amplitudes, $cn$, $sn$, $dn$ are Jacobi elliptic functions, $k$ is the modulus of the elliptic functions has been considered. The variation parameter $k$ from zero to 1 gives rise to a wide set of functions from trigonometric shapes to exponential pulse shapes modulating the field. The problem was reduced to the solution of general Heun' equation. The exact solution of the wave function was found at resonance for any $k$. It has been shown that the transition probability in this case does not depend on $k$. The present study may be useful for analysis interference experiments, improving magnetic spectrometers and the field of quantum computing.