Spin nutation effects in molecular nanomagnet$-$superconductor tunnel junctions

We study the spin nutation effects of the molecular nanomagnet on the Josephson current through a superconductor$|$molecular nanomagnet$|$superconductor tunnel junction. We explicitly demonstrate that due to the spin nutation of the molecular nanomagnet two oscillatory terms emerge in the $ac$ Josephson current in addition to the conventional $ac$ Josephson current. Some resonances occur in the junction due to the interactions of the transported quasiparticles with the bias voltage and molecular nanomagnet spin dynamics. The appearance of them indicate that the energy exchanged during these interactions is in the range of the superconducting energy gap. We also show that the spin nutation is able to convert the $ac$ Josephson current to a $dc$ one which is interesting for applications.