We have studied Josephson tunneling through a circularly polarized micron or submicron-size disk of a soft ferromagnetic material. Such a disk contains a vortex that exhibits rich classical dynamics and has recently been proposed as a tool to study quantum dynamics of the nanoscale vortex core. The change in the Josephson current that is related to a tiny displacement of the vortex core has been computed analytically and plotted numerically for permalloy disks used in experiments. It is shown that a Josephson junction with a magnetic disk in the vortex state can be an interesting physical system that may be used to measure the nanoscale motion of the magnetic vortex.
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