Oscillatory eigenmodes and stability of one and two arbitrary fractional vortices in long Josephson 0-k junctions

We investigate theoretically the eigenmodes and the stability of one and two arbitrary fractional vortices pinned at one and two k phase discontinuities in a long Josephson junction. In the particular case of a single k discontinuity, a vortex is spontaneously created and pinned at the boundary between the 0 and k regions. In this work we show that only two of four possible vortices are stable. A single vortex has an oscillatory eigenmode with a frequency within the plasma gap. We calculate this eigenfrequency as a function of the fractional flux carried by a vortex. For the case of two vortices, pinned at two k discontinuities situated at some distance a from each other, splitting of the eigenfrequencies occurs. We calculate this splitting numerically as a function of a for different possible ground states. We also discuss the presence of a critical distance below which two antiferromagnetically ordered vortices form a strongly coupled “vortex molecule” that behaves as a single object and has only one eigenmode.