  The splitting instability of a doubly-quantized vortex in the BEC-BCS
crossover of a superfluid Fermi gas is investigated by means of a low-energy
effective field theory. Our linear stability analysis and non-equilibrium
numerical simulations reveal that the character of the instability drastically
changes across the crossover. In the BEC-limit, the splitting of the vortex
into two singly-quantized vortices occurs through the emission of phonons,
while such an emission is completely absent in the BCS-limit. In the
crossover-regime, the instability and phonon emission are enhanced, and the
lifetime of a doubly-quantized vortex becomes minimal. The emitted phonon can
be observed as a spiraling pattern amplified due to the rotational
superradiance, known as a mechanism to carry away energy and angular momentum
from a spinning black hole. We also investigate the influence of temperature,
population imbalance, and three-dimensional effects.
