  Strongly-coupled quantum fields, such as multi-component atomic condensates,
optical fields and polaritons, are remarkable systems where the simple dynamics
of coupled oscillators can meet the intricate phenomenology of quantum fluids.
When the coupling between the components is coherent, not only the particles
number, but also their phase texture that maps the linear and angular momentum,
can be exchanged. Here, on a system of exciton-polaritons, we have realized a
so-called full-Bloch beam: a configuration in which all superpositions of the
upper and the lower polariton -- all quantum states of the associated Hilbert
space -- are simultaneously present at different points of the physical space,
evolving in time according to Rabi-oscillatory dynamics. As a result, the light
emitted by the cavity displays a peculiar dynamics of spiraling vortices
endowed with oscillating linear and angular momentum and exhibiting ultrafast
motion of their cores with striking accelerations to arbitrary speeds. This
remarkable vortex motion is shown to result from distortions of the
trajectories by a homeomorphic mapping between the Rabi rotation of the full
wavefunction on the Bloch sphere and Apollonian circles in the real space where
the observation is made. Such full-Bloch beams offer new prospects at a
fundamental level regarding their topological properties or in the
interpretation of quantum mechanics, and the Rabi-rotating vortices they yield
should lead to interesting applications such as ultrafast optical tweezers.
