  Time varying media can dramatically modify the emission of embedded sources
by producing time reversed waves refocusing on the source. Here we show that
such a back action can create an angular momentum by setting the source in a
spontaneous spin state. We experimentally implement this coupling using
self-propelled bouncing droplets sources coupled to the surface waves they emit
on a parametrically excited bath. The spin state dynamics result from a
self-organized interplay between the source motion and the time reversed waves.
The discrete stability analysis agrees with the experimental observations. In
addition, we show that these spin states provide a unique opportunity for an
experimental access to parameters enabling comparison and calibration of the
various existing models.
