Echo effect in accelerators

The echo effect has been known for many years in different fields of physics. Examples are the spin echo in solids, photon echo in solids and gases, plasma wave echo, and the echo in a liquid with gas bubbles. The media that exhibit the echo characteristically consist of (or contain in them) an ensemble of oscillators with different eigenfrequencies and negligibly small dissipation. An initial perturbation applied to such a medium excites oscillations (or waves) that slowly damp due to dephasing. This kind of damping differs essentially from that which would result from a dissipative mechanism; in particular, dephasing does not increase the entropy of the medium. It has the remarkable feature that even after the oscillations are completely damped out, the system keeps a "recollection" about them, and a special kind of a disturbance applied to the medium can transiently restore the oscillations in the form of an echo signal. The betatron echo in a hadron accelerator can be observed in a situation where the beam is injected off-center into the ring at time n=O (n is the time measured in the number of turns), causing its centroid to undergo betatron oscillations