Effect of Diffusion on Bunched Beam Echo

When a beam receives a dipole kick, its centroid signal decoheres due to the betatron tune spread in the beam. Long after the signal has decohered, however, a followup quadrupole kick to the beam brings a pronounced echo back to the centroid signal. This echo effect has been analyzed for the case of a bunched beam in Ref. (1). In this work, the perturbation calculation of Ref. (1) is extended to include a diffusion in betatron amplitude. The effect of diffusion on the magnitude of the echo is then parameterized and studied.