  Quantum Kerr-nonlinear oscillator is a paradigmatic model in cavity and
circuit quantum electrodynamics, and quantum optomechanics. We theoretically
study the echo phenomenon in a single impulsively excited ("kicked")
Kerr-nonlinear oscillator. We reveal two types of echoes, "quantum" and
"classical" ones, emerging on the long and short time-scales, respectively. The
mechanisms of the echoes are discussed, and their sensitivity to dissipation is
considered. These echoes may be useful for studying decoherence processes in a
number of systems related to quantum information processing.
