Role of Mode Interaction in the Amplification and Suppression of Echoes

Abstract

Phenomena similar to spin echo have recently been observed in a variety of physical systems whose only common feature is that they comprise large collections of oscillators distributed over some narrow frequency range. When these oscillators represent collective modes, any nonlinear interaction associated with echo generation also introduces coupling among the modes, thereby drastically altering the echo process. One consequence is the creation, under suitable conditions, of an unstable regime in which echo amplification of the type recently observed in a ferrimagnet can occur. The coupling must, however, be restricted to modes which are very close to each other in frequency. Specifically, we study in detail a multimode system in which oscillations are assumed to couple only if their frequency separation is within a range $\sigma$. If $\tau$ is the pulse interval, then echo behavior is characterized by the product $\sigma \tau$. For $\sigma \tau \ll 1$, echo processes are no different than in systems of isolated particles. For $\sigma \tau \sim 1$, in a conservative system, they are enhanced by energy transfer among modes and produce amplified echoes. Conditions for such behavior exist in nonuniform media when oscillation modes are quasilocalized in space. For $\sigma \tau \gg 1$, the echo process is suppressed because of phase mixing among interacting modes. This is typically the case for plane-wave modes in uniform media. If the nonlinearity originates in the dynamics of individual localized particles, all modes interact mutually on an equal basis and no echo is expected.