that can arise at times $2\tau $, $3\tau , \ldots $. The ‘response’ plotted on the vertical axis is the magnitude of the complex valued order parameter, Eq. (8). See Sec. IV for details of this computation.](fig1)

numerical simulation (see Sec. IV for details). In order for this phenomenon to occur, the system must have two fundamental attributes, nonlinearity and memory. Nonlinearity is necessary because the response seen in Fig. 1 is not the same as the sum of the responses to each of the individual stimulus pulses in the absence of the other pulse (which is simply the decay that occurs immediately after the individual stimuli, without the echo). Memory is necessary in the sense that the system state after the decay of the second pulse must somehow encode knowledge of the previous history even though the global average of the system state, as represented by the order parameter, is approximately the same as before the two pulses were applied.

Echo phenomena of this type, occurring in systems of many oscillators having a spread in their natural oscillation frequencies, have been known for a long time. The first example was the ‘spin echo’ discovered in 1950 by Hahn[@hahn50], where 