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We study echoes and what we call 'revival echoes' for a collection of atomsthat are described by a single quantum wavefunction and are confined in aweakly anharmonic trap. The echoes and revival echoes are induced by applyingtwo, successive temporally localized potential perturbations to the confiningpotential, one at time $t=0$, and a smaller one at time $t=\tau$. Pulse-likeresponses in the expectation value of position $<x(t)>$ are predicted at $t\approx n\tau$ ($n=2,3,...$) and are particularly evident at $t \approx 2\tau$.A novel result of our study is the finding of 'revival echoes'. Revivals (butnot echoes) occur even if the second perturbation is absent. In particular, inthe absence of the second perturbation, the response to the first perturbationdies away, but then reassembles, producing a response at revival times $mT_x$($m=1,2,...$). Including the second perturbation at $t=\tau$, we findtemporally localized responses, revival echoes, both before and after $t\approxmT_x$, e.g., at $t\approx m T_x-n \tau$ (pre-revival echoes) and at $t\approxmT_x+n\tau$, (post-revival echoes), where $m$ and $n$ are $1,2,...$ . Dependingon the form of the perturbations, the 'principal' revival echoes at $t \approxT_x \pm \tau$ can be much larger than the echo at $t \approx 2\tau$. We developa perturbative model for these phenomena, and compare its predictions to thenumerical solutions of the time-dependent Schr\"odinger Equation. The scalingof the size of the various echoes and revival echoes as a function of thesymmetry and size of the perturbations applied at $t=0$ and $t=\tau$ isinvestigated. We also study the presence of revivals and revival echoes inhigher moments of position, $<x^p (t)>$, $p>1$, and the effect of atom-atominteractions on these phenomena.

Echoes and revival echoes in systems of anharmonically confined atoms