Echoes and revival echoes in systems of anharmonically confined atoms

We study echoes and what we call “revival echoes” for a collection of atoms that are described by a single quantum wave function and are confined in a weakly anharmonic trap. The echoes and revival echoes are induced by applying two successive temporally localized potential perturbations to the confining potential, one at time t = 0, and a smaller one at time t = τ . Pulselike responses in the expectation value of position 〈x(t)〉 are predicted at t ≈ nτ (n = 2,3, . . .) and are particularly evident at t ≈ 2τ . While such echoes are familiar from previous work, a result of our study is the finding of revival echoes. Revivals (but not echoes) occur even if the second perturbation is absent. In particular, in the absence of the second perturbation, the response to the first perturbation dies away but then reassembles, producing a response at revival times mTx (m = 1,2, . . .). The existence of such revivals is due to the discreteness of the quantum levels in a weakly anharmonic potential, and has been well studied previously. If we now include the second perturbation at t = τ , we find temporally localized responses, revival echoes, both before and after t ≈ mTx [e.g., at t ≈ mTx − nτ (prerevival echoes) and at t ≈ mTx + nτ , (postrevival echoes)] where m and n are 1,2, . . .. One notable point is that, depending on the form of the perturbations, the “principal” revival echoes at t ≈ Tx ± τ can be much larger than the echo at t ≈ 2τ . We develop a perturbative model for these phenomena, and compare its predictions to the numerical solutions of the time-dependent Schrödinger equation. The scaling of the size of the various echoes and revival echoes as a function of the symmetry of the perturbations applied at t = 0 and t = τ , and of the size of the external perturbations is investigated. The quantum recurrence and revival echoes are also present in higher moments of position, 〈xp(t)〉, p > 1. Recurrences are present at t ≈ mTx/j , and dominant prerevival and postrevival echoes occur at fractional shifts of τ [i.e. t ≈ (mTx ± τ )/j ] where the m = 1,2, . . . and the integer values of j are determined by p. Additionally, we use the Gross-Pitaevskii equation to study the effect of atom-atom interactions on these phenomena. We find that echoes and revival echoes become more difficult to discern as the size of the second perturbation is increased and/or as the atom-atom interactions become stronger.