\psi ^* ({\bf{r}},t)\psi
    ({\bf{r'}},t)\rangle $ is the single atom density matrix for the atomic ensemble in real space. Impacts of $\varphi _i$ on the matter-wave dynamics require only relatively simple algebra, so we may drop $\varphi _i$ during the derivation and recover their values if necessary.

4.  [ A grating echo refers to the Bragg scattering of probe light pulse (the last graph in Fig. \[gratingechoTLIfigure\]) off the revived atomic density grating. The amplitude of the grating echo is evaluated under first order Born approximation. We assume the evolution of the atomic density grating is not perturbed by the scattered light during the probing interval. Using a slowly varying amplitude approximation, the coupling between $E_N^b $ and the probe light field $E_N^a $ around time $T_N $ is governed by the equation: $$(\partial _t  - c\partial _{x_{\bf{Q}} } )E_N^b  = i\alpha \omega
    N_{atom} \rho _{{\bf{r}},{\bf{r}}} (t)e^{i{\bf{Q}}_N  \cdot
    {\bf{r}}} E_N^a. \label{lightprop}$$ Here $x_{\bf{Q}} $ measures the distance along the direction ${\bf{Q}}_N $, and $\alpha $ is the dipole polarizability of ground state atoms at standing wave frequency $\omega$. If we ignore the light propagation retardation, the overlap between the Bragg-scattered field with the $E_N^b $ can be written as: $$E_N^b (T_N 