of $\frac{{2\pi }}{\lambda }$, with $\lambda$ the optical wavelength.]{}

2.  [ Standing wave pulses are in the Raman-Nath regime, e.g., the positions of atoms are considered frozen during each pulse duration. We thus can use Dirac delta functions to effectively describe the temporal profile of the pulses, e.g., $\Omega _i (t) = \theta _i
    \delta (t - T_i )$ that is characterized by its strength $\theta _i
    $ and the time of arrival $T_i $. $\theta _i $ is referred as the pulse area. ]{}

3.  The atomic ensemble is assumed to be dilute such that atom-atom interactions are ignored. The ensemble is assumed to be dilute in phase space as well so that the single atom picture is applied without conflicting with quantum statistics. We can thus use a Dirac ket $|\psi\rangle$ to label the matter-wave state, that obeys the single atom Schr[ö]{}dinger’s equation,

    $$i\partial _t |\psi\rangle  = (\frac{{{\bf{\hat p}}^2 }}{2} +
    \sum\limits_i {\Omega _i (t)\cos ({\bf{Q}}_i  \cdot {\bf{\hat r}} +
    \varphi _i )} )|\psi\rangle .\label{waveequ}$$

    Given a fixed amount of atoms participating in the interaction, the spatial density distribution of the atomic ensemble is given by $\rho({\bf{r}},t) = N_{atom} \rho _{{\bf{r}},{\bf{r}}} (t)$, where $\rho _{{\bf{r}},{\bf{r}}'} (t) =  \langle 