revived atomic density grating. The assumptions in the model are summarized below. For notational convenience, we set $\hbar = m = 1$.

1.  [2-level atoms are always in their adiabatic ground state, with any spontaneous light scattering ignored. Standing waves pulse $i$ with the time-dependent light shift [@opticalpumpingoperator] potential $V_i = \Omega _i (t)\cos ({\bf{Q}}_{\bf{i}} \cdot {\bf{r}}
    + \varphi _i )$ acts as a phase grating to atoms. Here ${\bf{Q}}_i =
    {\bf{k}}_i^a - {\bf{k}}_i^b $ is the k-vector of the standing wave. ${\bf{k}}_i^a $, ${\bf{k}}_i^b $ are the k-vectors of the traveling wave light fields, whose electric field amplitudes are referred to as $E_i^a$ and $E_i^b$. And $\varphi _i $ determines the position of the standing wave nodes in the lab frame. We assume the light fields have Gaussian beam profiles and the atomic sample is located at the center of each beam. The cross-sections of all the beams are characterized by a waist size $w_0$, that is much larger than the size of the atomic sample. We also assume all the ${\bf{k}}_i^a $ and ${\bf{k}}_i^b $ to have significant differences in propagation directions such that any linear combinations of ${\bf{Q}}_i $s have magnitude either zero or on the order 