matrix operator $\hat
\rho$. TLE corresponds to a revival of atomic density grating at a certain time t. Since the revived density grating is spatially periodic, we shall consider its particular $\bf Q$ component given by: $$\rho_{\bf Q}={\bf Tr}(\hat \rho(t) e^{i {\bf Q}\cdot \hat {\bf
r}})=W({\bf Q},0,t),\label{rhoQdefine}$$ where $W({\bf q},{\bf x},t)$ is the Weyl function defined in Eq. (\[weyldefine\]).

To express Eq. (\[rhoQdefine\]) in terms of an arbitrary initial condition described by $\hat \rho(0)$, one can propagate $\hat
\rho(0)$ forward in time by considering the interaction of atoms with a sequence of standing wave pulses. Alternatively, we shall define the time-dependent displacement operator $\hat D({\bf q},{\bf
x},t) = e^{i ({\bf q}\cdot \hat {\bf r}(t) - {\bf x}\cdot \hat {\bf
p}(t))}$ in the Heisenberg picture and propagate the displacement operator $\hat D({\bf Q},0,t)$ backward in time. Here we take the latter approach which is advantageous for two reasons: 1) It applies to general initial conditions. 2) As we will see, the equation of motion for $\hat D({\bf q},{\bf x},t)$ due to pulsed periodic potentials, during free-propagations and due to a quadratic potential can be integrated easily. Since our analysis is generally applied to different types of interferometers using TLE, we shall 