)  =  - i\eta \frac{{N_{atom} \alpha }}{{w_0 ^2 \lambda
    }}E_N^a (T_N )\int {d^3 {\bf{r}}} \rho ({\bf{r}},T_N)e^{i{\bf{Q}}_N
    \cdot {\bf{r}}}, \label{backamp}$$ where $\eta $ is a geometric factor of order unity depending on the overlap between the Gaussian mode $E_N^a $ and $E_N^b $. $E_N^b (T_N
    )$ in Eq. (\[backamp\]) gives the magnitude of the grating echo, and is considered to be the output of the TLI.]{}

The free space displacement diagram
-----------------------------------

### The displacement operator and the Weyl function \[sectionrule\]

We define $g = \eta \frac{{N_{atom} \alpha }}{{w_0 ^2 \lambda
}}E_N^a (T_N )$. Recalling Eq. (\[weyldefine\]), we define the displacement operator $\hat D({\bf{q}},{\bf{x}}) = e^{i({\bf{q}}
\cdot {\bf{\hat r}} - {\bf{x}} \cdot {\bf{\hat p}})}$ and let $W({\bf{q}}, {\bf{x}},t)={\bf Tr}(\hat \rho(t) \hat D({\bf{q}},
{\bf{x}}))$, so that Eq. (\[backamp\]) can be rewritten in a compact form using the Weyl function: $$E_N^b (T_N )  =  - ig W({\bf{Q}}_N ,0, T_N). \label{backamp2}$$

To calculate the TLI output $E_N^b (T_N )$ , we might integrate the matter-wave equation Eq. (\[waveequ\]) to express $\hat \rho
{{(T}}_{{N}} )$ in terms of $\hat \rho {{(T}}_{{1}} )$. Here instead we shall take the Heisenberg picture and integrate the equation of motion for the displacement operator $\hat D({\bf{Q}},{\bf{x}})$ (Appendix A) so 