still need to specify a particular $({\bf{q}}_1 ,{\bf{x}}_1)$ at time $T_1^-$. Clearly, $W({\bf{q}}_1 ,{\bf{x}}_1)$ needs to be non-zero for a non-vanishing contribution to Eq. (\[npulserule\]). In the next subsection, $W({\bf{q}}_1 ,{\bf{x}}_1 ,t=T_1^-)$ for a “uniform and broad” atomic sample is studied and we find that $({\bf{q}}_1 =0 ,{\bf{x}}_1 \approx 0)$ is required.

Momentum conservation and echo condition
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### Input displacement line\[sectionextensive\]

![Diagram restricting the displacement line at the input of the first scattering vertex \[Eq. (\[qconserve\]) and Eq. (\[xconserve1\])\]. The blue Gaussian on the left represents the matter-wave spatial correlation function due to Maxwell’s velocity distribution with coherent length $\lambda_c $ . Here ${\bf
q}_1=0$ and ${\mathbf{x}}_1 $ need to be within the coherent length of the atomic sample.[]{data-label="iniconditionfig"}](inicondition.eps "fig:"){width="2in"}\

This subsection considers the possible values for $({\bf{q}}_1
,{\bf{x}}_1)$ that determine the input displacement line in Fig. \[npulserulefig\]. We shall consider a “uniform and broad” atomic sample, specified by the phase-space distribution with the Wigner function: $$w_\beta  ({\bf{r}},{\bf{p}}) = g({\bf{r}} - \beta
{\bf{p}})f({\bf{p}}), \label{wigerbeta}$$ where $g({\bf{r}})$ is a spatial distribution function, assumed to be “uniform and broad”, e.g., uniformly distributed over many optical wavelengths $\lambda$. This requirement is more general than simply requiring a distribution over a large spatial 