extension. $f({\bf p})$ gives a velocity distribution.

The Weyl function evaluated at $({\bf{q}}_1 ,{\bf{x}}_1)$ is given by: $$\begin{array}{c}
W({\bf{q}}_1,  {\bf{x}}_1, t=T_1^-)=< e^{{\bf{q}}_1 \cdot {\bf{\hat
r}} - {\bf{x}}_1  \cdot {\bf{\hat p}}}  > _{T_1^- }\\  = \int {d^3
{\bf{r}}d^3 {\bf{p}}w_\beta ({\bf{r}},{\bf{p}})e^{i({\bf{q}}_1 \cdot
{\bf{r}} - {\bf{x}}_1 \cdot {\bf{p}})} }. \label{wignerbeta2}
\end{array}$$

Here ${\bf{q}}_1 $ can only have discrete values specified by Eq. (\[npulserule\]), which is either zero or much larger than the spatial frequency of $g({\bf{r}})$ according to the assumption (A1) in section \[sectionmodel1\]. A Non-vanishing integration in Eq. (\[wignerbeta2\]) over ${\bf{r}}$ thus requires the condition of [*momentum conservation*]{}: $${\bf{q}}_1  = \sum\limits_{j = 1}^N {n_j {\bf{Q}}_j }  = 0.
\label{qconserve}$$ The restriction from Eq. (\[qconserve\]) simplifies Eq. (\[wignerbeta2\]) and we end up with: $$\begin{array}{c}
W({\bf{q}}_1,  {\bf{x}}_1, t=T_1^-)
= \int {d^3 {\bf{p}}f({\bf{p}})e^{i{\bf{x}}_1  \cdot {\bf{p}}} }\\
={\emph F}({\bf x}_1)\\
= e^{ - 2(\frac{{{\bf{x}}_1 }}{{\lambda_c }})^2 },  \\
 \end{array} \label{wignerbeta3}$$ where ${\emph F}({\bf x})$ is the Fourier transform of $f({\bf p})$. The velocity distribution function $f({\bf p})$ specifies a coherence length $\lambda_c$. As a simple example, in the third line of Eq. (\[wignerbeta3\]) we consider an isotropic Maxwell distribution of atom velocity as $f({\bf{v}}) \propto e^{ -
\frac{{{\bf{v}}^2 }}{{2u^2 }}}$, and we define the coherence length 