this initial-condition-dependent requirement will be concretized with an “echo condition” for the displacement diagram.

### Echo condition and echo time

In the displacement diagram given by Fig. \[npulserulefig\], the displacement line connects the point $({\bf{q}}_1=0 ,{\bf{x}}_1\sim
0)$ at $t=T_1^-$ with the point $({\bf{q}}_N={\bf{Q}}_N
,{\bf{x}}_N=0)$ at $T_N$. If [$T_i $ , $i=1,...,N-1$]{} are fixed, then ${\bf{x}}_1 = {\bf{x}}_1 (T_N )$ is a function of the detecting-pulse arriving time $T_N$.

With a particular TLI setting described by the parameters $\{ (n_i
{\bf{Q}}_i ,T_i), i = 1,...,N\} $, if ${\bf{x}}_1 $ defined in Eq. (\[npulserule\]) is along the direction of ${\bf{Q}}_N $, then by varying the time $T_N$, ${\bf{x}}_1 $ can be arbitrarily close to zero. The [*echo condition*]{} is defined as follows: In a general TLI setup \[using a “uniform and broad” atomic sample specified by Eq. (\[wigerbeta\])\] with the first N-1 pulse parameters set to be $\{ n_i {\bf{Q}}_i ,T_i ,i = 1,...,N - 1\}$, a revival of atomic density grating with the k-vector ${\bf{Q}}_N $ requires the existence of an echo time $T^{echo} $ such that $${\bf{x}}_1 (T_{^N }  = T^{echo} ) = \sum\limits_{j = 2}^N {n_j
{\bf{Q}}_j } T_{j,1}  = 0. \label{xconserve2}$$

In the displacement diagram, an echo line 