both equations, given by $$N_{\min } (D) = D + 2. \label{Nmin}$$ For $N > N_{\min }$, more than one set of integer solutions for $n_i$ may satisfy both equations. The contributions from different pathways coherently add up and may interfere. An example illustrating this interference effect (a “balanced 4-pulse” configuration) will be given later in this section.

Interferometry observable
-------------------------

### Recoil phase

We now come to the pre-factor “$\prod\limits_{i=1}^{N-1}
J_{n_i}(2\theta_i \sin \frac{\bf Q_i \cdot x_i}{2})$” in Eq. (\[npulserule\]). We shall define: $$\phi_i^{recoil}=\frac{\bf Q_i \cdot x_i}{2} \label{equrecoilphase}$$ as the recoil phase at each scattering vertex in Fig. \[npulserulefig\], referring to the cumulative phase shift due to the kinetic energy of recoil effects. Clearly, $\phi_i^{recoil}$ follows $\bf x_i$ and increases linearly with $T$, so its slope gives a temporal periodicity of the grating echo amplitude in Eq. (\[npulserule\]). The periodicity of the grating echo amplitude was measured experimentally to determine the recoil frequency of atoms [@TLCahn97; @hbarBEC02; @littleg].

### Acceleration induced phase\[accsection\]

We can generalize Eq. (\[npulserule\]) in a constantly accelerating frame (Appendix A). In this case Eq. (\[wignerrule1\]) is modified to include a phase factor: $$W({\bf{q}},{\bf{x}},t)  =  W({\bf{q}},{\bf{x}} - {\bf{q}}t, 0) e^{i
\frac{m}{\hbar}({\bf{x}} - \frac{{{\bf{q}}t}}{2}) \cdot {\bf{a}}t},
\label{wignerrule1b}$$ where 