${\bf{a}}$ is the acceleration constant. Equation (\[npulserule\]) is modified accordingly; we have, $$E_N^b (T_N ) = - ig\sum\limits_{\{ n_i \} } [\prod\limits_{i = 1}^{N - 1} J_{n_i } (2\theta _i \sin \frac{{\bf Q }_i  \cdot {\bf{x}}_i }{2})]  W({\bf{q}}_1 , {\bf{x}}_1 , {T_1^- }) e^{i\frac{m}{\hbar}\int_{T_1 }^{T_N } {{\bf{x}}(t) \cdot {\bf{a}}dt} + \sum\limits_i {n_i \varphi _i } } ,
\label{npulseruleb}$$ where ${\bf{x}}(t)$ is given by the displacement line trajectory in the ${\bf{x}} - t$ diagram as shown in Fig. \[npulserulefig\]. In Eq. (\[npulseruleb\]) we also include the phases $\varphi_i$ for each standing wave pulse explicitly. Equation (\[npulseruleb\]) indicates that the displacement line $\bf x(t)$ in Fig. \[npulserulefig\] can be interpreted as the relative displacement between two diffraction path pair - external perturbations shift the relative phase between the two paths the same way as a Mach-Zehnder interferometer.

### Rotation induced phase or Sagnac phase

Due to the equivalent principle, it is always possible to choose an inertial frame to describe the atomic motion, and attribute the inertial forces to the motion of the standing wave field classically, as long as the space-time curvature is negligible over the extension of the atomic ensemble. With this in mind, consider Fig. \[gratingechoTLIfigure\] with grating vectors 