What does this equation mean?

The Galilei loop admits an exact, non-simulated numerical check of the whole chain, because $\Phi$[$\mathcal L_{\rm G}$] is precisely the recoil phase every stimulated-Raman atom interferometer already uses for its own calibration. Take a sodium atom, m=3.8175$\times$10^{-26}\,$\mathrm{kg}$ , driven by a two-photon stimulated-Raman transition on the 589\,$\mathrm{nm}$ line, effective wavevector $k_{\rm eff}$=2(2$\pi$/589\,$\mathrm{nm}$)=2.1335$\times$10^{7}\,$\mathrm{m^{-1}}$ [ 16 ] . The associated recoil velocity is $v_r$=$\hbar$ $k_{\rm eff}$/m=5.894\,$\mathrm{cm\,s^{-1}}$ . Hold the loop open for T=50\,$\mathrm{ms}$ , comparable to the short interrogation times of early stimulated-Raman interferometers, so the translation leg is b=$v_r$T=2.947\,$\mathrm{mm}$ . The loop area is $v_r$b=$v_r^2$T=1.737$\times$10^{-4}\,$\mathrm{m^2\,s^{-1}}$ , and the phase debt is $\Phi[\mathcal L_{\rm G}]=\frac{m\,v_rb}{\hbar}\approx6.29\times10^{4}\ \text{rad}$. This is an exact analytic evaluation of the formula derived above for stated, idealized parameters — not a simulation, not a measurement, and not the output of any code run for this article. It is offered as illustrative, and it carries a built-in consistency check: because $v_r$=$\hbar$ $k_{\rm eff}$/m by definition of a recoil kick, the same number equals $\hbar$ $k_{\rm eff}^2$T/m , the ordinary recoil phase used to extract $\hbar$/m in precision atom-interferometry measurements through the general path-integral formalism for atomic interferometry [ 14 ] . The two expressions are algebraically identical, so this is a check that the loop construction reproduces a quantity the field already measures by an independent route, not new physics. Inverting the formula returns $m_{\rm op}$[$\mathcal L_{\rm G}$]=$\hbar$$\Phi$/($v_r$b)=m exactly, because the phase was defined from m in the first place. That circularity is the point of this section: it isolates what an idealized, confound-free evaluation looks like, so the next two sections can show exactly what breaks the idealization.