# The Microfabricated Alkali Vapor Cell with High Hermeticity for Chip-Scale Atomic Clock
## 3 Results and Discussion
\[\gamma_{2}=\gamma_{Rb-wall}+\gamma_{Rb-BG}+\gamma_{se}=\left[\left(\frac{\left(2.405 \right)^{2}}{a^{2}}+\frac{p^{2}}{L^{2}}\right)D_{0}\frac{P_{0}}{P}+N_{0}\overline {v}_{r}s_{Rb-bg}\frac{P}{P_{0}}+n\overline{v}_{Rb}s_{se}\right] \tag{2}\]  
where \(\gamma_{Rb-wall}\) represents the effect of alkali-wall collisions, \(\gamma_{Rb-BG}\) stands for alkali-buffer gas collisions, and \(\gamma_{se}\) represents the contribution to the ground state coherence of the Rb-Rb spin-exchange collision. \(L\) and \(R\) are the length and the radius of the cell, respectively, and \(D_{0}\) is the diffusion constant of Rb atoms in abuffer gas at pressure \(P_{0}\) = 101.3 kPa. \(P\) is the pressure of the buffer gas. \(N_{0}\) is the Loschimdt constant, \(\overline{v}_{r}\) is the relative velocity of Rb atoms with respect to the buffer gas atoms, and \(\sigma_{Rb-bg}\) is the collisional cross-section between Rb and the buffer gas atoms. \(\sigma_{se}\) is the Rb spin-exchange cross-section. \(n\) the number density of Rb. The filled buffer gas is utilized to suppress the decoherence collisions of the atoms to narrow the FWHM of the CPT signal. This narrowing effect of the CPT signal can be explained by the equation above, but it is not linearly dependent on the pressure. This also explains why the narrowing effect of the buffer gas gradually weakened with the increasing pressure (50-70-90 Torr). Furthermore, the linewidth was broadened by the increasing optical power, which was helpful to increase the signal amplitude. Optical power must be carefully investigated, for it is a trade-off between the linewidth and signal-to-noise ratio. For relevant information in order to optimize the clock short term frequency stability, as shown in the following equation, we investigated the evolution of the ratio between the CPT contrast and the resonance linewidth as a function of the total laser intensity [24]. \[\sigma_{y}(\tau)\cong\frac{\Delta\
u}{\
u_{0}}\frac{1}{SNR}\tau^{-\frac{1}{2}} \tag{3}\]  
\[SNR=\frac{S}{N}=Contrast\times\sqrt{\frac{P_{out}}{2 nv_{opt}}} \tag{4}\]  
where \(\sigma_{y}(\tau)\) represents the shot noise limit, \(\Delta\
u\) stands for the linewidth of the CPT resonance, \(\
u_{0}\) = 6.835 GHz is the microwave frequency, and \(\tau\) represents the integration time. _Contrast_ is the ratio between the height of the resonance error signal and the background voltage at the output of photodiode. \(P_{out}\) and \(\
u_{opt}\) are the laser power and light frequency, respectively, and \(h\) is the Planck constant. Although the height of the CPT resonance increased with the laser power, the photodetector also received a higher background dc voltage. Therefore, the increase rate of contrast would gradually slow down as the power further increased. The ratio contrast/FWHM finally exhibited a maximum value corre  
Figure 8: CPT linewidth as a function of laser power at different pressure (50–70–90 Torr) and the linear fitting curve. sponding to a certain value of laser power, resulting an optimized short-term frequency stability. It is also necessary to investigate the effect the temperature on the ratio contrast/FWHM of the CPT 0-0 resonance. The number of the atoms was restricted by the small volume of the MEMS cell. Increasing the heating temperature would benefit the atomic density and enhance the CPT resonance. However, the amplitude of CPT resonance would drop with the further increase of temperature, which could be explained by the optical density effect. As shown in Figure 9, the ratio contrast/FWHM was found to be maximized for an optical power of 120 \(\upmu\)W and a temperature about 90 \({}^{\circ}\)C in the MEMS cell with 70 torr buffer gas.