# Lifetime Measurement of Cesium Atoms Using a Cold Rydberg Gas
## 3 Experimental Method and Results
### Interaction Induced Decay  
Due to the large orbital radius (\(\sim\)\(n^{2}\)), Rydberg atoms have a large collision cross-section with the ground atom and the Rydberg atom, which modify the decay rate and shorten the lifetime of the Rydberg atom. Moreover, the strong van der Waals interaction (\(\sim\)\(n^{11}\)) between Rydberg atoms will also affect the decay rate and lifetime of Rydberg atoms. In order to investigate the effect of the interaction between Rydberg atoms on the lifetime of Rydberg atoms, we do a series of measurements like as in Figure 2 but for different Rydberg densities by changing the power of the second excitation laser. Figure 3 shows the measurement of the lifetime (Figure 3a) and the decay rate (Figure 3b) as a function of the Rydberg atomic density for the \(|60D_{5/2}\rangle\) state. It is found that the measured lifetime \(\tau_{exp}\) decreases when the Rydberg density increases. For example, when the Rydberg atomic density \(\rho\) = 5.71 \(\times\) 10\({}^{9}\) cm\({}^{-3}\), the measured lifetime is 30.3 \(\upmu\)s, two times less than the calculation of \(\tau_{eff}\) = 76.3 \(\upmu\)s by the Equation (4), as shown by the black arrow. The measured lifetime increases as the Rydberg density decreases, and up to 75.2 \(\upmu\)s when Rydberg density \(\rho\) = 2.57 \(\times\) 10\({}^{9}\) cm\({}^{-3}\), as shown in the blue oval of Figure 3a. It is noted, from Figure 3a, that the lifetime does not further increase even we decrease the Rydberg density further, which means that when the Rydberg density is smaller than some value, \(\rho_{0}\), (\(\rho_{0}\) = 2.3\(\times\) 10\({}^{9}\) cm\({}^{-3}\) for the \(|60D_{5/2}\rangle\) state), the interaction collision between Rydberg atoms has a negligible effect for lifetime measurements, and measured lifetime approaches the calculation of \(\tau_{eff}\). The dependence of lifetime on the Rydberg atomic density can be attributed to the collision interaction induced decay; also, see the measured decay rates of Figure 3b. In order to explain the behavior of Figure 3, we define the total decay rate, \(\Gamma_{tol}\) = \(\Gamma_{sp}\) + \(\Gamma_{BBR}(T)\) + \(\sigma_{n}\
u\rho\), where \(\sigma_{n}\) is the cross section, \(\
u\) is a velocity of the atom, and \(\rho\) is the Rydberg atomic density, respectively. For the case of \(\rho\geqslant\rho_{0}\), the total decay rate linearly increases with the atomic density, but for the case of \(\rho<\rho_{0}\), the total decay rate does not change and approaches to the content, \(\Gamma_{eff}(T)\) = \(\Gamma_{sp}\) + \(\Gamma_{BBR}(T)\). Therefore, the \(\Gamma_{tol}\) is written as  
\[\left\{ \begin{array}{ll}\Gamma_{tol}=\Gamma_{eff}(T)+\sigma_{n}\
u\rho&( \rho\geqslant\rho_{0})\\ =\Gamma_{eff}(T)&(\rho<\rho_{0})\end{array} \right. \tag{6a}\]  
For the \(60|D_{5/2}\rangle\) state of the Figure 3, \(\rho_{0}\) \(\sim\)2.3 \(\times\) 10\({}^{9}\) cm\({}^{-3}\). It is noted that the \(\rho_{0}\) depends on the \(n\) used. The larger the \(n\) is, and the smaller \(\rho_{0}\) is. For obtaining the interaction cross-section, \(\sigma_{n}\), we use the Equation (6) to fit the measured decay rates of Figure 3b, as shown with the blue dashed line of Figure 3b. The collision cross-section for \(|60D_{5/2}\rangle\)\(\sigma_{n}\) = (\(5.11\pm 0.36\)) \(\times\) 10\({}^{-9}\) cm\({}^{2}\) is extracted from the  
Figure 3: (Color online) Measured lifetimes (**a**) and decay rate (**b**) of the \(|60D_{5/2}\rangle\) state as a function of the Rydberg atomic density. The dashed line in (**b**) is the fitting of Equation (6), from which the collision cross-section \(\sigma_{n}\) = (5.11 \(\pm\) 0.36) \(\times\) 10\({}^{-9}\) cm\({}^{2}\) is obtained. Blue ovals mark the measured lifetime (**a**) and decay rate (**b**) for the atomic density of \(\rho\) = 2.57 \(\times\) 10\({}^{9}\) cm\({}^{-3}\), and black arrows sign the measured data for the density of \(\rho\) = 5.71 \(\times\) 10\({}^{9}\) cm\({}^{-3}\).