# Lifetime Measurement of Cesium Atoms Using a Cold Rydberg Gas
## 3 Experimental Method and Results
### N Dependence of Lifetime  
We investigate the \(n\) scaling law of lifetime in this section. We measure the lifetime of Rydberg atoms with different principal quantum numbers \(n\) = 60-83 by changing the second laser frequency. In the Table 2 and Figure 4, we present the measurements and calculations of Rydberg lifetimes for \(|nS_{1/2}\rangle\) (Figure 4a) and \(|nD_{5/2}\rangle\) (Figure 4b) states. The measured data are the average of five independent measurements and the error bars display the standard error. We see that the measured lifetimes, \(\tau_{exp}\), agree well with the calculations, \(\tau_{eff}\), which shows that the influence of blackbody radiation at the room temperature is dominant, while the interaction induced decay has a negligible effect at the condition of the low Rydberg density. For comparison, we also display the calculations of spontaneous radiation lifetime \(\tau_{sp}\) (blue stars), and BBR lifetime \(\tau_{BBR}\) (pink triangles). Both the spontaneous radiation lifetime and BBR lifetime display increase with principal quantum number \(n\). From Figure 4, it is found that the spontaneous lifetime is larger than the BBR lifetime at 300 K for \(|nS_{1/2}\rangle\) Rydberg states, but less than BBR lifetime for \(|nD_{5/2}\rangle\) states. This phenomenon is attributed to the fact that \(nS\) and \(nD\) states have different quantum defects and energy level structures that lead to the different radial matrix elements, as shown with the Equation (1). For both \(nS\) and \(nD\) states, the room temperature BBR will enhance the decay of the Rydberg atom and shorten the lifetime. In order to obtain the \(n\)-scaling law, we use function, \(\tau=a\times(n-\delta_{L})^{b}\) with \(a\) and \(b\) the fitting parameters, to fit the measured and calculated lifetimes, see lines in Figure 4. The fitting parameters of \(b_{exp}\) = 2.55 \(\pm\) 0.02 and \(b_{theor}\) = 2.50 \(\pm\) 0.02 for \(|nD_{5/2}\rangle\), and \(b_{exp}\) = 2.30 \(\pm\) 0.01 and \(b_{theor}\) = 2.27 \(\pm\) 0.01 for \(|nS_{1/2}\rangle\) state, respectively. The measurements of \(b\) agree with the calculations, and the difference of scaling magnitude \(b\) is 3% for \(|nD_{5/2}\rangle\) \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \multirow{2}{*}{\(n\)} & \multicolumn{4}{c}{\(S_{1/2}\)} & \multicolumn{4}{c}{\(D_{5/2}\)} \\ \cline{2-10}  & \(\tau_{sp}\) & \(\tau_{BBR}\) & \(\tau_{eff}\) & \(\tau_{exp}\) & \(\tau_{sp}\) & \(\tau_{BBR}\) & \(\tau_{eff}\) & \(\tau_{exp}\) \\ \hline
60 & 226.9 & 164.1 & 95.2 & 95.4 \(\pm\) 2.7 & 124.2 & 197.6 & 76.3 & 75.7 \(\pm\) 0.7 \\
63 & 265.4 & 180.7 & 107.5 & 107.4 \(\pm\) 2.4 & 144.7 & 216.7 & 86.7 & 85.3 \(\pm\) 0.3 \\
65 & 293.3 & 192.2 & 116.1 & 115.7 \(\pm\) 1.8 & 159.4 & 229.9 & 94.1 & 94.6 \(\pm\) 2.4 \\
67 & 323.1 & 204.0 & 125.1 & 124.4 \(\pm\) 1.8 & 175.2 & 243.5 & 101.9 & 102.0 \(\pm\) 6.3 \\
70 & 371.6 & 222.4 & 139.1 & 138.6 \(\pm\) 2.4 & 200.7 & 264.5 & 114.1 & 115.4 \(\pm\) 2.2 \\
73 & 424.6 & 241.4 & 153.9 & 153.5 \(\pm\) 4.2 & 228.6 & 286.4 & 127.1 & 127.0 \(\pm\) 2.5 \\
76 & 482.5 & 261.2 & 169.5 & 169.5 \(\pm\) 2.4 & 259.0 & 309.1 & 140.9 & 141.5 \(\pm\) 2.7 \\
80 & 567.5 & 288.8 & 191.4 & 191.5 \(\pm\) 4.4 & 303.5 & 340.6 & 160.5 & 161.0 \(\pm\) 3.5 \\
83 & 637.5 & 310.3 & 208.7 & 209.8 \(\pm\) 1.0 & 340.0 & 365.2 & 176.1 & 176.6 \(\pm\) 1.8 \\ \hline \end{tabular}
\end{table} 
Table 2: Measurements of the lifetime \(\tau_{exp}\) and calculations of spontaneous lifetimes \(\tau_{sp}\), blackbody radiation lifetimes \(\tau_{BBR}\) and effective lifetimes \(\tau_{eff}\) (in \(\mu\)s) of \(|nS_{1/2}\rangle\) and \(|nD_{5/2}\rangle\) Rydberg states, error bar displays the standard error of five independent measurements. state and 1% for \(|nS_{1/2}\rangle\) state. The measured lifetime for both \(S\)- and \(D\)-type Rydberg state increases with the principal quantum number \(n\). In the Table 3, we present the comparison of our measurements and previous \(b\) value in literatures for different \(n\) range.