## 3 Analysis and Discussions  
The most advanced experiments demonstrate that collisions in a Rydberg atomic ensemble will yield ionization which will produce positive ions and electrons [15]. We know that, under the conditions of a weak electric field, the Rydberg level shift will be dominated by Stark shift. The dependence of the Rydberg level shift on the weak electric field can be measured by Rydberg EIT spectroscopy. We expand this idea and use a semi-classical model to explain the physical phenomena of the Rydberg level shift. In a thermal Rydberg atomic ensemble, the collisions between Rydberg atoms and ground-state atoms induce charged particles (positive ions and electrons), which cause avalanche ionization [15, 16, 17]. Here, the photo-ionization process is very weak, it can be ignored. In our system, the Rydberg atomic density is obtained by the combined effect of the ground-state cesium atomic density and the two-step Rydberg excitation \(R_{exc}\)  
Figure 2: (**a**) Energy level schematic of the ladder-type EIT of Cs atoms with Rydberg state (\(50\)S\({}_{1/2}\)). \(\Delta_{P}\) is the detuning of probe laser, \(\Delta_{C}\) is the detuning of coupling laser; (**b**) Typical Rydberg EIT signal. The 852-nm probe laser is detuned by \(\Delta_{P}\) from the Cs \(6S_{1/2}(F=4)\to 6P_{3/2}(F^{\prime}=5)\) transition (here, \(\Delta_{P}=0\)), while the 509-nm coupling laser is scanned across the Cs \(6P_{3/2}(F^{\prime}=5)\to 50S_{1/2}\) Rydberg transition. under the premise that the other experimental conditions (i.e., probe Rabi frequency and coupling Rabi frequency) remain unchanged, and ionization leads to the reduction of the Rydberg atomic density. In the theoretical model, positive ions and electrons will generate an inhomogeneous local electric field in the local space of the atomic vapor, and the local electric field interesting with Rydberg atoms will reduce the Rydberg energy and the cause a Rydberg level shift. We assume the electron density and the positive ion density to be equal, and both parameters are given by \(N_{\text{C}}\). The dependence of charged particles'density and Rydberg atomic density can be obtained by solving the following rate equations [15]:  
\[\dot{N}_{Ryd}=-N_{Ryd}N_{g}\sigma_{g}\overline{v}-N_{Ryd}N_{Ryd}\sigma_{Ryd} \overline{v}^{\prime}+R_{exc} \tag{1}\]  
\[\dot{N}_{C}=N_{Ryd}N_{g}\sigma_{g}\overline{v}+N_{Ryd}N_{Ryd}\sigma_{Ryd} \overline{v}^{\prime}-N_{\text{C}}\Gamma_{d} \tag{2}\]  
where \(N_{Ryd}\) and \(N_{\text{C}}\) are Rydberg atom density and induced density of charged particles, respectively. \(N_{g}\) is the ground-state atomic density; \(\sigma_{g}\)is cross-section for Rydberg-ground state atomic collisions, \(\sigma_{Ryd}\) is geometric cross-section of Rydberg atomic collisions [18; 19], \(\overline{v}\) is the mean relative velocity of Rydberg atoms colliding with ground-state atoms, and \(\overline{v}^{\prime}\) is the mean relative velocity of Rydberg atoms colliding with other Rydberg atoms. The expected value of \(\sigma_{g}\) is 0.06\(\sigma_{Ryd}\)[15], with \(\sigma_{Ryd}=\pi(a_{0}^{2}n^{*4})\)[15; 20] (where \(a_{0}\) is the Bohr radius) being the geometric cross-section of Rydberg state with an effective principal quantum number \(n^{*}\). \(R_{exc}=\Omega_{p}^{2}/(\Omega_{p}^{2}+\Omega_{c}^{2})\cdot N_{g}\Omega_{p} \Omega_{c}/\Delta\) is the Rydberg component in  
Figure 3: (**a**) At temperature of 40 \({}^{\circ}\)C, the Rydberg EIT signal is obtained by scanning the 509-nm coupling laser frequency with different 852-nm probe laser detuning; (**b**) Rydberg EIT signal intensity versus probe laser detuning for different atomic cell temperature; (**c**) The 852-nm probe laser detuning of the maximum intensity EIT signal changes with temperature (ground-state atomic density), including the three temperature corresponding data points in (**b**). Rabi frequency of 852-nm probe laser is \(\sim\)5.0 MHz, and that of the 509-nm coupling laser is \(\sim\)36.0 MHz. superposition dark state of EIT, where \(\Delta\) is two-photon detuning.