# Experimental Evaluation of the Blackbody Radiation Shift in the Cesium Atomic Fountain Clock
## 2 Theoretical Research on the Blackbody Radiation (BBR) Shift  
BBR refers to the phenomenon by which objects at certain temperatures radiate electromagnetic waves. This radiation field has a broad frequency spectrum and can cause changes in the atomic energy levels. In the atomic free flight zone of the cesium fountain clock, cold atoms interact with an equilibrium thermal radiation field of approximately 300 K, and the movement of the hyperfine energy level causes a frequency shift of the cesium fountain clock [13]. The thermal radiation emitted by a blackbody heated to a given temperature \(T\) has a spectrum given by [PERSON]'s law [12; 14]:  
\[\rho(\
u)=\frac{8\pi l\
u^{3}}{c^{3}}\frac{1}{e^{h\
u/k_{B}T}-1} \tag{1}\]  
where \(h\) is [PERSON]'s constant, \(c\) is the speed of light, and \(k_{B}\) is the [PERSON] constant. \(\rho(\
u)\) is the energy density of the BBR field, in J/Hz/m\({}^{3}\). BBR exhibits a very broad spectrum at room temperature [15]. Figure 1 shows the relationship between the spectral density of a BBR field \(\rho(\
u)\) and the frequency for several temperatures of the blackbody. The inset shown in Figure 1 is a partial enlargement with frequency in the range of 0 to 4 \(\times\) 10\({}^{10}\). \(\
u_{hf}\) denotes the hyperfine splitting frequency in the ground state, and \(\
u_{lef}\) denotes the lowest allowed electric dipole transition frequency. The superfine energy level of cesium atoms interacts with the BBR field, and the energy level shift is affected by the adjacent energy level. The magnetic dipole transition in the microwave frequency band and line D2 of the electrical dipole transition in the optical frequency band have the greatest impact on the energy level shift. The perturbation of the magnetic dipole transition on the ground-state energy level is mainly the second-order Zeeman effect of the BBR field on the transition \(\left|6^{2}S_{1/2},F=4,m_{F}=0\right>\rightarrow\left|6^{2}S_{1/2},F=3,m_{F}= 0\right>\)[16]. Assuming there is a pure two-level quantum system subject to random perturbations, for a temperature of approximately 300 K the influence of the BBR field near the resonance part can be ignored because the contributions to the first-order transition by the two sides of the resonance point offset each other. The energy of the BBR field is mainly concentrated in the frequency \(\
u\gg v_{hfs}\), where only the component of the random magnetic field in the direction of the C-field (the C-field is a static magnetic field produced by a long coil and used to provide aquantization axis) has an effect on the clock transition \(\left|4,0\right>\rightarrow\left|3,0\right>.\) Therefore, one third of the isotropic magnetic field induction energy contributes to the AC Zeeman frequency shift:  
\[ \begin{array}{ll}\frac{\Delta v_{hfs}}{v_{hfs}}&=\frac{\left(g_{j}+g_{j} \right)^{2}h_{F}^{2}}{3h^{2}}\int_{\
u>w_{hfs}}^{\infty}\frac{1}{v_{hfs}^{2} }\frac{g_{j}^{2}}{v_{hfs}^{2}-v^{2}}dv\\ &\cong-\mu_{0}\frac{\left(g_{j}+g_{j}\right)^{2}h_{F}^{2}}{3h^{2}}\int_{0}^{ \infty}\frac{e^{\left(\
u\right)}}{v^{2}}dv\\ &=-\frac{4}{9}\tau^{3}\mu_{0}\left(g_{j}+g_{j}\right)^{2}\mu_{B}^{2}\frac{ \left(kT\right)^{2}}{(zh)^{3}}\\ &=-1.304\times 10^{-17}\left(\frac{T}{300}\right)^{2}\end{array}  \tag{2}\]  
where \(B_{\
u}\) is the amplitude of the magnetic induction component that oscillates at the frequency \(\
u\), and \(g_{j}\) and \(g_{i}\) are the electron \(g\)-factor and nuclear \(g\)-factor, respectively. It can be observed that at room temperature, the frequency shift caused by the [PERSON] effect owing to the BBR magnetic field is of the order of \(10^{-17}\), which is negligible compared to the other perturbations considered here. It can be observed from Figure 1 that the energy of the BBR spectrum is concentrated around 20 THz, which is much smaller than the electronic dipole transition frequencies of 335 THz for line D1 and 352 THz for line D2. This means that the BBR electric field can be approximated as a weak, slowly changing, non-resonant AC electric field [17].