# Experimental Evaluation of the Blackbody Radiation Shift in the Cesium Atomic Fountain Clock
## 2 Theoretical Research on the Blackbody Radiation (BBR) Shift
The influence of the BBR field can be approximated as a second-order Stark effect produced by the electrostatic field, which depends on the root mean square value of the electric field strength, and can be solved by perturbation theory. Since the induced electric dipole moment is  
Figure 1: Spectral density of blackbody radiation (BBR) for six temperatures. directional, the tensor form of the polarizability must be used here. For the electric field acting on a single-electron atom, the energy change can be expressed as follows:  
\[\Delta E_{Stark}=-\sum_{i,j}K_{i,j}E_{i}E_{j}\quad i,j\in x,y,z \tag{3}\]  
where \(K_{i,j}\) is a 3 \(\times\) 3 matrix. Taking the direction of the C-field as the z axis, owing to the rotational symmetry between the x and y directions, \(K_{x,x}=K_{y,y}=K_{1}\) and \(K_{z,z}=K_{2}\) can be set, and the Stark shift caused by the electric field can be expressed as follows:  
\[ \begin{array}{ll}\Delta E_{Stark}&=-\Big{(}K_{1}E_{x}^{2}+K_{1}E_{y}^{2}+K_ {2}E_{z}^{2}\Big{)}\\ &=-E^{2}\big{[}K_{1}+(K_{2}-K_{1})\cos^{2}\theta\big{]}\end{array}  \tag{4}\]  
where the first part is the scalar polarizability independent of the direction of the magnetic field, and the second part is the direction-dependent tensor polarizability. The tensor part is zero when we average over all directions. Using spherical tensor analysis, the frequency shift of the transition sub-level of a cesium fountain clock caused by the [PERSON] effect in the BBR field is calculated, and the frequency shift of the clock transition can thus be obtained [18]:  
\[ \begin{array}{ll}\Delta v_{Stark}=\frac{1}{h}\big{(}\Delta E_{F=4,m_{F}=0}- \Delta E_{F=3,m_{F}=0}\big{)}\\ \hskip 14.226378 pt=-\frac{1}{2}E^{2}\Big{(}\frac{16}{7}\frac{\alpha_{10}}{h}+ \frac{1}{7}\frac{\alpha_{12}}{h}\Big{)}\end{array}  \tag{5}\]  
In this formula, the contributions of contact and spin dipole-dipole interaction correspond to the scalar polarizability \(\alpha_{10}\) and the tensor polarizability \(\alpha_{12}\), respectively. The average contribution of the tensor polarization rate to the frequency shift is zero. The scalar polarizability \(\alpha_{10}\) can be obtained by theoretical calculation and experimental measurement [19]. The DC Stark frequency shift caused by the BBR field is determined from the root mean square of the electric field intensity. After substituting this into Equation (3), the theoretical value of the impact of the BBR shift on the clock transition at temperature \(T\) can be obtained as follows:  
\[ \begin{array}{ll}\Delta v_{BBR}&=-\frac{8}{7}\frac{\alpha_{10}}{h}\big{<}E ^{2}(\
u)\\ &=-\frac{8\alpha_{10}}{7h}(831.9V/\text{m})^{2}\Big{(}\frac{T}{300}\Big{)}^{4 }\end{array}  \tag{6}\]  
A more accurate analysis of the BBR shift needs to consider the quantum characteristics of the BBR field. In 1981, [PERSON] et al. proposed an approximate method [20]. Taking into account the frequency spectrum distribution of the electronic dipole lines D1 and D2, the corrective term about \(T^{6}\) was obtained, which accounts for the BBR spectrum with respect to the DC Stark shift of the Cs clock transition, and the expression for the total BBR shift is obtained as follows [15]:  
\[\frac{\Delta v_{BBR}}{\
u_{0}}=\beta\Big{(}\frac{T}{T_{0}}\Big{)}^{4}\bigg{[} 1+\varepsilon\bigg{(}\frac{T}{T_{0}}\bigg{)}^{2}\bigg{]} \tag{7}\]  
where \(T_{0}\) = 300 K and \(\varepsilon\) = 1.4 \(\times\) 10\({}^{-2}\)[19], \(\beta\) can be obtained by theoretical calculation or experimental measurement; here, we take the direct measurement value in [21], \(\beta\) = (\(-\)1.43 \(\pm\) 0.09) \(\times\) 10\({}^{-14}\). The uncertainty of BBR shift describes the sensitivity of the BBR shift to temperature fluctuations. The following expression of the uncertainty of BBR shift can be derived:  
\[\bigg{(}\frac{\Delta v_{\text{BBR}}}{\
u_{0}}\bigg{)}^{\prime}=\bigg{[}6\beta \varepsilon\bigg{(}\frac{T}{T_{0}}\bigg{)}^{5}+4\beta\bigg{(}\frac{T}{T_{0}} \bigg{)}^{3}\bigg{]}\frac{\Delta T}{T_{0}} \tag{8}\]  
where \(\Delta T\) denotes the change of temperature.