## 3 Polarization Tensor in the Area of Evanescent Waves
\begin{split}\text{Im}\,\Pi^{(1)}_{00}(\omega,k;\Delta,T_{g})= \frac{8\alpha\hbar c}{v_{F}^{2}p(\omega,k)}\theta[\hbar cp(\omega,k)-\Delta] \int\limits_{u^{(-)}(\omega,k)}^{u^{(+)}(\omega,k)}\frac{du}{e^{\bar{p}(T_{g}) u}+1}\frac{(2 cu-\omega)^{2}-v_{F}^{2}k^{2}}{\sqrt{v_{F}^{2}k^{2}A(\omega,k; \Delta)-(2 cu-\omega)^{2}}},\\ \text{Im}\,\Pi^{(1)}(\omega,k;\Delta,T_{g})=\frac{8\alpha\hbar c (\omega,k)}{v_{F}^{2}}\theta[\hbar cp(\omega,k)-\Delta]\int\limits_{u^{(-)}( \omega,k)}^{u^{(+)}(\omega,k)}\frac{du}{e^{\bar{p}(T_{g})u}+1}\frac{(2 cu-\omega )^{2}+v_{F}^{2}k^{2}[1-A(\omega,k;\Delta)]}{\sqrt{v_{F}^{2}k^{2}A(\omega,k; \Delta)-(2 cu-\omega)^{2}}},\end{split}  \tag{22}\]  
which are valid over the entire region (9). Here, \(\theta(x)\) is the step function equal to 1 for \(x\geqslant 0\) and to 0 for \(x<0\). Next, we consider the polarization tensor in the interval (10). In this case, the first contributions to (8) are given by [52]  
\[\Pi^{(0)}_{00}(\omega,k;\Delta)=\frac{\alpha\hbar k^{2}}{\bar{p}( \omega,k)}\,\Psi(\omega,k,\Delta), \tag{23}\] \[\Pi^{(0)}(\omega,k;\Delta)=\alpha\hbar k^{2}\bar{p}(\omega,k)\, \Psi(\omega,k,\Delta),\]  
where  
\[\bar{p}^{2}(\omega,k)=\frac{v_{F}^{2}}{c^{2}}k^{2}-\frac{\omega^{2}}{c^{2}}\geqslant 0 \tag{24}\]  
and \(\Psi\) is defined as  
\[\Psi(\omega,k,\Delta)=2\bigg{\{}\frac{\Delta}{\hbar c\bar{p}(\omega,k)}+\left[ 1-\frac{\Delta^{2}}{\hbar^{2}c^{2}\bar{p}^{2}(\omega,k)}\right]\,\arctan\frac {\hbar c\bar{p}(\omega,k)}{\Delta}\bigg{\}}. \tag{25}\]  
Similar to the plasmonic interval (9), in the interval (10) the quantities \(\Pi^{(1)}_{00}\) and \(\Pi^{(1)}\) are the complex-valued functions. Here, we present their explicit expressions without separating the real and imaginary parts [52]  
\[ \begin{split}\Pi^{(1)}_{00}(\omega,k;\Delta,T_{g})=\frac{8\alpha \hbar c^{2}\bar{p}(\omega,k)}{v_{F}^{2}}\int\limits_{\frac{\Delta}{2 mc}}^{ \infty}\frac{dv}{e^{\bar{p}(\omega,k)}+1}\left\{\frac{\omega^{2}}{c^{2}}-\frac {1}{2}\sum\limits_{\lambda=\pm 1}\frac{\left[\bar{p}(\omega,k)v+\lambda\frac{\omega}{c} \right]^{2}+\frac{v_{F}^{2}k^{2}\Delta^{2}}{c^{4}\hbar^{2}\bar{p}^{2}(\omega,k )}}\right\},\\ \Pi^{(1)}(\omega,k;\Delta,T_{g})=\frac{8\alpha\hbar c^{2}\bar{p}( \omega,k)}{v_{F}^{2}}\int\limits_{\frac{\Delta}{2 mc}}^{\infty}\frac{dv}{e^{ \bar{p}(\omega,k)}+1}\left\{\frac{\omega^{2}}{c^{2}}-\frac{1}{2}\sum\limits_{ \lambda=\pm 1}\frac{\left[\bar{p}(\omega,k)v+\lambda\frac{\omega}{c}\right]^{2}+ \frac{v_{F}^{2}k^{2}\Delta^{2}}{c^{4}\hbar^{2}\bar{p}^{2}(\omega,k)}}{\sqrt{1 -v^{2}-2\lambda\frac{\omega}{c\bar{p}(\omega,k)}v+\frac{v_{F}^{2}k^{2}\Delta^{ 2}}{c^{4}\hbar^{2}\bar{p}^{2}(\omega,k)}}}\right\},\end{split}  \tag{26}\]where  
\[D(\omega,k,T_{g})=\frac{\hbar c\bar{p}(\omega,k)}{2k_{B}T_{g}}. \tag{27}\]  
For a calculation of the contribution \(F_{M}\) to the nonequilibrium dispersion force (2), which is given by (3), one also needs to obtain the values of the polarization tensor at the pure imaginary Matsubara frequencies. These are easily obtained from (8) and the respective expressions (23)-(27) found in the interval (10) where we put \(\omega=i\xi_{E,I}\). In this case, the definitions (24) and (27) take the form  
\[\bar{p}^{2}(i\xi_{E,I},k)\equiv\bar{p}_{I}^{2}(k)=\frac{v_{F}^{2}}{c^{2}}k^{2}+ \frac{\xi_{EJ}^{2}}{c^{2}},\qquad D(i\xi_{E,I},k,T_{g})=\frac{\hbar c\bar{p}_{ I}(k)}{2k_{B}T_{g}}. \tag{28}\]  
Thus, all expressions for the polarization tensor appearing in both contributions \(F_{M}\) and \(F_{r}\) to the nonequilibrium dispersion force through the reflection coefficients (4) are presented.