\tanh(\frac{\beta \hbar \omega}{2}) }.$$ Here $T_{12}$ is defined in the main text by Eq. (\[Transmisson\_p\]), evaluated at $v_1=0$, $\beta = 1/(k_b T)$. Finally, we make one approximation, that is to assume $\tanh(x) \approx x$ valid if frequency is small in comparison with temperature. Then, we find $$a_1 \approx e^2 n_1^2 \rho_D,$$ if we compare our formula with that of Jauho and Smith[@jauho-smith1993] (Eq. (5) and (27)) and that of Flensberg and Hu[@flensberg-hu1995] (Eq. (2) and (20)). Here $n_1$ is the carrier surface density and $\rho_D$ is the Coulomb drag coefficient. Because of the existence of the second term, $a_2$, we don’t have a simple interpretation of Joule heating due to Coulomb drag. Numerically, for the parameters used for Fig. 4(a) in the main texts, we find $a_1 = 5.5 \times 10^{-6}\,$Ws$^2$/m$^4$, while $a_2 = - 4.3 \times 10^{-6}\,$Ws$^2$/m$^4$. There is a cancellation effect, given a smaller overall $a$.

Numerical calculation of the NEGF conjunction system\[app-e\]
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For the 4-terminal junction double-layer system discussed in the main texts, wave-vector is not a good quantum number and we cannot study finite size transport in ${\bf k}$ space. As a result, we do calculation in real space with the electron Green’s functions $G_{jk}^{>,<,r}$ 