# How to formulate the master equation for three systems?

I have a three composite system of the form $$H_{\text{tot}}=H_{ab}\otimes H_c$$ where the system $$C$$ is behaving as the dissipator or the environment (I can model it as a thermal bath). And it is coupled only to system $$B$$ but not $$A$$. While $$A$$ is coupled with $$B$$ and entangles with it under time evolution. At $$t=0$$ is can take a composite state completely separable $$H_{\text{tot}}(0)=H_a\otimes H_b \otimes H_c$$. My objective is to solve the Master equation (more precisely the Lindbladian form) for $$\rho_{ab}.$$

But when I do that (with the partition as $$H_{ab}|H_c$$), the sub-system $$A$$ trivially disappears from the equations of motion. Because it does not couple with $$C$$ directly but only acts via $$B$$ indirectly. What is the right way to model this kind of interaction?

• You have a Hilbert space $$\mathcal{H}_A\otimes\mathcal{H}_B\otimes\mathcal{H}_C$$.
• The initial state is $$\rho_\text{tot}(0)=\rho_A\otimes\rho_B\otimes\rho_C$$.
• There is a Hamiltonian acting on the system of the form $$H_{\text{tot}}=H_{AB}\otimes\mathbb{I}_C+\mathbb{I}_A\otimes H_{BC}$$
• Instead of directly calculating the effect of the Hamiltonian evolution $$\rho_{\text{tot}}(t)=e^{-iH_{\text{tot}}}\rho_{\text{tot}}(0)e^{iH_{\text{tot}}}$$, you want to solve a Master equation $$\frac{d\rho_{AB}}{dt}=-i[H_{AB},\rho_{AB}]+\sum_nL_n\rho_{AB}L_n^\dagger-\frac12L_n^\dagger L_n\rho_{AB}-\frac12\rho_{AB}L_n^\dagger L_n$$
If so, then note that the presence of the Hamiltonian $$H_{AB}$$ means that the systems A and B should couple together.