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?


1 Answer 1


To be clear:

  • 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 $$ \frac{d\rho_{AB}}{dt}=-i[H_{AB},\rho_{AB}]+\sum_nL_n\rho_{AB}L_n^\dagger-\frac12L_n^\dagger L_n\rho_{AB}-\frac12\rho_{AB}L_n^\dagger L_n $$

Is this a correct summary of what you're wanting to do?

If so, then note that the presence of the Hamiltonian $H_{AB}$ means that the systems A and B should couple together.


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