2
$\begingroup$

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?

$\endgroup$
3
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.