# Can quantum systems interact with multiple environments of different types?

If it can, how can we write the Hamiltonian of the total System is it just (for example with N bath)

$$H_{tot} = H_{s} + H_{B_{1}} + H_{B_{2}} + ... + H_{B_{N}} + H_{I_{1}} + H_{I_{2}} + ... +H_{I_{N}}$$

And can environments be a different structure, for example a qubit modeled by two level system coupled to a bosonic bath (harmonic) and to a different two level system (spin) independent from the bosonic bath ? How it evolves (finding the Linbladian operator) ?

Appreciate if you leave a reference or paper related to this question. I am just started learning Open Quantum System for my undergraduate thesis.

• Unless there is some properties that are different for these different baths, you should be able to combine them all into one bath so that you end up with the same old situation. Aug 14, 2020 at 7:23

Can quantum systems interact with multiple environments of different types? If it can, how can we write the Hamiltonian of the total System is it just (for example with N bath) $$H_{tot} = H_{s} + H_{B_{1}} + H_{B_{2}} + ... + H_{B_{N}} + H_{I_{1}} + H_{I_{2}} + ... +H_{I_{N}}$$
Yes, in principle you can do whatever you want with quantum systems and reservoirs; you can connect them through different couplings and play around with different interaction-network topologies. The example you are proposing may be described by the Hamiltonian: $$H=H_{q1}+H_{q2}+H_B+H_{q1q2}+H_{q1B},$$ where $$H_{qj}$$ is the free Hamiltonian of qubit $$j$$, $$H_B$$ is the bath Hamiltonian, $$H_{q1q2}$$ introduces a qubit-qubit coupling and $$H_{q1B}$$ is the interaction Hamiltonian between qubit 1 and the bath. Note, however, that in such systems the action of the bath may "reach" qubit 2 as well, if they qubit-qubit interaction is not very weak. In these cases, so-called "global master equations" are required. On the contrary, if $$H_{q1q2}$$ has a very weak coupling energy, you may rely on a local master equation and the dissipative action of the bath will only affect qubit 1, while qubit 2 will follow a unitary dynamics. This kind of systems are very interesting and may display curious phenomena, for instance, if you are interested in a reference, have a look at this paper.