I am currently slightly confused on the different models that exists to treat the interaction of a two-level system with a radiation field.
Currently, I understood that there are several models.
Rabi Model
The interaction of a two-level system (quantum mechanical treated) with one mode only of a light field (classically treated in the dipole-approximation)
Jaynes-Cummings model
The interaction of a two-level system (quantum mechanical treated) with one mode only of a light field (quantum mechanically treated in the dipole-approximation)
Wigner-Weisskopf model
The interaction of a two-level system (quantum mechanical treated) with all modes of a radiationfield (free field) (quantum mechanically treated in the dipole-approximation)
Damped Jaynes-Cummings model
Here, the two level system interacts with a light mode, that is coupled to a reservoir.
I am confused. In all models (except for the Rabi-model) I apply the dipole-approximation. Fair enough. And all of them can be described with the total Hamiltonian $H=H_A+H_F+H_I$, whereas $H_A=\omega_0\sigma_+\sigma_-$ , $ H_F=\sum_k\omega_kb_k^\dagger b_k$ and $H_I=\sigma_+\otimes B+\sigma_-\otimes B^\dagger$ and $B=\sum_kg_kb_k$. Here, $\omega_0$ describes the transition energy between the two levels and $\omega_k$ the field-mode frequency, $g_k$ the coupling constants.
My question now is: This interaction hamiltonian implies interaction with all modes, so for the JCM I need to set all but one coupling strengths to zero. For the damped Jaynes-Cummings model, I set a Lorentz spectral density to model the reservoir-interaction the light mode interacts with. But why? This would imply my two-level system interacts with all modes, just not equally coupled. Also: How does the spectral density of a free field look like?
Another question: What is meant with "correlation time of the bath" or "typical time scales", when they talk about if and how a dynamics can be Markovian or not.
