Common subsystem-bath interaction operators?

Question

Im new to the field of quantum open systems and I wanted to know what are the most common operators for describing the subsystem-bath interaction. To narrow down a bit my question, say we have as a subsystem an EM field mode in a cavity interacting with outside EM field modes that serve as a sort of damping.

The total Hamiltonian of the system $H$ would look something like $$ H = H_S + H_B + H_I = \underbrace{(\omega_0a^{\dagger}a)\otimes1_B}_{\text{cavity}}+ \underbrace{1_S\otimes \left(\sum_i^N \omega_0 a^{\dagger}_ia_i\right)}_{\text{bath}}+ \underbrace{\sum_{\alpha}A_{\alpha}\otimes B_{\alpha}}_{\text{interaction}}, $$ and the evolution of the subsystem described by $\rho_S(t)=\text{Tr}_B\rho(t)$ can be approximated by the Lindbland equation.

What could $A_{\alpha}$ and $B_{\alpha}$ be?

I was thinking that maybe some linear combination of $a$ and $a^{\dagger}$ (what else could it be?..) so that it describes the spontaneous emission and absorption of the cavity mode which would ultimately lead to $\langle n(t) \rangle\equiv \text{Tr} (a a^{\dagger} \rho_S(t)) \longrightarrow$ thermal, as $t\rightarrow \infty$.

But I cant come up with a good form of $H_I$…

Answer 1

For the particular case specified in the OP (a cavity mode coupling to an external bath), the most standard form is probably the Gardiner-Collett Hamiltonian [see C. W. Gardiner and M. J. Collett Phys. Rev. A 31, 3761 (1985)] or a variation thereof.

Adapted to your specific example, the interaction Hamiltonian would read

$$H_\mathrm{int} = \sum_i^N g_i a_i^\dagger a + g_i^* a_i a^\dagger \,.$$

I'll leave it to you to extract the $A_\alpha$, $B_\alpha$ operators from that.

Note that this is only a simple model for the system-bath interaction. Even for this particular case, more general forms are required in the presence of extreme coupling strengths [see e.g. Frisk Kockum et al. Nature Reviews Physics 1, 19–40 (2019) for a review].

For different systems such as a two-level system coupling to a bath, different operators are required. One often finds that models with a similar bilinear form as the above give a good first approximation and are commonly used practically.