# Common subsystem-bath interaction operators?

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 $$H_I$$

• Did you check a textbook on open quantum systems? Commented Mar 4, 2021 at 18:24

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.

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

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.

• Thanks for your answer and for those refs! Would the physical interpretation of this interaction be simply the hopping of modes in and out of the cavity? Commented Mar 4, 2021 at 12:31
• @FriendlyLagrangian glad it was helpful! With regard to your question: yes, that's the idea. The $g_i$-coupling is then determined by the mirror. To get some more intuition for this model, you can consider the system-bath Hamiltonian of a two-level system coupled to the electromagnetic field. It has a very similar form (in the rot.-wave approx) which can be derived from the minimal coupling prescription. For the cavity case, a derivation has only been given more recently (see doi.org/10.1103/PhysRevA.67.013805 and doi.org/10.1103/PhysRevX.10.011008). Commented Mar 4, 2021 at 12:48
• @FriendlyLagrangian Modes don't hop. Excitations hop. What is hopping in and out of the cavity is e.g. a photon which leaves the cavity. Commented Mar 4, 2021 at 18:29
• @NorbertSchuch good point, I guess I’ve never got to know the difference between a mode and an excitation, specially for a free oscillator. Commented Mar 5, 2021 at 9:32
• One oscillator = one mode. Then, you can have as many excitation quanta on this oscillator as you want. When you couple two oscillators, it is not the oscillators which hop (obviously!), but the excitations (i.e. energy). Commented Mar 5, 2021 at 10:37