Describing small, NRQM systems purely in terms of photons

Is there a canonical way to describe an open, non-relativistic quantum system with density matrix $$\rho(t)$$ entirely in terms of the light that it emits and absorbs (and vice versa?) Or is it possible in general for a density matrix trajectory $$\rho(t)$$ to be induced by several (e.g. possibly contrived and time dependent) photon baths?

For Markovian systems, this is possible in a certain sense. If the quantum system is linearly coupled to the bath (which is usually the case, e.g. in light-matter interaction, in cavity-bath interaction etc.), one can write an input-output relation

$$\hat{b}_\mathrm{out}(t) = \hat{b}_\mathrm{in}(t) + i\kappa\hat{a}(t)$$

where $$b$$ are the bath operators, $$a$$ is the system operator and $$\kappa$$ the coupling between them.

This equation implies that if you are able to measure/specify the correlation function of the input and output operators, you are able to reconstruct the corresponding correlation function of the system operator.

• In this context, would the subscripts 'in' and 'out' refer to the bath operator before and after the system is introduced (e.g. as a perturbation), or should they be interpreted differently? In the terms you have introduced, my question might be rephrased as, "if $\hat b$ represents the time evolution of a photon field, then is the operator $\hat a$ well defined over a wide range of additional perturbations $ik_\gamma \hat a_\gamma(t)$, or is the operator $\hat a$ always dependent on other perturbations?"
– TLDR
Nov 16, 2020 at 1:05
• @TLDR For the physical meaning of the in- and out-operator check out my answer here: physics.stackexchange.com/a/534741/101770 They are essentially asymptotic degrees of freedom in the sense of scattering. So you can think of it as the radiation you would see "far away" from the system. Nov 16, 2020 at 10:31
• With regards to your second question: I am not 100% sure what $ik_\gamma \hat{a}_\gamma(t)$ signifies, but will attempt to answer anyway. My statement in the answer was that you can in principle reconstruct correlation functions of $\hat{a}(t)$ from the in- and out-operator correlation functions. If instead you want to calculate $\hat{a}(t)$ by solving the systems dynamics, you have to solve its equations of motions. Indeed, other perturbations become relevant then. However, the statement I made is general, at least for Markovian systems. Nov 16, 2020 at 10:34
• So should $\hat b(t)$ and $\hat a(t)$ be understood as Heisenberg picture operators, and $t$ as approaching infinity? My second question was whether the operator $\hat a$ is dependent on other perturbations that might act on the system ($ik_\gamma \hat a_\gamma(t)$ represents a sum in Einstein notation.) For instance, with (asymptotic?) system operators $\hat a(t)$ and $\hat c(t)$, would the 'output' operator in which both interactions are present be expressed as $\hat b_{\text{in}}(t) + i\kappa \hat a(t) + i\sigma \hat c(t)$, or something else that accounts for interactions?
– TLDR
Nov 16, 2020 at 22:23
• @TLDR The precise equations of motion are fixed by the Hamiltonian. I would suggest that you include a few example Hamiltonians in your questions to clarify which type of couplings/perturbations exactly you are thinking about. In the above comment, it is not clear to me what you mean. Nov 17, 2020 at 10:20