Photon scattering from a system of multilevel quantum emitters. I. Formalism

We introduce a formalism to solve the problem of photon scattering from a system of multilevel quantum emitters. Our approach provides a direct solution of the scattering dynamics. As such the formalism gives the scattered fields' amplitudes in the limit of a weak incident intensity. Our formalism is equipped to treat both multiemitter and multilevel emitter systems, and is applicable to a plethora of photon-scattering problems, including conditional state preparation by photodetection. In this paper, we develop the general formalism for an arbitrary geometry. In the following paper (part II) S. Das et al. [Phys. Rev. A 97, 043838 (2018)], we reduce the general photon-scattering formalism to a form that is applicable to one-dimensional waveguides and show its applicability by considering explicit examples with various emitter configurations.

Figure 1

Schematic of photon scattering from a generic system of emitters distributed in some dielectric medium with a spatially dependent electric permittivity $\epsilon \left(\vec{r}\right)$. The emitters can be either a simple two-level system with a decay rate $\mathrm{\Gamma }$ or have multiple levels. The emitters are assumed to consist of two separated subspaces: an excited-state manifold $M_e$ with excited states $|e_m\rangle (m=1,2,\cdots ,n)$ and corresponding decay rates ${\mathrm{\Gamma }}_m$, and a ground-state manifold $M_g$ with ground states $|g_m\rangle$. We assume the couplings between the two manifolds $V_{\pm }$ to be perturbative and model the excited states' decay by Lindblad operators ${\widehat{L}}_k$. The couplings within the excited and ground manifold are shown by the wiggly and straight arrow-headed lines, respectively. The mode operator for an incident field is represented by ${\widehat{\mathcal{E}}}_{\mathrm{in}}^{+}$, while the scattered outgoing field is given by the mode operator ${\widehat{\mathcal{E}}}_s^{+}$.