Theory of Three-Photon Transport Through a Weakly Coupled Atomic Ensemble

TL;DR

This work develops a diagrammatic scattering framework to analyze three-photon transport through weakly coupled, chiral atomic ensembles in a 1D waveguide. By leveraging Bethe Ansatz and Yudson representations, the authors derive exact $n$-photon S-matrices for single-atom scattering and extract connected (genuine) $n$-photon interactions, enabling analytic access to three-photon output states. The perturbative expansion in the small coupling parameter $eta$ yields explicit expressions for the outgoing three-photon wavefunction, the connected third-order correlation function $g_c^{(3)}$, and the third-order electric-field quadrature cumulant, with numerical validation against cascaded master equation simulations showing good agreement at moderate OD. The results reveal non-Gaussian light signatures arising from two- and three-photon processes and establish a practical framework for predicting and observing non-Gaussian photon transport in atomic ensembles, with potential extensions to higher-order photon inputs and varied input states.

Abstract

Understanding multi-photon interactions in non-equilibrium quantum systems is an outstanding challenge in quantum optics. In this work, we develop an analytical and diagrammatic framework to explore three-photon interactions in atomic ensembles weakly coupled to a one-dimensional waveguide. Taking advantage of the weak coupling, we use our diagrammatic framework to perform perturbation theory and calculate the leading-order contributions to the three-photon wavefunction, which would otherwise be intractable. We then compute the outgoing photon wavefunction of a resonantly driven atomic ensemble, with photon-photon interactions truncated up to three photons. Our formulation not only captures the individual transmission of photons but also isolates the connected S-matrix elements that embody genuine photon-photon correlations. Through detailed analysis, we obtain the analytic expressions of the connected third-order correlation function and the third-order electric-field-quadrature cumulant, which reveal non-Gaussian signatures emerging from the interplay of two- and three-photon processes. We also calculate the optical depth where non-Gaussian photon states can be observed. Numerical simulations based on a cascaded master equation validate our analytical predictions on a small-scale system. These results provide a formalism to further explore non-equilibrium quantum optics in atomic ensembles and extend this to the regime of non-Gaussian photon transport.

Figures (19)

• Figure 1: An array of $M$ chirally coupled two-level atoms (depicted as red circles) driven by an external coherent field $\ket{\alpha}$ producing a strongly correlated output photon state $|\text{out}\rangle$. Each atom couples to the waveguide (dark line) with a decay rate $\Gamma = \beta\Gamma_{\text{tot}}$ and to external loss channel with a decay rate $(1-\beta)\Gamma_{\text{tot}}$ (gray line). Without loss of generality, for theoretical convenience we model each loss channel as an auxiliary waveguide.
• Figure 2: Diagrammatic representation of ${{}_{22}\hat{S}_{p_1p_2,k_1k_2}}$. The first two terms represent the disconnected parts ${}_{11}\hat{S}_{p_1,k_1}\,{}_{11}\hat{S}_{p_2,k_2}$ and ${}_{11}\hat{S}_{p_1,k_2}\,{}_{11}\hat{S}_{p_2,k_1}$, while the last term represents the connected part $\hat{S}_{p_1p_2,k_1k_2}^C$. We have explicitly included the $\beta$-dependence of the connected part.
• Figure 3: Diagrammatic representation of ${}_{21}\hat{S}_{\slashed{p}_1p_2,k_1k_2}$. The first two terms represent the disconnected parts ${}_{10}\hat{S}_{\slashed{p}_1,k_1}\,{}_{11}\hat{S}_{p_2,k_2}$ and ${}_{10}\hat{S}_{\slashed{p}_1,k_2}\,{}_{11}\hat{S}_{p_2,k_1}$, while the last term represents the connected part $\hat{S}_{p_1p_2,k_1k_2}^C$. We have explicitly included the $\beta$-dependence of the connected part.
• Figure 4: Two examples of the concatenated diagrams.
• Figure 5: Concatenated diagrams of a few possible processes for unidirectional two-photon scattering. (a) two photon non-interacting transport through the waveguide. (b)-(d) each represents the sum of two-photon transport process where (b) two-photon interact exactly once at a single atom (c) two-photon interaction happening twice at two different atoms (d) two photons interact twice at two atoms before one photon is lost at the third atom. The order estimates below each diagram show their respective scaling behavior in the large optical depth regime with $\beta M=\mathcal{O}(1)$.