On-Demand Control of Input-State-Dependent Single-Photon Scattering in Multi-Mode Waveguides

TL;DR

This work develops a theoretical framework for on-demand control of single-photon transport in broadband, multi-mode waveguides using a driven Λ-type emitter. By deriving an exact multi-mode scattering matrix via the Lippmann-Schwinger formalism, it reveals two interference mechanisms—EIT for complete transmission and Fano resonance for complete reflection—whose balance is tunable through the driving field. A key finding is input-state-dependent scattering: in multi-mode regimes, a coherently prepared superposition state enables unit reflection through inter-mode interference, while single-mode inputs cannot achieve this, and EIT maintains robust transmission. The results point to versatile, state-controlled quantum photonic devices, including dual-frequency filters and mode-selective routers, with broad applicability to on-chip spectrometers.

Abstract

Precise control of a single photon transport in broadband, multi-mode waveguides is a fundamental challenge for scalable quantum networks. We propose a theoretical scheme for on-demand control of single-photon scattering using a driven $Λ$-type emitter coupled to a rectangular waveguide. By employing the Lippmann-Schwinger formalism, we derive the exact analytical scattering matrix and reveal two key interference mechanisms: electromagnetically induced transparency for complete transmission and Fano resonance for complete reflection. We demonstrate that the single-photon scattering is dynamically engineered by the driving field, enabling a switch between complete transmission and dual-frequency complete reflection. Crucially, in the multi-mode regime, we show that the scattering is governed by quantum interference between modes, making it critically dependent on the input photonic state. By preparing the photon in a specific coherent superposition state, the multi-mode interference is harnessed to achieve Fano resonance-mediated complete reflection. Conversely, a single-mode input suppresses complete reflection. This input-state-dependent scattering establishes a general framework for multi-mode quantum photonics, paving the way for broadband dual-frequency filters, multi-mode quantum routers, and on-chip spectrometers.

Figures (4)

• Figure 1: Schematic of a driven $\Lambda$-type emitter coupled to an infinite rectangular waveguide (cross-section $A=ab, a=1.5b$). The waveguide couples $\vert e\rangle\leftrightarrow\vert g\rangle$ with strength $g_{j,k}$ and detuning $\omega_{e}-\omega_{j,k}$, while a classical field drives $\vert e\rangle\leftrightarrow\vert f\rangle$ with Rabi frequency $\Omega$ and detuning $\delta$.
• Figure 2: Single-mode regime analysis. (a) Phase map showing complete reflection peak (CRP) and complete transmission peak (CTP) counts across varied driving field parameters $\Omega$ and $\delta$; (b) Reflectance $R$ as a function of $\omega$ and $\Omega$, where black thick curves indicate complete reflection at $\text{Re}[G(\omega)]=0$, and white dotted curves represent $\omega=\tilde{\nu}_{\pm}$; (c) $R$ versus $\omega$ for $(\delta,g)=(2,0.1),(0,0.1),(0,0.2)$ at $\Omega=1$ ; (d) $R$ versus $\omega$ for $\Omega=0,1,1.5,2$. In (b) and (d), $g=0.1$ and $\delta=0.5$. Other parameters include $a=1.5b$, with cutoff frequencies $\omega_{1}=3.78$, $\omega_{2}=7.02$, and $\omega_{e}=(\omega_{1}+\omega_{2})/2$. Frequencies are normalized to $c/b$.
• Figure 3: Reflectance $R(\omega)$ for different input states in (a) the two-mode and (b) the three-mode regimes. The common parameters are $a=1.5b$, $\Omega=0.5$, and $\delta=0$, with frequencies normalized to $c/b$. (a) Two-mode regime: the results are shown for the specific coherent superposition state (SCSS, $c_{j}\propto\rho_{j}g^{\ast}_{j,k_{j}}$), the single-mode state in the TM$_{11}$ mode (SMS1, $c_{1}=1$) and the TM$_{31}$ mode (SMS2, $c_{2}=1$), the equal-probability superposition state ($c_{1}=c_{2}=1/\sqrt{2}$), and dark state ($\sum_{j=1}^{j_{max}}c_{j}g_{j,k_{j}}=0$). Additional parameters are $g=0.1$, cutoff frequencies $\omega_{2}=7.02$ and $\omega_{3}=9.65$, and $\omega_{e}=(\omega_{2}+\omega_{3})/2$. (b) Three-mode regimes: the results are shown for the SCSS, SMS1, SMS2, the single-mode state in the TM$_{13}$ mode (SMS3, $c_{3}=1$), the equal-probability superposition state ($c_{1}=c_{2}=c_{3}=1/\sqrt{3}$), and dark state. Additional parameters are $g=0.05$, cutoff frequencies $\omega_{3}=9.65$ and $\omega_{4}=10.93$, and $\omega_{e}=(\omega_{3}+\omega_{4})/2$.
• Figure 4: Two-mode regime analysis. (a-c) Total reflectance $R(\omega)$ with varying $\Omega$ and $\delta$ for different input photon states: (a) the specific coherent superposition state (SCSS, $c_{j}\propto\rho_{j}g^{\ast}_{j,k_{j}}$), (b) the single-mode state in the TM$_{11}$ mode (SMS1, $c_{1}=1$), and (c) in the TM$_{31}$ mode (SMS2, $c_{2}=1$). (d)-(f) Mode-resolved reflectance $R_{i}(\omega)$ and transmittance $T_{i}(\omega)$ for modes $i=1$ (TM$_{11}$) and $i=2$ (TM$_{31}$) corresponding to the input states in (a-c), respectively, calculated at $\Omega=0.5$ and $\delta=0$. Other common parameters are $a=1.5b$, $g=0.1$, cutoff frequencies $\omega_{2}=7.02$, and $\omega_{3}=9.65$, $\omega_{e}=(\omega_{2}+\omega_{3})/2$, with frequencies normalized to $c/b$.