An asymptotic field approach for the control of dipole emission in integrated structures

TL;DR

This work introduces an asymptotic-field quantization framework to model spontaneous emission in arbitrary integrated photonic structures, enabling direct, channel-by-channel calculation of emission rates without Lorentzian or point-coupling assumptions. By grounding the method in stationary Maxwell solutions and Fermi's Golden Rule, it recovers standard results for waveguides and ring resonators and naturally incorporates backscattering and non-Lorentzian resonances. The authors demonstrate the framework with a tunable ring-based single-photon source and extend it to an interferometric design that deterministically routes emitted photons into specified output modes. The approach offers a general, physically transparent tool for designing and analyzing integrated quantum photonic devices across diverse material platforms, with broad potential for scalable quantum information and sensing applications.

Abstract

We present a general framework to model spontaneous emission in integrated photonic structures by exploiting quantization of the electromagnetic field in terms of asymptotic in/out modes. This approach allows for an efficient and physically meaningful calculation of the emission rate into each radiative channel of an arbitrary structure, without relying on approximations such as Lorentzian lineshapes or point-like system-bath coupling. We show that with this approach one can recover well-known results for dipole emission in waveguides or ring resonators, and that such results can be easily extended to include the effect of backscattering. Finally, as an application, we design a tunable integrated single-photon source that enables full control over both the emission rate and output mode. This flexibility makes our method particularly well-suited for the design and analysis of integrated single-photon sources in various material platforms.

Figures (10)

• Figure 1: A sketch of the kind of structure of interest. The red spot represents the dipole inside the interaction region.
• Figure 2: Sketch of the structures for enhanced dipole emission; (a) straight, single-mode waveguide. The red dot indicates the dipole, located at $\mathbf{r}_0 = (x_0, y_0, z_0)$. The two radiative channels of interest are called L (from the left) and R (from the right). The inset depicts the waveguide cross-section, with the dipole (the red dot with vertical arrows) emitting into the guided mode. (b) Ring resonator. The dipole is located at a longitudinal position $\zeta_0$ along the ring. The resonator supports both clockwise and counterclockwise modes. (c) Ring resonator with a localized scatterer, indicated as a brown mark at position $l_1$ along the ring. The scatterer introduces a coupling between the clockwise and counterclockwise modes, which affects both the total emission rate and the direction of the emitted photons.
• Figure 3: Contour plot of the normalized dipole emission rate $\Gamma_{\text{rb}}/\Gamma_{\text{wg}}$ in log scale as function of backscattering phase $\Delta$, and the reflectivity $\rho$. In particular, the emission can be completely suppressed when $\Delta = \pi/4 + m\pi$, where $m \in \mathbb{N}$. As expected, when $\rho\rightarrow 0$, one obtains the result for the backscatter-free ring. Parameters used: $\lambda= 630 nm$, length $l=300\pi\lambda_0=93.7$$\mu$m, and self-coupling parameter $\sigma=0.98$.
• Figure 4: The normalized dipole emission rate, $\Gamma_{\text{rb}}/\Gamma_{\text{wg}}$ (black solid line), and the normalized output probability, $P_{L(R)}=\Gamma_\mathrm{rb, L(R)}/\Gamma_\mathrm{rb}$ (red solid and dashed line, respectively), are shown as functions of the backscattering phase $\Delta$. Notably, while the overall dipole emission enhancement remains nearly constant, the probability of photon emission from the left or right port can be tuned by adjusting the dipole position within the ring. The parameters used in the simulation are: $\lambda_0 = 630\;\text{nm}$, length $l=300\pi\lambda_0=93.7$$\mu$m, reflectivity $\rho = 17 \times 10^{-3}$, and self-coupling parameter $\sigma=0.98$.
• Figure 5: An illustration of the proposed device is shown. The system comprises a main ring resonator (Main) with radius $R_m$, which is coupled to a Sagnac interferometer (with a $50:50$ beam splitter) via a self-coupling parameter $\sigma_{ms}$, and also coupled to an auxiliary ring (Aux) of radius $R_a$ with coupling strength $\sigma_{ma}$. This configuration allows for tunable interference between the clockwise and counterclockwise propagating modes, thereby enabling control over the emission properties of the dipole (indicated by the red spot) embedded within the main ring.