Photonic scattering in 2D waveguide QED: Quantum Goos-Hänchen shift

TL;DR

The paper develops a Green function framework to analyze single-photon scattering in two-dimensional waveguide QED with atomic lattices, revealing a quantum Goos–Hänchen (QGH) shift in backward scattering that is resonantly enhanced by subradiant states. The forward and backward scattering amplitudes are linked to the excitation Green function $G=(\omega - H_{eff})^{-1}$, enabling a concise description of how geometry, injection port, and frequency control photonic transport in 2D WQED. A complementary transfer-matrix method is also formulated and shown to agree with the Green function results for simple geometries, while the GF approach scales to large, complex networks. The work demonstrates that 2D WQED hosts tunable, geometry-driven quantum lateral shifts that can be engineered via injection conditions, offering a versatile platform for high-dimensional quantum photonics and potential applications in quantum sensing and information processing.

Abstract

Quantum emitters coupled to traveling photons in waveguides, known as waveguide quantum electrodynamics (WQED), offer a powerful platform for understanding light-matter interactions and underpinning emergent quantum technologies. While WQED has been extensively studied in one dimension, two-dimensional (2D) WQED remains largely unexplored, where novel photonic scattering phenomena unique to higher dimensions are expected. Here, we present a comprehensive scattering theory for 2D WQED based on the Green function method. We show that the mean displacement between emitted and injected photons serves as a quantum analogue of the Goos-Hänchen shift. When a photon is injected into a single off-centered port, the quantum Goos-Hänchen (QGH) shift can be enhanced in backward scattering under resonant conditions with subradiant states. When a photon is injected into the center port, there is no QGH shift due to the mirror symmetry of structure. However, for multiple-port injection with transverse momentum, the QGH shift is recovered and proportional to the derivative of phase with respect to transverse momentum. Unlike the classical Goos-Hänchen shift, these effects can be flexibly tuned by the injected photon's frequency. Our work provides a general framework for exploring and manipulating photonic scattering in complex WQED networks.

Figures (9)

• Figure 1: Schematics of a photon scattered by 2D atomic arrays coupled to waveguides. Two-level atoms with resonant frequency $\omega_0$ are placed in the nodes of crossing waveguides, and the waveguides are equally spaced. Quantum Goos-Hänchen shift is manifested by the mean position shifts ($\Delta P_{x}$,$\Delta P_{\bar{x}}$) in forward and backward scatterings, respectively.
• Figure 2: (a) The ratio of vertical and horizontal scatterings versus $\Gamma_y/\Gamma_x$. The frequency of injected photon is $\omega_0-0.078\Gamma_x$, the injection port is the first horizontal waveguide, and the other parameters are chosen as $d/c=\pi/(3\omega_0)$, $g_x=1$, $c=100$, and $N_x=100$. Red 'x’, blue '*', magenta solid dots, green '+' correspond to the numbers of waveguides along $y$ direction $N_y=3,4,5,6$, respectively. Black solid line denotes $(S_y+S_{\bar{y}})/(S_x+S_{\bar{y}}) \propto \Gamma_y/\Gamma_x$. (b) The forward (red) and backward (blue) scatterings along $x$-direction versus $N_x$ for $N_y=5$ and different injection ports. 'x’, star and cycle points denote the $y_0$, $y_1$ and $y_2$ ports, respectively.
• Figure 3: Total probability and quantum Goos-Hänchen shift in forward (a, c) and backward (b, d) scatterings for different injection frequencies and ports. The red solid, magenta solid, blue solid, green dashed, black dashed lines denote injection ports $y_2$, $y_1$, $y_0$, $y_{-1}$ and $y_{-2}$, respectively. The dashed-dot vertical dark lines indicate the frequencies of subradiant states. The parameters are chosen as $N_x=15$, $N_y=5$, $g_x=g_y=1$, $\omega_0 d/c=1$, and $\Gamma_x=\Gamma_y=0.01$.
• Figure 4: (a) The quantum Goos–Hänchen shift of reflected photon as a function of injection frequency and transverse momentum of Gaussian wavepacket. (b) The quantum Goos–Hänchen shift as a function of injection frequency with transverse momentum $k_y=0.1\pi$. All calculations are performed with $g_x=g_y=1$, $\omega_0d/c=1$, $N_x=15$, and $N_y=25$.
• Figure S1: (a) The relation between logarithm of the decay rate and inverse energy. Here, the dots represents eigenvalues obtained from exact diagonalization, and the black lines are given by Eq. \ref{['s28']}. Blue, yellow, green dots correspond to $\varphi=\pi/2$, $\pi/4$, $\pi/6$, respectively. (b) inverse participation ratio of eigenstates in different value of $\varphi$. (c) Symmetric scale-free localized states with the largest IPR (blue dots) with $\varphi=\pi/2$, and the fitting function (blue line) $|\psi_{loc}(j)|^2 \sim (e^{-\alpha F/N}+e^{\alpha F/N})^2$. The other parameters are chosen as $g_x=1$, $c=100$, and $N=600$.