Theory of the Collective Many-body Subradiance in Waveguide QED

Abstract

We present an analytical theory for the most subradiant modes in a finite one-dimensional emitter array coupled to either an ideal or a nonideal waveguide. Using an effective non-Hermitian Hamiltonian together with a Bragg-edge open-boundary ansatz, we derive compact eigenvalue expressions showing that the linewidths of the most subradiant states exhibit a universal N^{-3} scaling in both cases. However, in the deep-subwavelength regime, the decay rates display even-odd oscillations due to boundary interference. Furthermore, we demonstrate that the collective energy shift of the most subradiant state approaches a constant value that depends on the atomic separation, with the leading finite-size correction scaling as N^{-2}. These results unify the roles of Bragg-edge interference, finite-size effects, and near-field dipole-dipole interactions in shaping ultranarrow, strongly shifted subradiant resonances, providing a transparent framework beyond the ideal-waveguide limit and opening potential applications in subradiant spectroscopy and waveguide-QED-based sensing.

Figures (5)

• Figure 1: Schematic of an array of $N$ identical two-level atoms with lattice spacing $d$ coupled to a single-mode one-dimensional waveguide. Each atom decays into the guided mode at rate $\Gamma$ and into unguided free-space modes at rate $\gamma$.
• Figure 2: Benchmark of the ideal-waveguide asymptotic formulas against exact numerical diagonalization. (a),(b) Linewidths $\Gamma_\xi/\Gamma$ versus atom number $N$ for $\xi=1$ and $\xi=3$, respectively, for three lattice spacings: $d=0.02\lambda$ (solid blue lines and open circles), $d=0.1\lambda$ (dashed red lines and open triangles), and $d=0.25\lambda$ (dash-dotted green lines and open squares). (c) Linewidths $\Gamma_\xi/\Gamma$ versus $k_0 d$ at fixed $N=100$ for $\xi=1$ (solid blue line and open circles) and $\xi=3$ (dashed red line and open triangles). (d),(e) collective energy shifts $J_\xi/\Gamma$ versus $N$ for $\xi=1$ and $\xi=3$, respectively, for the same three spacings. (f) collective energy shifts $J_\xi/\Gamma$ versus $k_0 d$ at fixed $N=100$ for $\xi=1$ and $\xi=3$. In all panels, lines denote the analytical results, while symbols denote the numerical eigenvalues of the effective Hamiltonian. The linewidth data in (a)--(c) are compared with Eq. \ref{['eq:Gamma_zhang_main']}, and the collective-shift data in (d)--(f) are compared with Eq. \ref{['eq:J_ideal_main_equiv']}.
• Figure 3: Comparison between the numerical evaluation and the analytic formulas for the free-space decay rate. (a)–(c): Dimensionless prefactor $F^{(\mathrm{fs})}_{N+1}(\beta;\xi)$ as a function of $N$ for $d=0.01\lambda$, $0.02\lambda$, and $0.03\lambda$ with $\xi=1$, obtained from the discrete sum [Eq. \ref{['eq:F-def']}] (symbols) and the analytic expression [Eq. \ref{['eq:FN-fs-analytic']}] (solid lines). (d)–(f): Corresponding free-space linewidth $\Gamma^{(\mathrm{fs})}_\xi(N,d)$ computed from Eq. \ref{['eq:Gamma-fs-analytic']} (solid lines) and Eq. \ref{['eq:Gamma-fs-env']} (symbols). In all panels, $\gamma=0.1\Gamma$.
• Figure 4: Analytical decomposition of the most subradiant decay rates in the nonideal-waveguide case with $d=0.02\lambda$ and $\gamma=0.1\Gamma$. (a) $\Gamma_\xi/\Gamma$ versus $N$ for $\xi=1$. (b) $\Gamma_\xi/\Gamma$ versus $N$ for $\xi=3$. In each panel, the dash-dotted blue line denotes the guided-mode contribution $\Gamma_\xi^{(\mathrm{1D})}/\Gamma$ from Eq. \ref{['eq:Gamma-1D-final']}, the dashed orange line denotes the free-space contribution $\Gamma_\xi^{(\mathrm{fs})}/\Gamma$ from Eq. \ref{['eq:Gamma-fs-analytic']}, and the solid green line denotes the total analytical linewidth $\Gamma_\xi/\Gamma$ from Eq. \ref{['eq:Gamma-final']}. The red open circles are the exact numerical results. The figure shows that both the guided and nonguided channels inherit the same overall $(N+1)^{-3}$ subradiant envelope, while their different oscillatory structures combine to produce the full finite-size linewidth pattern.
• Figure 5: Comparison between the numerical eigenvalues of the effective Hamiltonian and the analytical asymptotic formulas for the collective energy shift in the nonideal-waveguide case with $\gamma=0.1\Gamma$. (a) collective energy shifts $J_\xi/\Gamma$ versus $N$ at fixed $d=0.02\lambda$ for $\xi=1$ (solid blue line and open circles) and $\xi=3$ (dashed red line and open squares). (b) $J/\Gamma$ versus $k_0d$ at fixed $N=100$, showing the thermodynamic-limit shift $J_\infty$ from Eq. \ref{['eq:Jinf-main']} (thick pink line), together with the finite-$N$ results for $\xi=1$ (solid blue line and open circles) and $\xi=3$ (dashed red line and open squares). (c) Deviation $(J_\xi-J_\infty)/\Gamma$ versus $N$ at fixed $d=0.02\lambda$ on a log--log scale for $\xi=1$ and $\xi=3$; the gray guide indicates the expected $(N+1)^{-2}$ scaling from Eq. \ref{['eq:Jxi-Jinf']}. In all panels, lines denote analytical predictions and symbols denote numerical eigenvalues.