Geometric control of maximal entanglement via bound states in the continuum

Abstract

Bound states in the continuum (BiCs) convert dissipative open systems into effectively closed quantum subspaces through destructive interference. We show that two identical giant atoms coupled to a one-dimensional waveguide support BICs that coincide with maximally entangled atomic states. Most importantly, entanglement is predominantly determined by the geometric design; the ratio of intra-atomic connection lengths fixes the concurrence, while the propagation phase between atoms selects a family of Bell-like states. We further analyze the dynamical stability of these maximally entangled BICs under exact time evolution, revealing a clear hierarchy of robustness against parameter perturbations. Our results establish an analytical bridge between symmetry, geometry, entanglement, and BICs in giant-atom waveguide platforms.

Figures (3)

• Figure 1: Schematic of two giant atoms coupled to a one-dimensional coupled-resonator waveguide through two connection sites in a braided configuration. Blue spheres represent the waveguide resonators, and orange spheres denote the giant atoms.
• Figure 2: (Left) Concurrence $\mathcal{C}(\lambda)$ as a function of the geometric ratio $\lambda = n_1/n_2$ (semi-log scale), characterizing the degree of bipartite entanglement between the two giant atoms. The concurrence is maximal at $\lambda=1$ ($n_1=n_2$). Insets illustrate representative connection geometries across different entanglement regimes. (Right) Polar representation $(x,y)=\lambda(\cos\theta,\sin\theta)$ of the fidelity $\mathcal{F}$ between the atomic BIC state and maximally entangled states $\ket{\Phi}=(\ket{e,g}+ e^{i\varphi}\ket{g,e})/\sqrt{2}$, as a function of $\lambda$ and the relative phase $\theta=k^{\star}\Delta x$. Colors encode $\mathcal{F}$ and solid black curves indicate selected iso-fidelity contours. High-fidelity regions around $\lambda=1$ correspond to maximally entangled BICs.
• Figure 3: (Left) Time evolution of the concurrence $C(t)$ for initial atomic states aligned with BICs in the symmetric (top) and antisymmetric (bottom) Bell states, $\ket{\Psi^{\pm}}$. The ideal geometric configuration $\lambda = 1$ and $k^{\star} = \pi/2$ (solid blue) yields long-lived entanglement, while evolution with $10\%$ detunings in the parameters $\lambda$, $\theta = k^{\star} \Delta x$, and $k^{\star}$ shows deviations from the ideal case. Dashed curves correspond to the Markovian approximation $C_{\pm}(t)=C_{\pm}(0)e^{-\Gamma^{\pm}(\Omega)t}$, where decay rates $\Gamma^{\pm}(\Omega)$ are given in Eq. \ref{['Rate']}. (Right) Polar heat maps of the normalized decay rates $\Gamma^{\pm}/\xi$ as a function of the geometric coordinates $(x,y)=\lambda(\cos\theta,\sin\theta)$. For even $n_2$, the vanishing of the giant-atom form factor at $k^{\star}$ suppresses angular interference, yielding purely concentric decay rings, whereas for odd $n_2$ residual interference produces strongly anisotropic decay landscapes. Simulation parameters: $\xi=1$, $g=0.1\xi$, $\Omega=\omega_c$. The waveguide is discretized with a large number of modes ($N_c = 4 \times 501$) to symmetrically resolve the resonant momenta $k = \pm k^\star$ ($k^{\star} = \pi/2$), ensuring convergence of the exact diagonalization.