Coherent Control of an Embedded Bound State without a Spectral Gap

Bound states in the continuum (BICs) can confine photonic excitations in open systems without conventional cavities or band gaps, making them promising candidates for long-lived quantum storage and single-photon control. Their use as quantum resources is limited, however, by two basic difficulties: they are dark to incident photons, and they are not protected by a spectral gap from the surrounding continuum. We address these difficulties in a giant atom coupled to a one-dimensional waveguide by separating radiative access from BIC-preserving manipulation. Atomic-frequency modulation dynamically breaks and restores the destructive-interference condition, allowing a mode-matched single photon to be captured into, and released from, the BIC. Symmetric coupling modulation, in contrast, preserves the BIC condition throughout the evolution and continuously changes the atomic and photonic weights of the stored state. This BIC-preserving deformation realizes a gapless adiabatic control problem: for a regular embedded continuum, the intrinsic leakage probability is proportional to the ramp rate rather than quadratically suppressed as in gapped systems. The linear scaling originates from near-resonant scattering states at the BIC energy and is independent of microscopic dispersion except through the prefactor. These results establish a coherent capture-control-release protocol for a populated BIC and identify the fundamental error law for manipulating quantum states that remain embedded in a gapless environment.