Photon-echo synchronization and quantum state transfer in short quantum links

The short quantum link regime, where the photon travel time $\tau$ is comparable to the emitter lifetime $1/\gamma$, is experimentally relevant but theoretically underexplored: existing few-mode descriptions lose validity as retardation and multimode effects become significant. Using a Delay Differential Equation (DDE) framework that admits exact analytical solutions from the single-mode cavity limit to the multimode waveguide continuum, we show that emitters coupled to a short link spontaneously lock into self-synchronized Rabi oscillations driven by coherent photon echoes, breaking the link's discrete time-displacement symmetry. The resulting spectral structure -- persistent quasi-dark states and vacuum Rabi splitting, including in the superstrong coupling regime -- enables efficient quantum state transfer (QST): benchmarking three protocols across the full $\gamma\tau$ parameter space, we find that STIRAP exploits the quasi-dark-state structure to achieve a quadratic infidelity floor $\mathcal{O}((\gamma\tau)^2)$, outperforming both SWAP (linear error $\mathcal{O}(\gamma\tau)$) and wavepacket engineering for $\gamma\tau \lesssim 1.44$, even in regimes where retardation cannot be neglected. These results establish photon-echo synchronization as an engineering resource for quantum state transfer, with DDE modeling providing the exact analytical predictions needed to design and optimize short-link experiments on current circuit-QED hardware.