Dynamical Regimes of Finite-Length Transmission Lines in Circuit Quantum Electrodynamics

We study the emergence of continuum, discrete-multimode, and single-mode regimes in finite-length transmission lines capacitively coupled to transmon qubits. We show that the appropriate description is selected by the hierarchy among the qubit frequency ${\omega }_q$, the characteristic transmission line frequency ${\omega }_{\mathrm{T}\mathrm{L}}$, and the characteristic coupling frequency ${\omega }_g$. In the long-line continuum limit, the transmission line acts as a structured reservoir described by a Drude–Lorentz spectral density; in the short-line limit, it reduces to an effective single-mode resonator; and, between these limits, it behaves as a discrete multimode coupler. This provides a unified cQED picture of the dynamical regimes of finite-length transmission lines in superconducting-circuit architectures.