Dynamical Regimes of Two Qubits Coupled through a Transmission Line

TL;DR

This paper analyzes how two superconducting qubits coupled via a finite transmission line exhibit different dynamical regimes, including Markovian and non-Markovian behaviors, depending on system parameters and line length.

Contribution

It provides a unified theoretical framework describing the transition between multimode, single-mode, and structured reservoir regimes in superconducting circuit QED.

Findings

The transmission line acts as a structured reservoir or a discrete mode depending on parameters.

Non-Markovian dynamics are identified in specific parameter regions.

A unified model describes different dynamical regimes in superconducting qubit systems.

Abstract

We investigate the reduced dynamics of two identical superconducting qubits capacitively coupled through a finite-length transmission line. Starting from circuit quantization, we derive a circuit Hamiltonian that naturally separates the line modes into even- and odd-parity sectors coupled to collective qubit operators. Depending on the hierarchy between the qubit frequency $\omega_q$, the mode spacing $\omega_{TL}$, and the coupling scale $\omega_g$, the line acts either as a structured reservoir or as a discrete few-mode coupler. In the long-line continuum limit, each sector is described by a Drude--Lorentz spectral density and the dynamics is solved with the hierarchical equations of motion. Using the Breuer--Laine--Piilo measure, we identify the parameter region in which the reduced dynamics exhibits non-Markovian relaxation. In the short-line limit, the continuum description breaks down and the dynamics becomes respectively multimode or single-mode. This establishes a unified cQED picture of the dynamical regimes of finite-length transmission lines in superconducting-circuit architectures.