Fundamental, topological and modular models for circuit QED

Superconducting circuits have become a leading platform for quantum computation and simulation due to their scalability and the precise control provided by Josephson junctions (JJs), their fundamental nonlinear element. Despite significant progress, several fundamental questions about their theoretical modelling remain unresolved, including how to accurately describe their behaviour across different energy scales. In particular, key issues involve understanding the spectra of quantum macroscopic circuit variables (e.g. the flux difference across a JJ), and the relationship between classical and quantum dynamics in long-distance superconducting networks. Resolving these open questions is crucial for understanding quantum many-body phenomena and for the design of distributed chiral networks.The FTMcQED project will address these challenges by focusing on two main areas. First, I will tackle the long-standing debate over extended vs. compact variable descriptions of flux and charge variables in superconducting circuits and explore its implications for many-body quantum systems (e.g., dissipative quantum phase transitions). This involves developing a geometrically and topologically consistent quantisation method to derive canonical quantum Hamiltonians for superconducting circuits, while properly accounting for parasitic effects. Second, I will construct modular, effective models for nonreciprocal, dissipative superconducting networks (e.g., waveguide QED) using electrical engineering techniques.To achieve these objectives, I will employ a combination of advanced analytical and numerical methods, enabling a systematic investigation of collective topological effects in both discrete and continuous models. This project aims to contribute to the fundamental understanding and development of new quantum devices, including broadband quantum-limited amplifiers and detectors, as well as novel families of superconducting qubits with enhanced noise suppression.