Lecture Notes on Quantum Electrical Circuits

TL;DR

This paper provides a pedagogical overview of the development and theoretical foundations of quantum electrical circuits, especially circuit-QED, for students and researchers in physics and electrical engineering.

Contribution

It offers a comprehensive, accessible introduction to the theory of quantum electrical circuits, including Hamiltonian and Lagrangian formulations, for both linear and nonreciprocal circuits.

Findings

Derivation of Hamiltonian and Lagrangian for quantum circuits

Application of classical network concepts to quantum circuits

Analysis methods for lossless and nonreciprocal circuits

Abstract

During the last 30 years, stimulated by the quest to build superconducting quantum processors, a theory of quantum electrical circuits has emerged and this theory goes under the name of circuit quantum electrodynamics or circuit-QED. The goal of the theory is to provide a quantum description of the most relevant degrees of freedom. The central objects to be derived and studied are the Lagrangian and the Hamiltonian governing these degrees of freedom. Central concepts in classical network theory such as impedance and scattering matrices can be used to obtain the Hamiltonian and Lagrangian description for the lossless (linear) part of the circuits. Methods of analysis, both classical and quantum, can also be developed for nonreciprocal circuits. These lecture notes aim at giving a pedagogical overview of this subject for theoretically-oriented Master or PhD students in physics and electrical engineering, as well as Master and PhD students who work on experimental superconducting quantum devices and wish to learn more theory.