During the last 30 years, stimulated by the quest to build superconducting quantum processors, a theory of quantum electrical circuits has emerged, which is called 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 comprehensive, theoretically oriented, overview of this subject for Master or PhD students in physics and electrical engineering.
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- Title
- Lecture Notes on Quantum Electrical Circuits
- Publisher
- TU Delft OPEN Publishing
- Author(s)
- Alessandro Ciani, Barbara M. Terhal, David P. DiVincenzo
- Published
- 2024-02-13
- Edition
- 1
- Format
- eBook (pdf, epub, mobi)
- Pages
- 172
- Language
- English
- ISBN-13
- 9789463668156
- License
- CC BY
- Book Homepage
- Free eBook, Errata, Code, Solutions, etc.
Introduction Overview Acknowledgements Lagrangians and Hamiltonians for electrical circuits Branch voltages and branch fluxes Inductive and capacitive branches The LC oscillator and its canonical quantization The Josephson junction and the Cooper pair box Mutual inductances and the ideal transformer Transformer Conservation laws in electrical circuit graphs Current and voltage sources Voltage sources Current sources External fluxes Time-dependent external fluxes External fluxes in superconductors and fluxoid quantization Applying canonical quantization Invertibility of the capacitance matrix Non-locality of capacitive interactions in the Hamiltonian Examples A pathological case? Two coupled flux qubits Flux qubit: replacing the inductor by two Josephson junctions The fluxonium qubit Circuits for two Cooper pair tunneling: TEXT qubit The Möbius-strip circuit Symmetries and forbidden transitions Symmetries: discussion of a tetrahedral qubit The transmon qubit, resonators and their coupling The CPB Hamiltonian and its spectrum From flux to phase CPB spectrum The anharmonic approximation Driving a transmon qubit Transmission lines and co-planar resonators Boundaries and resonators Input-output formalism: Heisenberg-Langevin equations Capacitively coupling a transmon to a resonator Perturbative analysis tools Linear networks and black-box quantization LTI networks Laplace and Fourier domain Single port Scattering matrix Additional properties of LTI networks Black-box quantization The energy-participation ratio approach to black-box quantization Single-port black-box quantization Lossy networks Single-port case Using the QuCAT software package Networks of transmon qubits in the dispersive regime Nonreciprocity Ports, terminals, and an important multi-terminal device The circulator and the gyrator Nonreciprocity in action Formalism of the circulator and the gyrator Admittance matrix of the gyrator The Lagrangian and Hamiltonian of the gyrator ``Magnetic field does no work" – gyrator as magnetic field Hamiltonian of the gyrator Noise, or all that can go wrong Lindblad master equation model A qubit circuit with a lossy circuit branch Noise sensitivity and protection Flux sweet spots Built-in protection Double protection? Models of superconducting amplifiers Exercises on amplifiers A review of canonical quantization Principle of minimal action and gauge invariance Legendre transformation: the Hamiltonian Poisson bracket and quantization Harmonic systems and beyond Normal modes of a harmonic system Diagonalization Eliminating high-frequency modes Illustration of the Born-Oppenheimer method Normal modes and Cauer's construction LC circuit shunted by a small admittance
