Geometrical description and Faddeev-Jackiw quantization of electrical networks

Abstract

In lumped-element electrical circuit theory, the problem of solving Maxwell's equations in the presence of media is reduced to two sets of equations, the constitutive equations encapsulating local geometry and dynamics of a confined energy density, and the Kirchhoff equations enforcing conservation of charge and energy in a larger, topological, scale. We develop a new geometric and systematic description of the dynamics of general lumped-element electrical circuits as first order differential equations, derivable from a Lagrangian and a Rayleigh dissipation function. Through the Faddeev-Jackiw method we identify and classify the singularities that arise in the search for Hamiltonian descriptions of general networks. The core of our solution relies on the correct identification of the reduced manifold in which the circuit state is expressible, e.g., a mix of flux and charge degrees of freedom, including the presence of compact ones. We apply our fully programmable method to obtain (canonically quantizable) Hamiltonian descriptions of nonlinear and nonreciprocal circuits which would be cumbersome/singular if pure node-flux or loop-charge variables were used as a starting configuration space. We also propose a specific assignment of topology for the branch variables of energetic elements, that when used as input to the procedure gives results consistent with classical descriptions as well as with spectra of more involved quantum circuits. This work unifies diverse existent geometrical pictures of electrical network theory, and will prove useful, for instance, to automatize the computation of exact Hamiltonian descriptions of superconducting quantum chips.

Accepted version. 37 pages and 16 figures. Comments are welcome

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Cited by in corpus (10)

• A Review of Design Concerns in Superconducting Quantum Circuits
• Lecture Notes on Quantum Electrical Circuits
• Exact quantization of nonreciprocal quasi-lumped electrical networks
• Stochastic theory of nonlinear electrical circuits in thermal equilibrium
• Viewing protected superconducting qubits through the lens of the cat qubit
• On "Consistent Quantization of Nearly Singular Superconducting Circuits"
• Improving the accuracy of circuit quantization using the electromagnetic properties of superconductors
• Systematic Construction of Time-Dependent Hamiltonians for Microwave-Driven Josephson Circuits
• Utilizing discrete variable representations for decoherence-accurate numerical simulation of superconducting circuits
• Microwave Circulation in an Extended Josephson Junction Ring