Symplectic geometry and circuit quantization

TL;DR

This paper introduces a symplectic geometry-based Hamiltonian approach to circuit quantization, enabling the quantization of complex circuits with nonlinear elements that traditional methods cannot handle.

Contribution

It presents a novel Hamiltonian formalism inspired by symplectic geometry that generalizes circuit quantization to nonlinear and driven circuits.

Findings

Provides a Hamiltonian formulation applicable to nonlinear circuit elements

Develops an efficient algorithm for circuit quantization

Enables quantization of circuits previously not quantizable with standard methods

Abstract

Circuit quantization is an extraordinarily successful theory that describes the behavior of quantum circuits with high precision. The most widely used approach of circuit quantization relies on introducing a classical Lagrangian whose degrees of freedom are either magnetic fluxes or electric charges in the circuit. By combining nonlinear circuit elements (such as Josephson junctions or quantum phase slips), it is possible to build circuits where a standard Lagrangian description (and thus the standard quantization method) does not exist. Inspired by the mathematics of symplectic geometry and graph theory, we address this challenge, and present a Hamiltonian formulation of non-dissipative electrodynamic circuits. The resulting procedure for circuit quantization is independent of whether circuit elements are linear or nonlinear, or if the circuit is driven by external biases. We explain how to re-derive known results from our formalism, and provide an efficient algorithm for quantizing circuits, including those that cannot be quantized using existing methods.