Geometrical description and Faddeev-Jackiw quantization of electrical networks

A. Parra-Rodriguez; I. L. Egusquiza

doi:10.22331/q-2024-09-09-1466

Geometrical description and Faddeev-Jackiw quantization of electrical networks

A. Parra-Rodriguez^1,2 and I. L. Egusquiza^3,4

¹Institut Quantique and Département de Physique, Université de Sherbrooke, Sherbrooke, Québec J1K 2R1, Canada
²Instituto de Física Fundamental IFF-CSIC, Calle Serrano 113b, 28006 Madrid, Spain
³Department of Physics, University of the Basque Country UPV/EHU, Apartado 644, 48080 Bilbao, Spain
⁴EHU Quantum Centre, University of the Basque Country UPV/EHU, Apartado 644, 48080 Bilbao, Spain

Published:	2024-09-09, volume 8, page 1466
Eprint:	arXiv:2304.12252v5
Doi:	https://doi.org/10.22331/q-2024-09-09-1466
Citation:	Quantum 8, 1466 (2024).

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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.

Featured image: Sketch of the geometrical reduction by the Faddeev-Jackiw method for electrical circuits. The initial manifold $\mathcal{M}_{2B}$ of branch charge and flux variables (one of each per lumped-element port) is first reduced to a submanifold $\mathcal{M}$ by solving the geometrical linear equations of the constraints encoded in matrix $\mathsf{F}$ (including Kirchhoff’s laws, and ideal transformers and nonreciprocal elements). In a second step, the Faddeev-Jackiw algorithm may be required to remove the zero-modes of the systematically constructed Lagrangian, to obtain classical Hamiltonian dynamics in a well-defined symplectic manifold. Finally, to perform canonical quantization, we further require that the symplectic manifold be diffeomorphic to a cotangent bundle.

Popular summary

Classical circuit theory often struggles with various types of singularities, primarily due to either parasitic limits of elements or nonreciprocal devices (effective elements vital for modeling nontrivial routing of quantum information), which can hinder accurate simulations and development. Our research introduces a new, systematic method to enhance classical circuit theory, making it particularly suitable for the nuanced demands of superconducting quantum technology. In this work, we have developed a geometric framework that extends the conventional use of Kirchhoff's laws and constitutive equations to provide a more comprehensive understanding of electric circuits, targeting the aforementioned issues.

Following the standard geometrical approach to analytical mechanics, we systematically construct first-order Lagrangians and Rayleigh dissipation functions. For dissipationless circuits, we apply the Faddeev-Jackiw reduction method to identify, classify, and, when possible, resolve the singularities commonly found in electric circuits, enabling us to derive canonically quantizable Hamiltonian dynamics. This procedure treats flux and charge variables on equal footing, such that the inclusion of nonreciprocal elements (flux-charge mixing constraints) is naturally implemented. In doing so, we have developed a programmable algorithm that overcomes the limitations of traditional node-flux or loop-charge configurations. Additionally, we have bolstered our method by proposing a specific topological assignment to the branch-variable manifolds, ensuring compatibility with both compact- and extended-variable perspectives, consistent with classical and quantum observations.

Looking ahead, this work opens the door to (i) solving the compact vs. extended-variable debate, (ii) extending the framework to quasi-lumped element circuits, and (iii) achieving a fully general quantum mechanical description of nonreciprocal and nonlinear dissipation. These developments will enhance the simulation and design processes for classical and quantum circuits, making them more reliable and scalable, while paving the way for significant advancements in practical applications such as quantum processors.

► BibTeX data

@article{ParraRodriguez2024geometrical,
  doi = {10.22331/q-2024-09-09-1466},
  url = {https://doi.org/10.22331/q-2024-09-09-1466},
  title = {Geometrical description and {F}addeev-{J}ackiw quantization of electrical networks},
  author = {Parra-Rodriguez, A. and Egusquiza, I. L.},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {8},
  pages = {1466},
  month = sep,
  year = {2024}
}

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