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.
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.
Following a consistent geometrical description previously introduced [], we present an exact method for obtaining canonically quantizable Hamiltonian descriptions of nonlinear, nonreciprocal quasilumped electrical networks. We identify and classify singularities arising in the quest for Hamiltonian descriptions of general quasilumped element networks via the Faddeev-Jackiw technique. We offer systematic solutions to cases previously considered singular—a major challenge in the context of canonical circuit quantization. The solution relies on the correct identification of the reduced classical circuit-state manifold, i.e., a mix of flux and charge fields and functions. Starting from the geometrical description of the transmission line, we provide a complete program including lines coupled to one-port lumped-element networks, as well as multiple lines connected to multiport nonreciprocal lumped-element networks, with intrinsic ultraviolet cutoff. On the way, we naturally extend the canonical quantization of transmission lines coupled through frequency-dependent, nonreciprocal linear systems, such as practical circulators. Additionally, we demonstrate how our method seamlessly facilitates the characterization of general nonreciprocal, dissipative linear environments. This is achieved by extending the Caldeira-Leggett formalism, using continuous limits of series of immittance matrices. We provide a tool in the analysis and design of electrical circuits and of special interest in the context of canonical quantization of superconducting networks. For instance, this work provides a solid ground for a precise nondivergent input-output theory in the presence of nonreciprocal devices, e.g., within (chiral) waveguide QED platforms.
We present a systematic canonical quantization procedure for lumped-element superconducting networks by using a redundant configuration-space description. The algorithm is based on an original, explicit, and constructive implementation of the symplectic diagonalization of positive semidefinite Hamiltonian matrices, a particular instance of Williamson's theorem. With it, we derive canonically quantized discrete-variable descriptions of passive causal systems. We exemplify the algorithm with representative singular electrical networks, a nonreciprocal extension for the black-box quantization method, as well as an archetypal Landau quantization problem.
In the quest to produce quantum technology, superconducting networks, working at temperatures just above absolute zero, have arisen as one of the most promising physical implementations. The precise analysis and synthesis of such circuits have required merging the fields of physics, engineering, and mathematics. In this dissertation, we develop mathematically consistent and precise Hamiltonian models to describe ideal superconducting networks made of an arbitrary number of lumped elements, such as capacitors, inductors, Josephson and phase-slip junctions, gyrators, etc., and distributed ones like transmission lines. We give formal proofs for the decoupling at high and low frequencies of lumped degrees of freedom from infinite-dimensional systems in different coupling configurations in models based on the effective Kirchhoff's laws. We extend the standard theory to quantize circuits that include ideal nonreciprocal elements all the way to their Hamiltonian descriptions in a systematic way. Finally, we pave the way on how to quantize general frequency-dependent gyrators and circulators coupled to both transmission lines and other lumped-element networks. We have explicitly shown, that these models, albeit ideal, are finite and present no divergence issues. We explain and dispel misunderstandings from the previous literature. Furthermore, we have demonstrated the usefulness of a redundant basis for performing separation of variables of the transmission line (1D) fields in the presence of point-like (lumped-element) couplings by time-reversal symmetry-breaking terms, i.e. nonreciprocal elements.
In this work, we introduce new methods for the quantization, decomposition, and extraction (from electromagnetic simulations) of lumped-element circuit models for superconducting quantum devices. Our flux-charge symmetric procedures center on the network matrix, which encodes the connectivity of a circuit's inductive loops and capacitive nodes. First, we use the network matrix to demonstrate a simple algorithm for circuit quantization, giving novel predictions for the Hamiltonians of circuits with both Josephson junctions and quantum phase slip wires. We then show that by performing pivoting operations on the network matrix, we can decompose a superconducting circuit model into its simplest equivalent "fundamental" form, in which the harmonic degrees of freedom are separated out from the Josephson junctions and phase slip wires. Finally, we illustrate how to extract an exact, transformerless circuit model from electromagnetic simulations of a device's hybrid admittance/impedance response matrix, by matching the lumped circuit's network matrix to the network topology of the physical layout. Overall, we provide a toolkit of intuitive methods that can be used to construct, analyze, and manipulate superconducting circuit models.
We develop a systematic procedure to quantize canonically Hamiltonians of light-matter models of transmission lines coupled through lumped linear lossless ideal nonreciprocal elements, that break time-reversal symmetry, in a circuit QED set-up. This is achieved through a description of the distributed subsystems in terms of both flux and charge fields. We prove that this apparent redundancy is required for the general derivation of the Hamiltonian for a wider class of networks. By making use of the electromagnetic duality symmetry in 1+1 dimensions, we provide unambiguous identification of the physical degrees of freedom, separating out the nondynamical parts. This doubled description can also treat the case of other extended lumped interactions in a regular manner that presents no spurious divergences, as we show explicitly in the example of a circulator connected to a Josephson junction through a transmission line. This theory enhances the quantum engineering toolbox to design complex networks with nonreciprocal elements.
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 nondissipative 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 rederive known results from our formalism, and provide an efficient algorithm for quantizing circuits, including those that cannot be quantized using existing methods. Published by the American Physical Society 2024