In this article, we introduce an algorithmic method to find the conservative energy and non-conservative power of a large class of maximally nonlinear electric circuits (including Josephson tunnel junctions, coherent quantum phase slips, and superconducting loops), based on the incidence matrix of the circuits' digraph. We consider two-port linear circuits with mostly holonomic constraints provided by either Maxwell-Kirchhoff's current rules or Maxwell-Kirchhoff's voltage rules. The circuit's independent variables, generally a superset of the degrees of freedom, are obtained from the solution space of Maxwell-Kirchhoff's current or voltage rules. The method does not require to find any Lagrangian. Instead, the circuit's classical or quantum Hamiltonian is obtained from the energy of the reactive (i.e., conservative) circuit elements by means of transformations complementary to Hamilton's equations. Dissipation (loss) is accounted for by using the Rayleigh dissipation function and defining generalized Poisson brackets--Poisson-Rayleigh brackets. Fluctuations (noise) are added as voltage or current sources characterized by bath modes. Non-conservative elements (e.g., noisy resistors) are included ab initio using the incidence-matrix method, without needing to treat them as separate elements. Finally, we show that in order to form a complete set of canonical coordinates, auxiliary (which could be parasitic in certain cases) circuit elements are required to find the Hamiltonian of circuits with an incomplete set of generalized velocities. In particular, we introduce two methods to eliminate the coordinates associated with the auxiliary elements by either Hamiltonian reduction or equation-of-motion reduction. We use auxiliary circuit elements to treat a maximally nonlinear circuit comprising simultaneously both a Josephson junction and a quantum phase slip.
In this article, we introduce an algorithmic method to find the conservative energy and non-conservative power of a large class of maximally nonlinear electric circuits (including Josephson tunnel junctions, coherent quantum phase slips, and superconducting loops), based on the incidence matrix of the circuits' digraph. We consider two-port linear circuits with mostly holonomic constraints provided by either Maxwell-Kirchhoff's current rules or Maxwell-Kirchhoff's voltage rules. The circuit's independent variables, generally a superset of the degrees of freedom, are obtained from the solution space of Maxwell-Kirchhoff's current or voltage rules. The method does not require to find any Lagrangian. Instead, the circuit's classical or quantum Hamiltonian is obtained from the energy of the reactive (i.e., conservative) circuit elements by means of transformations complementary to Hamilton's equations. Dissipation (loss) is accounted for by using the Rayleigh dissipation function and defining generalized Poisson brackets--Poisson-Rayleigh brackets. Fluctuations (noise) are added as voltage or current sources characterized by bath modes. Non-conservative elements (e.g., noisy resistors) are included ab initio using the incidence-matrix method, without needing to treat them as separate elements. Finally, we show that in order to form a complete set of canonical coordinates, auxiliary (which could be parasitic in certain cases) circuit elements are required to find the Hamiltonian of circuits with an incomplete set of generalized velocities. In particular, we introduce two methods to eliminate the coordinates associated with the auxiliary elements by either Hamiltonian reduction or equation-of-motion reduction. We use auxiliary circuit elements to treat a maximally nonlinear circuit comprising simultaneously both a Josephson junction and a quantum phase slip.
We investigate quantum synchronization phenomena in electrical circuits that incorporate specifically designed nonconservative elements. A dissipative theory of classical and quantized electrical circuits is developed based on the Rayleigh dissipation function. The introduction of this framework enables the formulation of a generalized version of classical Poisson brackets, which are termed Poisson-Rayleigh brackets. By using these brackets, we are able to derive the equations of motion for a given circuit. Remarkably, these equations are found to correspond to Kirchhoff's current laws when Kirchhoff's voltage laws are employed to impose topological constraints, and vice versa. In the quantum setting, the equations of motion are referred to as the Kirchhoff-Heisenberg equations, as they represent Kirchhoff's laws within the Heisenberg picture. These Kirchhoff-Heisenberg equations, serving as the native equations for an electrical circuit, can be used in place of the more abstract master equations in Lindblad form. To validate our theoretical framework, we examine three distinct circuits. The first circuit consists of two resonators coupled via a nonconservative element. The second circuit extends the first to incorporate weakly nonlinear resonators, such as transmons. Lastly, we investigate a circuit involving two resonators connected through an inductor in series with a resistor. This last circuit, which incidentally represents a realistic implementation, allows for the study of a singular system, where the absence of a coordinate leads to an ill-defined system of Hamilton's equations. To analyze such a pathological circuit, we introduce the concept of auxiliary circuit element. After resolving the singularity, we demonstrate that this element can be effectively eliminated at the conclusion of the analysis, recuperating the original circuit.
Abstract Quantum circuit theory is a powerful tool to describe superconducting circuits. In its language, quantum phase slips (QPSs) are considered to be the exact dual to the Josephson effect. This duality renders the integration of QPS junctions into a unified theoretical framework challenging. As we argue, different existing formalisms may be inconsistent, and the correct inclusion of time-dependent flux driving requires introducing a large number of auxiliary, nonphysical degrees of freedom. We resolve these issues by describing QPS junctions as inductive rather than capacitive elements, and reducing the Hilbert space to account for a compact superconducting phase. Our treatment provides an approach to circuit quantization exclusively in terms of node-flux-node variables, and eliminates spurious degrees of freedom. Finally, the inductive treatment reveals the possibility of a voltage-dependent renormalization of the QPS amplitude, by accounting for spatial variations of the electric field built up across the junction.
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.
Assuming only the solubility of Kirchhoff's laws, the conservation of energy, and the form of the constitutive relations, we present a flux-charge symmetric Hamiltonian formalism that always admits a symplectic reduction for nonsingular, well-posed circuits involving gyrators, inductors, and capacitors (LCG circuits) with arbitrary non-linearity. The formalism introduces a notion of circuit duality on non-linear non-reciprocal circuits and a family of point transformations that inserts, removes, and moves reciprocal elements across gyrators while preserving the Hamiltonian spectrum. With these transformations, we extend Tellegen's classic result that gyrators make reciprocal elements behave as their dual to non-linear reciprocal elements such as the Josephson junction and quantum phase slip. Furthermore, we present conditions under which non-reciprocal circuits can be transformed to dynamically equivalent reciprocal circuits.
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.
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