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.
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.
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.
The theory of circuit quantum electrodynamics has successfully analyzed superconducting circuits on the basis of the classical Lagrangian, and the corresponding quantized Hamiltonian describing these circuits.In many simplified versions of these networks, the modeling involves a singular Lagrangian that employs Kirchhoff's laws to eliminate inherent constraints of the system.In this work, we demonstrate the failure of such singular theories for the quantization of realistic, nearly singular superconducting circuits.Instead, we rigorously prove the validity of a perturbative analysis within the Born-Oppenheimer approximation.In particular, we find that the limiting behavior of the low-energy dynamics obtained from the regularized approach exhibits a fixed-point structure flowing to one of a few universal fixed points as parasitic capacitance values go to zero.This singular limit of the regularized analysis is, in many cases, completely unlike the singular theory.Consequently, we conclude that classical network synthesis techniques which build on Kirchhoff's laws must be critically examined prior to applying circuit quantization.
The theory of circuit quantum electrodynamics has successfully analyzed superconducting circuits on the basis of the classical Lagrangian, and the corresponding quantized Hamiltonian, describing these circuits. In many simplified versions of these networks, the modeling involves a Lagrangian that is singular, describing an inherently constrained system. In this work, we demonstrate the failure of the Dirac-Bergmann theory for the quantization of realistic, nearly singular superconducting circuits, both reciprocal and nonreciprocal. The correct treatment of nearly singular systems involves a perturbative Born-Oppenheimer analysis. We rigorously prove the validity of the corresponding perturbation theory using Kato-Rellich theory. We find that the singular limit of this regularized analysis is, in many cases, completely unlike the singular theory. Dirac-Bergmann, which uses the Kirchhoff's (and Tellegen's) laws to deal with constraints, predicts dynamics that depend on the detailed parameters of nonlinear circuit elements, e.g., Josephson inductances. By contrast, the limiting behavior of the low-energy dynamics obtained from the regularized Born-Oppenheimer approach exhibits a fixed point structure, flowing to one of a few universal fixed points as parasitic capacitance values go to zero.
Abstract With the knowledge of the new design rules in Chapter 7, we use this new insight to find the eigenvectors for the nano-vibrator problem, and then we use the same approach to examine the quantum LC circuit. While the usual approach is to use Kirchhoff’s laws to analyze a simple circuit classically, we first see that Hamilton’s equations can in fact be used, giving the same classical result. But then, using the new design rules and the knowledge of the total energy in the circuit, we identify a canonical coordinate and a conjugate momentum that have nothing to do with real space and motion of a particle of mass m. At the same time, consistent with the Schrödinger picture, we continue to see that the time evolution of an observable such as position, x(t), or current, i(t), is not part of the solution. Given that Hamilton’s equations give the same result as Kirchhoff’s law but the quantum solution does not, reinforces the idea that the quantum description is showing features that cannot be imagined with a viewpoint based on classical (i.e. non-quantum) analysis.
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.
Synchronization is a collective phenomenon in which interacting nonlinear oscillators develop a persistent relation between their phases and frequencies. In quantum systems, its description requires connecting the classical dynamics of limit-cycle oscillators with quantum fluctuations, dissipation, and the notion of phase. In this tutorial review, we offer a systematic framework for analyzing synchronization in classical and quantum dynamical systems. Starting from amplitude-phase dynamics, we introduce successively topics of relevance, such as phase locking, the Adler equation, Arnold tongues, and the Kuramoto model, and use these concepts to analyze coupled oscillators and their stability. We then formulate the corresponding description for quantum limit-cycle oscillators using Lindblad master equations and phase-space methods, with explicit examples of two and many coupled oscillators. We further discuss noise, synchronization measures, and the extreme quantum regime, including synchronization in spin systems. The emphasis throughout is on analytical derivations and a practical methodology for identifying synchronized states, determining their stability, and characterizing synchronization in quantum systems. The review is designed as a self-contained guide from which we hope that students can learn the essential concepts and techniques, while researchers can readily find the analytical tools and methodological insights needed to approach new synchronization problems.