In this work, we use Dirac’s constraint analysis (DCA) to solve superconducting quantum circuits (SQCs). The Lagrangian of an SQC reveals the constraints that are classified in a Hamiltonian framework, such that redundant variables can be removed to isolate the canonical degrees of freedom for subsequent quantization of the Dirac brackets. We demonstrate the robustness of DCA unlike certain other set of ideas like null vector and loop charge which are each applicable only to specific types of quantum circuits.
In this work, we use Dirac’s constraint analysis (DCA) to solve superconducting quantum circuits (SQCs). The Lagrangian of an SQC reveals the constraints that are classified in a Hamiltonian framework, such that redundant variables can be removed to isolate the canonical degrees of freedom for subsequent quantization of the Dirac brackets. We demonstrate the robustness of DCA unlike certain other set of ideas like null vector and loop charge which are each applicable only to specific types of quantum circuits.
In this work we apply Dirac's Constraint Analysis (DCA) to solve Superconducting Quantum Circuits (SQC). The Lagrangian of a SQC reveals the constraints, that are classified in a Hamiltonian framework, such that redundant variables can be removed to isolate the canonical degrees of freedom for subsequent quantization of the Dirac Brackets. We demonstrate the robustness of DCA unlike certain other set of ideas like null vector and loop charge which are each applicable only to specific types of quantum circuits.
Abstract In this work we exploit Dirac's Constraint Analysis (DCA) in Hamiltonian formalism to study different types of Superconducting Quantum Circuits (SQC) in a unified way. The Lagrangian of a SQC reveals the constraints, that are classified in a Hamiltonian framework, such that redundant variables can be removed to isolate the canonical degrees of freedom for subsequent quantization of the Dirac Brackets via a generalized Correspondence Principle. This purely algebraic approach makes the application of concepts such as graph theory, null vector, loop charge, etc that are in vogue, (each for a specific type of circuit), completely redundant. The universal validity of DCA scheme in SQC, proposed by us, is demonstrated by correctly re-deriving existing results for different SQCs, obtained previously exploiting different formalisms each applicable for a specific SQC. Furthermore, we have also analysed and predicted new results for a generic form of SQC - it will be interesting to see its validation in an explicit circuit implementation.
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.
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 quantum mechanics of superconducting circuits is derived by starting from a classical Hamiltonian dynamical system describing a dissipationless circuit, usually made of capacitive and inductive elements. However, standard approaches to circuit quantization treat fluxes and charges, which end up as the canonically conjugate degrees of freedom on phase space, asymmetrically. By combining intuition from topological graph theory with a recent symplectic geometry approach to circuit quantization, we present a theory of circuit quantization that treats charges and fluxes on a manifestly symmetric footing. For planar circuits, known circuit dualities are a natural canonical transformation on the classical phase space. We discuss the extent to which such circuit dualities generalize to nonplanar circuits.
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