Solving Superconducting Quantum Circuits in Dirac's Constraint Analysis Framework

TL;DR

This paper applies Dirac's Constraint Analysis to superconducting quantum circuits, providing a unified algebraic framework that simplifies analysis and re-derives known results while predicting new circuit behaviors.

Contribution

It introduces a universal Dirac's Constraint Analysis approach for superconducting circuits, eliminating the need for circuit-specific formalisms and enabling new predictions.

Findings

Successfully re-derived existing circuit results

Unified the analysis of different superconducting circuits

Predicted new behaviors for generic 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 {\it{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.