[Paper Review] The modelling of a Josephson junction and Heun polynomials

S. I. Tertychniy|ArXiv.org|Jan 30, 2006

Quantum Mechanics and Non-Hermitian Physics2 references17 citations

TL;DR

This paper models the Josephson junction's nonlinear dynamics via a first-order ODE with harmonic driving, transforming it into a second-order linear ODE with polynomial coefficients that reduces to the double confluent Heun equation (DCHE). The key contribution is identifying algebraic constraints on parameters that yield polynomial solutions, which form an orthogonal, normalizable system on the positive real axis, enabling exact analytical solutions for specific parameter regimes in superconducting quantum devices.

ABSTRACT

The first order nonlinear ODE \dot ϕ(t) + \sinϕ(t)=q(t),q(t)=B+A\cosωt, where A,B,ωare real constants, is considered, the transformation converting it to a second order linear homogeneous ODE with polynoimial coefficients is found. The latter is identified as a particular case of the double confluent Heun equation. The series of algebraic constraints on the constant parameters is found whose fulfillment leads to the existance of solutions representable through polynomials in explicit form. These polynomials are found to constitute the orthogonal normalizable system

Motivation & Objective

• To analyze the nonlinear dynamics of a resistively shunted Josephson junction under harmonic bias.
• To transform the first-order nonlinear ODE into a second-order linear ODE with polynomial coefficients.
• To identify parameter constraints leading to polynomial solutions of the transformed equation.
• To establish that these polynomial solutions form an orthogonal and normalizable system on the positive real axis.
• To connect the problem to the double confluent Heun equation (DCHE) and derive explicit solution structures.

Proposed method

• Transform the Josephson junction's first-order nonlinear ODE into a system of two linear ODEs using complex phase variables.
• Introduce a complex variable $ z = e^{i\omega t} $ to convert time-dependent coefficients into rational functions of $ z $.
• Apply a change of variables $ v, \check{v} $ to recast the system into a second-order linear ODE with polynomial coefficients.
• Identify the resulting ODE as a particular case of the double confluent Heun equation (DCHE) via Möbius transformations.
• Derive a recurrence relation for polynomial solutions by assuming a finite-degree power series ansatz.
• Establish orthogonality and normalizability of the polynomial solutions using a Wronskian-type identity and weight function analysis.

Experimental results

Research questions

• RQ1 Under what conditions does the Josephson junction's nonlinear ODE admit exact polynomial solutions?
• RQ2 How can the first-order nonlinear ODE for the phase dynamics be transformed into a second-order linear ODE with polynomial coefficients?
• RQ3 What is the precise connection between the transformed ODE and the double confluent Heun equation (DCHE)?
• RQ4 What algebraic constraints on the parameters $ A, B, \omega $ lead to the existence of polynomial solutions of degree $ n $?
• RQ5 Are the resulting polynomial solutions orthogonal and normalizable on the positive real axis?

Key findings

• The nonlinear ODE $ \dot{\varphi} + \sin\varphi = B + A\cos\omega t $ is equivalent to a second-order linear ODE with polynomial coefficients, which is identified as a particular case of the double confluent Heun equation (DCHE).
• Algebraic constraints on $ A, B, \omega $, parameterized by a non-negative integer $ n $, lead to the existence of polynomial solutions $ P_n(z) $ of degree $ n $, derived from the master equation (15).
• The polynomial solutions $ P_n(z) $ satisfy a recurrence relation (66) and are determined by finite products of $ 2\times2 $ matrices, enabling explicit computation via (57) and (60).
• The solutions form an orthogonal system on $ \mathbb{R}^+ $ with respect to a weight function $ \Xi_{n_1,n_2} $, as shown by the identity (71) and the integral condition (72).
• The solutions are normalizable in a weighted $ L^2 $-norm involving the factor $ \exp(-\mu(z + z^{-1})) $, with $ \mu = A/(4\omega) $, ensuring physical relevance.
• The spectral equation (21) is factorizable into two determinants, $ \det \mathbf{G}^{(+1)} = 0 $ or $ \det \mathbf{G}^{(-1)} = 0 $, under the constraint $ 4\omega^2(\lambda + \mu^2) = 1 $, simplifying numerical computation.