[Paper Review] The interrelation of the special double confluent Heun equation and the equation of RSJ model of Josephson junction revisited

S. I. Tertychniy|arXiv (Cornell University)|Nov 8, 2018

Quantum chaos and dynamical systems8 references4 citations

TL;DR

This paper establishes an explicit mathematical mapping between solutions of the RSJ model for overdamped Josephson junctions under sinusoidal bias and the special double confluent Heun equation (sDCHE). By leveraging holomorphic eigenfunctions of a linear operator on the punctured complex plane and analyzing their monodromy properties, the authors derive a matrix representation of the monodromy transformation, revealing a direct link between phase-locking behavior in the RSJ model and spectral properties of the sDCHE.

ABSTRACT

The explicit formulas for the maps interconnecting the sets of solutions of the special double confluent Heun equation and the equation of the RSJ model of overdamped Josephson junction in case of shifted sinusoidal bias are given. The approach these are based upon leans on the extensive application of eigenfunctions of certain linear operator acting on functions holomorphic on the universal cover of the punctured complex plane. The functional equation the eigenfunctions noted obey is derived, the matrix form of the monodromy transformation they infer is given.

Motivation & Objective

• To establish a rigorous mathematical correspondence between the RSJ model of Josephson junctions and the special double confluent Heun equation (sDCHE).
• To analyze the functional and monodromic properties of eigenfunctions of a specific linear operator acting on holomorphic functions over the punctured complex plane.
• To derive an explicit matrix representation of the monodromy transformation for sDCHE solutions, enabling the study of phase-locking behavior in the RSJ model.
• To provide a new criterion for phase-locking in Josephson junctions based on the vanishing of off-diagonal elements in the monodromy matrix.

Proposed method

• Complexification of the RSJ equation by mapping the phase variable φ(t) to Φ(z) = exp(iφ(t)) with z = exp(iωt), embedding the dynamics on the unit circle in the complex plane.
• Derivation of a Riccati-type differential equation (Eq. 2) for Φ(z), which is then transformed into a linear second-order ODE equivalent to the sDCHE.
• Introduction of a linear operator LC acting on holomorphic functions, defined via differentiation and inversion z ↦ z⁻¹, whose eigenfunctions are central to the analysis.
• Identification of eigenfunctions of LC as solutions to the sDCHE, with the operator preserving a domain invariant under z ↦ z⁻¹.
• Construction of the monodromy transformation via analytic continuation around z = 0, represented as a 2×2 matrix (Eq. 44) acting on a basis of sDCHE solutions.
• Explicit derivation of the monodromy matrix M in terms of values of eigenfunctions at preimages of z = -1 on the universal cover of C*.

Experimental results

Research questions

• RQ1 How can the solutions of the RSJ model for an overdamped Josephson junction under sinusoidal bias be systematically related to solutions of the special double confluent Heun equation (sDCHE)?
• RQ2 What functional and symmetry properties do the eigenfunctions of the operator LC possess, and how do they relate to the sDCHE?
• RQ3 What is the explicit form of the monodromy transformation for solutions of the sDCHE, and how does it encode dynamical behavior such as phase locking?
• RQ4 Can the phase-locking behavior in the RSJ model be characterized through spectral properties of the sDCHE, particularly via the monodromy matrix?

Key findings

• The monodromy matrix M for the sDCHE is explicitly derived in matrix form (Eq. 44), with diagonal elements real and equal, and off-diagonal elements purely imaginary.
• The determinant of the monodromy matrix is identically 1, consistent with the conservation of Wronskian in second-order linear ODEs.
• The eigenvalues of the monodromy matrix coincide if and only if one of the off-diagonal matrix elements vanishes, providing a new criterion for phase-locking in the RSJ model.
• The phase-locking behavior in the RSJ model corresponds to the degeneracy of the monodromy matrix eigenvalues, which is directly linked to the vanishing of off-diagonal terms in the matrix representation.
• The functional equation obeyed by the eigenfunctions of the operator LC is derived, and their analytic continuation across the Riemann surface of C* is fully characterized.
• The transformation φ(t) ↦ φ(t + 2π/ω) in the RSJ model corresponds to the monodromy action on sDCHE solutions, establishing a dynamical equivalence between time evolution and analytic continuation.