Classical resonance interactions and Josephson junction in macroscopic quantum dynamics

TL;DR

This paper demonstrates that the classical resonance dynamics of coupled Duffing oscillators are mathematically equivalent to the quantum dynamics of a boson Josephson junction, revealing insights into energy transfer, symmetry breaking, and quantum-like effects.

Contribution

It establishes a novel analogy between classical oscillator interactions and macroscopic quantum dynamics, providing solvable models and new perspectives on quantum phenomena.

Findings

Classical oscillator dynamics are equivalent to quantum Josephson junction equations.

The derived energy partition oscillator is solvable in quadratures and elementary functions.

Symmetry breaking bifurcations relate to macroscopic quantum self-trapping.

Abstract

It is shown that the classical dynamics of 1:1 resonance interaction between two identical linearly coupled Duffing oscillators is equivalent to the symmetric (non-biased) case of `macroscopic' quantum dynamics of two weakly coupled Bose-Einstein condensates. The analogy develops through the boson Josephson junction equations, however, reduced to a single conservative energy partition (EP) oscillator. The derived oscillator is solvable in quadratures, furthermore it admits asymptotic solution in terms of elementary functions after transition to the action-angle variables. Energy partition and coherency indexes are introduced to provide a complete characterization of the system dynamic states through the state variables of the EP oscillator. In particular, nonlinear normal and local mode dynamics of the original system associate with equilibrium points of such oscillator. Additional equilibrium points - the local modes - may occur on high energy level as a result of the symmetry breaking bifurcation, which is equivalent to the macroscopic quantum self-trapping effect in boson Josephson junction. Finally, since the Hamiltonian of EP oscillator is always quadratic with respect its linear momentum, the idea of second quantization can be explored without usual transition to the rigid pendulum approximation.