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arXiv:2110.15711 · European Journal of Physics

Study of geometric phase using classical coupled oscillators

Sharba Bhattacharjee, Biprateep Dey, Ashok Kumar Mohapatra

Published Jan 18, 2018• 3 citations
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Abstract

Abstract We illustrate the geometric phase associated with the cyclic dynamics of a classical system of coupled oscillators. We use an analogy between a classical coupled oscillator and a two-state quantum mechanical system to represent the evolution of the oscillator on an equivalent Hilbert space, which may be represented as a trajectory on the surface of a sphere. The cyclic evolution of the system leads to a change in phase, which consists of a dynamic phase along with an additional phase shift dependent on the geometry of the evolution. A simple experiment suitable for advanced undergraduate students is designed to study the geometric phase incurred during cyclic evolution of a coupled oscillator.

DOI
10.1088/1361-6404/aaa8a2

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We study the geometric phase factors underlying the classical and the corresponding quantum dynamics of a driven nonlinear oscillator exhibiting chaotic dynamics. For the classical problem, we compute the geometric phase factors associated with the phase space trajectories using Frenet-Serret formulation. For the corresponding quantum problem, the geometric phase associated with the time evolution of the wave function is computed. Our studies suggest that the classical geometric phase may be related to the the difference in the quantum geometric phases between two neighboring eigenstates.
It is shown that the state vector of a harmonic oscillator in a finite-dimensional Hilbert space changes sign only when the Hilbert space is of even dimension. The cyclic geometric phase for this finite-dimensional Hilbert-space harmonic oscillator is calculated. The effect of the finiteness of the Hilbert space on the dynamical and geometric phase change of a harmonic oscillator during a complete cycle is studied. In the limit of an infinite-dimensional case, the expressions reduce to the well known results.