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Physical Review A

Direct way to observe the geometric phase in nonlinear coupled oscillators

Wang Kaige, Jian Wang, Yang Guojian

Published Aug 1, 1997• 0 citations
Publisher

Abstract

We apply the geometric phase theory to the dissipative system of a set of nonlinear coupled oscillators. We found an area theorem for the geometric phase, which provides a direct and simple geometric method for observing the geometric phase of this system. The area theorem shows evidence that the geometric phase is related to certain dynamical features of the systems.

DOI
10.1103/physreva.56.1577

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We apply the geometric phase theory to the dissipative system of a set of nonlinear coupled oscillators. We found an area theorem for the geometric phase, which provides a direct and simple geometric method for observing the geometric phase of this system. The area theorem shows evidence that the geometric phase is related to certain dynamical features of the systems.
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Geometric phases of simple harmonic oscillator system are studied. Complete sets of "eigenfunctions" are constructed, which depend on the way of choosing classical solutions. For an eigenfunction, two different motions of the probability distribution function (pulsation of the width and oscillation of the center) contribute to the geometric phase which can be given in terms of the parameters of classical solutions. The geometric phase for a general wave function is also given. If a wave function has a parity under the inversion of space coordinate, then the geometric phase can be defined under the evolution of half of the period of classical motions. For the driven case, geometric phases are given in terms of Fourier coefficients of the external force. The oscillator systems whose classical equation of motion is Mathieu's equation are perturbatively studied, and the first term of nonvanishing geometric phase is calculated.
The geometric phase is a common phenomenon in a variety of physical systems, with significant applications in optics. We report the first experimental demonstration of the adiabatic geometric phase in nonlinear frequency conversion, wherein the coupling between the signal and idler frequencies constitutes the intrinsic two-level dynamics of the system. We observe a variety of effects associated with the geometric phase, including the adiabatic broadening of bandwidth, asymmetric transmission for opposite propagation directions, conjugation of the phase for orthogonal eigenstates, and the nonreciprocity associated with the pump field bias. Our work paves the way towards all-optically controlled geometric phase elements for wavefront shaping, isolation, guiding, and quantum optical applications, harnessing spectral and spatial correlations.
We show that the geometrical (Berry) phases discovered in Hamiltonian systems can also be defined as resulting from parallel transportation of vectors for nonlinear dissipative systems with cyclic attractors. If the nonlinear dissipative systems possess a certain kind of asymptotic solution defined in this Letter, the phase and amplitude accumulation of a geometrical type can be defined. Detuned one- and two-photon lasers showing periodic intensity pulsations are taken as examples of such systems.
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