Local and global bifurcations in magnetic resonance force microscopy

Abstract

The focus of this paper is on the investigation of local and global bifurcations in a continuum mechanics-based resonator model proposed for measurement of electron spin via magnetic resonance force microscopy (MRFM). The resonator model, derived using the extended Hamilton’s principle incorporating the Bloch equations for magnetization, is investigated analytically and numerically. Analysis of both adiabatic and non-adiabatic equilibrium configurations enables formulation of the dynamical system bifurcation structure and identification of the parameter space required for stable MRFM operation. A multiple-scales analysis of the limiting adiabatic model enables estimation of the local bifurcation thresholds for bistable solutions and prediction of the frequency shift that enables spin detection. Orbital instabilities of the adiabatic model reveal a global bifurcation structure where lengthy chaotic transients occur below a homoclinic jump-to-contact threshold which is determined via a Melnikov–Holmes analysis. Both local and global bifurcations are verified numerically in the non-adiabatic model and reveal a dense power spectra for the magnetic moments. The computation of the parameter space governing the model orbital instabilities enables a consistent estimation of robust MRFM operation conditions.