Utilizing a forced Van der Pol-Rayleigh-Helmholtz oscillator under heptamodal-frequency operations in Casimir force measurement

Abstract

In this paper, Casimir force sensitivity is investigated by utilizing a micro-cantilever under the driving forces in heptamodal operations. A novel forced Van der Pol-Rayleigh-Helmholtz nonlinear oscillator model is developed to describe the nonlinear dynamics of the micro-cantilever which is subject to the excitation and Casimir forces simultaneously. Demonstrating the effectiveness of the heptamodal operations, single- and tetramodal-frequency excitation schemes are also applied separately to resonate the micro-cantilever at the fundamental and higher eigenmodes. The oscillation observables of the externally driven micro-cantilever are determined in the presence of the Casimir forces in the separation distance range of 200–800 nm. Remarkable variations in amplitude ratio, phase shift, and frequency shift for different effective masses of the micro-cantilever are explored for the higher eigenmodes. In the current work, the AFM micro-cantilever exhibits the amplitude response of 0.82 nm to Casimir force at the fourth eigenmode for the separation distance ranging between 200 and 300 nm. The stable frequency shifts ranging between 10³ and 10⁶ Hz are also observed at the first four eigenmodes for larger separation distances (above around 500 nm). Moreover, the maximum phase shift response of around 150 degrees at the sixth eigenmode is achieved using heptamodal-frequency excitation of the lightest micro-cantilever (3.6 × 10⁻¹² kg) at the separation distance of 200 nm. Thus, implementing heptamodal-frequency excitation schemes has considerable potential to improve the phase shift sensitivity to Casimir forces when compared with other excitation schemes. Additionally, the parameters of the nonlinear oscillator significantly determine the patterns of the time-domain sensitivities to the external forces. Correspondingly, displacements of the micro-cantilever under the driving and Casimir forces at different eigenmodes are obtained to investigate diverse system nonlinearities. Furthermore, the virial and dissipated power are also determined for different effective masses of the micro-cantilever to explain the energy dissipation process in the measurement of Casimir forces. Therefore, in the present work, the observable responses and energy quantities for particular system nonlinearities are introduced to be utilized for nanometrological applications.