Spectral Analysis and Parameter Estimation in Fibre Levitated Optomechanics

Abstract

In levitated optomechanics, nano-scale objects are optically trapped so that their motion can be studied. These trapped nanoparticles are held in a 3D quadratic potential and act as damped harmonic oscillators; they are thermally and mechanically decoupled from the apparatus and their position is measured interfer-ometrically to picometre accuracy. These systems are well suited to sensing and metrology applications, as any external disturbance of the particle can be observed using the scattered trapping light.When examining the motion of a levitated nanoparticle, it’s position is recorded and used to estimate a power spectral density (PSD), from which state parameters can be estimated. In this thesis an experi-mental setup is presented, optimised for maximum collection of particle position information in 1D, using a fibre-based parabolic mirror trap and heterodyne measurement system in order to produce spectra with minimal noise and unwanted artefacts.A novel application of the Middleton expansion from RF engineering is used to generate a complete power spectrum that depends on the physical parameters of the system. This method treats the particle as a stochastic harmonic oscillator, phase modulated by a Gaussian random process with known PSD. We reproduce the PSD of intensity at a detector, a quantity that is sinusoidally dependent on particle posi-tion. This technique generates a single, full PSD using modified Bessel functions, and does not depend on assumptions about the relative phases of the interfered fields, highlighting the non-linear dependence of measured signal on position. Theoretical spectra are fitted to a measured PSD and the phase modulation depth is extracted; this is used to calculate the particle oscillation amplitude and, by an equipartition ar-gument, the centre of mass temperature to mass ratio. State parameters are tracked as environmental conditions change and an increase in centre of mass temperature as a function of decreasing background gas pressure is observed.