Thermometry of levitated nanoparticles in a hybrid electro-optical trap

There have been recent rapid developments in stable trapping of levitated nanoparticles in high vacuum. Cooling of nanoparticles, from phonon occupancies of 10(7) down to. similar or equal to 100-1000 phonons, have already been achieved by several groups. Prospects for quantum ground-state cooling seem extremely promising. Cavity-cooling without added stabilisation by feedback cooling remains challenging, but trapping at high vacuum in a cavity is now possible through the addition of a Paul trap. However, the Paul trap has been found to qualitatively modify the cavity output spectrum, with the latter acquiring an atypical 'split-sideband' structure, of different form from the displacement spectrum, and which depends on N, the optical well at which the particle localises. In the present work we investigate the N-dependence of the dynamics, in particular with respect to thermometry: we show that in strong cooling regions N greater than or similar to 100, the temperature may still be reliably inferred from the cavity output spectra. We also explain the N-dependence of the mechanical frequencies and optomechanical coupling showing that these may be accurately estimated. We present a simple 'fast-cavity' model for the cavity output and test all our findings against full numerical solutions of the nonlinear stochastic equations of motion for the system.