  Optical levitation of nanoscale particles has emerged as a platform for
precision measurement. Extremely low damping, together with optical
interferometric position detection, makes possible exquisite force measurement
and promises low-energy tests of fundamental physics. Essential to such
measurement is an understanding of the confidence with which parameters can be
inferred from spectra estimated from the indirect measurement provided by
interferometry. We present an apparatus optimized for sensitivity along one
motional degree of freedom, a theoretical model of the spectrum, and maximum
likelihood estimation. The treatment accounts for the sinusoidal dependence of
interferometric signal on particle position, and we use the technique to
extract thermodynamic quantities in a regime where simpler treatments are
confounded.
