  We characterize throughout the spectral range of an optical trap the nature
of the noise at play and the ergodic properties of the corresponding Brownian
motion of an overdamped trapped single microsphere, comparing experimental,
analytical and simulated data. We carefully analyze noise and ergodic
properties $(i)$ using the Allan variance for characterizing the noise and
$(ii)$ exploiting a test of ergodicity tailored for experiments done over
finite times. We derive these two observables in the low-frequency
Ornstein-Uhlenbeck trapped-diffusion regime and study analytically their
evolution towards the high-frequency Wiener free-diffusion regime, in a very
good agreement with simulated and experimental results. This leads to reveal
noise and ergodic spectral signatures associated with the distinctive features
of both regimes.
