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We characterize throughout the spectral range of an optical trap the natureof the noise at play and the ergodic properties of the corresponding Brownianmotion of an overdamped trapped single microsphere, comparing experimental,analytical and simulated data. We carefully analyze noise and ergodicproperties $(i)$ using the Allan variance for characterizing the noise and$(ii)$ exploiting a test of ergodicity tailored for experiments done overfinite times. We derive these two observables in the low-frequencyOrnstein-Uhlenbeck trapped-diffusion regime and study analytically theirevolution towards the high-frequency Wiener free-diffusion regime, in a verygood agreement with simulated and experimental results. This leads to revealnoise and ergodic spectral signatures associated with the distinctive featuresof both regimes.

Noise and ergodic properties of Brownian motion in an optical tweezer: looking at the crossover between Wiener and Ornstein-Uhlenbeck processes