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where Erf[ x ] = 2 / √ π $^{∫}$dx e − x 2 is the normalized Error Function (Erf[ ∞ ] = 1). This prob- ability distribution is plotted in the inset of figure 3 for the range of oscillation amplitudes a$_{os}$/a$_{th}$ = 0 − 2. For large oscillation amplitude a$_{os}$ >> a$_{th}$ the first term can be neglected, in addition we can assume Erf[ ... ] ≃ 1, and only very small angles contribute to the probability amplitude (sin ( ϕ ) ≃ ϕ ; cos ( ϕ ) ≃ 1) then, with ϕ$_{th}$ ≡ a$_{th}$/a$_{os}$ , we find 

<img src='content_image/332694.jpg'>

which is a normalized Gaussian probability distribution of the angle ϕ with variance Δ ϕ = ϕ$_{th}$/ √ 2 = a$_{th}$/ ( √ 2 a$_{os}$ ), in agreement with the high excitation limit of eq. 12. We note that the ratio ϕ$_{th}$ ≡ a$_{th}$/a$_{os}$ can be interpreted as the fluctuation of the phase due to thermal variation a$_{th}$/ √ 2 of the x-component when the y-component of the oscillation is fixed at a$_{os}$ (for a$_{os}$ >> a$_{th}$ thermal fluctuation of the y-component essentially gives no contribution to phase noise). For small oscillation amplitudes the second term in 26 is small and we find, to first order in 1 /ϕ$_{th}$ , 

<img src='content_image/332695.jpg'>

The angle probability distribution therefore becomes non-Gaussian and ultimately uniform (see inset of figure 3), as is expected for vanishing oscillation amplitude. The mean value 

