content_image/367187.jpg content_image/367188.jpg content_image/367189.jpg content_image/367190.jpg
Again, two matrices 

<img src='content_image/367187.jpg'>

<img src='content_image/367188.jpg'>

can be defined. The first matrix, M τ$_{Δ}$ ( t ), corresponds to a clockwise rotation with the frequency ω$_{Δ}$ and a delay angle ϕ$_{Δ}$ = − tan ( ω$_{Δ}$τ ) while the second matrix, M τ$_{Σ}$ ( t ), corresponds to a counter-clockwise rotation with the frequency ω$_{Σ}$ and a delay angle ϕ$_{Σ}$ = + tan ( ω$_{Σ}$τ ). 

## B. Appendix B 

For the calculation of the variation of the phase, we will assume that the variables x ( t ) and y ( t ) are Gaussian variables. At resonance, as discussed above 〈 x ( t ) 〉$_{τ}$ = 0 and 〈 y ( t ) 〉$_{τ}$ = Qa$_{exc}$ = a$_{os}$ , therefore their probability distributions are described by 

<img src='content_image/367189.jpg'>

These distributions are normalized and have the variance Δ x = Δ y = a$_{th}$/ √ 2 = $^{√}$kT / (2 c ). To calculate the distribution of the phase, first its probability function has to be calculated according to the general relation (see, for example, [38]), 

which in our case leads to 

<img src='content_image/367190.jpg'>

