Again, two matrices 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 ). 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 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 M τ$_{Δ}$ ( t ) = 1 2(1 + τ $^{2}$ω 2 $_{Δ}$)   cos( ω$_{Δ}$t ) − ω$_{Δ}$τ sin( ω$_{Δ}$t ) sin( ω$_{Δ}$t ) + ω$_{Δ}$τ cos( ω$_{Σ}$t ) − sin( ω$_{Δ}$t ) − ω$_{Δ}$τ cos( ω$_{Δ}$t ) cos( ω$_{Δ}$t ) − ω$_{Δ}$τ sin( ω$_{Δ}$t )   (24) M τ$_{Σ}$ ( t ) = 1 2(1 + τ $^{2}$ω 2 $_{Σ}$)   cos( ω$_{Σ}$t ) − ω$_{Σ}$τ sin( ω$_{Σ}$t ) sin( ω$_{Σ}$t ) + ω$_{Σ}$τ cos( ω$_{Σ}$t ) sin( ω$_{Σ}$t ) + ω$_{Σ}$τ cos( ω$_{Σ}$t ) − cos( ω$_{Σ}$t ) + ω$_{Σ}$τ sin( ω$_{Σ}$t )   (25) p$_{x}$ ( x ) = √ 1 πa 2 th e − ( x/a$_{th}$ ) 2 p$_{y}$ ( y ) = √ 1 πa 2 th e − (( y − a$_{os}$ ) /a$_{th}$ ) 2 p$_{ϕ}$ ( ϕ ) = ∫ ∫ dx dy p$_{x}$ ( x ) p$_{y}$ ( y ) δ ( ϕ − tan − $^{1}$( x/y )) B. Appendix B 16