where X ( ω ) and Y ( ω ) are the in-phase and out of phase components of the oscillation. Then the complex amplitude is A ( ω ) = X ( ω ) − iY ( ω ) and the time response to the driving source a$_{0}$ω 2 0 cos( ωt ) is a ( t ) = X ( ω ) cos( ωt ) + Y ( ω ) sin( ωt ) = | A ( ω ) | cos ( ωt + ϕ ( ω )) with a phase ϕ ( ω ). This phase describes the delay between the driving source and the response and is As described in the main text, to experimentally determine state of a harmonic oscillator, the measured deflection a ( t ) is multiplied with two reference signals cos ( ω$_{r}$t ) and sin ( ω$_{r}$t ) to obtain two quadrature signals with ω$_{Δ}$ = ω − ω$_{r}$ and ω$_{Σ}$ = ω + ω$_{r}$ . With the definitions the output of the two multiplication stages can be written in matrix notation as { x$_{q}$ ( t ) , y$_{q}$ ( t ) } = ( M$_{Δ}$ ( t ) + M$_{Σ}$ ( t )) { X ( ω ) , Y ( ω ) } . The corresponding time evolution can thus be decomposed into one vector rotating clockwise with the frequency ω$_{Δ}$ and another one rotating counter-clockwise with the frequency ω$_{Σ}$ . After the multiplication stage the two quadrature signals x$_{q}$ ( t ) and y$_{q}$ ( t ) are low-pass filtered. For a simple first order low pass the corresponding time domain signals are ϕ ( ω ) = − tan − 1 $^{(}$Y ( ω ) X ( ω )$^{)}$ = − π/ 2 + tan − 1 $^{(}$X ( ω ) Y ( ω )$^{)}$ (20) x$_{q}$ ( t ) = a ( t ) cos ( ω$_{r}$t ) = X ( ω ) 2 (cos( ω$_{Δ}$t ) + cos( ω$_{Σ}$t )) + Y ( ω 2 ) (sin( ω$_{Σ}$t ) + sin( ω$_{Δ}$t )) y$_{q}$ ( t ) = a ( t ) sin ( ω$_{r}$t ) = X ( ω ) 2 ( − sin( ω$_{Δ}$t ) + sin( ω$_{Σ}$t )) + Y ( ω ) 2 (cos( ω$_{Δ}$t ) − cos( ω$_{Σ}$t )) M$_{Δ}$ ( t ) = 1 2   cos( ω$_{Δ}$t ) sin( ω$_{Δ}$t ) − sin( ω$_{Δ}$t ) cos( ω$_{Δ}$t )   and M$_{Σ}$ ( t ) = 1 2   cos( ω$_{Σ}$t ) sin( ω$_{Σ}$t ) sin( ω$_{Σ}$t ) − cos( ω$_{Σ}$t )   〈 x$_{q}$ ( t ) 〉$_{τ}$ ≡ 1 τ ∫ 0 −∞ dξ x$_{q}$ ( t − ξ ) e ξ/τ (21) = X ( ω ) 2 cos( ω$_{Δ}$t ) − ω$_{Δ}$ τ sin( ω$_{Δ}$ t ) (1 + ω 2 $_{Δ}$τ $^{2}$) + Y ( ω ) 2 sin( ω$_{Δ}$t ) + ω$_{Δ}$ τ cos( ω$_{Δ}$ t ) (1 + ω 2 $_{Δ}$τ $^{2}$) + X ( ω ) 2 cos( ω$_{Σ}$t ) − ω$_{Σ}$ τ sin( ω$_{Σ}$ t ) (1 + ω 2 $_{Σ}$τ $^{2}$) + Y ( ω ) 2 sin( ω$_{Σ}$t ) + ω$_{Σ}$ τ cos( ω$_{Σ}$ t ) (1 + ω 2 $_{Σ}$τ $^{2}$) 〈 y$_{q}$ ( t ) 〉$_{τ}$ ≡ 1 τ ∫ 0 −∞ dξ y$_{q}$ ( t − ξ ) e ξ/τ (22) = X ( ω ) 2 − sin( ω$_{Δ}$t ) − ω$_{Δ}$τ cos( ω$_{Δ}$ t ) (1 + ω 2 $_{Δ}$τ $^{2}$) + Y ( ω ) 2 cos( ω$_{Δ}$t ) − ω$_{Δ}$ τ sin( ω$_{Δ}$ t ) (1 + ω 2 $_{Δ}$τ $^{2}$) + X ( ω ) 2 sin( ω$_{Σ}$t ) + ω$_{Σ}$ τ cos( ω$_{Σ}$ t ) (1 + ω 2 $_{Σ}$τ $^{2}$) + Y ( ω ) 2 − cos( ω$_{Σ}$t ) + ω$_{Σ}$ τ sin( ω$_{Σ}$ t ) (1 + ω 2 $_{Σ}$τ $^{2}$) (23) 15