content_image/339412.jpg content_image/339413.jpg content_image/339414.jpg content_image/339415.jpg content_image/339416.jpg content_image/339417.jpg
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 

<img src='content_image/339412.jpg'>

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 

<img src='content_image/339413.jpg'>

with ω$_{Δ}$ = ω − ω$_{r}$ and ω$_{Σ}$ = ω + ω$_{r}$ . With the definitions 

<img src='content_image/339414.jpg'>

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 

<img src='content_image/339415.jpg'>

<img src='content_image/339416.jpg'>

<img src='content_image/339417.jpg'>

