Noise ellipse

I have a squeezed and rotated vacuum as shown in the above figure. I calculated the symmetrized double sided noise spectrum for the $X$ and $Y$ quadratures, which are $S_X(\omega)$ and $S_Y(\omega)$. And also the cross correlated spectrum $S_{XY}(\omega)$. I knew that $S_X(\omega)$ and $S_Y(\omega)$ determine the noise in $X$ and $Y$ quadratures. What about the cross correlated spectrum $S_{XY}(\omega)$? How can I relate $S_{XY}(\omega)$ with the figure?


Consider the covariance matrix $$ \left(\begin{matrix} \langle X^2 \rangle & \langle XY \rangle \\ \langle XY \rangle & \langle Y^2 \rangle \end{matrix}\right) $$ (which, I understand, has as enties exactly your $S$es). Its eigenvectors and eigenvalues exactly characterize the ellipse: The eigenvectors are the principal axes of the ellipse, and the eigenvalue gives the variance along this axis (which should be the square of the size of your ellipse -- depending of course on what you plot).

(Note: The covariance matrix is defined with variances; I have used above that $\langle X\rangle = \langle Y\rangle = 0$.)

  • $\begingroup$ Thanks for your comment. But what I want to know is that: if I integrate $S_{XY}(\omega)$ over $\omega$ I will get the variance of the $X$ quadrature. Same is true for the $Y$ quadrature. My question is that: what will I get if I integrate $S_{XY}(\omega)$ ? $\endgroup$ – Mr. an Sep 3 '16 at 11:56
  • $\begingroup$ @Mr.an Could you start by giving the necessary details: What exactly do you mean by $S_{XY}(\omega)$ etc., and what do you want to know -- i.e., what would be a satisfactory answer to "what do I get"? (And please, edit these things into your question.) However, I still think that what I wrote should answer your question: The three numbers $S_X$, $S_Y$, and $S_{XY}$ together tell you the orientation and shape of the ellipse. (You can write down the explicit form of the eigenvectors and -values in terms of the $S$es, this might give you a more "explicit" connection.) $\endgroup$ – Norbert Schuch Sep 3 '16 at 12:29
  • $\begingroup$ @Mr.an Another perspective is that $S_{XY}$ measures the correlations in the fluctuations of $X$ and $Y$: If it is non-zero, their fluctuations are correlated. (If you take linear combinations of $X$ and $Y$ as given by the eigenvalues above, then their fluctuations are again uncorrelated.) $\endgroup$ – Norbert Schuch Sep 3 '16 at 12:54
  • $\begingroup$ Thanks again. Essentially I am doing some work that frequency dependent noise ellipse rotation, since this technique is useful for noise reduction in many areas. The spectra $S_X(\omega)$,$S_Y(\omega) $and $S_{XY}(\omega)$ are used to see the rotation of the ellipse. So I want to make a link between these spectra and the geometry above. Maybe my expression is not clear enough due to my English and also lack of understanding to the topic. :) $\endgroup$ – Mr. an Sep 3 '16 at 13:03
  • $\begingroup$ @Mr.an Why don't you edit your question that it is self-contained, defines the $S_{\cdot}(\omega)$, and clearly states what you are looking for and why? This will definitely increase your chances to get a satisfactory answer. Also note that comments can be deleted at any time, so they are definitely not the place for clarifications to the question. $\endgroup$ – Norbert Schuch Sep 3 '16 at 13:07

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