0
$\begingroup$

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?

$\endgroup$
1
$\begingroup$

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 $\langle X\rangle = \langle Y\rangle = 0$.)

$\endgroup$
  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.