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Does anyone know where the differential equation of covariance matrix comes from?

$$\frac{dC(t)}{dt}=AC(t)+C(t)A^T+D$$

where $C$ is the covariance matrix, $A$ is the drift matrix, and $D$ is the diffusion matrix. $A$ is obtained from the compact form of quantum Langevin equations as follows:

$$\frac{du(t)}{dt}=Au(t)+n(t)$$

I need the original paper if anyone knows.

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  • $\begingroup$ Have you tried using the definition of the covariance matrix $C(t)=\left<u(t)\cdot u(t)^T\right>$ ? $\endgroup$ – Frédéric Grosshans Jun 27 '19 at 13:16
  • $\begingroup$ @FrédéricGrosshans I haven't. You mean solving the compact form of QLEs and using the solutions as $u(t)$? $\endgroup$ – Ghaem Jun 27 '19 at 18:53
  • $\begingroup$ I mean writing $\frac{dC}{dt}=\frac{d}{dt}(uu^T)= \frac{du}{dt}u^T + u\left(\frac{du}{dt}\right)^T$ and expanding $\endgroup$ – Frédéric Grosshans Jun 27 '19 at 19:17
  • $\begingroup$ @FrédéricGrosshans The expansion gives $\dot{C}=AC+CA^T+\langle nu^T+un^T\rangle$ which $D \equiv \langle nu^T+un^T\rangle$, accordingly. But I see another definition for $D$ in papers: $D_{k,l} \delta(t-t')=[\langle n_{k}(t)n_{l}(t')+n_l(t')n_k(t)\rangle]/2$! $\endgroup$ – Ghaem Jun 28 '19 at 7:40
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    $\begingroup$ While I quite often use covariance matrices in quantum optics, I do not use them in a context with differntial equations, so I don’t know what the notation conventions are for this. I can’t help you more. Maybe links to the papers you’re referring to may help others to find what is missing $\endgroup$ – Frédéric Grosshans Jul 1 '19 at 13:18

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