# Where does differntial equation of covariance matrix come from?

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.

• Have you tried using the definition of the covariance matrix $C(t)=\left<u(t)\cdot u(t)^T\right>$ ? – Frédéric Grosshans Jun 27 '19 at 13:16
• @FrédéricGrosshans I haven't. You mean solving the compact form of QLEs and using the solutions as $u(t)$? – Ghaem Jun 27 '19 at 18:53
• 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 – Frédéric Grosshans Jun 27 '19 at 19:17
• @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: $D_{k,l} \delta(t-t')=[\langle n_{k}(t)n_{l}(t')+n_l(t')n_k(t)\rangle]/2$! – Ghaem Jun 28 '19 at 7:40
• 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 – Frédéric Grosshans Jul 1 '19 at 13:18