I cannot understand the derivation in Louis Garbe article (https://arxiv.org/abs/1910.00604) about how to obtain the covariance matrix equation from Fokker-Planck equation for the Wigner function in Appendix_Dissipative dynamics.
The Lindblad equation above into a Fokker-Planck equation for the Wigner function: \begin{equation} \frac{\partial W}{\partial t}(x,p)=-\omega_0p\frac{\partial W}{\partial x} - \omega_0(Xg^2-1)x\frac{\partial W}{\partial p} + \kappa(2W + \sum_{i=1}^2x_i\partial_iW + \sum_{i,j=1}^2\partial_i\sigma^L_{ij}\partial_jW), \end{equation} where $x_1=x$, $x_2=p$，and $$\sigma^L =\frac{1}{2}\begin{bmatrix}1 & 0 \\ 0 & 1+Xg^2\frac{\Gamma\omega_0}{\Omega\kappa}\end{bmatrix}$$
In article, an author say "Since this equation is quadratic in x and p, it can be solved by a Gaussian ansatz $ W=\frac{1}{\sqrt{\pi \text{det}(\sigma)}}\exp\left[-\frac{1}{2}x_i\left(\sigma^{-1}\right)_{ij}x_j\right] $. The displacement decays at a rate 2κ and will quickly reach 0. Thus, this function is entirely caracterised by the covariance matrix, which is described by the following equation: \begin{align} \partial_t\sigma = B\sigma+\sigma B^T- 2\kappa(\sigma - \sigma^L) \end{align} where \begin{align*} B =\begin{bmatrix} 0 & \omega_0 \\ \omega_0(Xg^2-1) & 0 \end{bmatrix} \end{align*}
In order to from F-P equation to covariance matrix equation, i insert Gaussian ansatz into F-P equation, ie \begin{align} \begin{split} \frac{\partial }{\partial t}(x,p)\frac{1}{\sqrt{\pi \text{det}(\sigma)}}\exp\left[-\frac{1}{2}x_i\left(\sigma^{-1}\right)_{ij}x_j\right]= &-\omega_0 \left(p\frac{\partial }{\partial x} - x\frac{\partial }{\partial p}\right)\frac{1}{\sqrt{\pi \text{det}(\sigma)}}\exp\left[-\frac{1}{2}x_i\left(\sigma^{-1}\right)_{ij}x_j\right]\\ &- \omega_0 X g^2 x\frac{\partial }{\partial p}\frac{1}{\sqrt{\pi \text{det}(\sigma)}}\exp\left[-\frac{1}{2}x_i\left(\sigma^{-1}\right)_{ij}x_j\right]\\ &+ 2 \kappa \frac{1}{\sqrt{\pi \text{det}(\sigma)}}\exp\left[-\frac{1}{2}x_i\left(\sigma^{-1}\right)_{ij}x_j\right]\\ &+ \kappa \left(x \frac{\partial }{\partial x} + p \frac{\partial }{\partial p}\right)\frac{1}{\sqrt{\pi \text{det}(\sigma)}}\exp\left[-\frac{1}{2}x_i\left(\sigma^{-1}\right)_{ij}x_j\right]\\ & + \frac12\kappa\left(\frac{\partial }{\partial x\partial x} + \frac{\partial }{\partial p\partial p} \right)\frac{1}{\sqrt{\pi \text{det}(\sigma)}}\exp\left[-\frac{1}{2}x_i\left(\sigma^{-1}\right)_{ij}x_j\right]\\ &+ \frac12 Xg^2\frac{\Gamma \omega_0}{\Omega}\frac{\partial }{\partial p\partial p}\frac{1}{\sqrt{\pi \text{det}(\sigma)}}\exp\left[-\frac{1}{2}x_i\left(\sigma^{-1}\right)_{ij}x_j\right] \end{split} \end{align} Then, simplify above equation, have \begin{align} \frac{\partial }{\partial t}(x,p)\frac{1}{\sqrt{\pi \text{det}(\sigma)}}\exp\left[-\frac{1}{2}x_i\left(\sigma^{-1}\right)_{ij}x_j\right] = &(-\omega_0 p + \kappa x)\frac{\partial }{\partial x}\frac{1}{\sqrt{\pi \text{det}(\sigma)}}\exp\left[-\frac{1}{2}x_i\left(\sigma^{-1}\right)_{ij}x_j\right]\\ &+(\omega_0 x - \omega_0 X g^2 x + \kappa p)\frac{\partial }{\partial p}\frac{1}{\sqrt{\pi \text{det}(\sigma)}}e^{-\frac{1}{2}x_i(\sigma^{-1})_{ij}x_j}\\ &+ 2 \kappa \frac{1}{\sqrt{\pi \text{det}(\sigma)}}\exp\left[-\frac{1}{2}x_i\left(\sigma^{-1}\right)_{ij}x_j\right]\\ & + \frac12\kappa\frac{\partial }{\partial x\partial x}\frac{1}{\sqrt{\pi \text{det}(\sigma)}}\exp\left[-\frac{1}{2}x_i\left(\sigma^{-1}\right)_{ij}x_j\right]\\ & + (\frac12\kappa + \frac12 Xg^2\frac{\Gamma \omega_0}{\Omega} ) \frac{\partial }{\partial p\partial p}\frac{1}{\sqrt{\pi \text{det}(\sigma)}}\exp\left[-\frac{1}{2}x_i\left(\sigma^{-1}\right)_{ij}x_j\right] \end{align} Next, evaluate the individual terms in the expression, have \begin{align} &\frac{\partial }{\partial t}(x)\frac{1}{\sqrt{\pi \text{det}(\sigma)}}\exp\left[-\frac{1}{2}x_i\left(\sigma^{-1}\right)_{ij}x_j\right]\\ = &\frac{\exp\left[-\frac{1}{2}x_i\left(\sigma^{-1}\right)_{ij}x_j\right] \left( -\frac{1}{2}x_i'(t)(\sigma(t)^{-1})_{ij} x_j (t) -\frac{1}{2}x_i(t) (\sigma(t)^{-1})_{ij} x_j'(t) -\frac{1}{2}x_i(t) (\sigma(t)^{-1})_{ij}'x_j (t) \right) }{\sqrt{\pi }\sqrt{\sigma_{11}(t) \sigma_{22}(t)-\sigma_{12}(t) \sigma_{21}(t)}}\\ &-\frac{\exp\left[-\frac{1}{2}x_i\left(\sigma^{-1}\right)_{ij}x_j\right] \left( \sigma_{11}'(t) \sigma_{22}(t) + \sigma _{11}(t) \sigma_{22}'(t) - \sigma_{12}'(t) \sigma_{21}(t) - \sigma _{12}(t) \sigma_{21}'(t) \right) }{2 \sqrt{\pi }\left(\sigma_{11}(t) \sigma_{22}(t)-\sigma_{12}(t) \sigma_{21}(t)\right)^{3/2}} \end{align} and \begin{align} &\frac{\partial }{\partial t}(x)\frac{1}{\sqrt{\pi \text{det}(\sigma)}}e^{-\frac{1}{2}x_i(\sigma^{-1})_{ij}x_j}\\ = &\frac{e^{-\frac{1}{2} x_i(t)(\sigma(t)^{-1})_{ij} x_j(t)} \left( -\frac{1}{2}x_i'(t)(\sigma(t)^{-1})_{ij} x_j (t) -\frac{1}{2}x_i(t) (\sigma(t)^{-1})_{ij} x_j'(t) -\frac{1}{2}x_i(t) (\sigma(t)^{-1})_{ij}'x_j (t) \right) }{\sqrt{\pi }\sqrt{\sigma_{11}(t) \sigma_{22}(t)-\sigma_{12}(t) \sigma_{21}(t)}}\\ &-\frac{e^{-\frac{1}{2} x_i(t)(\sigma(t)^{-1})_{ij} x_j(t)} \left( \sigma_{11}'(t) \sigma_{22}(t) + \sigma _{11}(t) \sigma_{22}'(t) - \sigma_{12}'(t) \sigma_{21}(t) - \sigma _{12}(t) \sigma_{21}'(t) \right) }{2 \sqrt{\pi }\left(\sigma_{11}(t) \sigma_{22}(t)-\sigma_{12}(t) \sigma_{21}(t)\right)^{3/2}} \end{align} and \begin{align} \frac{\partial }{\partial p}\frac{1}{\sqrt{\pi \text{det}(\sigma)}}e^{-\frac{1}{2}x_i(\sigma^{-1})_{ij}x_j} = &\frac{e^{-\frac{1}{2} x_i(\sigma^{-1})_{ij} x_j} \left( -\frac{1}{2}x_i'(\sigma^{-1})_{ij} x_j -\frac{1}{2}x_i (\sigma^{-1})_{ij} x_j' -\frac{1}{2}x_i (\sigma^{-1})_{ij}'x_j \right) }{\sqrt{\pi }\sqrt{\sigma_{11} \sigma_{22}-\sigma_{12} \sigma_{21}}}\\ &-\frac{e^{-\frac{1}{2} x_i(\sigma^{-1})_{ij} x_j} \left( \sigma_{11}' \sigma_{22} + \sigma _{11} \sigma_{22}' - \sigma_{12}' \sigma_{21} - \sigma _{12} \sigma_{21}' \right) }{2 \sqrt{\pi }\left(\sigma_{11} \sigma_{22}-\sigma_{12} \sigma_{21}\right)^{3/2}} \end{align} and \begin{align} &\frac{\partial }{\partial x \partial x}\frac{1}{\sqrt{\pi \text{det}(\sigma)}}e^{-\frac{1}{2}x_i(\sigma^{-1})_{ij}x_j}\\ = &\frac{e^{-\frac12 x_i\sigma^{-1}_{ij} x_j} \left(-\frac12 x_i (\sigma^{-1}_{ij})'x_j - \frac12 x_i'\sigma^{-1}_{ij} x_j -\frac12 x_i \sigma^{-1}_{ij} x_j'\right)^2}{\sqrt{\pi }\sqrt{\sigma_{11}\sigma_{22}-\sigma_{12}\sigma_{21}}}\\ &-\frac{-\frac12e^{-\frac12 x_i \sigma^{-1}_{ij} x_j} \left(\sigma_{11}'\sigma_{22} + \sigma _{11}\sigma_{22}' - \sigma_{12}'\sigma _{21} - \sigma_{12}\sigma_{21}'\right) \left( x_i (\sigma^{-1}_{ij})' x_j - x_i' \sigma^{-1}_{ij} x_j - x_i\sigma^{-1}_{ij} x_j'\right)}{\sqrt{\pi }\left(\sigma_{11}\sigma_{22} - \sigma_{12}\sigma_{21}\right)^{3/2}}\\ & + \frac{e^{-\frac{1}{2} x_i \sigma^{-1}_{ij} x_j} \left( -\frac{1}{2} x_i''\sigma^{-1}_{ij} x_j - \frac{1}{2} x_i (\sigma^{-1}_{ij})'' x_j - \frac{1}{2} x_i\sigma^{-1}_{ij} x_j'' - x_i'(\sigma^{-1}_{ij})'x_j - x_i(\sigma^{-1}_{ij})' x_j' - x_i'\sigma^{-1}_{ij} x_j'\right)}{\sqrt{\pi } \sqrt{\sigma _{11}\sigma_{22} - \sigma_{12} \sigma _{21}}}\\ &-\frac{\left(-2 \sigma _{12}' \sigma_{21}' + 2 \sigma _{11}' \sigma _{22}' + \sigma_{22}\sigma_{11}'' - \sigma_{21} \sigma_{12}'' - \sigma_{12}\sigma_{21}'' + \sigma _{11} \sigma _{22}''\right) e^{-\frac{1}{2} x_i\sigma^{-1}_{ij} x_j}}{2 \sqrt{\pi } \left(\sigma_{11}\sigma_{22} - \sigma_{12} \sigma_{21}\right)^{3/2}}\\ &+\frac{3 \left(\sigma_{22}\sigma_{11}' - \sigma_{21} \sigma_{12}' -\sigma _{12}\sigma _{21}'+\sigma _{11}\sigma _{22}'\right)^2 e^{-\frac{1}{2} x_i\sigma^{-1}_{ij} x_j}}{4 \sqrt{\pi } \left(\sigma _{11} \sigma _{22}-\sigma _{12} \sigma _{21}\right)^{5/2}} \end{align} et al. where we set \begin{align*} \sigma =\begin{bmatrix} \sigma_{11} & \sigma_{12} \\ \sigma_{21} & \sigma_{22}. \end{bmatrix} \end{align*} use $\partial_t x=0, \partial_x x=1,\partial_x (\sigma^{-1})_{11}=0,\partial_p x=0$ and simpliy above expression, have \begin{align} \begin{split} \frac{\partial}{\partial x}\frac{1}{\sqrt{\pi \text{det}(\sigma)}}\exp\left[-\frac{1}{2}x_1(\sigma^{-1})_{11}x_1\right] = &\frac{-\exp\left[-\frac{1}{2} x_1(\sigma^{-1})_{11} x_1\right] x_1\sigma^{-1}_{11} }{\sqrt{\pi }\sqrt{\sigma_{11} \sigma_{22}-\sigma_{12} \sigma_{21}}}\\ \frac{\partial}{\partial x}\frac{1}{\sqrt{\pi \text{det}(\sigma)}}\exp\left[-\frac{1}{2}x_2(\sigma^{-1})_{22}x_2\right] =&0\\ \frac{\partial}{\partial x}\frac{1}{\sqrt{\pi \text{det}(\sigma)}}\exp\left[-\frac{1}{2}x_1(\sigma^{-1})_{12}x_2\right] =&\frac{ -\frac{1}{2}\exp\left[-\frac{1}{2} x_1(\sigma^{-1})_{12} x_2\right](\sigma^{-1})_{12} x_2 }{\sqrt{\pi }\sqrt{\sigma_{11} \sigma_{22}-\sigma_{12} \sigma_{21}}}\\ \frac{\partial}{\partial p}\frac{1}{\sqrt{\pi \text{det}(\sigma)}}\exp\left[-\frac{1}{2}x_1(\sigma^{-1})_{11}x_1\right]=&0\\ \frac{\partial}{\partial p}\frac{1}{\sqrt{\pi \text{det}(\sigma)}}\exp\left[-\frac{1}{2}x_2(\sigma^{-1})_{22}x_2\right]=& \frac{-\exp\left[-\frac{1}{2} x_2(\sigma^{-1})_{22} x_2\right](\sigma^{-1})_{22} x_2 }{\sqrt{\pi }\sqrt{\sigma_{11} \sigma_{22}-\sigma_{12} \sigma_{21}}}\\ \frac{\partial}{\partial p}\frac{1}{\sqrt{\pi \text{det}(\sigma)}}\exp\left[-\frac{1}{2}x_1(\sigma^{-1})_{12}x_2\right]=& \frac{-\frac{1}{2}\exp\left[-\frac{1}{2} x_1(\sigma^{-1})_{12} x_2\right]x_1 (\sigma^{-1})_{12}}{\sqrt{\pi }\sqrt{\sigma_{11} \sigma_{22}-\sigma_{12} \sigma_{21}}}\\ \frac{\partial}{\partial x \partial x}\frac{1}{\sqrt{\pi \text{det}(\sigma)}}\exp\left[-\frac{1}{2}x_1(\sigma^{-1})_{11}x_1\right] = &\frac{\exp\left[-\frac12 x_1\sigma^{-1}_{11} x_1\right] \left(- \sigma^{-1}_{11} x_1 \right)^2}{\sqrt{\pi }\sqrt{\sigma_{11}\sigma_{22}-\sigma_{12}\sigma_{21}}} + \frac{\exp\left[-\frac{1}{2} x_1 \sigma^{-1}_{11} x_1\right] \left( -\sigma^{-1}_{11}\right)}{\sqrt{\pi } \sqrt{\sigma _{11}\sigma_{22} - \sigma_{12} \sigma _{21}}}\\ \frac{\partial}{\partial x \partial x}\frac{1}{\sqrt{\pi \text{det}(\sigma)}}\exp\left[-\frac{1}{2}x_2(\sigma^{-1})_{22}x_2\right] = & 0\\ \frac{\partial}{\partial x \partial x}\frac{1}{\sqrt{\pi \text{det}(\sigma)}}\exp\left[-\frac{1}{2}x_1(\sigma^{-1})_{12}x_2\right] = &\frac{\exp\left[-\frac12 x_1\sigma^{-1}_{12}x_2\right] \left(- \frac12 \sigma^{-1}_{12}x_2\right)^2}{\sqrt{\pi }\sqrt{\sigma_{11}\sigma_{22}-\sigma_{12}\sigma_{21}}}\\ \frac{\partial}{\partial p \partial p}\frac{1}{\sqrt{\pi \text{det}(\sigma)}}\exp\left[-\frac{1}{2}x_1(\sigma^{-1})_{11}x_1\right]= & 0\\ \frac{\partial}{\partial p \partial p}\frac{1}{\sqrt{\pi \text{det}(\sigma)}}\exp\left[-\frac{1}{2}x_2(\sigma^{-1})_{22}x_2\right]= & \frac{\exp\left[-\frac12 x_2\sigma^{-1}_{22} x_2\right] \left( - \sigma^{-1}_{22} x_2 \right)^2}{\sqrt{\pi }\sqrt{\sigma_{11}\sigma_{22}-\sigma_{12}\sigma_{21}}} + \frac{\exp\left[-\frac{1}{2} x_2 \sigma^{-1}_{22} x_2\right] \left( -\sigma^{-1}_{22}\right)}{\sqrt{\pi } \sqrt{\sigma _{11}\sigma_{22} - \sigma_{12} \sigma _{21}}}\\ \frac{\partial}{\partial p \partial p}\frac{1}{\sqrt{\pi \text{det}(\sigma)}}\exp\left[-\frac{1}{2}x_1(\sigma^{-1})_{12}x_2\right]= & \frac{\exp\left[-\frac12 x_1\sigma^{-1}_{12}x_2\right] \left(-\frac12 x_1 \sigma^{-1}_{12}\right)^2}{\sqrt{\pi }\sqrt{\sigma_{11}\sigma_{22}-\sigma_{12}\sigma_{21}}} \end{split} \end{align} Last, insert them back into the F-P equation and spilt equation into $\partial_t \sigma_{11},\partial_t \sigma_{22},\partial_t \sigma_{12}$, we have \begin{align} \begin{split} &\frac{\partial }{\partial t}(x,p)\frac{1}{\sqrt{\pi \text{det}(\sigma)}}e^{-\frac{1}{2}x_1(\sigma^{-1})_{11}x_1} = (-\omega_0 p + \kappa x)\frac{\partial }{\partial x}\frac{1}{\sqrt{\pi \text{det}(\sigma)}}e^{-\frac{1}{2}x_1(\sigma^{-1})_{11}x_1}\\ &+(\omega_0 x - \omega_0 X g^2 x + \kappa p)\frac{\partial }{\partial p}\frac{1}{\sqrt{\pi \text{det}(\sigma)}}e^{-\frac{1}{2}x_1(\sigma^{-1})_{11}x_1}\\ &+ 2 \kappa \frac{1}{\sqrt{\pi \text{det}(\sigma)}}e^{-\frac{1}{2}x_1(\sigma^{-1})_{11}x_1}\\ & + \frac12\kappa\frac{\partial }{\partial x\partial x}\frac{1}{\sqrt{\pi \text{det}(\sigma)}}e^{-\frac{1}{2}x_1(\sigma^{-1})_{11}x_1}\\ & + (\frac12\kappa + \frac12 Xg^2\frac{\Gamma \omega_0}{\Omega} ) \frac{\partial }{\partial p\partial p}\frac{1}{\sqrt{\pi \text{det}(\sigma)}}e^{-\frac{1}{2}x_1(\sigma^{-1})_{11}x_1}\\ \Rightarrow&\\ &\frac{e^{-\frac{1}{2} x_i(t)(\sigma(t)^{-1})_{ij} x_j(t)} \left( -\frac{1}{2}x_1(t) (\sigma(t)^{-1})_{11}'x_1 (t) \right) }{\sqrt{\pi }\sqrt{\sigma_{11}(t) \sigma_{22}(t)-\sigma_{12}(t) \sigma_{21}(t)}}\\ &-\frac{e^{-\frac{1}{2} x_1(t)(\sigma(t)^{-1})_{11} x_1(t)} \left( \sigma_{11}'(t) \sigma_{22}(t) + \sigma _{11}(t) \sigma_{22}'(t) - \sigma_{12}'(t) \sigma_{21}(t) - \sigma _{12}(t) \sigma_{21}'(t) \right) }{2 \sqrt{\pi }\left(\sigma_{11}(t) \sigma_{22}(t)-\sigma_{12}(t) \sigma_{21}(t)\right)^{3/2}}\\ = &(-\omega_0 p + \kappa x)\frac{-e^{-\frac{1}{2} x_1(\sigma^{-1})_{11} x_1} x_1\sigma^{-1}_{11} }{\sqrt{\pi }\sqrt{\sigma_{11} \sigma_{22}-\sigma_{12} \sigma_{21}}}\\ &+(\omega_0 x - \omega_0 X g^2 x + \kappa p)0\\ &+ 2 \kappa \frac{1}{\sqrt{\pi }\sqrt{\sigma_{11} \sigma_{22}-\sigma_{12} \sigma_{21}}}e^{-\frac{1}{2}x_1(\sigma^{-1})_{11}x_1}\\ & + \frac12\kappa\left(\frac{e^{-\frac12 x_1\sigma^{-1}_{11} x_1} \left(- \sigma^{-1}_{11} x_1 \right)^2}{\sqrt{\pi }\sqrt{\sigma_{11}\sigma_{22}-\sigma_{12}\sigma_{21}}} + \frac{e^{-\frac{1}{2} x_1 \sigma^{-1}_{11} x_1} \left( -\sigma^{-1}_{11}\right)}{\sqrt{\pi } \sqrt{\sigma _{11}\sigma_{22} - \sigma_{12} \sigma _{21}}}\right) \end{split} \end{align} simplify, have \begin{align} \partial_t\sigma(t) = -2(-\omega_0 p + \kappa x)/x_1\sigma_{11} + 4 \kappa\sigma_{11}^2/x_1^2 + \kappa -\kappa /x_1\sigma_{11} \end{align} we obtained the result of $\partial_t \sigma_{11}$ is different the result from covariance matrix equation \begin{align} \begin{split} \partial_t \sigma_{11} =& \omega_0\sigma_{21} + \omega_0\sigma_{12} - 2\kappa(\sigma_{11} - \frac12)\\ \partial_t \sigma_{12} =& \omega_0\sigma_{22} + \omega_0(Xg^2-1)\sigma_{11} - 2\kappa\sigma_{12}\\ \partial_t \sigma_{21} =& \omega_0\sigma_{22} + \omega_0(Xg^2-1)\sigma_{11} - 2\kappa\sigma_{21}\\ \partial_t \sigma_{22} =& \omega_0(Xg^2-1)\sigma_{12} + \omega_0(Xg^2-1)\sigma_{21} - 2\kappa(\sigma_{22} -\frac12 - \frac12Xg^2\frac{\Gamma\omega_0}{\Omega\kappa}) \end{split} \end{align}
I don't know if I miscalculated or missed something, or if I was thinking in the wrong way. I have no idea for this equation, it's been confusing me for a long time.
Any little help would be good for me. Any hint? Thank you very much!
