I have just started learning density matrix and quantum master equations, and I am given a problem set that asks to find the solution to the Lindblad equation with $H$, $L_+$, $L_-$, $L_z$, and $\rho(0)$ given in the matrix form.

The thing is, I know how to solve a system of linear differential equations such as Schrödinger equations to find a vector solution, but not a matrix equation with a matrix solution. Can anybody give me a hint or a link to the steps, because I think I am too dumb to find any, having searched the web and the library for about half the day.

  • 2
    $\begingroup$ The master equation is a system of linear, ordinary differential equations (ODEs) for the components of the density matrix (write them out explicitly!). If it helps, you could even collect these density matrix components into a vector... $\endgroup$ – Mark Mitchison Mar 20 '18 at 15:27
  • $\begingroup$ physics.stackexchange.com/q/115066 Im guessing what you mean by collect components into a vector is in the answer provided in the link above. Thanks anyways $\endgroup$ – Jin Mar 20 '18 at 15:34

We always solve the Lindblad form linear differential equations by numerical methods, such as fourth order Runge-Kutta method. If you want the steady state solution in analytic method, you can read this paper: PHYSICAL REVIEW A 92, 022116 (2015). I don't know if I solved your question.

  • $\begingroup$ $\uparrow$ Link? $\endgroup$ – Qmechanic May 18 '18 at 11:16

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