# Jump Method and the Lindblad Equation

I am studying the time evolution of a density matrix using the Lindblad equation. My initial density matrix is $\rho(0)=|\alpha\rangle\langle\alpha|$, where $|\alpha\rangle$ is a coherent state. Then I have to compute $\mbox{Tr}\rho^2$. Due to the easy numerical computation, I can do the same with the Stochastic Schrodinger equation using the Jump method. From the literature I understand that, in order to find the time evolution I have to take many different initial conditions !!!!

My question is about choosing $|\psi(0)\rangle=|\alpha\rangle$ as a wavefunction which corresponds to the initial density matrix. To make the average over different trajectory how can I take a different initial condition. I think that my initial wavefunction always should be $|\psi(0)\rangle=|\alpha\rangle$ since I take $\rho(0)=|\alpha\rangle\langle\alpha|$ !! Can anybody clarify my doubt ?? Could you also tell me about the evaluation of the density matrix at any instant from the stochastic wavefunction in order to compute $\mbox{Tr}\rho^2$.

For each trajectory $i$, at each point in time $t$, you will have some specific realisation of the stochastic wavefunction represented by the ket $\lvert\psi_i(t)\rangle$. If you perform $N$ trajectories in total, the density matrix at time $t$ is given by $$\rho(t) = \frac{1}{N}\sum_{i=1}^N \frac{\lvert\psi_i(t)\rangle\langle\psi_i(t)\rvert}{\langle\psi_i(t)\rvert\psi_i(t)\rangle},$$ where I have included an explicit normalisation factor in the denominator.