1
$\begingroup$

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 $\mbox{Tr}\rho^2$.

$\endgroup$
2
$\begingroup$

You only need to average over initial conditions if your initial state is mixed. Since your initial state is pure this is not necessary.

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.

$\endgroup$
  • $\begingroup$ You're welcome, I hope you find something cool in your numerical experiments. $\endgroup$ – Mark Mitchison Jun 3 '14 at 17:43
  • $\begingroup$ I am just trying to find the bipartite entanglement in a dissipative system, hoping that I will get something interesting . By the way, I have noticed that you are the student of M. B. Plenio. Actually I am following some of his articles !!! $\endgroup$ – Sijo Joseph Jun 3 '14 at 18:28

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.