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I have a density matrix $\rho$ in momentum representation at time $t=0$: \begin{equation} \langle p' |\rho(0) |p\rangle = \sum_{n=1}^{1000} p_n \Psi_n^*(p',0) \Psi_n(p,0) \end{equation}

resulting from a quantum jump calculation (the different $\Psi_n(p)$ are not orthogonal and have large overlaps)

I would like to calculate the time evolution of the position space distribution: $\langle x | \rho(t) |x \rangle$ with the free particle Hamiltonian $H = p^2/4$ without any source of decoherence. $x$ and $p$ are in units of $p_{\mathrm{zp}}$ and $x_\mathrm{zp}$, which are the zero point motions in some harmonic oscillator basis.

I can do that for each individual $\Psi_n(p)$ by calculating \begin{equation} \Psi_n(x,t) = \int dp e^{- \mathrm{i} p x/2} e^{- \mathrm{i} p^2/4} \Psi_n(p,0) \end{equation}

(ignoring pre factors)

and then calculating

\begin{equation} \langle x | \rho(t) |x \rangle = \sum_{n=1}^{1000} p_n |\Psi_n(x,t)|^2. \end{equation}

However, doing this takes an extremely long time with my computer as I have the $\Psi_n(p)$ only in numerical form on a discretized momentum space grid.

Is there a clever/much faster way of getting the numerical time evolution of $\langle x | \rho(t) |x \rangle$?

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  • $\begingroup$ How are the $\Psi_n(p)$ stored? $\endgroup$ – catalogue_number Jun 22 '20 at 13:03
  • $\begingroup$ As list in Wolfram Mathematica: {{$p_1$, $\Psi_n(p_1)$},{$p_2$, $\Psi_n(p_2)$},,..} $\endgroup$ – Luke Jun 22 '20 at 13:14
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I'm not sure why you've tagged the Wigner transform here - that's only useful for obtaining a quasiprobability x-p distribution.

I think you want the regular old Fourier transform, $$\Psi(x,t) = \frac{1}{2\pi} \int dp e^{ipx} \Psi(p,t)$$

As you probably know, the evolution of the (abstract) density operator is given by the Heisenberg equation, $$ i \hbar \frac{\partial \rho}{\partial t} = [H, \rho] $$. To work with this practically, we need to represent it in a basis - For the momentum space representation you have, that basis is essentially rectangular functions corresponding to the grid spacing $\Delta P$, $\phi_j(p) = \text{rect}_{\Delta P}(p-P_j)$ which are obviously orthogonal, but only approximately complete. (the normalisation is also a bit off, but that's just a prefactor) In this sense, you can now reinterpret a state $|\psi\rangle = \sum_j \psi_j |\phi_j\rangle$. Then the initial condition has the matrix representation $$ \rho_{ab}(t=0) = \sum_n w_n \Psi_n (p_a) \Psi^*_n(p_b)$$ for which the N^2-dimensional first-order ODE you must solve is $$ i \hbar \partial_t \rho_{ab}(t) = [\langle \phi_a | H | \phi_b \rangle, \rho_{ab}(t)]$$

Now, when it comes to actually answering your question - for grid spacing $\Delta P$, grid points P_b, I don't think there's anything faster than this (apart from maybe some trickery with the FFT...

$$ \langle x |\mathbb{1} \rho \mathbb{1}| x' \rangle = \int dp dq \langle x | p \rangle \langle p | \rho | q \rangle \langle q | x' \rangle \\ \approx \sum_a \sum_b \langle x | \phi_a \rangle \langle \phi_a | \rho | \phi_b \rangle \langle \phi_b | x' \rangle \\ = \sum_{ab} \rho_{ab}(t) \frac{e^{iP_a x-iP_b x'}}{\pi \Delta P} \text{sinc}(\Delta P x/2)^2 $$

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  • $\begingroup$ Thank you very much for your response! Is what you are saying equivalent with calculating the sum of my first equation on my momentum grid such that I get a large matrix $\rho_{ab}(t=0)$? I also do not quite understand how you derived the sinc function? $\endgroup$ – Luke Jun 22 '20 at 16:47
  • $\begingroup$ About your confusions: I tagged the Wigner transform because I thought it might be a way of solving my problem. The $\Psi_n$ are not orthogonal since they arise from different trajectories within the quantum trajectory formalism. My first equation is basically another way of writing: $\rho = \frac{1}{N} \sum_n^N \frac{|\Psi_n\rangle\langle \Psi_n |}{\langle \Psi_n |\Psi_n \rangle }$, where I average over many trajectories. $\endgroup$ – Luke Jun 22 '20 at 19:57
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    $\begingroup$ The sinc functions arose from $\langle x | \phi_a \rangle = \int dp \langle x | p \rangle \langle p | \phi_a \rangle = \frac{1}{2\pi} \int dp e^{ipx} \text{rect}_{\Delta P}(p-P_a)$ $\endgroup$ – catalogue_number Jun 23 '20 at 4:51

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