application of the operator marked over them. The states $\lvert
{\widetilde{\Phi}}_{\Xi}(t_k^{\pm})\rangle$ are the left/right limits of ${\lvert {\widetilde{\Phi}}_{\Xi}(t)\rangle}$ at $t_k$.[]{data-label="fig:tPhi"}](illustration.pdf){width=".7\textwidth"}

In , the reason for introducing $\eta(t)$ is the following property: $$\label{eq:prob_no_mark}
{\mathbb{P}}(\text{no mark exists in } (s_1, s_2)) =
  \exp \big( -[\eta(s_2) - \eta(s_1)] \big),$$ which helps us to realize the marked point process. In our implementation, the drawing of the marked point process and the evolution of the trajectory are simultaneously done. In detail, one trajectory ${\lvert {\widetilde{\Phi}}_{\Xi}(t)\rangle}$ can be obtained by the following steps:

1.  Set $t \leftarrow 0$, $\tilde{\eta} \leftarrow 0$, $|{\widetilde{\Phi}}_{\Xi}(0) \rangle \leftarrow \ket{\Psi(0)}$, $\mathit{flag}
      \leftarrow \mathit{false}$. Generate a random number $Y$ obeying the uniform distribution in $[0,1]$.

2.  Stop if $t$ is large enough. Otherwise, select a time step $\Delta t$, and solve $\eta(t + \Delta t)$ according to .

3.  If $\exp(\tilde{\eta} - \eta(t + \Delta t)) \leqslant 1 - Y$, then solve the equation of the mark time $\tilde{t̃}$ $$\exp(\tilde{\eta} - \eta(\tilde{t})) = 1 - Y$$ by interpolation of $\eta(t)$ in $[t, t + \Delta t]$, and set $\Delta t
      \leftarrow \tilde{t} - t$, $\mathit{flag} \leftarrow \mathit{true}$, $\tilde{\eta} = \eta(\tilde{t})$.

4.  Solve the equation to get $|{\widetilde{\Phi}}_{\Xi}
      (t+\Delta t) \rangle$. Set $t 