When solving dissipative quantum dynamics using quantum trajectories, it is assumed that for averaging over a sufficiently high number of \(N\) trajectories, the full density matrix can be recovered as.

\begin{equation} \Op{\rho}_N(t) = \frac{1}{N} \sum_{i=1}^N \KetBra{\Psi_i(t)}{\Psi_i(t)} \label{eq:rho_N} \end{equation}

Any expectation value for an observable \(\Op{A}\) with respect to the full density matrix can be calculated as the average of the expectation values from the individual trajectories:

\begin{equation} \Avg{\Op{A}}_{\rho_N} = \frac{1}{N} \sum_{i=1}^N \Avg{\Op{A}}_i\,; \qquad \Avg{\Op{A}}_i \equiv \Avg{\Op{A}}_{\Psi_i} \end{equation}

The variance of \(\Op{A}\) must be calculated as

\begin{equation} \begin{split} \var(\Op{A})_{\rho_N} = \Avg{\Op{A}^2}_{\rho_N} - \Avg{\Op{A}}_{\rho_N}^2\,. \end{split} \end{equation}

Note that the variance for the ensemble is in general not the average of the variances of all trajectories, unless the expectation value in each trajectory is approximately equal. See this pdf file for the full details (tex).


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