diff options
| -rw-r--r-- | xmds2/reports/4wm_with_perturbation_and_exact_analusis/4wm_with_perturbation_and_exact_analusis.tex | 152 |
1 files changed, 76 insertions, 76 deletions
diff --git a/xmds2/reports/4wm_with_perturbation_and_exact_analusis/4wm_with_perturbation_and_exact_analusis.tex b/xmds2/reports/4wm_with_perturbation_and_exact_analusis/4wm_with_perturbation_and_exact_analusis.tex index cb537a8..b30c0aa 100644 --- a/xmds2/reports/4wm_with_perturbation_and_exact_analusis/4wm_with_perturbation_and_exact_analusis.tex +++ b/xmds2/reports/4wm_with_perturbation_and_exact_analusis/4wm_with_perturbation_and_exact_analusis.tex @@ -15,106 +15,106 @@ approach} We consider the following system \begin{verbatim} - -------- |4> - \ - \ E3 -------- |3> - \ / \ - \ E2 / \ - \ / \ E1 - ------- |2> \ - \ - ------- |1> + --------------- |4> + \ \ + \ E3 \ -------- |3> + \ E4 \ / \ + \ \ / E2 \ + \ / \ E1 + |2> -------------- \ + \ \ + \ \ + ------------- |1> \end{verbatim} -Such system can exhibit slow and fast light behavior for the cases when -field 2 connecting $|$2$>$-$|$3$>$$|$is weak and treated as a probe field, field 1 -is strong and field 3 serve as a switch toggling slow (field 3 is off), -fast (field 3 is on and strong) light, and situations in between. +Our previous study showed that fast light condition can be achieved in +four-wave mixing condition when two of the fields $E_1$ and $E_3$ are +reasonably strong, in comparison to $E_2$ and $E_4$. -Below the examples of fast and slow propagation for the following set of parameters: +So we decided to check exact analysis vs the case of the perturbative +approach where we consider $E_2$ and $E_4$ to be too small to affect +populations and coherences governed by strong field only. Those are +$\rho_{11}, \rho_{22}, \rho_{33}, \rho_{13}, \rho_{24},$ and $\rho_{44}$. +Four simplicity we treat those as time and z-position independent. +Remaining density coefficient ($\rho_{12}, \rho_{14}, \rho_{23}$, and +$\rho_{34}$) depend on strength of $E_2$ and $E_4$ and had to be treated +fully with their fields, time and z dependence via PDE solver. Fields are +also treated fully as z and t dependent but clearly setting above +$\rho_{11}, \rho_{22}, \rho_{33}, \rho_{13}, \rho_{24},$ and $\rho_{44}$ as +constants at the beginning of the cell limits the precision of this +method. One might improve it by treating $\rho_{11}, \rho_{22}, \rho_{33}, +\rho_{13}, \rho_{24},$ and $\rho_{44}$ as function of z, $E_1$ and $E_3$ by +recalculating this dependence before running the rest of the solver. Since +time dependence is gone it will be very small computational burden. + + +Despite the simplicity the perturbative model describes the propagation +quite well. + +Below the examples of fast and slow propagation for the following common set of parameters: \begin{verbatim} -G3=3.4e7 // decay rate of level |3> in 1/s -G4=3.8e7 // decay rate of level |4> in 1/s -gt=3.1416e+04 // repopulation rate in 1/s +G3=1.89e7 // decay rate of level |3> in 1/s +G4=1.89e7 // decay rate of level |4> in 1/s +gt=6.28e4 // repopulation rate in 1/s z=1.5e-2 // cell length in m -N=1e10*(1e6); // number of particles per cubic m +N=1e9*(1e6); // number of particles per cubic m delta1=0; // field 1 one photon detuning delta2=0; // field 2 --------//--------- delta3=0; // field 3 --------//--------- +// Detuning of the fourth field $E_4$ +// is set by four-wave mixing condition. \end{verbatim} -\begin{figure}[h] - \begin{center} - \includegraphics[width=1.00\columnwidth]{slow_light/fields_propagation.pdf} - \end{center} - \caption{Propagation of the fields under slow light condition} - \label{fig:3fields_slow} -\end{figure} - -Slow light behavior is observed for the 1 $\backslash$mu S long Gaussian pulse and -the following fields parameters +\subsection{Slow light propagation} +To ensure uniform starting conditions scripts for exact and perturbative +approach are ran with the same command line parameters which set detunings +and field Rabi frequencies (in 1/s): \begin{verbatim} -E1=7.5e7; // field 1 Rabi frequency in 1/s -E2=1; // field 2 --------//----------- -E3=0; // field 3 --------//----------- +--delta1=0 --delta2=0 --delta3=0 \ +--E1o=1.9e7 --E2o=3.1e5 --E3o=0 --E4o=0 \end{verbatim} -fields 1 and 3 have no time dependence at the input of the cell. - -Propagation of all 3 fields is depicted at Fig.\ref{fig:3fields_slow} -and output beam fields profiles in comparison with input are shown at -Fig.\ref{fig:3fields_in_out_slow} -where it easy to see delayed pulse for the field 2 in the middle. +Notice that in this case fields $E_3$ and $E_4$ are zero since they are +governed by parameters E2o and E3o. +Results are shown in figure\ref{fig:3fields_slow}. \begin{figure}[h] - \begin{center} - \includegraphics[width=1.00\columnwidth]{slow_light/fields_before_after_cell.pdf} - \end{center} - \caption{Propagation of the fields under slow light condition} - \label{fig:3fields_in_out_slow} + \includegraphics[width=1.00\columnwidth]{slow_light_compared/fields_after_the_cell} + \caption{Slow light condition comparison. Fields are drawn at the + end of the cell. + \label{fig:3fields_slow} + } \end{figure} - -Fast light or negative group delay is observed for the similar conditions -but with field 3 switched on +\subsection{Fast light propagation} +To ensure uniform starting conditions scripts for exact and perturbative +approach are ran with the same command line parameters which set detunings +and field Rabi frequencies (in 1/s): \begin{verbatim} -E1=7.5e7; // field 1 Rabi frequency in 1/s -E2=1; // field 2 --------//----------- -E3=1.5e8; // field 3 --------//----------- +--delta1=0 --delta2=0 --delta3=0 \ +--E1o=1.9e7 --E2o=3.1e5 --E3o=3.8e7 --E4o=6.3e4 \end{verbatim} - -Propagation of all 3 fields is depicted at Fig.\ref{fig:3fields_fast} -and output beam fields profiles in comparison with input are shown at -Fig.\ref{fig:3fields_in_out_fast}. - +Results are shown in figure\ref{fig:3fields_fast}. \begin{figure}[h] - \begin{center} - \includegraphics[width=1.00\columnwidth]{fast_light/fields_propagation.pdf} - \end{center} - \caption{Propagation of the fields under fast light condition} + \includegraphics[width=1.00\columnwidth]{fast_light_compared/fields_after_the_cell} + \caption{Fast light condition comparison. Fields are drawn at the + end of the cell. \label{fig:3fields_fast} + } \end{figure} +Notice settling behavior for first half a micro second since our numerical +solver use initial conditions in assumption that strong fields are not +absorbed and precalculates only density elements for the ones affected by +the strong fields. -\begin{figure}[h] - \begin{center} - \includegraphics[width=1.00\columnwidth]{fast_light/fields_before_after_cell.pdf} - \end{center} - \caption{Propagation of the fields under fast light condition} - \label{fig:3fields_in_out_fast} -\end{figure} - -The advancement of the pulse is very small but it could be seen on zoomed -in Fig.\ref{fig:field2_in_out_fast}. +\section{Execution speed} +The perturbation approach is about 40\% faster since there is no need to +calculate full blown propagation equation for 6 density elements out of +required 10 (recall that we still propagate 4 fields in addition to this). +So at least 1000x200 (time x z) grid takes 0.6 seconds vs 1 second. There +are more samples along z but they are governed by adaptive solver and +hidden from user. -\begin{figure}[h] - \begin{center} - \includegraphics[width=1.00\columnwidth]{fast_light/probe_before_after_cell.pdf} - \end{center} - \caption{Propagation of the fields under fast light condition} - \label{fig:field2_in_out_fast} -\end{figure} -% LaTeX2e code generated by txt2tags 2.5 (http://txt2tags.sf.net) -% cmdline: txt2tags --target=tex report.t2t \end{document} |