perturbative report draft finalized

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}