Time-Dependent Solution of Density Matrix Equations for 3-Level Λ System

This notebook calculates the time dependent solution for the 3 - level lambda sytem and two pulses

Program

solbd[d_,dp_,c1_,c2_,g2_,g12_,g21_,ti_,tf_,b_,c_]:=NDSolve[{r11'[t]==-I  c1 Conjugate[r12[t]] Conjugate[f[t]]+I c1  f[t] r12[t]+g21 r22[t],r12'[t]==I c1 Conjugate[f[t] ]r11[t]+I d r12[t]-g12 r12[t]+I c2 g[t] r13[t]-I c1 Conjugate[f[t] ]r22[t],r13'[t]==-I c1  Conjugate[f[t]]Conjugate[r32[t]]+I d r13[t]-I dp r13[t]+I c2 g[t] r12[t],r22'[t]==I c1 Conjugate[r12[t] ]Conjugate[f[t]]+I c2 Conjugate[r32[t]] g[t]-I c1 f[t] r12[t]-I c2 g[t] r32[t]-g2 r22[t],r32'[t]==I c2  g[t] r33[t]+I dp r32[t]+I c1  f[t] Conjugate[r13[t]]-I c2  g[t] r22[t]-g12 r32[t],r33'[t]==-I c2 Conjugate[r32[t]] g[t]+I c2  g[t] r32[t]+(g2-g21) r22[t],r11[ti]==b,r12[ti]==0,r13[ti]==0,r33[ti]==c,r32[ti]==0,r22[ti]==1-b-c},{r11,r12,r13,r22,r32,r33},{t,ti,tf}]

***read in pulse functions****

f[t]:=Exp[-(t+1)^2]

g[t]:=Exp[-(t)^2]

solbd[0,0,1,1,1,.5,.5,-8,8,1,0]

Plot[Evaluate[Abs[r33[t]]^2/. %],{t,-8,8},PlotRange->All]

STIRAP

f[t]:=Exp[-(t)^2]

g[t]:=Exp[-(t+1.5)^2]

solbd[0,0,10,10,1,.5,.5,-8,8,1,0]

Plot[Evaluate[Abs[r33[t]]^2/. %],{t,-8,8},PlotRange->All]

Plot[Evaluate[Abs[r22[t]]^2/. %%],{t,-8,8},PlotRange->All]

close to STIRAP adiabatic switching, very little population in state 2