Unable to solve coupled equations

Question

I need to solve these coupled equations and plot the results as a function of CapitalGamma I tried to simplify all equations by using "Rationalize", as I found here for a similar problem, but I got nothing after long time of running. Could you please tell me what is the problem?!

s = Solve[{-0.264*^-6 \[Rho]12 - 
   I \[CapitalGamma] \[Rho]12 - (0.784615 + 
      13.983590 I) (-\[Rho]11 + \[Rho]22) - (4.511891*^-7 + 
      7.999390*^-6 I) \[Rho]12 (-\[Rho]11 + \[Rho]22) + (0.041176 \+ 0.729593 I) \[Rho]32 - (1.252607*^-7 + 
      2.219759*^-6 I) \[Rho]13 \[Rho]32 == 
  0 && -0.264*^-6 \[Rho]21 + 
   I \[CapitalGamma] \[Rho]21 - (0.784615 - 
      13.983590 I) (-\[Rho]11 + \[Rho]22) + (4.511891*^-7 + 
      7.999390*^-6 I) \[Rho]21 (-\[Rho]11 + \[Rho]22) + (0.041176 \- 0.729593 I) \[Rho]23 - (1.252607*^-7 - 
      2.219759*^-6 I) \[Rho]23 \[Rho]31 == 
  0 && -3.036*^-7 \[Rho]13 - (0.784615 + 
      13.983590 I) \[Rho]23 - (4.511891*^-7 + 
      7.999390*^-6 I) \[Rho]12 \[Rho]23 + (0.041176 + 
      0.729593 I) (-\[Rho]11 + \[Rho]33) - (1.252607*^-7 + 
      2.219759*^-6 I) \[Rho]13 (-\[Rho]11 + \[Rho]33) == 
  0 && -3.036*^-7 \[Rho]31 - (0.784615 - 
      13.983590 I) \[Rho]32 - (4.511891*^-7 - 
      7.999390*^-6 I) \[Rho]21 \[Rho]32 + (0.041176 - 
      0.729593 I) (-\[Rho]11 + \[Rho]33) - (1.252607*^-7 - 
      2.219759*^-6 I) \[Rho]31 (-\[Rho]11 + \[Rho]33) == 
  0 && (-0.0411768 + 0.729593 I) \[Rho]12 + (0.784615 + 
      13.983590 I) \[Rho]31 + (5.764498*^-7 + 
      5.779630*^-6 I) \[Rho]12 \[Rho]31 + (-3.96*^-8 - 
      I \[CapitalGamma]) \[Rho]32 == 
  0 && (0.784615 - 13.983590 I) \[Rho]13 - (0.041176 + 
      0.729593 I) \[Rho]21 + (5.764498*^-7 - 
      5.779630*^-6 I) \[Rho]13 \[Rho]21 + (-3.96*^-8 + 
      I \[CapitalGamma]) \[Rho]23 == 
  0 && -0.0528*^-5 \[Rho]11 - (0.784615 - 
      13.983590 I) \[Rho]12 + (0.0411768 - 
      0.729593 I) \[Rho]13 - (0.784615 + 13.983590 I) \[Rho]21 - 
   9.023782*^-7 \[Rho]12 \[Rho]21 + (0.0411768 + 
      0.729593 I) \[Rho]31 - (5.764498*^-7 + 
      5.779630*^-6 I) \[Rho]13 \[Rho]31 == 
  0 && -0.264*^-6 \[Rho]11 + (0.7846153 - 
      13.983590 I) \[Rho]12 + (0.7846153 + 
      13.983590 I) \[Rho]21 + 9.0237826*^-7 \[Rho]12 \[Rho]21 + 
   0.792*^-7 \[Rho]33 == 0 && (-0.041176 + 0.729593 I) \[Rho]13 - (0.0411768 + 0.7295932 I) \[Rho]31 + 2.505214*^-7 \[Rho]13 \[Rho]31 + 
   6.6*^-16 (400000000 \[Rho]11 - 120000000 \[Rho]33) == 
  0}, {\[Rho]12, \[Rho]13, \[Rho]31, \[Rho]21, \[Rho]11, \\[Rho]22, \[Rho]33, \[Rho]23, \[Rho]32}];Plot[\[Rho]12 /. s[[1]], {\[CapitalGamma], 0, 0.1}]

Answer 1

If Solve can't do the job or it takes to long, do it with NSolve, generate a list for certain values of Gamma and print it with ListLinePlot

First I rationalized the equations and a defined a solution-function with NSolve

rat[\[CapitalGamma]_] = Rationalize["equations", 0]

nsol[\[CapitalGamma]_] := 
   NSolve[rat[\[CapitalGamma]], {\[Rho]12, \[Rho]13, \[Rho]31, \
     \[Rho]21, \[Rho]11, \[Rho]22, \[Rho]33, \[Rho]23, \[Rho]32}, 
       WorkingPrecision -> 30]

Generate a table of solutions for Gamma from 1/200 to 1/10 (I ignored Gamma = 0, because it has a very diffent behavior.

tab1 = Table[{\[CapitalGamma] -> Gamma, nsol[Gamma]}, {Gamma, 1/200, 
1/10, 1/200}];

Pick out the corresponding values for the 18 solutions and print it (here done for [Rho]12, do the same for the other rho)

Table[{ListLinePlot[
   ta = Table[{\[CapitalGamma] /. tab1[[i, 1]], 
  Re[\[Rho]12 /. tab1[[i, 2, j]]]}, {i, 1, 20}], 
    Epilog -> Point[ta], PlotLabel -> Re, 
    AxesLabel -> {Gamma, \[Rho]12}], 
      ListLinePlot[
  ta = Table[{\[CapitalGamma] /. tab1[[i, 1]], 
  Im[\[Rho]12 /. tab1[[i, 2, j]]]}, {i, 1, 20}], 
    Epilog -> Point[ta], PlotLabel -> Im, 
    AxesLabel -> {Gamma, \[Rho]12}]}, {j, 1, 18}]