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Then use NDSolveValue to solve the system. Variable solArray contains the nine numerical solutions. First element in solArray, $\texttt{solArray[[1]][t]}$ is the first function $\rho_{1,1}$ and so forth. I use {t,0,1} as an example plotting the real components of the solutions in one plot then a single plot : For example, if you want the value of say $\rho_{2,1}(1/2)$ then use solArray[[4]][1/2].

{p11, p12, p13, p21, p22, p23, p31, p32, p33} = 
  NDSolveValue[
   Join[theEqns, theInit], {Subscript[\[Rho], 1, 1], Subscript[\[Rho],
     1, 2], Subscript[\[Rho], 1, 3], Subscript[\[Rho], 2, 1], 
    Subscript[\[Rho], 2, 2], Subscript[\[Rho], 2, 3], 
    Subscript[\[Rho], 3, 1], Subscript[\[Rho], 3, 2], 
    Subscript[\[Rho], 3, 3]}, {t, 0, 1}];
Length@solArray
Plot[Re@p11[t], {t, 0, 1}]

Edit: Added code to generate 3D plots of selected solution as function of the parameters below

Then use NDSolveValue to solve the system. Variable solArray contains the nine numerical solutions. First element in solArray, $\texttt{solArray[[1]][t]}$ is the first function $\rho_{1,1}$ and so forth. I use {t,0,1} as an example plotting the real components of the solutions in one plot then a single plot : For example, if you want the value of say $\rho_{2,1}(1/2)$ then use solArray[4][1/2].

{p11, p12, p13, p21, p22, p23, p31, p32, p33} = 
  NDSolveValue[
   Join[theEqns, theInit], {Subscript[\[Rho], 1, 1], Subscript[\[Rho],
     1, 2], Subscript[\[Rho], 1, 3], Subscript[\[Rho], 2, 1], 
    Subscript[\[Rho], 2, 2], Subscript[\[Rho], 2, 3], 
    Subscript[\[Rho], 3, 1], Subscript[\[Rho], 3, 2], 
    Subscript[\[Rho], 3, 3]}, {t, 0, 1}];
Length@solArray
Plot[Re@p11[t], {t, 0, 1}]

Edit: Added code to write DE as a function of parameters

First write the system as a function of parameters. Now the 9 equations are functions of the parameters of $\texttt{theEqnsF}$:

ClearAll["Global`*"]
theEqns2[\[Gamma]_, \[CapitalGamma]_, \[CapitalOmega]Ra_, \
\[CapitalOmega]Rb_, \[CapitalDelta]a_, \[CapitalDelta]b_] := \
{{Derivative[1][Subscript[\[Rho], 1, 1]][
      t] == \[Gamma] (-Subscript[\[Rho], 1, 1][t]) + 
      1/2 \[CapitalGamma] Subscript[\[Rho], 3, 3][t] - 
      I (1/2 \[CapitalOmega]Ra Subscript[\[Rho], 1, 3][t] - 
         1/2 \[CapitalOmega]Ra Subscript[\[Rho], 3, 1][t]) + \[Gamma]/
       2}, {Derivative[1][Subscript[\[Rho], 1, 2]][
      t] == -(\[Gamma] Subscript[\[Rho], 1, 2][t]) - 
      I (\[CapitalDelta]a Subscript[\[Rho], 1, 2][
           t] - \[CapitalDelta]b Subscript[\[Rho], 1, 2][t] - 
         1/2 \[CapitalOmega]Ra Subscript[\[Rho], 3, 2][t] + 
         1/2 \[CapitalOmega]Rb Subscript[\[Rho], 1, 3][
           t])}, {Derivative[1][Subscript[\[Rho], 1, 3]][t] == 
     1/2 (-((\[Gamma] + \[CapitalGamma]) Subscript[\[Rho], 1, 3][
             t]) - \[Gamma] Subscript[\[Rho], 1, 3][t]) - 
      I (\[CapitalDelta]a Subscript[\[Rho], 1, 3][t] + 
         1/2 \[CapitalOmega]Ra Subscript[\[Rho], 1, 1][t] - 
         1/2 \[CapitalOmega]Ra Subscript[\[Rho], 3, 3][t] + 
         1/2 \[CapitalOmega]Rb Subscript[\[Rho], 1, 2][
           t])}, {Derivative[1][Subscript[\[Rho], 2, 1]][
      t] == -(\[Gamma] Subscript[\[Rho], 2, 1][t]) - 
      I (-(\[CapitalDelta]a Subscript[\[Rho], 2, 1][
             t]) + \[CapitalDelta]b Subscript[\[Rho], 2, 1][t] + 
         1/2 \[CapitalOmega]Ra Subscript[\[Rho], 2, 3][t] - 
         1/2 \[CapitalOmega]Rb Subscript[\[Rho], 3, 1][
           t])}, {Derivative[1][Subscript[\[Rho], 2, 2]][
      t] == \[Gamma] (-Subscript[\[Rho], 2, 2][t]) + 
      1/2 \[CapitalGamma] Subscript[\[Rho], 3, 3][t] - 
      I (1/2 \[CapitalOmega]Rb Subscript[\[Rho], 2, 3][t] - 
         1/2 \[CapitalOmega]Rb Subscript[\[Rho], 3, 2][t]) + \[Gamma]/
       2}, {Derivative[1][Subscript[\[Rho], 2, 3]][t] == 
     1/2 (-((\[Gamma] + \[CapitalGamma]) Subscript[\[Rho], 2, 3][
             t]) - \[Gamma] Subscript[\[Rho], 2, 3][t]) - 
      I (\[CapitalDelta]b Subscript[\[Rho], 2, 3][t] + 
         1/2 \[CapitalOmega]Ra Subscript[\[Rho], 2, 1][t] + 
         1/2 \[CapitalOmega]Rb Subscript[\[Rho], 2, 2][t] - 
         1/2 \[CapitalOmega]Rb Subscript[\[Rho], 3, 3][
           t])}, {Derivative[1][Subscript[\[Rho], 3, 1]][t] == 
     1/2 (-((\[Gamma] + \[CapitalGamma]) Subscript[\[Rho], 3, 1][
             t]) - \[Gamma] Subscript[\[Rho], 3, 1][t]) - 
      I (-(\[CapitalDelta]a Subscript[\[Rho], 3, 1][t]) - 
         1/2 \[CapitalOmega]Ra Subscript[\[Rho], 1, 1][t] + 
         1/2 \[CapitalOmega]Ra Subscript[\[Rho], 3, 3][t] - 
         1/2 \[CapitalOmega]Rb Subscript[\[Rho], 2, 1][
           t])}, {Derivative[1][Subscript[\[Rho], 3, 2]][t] == 
     1/2 (-((\[Gamma] + \[CapitalGamma]) Subscript[\[Rho], 3, 2][
             t]) - \[Gamma] Subscript[\[Rho], 3, 2][t]) - 
      I (-(\[CapitalDelta]b Subscript[\[Rho], 3, 2][t]) - 
         1/2 \[CapitalOmega]Ra Subscript[\[Rho], 1, 2][t] - 
         1/2 \[CapitalOmega]Rb Subscript[\[Rho], 2, 2][t] + 
         1/2 \[CapitalOmega]Rb Subscript[\[Rho], 3, 3][
           t])}, {Derivative[1][Subscript[\[Rho], 3, 3]][
      t] == -((\[Gamma] + \[CapitalGamma]) Subscript[\[Rho], 3, 3][
          t]) - I (-(1/2) (\[CapitalOmega]Ra Subscript[\[Rho], 1, 3][
             t]) + 1/2 \[CapitalOmega]Ra Subscript[\[Rho], 3, 1][t] - 
         1/2 \[CapitalOmega]Rb Subscript[\[Rho], 2, 3][t] + 
         1/2 \[CapitalOmega]Rb Subscript[\[Rho], 3, 2][t])}} // Flatten

theInit = {{Subscript[\[Rho], 1, 1][0] == 
     1}, {Subscript[\[Rho], 1, 2][0] == 
     0}, {Subscript[\[Rho], 1, 3][0] == 
     0}, {Subscript[\[Rho], 2, 1][0] == 
     0}, {Subscript[\[Rho], 2, 2][0] == 
     0}, {Subscript[\[Rho], 2, 3][0] == 
     0}, {Subscript[\[Rho], 3, 1][0] == 
     0}, {Subscript[\[Rho], 3, 2][0] == 
     0}, {Subscript[\[Rho], 3, 3][0] == 0}} // Flatten

Can now supply the system function $\texttt{theEqnsF}$ to NDSolveValue where now must supply values for the parameters:

   (* theEqns[\[Gamma],\[CapitalGamma],\[CapitalOmega]Ra,\[CapitalOmega]\
Rb,\[CapitalDelta]a,\[CapitalDelta]b] *)
{p11, p12, p13, p21, p22, p23, p31, p32, p33} = 
  NDSolveValue[
   Join[theEqnsF[0.1, 1, 1, 8, 1, 0.5], 
    theInit], {Subscript[\[Rho], 1, 1], Subscript[\[Rho], 1, 2], 
    Subscript[\[Rho], 1, 3], Subscript[\[Rho], 2, 1], 
    Subscript[\[Rho], 2, 2], Subscript[\[Rho], 2, 3], 
    Subscript[\[Rho], 3, 1], Subscript[\[Rho], 3, 2], 
    Subscript[\[Rho], 3, 3]}, {t, 0, 1}];
Plot[Re@p33[t], {t, 0, 1}]

Now generate a series of plots for $\texttt{p33}$ when $\Delta b$ varies from 0.1 to 5:

plotTable = Table[
   (* theEqns[\[Gamma],\[CapitalGamma],\[CapitalOmega]Ra,\
\[CapitalOmega]Rb,\[CapitalDelta]a,\[CapitalDelta]b] *)
   {p11, p12, p13, p21, p22, p23, p31, p32, p33} = 
    NDSolveValue[
     Join[theEqns[0.1, 1, 1, 8, 1, currentB], 
      theInit], {Subscript[\[Rho], 1, 1], Subscript[\[Rho], 1, 2], 
      Subscript[\[Rho], 1, 3], Subscript[\[Rho], 2, 1], 
      Subscript[\[Rho], 2, 2], Subscript[\[Rho], 2, 3], 
      Subscript[\[Rho], 3, 1], Subscript[\[Rho], 3, 2], 
      Subscript[\[Rho], 3, 3]}, {t, 0, 1}];
   ParametricPlot3D[{t, currentB, Re@p33[t]}, {t, 0, 1}],
   {currentB, 0.1, 5, 0.1}
   ];
selectedFunction = "p33";
Show[plotTable, BoxRatios -> {1, 1, 1}, PlotRange -> All, 
 AxesLabel -> {Style["t", 14], , Style["\[CapitalDelta]b", 14], 
   Style[selectedFunction, 14]}]

Finally generate a table of values for $\texttt{p33}$ in the form needed for the function ListPlot3D and generate a smooth function of $\texttt{p33}$ over the indicated range:

plotTable = Table[
   (* theEqns[\[Gamma],\[CapitalGamma],\[CapitalOmega]Ra,\
\[CapitalOmega]Rb,\[CapitalDelta]a,\[CapitalDelta]b] *)
   {p11, p12, p13, p21, p22, p23, p31, p32, p33} = 
    NDSolveValue[
     Join[theEqns[0.1, 1, 1, 8, 1, currentB], 
      theInit], {Subscript[\[Rho], 1, 1], Subscript[\[Rho], 1, 2], 
      Subscript[\[Rho], 1, 3], Subscript[\[Rho], 2, 1], 
      Subscript[\[Rho], 2, 2], Subscript[\[Rho], 2, 3], 
      Subscript[\[Rho], 3, 1], Subscript[\[Rho], 3, 2], 
      Subscript[\[Rho], 3, 3]}, {t, 0, 1}];
           if = p33;
   Table[{t, currentB, if[t]} // N, {t, 0, 1, 1/50}],
   {currentB, 0.1, 5, 0.1}
   ];

ListPlot3D[Flatten[plotTable, 1], BoxRatios -> {1, 1, 1}, 
 AxesLabel -> {Style["t", 14], , Style["\[CapitalDelta]b", 14], 
   Style[selectedFunction, 14]}]

Also, if the array of functions is a bit awkward to deal with, you can individually specify each function as the following and then use them separately (have to use different names on the left though):

{p11, p12, p13, p21, p22, p23, p31, p32, p33} = 
  NDSolveValue[
   Join[theEqns, theInit], {Subscript[\[Rho], 1, 1], Subscript[\[Rho],
     1, 2], Subscript[\[Rho], 1, 3], Subscript[\[Rho], 2, 1], 
    Subscript[\[Rho], 2, 2], Subscript[\[Rho], 2, 3], 
    Subscript[\[Rho], 3, 1], Subscript[\[Rho], 3, 2], 
    Subscript[\[Rho], 3, 3]}, {t, 0, 1}];
Length@solArray
Plot[Re@p11[t], {t, 0, 1}]

Then use NDSolveValue to solve the system. Variable solArray contains the nine numerical solutions. solArray[1][t] is the first function $\rho_{1,1}$ and so forth. I use {t,0,1} as an example plotting the real components of the solutions in one plot then a single plot : For example, if you want the value of say $\rho_{2,1}(1/2)$ then use solArray[[4]][1/2].

Then use NDSolveValue to solve the system. Variable solArray contains the nine numerical solutions. First element in solArray, $\texttt{solArray[[1]][t]}$ is the first function $\rho_{1,1}$ and so forth. I use {t,0,1} as an example plotting the real components of the solutions in one plot then a single plot : For example, if you want the value of say $\rho_{2,1}(1/2)$ then use solArray[[4]][1/2].