I want to use solution of coupled differential equation further in the code can you help me in this regard. Proper code is pasted below, The solution of differential equation named as n1, n2 ms, mc, m1, m2 are the elements of a matrix , so i want these solution directly in the matrix.

Clear["Global`*"]
\[Mu] = 1; \[Tau] = 0.45;
d1 = 1/(4 \[Mu]^2 (1 - 2 \[Tau])) + (1 - 2 \[Tau])/4; 
 d2 = 1/(4 \[Mu]^2 (1 - 2 \[Tau])) - (1 - 2 \[Tau])/4;

 \[Phi] = 0; \[Gamma] = 20; \[CapitalOmega] = 1000;
  \[Theta] = 0;

G = 101.7; 
\[Alpha]11 = 0; \[Alpha]22 = 0; \[Alpha]11b = 0; \[Alpha]22b = 0;
\[Alpha]12 = 2*I*G*\[Gamma]/\[CapitalOmega];
\[Alpha]21 = 2*I*G*\[Gamma]/\[CapitalOmega];
\[Alpha]12b = -2*I*G*\[Gamma]/\[CapitalOmega];
\[Alpha]21b = -2*I*G*\[Gamma]/\[CapitalOmega];

\[Kappa]1 = 1.5;
\[Kappa]2 = 1.5;
solA = DSolve[{n1'[
      t] == (1/2) (\[Alpha]11 + \[Alpha]11b - 4 \[Kappa]1)*
       n1[t] + (1/2) E^(I \[Phi]) \[Alpha]12b*
       ms[t] + (1/2) E^(-I \[Phi]) \[Alpha]12*mss[t] + 
      1/2*(\[Alpha]11 + \[Alpha]11b), 
    n2'[t] == (1/2) (\[Alpha]22 + \[Alpha]22b - 4 \[Kappa]2) n2[
        t] + (1/2) E^(I \[Phi]) \[Alpha]21*
       ms[t] + (1/2) E^(-I \[Phi]) \[Alpha]21b*mss[t] + 
      1/2*(\[Alpha]22 + \[Alpha]22b), 
    ms'[t] == (1/2) E^(-I \[Phi]) \[Alpha]21b*
       n1[t] + (1/2) E^(-I \[Phi]) \[Alpha]12*
       n2[t] + (1/
         2) (\[Alpha]11 + \[Alpha]22b - (2 (\[Kappa]1 + \[Kappa]2)))*
       ms[t] + (1/2) (\[Alpha]12 + \[Alpha]21b) E^(-I \[Phi]), 
    mss'[t] == (1/2) E^(I \[Phi]) \[Alpha]21*n1[t] + (1/2) E^(
       I \[Phi]) \[Alpha]12b*
       n2[t] + (1/
         2) (\[Alpha]11b + \[Alpha]22 - (2 (\[Kappa]1 + \[Kappa]2)))*
       mss[t] + (1/2) (\[Alpha]12b + \[Alpha]21) E^(I \[Phi]), 
    n1[0] == d1 - 0.5 , n2[0] == 0, ms[0] == 0, mss[0] == 0}, {n1, n2,
     ms, mss}, {t}];
solB = DSolve[{m1'[t] ==  (\[Alpha]11 - 2 \[Kappa]1)*m1[t] + 
      E^(-I \[Phi]) \[Alpha]12*mc[t], 
    m2'[t] == (\[Alpha]22 - 2 \[Kappa]2) m2[t] + 
      E^(I \[Phi]) \[Alpha]21*mc[t], 
    mc'[t] == 
     1/2 E^(I \[Phi]) \[Alpha]21*m1[t] + 
      1/2 E^(-I \[Phi]) \[Alpha]12*m2[t] + 
      1/2 (\[Alpha]11 + \[Alpha]22 - 2 (\[Kappa]1 + \[Kappa]2))*mc[t],
     m1[0] == -d2 , m2[0] == 0, mc[0] == 0}, {m1, m2, mc}, {t}];
Vt = ( {
    {n1 + 0.5, -m1, ms, -mc},
    {-Conjugate[m1], n1 + 0.5, -Conjugate[mc], Conjugate[ms]},
    {Conjugate[ms], -mc, n2 + 0.5, -m2},
    {-Conjugate[mc], ms, -Conjugate[m2], n2 + 0.5}
   } ); 
Vtq = Chop[(Vt + Vt\[Transpose])/2];
Aq = ( {
    {n1 + 0.5, -m1},
    {-Conjugate[m1], n1 + 0.5}
   } );
Bq = ( {
    {n2 + 0.5, -m2},
    {-Conjugate[m2], n2 + 0.5}
   } );
Cq = ( {
    {ms, -mc},
    {-Conjugate[mc], mss}
   } );
sigma = Det[Aq] + Det[Bq] - 2 Det[Cq];
\[Xi] = Abs[Evaluate[Sqrt[(sigma - Sqrt[sigma^2 - 4*Det[Vtq]])/2]]];
Eme = Max[0, -Log2[2 \[Xi]]];

lg = LineLegend[{Directive[Black, AbsoluteThickness[3]], 
    Directive[Blue, Dashed, AbsoluteThickness[3]], 
    Directive[Red, AbsoluteThickness[3]]}, {"\[CapitalOmega]=1000"}];
Plot[Eme, {t, 0, 2}, 
 FrameTicks -> {{{0, 0.03, 0.2, 0.4, 0.6, 0.8, 1, 1.2, 1.4, 1.6, 1.8, 
     2}, None}, {{{0, 0}, {0.5, 50}, {1, 100}, {1.5, 150}, {2, 200}}, 
    None}}, PlotStyle -> {{Black, Thickness -> .007}, {Blue, 
    Thickness -> 0.007, Dashed}, {Red, Thickness -> 0.007}}, 
 Frame -> True, FrameTicksStyle -> 14, Axes -> False, 
 GridLines -> Automatic, GridLinesStyle -> Directive[Orange, Dashed], 
 PlotRange -> All, 
 PlotLegends -> Placed[lg, {Scaled[{0.67, 0.8}], {0, 0.2}}], 
 FrameLabel -> {Style["G t", FontSize -> 18, Bold], 
   Style["\!\(\*SubscriptBox[\(E\), \(N\)]\)", FontSize -> 18, Bold]}]

With correct syntax:

nl = Max[0, -(1/Log[2]) Log[
     Sqrt[2] \[Sqrt](0.5 - 6.12563 E^(-6. t) Cos[2.034 t]^4 + 
         0.25 (E^(-3. t)) (4.05 - 4.05 Cos[4.068 t]) + 
         0.0625 (E^(-3. t)) (4.05 - 4.05 Cos[4.068 t])^2 + 
         0.25 (E^(-3. t)) (4.05 + 4.05 Cos[4.068 t]) + 
         0.0625 (E^(-3. t)) (4.05 + 4.05 Cos[4.068 t])^2 - 
         6.12563 E^(-6. t) Sin[2.034 t]^4 - (4.07531 + 
            0. I) E^(-6. t) Sin[
            4.068 t]^2 - \[Sqrt](-4. E^(-12. t) (0.0625 E^(6. t) (2. \
+ (E^(-3. t)) (4.05 - 4.05 Cos[4.068 t]))^2 (-6.12563 Cos[
                    2.034 t]^4 + 
                  0.0625 E^(6. t) (2. + (E^(-3. t)) (4.05 + 
                    4.05 Cos[4.068 t]))^2) - 
               6.12563 (-6.12563 Cos[2.034 t]^4 + 
                  0.0625 E^(6. t) (2. + (E^(-3. t)) (4.05 + 
                    4.05 Cos[4.068 t]))^2) Sin[2.034 t]^4) + (0.5 - 
               6.12563 E^(-6. t) Cos[2.034 t]^4 + 
               0.25 (E^(-3. t)) (4.05 - 4.05 Cos[4.068 t]) + 
               0.0625 (E^(-3. t)) (4.05 - 4.05 Cos[4.068 t])^2 + 
               0.25 (E^(-3. t)) (4.05 + 4.05 Cos[4.068 t]) + 
               0.0625 (E^(-3. t)) (4.05 + 4.05 Cos[4.068 t])^2 - 
               6.12563 E^(-6. t) Sin[2.034 t]^4 - (4.07531 + 
                  0. I) E^(-6. t) Sin[4.068 t]^2)^2))]];
Plot[nl, {t, 0, 2}]

Works fine.