How to solve complex equations and plot the graph?

Question

I am trying to solve these equations and want to plot the value of $a^{dagger}a$ versus P0. I have tried this code but this gives me an error like "Conjunction::argr: Conjunction called with 1 argument; 2 arguments are expected. >>". If there is another method to solve this I don't know. If anybody can solve this will be appreciated.
delta = 4;
g0 = 1;
Pb = 2;
KL = 1;
kb = 0.2;
ome1 = 1;
gma = 0.02;
nth = 2;
Lf = Conjugate[a1]*b1;
Bf = a1*b1;
NL = Conjugate[a1]*a1;
sol = NSolve[{I*P0*(Conjugate[a1] - a1) + 
      I*Pb*(Lf - Conjugate[Lf] + Conjugate[Bf] - Bf) - KL*NL - 
      kb*(Lf*a1 + Conjugate[Lf]*Conjugate[a1] + Bf*Conjugate[a1] + 
         Conjugate[Bf]*
          a1 + (NL - 2*Abs[a1]^2)*(b1 + Conjugate[b1])) == 0, 
    I*g0*((NL - 2*Abs[a1]^2)*(Conjugate[b1] - b1) + 
         Conjugate[Lf]*Conjugate[a1] - Lf*a1 + Conjugate[Bf]*a1 - 
         Bf*Conjugate[a1]) + 
      I*Pb*(Conjugate[Bf] - Lf + Conjugate[Lf] - Bf) - 
      gma*Conjugate[b1]*b1 + gma*nth == 
     0, -I*delta*Lf - I*ome1*Lf + 
      I*g0*((2*NL - Abs[a1]^2 - Conjugate[b1]*b1 + 2*Abs[b1]^2 + 
            b1^2)*Conjugate[a1] - (Conjugate[Bf] + 2*Lf)*b1 - 
         Lf*Conjugate[b1]) - I*P0*b1 == 0, 
    I*delta*Bf - I*ome1*Bf + 
      I*g0*((2*NL - Abs[a1]^2 + Conjugate[b1]*b1 - 2*Abs[b1]^2)*
          a1 + (2*Bf - a1*b1 + Conjugate[Lf])*b1 + Bf*Conjugate[b1]) +
       I*P0*b1 + 
      I*Pb*(NL + Conjugate[b1]*b1 + a1^2 + b1^2 + 1) - (KL + gma)*
       Bf/2 - kb/
        2*((2*Bf - a1*b1 + Conjugate[Lf])*
          b1 + (Conjugate[b1]*b1 - 2*Abs[b1]^2)*a1 + 
         Bf*Conjugate[b1]) == 0, 
    I*delta*a1 + I*delta*a1 + I*g0*(Conjugate[Lf] + Bf) + I*P0 + 
      I*Pb*(b1 + Conjugate[b1]) - KL*a1/2 - 
      kb*(Conjugate[Lf] + Bf)/2 == 
     0, -I*ome1*b1 + I*g0*NL + I*Pb*(a1 + Conjugate[a1]) - gma*b1/2 ==
      0}, {a1, b1}];
Plot[{Evaluate[Conjugate[a1]*a1] /. sol}, {P0, 0, 30}]

Bob Hanlon · Answer

Clear["Global`*"]

Use exact values for the constants to facilitate simplification.
delta = 4;
g0 = 1;
Pb = 2;
KL = 1;
kb = 2/10;
ome1 = 1;
gma = 2/100;
nth = 2;
Lf = Conjugate[a1]*b1;
Bf = a1*b1;
NL = Conjugate[a1]*a1;

eqns = And @@ 
   Simplify[{I*P0*(Conjugate[a1] - a1) + 
       I*Pb*(Lf - Conjugate[Lf] + Conjugate[Bf] - Bf) - KL*NL - 
       kb*(Lf*a1 + Conjugate[Lf]*Conjugate[a1] + Bf*Conjugate[a1] + 
          Conjugate[Bf]*a1 + (NL - 2*Abs[a1]^2)*(b1 + Conjugate[b1])) == 0, 
     I*g0*((NL - 2*Abs[a1]^2)*(Conjugate[b1] - b1) + 
          Conjugate[Lf]*Conjugate[a1] - Lf*a1 + Conjugate[Bf]*a1 - 
          Bf*Conjugate[a1]) + I*Pb*(Conjugate[Bf] - Lf + Conjugate[Lf] - Bf) -
        gma*Conjugate[b1]*b1 + gma*nth == 
      0, -I*delta*Lf - I*ome1*Lf + 
       I*g0*((2*NL - Abs[a1]^2 - Conjugate[b1]*b1 + 2*Abs[b1]^2 + b1^2)*
           Conjugate[a1] - (Conjugate[Bf] + 2*Lf)*b1 - Lf*Conjugate[b1]) - 
       I*P0*b1 == 0, 
     I*delta*Bf - I*ome1*Bf + 
       I*g0*((2*NL - Abs[a1]^2 + Conjugate[b1]*b1 - 2*Abs[b1]^2)*
           a1 + (2*Bf - a1*b1 + Conjugate[Lf])*b1 + Bf*Conjugate[b1]) + 
       I*P0*b1 + 
       I*Pb*(NL + Conjugate[b1]*b1 + a1^2 + b1^2 + 1) - (KL + gma)*Bf/2 - 
       kb/2*((2*Bf - a1*b1 + Conjugate[Lf])*
           b1 + (Conjugate[b1]*b1 - 2*Abs[b1]^2)*a1 + Bf*Conjugate[b1]) == 0, 
     I*delta*a1 + I*delta*a1 + I*g0*(Conjugate[Lf] + Bf) + I*P0 + 
       I*Pb*(b1 + Conjugate[b1]) - KL*a1/2 - kb*(Conjugate[Lf] + Bf)/2 == 
      0, -I*ome1*b1 + I*g0*NL + I*Pb*(a1 + Conjugate[a1]) - gma*b1/2 == 0}];

Convert a1 and b1 to Cartesian form so that all unknowns are real.
eqns2 = Assuming[Element[{x1, y1, x2, y2, P0}, Reals],
  (eqns /. {a1 -> x1 + I*y1, b1 -> x2 + I*y2}) // 
   ComplexExpand // FullSimplify]

(* False *)

This indicates that the equations are inconsistent. Further, there are too many equations for the number of unknowns, so the system is overdetermined.

How to solve complex equations and plot the graph?

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