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I'm writing a Runge-Kutta algorithm for solving a system of coupled differential equations. My code works fine when defined as a module, but when I try to compile it, I get the following error:

CompiledFunction::cfte: Compiled expression {{0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.},{-0.00157061-0.49998 I,-0.00141364+5.92154*10^-6 I,1.25575*10^-8+2.66469*10^-6 I,3.76726*10^-9+0. I,0. +0. I,0. +0. I,0. +0. I,0. +0. I,0. +0. I,0. +0. I,0. +0. I,0. +0. I,0. +0. I,0. +0. I,0. +0. I,0. +0. I,0. +0. I,0. +0. I,0. +0. I,0. +0. I},{0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.},<<45>>,{0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.},{0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.},<<951>>} should be a rank 2 tensor of machine-size real numbers.
CompiledFunction::cfexe: Could not complete external evaluation; proceeding with uncompiled evaluation.

I would appreciate help identifying where I'm going wrong in trying to compile the code. I specify that the output should be complex, but I still get the error and can't get it to compile. Here is my code:

(*Define Parameters*)
Nsim = 20;
\[Epsilon] = N[50.0];
g = N[0.4];
\[Omega]c = N[2 \[Pi] * 7.6];
\[Omega]d = N[2 \[Pi]*7.5];
\[Kappa] = N[2 \[Pi]*1.6/1000];
J = N[2 \[Pi]*0.09];
\[CapitalOmega] = N[2 \[Pi]*13];
\[CapitalGamma] = N[2 \[Pi]*10/1000];
AMat = SparseArray[{{1, 1} -> \[Omega]c - \[Omega]d - I \[Kappa]/2, {Nsim, Nsim} -> \[Omega]c - \[Omega]d - I \[Kappa]/2, Band[{1, 1}] -> \[Omega]c - \[Omega]d - I \[Kappa]/2, Band[{1, 2}] -> J, Band[{2, 1}] -> J}, {Nsim, Nsim}];
BMat = SparseArray[{Band[{1, 1}] -> \[CapitalOmega] - \[Omega]d - I \[CapitalGamma]/2}, {Nsim, Nsim}];
DriveArray = -I \[Epsilon]*UnitVector[Nsim, 1];
U[\[Beta]_] = -0.35 Abs[\[Beta]]^2 \[Beta];
F\[Alpha][x_] := -I AMat.x 
F\[Beta][x_] := -I BMat.x - I U[x]

(*Define Compiler*)

RK4[fn\[Alpha]_, fn\[Beta]_] := With[{f\[Alpha] = fn\[Alpha], f\[Beta] = fn\[Beta]},

            Compile[{{\[Alpha]0, _Complex, 1}, {\[Beta]0, _Complex, 1}, {tf, _Real}, {n, _Integer}},

            Module[{dt, \[Alpha], \[Beta], Nsim, k1\[Alpha], k2\[Alpha], k3\[Alpha], k4\[Alpha], k1\[Beta], k2\[Beta], k3\[Beta], k4\[Beta]},

                Nsim = Length@\[Alpha]0;
                dt = N[tf/(n - 1)];
                \[Alpha] = ConstantArray[0.0, {n, Nsim}];
                \[Beta] = ConstantArray[0.0, {n, Nsim}];
                \[Alpha][[1]] = \[Alpha]0;
                \[Beta][[1]] = \[Beta]0;

                Do[{
  k1\[Alpha] = f\[Alpha]@\[Alpha][[nc]] - I g \[Beta][[nc]] + DriveArray,
  k1\[Beta] = f\[Beta]@\[Beta][[nc]] - I g \[Alpha][[nc]],

  k2\[Alpha] = f\[Alpha]@(\[Alpha][[nc]] + 0.5*dt*k1\[Alpha]) - I g (\[Beta][[nc]] + 0.5*dt*k1\[Beta]) + DriveArray,
  k2\[Beta] = f\[Beta]@(\[Beta][[nc]] + 0.5*dt*k1\[Beta]) - I g (\[Alpha][[nc]] + 0.5*dt*k1\[Alpha]),

  k3\[Alpha] = f\[Alpha]@(\[Alpha][[nc]] + 0.5*dt*k2\[Alpha]) - I g (\[Beta][[nc]] + 0.5*dt*k2\[Beta]) + DriveArray,
  k3\[Beta] = f\[Beta]@(\[Beta][[nc]] + 0.5*dt*k2\[Beta]) - I g (\[Alpha][[nc]] + 0.5*dt*k2\[Alpha]),

  k4\[Alpha] = f\[Alpha]@(\[Alpha][[nc]] + dt*k3\[Alpha]) - I g (\[Beta][[nc]] + dt*k3\[Beta]) + DriveArray,
  k4\[Beta] = f\[Beta]@(\[Beta][[nc]] + dt*k3\[Beta]) - I g (\[Alpha][[nc]] + dt*k3\[Alpha]),

  \[Alpha][[nc + 1]] = \[Alpha][[nc]] + dt*1/6*(k1\[Alpha] + 2 k2\[Alpha] + 2 k3\[Alpha] + k4\[Alpha]),
  \[Beta][[nc + 1]] = \[Beta][[nc]] + dt*1/6*(k1\[Beta] + 2 k2\[Beta] + 2 k3\[Beta] + k4\[Beta])
                   }, {nc, 1, n - 1}];

{\[Alpha], \[Beta]}],
{{\[Alpha], _Complex, 2}, {\[Beta], _Complex, 2}, {k1\[Alpha], _Complex, 1}, {k1\[Beta], _Complex, 1}, {k2\[Alpha], _Complex, 1}, {k2\[Beta], _Complex, 1}, {k3\[Alpha], _Complex, 1}, {k3\[Beta], _Complex, 1}, {k4\[Alpha], _Complex, 1}, {k4\[Beta], _Complex, 1}}, 
CompilationTarget -> "C"]]

(*Initialize System*)
runRK4 = RK4[F\[Alpha], F\[Beta]];
\[Alpha]0 = ConstantArray[0.0 , Nsim];
\[Beta]0 = ConstantArray[0.0, Nsim];
Tsim = 10.0; 
\[CapitalDelta]t = 0.01;
n = Tsim/\[CapitalDelta]t + 1 // Round;

(*Run*)
{\[Alpha]temp, \[Beta]temp} = runRK4[\[Alpha]0, \[Beta]0, Tsim, n];
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You initialized \[Alpha] and \[Beta] as real arrays. This is how you can make them complex:

\[Alpha] = Table[0. I, {n}, {Nsim}];
\[Beta] = Table[0. I, {n}, {Nsim}];

Also notice that I used Table instead of ConstantArray because the latter ist not compilable. However, several other parts of your code are not compilable, e.g. the SparseArrays.

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