Magnus expansion applied to a quantum two-level system with rectangular step potential in the interaction picture.
We compute 16 terms in the Magnus series and compare with the exact probability transition.

Algebra of sigma Pauli matrices

Cmt[x_,x_]:=0;
Cmt[a___, 0,     b___] := 0;
Cmt[a___, x_+y_, b___] :=    Cmt[a,x,b]+Cmt[a,y,b];
Cmt[a___, t^n_ x_,  b___] := t^n Cmt[a,x,b];
Cmt[a___, n_ x_Cmt, b___] := n   Cmt[a,x,b];
Cmt[a___, x_  y_,  b___] :=  Cmt[a, Expand[x y], b];
Cmt[a___, n_ sig[j_Integer], b___] := n *
     Cmt[a,sig[j],b];
Cmt[sig[i_],sig[j_]]:= - Cmt[sig[j],sig[i]] /; i > j
Cmt[z_Cmt,sig[i_]]:= -Cmt[sig[i],z];
Cmt[Cmt[sig[i_],sig[j_]],z_Cmt]:=
  -Cmt[Cmt[sig[j],z],sig[i]] - 
   Cmt[Cmt[z,sig[i]],sig[j]];
Cmt[z_Cmt,Cmt[sig[i_],sig[j_]]]:=
  -Cmt[sig[i],Cmt[sig[j],z]] -
   Cmt[sig[j],Cmt[z,sig[i]]];
  
Cmt[sig[1],sig[2]]:= 2 I sig[3];
Cmt[sig[1],sig[3]]:= - 2 I sig[2];
Cmt[sig[2],sig[3]]:= 2 I sig[1];

Coefficient matrix (Hamiltonian)

A[t_]:= -(I/hbar)V0 (Cos[om t] sig[1] - Sin[om t] sig[2]);
omg[1][t_] := Simplify[Integrate[A[t], t]];

First term in the Magnus expansion

ome[1][t_]:= Collect[omg[1][t] - (omg[1][t] /. t -> 0),{sig[1],sig[2],sig[3]}];

ome[1][t]

Recurrences

ss[n_Integer,1][t_]:= ss[n,1][t] = Simplify[
   Cmt[ome[n-1][t],A[t]]];
ss[n_Integer,j_Integer][t_]:= ss[n,j][t]=Sum[
   Simplify[Cmt[ome[m][t],ss[n-m,j-1][t]]], 
   {m,1,n-j}];
omg[n_Integer][t_]:= omg[n][t]=Simplify[Sum[ 
   (BernoulliB[j]/j!) * (
  Integrate[ss[n,j][t],t]), {j,1,n-1}]];
ome[n_Integer][t_]:= ome[n][t] = Collect[Simplify[omg[n][t] - 
    (omg[n][t] /. t->0)],{sig[1],sig[2],sig[3]}];

Results from term 2 up to term 16

ome[2][t]

ome[3][t]

ome[4][t]

ome[5][t]

ome[6][t]

ome[7][t]

ome[8][t]

ome[9][t];

ome[10][t];

Timing[ome[11][t];]

Timing[ome[12][t];]

Timing[ome[13][t];]

Timing[ome[14][t];]

Timing[ome[15][t];]

Timing[ome[16][t];]

Magnus approximations (up to p=16)

omex[j_]:=Collect[Simplify[Sum[ome[n][t],{n,1,j}]],{sig[1],sig[2],sig[3]}];

omex[1]

Do[omex[j]=Collect[Simplify[omex[j-1]+ome[j][t]],{sig[1],sig[2],sig[3]}],{j=2,16,1}]

omex[2]

Coefficient[omex[2]/I,sig[1]]

regla1={V0->g hbar/t,om->xi/t};

ww[i_]:=ww[i]=Table[Simplify[Coefficient[(omex[i]/I)/.regla1,sig[j]]],{j,1,3}];

Table[ww[i],{i,1,16}];

ww[5]

Exact  transition probability

probex=(4g^2/(4 g^2+xi^2))Sin[Sqrt[g^2+xi^2/4]]^2;

probex

Transition probability computed with the Magnus expansion (up to p =16) and difference with the exact result

diff[j_]:=Module[{vn,promag,vp1},vp1=ww[j]^2;vn=Sqrt[Total[vp1]];promag=(Sin[vn]/vn)^2 * Total[Take[vp1,2]];
Abs[(probex-promag)]/.{g->0.5,xi->1}
];

diff[2]

differences=Table[diff[j],{j,1,15,2}];