σ0=
;σ1=
;σ2=
;σ3=
;s0=
;s1=
;is=KroneckerProduct[s0,s1,s1];ρinitial=is.Transpose[is];w=10;Γ=0.1;(*SampleH(t):time-dependentσxonallqubits*)H[t_]:=(Sin[wt](KroneckerProduct[σ1,σ1,σ0]+KroneckerProduct[σ2,σ2,σ0]+KroneckerProduct[σ3,σ3,σ0])+Cos[wt](KroneckerProduct[σ0,σ1,σ1]+KroneckerProduct[σ0,σ2,σ2]+KroneckerProduct[σ0,σ3,σ3])+0.5(KroneckerProduct[σ1,σ0,σ1]+KroneckerProduct[σ2,σ0,σ2]+KroneckerProduct[σ3,σ0,σ3]));;(*Initializesymbolictime-dependent8x8densitymatrixρ(t)*)ρ=Array[r,{n,n}][t];(*Commutatorfunction*)Comm[A_,B_]:=A.B-B.A;(*Definethemasterequation:dρ/dt=-i[H,ρ]-Γ/2[H,[H,ρ]]*)drho=-IComm[H[t],ρ]-(Γ/2)Comm[H[t],Comm[H[t],ρ]];(*Generatedifferentialequationsforeachmatrixelement*)eqns=Flatten[Table[D[ρ[[i,j]],t]drho[[i,j]],{i,n},{j,n}]];(*Initialconditionsfromρ(0)*)initConds=Flatten[Table[r[i,j][0]ρinitial[[i,j]],{i,n},{j,n}]];(*Solvenumerically*)Tmax=10;sol=NDSolve[Join[eqns,initConds],Flatten[ρ],{t,0,Tmax},MaxStepsInfinity];(*Reconstructρ(t)asmatrixfunction*)ρsol[t_]:=Evaluate[Table[r[i,j][t]/.First[sol],{i,n},{j,n}]];
1 | 0 |
0 | 1 |
0 | 1 |
1 | 0 |
0 | -I |
I | 0 |
1 | 0 |
0 | -1 |
1 |
0 |
0 |
1 |
1
2