jz=DiagonalMatrix[{-1,0,1}]

jz//MatrixForm

jp=({{0, 0, 0}, {Sqrt[2], 0, 0}, {0, Sqrt[2], 0}})

jm=Transpose[jp]

(jx=(jp+jm)/2)//MatrixForm

(jy=(jp-jm)/(2I))//MatrixForm

Eigenvalues[jy]

jx.jy-jy.jx//MatrixForm

(H=e0 jz+v Cos[ω t]jx)//MatrixForm