Interaction of Ammonia with sinusoidal electric field

h=({{-A, η Cos[ω t]}, {η Cos[ω t], A}})

guess for transformation to find an effective time-independent Hamiltonian , ψ=W(t)χ

W=DiagonalMatrix[{(E^(I ω t/2)),(E^(-I ω t/2))}]

effective Hamiltonian

Heff=Inverse[W].(h.W-I ℏ (∂_t)W)//Simplify

Heff//MatrixForm

Heff=Heff//TrigToExp//Simplify

%//MatrixForm

note that this is the sum of a constant part and a rapidly varying part, which we assume averages to zero

Heff=Heff/.Exp[a_]->0

we now have an effective time-independent Hamiltonian, which we are now skilled at analyzing...

{es,kets}=Eigensystem[Heff]

bras=Inverse[kets\[Transpose]]

define outer product

a_⊗b_:=KroneckerProduct[a,b]

Sum[kets[[i]]⊗bras[[i]],{i,1,2}]//Simplify//MatrixForm

U=Sum[(E^(-I es[[i]]t/ℏ))kets[[i]]⊗bras[[i]],{i,1,2}]//Simplify

{{((E^(-((I t (Sqrt[4 (A^2)+(η^2)-4 A ω ℏ+(ω^2) (ℏ^2)]))/(2 ℏ)))) (-2 A+ω ℏ+(Sqrt[4 (A^2)+(η^2)-4 A ω ℏ+(ω^2) (ℏ^2)])+(E^((I t (Sqrt[4 (A^2)+(η^2)-4 A ω ℏ+(ω^2) (ℏ^2)]))/ℏ)) (2 A-ω ℏ+(Sqrt[4 (A^2)+(η^2)-4 A ω ℏ+(ω^2) (ℏ^2)]))))/(2 (Sqrt[4 (A^2)+(η^2)-4 A ω ℏ+(ω^2) (ℏ^2)])),-(((E^(-((I t (Sqrt[4 (A^2)+(η^2)-4 A ω ℏ+(ω^2) (ℏ^2)]))/(2 ℏ)))) (-1+(E^((I t (Sqrt[4 (A^2)+(η^2)-4 A ω ℏ+(ω^2) (ℏ^2)]))/ℏ))) η)/(2 (Sqrt[4 (A^2)+(η^2)-4 A ω ℏ+(ω^2) (ℏ^2)])))},{-(((E^(-((I t (Sqrt[4 (A^2)+(η^2)-4 A ω ℏ+(ω^2) (ℏ^2)]))/(2 ℏ)))) (-1+(E^((I t (Sqrt[4 (A^2)+(η^2)-4 A ω ℏ+(ω^2) (ℏ^2)]))/ℏ))) η)/(2 (Sqrt[4 (A^2)+(η^2)-4 A ω ℏ+(ω^2) (ℏ^2)]))),((E^(-((I t (Sqrt[4 (A^2)+(η^2)-4 A ω ℏ+(ω^2) (ℏ^2)]))/(2 ℏ)))) (2 A-ω ℏ+(Sqrt[4 (A^2)+(η^2)-4 A ω ℏ+(ω^2) (ℏ^2)])+(E^((I t (Sqrt[4 (A^2)+(η^2)-4 A ω ℏ+(ω^2) (ℏ^2)]))/ℏ)) (-2 A+ω ℏ+(Sqrt[4 (A^2)+(η^2)-4 A ω ℏ+(ω^2) (ℏ^2)]))))/(2 (Sqrt[4 (A^2)+(η^2)-4 A ω ℏ+(ω^2) (ℏ^2)]))}}

$Assumptions={η∈Reals}

time evolution matrix when the Bohr condition is satisfied

U/.ω->2A/ℏ//Simplify//ExpToTrig//Simplify

back to general case, calculate amplitude to change state in this process

{0,1}.U.{1,0}//Simplify

%//ExpToTrig//Simplify

define θ=ηt/ℏ, δ=ℏ(ω-ω0)/η, ℏω0=2A

Plot[(((Sin[(θ/2)Sqrt[1+(δ^2)]])^2)/(1+(δ^2)))/.θ->π,{δ,-3π,3π},PlotRange->All]

so linewidth is about δ=(Sqrt[3])/2=ℏΔω/η=Δωt/π->ℏΔω=2π/(Sqrt[3])(Uncertainty Principle)

Plot[(((Sin[(θ/2)Sqrt[1+(δ^2)]])^2)/(1+(δ^2)))/.θ->100,{δ,-3π,3π},PlotRange->All]