ladder operator for quantum harmonic oscillator

[saaru darshini]

 How do I represent ak ∣n1 n2 n3 ,,,,nn⟩ = √nk-1 ∣n1 n2 n3 ,,,,nn⟩ and  ak dagger ∣n1 n2 n3 ,,,,nn⟩= √nk-1 ∣n1 n2 n3 ,,,,nn⟩ in qutip and where can i find more about representing and solving i ℏ U'(t)=(Ha+Hf+Hi(t))U(t),t>=0 such equations where Ha is hermitian and NxN and Hi tilde(t)= ∑k=1N (Xk tilde(t) tensor ak.e^-iwkt + Xk dagger tilde(t) tensor ak dagger.e^iwkt)and  Hi∼​(t)=e^it(Ha​+Hf​)Hi​(t)e^−it(Ha​+Hf​)