For a harmonic oscillator of frequency \(\omega\), calculate the explicit time-dependent \(a(t), a^{\dagger}(t)\) operators in the picture with \(\hat{W}(t)=\exp \left[2 i \omega \hat{a}^{\dagger} \hat{a}\left(t-t_{0}\right)\right]\), and the explicit differential equations for a general operator and state.