single-particle evolution operator, $\hat
U_0^{(1)} (t) = \mathcal T \exp -i \int_0^t \hat H_0^{(1)}$, we can introduce the time dependent single-particle basis ${|{\phi_\alpha(t)}\rangle} = \hat U_0^{(1)} (t) {|{\phi_\alpha}\rangle}$ that defines a set of rotating atomic oscillators. Then we can define time dependent occupation number states analogously to Eq. , and the creation and annihilation operators are all rotating accordingly in the Schrödinger picture, for example, $$\begin{gathered}
  \label{eq:a_Sch(t)}
  \hat a_\alpha (t) =
  \hat U_0 (t) \, \hat a_\alpha \, \hat U_0^\dag(t).\end{gathered}$$ The rotating occupation number states satisfy the interaction free Schrödinger equation. In the absence of $\hat H_1$, the state of the rotating atomic oscillators and the number of excitations remain unchanged.

In the Heisenberg picture, the free Hamiltonian disappears from the equation of motion of the rotating atomic oscillator operators. For example, $\hat a_{\alpha H} (t) = \hat U^\dag (t) \hat a_\alpha (t)
\hat U(t)$ satisfies $$\begin{gathered}
  \label{eq:a-eq-motion}
  \frac d{dt} \hat a_{\alpha H} (t) =
  i \big[ \hat H_{1H}(t), \hat a_{\alpha H} (t) \big].\end{gathered}$$ To solve Eq. , we need to express $\hat H_1$ in terms of the rotating oscillator variables. We assume that the interaction does not bring the system far from the rotating reference state $\bigotimes^N {|{\phi_0(t)}\rangle}$. Then 