# Rotating Frame with degenerate levels

I'm working with a angular momentum transition J=0 -> J=1 with no applied magnetic field; so, the upper level has degeneracy 3. This atom is coupled with an electric field propagatin in the z-direction and produces dipole allowed transitions. Redefining the energy, the unperturbed Hamiltonian can be written as:

$$H = \sum_{i=-1,0,1} \hbar \omega_0 |m_i \rangle\langle m_i |$$

The interaction matrix V is:

$$\left( \begin{array}{cccc} 0 & \Omega _1{}^{\ast} \left(e^{-i t w}+e^{i t w}\right) & 0 & \Omega _2{}^{\ast} \left(e^{-i t w}+e^{i t w}\right) \\ \Omega _1 \left(e^{-i t w}+e^{i t w}\right) & \omega_0 & \Omega _3{}^{\ast} \left(e^{-i t w}+e^{i t w}\right) & 0 \\ 0 & \Omega _3\left(e^{-i t w}+e^{i t w}\right) & \omega_0 & \Omega _4{}^{\ast} \left(e^{-i t w}+e^{i t w}\right) \\ \Omega _2 \left(e^{-i t w}+e^{i t w}\right) & 0 & \Omega _4 \left(e^{-i t w}+e^{i t w}\right) & \omega_0 \end{array} \right)$$ where $\Omega_i$ are the Rabi frequencies.

Then, I can transform this sistem to the interaction picture (the rotating frame) using the following transformation:

$$U = \left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & e^{i t w} & 0 & 0 \\ 0 & 0 & e^{i t w} & 0 \\ 0 & 0 & 0 & e^{i t w} \end{array} \right)$$

so my new interaction matrix V still has fast rotating terms:

$$\left( \begin{array}{cccc} 0 & \Omega _1{}^c & 0 & \Omega _2{}^c \\ \Omega _1 & \omega_0-w & \left(e^{-i t w}+e^{i t w}\right)\Omega _3{}^c & 0 \\ 0 & \Omega _3\left[e^{-i t w}+e^{i t w}\right] & \omega_0-w & \left(e^{-i t w}+e^{i t w}\right) \Omega _4{}^c \\ \Omega _2 & 0 & \left(e^{-i t w}+e^{i t w}\right) \Omega _4 & \omega_0-w \end{array} \right)$$

What transformation should I use to eliminate the explicit time dependence?

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