Rotating wave approximation

[KFC]

In a text, it introduces an rotating frame and applies it on evolution of density matrix of two-level system. In the original frame, the first diagonal element of the time-derivitative of density matrix gives

$$\frac{d\rho_{11}}{dt} = i e^{i\omega_r t} K \rho_{21} - i e^{-i\omega_r t} K^* \rho_{12}^*$$

In rotating frame of freq $$\omega_r$$, it gives

$$\frac{d\rho_{11}}{dt} = i K \rho_{21} - i K^* \rho_{12}^*$$

I don't really understand how to get above result. In my opinion, I will factor out $$e^{i\omega_r t}$$ such that

$$\frac{d\rho_{11}}{dt} = i e^{i\omega_r t}\left[ K \rho_{21} - e^{-i2\omega_r t} K^* \rho_{12}^*\right]$$

and now in the rotating frame, it gives
$$\frac{d\rho_{11}}{dt} = i \left[ K \rho_{21} - e^{-i2\omega_r t} K^* \rho_{12}^*\right]$$

But there are some extra term $$e^{-i2\omega_r t}$$ left, I know this is not the correct result but how to get the correct one?

By the way, in the text, seems like it only consider the off-diagonal density matrix element is of the form $$e^{i\omega_r t}\rho_{12}$$ or $$e^{-i\omega_r t}\rho_{21}$$ but let all diagonal term unchanged (i.e. no $$e^{\pmi\omega_r t}$$), why is that?

 

[DrClaude]

The first step in the RWA is to go to the rotating reference frame, which is a unitary transform. This is not the same a removing a global complex phase.

https://arxiv.org/abs/1301.3585