Rotating wave approximation

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KFC

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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

[tex]\frac{d\rho_{11}}{dt} = i e^{i\omega_r t} K \rho_{21} - i e^{-i\omega_r t} K^*
\rho_{12}^*[/tex]

In rotating frame of freq [tex]\omega_r[/tex], it gives

[tex]\frac{d\rho_{11}}{dt} = i K \rho_{21} - i K^* \rho_{12}^*[/tex]

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

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

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

But there are some extra term [tex]e^{-i2\omega_r t}[/tex] 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 [tex]e^{i\omega_r t}\rho_{12}[/tex] or [tex]e^{-i\omega_r t}\rho_{21}[/tex] but let all diagonal term unchanged (i.e. no [tex]e^{\pmi\omega_r t}[/tex]), why is that?
 

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