- #1
KFC
- 477
- 4
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^* <br /> \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?
\frac{d\rho_{11}}{dt} = i e^{i\omega_r t} K \rho_{21} - i e^{-i\omega_r t} K^* <br /> \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?