stephen8686
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- TL;DR
- I do not understand the optical bloch equations and the Lindblad master equation. Are they not stating the same thing?
I am studying two level atoms interacting with fields in order to study Dicke Superradiance.
From Loudon's book, the Optical Bloch Equations for a two level atom interacting with a field say (with rotating wave approx):
$$\frac{d\rho_{22}}{dt}=- \frac{d\rho_{11}}{dt} = -\frac{1}{2} iV(e^{i\Delta \omega t}\rho_{12}-e^{-i\Delta \omega t}\rho_{21})$$
and
$$ \frac{d\rho_{12}}{dt}= \frac{d\rho_{21}^*}{dt} = \frac{1}{2}iVe^{-i\Delta\omega t}(\rho_{11}-\rho_{22}) $$
I have also seen the Lindblad master equation, given by:
$$ \frac{d}{dt}\hat{\rho} = \frac{1}{i\hbar} [\hat{H},\hat{\rho}] + \kappa \hat{L}[\hat{a}]\hat{\rho} + \sum^N_{j=1} \gamma \hat{L}[\hat{\sigma}_j^-]\hat{\rho} + \frac{1}{2T_2}\hat{L}[\hat{\sigma}^z_j]\hat{\rho}+ w\hat{L}[\hat{\sigma}_j^+]\hat{\rho} $$
So if the optical bloch equations already give the time evolution for all of the elements of the density operator matrix, why is the master equation important? Is it just easier to implement because it is one equation rather than two coupled ones, or is there a more important difference between these two approaches?
From Loudon's book, the Optical Bloch Equations for a two level atom interacting with a field say (with rotating wave approx):
$$\frac{d\rho_{22}}{dt}=- \frac{d\rho_{11}}{dt} = -\frac{1}{2} iV(e^{i\Delta \omega t}\rho_{12}-e^{-i\Delta \omega t}\rho_{21})$$
and
$$ \frac{d\rho_{12}}{dt}= \frac{d\rho_{21}^*}{dt} = \frac{1}{2}iVe^{-i\Delta\omega t}(\rho_{11}-\rho_{22}) $$
I have also seen the Lindblad master equation, given by:
$$ \frac{d}{dt}\hat{\rho} = \frac{1}{i\hbar} [\hat{H},\hat{\rho}] + \kappa \hat{L}[\hat{a}]\hat{\rho} + \sum^N_{j=1} \gamma \hat{L}[\hat{\sigma}_j^-]\hat{\rho} + \frac{1}{2T_2}\hat{L}[\hat{\sigma}^z_j]\hat{\rho}+ w\hat{L}[\hat{\sigma}_j^+]\hat{\rho} $$
So if the optical bloch equations already give the time evolution for all of the elements of the density operator matrix, why is the master equation important? Is it just easier to implement because it is one equation rather than two coupled ones, or is there a more important difference between these two approaches?