# Jaynes-Cummings Density Operator Evolution

• A
stephen8686
TL;DR Summary
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?

I think your first set of equations doesn't contain spontaneous emission while the Lindblad equation does. Compare your notes with this:
https://www.ifsc.usp.br/~strontium/Teaching/Material2020-1 SFI5814 Atomicamolecular/Anderson - Monograph - Bloch equations.pdf

Generally speaking, the Lindblad equation is used for open quantum systems and includes terms which lead to decoherence and dissipation. If you start with a pure state, this kind of time evolution usually leads to a mixed state. This can't happen under the unitary dynamics of isolated quantum systems which is represented by the first term of the RHS of the Lindblad equation you have written (the commutator term).