Redfield Equation

The Redfield equation is the Markovian Master Equation $\frac{d}{dt}{\rho }_S^I(t)=-{\sum }_{\alpha }([S_{\alpha }^I(t),B_{\alpha }^I(t){\rho }_S^I(t)]+[{\rho }_S^I(t)C_{\alpha }^I(t),S_{\alpha }^I(t)])$, where

• $[\cdot ,\cdot ]$ is the Commutator
• $B_{\alpha }^I(t)=\frac{1}{{\hslash }^2}{\sum }_{\beta }{\int }_0^t{C}_{\alpha \beta }(t-s)S_{\beta }^I(s)ds$
• $C_{\alpha }^I(t)=\frac{1}{{\hslash }^2}{\sum }_{\beta }{\int }_0^t{C}_{\beta \alpha }(s-t)S_{\beta }^I(s)ds$
• $C_{\alpha \beta }(\tau )$ is the 2-time correlation function.
• $S_{\alpha }^I(s)$ is the .. Help me ❓.
• The upper index $I$ represents that we are working in the Interaction Picture.

It is unclear whether the Redfield equation is assumed, or derived. Help me ❓

By applying the Markov Approximation and moving the equation back to the Schrodinger picture, we get the Master equation:

$\frac{d}{dt}{\rho }_S^I(t)=-\frac{i}{\hslash }[H_S,{\rho }_S(t)]+\frac{2}{\hslash }{\sum }_{\alpha \beta }{C}_{\alpha \beta }(S_{\beta }\rho S_{\alpha }-\frac{1}{2}\{S_{\alpha }{S}_{\beta }^{†},{\rho }_S(t)\})$, where

• $\{\cdot ,\cdot \}$ is the anti-commutator.
• $C_{\alpha \beta }$ are elements of the Kossohowski Matrix.