2-time correlation function

$C_{\alpha \beta }:={\mathrm{Tr}}^E[E_{\alpha }^I(t)E_{\beta }^I(s){\rho }_B]$ where

• $H_E$ is the Hamiltonian describing the evolution of the environment.
• $H_{int}:={\sum }_{\alpha }{S}_{\alpha }⨂E_{\alpha }$ is the Hamiltonian describing the interaction terms between the environment and the Reduced system, where $S_{\alpha }$ acts on our system and $E_{\alpha }$ acts on our environment
• $E_{\alpha }^I(t):=\exp (\frac{i{H}_{e}t}{\hslash })E_{\alpha }\exp (\frac{-i{H}_{e}t}{\hslash })$ describes the time evolution of our environment Operator $E_{\alpha }$ under the Interaction Picture.
• ${\rho }_B$ is the Density Operator associated with the Thermal bath
• ${\mathrm{Tr}}^E$ is the Partial Trace, tracing over the environment.