Effective Density Matrix of the System

We wish to obtain an equation of motion for the effective density matrix of the system at time $t>0$,

$${\rho}(t)\equiv {\rm Tr}_B [\chi(t)].$$ (11)

This object is sufficient to calculate expectation values of system operators $A_S$:

$$\langle A_S \rangle_t \equiv {\rm Tr_{total}} [{\chi}(t) A_S]$$

$$= {\rm Tr}_S \left[{\rm Tr}_B {\chi}(t) \right]{A}_S={\rm Tr}_S\left[ {\rho}(t){A}_S\right].$$ (12)

Now use

$${\rm Tr}_B [\tilde{\chi}(t)] = {\rm Tr}_B e^{iH_0t} {\chi}(t)e^{-iH_0t}$$

$$= e^{iH_S t}\left({\rm Tr}_B e^{iH_Bt} {\chi}(t)e^{-iH_Bt}\right)e^{-iH_St}= e^{iH_S t} {\rho}(t)e^{-iH_St}$$

$$\equiv \tilde{\rho}(t).$$ (13)

Note that the interaction picture $\rho(t)\leftrightarrow \tilde{\rho}{(t})$ involves only the free System Hamiltonian $H_S$ and not $H_0$,

$$\tilde{\rho}(t)\equiv e^{iH_S t} \rho(t) e^{-iH_St}.$$ (14)

Using

$$\tilde{A}_S(t)\equiv e^{iH_0t} A_S e^{-iH_0t} = e^{iH_St} A_S e^{-iH_St}$$ (15)

for system operators, one has

$$\langle A_S \rangle_t = {\rm Tr}_S\left[ \tilde{\rho}(t)\tilde{A}_S(t)\right] = {\rm Tr}_S\left[ {\rho}(t){A}_S(t)\right].$$ (16)