Interaction Picture

We define an interaction picture by writing

$$H \equiv H_0+V,\quad H_0\equiv H_S+H_B, \quad V\equiv H_{SB}$$ (4)

with the Hamiltonian $H_0$ describing the time evolution of the uncoupled system and bath, and the perturbation $V$ describing the interaction $H_{SB}$.

We define $\chi(t)$ as the total density matrix (system + bath) which obeys the Liouville-von-Neumann equation ,

$$\frac{d}{dt}\chi(t)=-i[H,\chi(t)] \leadsto \chi(t)= e^{-iHt} \chi(t=0) e^{iHt},$$ (5)

where we start with the initial condition $\chi(t=0)$ at time $t=0$. In the interaction picture,

$$\tilde{\chi}(t) \equiv e^{iH_0t} \chi(t) e^{-iH_0t}$$ (6)

$$\tilde{A}(t) \equiv e^{iH_0t} A e^{-iH_0t}.$$ (7)

The equation of motion for the density operator in the interaction picture becomes

$$\frac{d}{dt} \tilde{\chi}(t) = i [H_0,\tilde{\chi}(t)] + e^{iH_0t} \frac{d}{dt}\chi(t) e^{-iH_0t}$$

$$= i [H_0,\tilde{\chi}(t)] -i e^{iH_0t} [H,\chi(t)] e^{-iH_0t}$$

$$= i [H_0,\tilde{\chi}(t)] -i e^{iH_0t} [H_0+V,\chi(t)] e^{-iH_0t}$$

$$= i [H_0,\tilde{\chi}(t)] -i [H_0+\tilde{V}(t),\tilde{\chi}(t)]$$

$$= -i [\tilde{V}(t),\tilde{\chi}(t)].$$ (8)

In integral form, this can be written as

$$\tilde{\chi}(t) = {\chi}(t=0) -i \int_0^t dt'[\tilde{V}(t'),\tilde{\chi}(t')]$$ (9)

which we insert into Eq. (7.8) to obtain

$$\fbox{$ \begin{array}{rcl} \displaystyle \frac{d}{dt} \tilde{\chi... ...] -\int_0^t dt'[\tilde{V}(t),[\tilde{V}(t'),\tilde{\chi}(t')]]. \end{array}$\ }$$ (10)

Up to here, everything is still exact.