Born Approximation

In the interaction picture,

$$\tilde{\chi}(t') = R_0 \otimes \tilde\rho(t=0) \mbox{\rm to zeroth order in $V$} .$$ (19)

The Born approximation in the equation of motion Eq.(7.17) consists in

$$\tilde{\chi}(t') \approx R_0 \otimes \tilde\rho(t') .$$ (20)

This means one assumes that for all times $t'>0$, the total density matrix remains a product of the initial bath density matrix $R_0$ and the system density matrix $\tilde\rho(t')$. Intuitively, one argues that this is justified when the bath is `very large' and the coupling $H_{SB}$ `weak', so that the back-action of the system onto the bath can be neglected. In practice, one usually assumes a thermal equilibrium for the bath,

$$R_0 = \frac{e^{-\beta H_B}}{{\rm Tr}e^{-\beta H_B}},$$ (21)

where $\beta=1/k_BT$ with $T$ the bath equilibrium temperature.

Remark: A more detailed analysis of the Born approximation and alternative approximations can be done within the framework of the Projection Operator formalism.

Within the Born approximation, with Eq. (7.20), (7.18), and (7.17), one obtains a closed integro-differential equation for the reduced density operator $\tilde{\rho}(t)$ of the system in the interaction picture,

$$\fbox{$ \begin{array}{rcl} \displaystyle \frac{d}{dt} \tilde{\rho... ...r}_B [\tilde{V}(t),[\tilde{V}(t'),R_0 \otimes \tilde\rho(t')]]. \end{array}$\ }$$ (22)

Remark: Eq.(7.22) is exact up to second order in the perturbation $V$: set $\tilde\rho(t')=\rho(0)$ on the r.h.s. of Eq.(7.22). Since $\tilde\rho(t')$ in the double commutator on the r.h.s. of Eq.(7.22) depends on $V$, Eq.(7.22) is to infinite order in $V$ though not exact. Diagrammatically this corresponds to a summation of an infinite series of diagrams. It is non-trivial to make this statement more precise, but roughly speaking these diagrams contain certain vertex corrections as can be seen from the fact that $\rho(t)$ is a density matrix and not a wave function.