Explicit Form of Master Equation

The equation of motion Eq.(7.20) is pretty useless unless one specifies at least some more details for the interaction Hamiltonian $V\equiv H_{SB}$. Denoting system operators by $S'_j$ and bath operators by $B_k$, the most general form of $V$ is

$$V=\sum_{jk}c_{jk} S'_j\otimes B_k \equiv \sum_k S_k \otimes B_k,$$ (23)

where we have re-defined the sum over $j$ as a new system operator ( $\rightarrow$ similarity to Schmid-decomposition).

Remark: Note that $S_k$ and $B_k$ need not necessarily be hermitian. Inserting Eq.(7.23) into Eq.(7.22), we have

$$\frac{d}{dt} \tilde{\rho}(t) = -i \sum_k{\rm Tr}_B [\tilde{S}_k(t) \tilde{B}_k(t), R_0 \rho(t=0)]$$

$$- \int_0^t dt'\sum_{kl}{\rm Tr}_B[\tilde{S}_k(t) \tilde{B}_k(t), [\tilde{S}_l(t') \tilde{B}_l(t'),R_0 \tilde{\rho}(t')]].$$

To simplify things, we will assume

$${\rm Tr}_B \tilde{B}_k(t) R_0 =0$$ (24)

from now on. This is no serious restriction. We furthermore introduce the bath correlation functions

$$C_{kl}(t,t')\equiv {\rm Tr}_B \left[\tilde{B}_k(t)\tilde{B}_l(t')R_0\right].$$ (25)

Assumption 1:

$$[R_0,H_B]=0 .$$ (26)

This means

$$C_{kl}(t,t')\equiv C_{kl}(t-t').$$ (27)

We then have

$$\frac{d}{dt} \tilde{\rho}(t) = -\int_0^t dt'\sum_{kl}\Big[C_{kl}(t-t') \left\{ \tilde{S}_k(t)\ti... ...(t') \tilde{\rho}(t') - \tilde{S}_l(t') \tilde{\rho}(t') \tilde{S}_k(t)\right\}$$

$$+ C_{lk}(t'-t)\left\{ \tilde{\rho}(t')\tilde{S}_l(t') \tilde{S}_k(t) - \tilde{S}_k(t)\tilde{\rho}(t') \tilde{S}_l(t') \right\}\Big].$$ (28)

Assumption 2a (Markov approximation): the bath correlation function $C_{kl}(\tau)$ is strongly peaked around $\tau\equiv t-t'=0$ with a peak width $\delta \tau \ll \gamma^{-1}$, where $\gamma$ is a `typical rate of change of $\tilde{\rho}(t')$.' Note that the condition $\delta \tau \ll \gamma^{-1}$ can only be checked after the equation of motion for $\tilde{\rho}(t)$ has been solved. In the interaction picture, one then replaces $\tilde{\rho}(t')\to \tilde{\rho}(t)$ under the integral to obtain

$$\frac{d}{dt} \tilde{\rho}(t) = -\int_0^t dt'\sum_{kl}\Big[C_{kl}(t-t') \left\{ \tilde{S}_k(t)\ti... ..._l(t') \tilde{\rho}(t) - \tilde{S}_l(t') \tilde{\rho}(t) \tilde{S}_k(t)\right\}$$

$$+ C_{lk}(t'-t)\left\{ \tilde{\rho}(t)\tilde{S}_l(t') \tilde{S}_k(t) - \tilde{S}_k(t)\tilde{\rho}(t) \tilde{S}_l(t') \right\}\Big].$$ (29)

The important fact is that this approximation is carried out in the interaction (and not in the original Schrödinger) picture: in the interaction picture, the only relevant time-scale the change of the density matrix is $\gamma^{-1}$ and not (the usually much faster) timescales from the free evolution with $H_S$. In fact, one now transforms back into the Schrödinger picture, using Eq.(7.14),

$$\frac{d}{dt} \tilde{\rho}(t) = i[H_S,\tilde{\rho}(t)] + e^{iH_S t} \frac{d}{dt}\rho(t) e^{-iH_St}$$

$$\leadsto \frac{d}{dt}\rho(t) = -i[H_S,{\rho}(t)] + e^{-iH_S t} \frac{d}{dt}\tilde\rho(t) e^{iH_St}$$ (30)

which leads to

$$\leadsto \frac{d}{dt}\rho(t) = -i[H_S,{\rho}(t)]$$

$$- \int_0^t dt'\sum_{kl}\Big[C_{kl}(t-t') \left\{ e^{-iH_S t}\tilde{... ...ilde{\rho}(t) - \tilde{S}_l(t') \tilde{\rho}(t) \tilde{S}_k(t)\right\}e^{iH_St}$$

$$+ C_{lk}(t'-t)\left\{e^{-iH_S t} \tilde{\rho}(t)\tilde{S}_l(t') \tilde{S}_k(t) - \tilde{S}_k(t)\tilde{\rho}(t) \tilde{S}_l(t') \right\}e^{iH_St}\Big]$$

$$= -i[H_S,{\rho}(t)]$$

$$- \int_0^t dt'\sum_{kl}\Big[C_{kl}(t-t') \left\{ {S}_k\tilde{S}_l(t'-t) {\rho}(t) - \tilde{S}_l(t'-t) {\rho}(t) {S}_k\right\}$$

$$+ C_{lk}(t'-t)\left\{ {\rho}(t)\tilde{S}_l(t'-t) {S}_k - {S}_k{\rho}(t) \tilde{S}_l(t'-t) \right\}\Big].$$ (31)

Assumption 2b (Markov approximation): the integral over $t'$ can be carried out to $t=\infty$. This in fact is completely consistent with assumption 2a (see above). Defining

$$D_k\equiv \lim_{t\to\infty}\int_0^{t} d\tau\sum_lC_{kl}(\tau)\til... ...iv \lim_{t\to\infty}\int_0^{t} d\tau\sum_lC_{lk}(-\tau)\tilde{S}_l(-\tau),\quad$$ (32)

we can write

$$\fbox{$ \begin{array}{rcl} \displaystyle \frac{d}{dt}\rho(t) &=& ... ... {\rho}(t) {S}_k+ {\rho}(t)E_k {S}_k - {S}_k{\rho}(t) E_k \Big].\end{array}$\ }$$ (33)