Decomposition into Histories

We may write the Master equation Eq.(7.143) as

$$\frac{d}{dt}\rho(t) = \left( {\cal L}_0 + {\cal L}_1 \right) \rho(t).$$ (143)

This can be formally solved as follows: we define

$$\bar{\rho}(t) \equiv e^{-{\cal L}_0 t} \rho(t),\quad \bar{{\cal L}}_1(t) \equiv e^{-{\cal L}_0 t} {\cal L}_1 e^{{\cal L}_0 t}$$ (144)

$$\leadsto \frac{d}{dt} \bar{\rho}(t) = -{\cal L}_0\bar{\rho}(t) + e^{-{\cal L}_0 t} \left( {\cal L}_0 + ... ...L}_1 \right) e^{{\cal L}_0 t} \bar{\rho}(t) = \bar{{\cal L}}_1(t) \bar{\rho}(t)$$

$$\leadsto\bar{\rho}(t) = \rho(0) + \int_{0}^{t}dt_1 \bar{{\cal L}}_1(t_1)\bar{\rho}(t_1)$$

$$= \rho(0) + \int_{0}^{t}dt_1 \bar{{\cal L}}_1(t_1) \rho(0) + \int_{... ...1 \int_{0}^{t_1}dt_2 \bar{{\cal L}}_1(t_1) \bar{{\cal L}}_1(t_2)\bar{\rho}(t_2)$$

$$...$$

$$= \rho(0)+ \sum_{n=1}^{\infty} \int_{0}^{t} dt_1... \int_{0}^{t_n}dt_n \bar{{\cal L}}_1(t_1)... \bar{{\cal L}}_1(t_n)\rho(0).$$ (145)

Transforming back to $\rho(t)$, we can explicitely write this as

$${\rho}(t) = e^{{\cal L}_0 t} \rho(0)$$

$$+ \sum_{n=1}^{\infty} \int_{0}^{t} dt_1... \int_{0}^{t_n}dt_n e^{{\... ...e^{{\cal L}_0 t_2}... e^{-{\cal L}_0 t_n} {\cal L}_1 e^{{\cal L}_0 t_n} \rho(0)$$

$$= e^{{\cal L}_0 t} \rho(0)$$

$$+ \sum_{n=1}^{\infty} \int_{0}^{t} dt_1... \int_{0}^{t_n}dt_n \unde... ..._2-t_3)}... e^{{\cal L}_0 (t_{n-1}-t_n)} {\cal L}_1 e^{{\cal L}_0 t_n} \rho(0)}$$

$$\equiv e^{{\cal L}_0 t} \rho(0) + \sum_{n=1}^{\infty} \int_{0}^{t} dt_1... \int_{0}^{t_n}dt_n \underline{\rho_c(t;t_1,...,t_n)},$$ (146)

where we defined the un-normalised, conditioned `density matrix' $\rho_c(t;t_1,...,t_n)$ at time $t$ with $n$ quantum jumps occuring at times $t_1,...,t_n$. This object (the underlined term in Eq.(7.150)) indeed corresponds to the original density matrix $\rho(0)$, `freely' time-evolved with the effective Hamiltonian $H_{\rm eff}$ during the time intervals $(0,t_n]$, $(t_n,t_{n-1}]$,... interrupted by $n$ `jumps' at times $t_n$, $t_{n-1}$, ..., $t_1$. The total density matrix $\rho(t)$ at time $t$ then is the sum over all possible `trajectories' with $n=0,...,\infty$ jumps occuring in between a `free', effective time evolution.