Weak-coupling master equation for arbitrary initial conditions

The structure of the initial system-environment state is fundamental to determining the nature and characteristics of the evolution of such an open quantum system. The usual assumption is to consider that the initial system-environment state is separable. Here, we go beyond this simple case and derive the evolution equations, up to second order in a weak-coupling expansion, that describe the evolution of the reduced density matrix of the system for any arbitrary system-environment initial state. The structure of these equations allows us to determine the initial conditions for which the evolution of the reduced density matrix can be written in terms of a set of Lindblad-like equations, once considering the Markov and secular approximations. Moreover, we show that for initial states belonging to a subset of separable states such Lindblad-like equations become actual Lindblad equations and that the evolution of ${\rho }_s$ thereby obtained is also trace and positive preserving.

Figure 1

Schematic view of the V-type three-level system we are analyzing in this section.

Figure 2

Evolution of the population $P_1\left(t\right)={\mathrm{Tr}}_S\left\{{\rho }_s\left(t\right)\right|e_1\rangle \langle e_1\left|\right\}$ (top panel) and von Neumann entropy (bottom panel) for an entangled initial state (45) (blue solid line and red dashed lines corresponding to laser intensities $\epsilon =0$ and $\epsilon =0.012$, respectively) and an initially decorrelated pure state (52) (blue dotted and red dotted-dashed corresponding to $\epsilon =0$ and $\epsilon =0.012$, respectively). Other parameters are $\sigma =\sqrt{0.1}$ in Eq. (46), $\mathcal{A}=\sqrt{0.3},\mathcal{B}=\sqrt{0.3}$, and $\mathcal{C}=\sqrt{0.4}$. The frequency transitions are ${\omega }_1=1$ and ${\omega }_2=0.5$, and the parameters for the spectral density $J\left(\omega \right)$ are $\alpha =0.01,{\omega }_c=5$. The frequency window is chosen up to ${\omega }_{\mathrm{m}}=100$. All frequencies are in units of ${\omega }_s=1$.

Figure 3

Evolution of the population $P_j\left(t\right)={\mathrm{Tr}}_S\left\{{\rho }_s\left(t\right)\right|e_j\rangle \langle e_j\left|\right\},j=1,2$ for an entangled initial state (45) (blue solid and red dashed corresponding to $\epsilon =0$ and $\epsilon =0.012$, respectively) and for the corresponding separable initial state (55) (blue dotted and red dot-dashed lines corresponding to $\epsilon =0$ and $\epsilon =0.012$, respectively). The other parameters are the same as in Fig. 2.