*2.1. Open Quantum Systems*

As usual, we start with a closed quantum system (see Figure 1a). This system is represented by a pure state, |Ψ(*t*), which evolves unitarily according to the time-dependent Schrödinger equation:

$$i\hbar \frac{\partial |\Psi(t)\rangle}{\partial t} = \hat{H}|\Psi(t)\rangle. \tag{1}$$

Finding a solution to Equation (1) is inaccessible for most practical scenarios due to the large number of degrees of freedom involved. Therefore, it is a common practice to partition the system into two subsets of degrees of freedom, viz., open system and environment [6]. The open system can be described by a reduced density matrix:

$$\boldsymbol{\rho}\_{\rm sys}(t) = \text{Tr}\_{\rm env} \left[ \langle \Psi(t) \rangle \langle \Psi(t) \rangle \right],\tag{2}$$

where Trenv denotes the trace over the environment degrees of freedom. The reduced density matrix *ρ* ˆ sys can be shown to obey, in most general circumstances, a non-Markovian master equation [30,31]:

$$\frac{\partial \mathfrak{d}\_{\rm sys}(t)}{\partial t} = -i \left[ \hat{H}\_{\rm int}(t), \mathfrak{d}\_{\rm sys}(t) \right] + \int\_{t\_0}^{t} \hat{\mathcal{K}}(t, t') \mathfrak{d}\_{\rm sys}(t') dt',\tag{3}$$

where *H* ˆ int(*t*) is a system Hamiltonian operator in some interaction picture and Kˆ (*<sup>t</sup>*,*<sup>s</sup>*) is the "memory kernel" superoperator, which operates on the reduced state *ρ*ˆsys(*t*) and represents how the environment affects the system. If the solution to Equation (3) is known then the expectation value of any observable *A* ˆ of the system can be evaluated as:

$$
\langle \vec{A}(t) \rangle = \text{Tr}\_{\text{sys}}[\not p\_{\text{sys}}(t)\vec{A}].\tag{4}
$$

Unfortunately, solving Equation (3) is not an easy task. The effect of K ˆ (*<sup>t</sup>*,*<sup>s</sup>*) on *ρ*ˆsys(*t*) cannot be explicitly evaluated in general circumstances. Moreover, even if the explicit form of K ˆ (*<sup>t</sup>*,*<sup>s</sup>*) is known, the solution to Equation (3) is very demanding as the density matrix *ρ*ˆsys(*t*) scales very poorly with the number of degrees of freedom of the open system. Finally, if one is aiming at computing multi-time correlations functions, then it is necessary to incorporate the effect (backaction) of the successive measurements on the evolution of the reduced density matrix, which is, in general non-Markovian regimes, a very complicated task both from the practical and conceptual points of view.
