**1. Introduction**

Few-level open systems stand as everyday 'lab rats' in corner stone experiments and future technologies in nanoelectronics [1] and quantum optics [2]. Generically, they are electronic systems with a discrete spectrum (e.g., artificial atoms [3], nanowires or superconducting qubits [4]) connected to particle reservoirs or embedded in bosonic baths. Depending on the nature of the environment (i.e., fermionic or bosonic) to which the open systems are coupled, their theoretical investigation started with two toy-models, namely the single-level Hamiltonian of quantum transport and the Jaynes–Cummings (JC) Hamiltonian of a two-level system (TLS).

Surprisingly or not, studying the sequential tunneling transport regime or the optical properties of quantum emitters eventually boils down to solve formally similar Markovian master equations (MEs) for the so called reduced density operator (RDO). The latter defines the non-unitary evolution of the small system in the presence of the infinite degrees of freedom of the reservoirs. Such MEs are derived by tracing out the reservoir's degrees of freedom and are known from the early days of condensed matter and quantum optics (see the seminal works of Bloch [5], Wangsness [6] and Redfield [7]). The master equation cleverly bypasses the fact that the Liouville–von-Neumann (LvN) equation of the coupled systems (i.e., the open system and the reservoirs) is impossible to solve and takes advantage of the fact that all observables associated to the small and open system can be calculated as statistical averages w.r.t. the RDO.

Indeed, the RDO associated to the Jaynes–Cummings model has been a central object in quantum optics [8,9] (e.g., in the study of lasing and for the calculation of photon correlation functions). In this context the master equation (ME) approach goes as follows: (i) one studies an atomic few-level system

whose eigenvalues and eigenfunctions are supposed to be known; (ii) the dissipation in the system (e.g., cavity losses or various non-radiative recombination processes) is included through the so called 'jump' operators; (iii) the occupation of atomic levels changes due to photon emission or absorption, but the particle number is conserved as the system is not coupled to particle reservoirs; (iv) under the Markov approximation the ME acquires a Lindblad form, usually solved in the steady-state regime.

The above scenario changes when one aims to derive a quantum master equation describing transport phenomena. (i) The Coulomb interaction effects on the spectrum and eigenstates of the system cannot be always neglected, especially for confined systems like quantum dots or nanowires; this requires a many-body derivation of the master equation; (ii) The tunneling between source/drain probes prevents the charge conservation in the central system and the main quantity of interest is the electronic current; (iii) Finally, the steady-state regime does not cover the whole physics and cannot even be guaranteed in general; moreover, the validity of the rotating-wave (RWA) and Markov approximations must be established more carefully [10,11]. In fact it turns out that when applied to transport processes the master equation must rather be solved in its non-Markovian version.

Such generalized master equations which take into account the memory effects have been mostly derived and implemented for time-dependent transport in non-interacting [12,13] and interacting [14] quantum dots, nanowires, and rings [15]. It turns out that the generalized master equation (GME) method is a valuable tool for modeling and monitoring the dynamics of specific many-body states as well as for investigating time-dependent propagation along a sample [16] or capturing charge sensing effects [17] and counting statistics in electrostatically parallel QDs [18]. In particular, Harbola et al. [19] showed that a Lindblad form of the quantum master equation is still recovered in the high bias limit and by assuming the RWA.

Since then, a lot of theoretical work has been done to improve and refine the quantum master equation formalism. A formally exact memory-kernel for the Anderson model was derived and calculated using real-time path integral Monte Carlo methods [20]. A hierarchical quantum master equation approach with good convergence at not too low temperature was put forward by Härtle et al. [21]. As for molecular transport calculations one can rely on the GME written in terms of the many-body states of the isolated molecule [22,23]. A recent review on non-markovian effects in open systems is also available [24].

As we shall see below the implementation of GME approach to many-level systems with specific geometries poses considerable technical difficulties. These are related to the many-body structure of the central interacting system, to the accurate description of the contact regions and, more importantly, to the evaluation of the non-Markovian kernels which become complicated objects once we go beyond non-interacting single-level models.

A second useful extension of the ME method emerged in the context of cavity quantum electrodynamics. Here the system under study is a hybrid one, as the electronic system is still coupled to source/drain reservoirs (i.e., leads) but also interacts with a quantum cavity mode, the latter being subjected to dissipation into leaky modes described by a bosonic bath. Such systems are currently used in state-of-the-art measurements in cavity quantum electrodynamics [25–29]. Again, the many-body nature of the problem is essential, as the electron-photon coupling leads to the formation of dressed states whose dynamics in the presence of both particle and dissipative bosonic reservoirs is far from being trivial. The relevant reduced density operator now acts in the many-body electron-photon Fock space and describes the dynamics of dressed-states. This fact brings new technical difficulties in the derivation [30,31] and implementation of ME [32,33]. Let us also mention here recent studies on ground state electroluminescence [34,35] and on cavity enhanced transport of charge [36].

In view of the abovementioned comments, the aim of this work is: (i) to briefly review the development of the generalized master equation approach to time-dependent many-body transport in the presence of both fermionic and bosonic environments and (ii) to illustrate in a unified framework how the method really works, from formal technicalities to numerical implementation. In Section 2 we shall therefore derive a non-Markovian master equation which describes the dynamics and the

transport properties of rather general 'hybrid' system consisting in an electronic component *S*1 which is connected to particle reservoirs (i.e., leads) and a second subsystem *S*2. The latter, although not coupled to particle reservoirs, interacts with system *S*1 or with some leaky modes described as bosonic baths. Then we specialize this master equation to several systems of interest. More precisely, in Section 3 we recall GME results on transient charging of excited states and Coulomb-coupled quantum dots. Section 4 deals with thermoelectric transport. Applications to transport in cavity quantum electrodynamics are collected in Sections 5 and 6. We conclude in Section 7.
