**7. Summary**

It goes without saying that as transport experiments at nanoscale become more involved the formal tools must be suitably extended or adapted. In particular, the unavoidable charging and correlation effects at finite bias pushed the theoretical calculations from the very convenient single-particle (or at most mean-field) Landauer–Büttiker picture to the complicated many-body perturbation theory of the non-equilibrium Keldysh–Green's functions [117].

Here we summarized some results on time-dependent transport in open interacting systems which argue for the similar idea that if one looks for transient effects and dynamics of excited states the simple rate equation approach must be extended to the non-markovian generalized master equation.

The GME we used in all examples is constructed and solved w.r.t the exact many-body states of the central open system and can be therefore implemented numerically without major changes to study both Coulomb-interacting and hybrid systems where the fermion-boson interaction is crucial, like QD-cavity systems or nano-electromechanical systems. A consistent derivation of the GME the full knowledge of the eigenvalues and eigenfunctions of complicated interacting Hamiltonian (e.g., cavity-coupled systems must be described by 'dressed' states). With very few exceptions coming from quantum optics (i.e., the Jaynes-Cummings model for two-level or Λ and *V* three-level systems) such a task can only be achieved via numerically exact diagonalization of large matrices, especially for elecron-photon systems. To bypass this difficulty we proposed and succesfully used a stepwise diagonalization procedure.

The dynamics of excited states in a quantum wire, the onset of current-current correlations for a pair of electrostatically coupled quantum dots and thermoelectric effects were presented within a simple lattice model which however captures the relevant physics.

When turning to QED-cavity system we developed the GME within a continuous model which accounts for the geometrical details of the sample and of the contact regions. Moreover, the calculations were performed by taking into account both the paramagnetic and diamagnetic contributions to the electron-photon coupling and without relying on the rotating-wave approximation. This is an important step beyond the Jaynes–Cummings model. Also, the number of many-body stated needed in the calculations increased considerably. Thus, the accuracy of the stepwise numerical diagonalization had to be carefully discussed. Finally, for systems with long relaxation time a markovian version of GME was proposed and implemented via a clever vectorization procedure.

We end this review by pointing out possible improvements of the GME method and some of its future applications. At the formal level, perhaps the most challenging upgrade is the inclusion of time-dependent potentials describing laser pulses or microwave driving signals. Provided this is succesfully achieved, one could study transport through driven nano-electromechanical systems (NEMS) or the physics of Floquet states emerging in strongly driven systems [118,119]. Let us mention here that at least for closed systems (i.e., not connected to particle reservoirs) studies based on Floquet master equations for two-level system are already available [120,121]. As for more immediate applications we aim at the theoretical modeling of transport in Tavis–Cummings systems, motivated by the recent observation of state readout in a system of distant coupled quantum dots individually connected to a pair of leads and interacting via cavity photons [29].

**Author Contributions:** all the authors have a similar contribution to the paper in its concept, research, and manuscript preparation.

**Funding:** This work was partially supported by the Research Fund of the University of Iceland, the Icelandic Research Fund, gran<sup>t</sup> no. 163082-051, the Icelandic Instruments Fund, and Reykjavik University, gran<sup>t</sup> no. 815051. Some of the computations were performed on resources provided by the Icelandic High Performance Computing Centre at the University of Iceland. V.M. also acknowledge financial support from CNCS-UEFISCDI gran<sup>t</sup> PN-III-P4-ID-PCE-2016-0084 and from the Romanian Core Program PN19-03 (contract no. 21 N/08.02.2019)

**Conflicts of Interest:** The authors declare no conflict of interest.
