\section{Introduction}
\label{sec:sec0001}

Quantum information science has paved the way for a comprehensive understanding of quantum phenomena as resources, which in turn may be consumed to perform tasks that are not possible in the classical world~\cite{RevModPhys.91.025001}. This is the case of entanglement and quantum coherence, both concepts being rigorously formulated under the framework of resource theories~\cite{PhysRevLett.122.120503,PhysRevResearch.2.012035,PhysRevLett.119.140402,PhysRevLett.116.120404}. Indeed, the former is an essential resource to enhance precision of phase estimation tasks in quantum me\-tro\-lo\-gy~\cite{PhysRevLett.102.100401}, also useful for quantum key distribution~\cite{PhysRevLett.67.661,PhysRevLett.68.557}, while the later finds applications in quantum optics~\cite{RevModPhys.37.231}, quantum thermodynamics~\cite{PhysRevA.93.052335}, and many-body physics~\cite{PhysRevB.93.184428}. In this context, electronic and spin states have emerged very promising platform for designing high-sensitive devices with applicability from material science to biochemistry~\cite{RevModPhys.89.035002,PhysRevLett.126.170404,DeMille990,natrevmats.2017.88,Aslam67,nature25781,annurev-physchem-040513-103659}. Hence, designing physical system where quantum correlation resources can be produced in a controllable fashion becomes imperative.

The ubiquitous idea proposed by Ettore Majorana~\cite{Majorana_1937} regarding particles that constitutes their own antiparticles, nowadays called Majorana fermions, opened new avenues in several areas of  physics, including quantum computation~\cite{PhysRevB.97.205404,PhysRevX.6.031016}. Unlike ordinary Dirac fermions such as electrons and protons, i.e., spin-${1}/{2}$ charged particles described by complex fields, Majorana fermions are spin-${1}/{2}$ neutral particles whose real field equations remains invariant to charge conjugation symmetry~\cite{NaturePhys.5.614.2009}. Over the past eighty years, pro\-bing the signature of Majorana fermions still remains an experimental challenge for the high energy physics community. Quite recently, Majorana fermions has been predicted as zero-mode quasi-particle excitations in quantum many-body systems, and its signatures experimentally observed in some solid-state setups~\cite{PhysRevLett.104.040502,PhysRevB.81.125318}. Indeed, Majorana fermions play an important role as quasi-particle excitation in prototypical models for fault-tolerant topological quantum computation~\cite{KITAEV20032}, possibly manipulated in topological superconductors~\cite{PhysRevB.61.10267}, or fractional quantum Hall systems~\cite{MOORE1991362}, and even in driven dissipative devices~\cite{PhysRevB.102.134501,PhysRevLett.125.147701}. Noteworthy, the interest in Majorana fermions relies on their exotic properties, such as non-abelian statistics, thus differing quite radically from the original conventional electrons that condense into the superconducting state~\cite{PhysRevLett.98.237002}. Despite the intense debate regarding some experimental results in the recent years, the hope for rea\-li\-zing quantum devices based on  Majorana fermions is still alive~\cite{d41586-021-00612-z,d41586-021-00954-8}.

Motivated by the potential applicability of Majorana fermions (MFs), in this work we study the dynamics of two fermionic systems, one composed of MFs, and the other comprising regular fermions (RFs). Both systems are mediated by a quantum dot (QD) in an experimentally feasible solid-state setup. From an open quantum dynamics approach, we investigate the behavior of the quantum resources of these fermionic systems combined to the QD, i.e., quantum coherence and entanglement, thus showing these quantities are somehow correlated. By consi\-de\-ring initial states with different fermionic occupations, the results show the two-body marginal states of each system exhibit nonzero values of concurrence and quantum coherences du\-ring the nonunitary dynamics. Moreover, the time evolution of such quantum resources behave quite distinctly when the QD is coupled to MFs as compared to the case in which it is coupled to RFs. While in the former configuration the quantum resources evolves quite similarly regardless of temperature, in the latter they are rather distinct. This is a striking feature in favor of the physical setup with MFs as the results point towards the possibility of designing quantum sensors that work in wide range of temperatures. The results presented here may be also relevant for the use of quantum correlation measures to probe non-local features of Majorana fermions as proposed by You {\it et al}~\cite{SciRep_You_2014}.

\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.8]{FIG_01.pdf}
\caption{(Color online) Schematic representation of the physical setups discussed in the paper. In the upper panel $(a)$, the non-local Majorana fermions appears as edge modes in the ends of two nanowires of length $L$. The right-most MFs labelled as ${\gamma_{B_{1}}}$ and ${\gamma_{B_{2}}}$ are coupled to a single-level quantum dot (QD), with $\lambda_1$ and $\lambda_2$ being the coupling parameters of the QD to its $j$-th neighboring MFs. In turn, the QD interacts with a fermionic reservoir at finite temperature. The lower panel $(b)$ describes the system of two electronic $f$-orbitals, i.e., regular fermions, which in turn hybridizes with the electronic orbitals of the quantum dot by means of coupling constants $\lambda_1$ and $\lambda_2$. The system of RFs$+$QD is allowed to interact to a bath of free fermions, thus modeling an open quantum system.}
\label{fig:fig00001}
\end{center}
\end{figure}

The paper is organized as follows. In Sec.~\ref{sec:sec0002} we introduce the details of the physical models. In Sec.~\ref{sec:sec0003} we describe the open system dynamics due to the coupling of the physical models with a dissipative environment. Such environment depends on the ohmicity parameter in a way that we can set both the Markovian and non-Markovian regimes. In Sec.~\ref{sec:sec0004n} we analyze the dynamics of en\-tan\-glement and quan\-tum co\-he\-rences in the system of MFs$+$QD and RFs$+$QD under the influence of the environment in both regimes, at zero and finite temperatures. In particular, Sec.~\ref{sec:sec0004a} presents the numerical analysis of occupations for a separable initial state with single-fermion occupation, while in Sec~\ref{sec:sec0004b} we discuss the dynamics of concurrence and $\ell_1$-norm of coherence for entangled initial states with single-fermion and two-fermion occupations. Focusing on the system of MFs$+$QD, we discuss in Sec.~\ref{sec:sec0005} the role of non-locality of the Majorana bound states in the dynamics of occupations and quantum resources. Finally, in Sec.~\ref{sec:conclusions} we summarize our conclusions.

\section{The Model}
\label{sec:sec0002}
In this section we will investigate the spectral properties of the two different models depicted in Fig.~\ref{fig:fig00001}: the first one consist of non-local Majorana bound states, while the second comprises canonical regular fermions. In both systems, the two fermionic species couples to a single level quantum dot (QD). Throughout this work we assume a system of spinless fermions, which can be experimentally realized by applying a strong magnetic field. In fact, spinless fermion is an important condition for the emergence of Majorana bound states in topological superconductors~\cite{MOORE1991362,PhysRevB.81.125318,0034-4885/75/7/076501}. Unless other\-wise stated, the coupling to the reservoir is turned-off, and thus both MFs$+$QD and RFs$+$QD setups can be understood as closed quantum systems. The dynamics of the open quantum system will be investigated in Sec.~\ref{sec:sec0003}.

\subsection{Majorana fermions}
\label{subsec:sec0002a}
For the case of Majorana fermions, we consider two wires with finite length $L$, supporting Majorana modes in their ends. Figure~\ref{fig:fig00001}(a) depicts this physical setting. Due to the finiteness of $L$, we shall expect a non-zero overlap between the two MFs wave functions at the ends of each nanowire. The Hamiltonian of the paired Majorana fermions is given by
\begin{equation}
 \label{eq:eq000001}
{H_{MF}} = \frac{i}{2}\, {\sum_{j = 1,2}} \, {\epsilon_{j}}\, {\gamma_{A_j}}{\gamma_{B_j}} ~,
\end{equation}
where $\gamma_{{A_j},{B_j}}$ are the two MF operators satisfying the Clifford algebra $\{ {\gamma_{X_j}}, {\gamma_{Y_l}}\} = 2 \,{\delta_{X,Y}}\,{\delta_{j,l}}$, and ${\epsilon_j}  \sim e^{-L/{\xi_j}}$ stands for the hybridization between each pair of MFs edge modes due to the overlap of their wavefunctions, with $\xi_j$ being the effective coherence length~\cite{Wang_2013}. Importantly, the coupling energy $\epsilon_j$ decreases exponentially with the length $L$ of the nanowires, thus vanishing in the case of $L \gg {\xi_j}$. Furthermore, the MF pair becomes equivalent to a single zero-energy regular fermion (RF) in the asymptotic regime $L \rightarrow \infty$. Next, we will consider the single level non-interacting quantum dot (QD) modeled by the Hamiltonian
\begin{equation}
 \label{eq:eq000002}
{H_{QD}} = {\epsilon_d}\, {\hat{n}_d} ~,
\end{equation}
where $\epsilon_d$ is the energy of the QD, ${\hat{n}_d} = {d^{\dagger}}d$ is the electron number operator, while $d$ ($d^{\dagger}$) is the electron annihilation (creation) operator. The coupling between the QD and the pair of right-most Majorana edge modes, being labelled as ${\gamma_{B_{1}}}$ and ${\gamma_{B_{2}}}$, is described by the tunneling Hamiltonian
\begin{equation}
 \label{eq:eq000003}
{H_{MD}} = {\sum_{j = 1,2}}\, {\lambda_j}({d^{\dagger}} - d)\, {\gamma_{Bj}} ~,
\end{equation}
where the real parameters $\{ \lambda_j \}_{j = 1,2}$ characterize the coupling of the QD to its $j$-th neighboring MFs.

Remarkably, Majorana operators can be mapped onto RFs creation and annihilation operators as ${\gamma_{A_j}} = i({f_j} - {f_j^{\dagger}})$, and ${\gamma_{B_j}} = {f_j} + {f_j^{\dagger}}$, where $f_j$ ($f_j^{\dagger}$) is the RF annihilation (creation) operator fulfilling the anticommutation relations $\{{f_j}, {f_l^{\dagger}}\} = {\delta_{jl}}$, and $\{{f_j},{f_l}\} = \{{f_j^{\dagger}}, {f_l^{\dagger}}\}  = 0$. Noteworthy, RFs are non-local in the sense that they are composed by Majorana modes living far apart from each other. Next, it can be readily shown that the Hamiltonian of paired MFs in Eq.~\eqref{eq:eq000001} can be recasted as
\begin{equation}
\label{eq:eq000004}
{H_{MF}} = {\sum_{j = 1,2}} \, {\epsilon_j}\left({\hat{n}_j} - \frac{1}{2}\right) ~,
\end{equation}
where ${\hat{n}_j} = {f^{\dagger}_j}{f_j}$ is the RFs number operator. Similarly, the Hamiltonian in Eq.~\eqref{eq:eq000003} which models the QD-MFs coupling is rewritten as
\begin{equation}
\label{eq:eq000005}
{H_{MD}} = {\sum_{j = 1,2}}{\lambda_j}({d^{\dagger}}{f_j} + {f_j^{\dagger}}d  + {d^{\dagger}}{f_j^{\dagger}} + {f_j}d) ~.
\end{equation}
Finally, combining Eqs.~\eqref{eq:eq000002},~\eqref{eq:eq000004}, and~\eqref{eq:eq000005}, the total Hamiltonian of the system read as
\begin{equation}
\label{eq:eq000006}
{H_M} = {\epsilon_d}{\hat{n}_d} + {\sum_{j = 1,2}}{\epsilon_j}\left({\hat{n}_j} - \frac{1}{2}\right) + {\sum_{j = 1,2}}{\lambda_j}({d^{\dagger}}{f_j} + {d^{\dagger}}{f_j^{\dagger}} + \text{H.c.}) ~.
\end{equation}

\subsection{Regular fermions}
\label{subsec:sec0002b}

For the system composed of regular fermions, we assume the QD is coupled to two other electron orbitals as depicted in Fig.~\ref{fig:fig00001}(b). These orbitals could be thought as two other quantum dots. For the sake of clariness, we will continue using the same notation of $\{ {f_j} \}_{j = 1,2}$ operators for these orbitals. In this case, the Hamiltonian of the system is written as
\begin{equation}
\label{eq:eq000007}
{H_R} = {\epsilon_d}{\hat{n}_d} + {\sum_{j = 1,2}}{\epsilon_j}\left({\hat{n}_j} - \frac{1}{2}\right) + {\sum_{j = 1,2}}{\lambda_j}({d^{\dagger}}{f_j} + {f_j^{\dagger}}{d}) ~,
\end{equation}
where the first term describes the QD, the second describes the electron orbitals, and the third accounts for the hybridization between the QD and the $f$-orbitals. Similarly to the previous case, here we have ${\hat{n}_d} = {d^{\dagger}}d$, and ${\hat{n}_j} = {f^{\dagger}_j}{f_j}$ for $j = \{1,2\}$. Note that, for convenience, the energy of the $f$-orbitals are shifted by constant $(-1/2)\sum_{j=1,2}\epsilon_j$ which does not change the dynamics of the quantum states.

\subsection{Generalized Hamiltonian}
\label{subsec:sec0002c}

Next, we will present a generalized Hamiltonian that encompasses both MFs and RFs systems. From Eqs.~\eqref{eq:eq000006} and~\eqref{eq:eq000007} we see that, apart from the terms ${d^{\dagger}} {f_j^{\dagger}} + {f_j}\, d$ inherited from the tunnel coupling of the QD and MFs, both Hamiltonians have the same form. It is therefore convenient to define a generic Hamiltonian that comprises both MFs and RFs systems as
\begin{equation}
\label{eq:eq000008}
{H_{M,R}} = {\epsilon_d}{\hat{n}_d} + {\sum_{j = 1}^2}\, {\epsilon_j}\left({\hat{n}_j} -  \frac{1}{2}\right) + {\sum_{j = 1,2}}({\lambda_j}\,{d^{\dagger}}{f_j} + {\widetilde{\lambda}_j}\,{d^{\dagger}}{f_j^{\dagger}} + {\rm H.c.}) ~.
\end{equation}
In special, the Hamiltonian describing the MFs system [see Eq.~\eqref{eq:eq000006}] is recovered by choosing ${\widetilde{\lambda}_j} = {\lambda_j}$, while that one modeling the RFs system [see Eq.~\eqref{eq:eq000007}] is obtained by setting ${\widetilde{\lambda}_j} = 0$.

In the following we will discuss the matrix representation of the generalized Hamiltonian with respect to the basis $\{ |{n_1},{n_2}, {n_d}\rangle\}$, where the index ${n_{1,2,d}} = \{ 0,1 \}$ assigns the occupation number in the single-particle states created with the operators $f^{\dagger}_1$, $f^{\dagger}_2$, and $d^{\dagger}$ acting on the vacuum state $\ket{\tilde 0}\equiv |{0_1},{0_2},{0_d}\rangle$. To study in details the Hamiltonian in Eq.~\eqref{eq:eq000008} we fix the basis ordering $\{ \ket{\tilde 0}$, ${f_1^{\dagger}} {d^{\dagger}}\ket{\tilde 0}$, ${f_2^{\dagger}} {d^{\dagger}}\ket{\tilde 0}$, ${f_1^{\dagger}}{f_2^{\dagger}}\ket{\tilde 0}, {d^{\dagger}}\ket{\tilde 0}$, ${f_1^{\dagger}}\ket{\tilde 0}$, ${f_2^{\dagger}}\ket{\tilde 0}$, ${f_1^{\dagger}} {f_2^{\dagger}} {d^\dagger} \ket{\tilde 0} \}$. Note the first four states in such basis are eigenvectors of the operator $\hat N = {\hat{n}_1} + {\hat{n}_2} + {\hat{n}_d}$ with even eigenvalues, while for the last four states the eigenvalues are odd. We clearly see that all the processes described by the generic Hamiltonian either conserves the number of particle (for the RFs system), or change it by two particles (for the MFs system). Onto this basis the Hamiltonian in Eq.~\eqref{eq:eq000008} takes the form
\begin{equation}
\label{eq:eq000010}
H = \frac{1}{2}\left(\mathbb{I} + {\sigma_z} \right) \otimes {\mathcal{H}_0} + \frac{1}{2}\left(\mathbb{I} - {\sigma_z}\right)\otimes {\mathcal{H}_1} ~,
\end{equation}
where $\mathbb{I}$ is the $2\times 2$ identity matrix and  $\sigma_z$ is the Pauli matrix, with the Hermitian blocks
\begin{equation}
\label{eq:eq000011}
{\mathcal{H}_0} = \left[\begin{matrix} -{\epsilon_+} & -{\widetilde\lambda_1} & -\widetilde\lambda_2 & 0\\
- {\widetilde\lambda_1} & {\epsilon_d} + {\epsilon_-} & 0 & {\lambda_2} \\
-{\widetilde\lambda_2} & 0 & {\epsilon_d} - {\epsilon_-} & -{\lambda_1} \\
0 & {\lambda_2} & -{\lambda_1} & {\epsilon_+} \end{matrix}\right] ~,
\end{equation}
and
\begin{equation}
\label{eq:eq000012}
{\mathcal{H}_1} = \left[\begin{matrix} {\epsilon_d} - {\epsilon_+} & {\lambda_1} & {\lambda_2} & 0\\
{\lambda_1} & {\epsilon_-} & 0 & - {\widetilde{\lambda}_2} \\
{\lambda_2} & 0 & - {\epsilon_-} & {\widetilde{\lambda}_1} \\
0 & - {\widetilde{\lambda}_2} & {\widetilde{\lambda}_1} & {\epsilon_d} + {\epsilon_+}
\end{matrix}\right] ~,
\end{equation}
where we have defined ${\epsilon_{\pm}}  = ({\epsilon_1} \pm {\epsilon_2})/2$.
\begin{table}[!t]
\caption{Eigenvectors and eigenvalues of the Hamiltonian in Eq.~\eqref{eq:eq000010} for the case of Majorana fermions (${\widetilde{\lambda}_j} = {\lambda_j}$). Here we have defined the parameters $\xi_{\pm}=(\epsilon_d\pm\epsilon)/2$, and ${\Delta_{\pm}} = \sqrt{{\xi_{\pm}^2} + {\lambda^2_1}+\lambda_2^2}$, while ${b_{\mu\nu}^{-1}} = {\sqrt{\,2{\Delta_{\mu}}({\Delta_{\mu}} + \nu{\xi_{\mu}})}}$, and ${c_{\mu\nu}} = ({\xi_{\mu}} + \nu{\Delta_{\mu}}){b_{\mu\nu}}$, with $\mu, \nu=\{+,- \}$.}
\begin{center}
\begin{tabular}{ll}
\hline\hline
Eigenstate & Energy  \\
\hline
$|{E_1}\rangle = \left({b_{++}}({\lambda_2}{f_2^{\dag}} + {\lambda_1}{f_1^\dag}){d^\dag} + {c_{++}} \right)\ket{\tilde 0}$ & ${E_1} = {\xi_-} - {\Delta_+}$ \\
$|{E_2}\rangle = \left({b_{--}}({\lambda_1} {f_1^\dag} + {\lambda_2}{f_2^\dag}) + {c_{--}}{d^\dag}\right)\ket{\tilde 0}$ & ${E_2} = {\xi_-} - {\Delta_-}$  \\
$|{E_3}\rangle = \left({b_{+-}} ({\lambda_1}{f^\dag_2} - {\lambda_2}{f^\dag_1}) + {c_{+-}}{f_1^\dag}{f_2^\dag}{d^\dag}\right)\ket{\tilde 0} $ & ${E_3} = {\xi_+} - {\Delta_+}$  \\
$|{E_4}\rangle = \left({b_{-+}}({\lambda_1}{f^\dag_2} - {\lambda_2}{f^\dag_1}){d^\dag} + {c_{-+}}{f^{\dag}_1}{f^{\dag}_2}\right)\ket{\tilde 0}$ & ${E_4} = {\xi_+} - {\Delta_-}$  \\
$|{E_5}\rangle = \left({b_{-+}}({\lambda_2}{f^\dag_2} + {\lambda_1}{f^\dag_1}) + {c_{-+}}{d^\dag}\right)\ket{\tilde 0}$ & ${E_5} = {\xi_-} + {\Delta_-}$ \\
$|{E_6}\rangle = \left({b_{+-}}({\lambda_1}{f^\dag_1} + {\lambda_2}{f^\dag_2}){d^\dag} + {c_{+-}}\right)\ket{\tilde 0}$ & ${E_6} = {\xi_-} +{\Delta_+}$ \\
$|{E_7}\rangle = \left({b_{--}}({\lambda_2} {f^\dag_1} - {\lambda_1}{f^\dag_2}){d^\dag} - {c_{--}}{f^\dag_1}{f^\dag_2}\right)\ket{\tilde 0}$ & ${E_7} = {\xi_+} + {\Delta_-}$ \\
$|{E_8}\rangle = \left({b_{++}}({\lambda_1}{f^\dag_2} - {\lambda_2}{f^\dag_1}) + {c_{++}}{f^\dag_1}{f^\dag_2}{d^\dag}\right)\ket{\tilde 0}$ & ${E_8} = {\xi_+} + {\Delta_+}$ \\
\hline\hline
\end{tabular}
\label{tab:tab000001}
\end{center}
\end{table}

Next, we will discuss the spectral properties of the Hamiltonian in Eq.~\eqref{eq:eq000010} for the case of MFs and RFs. For simplicity, from now on we will focus on the particular case of ${\epsilon_1} = {\epsilon_2} = \epsilon$, which implies ${\epsilon_-} = 0$ and ${\epsilon_+} = \epsilon$. For the system of MFs, we set ${\widetilde{{\lambda}_j}} = {{\lambda}_j}$ in Eqs.~\eqref{eq:eq000011} and~\eqref{eq:eq000012}. The set of eigenstates ${\{|{E_j}\rangle\}_{j = 1,\ldots,8}}$ and energies ${\{ {E_j} \}_{j = 1,\ldots,8}}$ of the Hamiltonian for MFs are listed in Table~\ref{tab:tab000001}. It is worth of mentioning that ${E_1}$ corresponds to the ground state of the system. Importantly, the set of states $\{|{E_j}\rangle \}_{j = 1,\ldots,4}$ exhibits even parity respective to the occupation number, while the set $\{|{E_j}\rangle \}_{j = 5,\ldots,8}$ belongs to the odd parity sector. Particularly, for the asymptotic case $L \gg \xi_{1,2}$ in which the MFs e\-ner\-gies becomes negligible, i.e., ${\epsilon_1} = {\epsilon_2} = \epsilon \approx 0$, we thus have that ${\xi_+} = {\xi_-} = {\epsilon_d}/2$. As a consequence, the energy spectrum will collapse into the two four-fold degenerated energy levels ${\epsilon_d}/2 \pm \sqrt{ {({{\epsilon}_d}/2)^2} + {{\lambda}^2_1}+ {{\lambda}_2^2} }$.

For the system of RFs we set the parameter $\widetilde\lambda_j=0$ into Eq.~\eqref{eq:eq000010}. On the one hand, from Eq.~\eqref{eq:eq000011} it follows that $\mathcal{H}_0$ exhibits a one-dimensional matrix block corresponding to the occupation quantum number $N = 0$, and also a three-dimensional block respective to $N = 2$. On the other hand, $\mathcal{H}_1$ in Eq.~\eqref{eq:eq000012} presents a three-dimensional matrix block regarding to $N = 1$, and a one-dimensional block for the $N = 3$ parity sector. The set of eigenstates ${\{|{E_j}\rangle\}_{j = 1,\ldots,8}}$ and energies ${\{ {E_j} \}_{j = 1,\ldots,8}}$ of the Hamiltonian for RFs are listed in Table~\ref{tab:tab000002}.

\section{Dynamics of the open quantum system}
\label{sec:sec0003}

In this section we will describe the dynamics of both the systems of MFs$+$QD and RFs$+$QD undergoing the dissipative effects from the coupling to a fermionic reservoir $\mathcal{B}$ that is initialized at the equilibrium state (see Fig.~\ref{fig:fig00001} for details). In this analysis, we set ${H_S} = {H_{M,R}}$ being the generalized Hamiltonian of the systems [see Eq.~\eqref{eq:eq000008}], which recovers either the Hamiltonian of MFs$+$QD [see Eq.~\eqref{eq:eq000006}] or RFs$+$QD [see Eq.~\eqref{eq:eq000007}] depending on adjusting of the physical parameters. In turn, the QD is weakly coupled to an environment of free fermions at finite temperature, with Hamiltonian ${H_{\mathcal{B}}} = {\sum_k}\, {\varepsilon_k}{c_k^{\dagger}}{c_k}$, where ${c_k^{\dagger}}$ (${c_k}$) is the creation (annihilation) operator respective to the $k$-th fermionic mode, while ${\varepsilon_k}$ is the energy~\cite{PhysRevB.101.155134,PhysRevE.102.012136}. The Hamiltonian modeling the inte\-rac\-tion between the system and the bath $\mathcal{B}$ is given by ${H_I} = {\sum_k} \, {g_k} ({d^{\dagger}}{c_k} + d{c_k^{\dagger}})$, where $g_k$ is the coupling strength. Hence, the Hamiltonian of the joint system read as $H := {H_S} + {H_{\mathcal{B}}} + {H_I}$.
\begin{table}[t]
\caption{Eigenstates and energies of the Hamiltonian in Eq.~\eqref{eq:eq000010} for the case of regular fermions (${\widetilde{\lambda}_j} = 0$). Here we have defined the parameters $\xi_{\pm}=(\epsilon_d\pm\epsilon)/2$, and ${\Delta_{\pm}} = \sqrt{{\xi_{\pm}^2} + {\lambda^2_1}+\lambda_2^2}$, while ${b_{\mu\nu}^{-1}} ={\sqrt{\, 2{\Delta_{\mu}}({\Delta_{\mu}} + \nu{\xi_{\mu}})}}$, and ${c_{\mu\nu}} = ({\xi_{\mu}} + \nu{\Delta_{\mu}}){b_{\mu\nu}}$, with $\mu, \nu=\{+,- \}$.}
\begin{center}
\begin{tabular}{ll}
\hline\hline
Eigenstate & Energy  \\
\hline
$|{E_1}\rangle = \ket{\tilde 0}$ & ${E_1} = - \epsilon$ \\
$|{E_2}\rangle = \frac{1}{\sqrt{ {\lambda_1^2} + {\lambda_2^2}}}\, ({\lambda_1}{f_2^{\dagger}} - {\lambda_2}{f_1^{\dagger}})\ket{\tilde 0}$ & ${E_2} = 0$  \\
$|{E_3}\rangle = \frac{1}{\sqrt{ {\lambda_1^2} + {\lambda_2^2}}}\, ({\lambda_1}{f_1^\dag} + {\lambda_2}{f_2^{\dagger}}){d^\dag}\ket{\tilde 0}$ & ${E_3} = {\epsilon_d}$  \\
$|{E_4}\rangle = {f_1^{\dagger}}{f^\dag_2}{d^\dag}\ket{\tilde 0} $ & ${E_4} = {\epsilon_d} + \epsilon$  \\
$|{E_5}\rangle = \left({b_{--}}({\lambda_1}{f^\dag_1} + {\lambda_2}{f^\dag_2}) + {c_{--}}{d^\dag}\right)\ket{\tilde 0}$ & ${E_5} = {\xi_-} - {\Delta_-}$ \\
$|{E_6}\rangle = \left({b_{-+}}({\lambda_1}{f^\dag_2} - {\lambda_2}{f^\dag_1}){d^\dag} + {c_{-+}}{f_1^{\dagger}}{f_2^{\dagger}}\right)\ket{\tilde 0}$ & ${E_6} = {\xi_+} - {\Delta_-}$ \\
$|{E_7}\rangle = \left({b_{-+}}({\lambda_1} {f^\dag_1} + {\lambda_2}{f^\dag_2}) + {c_{-+}}{d^\dag}\right)\ket{\tilde 0}$ & ${E_7} = {\xi_-} + {\Delta_-}$ \\
$|{E_8}\rangle = \left({b_{--}}({\lambda_2}{f^\dag_1} - {\lambda_1}{f^\dag_2}){d^{\dagger}} - {c_{--}}{f^\dag_1}{f^\dag_2}\right)\ket{\tilde 0}$ & ${E_8} = {\xi_+} + {\Delta_-}$ \\
\hline\hline
\end{tabular}
\label{tab:tab000002}
\end{center}
\end{table}

\subsection{Density matrix formalism}

To obtain the dynamics of the physical quantities of the system, we employ the well-known density matrix formalism, within which the dynamics of the system can be obtained by tracing out the environmental degrees of freedom. The re\-sul\-ting quantum master equation can be written as~\cite{PhysRevA.60.91,RevModPhys.89.015001}
\begin{align}
\label{eq:eq000013}
\frac{d{\rho_S}(t)}{dt} &= - i[{H_S} ,{\rho_S} (t) ] + {\int_0^t} d\tau  {{\alpha}^+}(t - \tau) [({V_{\tau - t}} \, {d^{\dagger}}) {\rho_S} (t) ,d] \nonumber\\
&+{\int_0^t} d\tau {{\alpha}^-}(t - \tau)[({V_{\tau - t}} \, d) \, {\rho_S} (t), {d^{\dagger}}] + \text{H.c.} ~,
\end{align}
in which ${V_{\tau - t}}\bullet = {e^{ i (\tau - t) {H_S}}} \bullet \, {e^{- i (\tau - t) {H_S}}}$, and the correlation functions read as
\begin{equation}
\label{eq:eq000014}
{\alpha^+} (t) = {\int_0^{\infty}} d\omega J(\omega) {N_F}(\omega){e^{i\omega t}} ~,
\end{equation}
and
\begin{equation}
\label{eq:eq000015}
{\alpha^-} (t) = {\int_0^{\infty}}  d\omega J(\omega)({N_F}(\omega) + 1){e^{-i\omega t}} ~.
\end{equation}
Here, ${N_F}({\omega}) = {{[\exp(\beta{\omega})+1]}^{-1}}$ is the Fermi-Dirac distribution describing the fermionic reservoir at a given temperature $T= ({k_B}\beta)^{-1}$, and $J(\omega) = {g^2}(\omega)\, {\left|{ {\partial {\omega}(k)}/{\partial k}} \right|}^{-1}$ is the spectral density of the environment, in which $g(\omega)$ is the density of states of the bath. Hereafter we set Boltzmann's and Plank's constants to the unity, i.e., $k_B = \hbar = 1$. Furthermore, we have implicitly assumed the chemical potential of the bath to be zero.

\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.9]{FIG_02.pdf}
\caption{(Color online) Spectral density $J(\omega)$ for $s = 1$ ({\it ohmic} case), coupling strength $\gamma = 0.05$, and cut-off frequencies ${\omega_c} = 10$ (red solid line), ${\omega_c} = 30$ (blue dashed line), and ${\omega_c} = 50$ (black dotted line).}
\label{fig:fig00002}
\end{center}
\end{figure}
Overall, the behavior of a system can be described by considering an accurate model for the spectral density at low frequencies. Therefore, from now on we will focus on the affect of a generic bath on the system as described the spectral density~\cite{RevModPhys.59.1,CALDEIRA1983587,Weiss_2008_book}
\begin{equation}
\label{eq:eq000016}
J(\omega) = \gamma \, {{\omega}^s} \, {{\omega}_c^{1 - s}} {e^{-\omega/{{\omega}_c}}} ~,
\end{equation}
for all $s > 0$, where $\gamma$ is the coupling strength of the system and the environment. The exponential factor in Eq.~\eqref{eq:eq000016} provides a smooth cut-off for the spectral density, which is modulated by the frequency $\omega_c$. The environment can be classified as \textit{sub-ohmic} ($0 < s < 1$), \textit{ohmic} ($s=1$), and \textit{super-ohmic} ($s>1$)~\cite{CALDEIRA1983587,RevModPhys.59.1,Weiss_2008_book}. The frequency $\omega_c$ describes the decaying time scale of the environment as ${\tau_c}\sim {\omega_c^{-1}}$. The Markov limit would correspond to the case ${{\tau}_c} \ll {1/{\Gamma}}$, i.e., when the environment correlation time $\tau_c$ is much smaller than the typical dissipation time scale of the system given by $\Gamma \sim {\int_0^{\infty}} dx\, {{\alpha}^-}(t - x)$. For clarity, in Fig.~\ref{fig:fig00002} we show the spectral density $J(\omega)$ in the ohmic case $s = 1$, for ${\omega_c} = 10$ (non-Markovian) and ${\omega_c} = 50$ (Markovian), also setting $\gamma = 0.05$. Generically speaking, the larger $\omega_c$ is the more Markovian is the dynamics of the system.

To solve the master equation, one may recast the marginal state ${{\rho}_S}(t)$ in terms of the occupation number basis $\{|{n_1},{n_2},{n_d}\rangle\}$ as
\begin{equation}
\label{eq:eq000017}
{{\rho}_S}(t) = {\sum_{ \mathbf{k}, \mathbf{m} }} \, {{A}^{ {k_1}, \, {k_2}, \, {k_d}}_{ {m_1}, \, {m_2},\, {m_d} }}(t)\,  |{k_1},{k_2},{k_d}\rangle\langle{m_1},{m_2},{m_d}| ~,
\end{equation}
where $\mathbf{k} = ({k_1},{k_2},{k_d} )$, $\mathbf{m} = ( {m_1},{m_2},{m_d})$, with ${k_j} = \{0,1\}$, and ${m_j} = \{0,1\}$ for $j = \{1,2,d\}$, while
\begin{equation}
\label{eq:eq000018}
{{A}^{ {k_1},\,{k_2},\, {k_d}}_{ {m_1},\, {m_2},\, {m_d} }}(t) = \langle{k_1},{k_2},{k_d}|{{\rho}_S}(t)|{m_1},{m_2},{m_d}\rangle ~.
\end{equation}
Plugging Eq.~\eqref{eq:eq000018} into Eq.~\eqref{eq:eq000013}, we obtain a set of coupled differential equations for the time-dependent coefficients $\{ {{A}^{ {k_1}, \, {k_2}, \, {k_d}}_{ {m_1}, \, {m_2},\,{m_d} }}(t) \}_{\mathbf{k}, \mathbf{m}}$, whose solution fully characterize the reduced density matrix ${{\rho}_S}(t)$, for a given initial state ${{\rho}_S}(0)$ of the system. The analytical solution is rather complicated and far from trivial. We refer to Appendix~\ref{sec:appendix0A} for details on simplifying the master equation.

The density matrix ${{\rho}_S}(t)$ stores information about the evolution of the system. The solution of the master equation in Eq.~\eqref{eq:eq000013}, also using Eq.~\eqref{eq:eq000017}, allow us to study the occupation numbers $\{ \langle{\hat{n}_{\mu}}\rangle\}_{\mu = 1,2,d}$ of the QD and the two fermions, which can be written as
\begin{equation}
\label{eq:eq000019}
\langle{\hat{n}_{\mu}}\rangle = {\sum_{{n_1},\,{n_2},\, {n_d}}}\, {n_{\mu}} \, {{A}^{ {n_1}, \, {n_2}, \, {n_d}}_{ {n_1}, \, {n_2},\, {n_d} }} \, (t) ~,
\end{equation}
where ${\hat{n}_d} = {d^{\dagger}}d$, ${\hat{n}_1} = {f_1^{\dagger}}{f_1}$, and ${\hat{n}_2} = {f_2^{\dagger}}{f_2}$ are the fermion number operator.

\subsection{Entanglement and quantum coherences}
\label{sec:sec0004}

Here we will introduce the minimal theoretic framework to study the role of entanglement and quantum cohe\-ren\-ces in both the systems of MFs and RFs, with respect to the two-body reduced states
\begin{equation}
\label{eq:eq000020}
{\rho_{jl}}(t) = {\sum_{ {k_j}, \, {k_l} }}~{\sum_{ {m_j}, \, {m_l} }}~{{A}^{ {k_j},\, {k_l}}_{ {m_j},\, {m_l} }}(t) \, |{k_j},{k_l}\rangle\langle{m_j},{m_l}| ~,
\end{equation}
with $j,l = \{1,2,d\}$ and $j \neq l$, ${k_j} = \{0,1\}$, and ${m_j} = \{0,1\}$, where
\begin{equation}
\label{eq:eq000021}
{{A}^{ {k_j},\, {k_l}}_{ {m_j},\, {m_l} }}(t) = {\sum_{\substack{{k_y} : \, y \neq j \neq l }}}~{{A}^{ {k_1},\, {k_2},\, {k_d}}_{ {m_1},\, {m_2},\, {k_d} }}(t) ~.
\end{equation}
For our purposes, we will address quantum correlations according to the concurrence, a bipartite entanglement quantifier~\cite{RevModPhys.81.865}, while for quantum coherences our analysis is based on the so-called $\ell_1$-norm of coherence~\cite{RevModPhys.89.041003}.  The concurrence is defined as ~\cite{PhysRevLett.78.5022,PhysRevLett.80.2245}
\begin{equation}
\label{eq:eq000022}
\text{Conc}[\rho] = \max(0,{\varpi_1} - {\varpi_2} - {\varpi_3} - {\varpi_4}) ~,
\end{equation}
where $\{ {\varpi_j} \}_{j = 1,\ldots,4}$ are the eigenvalues in decreasing order of the matrix
\begin{equation}
\label{eq:eq000023}
R[\rho] := \sqrt{ \sqrt{\rho}\, ({\sigma_y}\otimes{\sigma_y}){\rho^*}({\sigma_y}\otimes{\sigma_y})\sqrt{\rho} } ~.
\end{equation}
Opposite to entanglement, quantum coherence is a basis dependent quantity, and thus its formulation requires to fix some preferred basis states. Hereafter we will adopt the reference basis $\{ | {n_1},{n_2}\rangle, |{n_1},{n_d}\rangle, |{n_2},{n_d}\rangle \}$, with ${n_j} = \{0,1\}$ for $j = \{1,2,d\}$. To characterize the quantum coherence stored in the marginal states of Eq.~\eqref{eq:eq000020}, we consider the so-called ${{\ell}_1}$-norm of coherence, i.e., a monotonic distance-based quantifier of coherence written as~\cite{BCP2014}
\begin{equation}
\label{eq:eq000024}
{\mathcal{C}_{\ell_1}}[{\rho_{jl}}(t)] = {\sum_{\substack{j\neq l ; \, {\mathbf{n}_{jl}} \neq {\mathbf{k}_{jl}}  }}} \, |\langle{k_j},{k_l}|\, {\rho_{jl}}(t) |{n_j},{n_l}\rangle|  ~,
\end{equation}
where ${\mathbf{n}_{jl}} := ({n_j},{n_l} )$, and ${\mathbf{k}_{jl}} := ({k_j},{k_l} )$, with ${n_j} = \{0,1\}$, and ${k_j} = \{0,1\}$, for $j,l = \{1,2,d\}$. In what follows we present our numerical results obtained from the equations derived above.

\section{Numerical results}
\label{sec:sec0004n}

To obtain our numerical results we set the coupling strength $\gamma = 0.05$, $\lambda_2=2\lambda_1=0.2$ and $\epsilon_d=0.5$. Bearing in mind that the chemical potential of the bath is zero, this fixed value of $\epsilon_d$ serves as an energy reference for our calculations. Otherwise stated, we will also use $\epsilon=0.5$. Moreover, we will assume a bath characterized by $s=1$. Our analysis will consider either Markovian and non-Markovian baths, for which we use ${\omega_c} = 10$ and  ${\omega_c} = 50$, respectively. The results will also be shown for zero temperature ($T=0$) and finite temperature ($\beta = 1$). We will consider initial states within the Hilbert subspace of the system characterized by occupation $N=1$ (odd) and $N=2$ (even).

\begin{figure}[t!]
\begin{center}
\includegraphics[scale=0.925]{FIG_03.pdf}
\caption{(Color online) Populations of the subsystem of MFs$+$QD (black solid line), and RFs$+$QD (red dotted line), where $\langle \bullet \rangle = \text{Tr}(\bullet \, {\rho_S}(t))$, with ${\rho_S}(t)$ being the reduced density matrix obtained from the master equation in Eq.~\eqref{eq:eq000013}. The system of MFs$+$QD (RFs$+$QD) is initialized at the state ${{\rho}_S}(0) = |\tilde{1}\rangle\langle\tilde{1}|$, with $|\tilde{1}\rangle = {d^{\dagger}}\ket{\tilde 0}$, and coupled to a fermionic reservoir at zero temperature $(T = 0)$. Here we set the coupling strength $\gamma = 0.05$, $s = 1$, and cut-off frequencies ${\omega_c} = 10$ (left panels), and ${\omega_c} = 50$ (right panels).}
\label{fig:fig00003}
\end{center}
\end{figure}
\begin{figure}[t!]
\begin{center}
\includegraphics[scale=0.925]{FIG_04.pdf}
\caption{(Color online)
Populations of the subsystem of MFs$+$QD (black solid line), and RFs$+$QD (red dotted line), where $\langle \bullet \rangle = \text{Tr}(\bullet \, {\rho_S}(t))$, with ${\rho_S}(t)$ being the reduced density matrix obtained from the master equation in Eq.~\eqref{eq:eq000013}. The system of MFs$+$QD (RFs$+$QD) is initialized at the pure state ${{\rho}_S}(0) = |\tilde{1}\rangle\langle\tilde{1}|$, with $|\tilde{1}\rangle = {d^{\dagger}}\ket{\tilde 0}$, and coupled to a fermionic reservoir at finite temperature $(\beta = 1)$. Here we set the coupling strength $\gamma = 0.05$, $s = 1$, and cut-off frequencies ${\omega_c} = 10$ (left panels), and ${\omega_c} = 50$ (right panels)
}
\label{fig:fig00004}
\end{center}
\end{figure}

\begin{figure*}[!t]
\begin{center}
\includegraphics[scale=0.95]{FIG_05.pdf}
\caption{(Color online) Comparison between concurrence (black solid line) and $\ell_1$-norm of coherence (red dashed line) for the non-Markovian dynamics (${\omega_c} = 10$) of MFs and RFs. Panels (a),~(b),~(e),~(f),~(i), and~(j) refers to the case of zero temperature ($T = 0$), while panels (c),~(d),~(g),~(h),~(k), and~(l) show the curves for the bath at finite temperature ($\beta = 1$). Here we choose the initial state of the system MFs$+$QD (RFs$+$QD) given by ${{\rho}_S}(0) = |\tilde{1}\rangle\langle\tilde{1}|$, with $|\tilde{1}\rangle = {d^{\dagger}}\ket{\tilde 0}$, and $\gamma = 0.05$, $s = 1$, $\epsilon = 0.5$, ${{\epsilon}_d} = 0.5$, and ${\lambda_2} = 2{\lambda_1} = 0.2$.}
\label{fig:fig00005}
\end{center}
\end{figure*}

\begin{figure*}[!t]
\begin{center}
\includegraphics[scale=0.95]{FIG_06.pdf}
\caption{(Color online) Comparison between concurrence (black solid line) and $\ell_1$-norm of coherence (red dashed line) for the Markovian dynamics (${\omega_c} = 50$) of MFs and RFs. Panels (a),~(b),~(e),~(f),~(i), and~(j) refers to the case of zero temperature ($T = 0$), while panels (c),~(d),~(g),~(h),~(k), and~(l) show the curves for the bath at finite temperature ($\beta = 1$). Here we choose the initial state of the system MFs$+$QD (RFs$+$QD) given by ${{\rho}_S}(0) = |\tilde{1}\rangle\langle\tilde{1}|$, with $|\tilde{1}\rangle = {d^{\dagger}}\ket{\tilde 0}$, and $\gamma = 0.05$, $s = 1$, $\epsilon = 0.5$, ${{\epsilon}_d} = 0.5$, and ${\lambda_2} = 2{\lambda_1} = 0.2$.}
\label{fig:fig00006}
\end{center}
\end{figure*}

\begin{figure*}[ht]
\begin{center}
\includegraphics[scale=0.95]{FIG_07.pdf}
\caption{(Color online) Comparison between concurrence (black solid line) and $\ell_1$-norm of coherence (red dashed line) for the non-Markovian dynamics (${\omega_c} = 10$) of MFs and RFs. Panels (a),~(b),~(e),~(f),~(i), and~(j) refers to the case of zero temperature ($T = 0$), while panels (c),~(d),~(g),~(h),~(k), and~(l) show the curves for finite temperature ($\beta = 1$). Here we choose the initial state of the system MFs$+$QD (RFs$+$QD) given by ${{\rho}_S}(0) = |\tilde{+}\rangle\langle\tilde{+}|$, with $|\tilde{+}\rangle := \frac{1}{\sqrt{2}}\, ({f_1^{\dagger}} + {f_2^{\dagger}})\ket{\tilde 0}$, and $\gamma = 0.05$, $s = 1$, $\epsilon = 0.5$, ${{\epsilon}_d} = 0.5$, and ${\lambda_2} = 2{\lambda_1} = 0.2$.}
\label{fig:fig00007}
\end{center}
\end{figure*}

\begin{figure*}[!t]
\begin{center}
\includegraphics[scale=0.95]{FIG_08.pdf}
\caption{(Color online) Comparison between concurrence (black solid line) and $\ell_1$-norm of coherence (red dashed line) for the Markovian dynamics (${\omega_c} = 50$) of MFs and RFs. Panels (a),~(b),~(e),~(f),~(i), and~(j) refers to the case of zero temperature ($T = 0$), while panels (c),~(d),~(g),~(h),~(k), and~(l) show the curves at finite temperature ($\beta = 1$). Here we choose the initial state of the system MFs$+$QD (RFs$+$QD) given by ${{\rho}_S}(0) = |\tilde{+}\rangle\langle\tilde{+}|$, with $|\tilde{+}\rangle := \frac{1}{\sqrt{2}}\, ({f_1^{\dagger}} + {f_2^{\dagger}})\ket{\tilde 0}$, and $\gamma = 0.05$, $s = 1$, $\epsilon = 0.5$, ${{\epsilon}_d} = 0.5$, and ${\lambda_2} = 2{\lambda_1} = 0.2$.}
\label{fig:fig00008}
\end{center}
\end{figure*}

\begin{figure*}[!t]
\begin{center}
\includegraphics[scale=0.95]{FIG_09.pdf}
\caption{(Color online) Comparison between concurrence (black solid line) and $\ell_1$-norm of coherence (red dashed line) for the non-Markovian dynamics (${\omega_c} = 10$) of MFs and RFs. Panels (a),~(b),~(e),~(f),~(i), and~(j) refers to the case of zero temperature ($T = 0$), while panels (c),~(d),~(g),~(h),~(k), and~(l) show the curves at finite temperature ($\beta = 1$). Here we choose the initial state of the system MFs$+$QD (RFs$+$QD) given by ${{\rho}_S}(0) = |{W}\rangle\langle{W}|$, with $|{W}\rangle := \frac{1}{\sqrt{3}}\, ({d^{\dagger}} + {f_1^{\dagger}} + {f_2^{\dagger}})\ket{\tilde 0}$, and $\gamma = 0.05$, $s = 1$, $\epsilon = 0.5$, ${{\epsilon}_d} = 0.5$, and ${\lambda_2} = 2{\lambda_1} = 0.2$.}
\label{fig:fig00009}
\end{center}
\end{figure*}

\begin{figure*}[!t]
\begin{center}
\includegraphics[scale=0.95]{FIG_10.pdf}
\caption{(Color online) Comparison between concurrence (black solid line) and $\ell_1$-norm of coherence (red dashed line) for the Markovian dynamics (${\omega_c} = 50$) of MFs and RFs. Panels (a),~(b),~(e),~(f),~(i), and~(j) refers to the case of zero temperature ($T = 0$), while panels (c),~(d),~(g),~(h),~(k), and~(l) show the curves at finite temperature ($\beta = 1$). Here we choose the initial state of the system MFs$+$QD (RFs$+$QD) given by ${{\rho}_S}(0) = |{W}\rangle\langle{W}|$, with $|{W}\rangle := \frac{1}{\sqrt{3}}\, ({d^{\dagger}} + {f_1^{\dagger}} + {f_2^{\dagger}})\ket{\tilde 0}$, and $\gamma = 0.05$, $s = 1$, $\epsilon = 0.5$, ${{\epsilon}_d} = 0.5$, and ${\lambda_2} = 2{\lambda_1} = 0.2$.}
\label{fig:fig00010}
\end{center}
\end{figure*}

\begin{figure*}[t]
\begin{center}
\includegraphics[scale=0.95]{FIG_11.pdf}
\caption{(Color online) Comparison between concurrence (black solid line) and $\ell_1$-norm of coherence (red dashed line) for the non-Markovian dynamics (${\omega_c} = 10$) of MFs and RFs at zero ($T = 0$) and finite ($\beta = 1$) temperatures. Here we choose the initial state of the system MFs$+$QD (RFs$+$QD) given by ${{\rho}_S}(0) = |\tilde{\phi}\rangle\langle\tilde{\phi}|$, with $|\tilde{\phi}\rangle := \frac{1}{\sqrt{2}}(\mathbb{I} + {f_1^{\dagger}}{f_2^{\dagger}})\ket{\tilde 0}$, and $\gamma = 0.05$, $s = 1$, $\epsilon = 0.5$, ${{\epsilon}_d} = 0.5$, and ${\lambda_2} = 2{\lambda_1} = 0.2$.}
\label{fig:fig00011}
\end{center}
\end{figure*}

\begin{figure*}[!ht]
\begin{center}
\includegraphics[scale=0.95]{FIG_12.pdf}
\caption{(Color online) Comparison between concurrence (black solid line) and $\ell_1$-norm of coherence (red dashed line) for the Markovian dynamics (${\omega_c} = 50$) of MFs and RFs at zero ($T = 0$) and finite ($\beta = 1$) temperatures. Here we choose the initial state of the system MFs$+$QD (RFs$+$QD) given by ${{\rho}_S}(0) = |\tilde{\phi}\rangle\langle\tilde{\phi}|$, with $|\tilde{\phi}\rangle := \frac{1}{\sqrt{2}}(\mathbb{I} + {f_1^{\dagger}}{f_2^{\dagger}})\ket{\tilde 0}$, and $\gamma = 0.05$, $s = 1$, $\epsilon = 0.5$, ${{\epsilon}_d} = 0.5$, and ${\lambda_2} = 2{\lambda_1} = 0.2$.}
\label{fig:fig00012}
\end{center}
\end{figure*}

\subsection{Dynamics of occupations of the single-fermion initial state}
\label{sec:sec0004a}

As we have discussed already, we are mainly interested in the dynamics of quantum coherences and correlations, before doing so, we will briefly discuss how the occupations evolve in time for a simple case of $N=1$. For this, we will consider the initial state ${{\rho}_S}(0) = |\tilde{1}\rangle\langle\tilde{1}|$, with $|\tilde{1}\rangle = {d^{\dagger}}\ket{\tilde 0}$ representing the nonzero fermionic occupation in the QD at time $t = 0$.

The dynamics of the populations $\langle f^\dagger_1 f_1\rangle$, $\langle f^\dagger_2 f_2\rangle$ and $\langle d^\dagger d\rangle$ in both MFs$+$QD (black solid line) and RFs$+$QD (red dotted line) setups are shown in Figs.~\ref{fig:fig00003} ($T = 0$) and~\ref{fig:fig00004} ($\beta=1$). Left and right panels refer to cutoff frequencies ${\omega_c} = 10$ (non-Markovian), and ${\omega_c} = 50$ (Markovian), respectively. For $T = 0$, the non-Markovian dynamics ($\omega_c = 10$) of the average occupation $\langle{d^{\dagger}}{d}\rangle$ of the QD decreases and exhibits damped oscillations in both the systems of RFs$+$QD and MFs$+$QD, going to zero in the former case, while the latter reaches a nonzero stationary value [see Fig.~\ref{fig:fig00003}(a)]. The long-time behavior of the occupation vanishes for the RFs$+$QD system because all the levels are above the chemical potential of the system, in which case the fermion leaks into the bath. In contrast, the occupations $\langle{f_j^{\dagger}}{f_j}\rangle$ will grow and oscillate with damped amplitudes, thus approaching a sta\-tio\-na\-ry value which is asymptotically zero for RFs and nonzero for MFs [see Figs.~\ref{fig:fig00003}(c), and~\ref{fig:fig00003}(e)]. Here, the finite occupation of the MFs in the long-time regime results from the terms proportional to $\widetilde\lambda_j$ in the Hamiltonian [see Eq.~\eqref{eq:eq000008}] acting as source of charge for the system.

At finite temperature ($\beta = 1$), the occupations will behave quite similarly to the case of zero temperature, except that populations of the QD and both fermions will asymptotically converge to nonzero values at later times of the dynamics [see  Figs.~\ref{fig:fig00004}(a), ~\ref{fig:fig00004}(c), and~\ref{fig:fig00004}(e)]. Here, the occupation is mainly provided by thermal excitation, since $T=1$ is of the order of the energy of the levels $\epsilon=\epsilon_d=0.5$ in this case.

In the Markovian dynamics of the occupations ($\omega_c = 50$) shown in the right panels of Figs.~\ref{fig:fig00003} and~\ref{fig:fig00004}, the population of the QD decreases quite smoothly vanishing at large $t$ for zero tem\-pe\-ra\-ture [see Fig.~\ref{fig:fig00003}(b)], while reaches a stationary nonzero value for finite temperature ($\beta = 1$) [Fig.~\ref{fig:fig00004}(b)]. Interestingly, the occupation of the QD for both the systems of MFs and RFs show similar behaviors, with a negligible difference. This is because the temperature used is large enough to smooth out any effect of the non-conserving term of the Hamiltonian into the QD occupation. Moreover, the occupations $\langle{f_1^{\dagger}}{f_1}\rangle$ and $\langle{f_2^{\dagger}}{f_2}\rangle$ exhibit nonzero values that increases and reach stationary values for a wide time window in the system of MFs, for both temperatures [see Figs.~\ref{fig:fig00003}(d),~\ref{fig:fig00003}(f),~\ref{fig:fig00004}(d), and~\ref{fig:fig00004}(f)]. However, the occupations of RFs increase initially, but invert this behavior vanishing eventually at later time for $T = 0$ [see Figs.~\ref{fig:fig00003}(d), and~\ref{fig:fig00003}(f)].  At finite temperature  ($\beta = 1$)  these occupations saturate at a nonzero value for longer times [see Figs.~\ref{fig:fig00004}(d), and~\ref{fig:fig00004}(f)]. The insets clearly show fluctuations in the amplitudes of the fermionic occupations at earlier times, which in turn are smoothly suppressed as $t$ increases.

\subsection{Dynamics of coherence and correlations}
\label{sec:sec0004b}

Let us now turn our attention to the numerical analysis for entanglement and quantum coherence for different classes of initial pure states for the systems of MFs$+$QD and RFs$+$QD.

\subsubsection{Single-fermion initial state}

\paragraph{Separable initial state ---} Similar to the previous section, here we consider the initial state  ${{\rho}_S}(0) = |\tilde{1}\rangle\langle\tilde{1}|$, with $|\tilde{1}\rangle = {d^{\dagger}}\ket{\tilde 0}$. Importantly, this initial state is completely uncorrelated, and also incoherent with respect to the reference basis of states $\{ |{n_1},{n_2},{n_d}\rangle\}$.

In Figs.~\ref{fig:fig00005} and~\ref{fig:fig00006} we compare the concurrence and the $\ell_1$-norm of coherence for both systems of MFs$+$QD and RFs$+$QD for cutoff frequencies ${\omega_c} = 10$ and ${\omega_c} = 50$, respectively.  In Fig.~\ref{fig:fig00005}, note the reduced density matrix ${\rho_{12}}(t)$ [see Eq.~\eqref{eq:eq000020}] exhibits nonzero values for concurrence and $\ell_1$-norm of coherence, which in turn coincides in both systems of MFs and RFs, regardless  the temperature [see Figs.~\ref{fig:fig00005}(a),~\ref{fig:fig00005}(b),~\ref{fig:fig00005}(c), and~\ref{fig:fig00005}(d)]. In contrast, for marginal states ${\rho_{1d}}(t)$ and ${\rho_{2d}}(t)$ [see Eq.~\eqref{eq:eq000020}], the dynamics of entanglement and quantum co\-he\-ren\-ce will coincide only in the system of RFs, at zero tem\-pe\-ra\-ture [see Fig.~\ref{fig:fig00005}(f) and~\ref{fig:fig00005}(j)], thus behaving quite diffe\-ren\-tly in the other scenarios. Indeed, at finite temperature ($\beta = 1$), note that concurrence starts growing but  goes suddenly to zero for MFs and RFs, while the quantum coherence exhibits oscillations that are suppressed until approaching a nonzero constant value [see Figs.~\ref{fig:fig00005}(g),~\ref{fig:fig00005}(h),~\ref{fig:fig00005}(k), and~\ref{fig:fig00005}(l)]. In Figs.~\ref{fig:fig00005}(e) and~\ref{fig:fig00005}(i), concurrence exhibits a revival after suddenly going to zero, and then saturates at a constant value for longer times of the dynamics.

Figures~\ref{fig:fig00006}(a),~\ref{fig:fig00006}(b),~\ref{fig:fig00006}(c), and~\ref{fig:fig00006}(d) show that concurrence and quantum coherence of the marginal state ${\rho_{12}}(t)$ coincide for both species of MFs and RFs, regardless the temperature, for the Markovian frequency $\omega_c = 50$. However, for states ${\rho_{1d}}(t)$ and ${\rho_{2d}}(t)$, entanglement and quantum co\-he\-ren\-ce will coincide exclusively for RFs, at zero tem\-pe\-ra\-ture, also exhi\-bi\-ting a highly oscillating behavior [see Fig.~\ref{fig:fig00006}(f) and~\ref{fig:fig00006}(j)]. Indeed, for finite temperature ($\beta = 1$), note that concurrence starts increasing and dropping to zero suddenly for MFs [see Figs.~\ref{fig:fig00006}(g), and~\ref{fig:fig00006}(k)], while it exhibits revivals whose amplitude mostly decreases in the system of RFs [see Figs.~\ref{fig:fig00006}(h), and~\ref{fig:fig00006}(l)]. In addition, quantum cohe\-ren\-ce exhibits a highly oscillating regime, saturating at a nonzero constant value for longer times [see Figs.~\ref{fig:fig00006}(g),~\ref{fig:fig00006}(h),~\ref{fig:fig00006}(k), and~\ref{fig:fig00006}(l)]. In Figs.~\ref{fig:fig00006}(e) and~\ref{fig:fig00006}(i), concurrence exhibits a revival after suddenly going to zero, also experiencing fluctuations in its amplitude that are suppressed as starts decreasing, and then approaches zero at later times. We emphasize this behavior is strikingly different from the non-Markovian case ($\omega_c = 10$) in Figs.~\ref{fig:fig00005}(e) and~\ref{fig:fig00005}(i), in which both concurrence and quantum coherence remains nonzero for longer times of the dynamics. An important message we can take from the result from Figs.~\ref{fig:fig00005} and~\ref{fig:fig00006} is that the quantum correlations and coherences depend strongly on the temperature for RFs, but are qualitatively similar for MFs at both zero and finite temperatures.

\paragraph{Superposition initial state ---} Here we set the input state ${{\rho}_S}(0) = |\tilde{+} \rangle\langle\tilde{+}|$, with $|\tilde{+}\rangle := \frac{1}{\sqrt{2}}\, ({f_1^{\dagger}} + {f_2^{\dagger}})\ket{\tilde 0}$, in which the QD is entangled with one of the two fermionic species, also exhi\-bi\-ting nonzero coherence regarding to the same subspa\-ce. For this case, Figs.~\ref{fig:fig00007} and~\ref{fig:fig00008} show the concurrence and the $\ell_1$-norm of coherence for both systems of MFs$+$QD and RFs$+$QD, for the cutoff frequencies ${\omega_c} = 10$ and ${\omega_c} = 50$, respectively.

In Fig.~\ref{fig:fig00007}, for the case of RFs at zero temperature, we first note that the dynamics of entanglement and quantum co\-he\-ren\-ce are identical for each of two-body reduced state of the system [see Figs.~\ref{fig:fig00007}(b),~\ref{fig:fig00007}(f), and~\ref{fig:fig00007}(j)]. In opposite, for MFs at zero temperature, concurrence of state ${\rho_{12}}(t)$ exhibits a revival after going to zero, and then decreases until completely va\-ni\-shing, while the quantum coherence oscillates until reaches a stationary value [see Fig.~\ref{fig:fig00007}(a)]. In addition, Figs.~\ref{fig:fig00007}(e) and~\ref{fig:fig00007}(i) show the marginal states ${\rho_{1d}}(t)$ and ${\rho_{2d}}(t)$ have nonzero oscilla\-ting values of quantum coherence that approaches a stationary value for a wide time window. In turn, concurrence of the state ${\rho_{1d}}(t)$ increases and suddenly goes to zero for short times [see Fig.~\ref{fig:fig00007}(e)], while for the state ${\rho_{2d}}(t)$ the concurrence shows revivals after going suddenly to zero, and then oscillates and remains finite for longer times [see Fig.~\ref{fig:fig00007}(i)]. Next, moving to the case of finite temperature ($\beta = 1$), the concurrence of state ${\rho_{12}}(t)$ decreases and goes to zero in both systems of MFs and RFs, also showing an intermediate revival before it completely vanishing in the former setup, while quantum coherence starts decreasing and asymptotically converges to a stationary value [see Figs.~\ref{fig:fig00007}(c) and~\ref{fig:fig00007}(d)]. Furthermore, Figs.~\ref{fig:fig00007}(g),~\ref{fig:fig00007}(h),~\ref{fig:fig00007}(k) and~\ref{fig:fig00007}(l) show that states ${\rho_{1d}}(t)$ and ${\rho_{2d}}(t)$ have zero valued concurrence, except for a narrow peak that appears at short times of the dynamics, while the $\ell_1$-norm of coherence shows damped oscillations and then saturates to a fixed value.

The dynamics in the Markovian regime, with $\omega_c = 50$, for the same initial state as shown in Fig.~\ref{fig:fig00008}. We first note that the curves of entanglement and quantum coherence are identical for the two-body reduced states of the system of RFs at zero temperature [see Figs.~\ref{fig:fig00008}(b),~\ref{fig:fig00008}(f), and~\ref{fig:fig00008}(j)]. In particular, we highlight the damping of rapid oscillations that appears in both quantities for states ${\rho_{1d}}(t)$ and ${\rho_{2d}}(t)$ encoding information about two fermions and QD. For MFs at zero temperature, concurrence and quantum coherence of state ${\rho_{12}}(t)$ decays exponentially and saturates into a stationary value [see Fig.~\ref{fig:fig00008}(a)]. Furthermore, Figs.~\ref{fig:fig00008}(e) and~\ref{fig:fig00008}(i) show the marginal states ${\rho_{1d}}(t)$ and ${\rho_{2d}}(t)$ have nonzero damped oscillating va\-lues of concurrence and quantum coherence, with the former going to zero and the later approaching a fixed nonzero value. For finite temperature ($\beta = 1$), Figs.~\ref{fig:fig00008}(c) and~\ref{fig:fig00008}(d) show the concurrence of state ${\rho_{12}}(t)$ decreases exponentially in the system of MFs, while abruptly goes to zero in the case of RFs. In turn, quantum coherence of MFs starts decreasing and asymptotically converges to a stationary value, and for RFs the $\ell_1$-norm of coherence exhibits a revival after suddenly going to zero and then asymptotically approaches a stationary value. Figs.~\ref{fig:fig00008}(g),~\ref{fig:fig00008}(h),~\ref{fig:fig00008}(k) and~\ref{fig:fig00008}(l) show the states ${\rho_{1d}}(t)$ and ${\rho_{2d}}(t)$ have nonzero quantum coherences with oscillation damping in the amplitudes, and also nonzero entanglement signaled by a narrow peak of concurrence that survives only for a short time window of the dynamics.

\paragraph{Werner initial state ---} We now consider the time evolution of the dynamics of a third type of single-fermion initial state. We choose ${{\rho}_S}(0) = |{W}\rangle\langle{W}|$, where $|{W}\rangle := \frac{1}{\sqrt{3}}\left({d^{\dagger}} + {f_1^{\dagger}} + {f_2^{\dagger}} \right)\ket{\tilde 0}$ is the Werner state that fully correlates the two fermions and the QD, also exhibiting nonzero quantum coherence in all Hilbert subspaces. Figures~\ref{fig:fig00009} and~\ref{fig:fig00010} show the plots of concurrence and the $\ell_1$-norm of coherence for both the systems of MFs$+$QD and RFs$+$QD, for the cutoff frequencies ${\omega_c} = 10$ and ${\omega_c} = 50$, respectively.

In Fig.~\ref{fig:fig00009}, for the case of RFs at zero temperature, values of concurrence and $\ell_1$-norm of coherence coincides for each of two-body reduced state of the system [see Figs.~\ref{fig:fig00009}(b),~\ref{fig:fig00009}(f), and~\ref{fig:fig00009}(j)]. Conversely, for MFs at zero temperature, concurrence of states ${\rho_{jd}}(t)$ exhibit revivals after going to zero, while it reaches nonzero asymptotic values for the two-body states that mix both the QD and fermionic degrees of freedom [see Figs.~\ref{fig:fig00009}(e), and~\ref{fig:fig00009}(i)], and also vanishes at later times for the reduced state of two fermions [see Fig.~\ref{fig:fig00009}(a)]. In addition, for MFs at finite temperature, concurrence of the state ${\rho_{12}}(t)$ exhi\-bits revivals and asymptotically goes to zero [see Fig.~\ref{fig:fig00009}(c)], while abruptly vanishes for states ${\rho_{jd}}(t)$ [see Figs.~\ref{fig:fig00009}(g), and~\ref{fig:fig00009}(k)]. In turn, quantum coherence show fluctuations in its amplitudes and approaches stationary values, regardless of the temperature [see Figs.~\ref{fig:fig00009}(a),~\ref{fig:fig00009}(c),~\ref{fig:fig00009}(e),~\ref{fig:fig00009}(g),~\ref{fig:fig00009}(i), and~\ref{fig:fig00009}(k)]. For the system of RFs, $\ell_1$-norm of coherence exhibits damped oscillations and vanishes asymptotically for the states ${\rho_{jd}}(t)$ [see Fig.~\ref{fig:fig00009}(h), and~\ref{fig:fig00009}(l)], while for the state ${\rho_{12}}(t)$ it reaches nonzero stationary value [see Fig.~\ref{fig:fig00009}(d)].

In the Markovian regime (${\omega_c} = 50$), at zero temperature ($T = 0$), Figs.~\ref{fig:fig00010}(b),~\ref{fig:fig00010}(f), and~\ref{fig:fig00010}(j) show the concurrence and quantum coherence of all states behave identically in the system of RFs, experiencing rapid oscillations at earlier times. In contrast, for the system of MFs, both quantities takes different values and exhibit non-monotonic decays with rapid oscillations that are suppressed at later times [see Figs.~\ref{fig:fig00010}(a),~\ref{fig:fig00010}(e), and~\ref{fig:fig00010}(i)]. Next, moving to the case of finite temperature ($\beta = 1$), the concurrence of state ${\rho_{jd}}(t)$ decreases and suddenly goes to zero in both systems of MFs and RFs, while quantum coherence show fluctuations in its amplitudes and asymptotically converges to a stationary value [see Figs.~\ref{fig:fig00010}(g),~\ref{fig:fig00010}(h),~\ref{fig:fig00010}(k), and~\ref{fig:fig00010}(l)]. In addition, for the marginal state ${\rho_{12}}(t)$, both quantum resources monotonically decays and approaches stationary values [Figs.~\ref{fig:fig00010}(c), and~\ref{fig:fig00010}(d)]. We point out that quantum coherence converges to a finite value in the system MFs, while it goes to zero for the reduced state of two fermions. This show that quantum co\-he\-ren\-ce of the two-body state of MFs is a more robust quantum resource than that of RFs.

\subsubsection{Two-fermions initial state}

We now show the dynamics of correlation and quantum coherences in the system starting from an initial state containing two fermions.  Here we choose the initial state ${{\rho}_S}(0) = |\tilde{\phi}\rangle\langle\tilde{\phi}|$, where $|\tilde{\phi}\rangle := \frac{1}{\sqrt{2}}(\mathbb{I} + {f_1^{\dagger}}{f_2^{\dagger}})\ket{\tilde 0}$. Noteworthy, this initial state has even parity respective to the occupation number of fermions, also exhi\-bi\-ting nonzero correlations and quantum coherences between the two fermionic subspaces. In Figs.~\ref{fig:fig00011} and~\ref{fig:fig00012} we compare the concurrence and the $\ell_1$-norm of coherence of the systems of MFs$+$QD and RFs$+$QD, for the cutoff frequencies ${\omega_c} = 10$ and ${\omega_c} = 50$, respectively. We shall begin discussing Fig.~\ref{fig:fig00011} for the case of non-Markovian dynamics (${\omega_c} = 10$). For MFs at zero temperature, Fig.~\ref{fig:fig00011}(a) show that $\ell_1$-norm of coherence of ${\rho_{12}}(t)$ decreases and slowly, reaching a finite stationary value, while concurrence oscillates and suddenly goes to zero. In addition, Figs.~\ref{fig:fig00011}(e) and~\ref{fig:fig00011}(i) show the oscillation patterns for quantum coherence of states ${\rho_{1d}}(t)$ and ${\rho_{2d}}(t)$, with the latter saturating faster to a finite stationary value than the former. Interestingly, concurrence of state ${\rho_{1d}}(t)$ goes to zero after displaying a few revivals [see Fig.~\ref{fig:fig00011}(e)], while for the state ${\rho_{2d}}(t)$ concurrence starts increasing and exhibits periodic oscillations around a stationary value for larger times [see Fig.~\ref{fig:fig00011}(i)]. For RFs at zero temperature, Figs.~\ref{fig:fig00011}(f) and~\ref{fig:fig00011}(j) show that both quantum resources for states ${\rho_{jd}}(t)$ oscillates and vanishes for larger times. In particular, note that concurrence and quantum coherence of state ${\rho_{1d}}(t)$ oscillates appro\-xi\-ma\-te\-ly with the same period [see Fig.~\ref{fig:fig00011}(f)], while the former goes to zero faster than the latter for the reduced state ${\rho_{2d}}(t)$ [see Fig.~\ref{fig:fig00011}(j)]. For the two-body state ${\rho_{12}}(t)$, Fig.~\ref{fig:fig00011}(b) show that concurrence exhibits a revival after suddenly going to zero, while quantum coherence decay to a stationary value. Interestingly, both resources asym\-pto\-ti\-cally reaches the same numerical value for later times of the dynamics. For MFs at finite temperature, Fig.~\ref{fig:fig00011}(g) and~\ref{fig:fig00011}(k) show the $\ell_1$-norm of coherence of states ${\rho_{jd}}(t)$ starts in\-crea\-sing and slowly reaches a finite stationary value, while concurrence shows narrow peaks at the earlier times of the dynamics. In Fig.~\ref{fig:fig00011}(c), concurrence of state ${\rho_{12}}(t)$ goes abruptly to zero, while $\ell_1$-norm of coherence decreases slowly. For RFs at finite temperature, Fig.~\ref{fig:fig00011}(d) show the quantum coherence of state ${\rho_{12}}(t)$ decreases and remains finite, while concurrence suddenly goes to zero. In Fig.~\ref{fig:fig00011}(h), the concurrence of state ${\rho_{1d}}(t)$ show a narrow peak at earlier times and suddenly goes to zero, while in Fig.~\ref{fig:fig00011}(l) concurrence of ${\rho_{2d}}(t)$ is zero at all times of the dynamics. For both two-body reduced state, the $\ell_1$-norm of coherence shows damped oscillations and approaches zero.

Next, let us comment on Fig.~\ref{fig:fig00012} for the case of Markovian dynamics (${\omega_c} = 50$). For MFs at zero temperature, Fig.~\ref{fig:fig00012}(a) show that $\ell_1$-norm of coherence of ${\rho_{12}}(t)$ decreases exponentially, while concurrence suddenly goes to zero. In addition, Figs.~\ref{fig:fig00012}(e) and~\ref{fig:fig00012}(i) show the concurrence of states ${\rho_{1d}}(t)$ and ${\rho_{2d}}(t)$, with the former going to zero faster than the latter. In both cases, quantum coherence oscillates and approaches a stationary value for larger times. For RFs at zero temperature, Figs.~\ref{fig:fig00012}(f) and~\ref{fig:fig00012}(j) show that both quantum resources for states ${\rho_{jd}}(t)$ display rapid oscillations that are suppressed for later times. In contrast, for the two-body state ${\rho_{12}}(t)$, Fig.~\ref{fig:fig00012}(b) show that concurrence exhibits a revival after dropping suddenly to zero, while quantum coherence decays  monotonically to a stationary value. We point out that both resources starts saturating to the same numerical value for later times of the dynamics. For MFs at finite temperature, Fig.~\ref{fig:fig00012}(g) and~\ref{fig:fig00012}(k) show the states ${\rho_{jd}}(t)$ have zero valued concurrence, except for a narrow peak at the earlier times of the dynamics, while the $\ell_1$-norm of coherence shows damped oscillations and then saturates to a stationary value. In Fig.~\ref{fig:fig00012}(c), concurrence of state ${\rho_{12}}(t)$ abruptly goes to zero, while $\ell_1$-norm of coherence of state ${\rho_{12}}(t)$ monotonically decreases and va\-ni\-shes for larger times. For RFs at finite temperature, Fig.~\ref{fig:fig00012}(d) show the quantum coherence of state ${\rho_{12}}(t)$  decreases monotonically and remains finite, while concurrence suddenly goes to zero. For the two-body reduced states ${\rho_{jd}}(t)$, Figs.~\ref{fig:fig00012}(h) and~\ref{fig:fig00012}(l) show the $\ell_1$-norm of coherence displays oscillations that are rapidly damped, while concurrence is zero at all times of the dynamics.

Before closing this section, as a final remark is in order. Apart from the asymptotic case shown in Figs.~\ref{fig:fig00011}(b) and~\ref{fig:fig00012}(b), note that concurrence and quantum coherence display different dynamical behaviors, i.e., the two quantum resources do not coincide at any time of the dynamics. This feature somehow suggest a parity fingerprint that is related to the global occupation of the initial state. On the one hand, for initial states with odd parity in the occupation number, quantum coherence and concurrence coincide for some of the marginal states of the system [see Figs.~\ref{fig:fig00005},~\ref{fig:fig00006}, ~\ref{fig:fig00007},~\ref{fig:fig00008},~\ref{fig:fig00009},~\ref{fig:fig00010}]. On the other hand, Figs.~\ref{fig:fig00011} and~\ref{fig:fig00012} show the quantum resources behave quite differently at all times of the dynamics when the system is initialized in a probe state with even parity in the occupation number.

\section{Dynamics in the non-local regime of the Majorana fermions}
\label{sec:sec0005}

In this section we will discuss possible role of non-locality of Majorana fermions in the dynamics of occupations and quantum resources. The physical system of MFs$+$QD comprises a pair of Majorana bound states that arises at the ends of two superconductor nanowires of length $L$ [see Figure~\ref{fig:fig00001}(a)]. In turn, for large values of $L$, i.e., taking the MFs far from apart each other, it is well known the couplings $\epsilon_j \approx 0$ ($j = 1,2$) between them will become negligible, mainly because such energies decreases exponentially with the length of the nanowires. Hence, in our setting, the smaller the coupling $\epsilon$, the more non-local should be the pair of MFs. In the following, we will investigate the non-locality of MFs by setting $\epsilon = 0.0005$. For simplicity, the system of MFs$+$QD is initialized in the fully uncorrelated initial state ${{\rho}_S}(0) = |\tilde{1}\rangle\langle\tilde{1}|$, with $|\tilde{1}\rangle = {d^{\dagger}}\ket{\tilde 0}$. Moreover, we set $\gamma = 0.05$, $s = 1$, ${{\epsilon}_d} = 0.5$, and ${\lambda_2} = 2{\lambda_1} = 0.2$.
\begin{figure}[h!]
\begin{center}
\includegraphics[scale=0.925]{FIG_13.pdf}
\caption{(Color online) Populations of the subsystem of MFs$+$QD with non-local Majorana fermions ($\epsilon = 0.0005$), where $\langle \bullet \rangle = \text{Tr}(\bullet \, {\rho_S}(t))$, for both the non-Markovian (${\omega_c} = 10$) and Markovian dynamics (${\omega_c} = 50$), at zero ($T = 0$, right panels) and finite ($\beta = 1$, left panels) temperatures. The system of MFs$+$QD is initialized at the state ${{\rho}_S}(0) = |\tilde{1}\rangle\langle\tilde{1}|$, with $|\tilde{1}\rangle = {d^{\dagger}}\ket{\tilde 0}$. Here we set $\gamma = 0.05$, $s = 1$, ${{\epsilon}_d} = 0.5$, and ${\lambda_2} = 2{\lambda_1} = 0.2$.}
\label{fig:fig00013}
\end{center}
\end{figure}

\begin{figure*}[t!]
\begin{center}
\includegraphics[scale=0.95]{FIG_14.pdf}
\caption{(Color online) Plot of concurrence (black solid line) and $\ell_1$-norm of coherence (red dashed line) of the subsystem of MFs$+$QD with non-local Majorana fermions ($\epsilon = 0.0005$), for both the non-Markovian (${\omega_c} = 10$) and Markovian dynamics (${\omega_c} = 50$), at zero ($T = 0$) and finite ($\beta = 1$) temperatures. The system is initialized in the uncorrelated state ${{\rho}_S}(0) = |\tilde{1}\rangle\langle\tilde{1}|$, with $|\tilde{1}\rangle = {d^{\dagger}}\ket{\tilde 0}$, with $\gamma = 0.05$, $s = 1$, ${{\epsilon}_d} = 0.5$, and ${\lambda_2} = 2{\lambda_1} = 0.2$.}
\label{fig:fig00014}
\end{center}
\end{figure*}

Figure~\ref{fig:fig00013} shows the dynamics of the occupations in the system of MFs$+$QD. In the non-Markovian regime ($\omega_c = 10$), note the occupation of the QD and both the MFs take nonzero values and oscillates around stationary values, regardless of the temperature. For the case of Markovian dynamics ($\omega_c = 50$), we point out the occupation $\langle{d^{\dagger}}{d}\rangle$ mostly decreases, while $\{ \langle{f_j^{\dagger}}{f_j}\rangle\}_{j = 1,2}$ starts growing and approaches stationary values at later times of the dynamics, also regardless of the temperature of the system. In particular, note the occupation of the QD goes to zero at zero temperature [see Fig.~\ref{fig:fig00013}(a)], while it remains a nonzero constant for finite temperature ($\beta = 1$) [see Fig.~\ref{fig:fig00013}(b)].

In Fig.~\ref{fig:fig00014} we plot the concurrence and the $\ell_1$-norm of coherence for the system of MFs$+$QD for the cutoff frequencies ${\omega_c} = 10$ and ${\omega_c} = 50$, at zero ($T = 0$) and finite temperature ($\beta = 1$). Note both the entanglement and quantum coherence of the reduced state of fermions ${\rho_{12}}(t)$ takes nonzero values, and also coincide for all times of the dynamics, regardless of temperature and cutoff frequencies [see Figs.~\ref{fig:fig00014}(a),~\ref{fig:fig00014}(b),~\ref{fig:fig00014}(c), and~\ref{fig:fig00014}(d)]. Noteworthy, such quantum resources are robust against the environmental decohering effects and asymptotically converge to stationary values. Next, we point out the marginal states $\{ {\rho_{jd}}(t) \}_{j = 1,2}$ have nonzero oscillating quantum coherences that approach stationary values at later times, regardless the temperature, for both the non-Markovian ($\omega_c = 10$) and Markovian ($\omega_c = 50$) settings. However, the concurrence of such states is a zero valued quantity, except for narrow peaks that appears at earlier times of the dynamics as shown in Figs.~\ref{fig:fig00014}(e),~\ref{fig:fig00014}(g),~\ref{fig:fig00014}(i),~\ref{fig:fig00014}(k) ($T = 0$, $\omega_c = 10$), and Figs.~\ref{fig:fig00014}(h),~\ref{fig:fig00014}(l) ($\beta = 1$, $\omega_c = 50$). In particular, for $\beta = 1$ and $\omega_c = 10$, note that he concurrence of ${\rho_{1d}}(t)$ and ${\rho_{2d}}(t)$ shows an oscillation pattern that is smoothly suppressed to zero [see Figs.~\ref{fig:fig00014}(f),~\ref{fig:fig00014}(j)].

\section{Conclusions}
\label{sec:conclusions}

In this work we have investigated the dynamics of quantum resources  in a tripartite fermionic system coupled to an external reservoir. Two classes of systems were considered: (i) a two-level quantum dot coupled to two regular fermion levels (RFs$+$QD), and (ii) a quantum dot coupled to two pairs of Majorana fermions (MFs$+$QD). In both cases, the quantum dot is coupled to a fermionic reservoir. Invoking a quantum master equation approach that includes a memory kernel allowing study both the Markovian to the non-Markovian regimes, at zero and finite temperatures, we analyze the time evolution of the quantum resources of the systems, namely, pairwise entanglement and quantum coherence, quantified by concurrence and ${{\ell}_1}$-norm, respectively.

In general, we observe a clear distinction of the dynamics in each system, depending on the initial state and the pa\-ri\-ty of the global occupation of such state. On the one hand, for a fully uncorrelated incoherent initial state with a single fermion in the system, we observe that entanglement and coherence are generated by the dynamics at both the zero and finite temperatures in the MFs$+$QD system. On the other hand, in the RFs$+$QD these quantities are generated but survive only during certain time window, vanishing at later times of the dynamics at zero temperature, and saturating at a small value at finite temperature. Overall, these features are observed in both the Markovian and non-Markovian dissipation regimes.

For entangled and coherent single-fermion initial states, the quantum resources decrease but saturate to finite stationary values for the MFs$+$QD system regardless of temperature, for both the Markovian and non-Markovian regimes. On the other hand, for the RFs$+$QD system, the quantum coherence settle down at a finite value at nonzero temperature. Similar features are observed for choosing an initial state with two fermions in the system. We discussed the physical setting in which the Majorana fermions in each pair are almost fully decoupled from each other, i.e., typically a long topological nanowire. We observe qualitatively similar behavior, showing the robustness of our results against changes in the to\-po\-lo\-gical superconductor hosting the Majorana fermions. Finally, we highlight the correlated behavior of the quantum resources at both zero and finite temperatures in the MFs$+$QD system, which is not observed in the RFs$+$QD case. Importantly, this might be seen as a signature of the property called genuine and distributed correlated coherence, which in turn witness the amount of quantum coherence that is contained within the correlations of the tripartite system~\cite{PhysRevA.94.022329,JMathPhys_51_414013}. This suggests a potential applicability of the MFs in designing quantum sensors capable of working in a wider range of temperatures, which may be relevant in quantum metrology.

\begin{acknowledgments}
We thank In\'{e}s de Vega for fruitful conversations. The authors acknowledge the financial support from the Brazilian ministries MEC and MCTIC. The project was funded by Brazilian funding agencies CNPq (Grants 307028/2019-4 and 305738/2018-6), FAPESP (Grant No 2017/03727-0), Coordena\c{c}\~{a}o de Aperfei\c{c}oamento de Pessoal de N\'{i}vel Superior -- Brasil (CAPES) (Finance Code 001), and by the Brazilian National Institute of Science and Technology of Quantum Information (INCT/IQ).
\end{acknowledgments}

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\section*{Appendix}

\section{Details on the Master equation}
\label{sec:appendix0A}

The reduced dynamics of the system MFs$+$QD is described by the quantum master equation
\begin{align}
\label{eq:eq000024}
\frac{d{\rho_S}(t)}{dt} &= - i\, [{H_S},{\rho_S} (t) ] + {\int_0^t} d\tau  {{\alpha}^+}(t - \tau) [({V_{\tau - t}} \, {d^{\dagger}}) {\rho_S} (t) ,d] \nonumber\\
&+ {\int_0^t} d\tau {{\alpha}^-}(t - \tau)[({V_{\tau - t}} \, d) \, {\rho_S} (t),{d^{\dagger}}] + \text{H.c.} ~,
\end{align}
with ${H_S} = {H_1} + {H_2} + {H_3}$, where ${H_1} = {\epsilon_d}{\hat{n}_d}$ is the QD Hamiltonian, ${H_2} = {\sum_{j = 1}^2}\, {\epsilon_j}\left({\hat{n}_j} - {1}/{2}\right)$ is the MFs (RFs) Hamiltonian, and ${H_3} = {\sum_{j = 1,2}}({\lambda_j}\,{d^{\dagger}}{f_j} + {\widetilde{\lambda}_j}\,{d^{\dagger}}{f_j^{\dagger}} + {\rm H.c.})$ stands for the Hamiltonian modeling the MFs$+$QD (RFs$+$QD) coupling. To clarify, by setting ${\widetilde{\lambda}_j} = {\lambda_j}$ one obtains the Hamiltonian of the system MFs$+$QD, while the Hamiltonian of the system RFs$+$QD is recovered imposing that ${\widetilde{\lambda}_j} = 0$. In the former case, the e\-ner\-gies $\{ {E_j} \}_{j = 1,\ldots,8}$ and the eigenstates $\{ | {E_j}\rangle\}_{j = 1,\ldots, 8}$ of the system MFs$+$QD have been presented in Table~\ref{tab:tab000001}, while for the latter the spectral decomposition of the Hamiltonian for RFs$+$QD is given in Table~\ref{tab:tab000002} in the main text. In Eq.~\eqref{eq:eq000024} we have introduced the operator ${V_{\tau - t}}\, {d^{\dagger}} = {e^{ i (\tau - t) {H_S}}} {d^{\dagger}} \, {e^{- i (\tau - t) {H_S}}}$, which in turn can be written more conveniently as
\begin{align}
\label{eq:eq000025}
{{V}_{\tau - t}}\,{d^{\dagger}} = {\sum_{j,l = 1}^8} \, {{e}^{i(\tau - t)({{E}_j} - {{E}_l})}} \,  \langle{E_j}|{{d}^{\dagger}}|{E_l}\rangle \, |{E_j}\rangle\langle{E_l}| ~,
\end{align}
with
\begin{equation}
\label{eq:eq000026}
 \langle{E_j}|{{d}^{\dagger}}|{E_l}\rangle = {\sum_{{k_1}, \, {k_2}}}  \, {{(-1)}^{ {{k}_1} + {{k}_2} }} \langle{E_j}|{{k}_1},{{k}_2}, 1\rangle\langle{{k}_1},{{k}_2},0|{E_l}\rangle ~.
\end{equation}
Next, the system-bath correlation functions are given by
\begin{equation}
\label{eq:eq000027}
{\alpha^+} (t) = {\int_0^{\infty}} d\omega \, J(\omega) {N_F}(\omega) \, {e^{i\omega t}} ~,
\end{equation}
and
\begin{equation}
\label{eq:eq000028}
{\alpha^-} (t) = {\int_0^{\infty}} d\omega \, J(\omega)({N_F}(\omega) + 1)\, {e^{-i\omega t}} ~,
\end{equation}
with ${N_F}({\omega}) = {{[\exp(\beta{\omega})+1]}^{-1}}$ being the Fermi-Dirac distribution related to the fermionic reservoir, and $J(\omega) = \gamma \, {{\omega}^s} \, {{\omega}_c^{1 - s}} {e^{-\omega/{{\omega}_c}}}$ is the spectral density of the environment. For all $s > 0$ and finite temperature $0 < T < \infty$ (i.e., $\infty > \beta > 0$), it is straightforward to verify that
\begin{equation}
\label{eq:eq000029}
{{{{\alpha}^+}(t)}} =  \frac{\gamma}{4\, {\beta^2}} \, {(2\beta{\omega_c})^{1 - s}} \, \Gamma(1 + s)  \left[ {{\xi}_{1 + s}}\left( z(t)\right) - {{\xi}_{1 + s}}\left( z(t) + 1/2 \right) \right] ~,
\end{equation}
with
\begin{equation}
\label{eq:eq000030}
z(t) := \frac{1 + \beta{{\omega}_c} - i{\omega_c}t}{2\beta{{\omega}_c}} ~,
\end{equation}
and
\begin{equation}
\label{eq:eq000031}
{{{{\alpha}^-}(t)}} =  {{{{\alpha}^+}(t)}^*} + \frac{\gamma\, {{\omega}_c^2}\, \Gamma(1 + s)}{ {\left( 1 + i{{\omega}_c} t \right)}^{1 + s} }   ~,
\end{equation}
where ${{\xi}_{1 + s}}(x)$ defines the generalized Riemann zeta function as ${{\xi}_{1 + s}}(x) = {\sum_{k = 0}^{\infty}} \, { {{(x + k)}^{- 1 - s}} } $. Noteworthy, for the asymptotic case of zero temperature $T \rightarrow 0$ (i.e., $\beta \rightarrow \infty$), one gets ${\lim_{\beta \rightarrow \infty}} \, {{{{\alpha}^+}(t)}} = {\lim_{\beta \rightarrow \infty}} \, {{{{\alpha}^+}(t)}^*} = 0$, and thus
\begin{equation}
\label{eq:eq000032}
{\lim_{\beta \rightarrow \infty}} \, {{\alpha^-}(t)} = \frac{\gamma\, {{\omega}_c^2}\, \Gamma(1 + s)}{ {\left( 1 + i{{\omega}_c}t \right) }^{1 + s} }   ~.
\end{equation}

In the following we will discuss how to solve the master equation in Eq.~\eqref{eq:eq000024} for the reduced density matrix ${{\rho}_S}(t)$. To do so, we point out the marginal state ${{\rho}_S}(t)$ can be written in terms of the occupation number basis states $\{|{n_1},{n_2},{n_d}\rangle\}$ as follows
\begin{equation}
\label{eq:eq000033}
{{\rho}_S}(t) = {\sum_{ \mathbf{k}, \mathbf{m} }} \, {{A}^{ {k_1}, \, {k_2}, \, {k_d}}_{ {m_1}, \, {m_2},\, {m_d} }}(t)\,  |{k_1},{k_2},{k_d}\rangle\langle{m_1},{m_2},{m_d}| ~,
\end{equation}
where $\mathbf{k} = ({k_1},{k_2},{k_d} )$, $\mathbf{n} = ( {m_1},{m_2},{m_d})$, with ${k_j} = \{0,1\}$, and ${m_j} = \{0,1\}$ for $j = \{1,2,d\}$, while
\begin{equation}
\label{eq:eq000034}
{{A}^{ {k_1},\,{k_2},\, {k_d}}_{ {m_1},\, {m_2},\, {m_d} }}(t) = \langle{k_1},{k_2},{k_d}|{{\rho}_S}(t)|{m_1},{m_2},{m_d}\rangle ~.
\end{equation}
Next, we will substitute Eq.~\eqref{eq:eq000033} into Eq.~\eqref{eq:eq000024}, and also project the aforementioned master equation onto the occupation number basis, thus obtaining a set of coupled differential equations for the time-dependent coefficients $\{ {{A}^{ {k_1}, \, {k_2}, \, {k_d}}_{ {m_1}, \, {m_2},\,{m_d} }}(t) \}_{\mathbf{k}, \mathbf{m}}$. By proceeding in this way, one gets
\begin{align}
\label{eq:eq000035}
& \frac{d}{dt}{{A}^{ {\ell_1},\, {\ell_2},\,{\ell_d}}_{ {n_1},\,{n_2},\,{n_d} }}(t) = - i\, \langle{{\ell}_1},{{\ell}_2},{{\ell}_d}|[{ {{H}_S}, {{\rho}_S}(t)}]|{{n}_1},{{n}_2},{{n}_d}\rangle \nonumber\\
& + {\int_0^t}\, d\tau \, {{\alpha}^+}(t - \tau) \, \langle{{\ell}_1},{{\ell}_2},{{\ell}_d}|[{ ({{V}_{\tau - t}}\,{{d}^{\dagger}}) \, {{\rho}_S}(t), d }]|{{n}_1},{{n}_2},{{n}_d}\rangle \nonumber\\
&+ {\int_0^t}\, d\tau \, {{\alpha}^-}(t - \tau) \, \langle{{\ell}_1},{{\ell}_2},{{\ell}_d}|[{ ({{V}_{\tau - t}}\,{d}) \, {{\rho}_S}(t), {{d}^{\dagger}} }]|{{n}_1},{{n}_2},{{n}_d}\rangle + \text{H.c.} ~.
\end{align}
The set of differential equations given in Eq.~\eqref{eq:eq000035} are too complicated to be solved analytically, and thus we require some numerical simulations to fully characterize the marginal state ${\rho_S}(t)$. We point out that is possible to simplify the calculations with help of analytical expressions for each contribution that appears in the right-hand side of Eq.~\eqref{eq:eq000035}. In this regard, the matrix elements of the sum of Hamiltonians $H_1 + H_2$ respective to the occupation number basis are given by
\begin{widetext}
\begin{align}
\label{eq:eq000036}
\langle{{\ell}_1},{{\ell}_2},{{\ell}_d}|[{ {H_1} + {H_2}, {{\rho}_S}(t)}]|{{n}_1},{{n}_2},{{n}_d}\rangle = {\sum_{j = \{1,2,d\}}} {{\epsilon}_j} \, ({{\delta}_{{{\ell}_j},1}} {{\ell}_j} - {{\delta}_{{{n}_j},1}} {{n}_j}) \, {A^{{\ell_1},\,{\ell_2},\,{\ell_d}}_{{n_1},\, {n_2},\, {n_d}}}(t) ~,
\end{align}
while the matrix element for the coupling Hamiltonian $H_3$ read as
\begin{align}
\label{eq:eq000037}
\langle{{\ell}_1},{{\ell}_2},{{\ell}_d}|[{ {{H}_3}, {{\rho}_S}(t)}]|{{n}_1},{{n}_2},{{n}_d}\rangle &= {\lambda_1} {\sum_{j = 0,1}}\, {(-1)^j} \left[  {{(-1)}^{{\ell_2}}} \, {\mathcal{C}_{{\ell_1},\, {\ell_d},j}} \, {A^{{\ell_1} - 2j + 1,\, {\ell_2},\, {\ell_d} + 2j - 1}_{{n_1},\,{n_2},\,{n_d}}} (t) - {{(-1)}^{ {n_2}}} \, {\mathcal{C}_{{n_1},\, {n_d}, j}} \, {A_{{n_1} - 2j + 1,\, {n_2},\, {n_d} + 2j - 1}^{{\ell_1},\,{\ell_2},\,{\ell_d}}} (t) \right] \nonumber\\
& + {\widetilde{\lambda}_1} {\sum_{j = 0,1}}\, {(-1)^j} \left[ {(-1)^{\ell_2}} \, {\widetilde{\mathcal{C}}_{{\ell_1},\, {\ell_d},j}} \, {A^{{\ell_1} + 2j - 1,\, {\ell_2},\, {\ell_d} + 2j - 1}_{{n_1},\,{n_2},\,{n_d}}} (t) -  {{(-1)}^{{n_2}}} \, {\widetilde{\mathcal{C}}_{{n_1},\, {n_d},j}} \, {A_{{n_1} + 2j - 1,\, {n_2},\, {n_d} + 2j - 1}^{{\ell_1},\,{\ell_2},\,{\ell_d}}} (t) \right] \nonumber\\
& + {\lambda_2} {\sum_{j = 0,1}}\, {(-1)^j} \left[ {\mathcal{C}_{{\ell_2},\, {\ell_d},j}} \, {A^{{\ell_1}, \, {\ell_2} - 2j + 1,\, {\ell_d} + 2j - 1}_{{n_1},\,{n_2},\,{n_d}}} (t) -  {\mathcal{C}_{{n_2},\, {n_d},j}} \, {A_{{n_1}, \, {n_2} - 2j + 1,\, {n_d} + 2j - 1}^{{\ell_1},\,{\ell_2},\,{\ell_d}}} (t) \right] \nonumber\\
& + {\widetilde{\lambda}_2} {\sum_{j = 0,1}}\, {(-1)^j} \left[ {\widetilde{\mathcal{C}}_{{\ell_2},\, {\ell_d},j}}  \, {A^{{\ell_1},\, {\ell_2} + 2j - 1,\, {\ell_d} + 2j - 1}_{{n_1},\,{n_2},\,{n_d}}} (t) - {\widetilde{\mathcal{C}}_{{n_2},\, {n_d},j}}  \, {A_{{n_1},\, {n_2} + 2j - 1,\, {n_d} + 2j - 1}^{{\ell_1},\,{\ell_2},\,{\ell_d}}} (t) \right] ~,
\end{align}
\end{widetext}
where we have defined the time-independent coefficients
\begin{equation}
\label{eq:eq000038}
{\mathcal{C}_{x,y,j}} = {\delta_{x,j}} \, {\delta_{y, 1 - j}} \, {(-1)^x} \sqrt{(x + 1 - j)(y + j)} ~,
\end{equation}
and
\begin{equation}
\label{eq:eq000039}
{\widetilde{\mathcal{C}}_{x,y,j}} = {\delta_{x,1 - j}} \, {\delta_{y, 1 - j}} \, {(-1)^x} \sqrt{(x + j)(y + j)} ~.
\end{equation}
We point out that Eq.~\eqref{eq:eq000037} will reduce to a simpler form for the case of Majorana fermions (${\widetilde{\lambda}_j} = {\lambda_j}$), and also for regular fermions (${\widetilde{\lambda}_j} = 0$). Moreover, one may verify that
\begin{align}
\label{eq:eq000040}
& {\int_0^t}\, d\tau \, {{\alpha}^+}(t - \tau) \, \langle{{\ell}_1},{{\ell}_2},{{\ell}_d}|[{ ({{V}_{\tau - t}}\,{{d}^{\dagger}}) \, {{\rho}_S}(t), d }]|{{n}_1},{{n}_2},{{n}_d}\rangle = \nonumber\\
=& {\sum_{j,l = 1}^8}\, {{\texttt{G}}_{jl}^{+}}(t) \, \langle{E_j}|\, {{d}^{\dagger}}|{E_l}\rangle \left[ {\delta_{{n_d} , 1}} {{(-1)}^{ {{n}_1} + {{n}_2} }} \sqrt{n_d} \, {\mathcal{Y}^{{\ell_1},\,{\ell_2},\,{\ell_d}}_{{n_1},\, {n_2},\, {n_d} - 1}}(j,l,t) \right. \nonumber\\
&\left. - \, {\delta_{{\ell_d} , 0}} \, {{(-1)}^{ {{\ell}_1} + {{\ell}_2} }} \sqrt{{{\ell}_d} + 1} \, {\mathcal{Y}^{{\ell_1},\,{\ell_2},\,{\ell_d} + 1}_{{n_1},\, {n_2},\, {n_d}}}(j,l,t) \right]~,
\end{align}
and
\begin{align}
\label{eq:eq000041}
& {\int_0^t}\, d\tau \, {{\alpha}^-}(t - \tau) \, \langle{{\ell}_1},{{\ell}_2},{{\ell}_d}|[{ ({{V}_{\tau - t}}\,{d}) \, {{\rho}_S}(t), {{d}^{\dagger}} }]|{{n}_1},{{n}_2},{{n}_d}\rangle = \nonumber\\
& = {\sum_{j,l = 1}^8}\, {{\texttt{G}}_{jl}^{-}}(t) \, \langle{E_j}|{d}|{E_l}\rangle \left[ {\delta_{{n_d} , 0}} \, {{(-1)}^{ {{n}_1} + {{n}_2} }} \sqrt{{{n}_d} + 1} \, {\mathcal{Y}^{{\ell_1},\,{\ell_2},\,{\ell_d}}_{{n_1},\, {n_2},\, {n_d} + 1}}(j,l,t) \right. \nonumber\\
&\left. - \, {\delta_{{\ell_d} , 1}} {{(-1)}^{ {{\ell}_1} + {{\ell}_2} }} \sqrt{{{\ell}_d}} \, {\mathcal{Y}^{{\ell_1},\,{\ell_2},\,{\ell_d} - 1}_{{n_1},\, {n_2},\, {n_d}}}(j,l,t) \right] ~,
\end{align}
where we define
\begin{equation}
\label{eq:eq000042}
{{\texttt{G}}_{jl}^{\pm}}(t) := {\int_0^t}\, d\tau \, {{\alpha}^{\pm}}(t - \tau) \, {{e}^{i(\tau - t)({{E}_j} - {{E}_l})}} ~,
\end{equation}
and
\begin{equation}
\label{eq:eq000043}
{\mathcal{Y}^{{\ell_1},\,{\ell_2},\,{\ell_d}}_{{n_1},\, {n_2},\, {n_d}}}(j,l,t) := {\sum_{ {k_1},\, {k_2}, \, {k_d}}}  \langle{{\ell}_1},{{\ell}_2},{{\ell}_d}|{E_j}\rangle\langle{E_l}|{k_1},{k_2},{k_d}\rangle \, {A^{{k_1},\,{k_2},\,{k_d}}_{{n_1},\, {n_2},\, {n_d}}}(t) ~,
\end{equation}
with the matrix element $ \langle{E_j}|{d^{\dagger}}|{E_l}\rangle$ explicitly defined in Eq.~\eqref{eq:eq000026}.