\section{Introduction}
Batteries as the portable energy storage devices play a significant role in our modern life. Nearly all aspects of this new
era from household appliance to medical instruments, navigation, transportation and so on depend, in a way, on the performance of these systems.A typical battery contains one or more electrochemical cells in which an energy conversion occurs from chemical type to
electrical one, by means of the so-called reduction-oxidation reactions ~\cite{schmidt2018batteries,weber2011redox,pan2015redox}.\\
On the other hand, quantum effects become important as the size scale of electronic devices is quickly decreasing. So there may be an advantage from such quantum effects upon which one may build devices that show a better performance than their classical analogous ~\cite{geppert2000quantum,joulain2016quantum,mcrae2019graphene,shukla2008nonlinear,shen2014quantum,lvovsky2009optical,hedges2010efficient,dennis2002topological,ladd2010quantum,valiev2005quantum,divincenzo1999quantum}. One of such devices can be quantum storage system or quantum battery (QB).\\
QB is a collection of one or more quantum system(s) (mostly a two-level one) with the ability of energy storage. The notion of QB has been brought into the spotlight zone after the work by Alicki and Fannes ~\cite{Alicki:13}.\\
Generally, the study of QB performance has been divided into two distinct scenarios: charging and (self) discharging. The latter indicates a situation in which QB loses its energy due to interaction with surrounding environment ~\cite{arjmandi2022enhancing,Kamin:20-1,Santos:21b}. Actually, this is also prevalent among the traditional batteries ~\cite{conway1997diagnostic,Ricketts:00}. In the former, usually an external charging field acts as a charger in order to store energy in QB ~\cite{PRB2019Batteries,PRL_Andolina}.\\
Some figures of merit are proposed by which one can evaluate the useful capabilities and performance of a special QB. The most studied quantities are stored energy ~\cite{chang2021optimal,PRE2019Batteries}, ergotropy ~\cite{tabesh2020environment,Le:18,PRL_Andolina}, charging power ~\cite{Crescente:20,Andolina:19,ghosh2021fast} and so on.\\
Firstly, Alicki and Fannes claimed that global entangling operations end in an increment of energy extractable from QB ~\cite{Alicki:13}. However, this idea appeared to be imperfect since Hovhannisyan et al. ~\cite{PRL2013Huber} proved that the optimal energy extraction can be achieved using indirect consecutive permutation operations by which no entanglement is dynamically generated. But it is worth noting that the performance of such operations is a time consuming process. In fact, it is argued that collective charging operations (with entanglement creation) might offer a considerable speed up to charging process compared to parallel charging scenario in which each cell of QB is individually charged ~\cite{Binder:15}.\\
Charging power can also be enhanced by disordered interactions between quantum cells of QB ~\cite{ghosh2020enhancement}. Moreover, the effect of disorder and localization on QBs is studied ~\cite{Andolina:19-2}. It is shown that the batteries which are in many-body localized phase gain more stability and shorter optimal time scale of charging process, in comparison with those being the ergodic and Anderson localized phases.\\
 The role of quantum entanglement and coherence for two and three-cell QB are investigated in ~\cite{Kamin:20-2}. The authors have shown that while entanglement seems to be effectless or even destructive, quantum coherence presents a qualitative relation with the efficiency of the QB. However, more efforts are required to address the impact of quantum correlations in this field since a universal relation between the performance of quantum battery and its correlation contents has not yet been found ~\cite{araya2019geometrical}.\\
As stated before, energy storage in QB can be done using an external charging field. Andolina et al. considered three well-known quantum optical states Fock, coherent and squeezed vacuum as the initial state of the charger ~\cite{PRL_Andolina}. They confirmed that the coherent state is optimal in the sense of energy deposition and extraction because it results in a smaller amount of charger-battery entanglement.\\
In this paper, we consider a QB including four two-level quantum cells (qubit) interacting with one or two photonic cavities as charger(s), each of which prepared in a coherent initial state. The results reveal that, regarding the charging power of the QB, employing two independent chargers is significantly superior to single charger case, while the average photon number is kept the same for both cases. Furthermore, quantum correlated bipartite chargers eg. semi bell states result in more amount of ergotropy in comparison with uncorrelated one. Also the results show that the generated quantum correlations measured by total mutual information between QB and chargers and the quantum consonance between the qubits of the QB play a positive role as expected from the relation with amount of available ergotropy in spite of the previous work about the entanglement between QB and Chargers ~\cite{PRL_Andolina}.\\
This paper is organized as follows. The next section presents the physical model and its dynamics. Section III is dedicated to definition of some figures of merit and concepts which are used in this paper. In section IV the results of different charging scenarios and some discussions can be found. Finally, section V concludes the paper.
\section{Model}
In this paper, we consider an array of identical two-level quantum systems (qubit) as a QB, interacting with one or two independent single-mode photonic cavities as the charging device ~\cite{dicke1954coherence,tavis1968exact,tavis1969approximate}:

\begin{equation}
H=H_{B}+H_{C}+H_{I},\label{eq1}
\end{equation}
where
\begin{subequations}\label{eq:litdiff}
\begin{align}
H_{B} &=\frac{\omega_{0}}{2}\sum_{i}^{N_{B}}{\sigma_{i}^{z}} \label{eq2(a)}\\
H_{C} &=\sum_{j}^{N_{C}}{\omega_{j} a_{j}^{\dagger} a_{j}} \label{eq2(b)}\\
H_{I} &=\sum_{i,j}{J_{ij}(\sigma_{i}^{+} a_{j}+\sigma_{i}^{-} a_{j}^{\dagger})} \label{eq2(c)}
\end{align}
\end{subequations}
are battery, charger and interaction Hamiltonians, respectively. In these equations, $N_{B}$ and $N_{C}$ are the number of quantum cells of the QB and the chargers respectively, $\sigma_{i}^{k} (k=x,y,z)$ denotes the well-known Pauli matrices and $\sigma_{i}^{\pm}$ are raising and lowering operators for ith qubit and $a_{j}^{\dagger}$ ($a_{j}$) are creation and annihilation operators of jth charger. $\omega_{0}$ is the energy splitting between the ground and excited states of each qubit while $\omega_{j}$ is the frequency of photons in each cavity and $J_{ij}$ is the coupling strengths between ith qubit and jth charger.  Here, we consider the resonant condition $\omega_{j}=\omega_{0}$ and also assume that the coupling strengths are independent of battery and chargers i.e. $J_{ij}=J$. Suppose the QB and chargers undergo a unitary evolution, $U=exp(-iHt)$ where for simplicity we set $\hbar = 1$:
\begin{align}
\rho_{B/C}(t)=Tr_{C/B}(U \rho_{B}(0)\otimes\rho_{C}(0) U^{\dagger} ),\label{eq3}
\end{align}
where $\rho_{B}(0)$ and $\rho_{C}(0)$ are initial states of battery and charger(s) respectively. Thorough this paper, we focus on an initial 4-cell empty battery in which all qubits are prepared in their ground states so $\rho_{B}(0)= \ket{g} ^{\otimes 4} \bra{g}$. Meanwhile, we set each charger to be in a coherent optical state $\ket{\alpha}$ where $\alpha$ is a complex number identifying the average number of embedded photons in the cavity as $\overline{n}=|\alpha|^2$ ~\cite{Scully:Book}. In fact, it is shown that coherent state is the optimal state to charge a quantum battery since it causes less amount of battery-charger entanglement ~\cite{PRL_Andolina}.
\section{Figures of merit}\label{sec3}
In this section, we provide some figures of merit by which one can evaluate the performance of a QB.\\
\subsection{Thermodynamical performance}
 Assume a quantum battery with density matrix $\rho$ and Hamiltonian $H$. Then its total amount of stored energy is simply given by:
\begin{align}
E(\rho,H)=Tr(\rho H).\label{eq3}
\end{align}
Mostly, due to different reasons, one is not able to entirely extract this energy through unitary operations. So we need to consider ergotropy i.e. the maximum amount of energy which can be unitarily extracted ~\cite{Allahverdyan:04}. It is defined as the difference between internal energies of quantum state $\rho$ and its corresponding passive state $\eta$:
\begin{align}
\mathcal E(\rho,H)=E(\rho,H) - E(\eta,H).\label{eq5}
\end{align}

\begin{figure}[t!]
\includegraphics[width=\linewidth]{schematic.eps}
\caption{Schematic figure of the three charging schemes: (a) single, (b) double uncorrelated and (c) double correlated chargers.}
\label{Fig}
\end{figure}

If $\eta$ is passive, it provides no energy by means of unitary operations. Passive state of a quantum system possesses non-increasing population with respect to its Hamiltonian $H$ and also $[H,\eta]=0$. Consider the spectral decompositions of the battery state and Hamiltonian as $\rho = \sum_{i=1}^{d} {p_{i} \ket{p_{i}}\bra{p_{i}}}$ and $H = \sum_{i=1}^{d} {\epsilon_{i} \ket{\epsilon_{i}}\bra{\epsilon_{i}}}$ so that $p_{1}\geq p_{2}\geq . . . \geq p_{d}$ and $\epsilon_{1}\leq \epsilon_{2}\leq . . . \leq \epsilon_{d}$ where $d$ denotes the dimension of the Hilbert space. Then the passive state is $\eta=\sum_{i=1}^{d} {p_{i} \ket{\epsilon_{i}}\bra{\epsilon_{i}}}$ and ergotropy can be written as
\begin{align}
\mathcal E(\rho,H)=\sum_{i,j}^{d} {p_{i} \epsilon_{j} (|\langle p_{i}|\epsilon_{j}\rangle|^{2} - \delta_{i,j})}.\label{eq6}
\end{align}
where $\delta_{i,j}$ is the Kronecker delta function. As ergotropy is the maximum amount of extractable energy using unitary operations so the most successful extracting operations are those which transform the quantum state $\rho$ to the passive state $\eta$.\\
Another important figure of merit that determines how fast a quantum battery can be charged is power:
\begin{align}
P(t)=\frac{E(\rho,H)}{t},\label{eq8}
\end{align}
where $t$ denotes the charging time.
\subsection{Correlations}
As previously mentioned, the most efforts are dedicated to the role of quantum entanglement in the performance of quantum batteries. However, entanglement is not the only quantifier of quantum correlations. In this paper, we consider quantum consonance which estimates the global coherence in quantum systems and for a general multipartite density matrix is defined by ~\cite{pei2012using}
\begin{align}
C(\rho)=\sum_{k_{1}k_{2}...k_{n},l_{1}l_{2}...l_{n}}{|\rho_{k_{1}k_{2}...k_{n},l_{1}l_{2}...l_{n}}^{c}\prod_{m}{(1-\delta_{k_{m},l_{m}})}|},
\end{align}

\begin{figure*}[t!]
	\centering
		\includegraphics[width=1.0\linewidth]{Figure_1.eps}
	\caption{Charging power of the QB with one ($N_{C}=1$) and two ($N_{C}=2$) independent chargers.}
	\label{fig:Figure_1}
\end{figure*}

where $\rho^{c}=U \rho U^{\dagger}$ is the transformed density matrix by using some unitary operations which eliminate the local coherence of $\rho$.\\
Moreover, one may want to know how much the battery and charger(s) become correlated during the charging process. It can be quantified by battery-charger mutual information. It evaluates the total amount of correlations (classical+quantum) between two parts. Let us consider $\rho_{BC}$ to be the joint density matrix of battery and charger. Then mutual information is given by ~\cite{mahdian2016comparison,Ollivier:01}
\begin{align}
I(\rho_{BC})=S(\rho_{\rho_{B}})+S(\rho_{\rho_{C}})-S(\rho_{\rho_{BC}}),\label{eq8}
\end{align}
in which $S(\rho)=Tr(\rho log \rho)$ is the Von Neumann entropy. 
\section{Two scenarios: charging with single or double photonic cavities }\label{sec4}
As we previously mentioned, we consider each charger to be in coherent state $\ket{\alpha}$. In this section, we investigate two distinct charging scenarios (Fig. 1): charging with $i)$ one single-mode cavity $\ket{\alpha}$ and $ii)$ two single-mode cavities $\{\ket{\alpha_{1}},\ket{\alpha_{2}}\}$ with different configurations (correlated or uncorrelated).

\begin{figure}[h!]
	\centering
		\includegraphics[width=0.4\textwidth]{Figure_2.eps}
	\caption{Ergotropy of the QB with initially correlated and uncorrelated chargers.}
	\label{fig:Figure_2}
\end{figure}

\subsection{Boosting the charging power}
Now, we compare two charging cases in the first of which there is only one charger ($N_{C}=1$) in state $\ket{\alpha}$, while the second case contains two uncorrelated chargers ($N_{C}=2$) in state $\ket{\alpha_{1}}\ket{\alpha_{2}}$. For an unbiased comparison between these charging schemes we consider an equal average number of photons in both cases ($|\alpha|^{2}=|\alpha_{1}|^{2}+|\alpha_{2}|^{2}$). Fig. 2 shows the charging power for these two charging protocols with $ \alpha=0.5, 1.5,$ and $ 2.5$. Clearly, in all cases, charging with two chargers appears to have a better performance in terms of power, regardless of field intensities. It is an interesting result: though the average number of photons in both charging approaches is the same, it is possible to charge the battery with a higher rate if one use two cavities. Moreover, raising the average number of embedded photons in the cavities leads to the growth of maximum power. In fact, the gain in ability of faster charging in double cavities compared to the single cavity can be up to $76$ percent for $\alpha=2.5$.\\
 In addition, the double charger case needs a shorter time scale to reach the maximum power compared to the single charger scheme for which the peaks take longer times to emerge.\\ 
 We should mention that there is no considerable difference between the single and double chargers schemes in terms of energy injection to the quantum battery. So here the results for internal energy and ergotropy are not presented.\\
In what follows, we want to see if different configurations of chargers with or without correlations play any significant role in the charging of the QB.
\subsection{Correlated and uncorrelated chargers}
One may ask if the existence of correlations between chargers can result in a considerable effect during the charging process. So here we aim at addressing this question. We consider two double chargers schemes. In the first one, the state of two chargers is a product state $\ket{\psi}=\ket{\alpha_{1}}\ket{\alpha_{2}}$, while in the second, the two chargers are prepared to be in an entangles semi Bell state ~\cite{wang2001multipartite,chatterjee2021quantifying}

\begin{figure}[h!]
	\centering
		\includegraphics[width=0.4\textwidth]{Figure_3.eps}
	\caption{Ratio of extractable energy of the QB with initially correlated and uncorrelated chargers.}
	\label{fig:Figure_3}
\end{figure}

\begin{equation}
\ket{\phi^{+}}=\frac{1}{\sqrt{N_{+}}}(\ket{\alpha_{1}}\ket{\alpha_{1}}+\ket{\alpha_{2}}\ket{\alpha_{2}}),\label{eq8}
\end{equation}
where $N_{+}=2(1+e^{-2(|\alpha_{1}|^{2}+|\alpha_{2}|^{2})})$ is normalization constant. Through the rest of paper, we set $\alpha_{1}=-\alpha_{2}=\alpha$. In Appendix. A we provide the results of other choices of semi Bell states to prove the universality of our results.

\begin{figure}[h!]
	\centering
		\includegraphics[width=0.4\textwidth]{Figure_4.eps}
	\caption{Quantum consonance of the QB with initially correlated and uncorrelated chargers.}
	\label{fig:Figure_4}
\end{figure}
$$
$$Fig. 3 shows the ergotropy of quantum battery versus time for the two initial states $\ket{\psi}$ and $\ket{\phi^{+}}$ of the cavities. A significant capability of the semi Bell state of the cavities in charging the quantum battery can be inferred when $\alpha$ is large enough, compared to the uncorrelated state $\ket{\psi}$ (Fig. 3c). In fact, when the cavities share no initial correlation, raising the intensities of the charging field reduces the overall stored ergotropy in the quantum battery, though there is no considerable difference between the two types of chargers for smaller values of $\alpha$ (Fig. 3a). So if the preparation of low-intensity fields is not available the semi Bell state appears to be the best choice as an initial state of the chargers. In addition, it can be qualitatively seen that the maximum amount of ergotropy takes a shorter time to be deposited in the quantum battery (Fig. 3b and 3c) if one uses an initially correlated state for the photonic cavities. Therefore, this kind of charging scheme is also able to provide some advantages in terms of charging power.\\
The ratio of ergotropy and internal energy i.e. $\Gamma=\frac{\mathcal E(\rho,H)}{E(\rho,H)}$ is also an interesting quantity since it determines what fraction of the total stored energy is extractable. In Fig. 4 this ratio is depicted versus time for the two types cavity initial states. A similar behavior to ergotropy can be seen from this figure. While the ratios obtained by the two charging settings are almost the same for $\alpha=0.5$, increasing the intensity of charging fields results in the distinction of the extractable energy ratio deposited in the quantum battery by the correlated cavities.\\
The dynamics of generated quantum consonance in the quantum battery is shown in Fig. 5. As stated before, the battery is initially prepared to be in its pure ground state $\rho_{B}(0)=\ket{g}^{\otimes 4}\bra{g}$ without any local and global coherence or correlations between qubits. Then, QC is generated in the QB due to the qubits coupling and also interaction with the chargers. Generally, both types of cavity's initial states are able to create QC among qubits of the QB. However, if one prepares the correlated state $\ket{\phi^{+}}$ as the chargers, the larger amount of QC appears in the QB similar to the stored ergotropy.\\

\begin{figure}[h!]
	\centering
		\includegraphics[width=0.4\textwidth]{Figure_5.eps}
	\caption{Battery-charger mutual information with initially correlated and uncorrelated chargers.}
	\label{fig:Figure_5}
\end{figure}
$$
$$Finally, we are interested in the total amount of correlation between the QB and chargers i.e., mutual information. Actually, It is known that the battery-charger entanglement plays a destructive role in terms of ergotropy storage since it causes the purity of the QB state to be reduced ~\cite{PRL_Andolina}. However, this is not the case for battery-charger mutual information as shown by Fig. 6. While there is complete accordance  between $\ket{\phi^{+}}$ and $\ket{\psi}$ regarding the creation of mutual information between the QB and chargers (similar to the stored ergotropy) for $\alpha=0.5$, the quantities differ from each other when $\alpha$ is increased, mostly with superiority of the latter. By comparing Fig. 3 and Fig. 6 one can highly confirm that the generated mutual information between QB and chargers positively contributes to the deposition of ergotropy in the QB. To a certain extent, whenever the QB and chargers are more correlated (quantum or classical), larger amounts of ergotropy is stored in the QB.

\begin{figure}[h!]
	\centering
		\includegraphics[width=0.6\textwidth]{Figure_6.eps}
	\caption{(a) Ergotropy, (b) the ratio of extractable energy, (c) quantum consonance and (d) battery-charger mutual information for $\ket{\psi}$, $\ket{\zeta}$ and $\ket{\phi^{-}}$.}
	\label{fig:Figure_6}
\end{figure}

$$
$$\section{Conclusions}
In this paper, we consider a 4-cell QB which can interact with one or two photonic cavities as chargers. We demonstrate that the simultaneous utilization of two independent chargers in coherent states considerably increases the charging power of the QB. This result is achieved while the single and double chargers scenarios contain the same average number of photons.\\
Moreover, It is shown that when the chargers are initially correlated namely being in a semi Bell state and by using a sufficient amount of fields intensity, more ergotropy and the ratio of useful extractable energy are stored in the QB in comparison to the case in which the chargers are in a product state without any correlations. We make this more evident by preparing the results for various types of semi Bell states. These results provide a good motivation for considering the role of different correlations between the chargers in the energy storage.\\
In addition, we show that the generation of mutual information between the QB and chargers has no opposition with ergotropy deposition. This result is also interesting since it is previously shown that battery-charger entanglement plays a destructive role in ergotropy storage ~\cite{PRL_Andolina}.\\
Finally, we see that non-local coherence lies on the state of the battery has a positive qualitative relation with the performance of the QB.
\section{Apendix A: universality of the results for other types of semi Bell states}
In this section, we provide complementary materials to prove the universality of our results. We consider other types of semi Bell states as the initial state of the photonic cavities namely, $\ket{\phi^{-}}=\frac{1}{\sqrt{N_{-}}}(\ket{\alpha}\ket{\alpha}-\ket{-\alpha}\ket{-\alpha})$ and $\ket{\zeta}=\frac{1}{\sqrt{\kappa}}(\ket{\alpha}\ket{0}+\ket{0}\ket{\alpha})$ with $N_{-}=2(1-e^{-4|\alpha|^{2}})$ and $\kappa=2(1-e^{-2|\alpha|^{2}})$ ~\cite{mishra2016quantum}. As shown by Fig. 7a and 7c, both $\ket{\phi^{-}}$ and $\ket{\zeta}$ store higher amounts of ergotropy and QC in the QB compared to $\ket{\psi}$, with $\ket{\phi^{-}}$ being the best choice. In the other words, when the chargers are initially correlated, one can see better performance of the QB regarding the stored ergotropy and generated quantum correlation. This is even the case if one considers the useful extractable energy $\Gamma$ (Fig. 7b). Here, $\ket{\zeta}$ initially deposits a larger value of $\Gamma$, though in the steady limit $\ket{\phi^{-}}$ and $\ket{\zeta}$ have no considerable advantage over each other and both result in greater $\Gamma$ compared to uncorrelated initial chargers.




