Optimal state for a Tavis-Cummings quantum battery via Bethe ansatz method

Large-capacity and high-density batteries have formed an inevitable requirement with the development of industrial technology, and high-density batteries inevitably require the miniaturization of battery cells. When the battery size is small enough that its dynamics are described by quantum mechanics, it is necessary to consider the impact of quantum effects on batteries. Based on this, a series of quantum batteries have been studied in recent years [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], including Dicke quantum battery [10, 11], spin-chain quantum batteries [12, 13], Random quantum batteries [14], Sachdev-Ye-Kitaev Batteries [15], topological batteries [16], and so on. In considering the impact of quantum entanglement on quantum batteries, Alicki and Fannes found that global entangling operations could extract more work from a quantum battery than local operations [1]. Hovhannisyan et al. found that a series of $N$ global entangling operations can extract the maximum work without creating any entanglement in the quantum battery [2]. In studying the direction of the dissipative quantum battery, F. Barra found that a unitary evolution with a single globally conserved quantity can extract work from the thermodynamic equilibrium state, and this process does not violate the second law of thermodynamics [17]. J. Q. Quach and W. J. Munro use the dark states to charge and stabilize open quantum batteries. They found that the superextensive capacity and power density are correlated with entanglement, and the stored energy of the battery is stable without the need to continually access the battery [18]. Since the decoherence of the quantum system will cause the aging of the quantum battery [19], S. Y. Bai and J. H. An proposed to use Floquet engineering to suppress the decoherence of the quantum system to prevent the aging of the quantum battery [20].

In Ref. [9], the authors investigated the effect of three different cavity field initial states on the performance of the T-C quantum battery using the numerical or approximate method without considering other cavity field initial states. In this paper, we analytically solve the T-C model by the Bethe ansatz method to find the cavity field initial state that leads to the best performance of the T-C quantum battery. The Bethe ansatz (BA) is an exact method for the calculation of eigenenergies and eigenstates in quantum many-body systems, and it is a particular form of the wave function and was introduced by H. Bethe to solve the one-dimensional spin-1/2 Heisenberg model [21]. In the present, the method has been extended to other models in one dimension. Many other quantum many-body systems are known to be solvable by some variant of the Bethe ansatz [22, 23, 24, 25, 26, 27]. C. N. Yang used Bethe ansatz to study the Fermi problem and found that the prerequisite for the Fermi system to be solved is that it must satisfy a set of equations, which we now call Yang-Baxter equations [28, 29, 30]. The Yang–Baxter equation provides a set of relations that can be solved to yield new integrable models. The BA can be constructed by using the Yang-Baxter algebra of the transfer matrix to generate the wavefunctions by applying quasi-particle creation operators to a pseudo-vacuum state. The BA is the quantum version of the Inverse Scattering Method [31]. Based on the Bethe ansatz method (BAM), N. M. Bogoliubov studies a series of integrable models in quantum optics [32, 33, 34]. After constructing the BA of the system, the urgent problem is to solve the Bethe ansatz equation, a system of multivariate higher-order equations. Generally speaking, it is impossible to directly solve the Bethe ansatz equation for a large particle number system on a desktop computer. Now there are some software and methods for solving this equation, such as Bertini software [35], BertiniLab [36], Homotopy continuation method [37, 38], and so on. However, using the above methods or software on a personal desktop computer cannot solve the Bethe ansatz equation of the large particle number system. In this paper, we propose a way to quickly solve the Bethe ansatz equation on a personal desktop computer.

In this paper, we use $N$ identical two-level atoms as the stored energy device and the cavity field as the energy supply device to study which cavity field initial state can improve the stored energy capacity and the average charging power of the QB. Here, the interaction between the atomic ensemble and the optical field is described by the Tavis-Cummings (T-C) model [39], so this stored energy device is called the T-C quantum battery (TCQB). We use the BAM to obtain the eigenvalues and eigenstates of the T-C Hamiltonian and then study the relationship between the probability distribution of the cavity field initial state in the number states and the performance of the TCQB. We propose an equal probability and equal expected value splitting method, which can help us find the optimal initial state of the TCQB. At the same time, we discovered two novel phenomena. These two phenomena with rich physical meaning can be described by two empirical inequalities, and their physical meaning implies that they have a wide range of applications in experiments. Finally, due to the unavoidable interaction between the cavity QED system and its environment in the experiment, which has been studied in spin squeezing [40] and superradiation [41, 42], we investigate the effects of the decay of the optical field and the collective dephasing of the atoms on the performance of the TCQB.

As shown in Fig. 1, our system consists of a single-mode cavity field and $N$ identical two-level atoms. the cavity field and the atoms are coupled through electric dipole interaction, of which there is no interaction between the $N$ identical two-level atoms, and this model can be described by Dicke Hamiltonian ($\hbar=1$) [43, 44]

Here, $\omega_{c}$ and $\omega_{a}$ are the eigenfrequency of the cavity field and the transition frequency between the two energy levels of the atom, respectively. $\hat{a}^{\dagger}$ and $\hat{a}$ respectively represent the creation and annihilation operators of bosons, which satisfy the commutation relations $[\hat{a},\hat{a}^{\dagger}]=1$, $[\hat{a},\hat{a}]=[\hat{a}^{\dagger},\hat{a}^{\dagger}]=0$. $\hat{J}_{\alpha}=\Sigma_{i=1}^{N}\hat{\sigma}_{\alpha}^{i}/2$ in terms of the Pauli matrices $\hat{\sigma}_{\alpha}^{i}(\alpha=x,y,z)$ of the $i$-th atom is the collective angular momentum operator for the spin ensemble consisting of $N$ identical two-level atoms, these operators ${\hat{J}_{x},\hat{J}_{y},\hat{J}_{z}}$, satisfy the commutation relations of SU(2) algebra and $\hat{J}_{\pm}=\hat{J}_{x}\pm i\hat{J}_{y}$.

However, the Dicke model is not integrable, and its precise analytical solution cannot be obtained. The rotating-wave approximation (RWA) was introduced to overcome this problem. In the regime of near resonance $\omega_{c}\simeq\omega_{a}$ and relatively weak coupling $g\ll\min\{\omega_{c},\omega_{a}\}$, the counter-rotating wave term can be ignored, and the Dicke model can be reduced to following T-C model

The exact analytical solution of the T-C model exists. Under near resonance and weak coupling conditions, the Dicke model and the T-C model have the same dynamics in a short time. The primary purpose of our work is to find the optimal initial state of the cavity field so that the TCQB can achieve greater stored energy capacity and greater average charging power. Next, we will investigate the best initial state to achieve our goal by using the BAM.

In Appendix A, we already know the eigenstates and eigenvalues of $\hat{H}_{TC}$, and now we study the charging problem of the TCQB in the eigen-representation of $\hat{H}_{TC}$. Let us start to consider the dynamic evolution of the system, the initial state of the system assume is

where $\left|J,-J\right\rangle$ is the initial state of the atom ensemble, which means that all atoms are in the ground state. $\left|\psi_{0}\right\rangle$ is an arbitrary initial state of the cavity field. Then the state of the system at time $t$ is

Inserting the following unit operator in front of the initial state in Eq. (4)

where $N_{\sigma}$ is the normalization coefficient of the $\sigma$-th eigenstate $\left|\Phi_{J,M}\left(\{\lambda^{\sigma}\}\right)\right\rangle$. Then, for a particular $M$, we can obtain the state of the system at time $t$

where $C\left(\Phi_{J,M}^{\sigma},\Phi(0)\right)=\left\langle\Phi_{J,M}\left(\{\lambda^{\sigma}\}\right)\right|\left.\Phi(0)\right\rangle$, it is the inner product between the eigenstate of Hamiltonian $\hat{H}_{TC}$ and the initial state of the system, and its specific expression is

The calculation of the above equation is shown in Appendix B. Here, $C(M,\psi_{0})=\left\langle M|\psi_{0}\right\rangle$, and $(\left\langle M\right|)^{\dagger}$ is the number state which is the eigenstate of the number operator $\hat{a}^{\dagger}\hat{a}$.

The energy acquired by the atomic ensemble from the cavity field at time $t$ is

and the average charging power

Since $\hat{M}=\hat{J}_{z}+\hat{a}^{\dagger}\hat{a}$, we can rewrite the above equation as follows

Here we use $\left\langle\Phi(0)\right|\hat{M}\left|\Phi(0)\right\rangle$ instead of $\left\langle\Phi(t)\right|\hat{M}\left|\Phi(t)\right\rangle$ for $\hat{M}$ is a conserved quantity. Substituting the initial state $\left|\Phi(0)\right\rangle$ into the above equation, then

where $\bar{n}_{0}=\left\langle\Phi(0)\right|\hat{a}^{\dagger}\hat{a}\left|\Phi(0)\right\rangle$ is the average photon number in the initial state. The physical meaning of the above equation is well understood. Since the total number of excitations is conserved, the total energy acquired by all atoms must be equal to the energy lost by the cavity field. $\bar{n}_{0}$ is determined by the initial state of the cavity field, so we just need to obtain the average photon number in the cavity field at time $t$, and it is

Substituting Eq. (6) into the above equation, then we obtain

where

In summary, the energy acquired by the atomic ensemble at time $t$ is

The average photon number of the initial state can be expanded in the number state space

then

and the average charging power

here $F(M,t)=M-f(M,t)$, $\left|C(M,\psi_{0})\right|^{2}=\left|\left\langle M|\psi_{0}\right\rangle\right|^{2}$ is the probability distribution of the initial state $\left|\psi_{0}\right\rangle$ in the number states, and $F(M,t)$ is a binary function that varies with $M$ and $t$. From the above equation, we can see that the energy obtained by the atomic ensemble from the cavity field at time $t$ is actually an expected value of the function $F(M,t)$ under the probability distribution $\left|C(M,\psi_{0})\right|^{2}$. If we choose the initial state of the cavity field as the number state $\left|\psi_{0}\right\rangle=\left|M\right\rangle$, then

Here, we refer to $F(M,t)$ as the number-state stored energy. In fact, as long as we know the probability distribution of the optical field initial state in the number states, we can immediately know the stored energy and the average charging power of the TCQB at any time. In this way, we no longer need to solve the system dynamics when considering the influence of the other optical field initial state on the performance of the TCQB.

Next, to get the specific expression of $F(M,t)$, we only need to solve the BAE to obtain the solution of $\{\lambda^{\sigma}\}$. But generally speaking, it is challenging to solve the BAE. Especially when $M$ is huge, we will face a high-order, multi-variable system of multiple equations, and the solution process is complicated. Fortunately, we have discovered a quick way to solve these equations called a trial solution method. The core idea is to use the value of the former set of solutions of BAE as the trial solution of the latter set of BAE. We consider a system with $10$ atoms to give a simple example. The Bethe ansatz equation is a two-variable quadratic equation system when $M=2$ and its three sets of solutions are easy to obtain. Here we assume that one set of the solution is $\lambda_{1}$, $\lambda_{2}$. If we need to solve the BAE when $M=3$, then we only need to take any one of $\{\lambda_{1},\lambda_{2}\}$, $\lambda_{1}$, $\lambda_{2}$, and these three numbers $\{\lambda_{1},\lambda_{1},\lambda_{2}\}$ (or $\{\lambda_{2},\lambda_{1},\lambda_{2}\}$) are used as the trial solution of the three-variable BAE when $M=3$, and then use Matlab to numerically solve these equations, We will quickly get the solutions.

When $M$ becomes very large, the situation will start to become complicated. For example, when $M=8$, we will get $9$ sets of solutions, and we choose any one of them $\{\lambda_{1},\lambda_{2},\cdots,\lambda_{8}\}$, which contains $8$ values. When we need to solve the solution of BAE when $M=9$, we choose any one of $\{\lambda_{1},\lambda_{2},\cdots,\lambda_{8}\}$, and then form a set of trial solutions containing $9$ values with $\{\lambda_{1},\lambda_{2},\cdots,\lambda_{8}\}$ , we will get all the solutions. However, we may not find all the solutions. Namely, the number of solutions is not $10$. At this time, we need to randomly select $7$ values from $\{\lambda_{1},\lambda_{2},\cdots,\lambda_{8}\}$, and then randomly select $2$ values from $\{\lambda_{1},\lambda_{2},\cdots,\lambda_{8}\}$ to form a trial solution with $9$ values. If we find a new solution, but the number of solutions is still not $10$, then We respectively randomly select $6$ and $3$ values from $\{\lambda_{1},\lambda_{2},\cdots,\lambda_{8}\}$ to form a set of trial solutions with $9$ values, and so on. Finally, we will obtain all the solutions.

In Fig. 2, we compare the running time required to solve the Bethe ansatz equation for different values of $M$ by our trial solution method and by the Bertini software when $N=10$, and we find that the trial solution method has a significant advantage. When $M\geq 8$, we did not use this software to solve the Bethe ansatz equation because the time required to solve the equation using the Bertini software was too long. Because these solutions of the BAE are too lengthy, in Appendix C, we give some solutions of the BAE. We can get the corresponding binary function $F(M,t)$ through these solutions. Also, because these functions are cumbersome, we only list a few in Appendix C.

In Fig. 3(a), the energy obtained by the atomic ensemble from the cavity field changes with time for different $M$ are respectively drawn. The marked lines with different colors represent numerical solutions. The unmarked lines with different colors denote analytical solutions obtained with BAM. In Appendix C, we give some concrete expressions of function $F(M,t)$, and we ignore some high-order small quantities in the analytical solution. The quantum toolbox QuTIP is used for numerical solutions [45]. It is worth noting that all marked lines are obtained by the numerical solution and all unmarked lines are obtained by the analytical solution should be strictly consistent in the power-related graphs. But we will find a little bit of deviation in some parts of the figure because we have already ignored some smaller values in $F(M,t)$. In Fig. 3(c), we have plotted the change of $F(M,t)$ with the initial average photon number $M$ at different times. From Fig. 3(a) and Fig. 3(c), we can intuitively see that $F(M,t)$ is proportional to $M$ in the time range we consider. Therefore, when the initial state of the cavity field is a number state, the number state with a larger initial average photon number can enable the TCQB to obtain more energy. Similarly, in Fig. 3(b), we can also see that the more the initial average photon number can improve the average charging power of the TCQB when the cavity field is in a number state in the time frame we consider. It is important to note that the time range we consider is from time $0$ to the time when the quantum battery first has the maximum stored energy in this article.

To verify whether the evolution of $E(t)$ and $P(t)$ with time $t$ in an arbitrary state of the initial cavity field is indeed the expected values of $F(M,t)$ and $F(M,t)/t$ under the probability distribution of the cavity filed initial state in the number states. We have studied the change of the stored energy and the average charging power of the TCQB with time when the initial state is the coherent state. If the initial state is coherent state $\left|\alpha\right\rangle$, then the stored energy and the average charging power is

where $p(M)=\left|\left\langle M|\alpha\right\rangle\right|^{2}$. In Fig. 4(a), we respectively draw the probability distribution of the coherent state in the number states when $\alpha=\sqrt{4}$ and $\alpha=\sqrt{6}$. According to these probability distributions and the function $F(M,t)$, the variations of the stored energy $E(t)$ and the average charging power $P(t)$ of the atomic ensemble with time $t$ are drawn respectively in Fig. 4(b) and Fig. 4(c). Indeed, the stored energy and the average charging power of the TCQB are the expected values of $F(M,t)$ and $F(M,t)/t$ under the probability distribution $p(M)$, respectively. In fact, not only for the coherent state but for any state, by finding its probability distribution in the number states and the function $F(M,t)$, we can draw the changes of the stored energy $E(t)$ and the average charging power $P(t)$ with time $t$. From here, we can also see that the stored energy capacity and the average charging power of the TCQB are proportional to the initial average photon number. This is very easy to understand because, for the same type of state, the probability distribution of the state with more initial average photons is to the right in the number states. For example, when $\alpha=\sqrt{6}$, the probability distribution of the coherent state in the number states is more to the right than the probability distribution when $\alpha=\sqrt{4}$. We have proved that $F(M,t)$ increases as $M$ increases. Therefore, the larger the initial average photon number, the greater the stored energy capacity and the higher average charging power of the TCQB under the same type of the initial state.

We found the relationship between the stored energy and the average charging power of the TCQB and the probability distribution of the optical field initial state in the number states. However, for an arbitrary cavity field initial state, it is uncertain whether the more the initial average photon number, the greater the stored energy capacity and the higher the charging power of the TCQB. In Appendix D, under the constraint of equal average photon number, we found the optimal initial state using equal probability and equal expected value splitting method. To more easily understand our proposed equal probability and equal expected value splitting method, in the following, we use a concrete example to illustrate how to discover the optimal initial state.

According to the equal probability and equal expected value splitting method, we assume that the initial average photon number $\bar{n}=10$. To prove that the number state is the optimal initial state for the maximum stored energy capacity and the maximum average charging power of the TCQB under the same initial average photon number, we use the number state $\left|10\right\rangle$ as the reference state and compare it with the following state

We can give a set of probability distributions $\{p(0)=4/45,p(8)=1/4,p(9)=1/9,p(10)=1/10,p(12)=1/4,p(15)=1/5\}$ that satisfy the constraints. It is easy to verify that the number state $\left|10\right\rangle$ and the state $\left|\phi\right\rangle$ have the same average photon number. The probability distribution of the above two optical states in the number states is shown in Fig. 5(a). Our purpose is to compare the magnitude of the stored energy and the average charging power of the TCQB in these two initial states. The differences between them are as follows

Using the equal probability and equal expected value splitting method, we divide some probabilities into two parts $p^{1}(i)$ and $p^{2}(i)$ (where $i=2,8,9$), and divide the probability $p_{r}(10)$ of the number state $\left|10\right\rangle$ into three parts $p_{r}^{0}(10)$, $p_{r}^{1}(10)$, and $p_{r}^{2}(10)$. They satisfy the following relationships

In Appendix D, we give a general solution of the probability of satisfying all the above equations, and the above separated probabilities are all shown in Fig. 5(b). Substituting Eq. (25a)-Eq. (25h) into Eq. (23) gives

Then we substitute the $p(12)$ and $p(15)$, which are obtained from Eq. (25d)-Eq. (25h), into the above equation yields

Before going further, it is important to note that the time range we discuss is from time $0$ to the time when the TCQB reaches its maximum value for the first time. In Fig. 3(c) and Fig. 6, we can find that the average slope of $F(M,t)$ at any time decreases or remains constant as $M$ increases in the time range we consider. And since $10>i$, $p^{1}(i)>0$, and $p^{2}(i)>0$, $\varDelta E(t)\geq 0$ and $\varDelta P(t)\geq 0$, this indicates that the number state $\left|10\right\rangle$ enables the TCQB to have a larger stored energy capacity and a higher average charging power compared to $\left|\phi\right\rangle$. In Fig. 5c, we plot the variations of the stored energy and the average charging power of the TCQB with time $t$ for different cavity field initial states. From the actual dynamical evolution of the TCQB, we can obtain the same conclusion as above. In Appendix D, we give complete proof that the optimal initial state that enables the TCQB to have the maximum stored energy capacity and the highest average charging power is the number state $\left|\bar{n}\right\rangle$ when the initial average photon number $\bar{n}$ is an integer. However, when the initial average photon number $\bar{n}$ is a non-integer, the optimal initial state is $\sqrt{1-(\bar{n}_{0}-[\bar{n}_{0}])}\left|[n_{0}]\right\rangle+\sqrt{\bar{n}_{0}-[\bar{n}_{0}]}\left|[\bar{n}_{0}]+1\right\rangle$, which is a superposition of the number states $\left|[n_{0}]\right\rangle$ and $\left|[n_{0}]+1\right\rangle$, where $[\bar{n}]$ indicates the integral part of $\bar{n}$.

When we study the ratio of the energy obtained by the TCQB from the different number-state optical fields as a function of time $t$, we find two novel phenomena, which are summarized as follows using two empirical inequalities

The above two inequalities hold when $M\geq m\geq m_{0}$ and $t\leq t_{max}$ ($t_{max}$ represents the time when $F(M,t)$ takes the first maximum value, and show in Fig. 7(b). The physical meaning of the above two inequalities is very clear. Inequality (28) shows that the stored energy of the TCQB is very sensitive to the number-state cavity field. When there is no coupling between the atomic ensemble and the number-state cavity field, we cannot obtain information about the cavity field through the stored energy of the TCQB. However, when there is a coupling between the atomic ensemble and the number-state cavity field, we can obtain the information of the initial average photon number of the cavity field by measuring the stored energy of the TCQB. Moreover, the shorter the coupling time between the atomic ensemble and the number-state cavity field, the more accurate we can obtain the information of the cavity field. As a simple example, the atomic ensemble with all atoms in the ground state is coupled with a known number-state cavity field $\left|m\right\rangle$, and then we can obtain the stored energy $E(m)$ of the atomic ensemble. Nextly, We use another atomic ensemble with the same initial properties to couple with an unknown number-state cavity field $\left|M\right\rangle$, and obtain the stored energy $E(M)$ of the atomic ensemble by measuring, then we can immediately get the average photon number of the unknown number-state cavity field, i.e., $M=mE(M)/E(m)$ (This equation holds when the coupling time is short, and the shorter the time, the more accurate the result). Inequality (29) shows that the measurement is accurate even for long-term coupling when $m$ is not much different from $M$. Also, we can see from Fig. 7(b) that the TCQB can be fully charged when the initial average photon number is large enough.

Above, we have studied the optimal initial state of the TCQB in the process of unitary evolution. However, in practical applications, due to the interaction between the TCQB system and its environment, the impact of the decoherence on the performance of the TCQB must be considered. Based on this practical problem, we will study the effects of the collective dephasing of $N$ identical two-level atoms and the decay of the optical field on the performance of the TCQB when the TCQB is coupled with the environment. Under Markov approximation, Born approximation, and rotating wave approximation, the dynamics of the density matrix $\hat{\rho}$ of the TCQB is described by the following Lindblad master equation[47]

where the Lindblad superoperators are defined by $\mathcal{L}_{\hat{A}}[\hat{\rho}]=2\hat{A}\hat{\rho}\hat{A}^{\dagger}-\hat{A}^{\dagger}\hat{A}\hat{\rho}-\hat{\rho}\hat{A}^{\dagger}\hat{A}$, $\gamma_{\Phi}$ and $\kappa$ are the collective dephasing of the atoms and the decay of the cavity field, respectively. Generally, it is not easy to solve Eq. (30) analytically, but the very friendly QuTIP [45] and PIQS [46] packages make it very convenient to solve the above density matrix. Then we can get the density matrix of the TCQB at any time. In Fig. 8, the time evolution of the stored energy of the TCQB, Fig. 8(a), and the average charging power of the TCQB, Fig. 8(b), are shown under the dynamics of Eq. (30) for different system parameters.

In Fig. 8(a) and Fig. 8(b), at the initial time, $N$ identical two-level atoms are all in the ground state, and the light field is in the number state $\left|M=10\right\rangle$. We compared the effects of the different collective dephasings of the atoms and the different decay intensities of the optical field on the stored energy and the average charging power of the TCQB. We found that the collective dephasing of the atoms or the decay of the optical field can destroy the maximum stored energy and the maximum average charging power of the TCQB, and this destruction increases as the collective dephasing or decay increases. As photons continue to flow into the environment, the absorption and radiation of photons by the atoms no longer remain in balance. Finally, due to the complete leakage of the energy, the stored energy of the TCQB will also be exhausted. Especially under the limit of bad cavity ($\kappa\gg g$), the energy loss of the TCQB is faster, as shown by the yellow and brown dash-dot lines in Fig. 8(a). Since the decay of the cavity field destroys the performance of the TCQB, we need to avoid it as much as possible in practice. However, the difference is that although the collective dephasing of the atoms will destroy the maximum stored energy and the maximum average charging power of the TCQB, without the decay of the cavity field, the collective dephasing of the atoms will accelerate the evolution of the atoms to reach a steady state, which means the absorption and radiation of the photons by the atoms reach a dynamic balance, as shown by the green and sky-blue dashed lines in Fig. 8(a), and the solid black line represents the above-mentioned balance line.

In this work, we mainly study the influence of the cavity field initial state on the stored energy and the average charging power of the T-C quantum battery. The purpose is to find the optimal initial state of the cavity field that enables the T-C quantum battery to have the maximum stored energy and the maximum average charging power. We use the Bethe ansatz method to find that the stored energy and the average charging power of the T-C quantum battery are closely related to the probability distribution of the cavity field initial state in the number states. We have defined a quantity called number-state stored energy. Through this quantity, we only need to know the probability distribution of the cavity field initial state in the number states to obtain the stored energy and the average charging power of the T-C quantum battery at any time. When solving the Bethe ansatz equation, we found a quick way to solve this equation. When studying the influence of the initial average photon number on the performance of the T-C quantum battery, we found that the larger the average photon number, the greater stored energy capacity and the higher average charging power of the T-C quantum battery. When studying the effect of the initial state of the different cavity fields on the T-C quantum battery under the same average photon number, we proposed an equal probability and equal expected value splitting method. In this way, we prove two inequalities that can be reduced to Jensen’s inequalities. By these two inequalities we proved, and the average slope of the number state stored energy of the T-C quantum battery decreases as the initial photon number increases, we found the optimal initial state of the T-C quantum battery. Meanwhile, We found two novel phenomena, which imply the super-sensitivity of the stored energy of the T-C quantum battery to the number-state cavity field, and we have pointed out the potential applications of these two phenomena. Finally, we considered the effect of the decoherence on the performance of the T-C quantum battery. We find that both the collective dephasing of the atoms and the decay of the cavity field can reduce the maximum stored energy of the T-C quantum battery. However, the collective dephasing of the atoms can accelerate the evolution of the atoms to reach the steady state in the absence of the decay of the cavity field.