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Recently, quantum battery based on various physical models from quantum optics model to spin model and its enhancement of charging performance have attracted increasing interest. It has been demonstrated that quantum entanglement is beneficial to the speedup of work extraction. In this paper, by an exact diagonalization approach, we investigate the charging performance of the field intensity-dependent Dicke model (also called intensity-dependent Dicke model) quantum battery, which consists of N qubits collectively interacting with a single-mode cavity. The considered intensity-dependent Dicke model is a generalized Dicke model with a nonlinear-coupling fashion and different weights of energy conserved term and non-conserved term. Firstly, we consider the influences of energy non-conserved term (also called anti-rotating wave term) on the maximum stored energy and maximum charging power in quantum battery. It is shown that the maximum stored energy is not very sensitive to the increase of the weight of energy non-conserved term, but the maximum charging power undergoes a significant change with the increase of the weight of energy non-conserved term. We also show that the maximum charging power increases monotonically with the increase of coupling constant between qubits and cavity, but the maximum stored energy is not monotonically related to the increase of coupling constant. Then, we further examine in detail the characteristics of the maximum stored energy, charging time, energy quantum fluctuation and maximum charging power in the quantum battery under the same weight between energy conserved term and non-conserved term. By comparing the charging performances of quantum battery based on the single-photon-Dicke model with those based on the two-photon-Dicke model, we find that the performances, specifically, the charging time and maximum charging power of the intensity-dependent Dicke quantum battery are better than those of single-photon Dicke quantum battery, but weaker than those of two-photon Dicke quantum battery. Of particular interest is that the relationship of maximum charging power with large quantum cell number in intensity-dependent Dicke quantum battery has the same form as that in the two-photon Dicke quantum battery, i.e. their maximum values of charging power are both proportional to the large quantum cell number squared, specifically,
$ P_{\mathrm{max}}^{\mathrm{ID}}\propto N^2 $ and$ P\mathrm{_{max}^{2ph}}\propto N^2 $ , which are consistent with the upper bound given by the paper (Gyhm J, Šafránek D, Rosa D2022 Phys. Rev. Lett. 128 140501 ). It is worthwhile to mention that Dou et al. (Dou F Q, Zhou H, Sun J A2022 Phys. Rev. A 106 032212 ) showed that using the quantum advantage of maximum charging power in the quantum battery based on cavity Heisenberg-spin-chain model$ P\mathrm{_{max}}\propto N^2 $ can be obtained. Therefore, this study of the charging performance based on the intensity-dependent Dicke quantum battery may provide an alternative approach to the further research on quantum battery.[1] Alicki R, Fannes M 2013 Phys. Rev. E 87 042123
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[2] Hovhannisyan K V, Perarnau Llobet M, Huber M, Acin A 2013 Phys. Rev. Lett. 111 240401
Google Scholar
[3] Zhang Y Y, Yang T R, Fu L, Wang X 2019 Phys. Rev. E 99 052106
Google Scholar
[4] Ferraro D, Campisi M, Andolina G M, Pellegrini V, Polini M 2018 Phys. Rev. Lett. 120 117702
Google Scholar
[5] Quach J Q, McGhee K E, Ganzer L, Rouse D M, Lovett B W, Gauger E M, Keeling J, Cerullo G, Lidzey D G, Virgili T 2022 Sci. Adv. 8 eabk3160
Google Scholar
[6] Crescente A, Carrega M, Sassetti M, Ferraro D 2020 Phys. Rev. B 102 245407
Google Scholar
[7] Lu W J, Chen J, Kuang L M, Wang X 2021 Phys. Rev. A 104 043706
Google Scholar
[8] Dou F Q, Lu Y Q, Wang Y J, Sun J A 2022 Phys. Rev. B 105 115405
Google Scholar
[9] Dou F Q, Zhou H, Sun J A 2022 Phys. Rev. A 106 032212
Google Scholar
[10] Le T P, Levinsen J, Modi K, Parish M M, Pollock F A 2018 Phys. Rev. A 97 022106
Google Scholar
[11] Ghosh S, Chanda T, Sen(De) A 2020 Phys. Rev. A 101 032115
Google Scholar
[12] Huangfu Y, Jing J 2021 Phys. Rev. E 104 024129
Google Scholar
[13] Zhao F, Dou F Q, Zhao Q 2022 Phys. Rev. Research 4 013172
Google Scholar
[14] Yao Y, Shao X Q 2022 Phys. Rev. E 106 014138
Google Scholar
[15] Peng L, He W B, Chesi S, Lin H Q, Guan X W 2021 Phys. Rev. A 103 052220
Google Scholar
[16] Liu J X, Shi H L, Shi Y H, Wang X H, Yang W L 2021 Phys. Rev. B 104 245418
Google Scholar
[17] Santos A C, Saguia A, Sarandy M S 2020 Phys. Rev. E 101 062114
Google Scholar
[18] Dou F Q, Wang Y J, Sun J A 2020 Europhys. Lett. 131 43001
Google Scholar
[19] Zheng R H, Ning W, Yang Z B, Xia Y, Zheng S B 2022 New. J. Phys. 24 063031
Google Scholar
[20] Binder F C, Vinjanampathy S, Modi K Goold J 2015 New J. Phys. 17 075015
Google Scholar
[21] Campaioli F, Pollock F A, Binder F C, Céleri L, Goold J, Vinjanampathy S, Modi K 2017 Phys. Rev. Lett. 118 150601
Google Scholar
[22] Gyhm J, Šafrǎnek D, Rosa D 2022 Phys. Rev. Lett. 128 140501
Google Scholar
[23] Shi H L, Ding S, Wan Q K, Wang X H, Yang W L 2022 Phys. Rev. Lett. 129 130602
Google Scholar
[24] Yu W L, Zhang Y, Li H, Wei G F, Han L P, Tian F, Zou J 2023 Chin. Phys. B 32 010302
Google Scholar
[25] Jaynes E T, Cummings F W 1963 Proc. IEEE 51 89
Google Scholar
[26] Rabi I I 1936 Phys. Rev. 49 324
Google Scholar
[27] Braak D 2011 Phys. Rev. Lett. 107 100401
Google Scholar
[28] Chen Q, Wang C, He S, Wang K 2012 Phys. Rev. A 86 023822
Google Scholar
[29] Dicke R H 1954 Phys. Rev. 93 99
Google Scholar
[30] Buck B, Sukumar C V 1981 Phys. Lett. A 81 132
Google Scholar
[31] Ng K M, Lo C F, Liu K L 2000 Phys. A: Stat. Mech. Appl. 275 463
Google Scholar
[32] Duan L, Xie Y F, Braak D, Chen Q H 2016 J. Phys. A: Math. Theor. 49 464002
Google Scholar
[33] Lo C F 2020 Sci. Rep. 10 18761
Google Scholar
[34] Liu X Y, Ren X Z, Wang C, Gao X L, Wang K L 2020 Commun. Theor. Phys. 72 065502
Google Scholar
[35] Valverde C, Gonalves V G, Baseia B 2016 Phys. A: Stat. Mech. Appl. 446 171
Google Scholar
[36] He W B, Chesi S, Lin H Q, Guan X W 2019 Phys. Rev. B 99 174308
Google Scholar
[37] Andolina G M, Farina D, Mari A, Pellegrini V, Giovannetti V 2018 Phys. Rev. B 98 205423
Google Scholar
[38] Friis N, Huber M 2018 Quantum 2 61
Google Scholar
[39] Johansson J, Nation P, Nori F 2013 Comput. Phys. Commun. 184 1234
Google Scholar
[40] Crescente A, Carrega M, Sassetti M, Ferraro D 2020 New J. Phys. 22 063057
Google Scholar
[41] 刘雪莹, 成书杰, 高先龙 2022 71 134203
Google Scholar
Liu X Y, Cheng S J, Gao X L 2022 Acta Phys. Sin. 71 134203
Google Scholar
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图 1 基于依赖强度Dicke模型的量子电池中(a)最大存储能量
$E_{\max}^{{\rm{ID}}}$ (以$\omega_{\rm{a}}$ 为单位)和(b)最大充电功率$P_{\max}^{{\rm{ID}}}$ (以$\omega_{\rm{a}}^2$ 为单位)随权重参数$\xi$ 和耦合常数g的变化. 在数值计算的过程中, 电池单元数被设定为$N=10$ Figure 1. (a) Stored energy
$E_{\max}^{{\rm{ID}}}$ (in units of$\omega_{\rm{a}}$ ) and (b) maximum average charging power$P_{\max}^{{\rm{ID}}}$ (in units of$\omega_{\rm{a}}^2$ ) as a function of the parameters$\xi$ and g for the intensity-dependent Dicke quantum battery, where the number of quantum cell$N = 10$ is chosen in the calculation
图 2 在不同的耦合常数
$g = 0.005 \to 0.1 \to 0.5$ 下, 基于依赖强度Dicke模型的量子电池中存储能量$E^{{\rm{ID}}}(t)$ (以$\omega_a$ 为单位)、能量量子涨落$\varSigma^{{\rm{ID}}}(t)$ (以$\omega_{\rm{a}}$ 为单位), 以及平均充电功率$P^{{\rm{ID}}}(t)$ (以$\omega_{\rm{a}}^2$ 为单位) 随无量纲时间参数$\omega_{\rm{a}} t$ 的演化, 量子电池中的量子单元数被设定为$N = 10$ Figure 2. Stored energy
$E^{{\rm{ID}}}(t)$ (in units of$\omega_{\rm{a}}$ ), its fluctuation$\varSigma^{{\rm{ID}}}(t)$ (in units of$\omega_{\rm{a}}$ ), and average charging power$P^{{\rm{ID}}}(t)$ (in units of$\omega_{\rm{a}}^2$ ) versus the dimensionless quantity$\omega_{\rm{a}} t$ for the intensity-dependent Dicke quantum battery in the different couplings, where quantum cell is set to$N = 10$ in the calculation
图 3 在不同的量子单元数(
$N = 8 \to 10 \to 12 \to 14$ )下, 基于依赖强度Dicke模型的量子电池中(a)最大存储能量$E_{\max}^{{\rm{ID}}}$ (以$ \omega_{\rm{a}}$ 为单位)和(b)最大充电功率$P_{\max}^{{\rm{ID}}}$ (以$g \omega_{\rm{a}}^2$ 为单位)随耦合常数g的变化Figure 3. (a) The maximum stored energy
$E_{\max}^{{\rm{ID}}}$ (in units of$\omega_{\rm{a}}$ ) and (b) maximum charging power$P_{\max}^{{\rm{ID}}}$ (in units of$\omega_{\rm{a}}^2$ ) versus the couplings constant g for the intensity-dependent Dicke model in the different quantum cells$N = 8 \to 10 \to 12 \to 14$ 
图 4 在不同的耦合常数
$g = 0.005 \to 0.1 \to 0.5$ 下, 基于依赖强度Dicke模型的量子电池中(a)最大存储能量$E_{\max}^{{\rm{ID}}}$ (以$\omega_a$ 为单位)和(b)能量量子涨落$\varSigma^{{\rm{ID}}}$ (以$\omega_a$ 为单位)随量子单元数N的变化Figure 4. (a) The maximum stored energy
$E_{\max}^{{\rm{ID}}}$ (in units of$\omega_a$ ) and (b) its quantum fluctuation$\varSigma^{{\rm{ID}}}$ (in units of$\omega_a$ ) versus the number of quantum cells N for the intensity-dependent Dicke model in the different couplings$g = 0.005 \to 0.1 \to 0.5$ .
图 5 在不同的耦合常数(
$g = 0.005 \to 0.1 \to 0.5$ )下, 基于依赖强度Dicke模型和双光子Dicke模型的量子电池的最大充电功率$P_{\max}^{{\rm{ID}}}$ 和$ P_{\max}^{{\rm{2ph}}}$ (以$g \omega_{\rm{a}}^2$ 为单位)随量子单元数N的变化, 量子单元数$N \in [1, 30]$ Figure 5. Comparison of maximum charging power
$P_{\max}^{{\rm{ID}}}$ and$P_{\max}^{{\rm{2ph}}} $ (in units of$g \omega_{\rm{a}}^2$ ) versus the number of qubits N for the intensity-dependent Dicke model and two-photon Dicke model in the different couplings$g = 0.005 \to 0.1 \to 0.5$ , quantum cells$N \in [1, 30]$ 表 1 在不同耦合常数
$g = 0.005 \to 0.1 \to 0.5$ 下, 基于依赖强度、单光子和双光子3种Dicke模型的量子电池中最大存储能量$E_{\max}(t_{\rm{E}})$ (以$\omega_{\rm{a}}$ 为单位)、能量量子涨落$ \bar{\varSigma} \equiv \varSigma (t_{\rm{E}})$ (以$\omega_{\rm{a}}$ 为单位)及充电时间$t_{\rm{E}}$ (以$\omega_{\rm{a}}^{-1}$ 为单位)等充电性能参数的比较. 在数值计算中, 量子单元数被设定为$N = 10$ Table 1. Comparisons of the maximum stored energy
$E_{\max}(t_{\rm{E}})$ (in unit of$\omega_{\rm{a}}$ ), its fluctuations$\bar{\varSigma} \equiv \varSigma (t_{\rm{E}})$ (in units of$\omega_{\rm{a}}$ ) and corresponding charging time$t_{\rm E}$ (in units of$\omega_{\rm{a}}^{-1}$ ) for the intensity-dependent, single photon, and two-photon Dicke models in the different couplings$ g = 0.005 \to 0.1 \to 0.5$ , and quantum cells$N = 10$ in the calculationDicke 1ph Dicke ID Dicke 2ph $E_{\max}$ $\bar{\varSigma}$ $\omega_{\rm{a} } t_{\rm{E}}$ $E_{\max}$ $\bar{\varSigma}$ $ \omega_{\rm{a}}\rm{\mathit{t}_E} $ $E_{\max}$ $\bar{\varSigma}$ $\omega_{\rm{a} } t_{\rm{E}}$ $g=0.005$ 8.861 1.194 127.353 7.313 1.577 40.760 7.095 1.410 19.173 $g=0.1$ 7.931 1.390 5.428 6.408 3.381 0.875 6.815 3.523 0.462 $g=0.5$ 6.766 3.472 0.585 6.973 3.547 0.183 6.994 3.554 0.094 
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表 2 在不同耦合常数
$g = 0.005 \to 0.1 \to 0.5$ 下, 基于依赖强度、单光子和双光子3种Dicke模型的量子电池中最大充电功率$P_{\max}(t_{\rm{P}})$ (以$\omega_{\rm{a}}^2$ 为单位)和充电时间$t_{\rm{P}}$ (以$\omega_{\rm{a}}^{-1}$ 为单位)等充电性能参数上的比较, 量子单元数被设定为$N = 10$ Table 2. Comparisons of the maximum average charging power
$P_{\max}(t_{\rm{P}})$ (in units of$\omega_{\rm{a}}^2$ ) and corresponding charging time$t_{\rm{P}}$ (in units of$\omega_{\rm{a}}^{-1}$ ) for the intensity-dependent, single photon, and two-photon Dicke models in the different couplings$ g = 0.005 \to 0.1 \to 0.5$ , and quantum cell$N = 10$ Dicke 1ph Dicke ID Dicke 2ph $P_{\max}^{{\rm{1 ph}}}$ $\omega_{\rm{a} } t_{\rm{P}}$ $P_{\max}^{\rm{ID}}$ $\omega_{\rm{a} } t_{\rm{P }}$ $P_{\max}^{{\rm{2 ph}}}$ $\omega_{\rm{a} } t_{\rm{P}}$ $g=0.005$ 0.093 68.347 0.255 19.033 0.496 9.482 $g=0.1$ 1.882 1.775 8.421 0.634 17.008 0.334 $g=0.5$ 13.370 0.420 43.891 0.132 85.176 0.069
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表 3 在不同耦合常数(
$g = 0.005 \to 0.1 \to 0.5$ )下, 基于依赖强度Dicke模型中最大充电功率$P_{\max }^{{\rm{ID}}, 2 {\rm{ph}}} \propto {\rm{ }}aN^b$ (以$\omega_{\rm{a}}^2$ 为单位)和双光子耦合Dicke模型中最大充电功率$P_{\max }^{{\rm{2 ph}}} \propto {\rm{ }} \alpha N^\beta$ (以$\omega_{\rm{a}}^2$ 为单位)的比较Table 3. Comparisons of the maximum average charging power
$P_{\max }^{\rm{ID}} \propto {\rm{ }}aN^b$ and$P_{\max }^{{\rm{2 ph}}} \propto {\rm{ }} \alpha N^\beta$ (in units of$\omega_{\rm{a}}^2$ ) for the intensity-dependent and two-photon Dicke models in the different couplings$g = 0.005 \to 0.1 \to 0.5$ Dicke ID Dicke 2ph a b $\alpha$ $\beta$ $g=0.005$ 0.51 2.00 0.92 2.04 $g=0.1$ 0.65 2.09 1.31 2.09 $g=0.5$ 0.97 1.96 1.75 1.98
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[1] Alicki R, Fannes M 2013 Phys. Rev. E 87 042123
Google Scholar
[2] Hovhannisyan K V, Perarnau Llobet M, Huber M, Acin A 2013 Phys. Rev. Lett. 111 240401
Google Scholar
[3] Zhang Y Y, Yang T R, Fu L, Wang X 2019 Phys. Rev. E 99 052106
Google Scholar
[4] Ferraro D, Campisi M, Andolina G M, Pellegrini V, Polini M 2018 Phys. Rev. Lett. 120 117702
Google Scholar
[5] Quach J Q, McGhee K E, Ganzer L, Rouse D M, Lovett B W, Gauger E M, Keeling J, Cerullo G, Lidzey D G, Virgili T 2022 Sci. Adv. 8 eabk3160
Google Scholar
[6] Crescente A, Carrega M, Sassetti M, Ferraro D 2020 Phys. Rev. B 102 245407
Google Scholar
[7] Lu W J, Chen J, Kuang L M, Wang X 2021 Phys. Rev. A 104 043706
Google Scholar
[8] Dou F Q, Lu Y Q, Wang Y J, Sun J A 2022 Phys. Rev. B 105 115405
Google Scholar
[9] Dou F Q, Zhou H, Sun J A 2022 Phys. Rev. A 106 032212
Google Scholar
[10] Le T P, Levinsen J, Modi K, Parish M M, Pollock F A 2018 Phys. Rev. A 97 022106
Google Scholar
[11] Ghosh S, Chanda T, Sen(De) A 2020 Phys. Rev. A 101 032115
Google Scholar
[12] Huangfu Y, Jing J 2021 Phys. Rev. E 104 024129
Google Scholar
[13] Zhao F, Dou F Q, Zhao Q 2022 Phys. Rev. Research 4 013172
Google Scholar
[14] Yao Y, Shao X Q 2022 Phys. Rev. E 106 014138
Google Scholar
[15] Peng L, He W B, Chesi S, Lin H Q, Guan X W 2021 Phys. Rev. A 103 052220
Google Scholar
[16] Liu J X, Shi H L, Shi Y H, Wang X H, Yang W L 2021 Phys. Rev. B 104 245418
Google Scholar
[17] Santos A C, Saguia A, Sarandy M S 2020 Phys. Rev. E 101 062114
Google Scholar
[18] Dou F Q, Wang Y J, Sun J A 2020 Europhys. Lett. 131 43001
Google Scholar
[19] Zheng R H, Ning W, Yang Z B, Xia Y, Zheng S B 2022 New. J. Phys. 24 063031
Google Scholar
[20] Binder F C, Vinjanampathy S, Modi K Goold J 2015 New J. Phys. 17 075015
Google Scholar
[21] Campaioli F, Pollock F A, Binder F C, Céleri L, Goold J, Vinjanampathy S, Modi K 2017 Phys. Rev. Lett. 118 150601
Google Scholar
[22] Gyhm J, Šafrǎnek D, Rosa D 2022 Phys. Rev. Lett. 128 140501
Google Scholar
[23] Shi H L, Ding S, Wan Q K, Wang X H, Yang W L 2022 Phys. Rev. Lett. 129 130602
Google Scholar
[24] Yu W L, Zhang Y, Li H, Wei G F, Han L P, Tian F, Zou J 2023 Chin. Phys. B 32 010302
Google Scholar
[25] Jaynes E T, Cummings F W 1963 Proc. IEEE 51 89
Google Scholar
[26] Rabi I I 1936 Phys. Rev. 49 324
Google Scholar
[27] Braak D 2011 Phys. Rev. Lett. 107 100401
Google Scholar
[28] Chen Q, Wang C, He S, Wang K 2012 Phys. Rev. A 86 023822
Google Scholar
[29] Dicke R H 1954 Phys. Rev. 93 99
Google Scholar
[30] Buck B, Sukumar C V 1981 Phys. Lett. A 81 132
Google Scholar
[31] Ng K M, Lo C F, Liu K L 2000 Phys. A: Stat. Mech. Appl. 275 463
Google Scholar
[32] Duan L, Xie Y F, Braak D, Chen Q H 2016 J. Phys. A: Math. Theor. 49 464002
Google Scholar
[33] Lo C F 2020 Sci. Rep. 10 18761
Google Scholar
[34] Liu X Y, Ren X Z, Wang C, Gao X L, Wang K L 2020 Commun. Theor. Phys. 72 065502
Google Scholar
[35] Valverde C, Gonalves V G, Baseia B 2016 Phys. A: Stat. Mech. Appl. 446 171
Google Scholar
[36] He W B, Chesi S, Lin H Q, Guan X W 2019 Phys. Rev. B 99 174308
Google Scholar
[37] Andolina G M, Farina D, Mari A, Pellegrini V, Giovannetti V 2018 Phys. Rev. B 98 205423
Google Scholar
[38] Friis N, Huber M 2018 Quantum 2 61
Google Scholar
[39] Johansson J, Nation P, Nori F 2013 Comput. Phys. Commun. 184 1234
Google Scholar
[40] Crescente A, Carrega M, Sassetti M, Ferraro D 2020 New J. Phys. 22 063057
Google Scholar
[41] 刘雪莹, 成书杰, 高先龙 2022 71 134203
Google Scholar
Liu X Y, Cheng S J, Gao X L 2022 Acta Phys. Sin. 71 134203
Google Scholar
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