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The Role of Quantum Discord in Quantum Battery
DOI: 10.12677/MP.2021.116015, PDF, HTML, XML, 下载: 181  浏览: 397

Abstract: In this paper, we use a two-qubit system to simulate the quantum battery charger model. By accu-rately solving that two identical two-level atoms are in a single-mode cavity field system, the effects of quantum discord on energy conversion and ergotropy in the quantum battery under different average photon numbers are discussed. The results show that the quantum discord provides the power for the transmission of low energy to high energy in the quantum battery system, and the ergotropy will change with the increase of the number of photons. When the number of photons is small, the evolution of the system is greatly affected by quantum discord. With the increase of the number of photons, the weight of the number of photons in the evolution becomes larger and larger, resulting in a more complex evolution of the system.

1. 引言

2. 理论描述

2.1. 量子失协

$I\left(a:b\right)=H\left(a\right)+H\left(b\right)-H\left(a,b\right)$ (1)

$J\left(a,b\right)=H\left(a\right)-H\left(a|b\right)=H\left(b\right)-H\left(b|a\right)$ (2)

$I\left({\rho }_{ab}\right)=S\left({\rho }_{a}\right)+S\left({\rho }_{b}\right)-S\left({\rho }_{ab}\right)=S\left({\rho }_{a}\right)-S\left(a|b\right)$ (3)

$D\left({\rho }_{ab}\right)=I\left({\rho }_{ab}\right)-{C}^{\prime }\left({\rho }_{ab}\right)$ (4)

${C}^{\prime }\left({\rho }_{ab}\right)=S\left({\rho }_{a}\right)-\underset{\left\{{\Pi }_{k}\right\}}{\mathrm{min}}\left[S\left({\rho }_{ab}|\left\{{\Pi }_{k}\right\}\right)\right]$ (5)

$D\left({\rho }_{ab}\right)=S\left({\rho }_{b}\right)-S\left({\rho }_{ab}\right)+\underset{\left\{{\Pi }_{k}\right\}}{\mathrm{min}}S\left({\rho }_{ab}|\left\{{\Pi }_{k}\right\}\right)$ (6)

2.2. 理论模型

$H={H}_{0}+{H}_{1}$ (7)

${H}_{0}=\frac{1}{2}\omega {\sigma }_{z}^{A}+\frac{1}{2}\omega {\sigma }_{z}^{B}+\nu {a}^{+}a$ (8)

${H}_{1}=g{\mu }_{1}\left({a}^{+}{\sigma }_{-}^{A}+a{\sigma }_{+}^{A}\right)+g{\mu }_{2}\left({a}^{+}{\sigma }_{-}^{B}+a{\sigma }_{+}^{B}\right)$ (9)

$\frac{\text{d}\rho \left(t\right)}{\text{d}t}=-\frac{i}{\hslash }\left[{H}_{I},\rho \left(t\right)\right]$ (10)

$\rho \left(t\right)=U\rho \left(0\right){U}^{+}$ (11)

$\begin{array}{c}U=\mathrm{exp}\left(-i{H}_{I}t/\hslash \right)\\ =\mathrm{exp}\left[-i\lambda t\left(a{\sigma }_{+}^{A}+{a}^{+}{\sigma }_{-}^{A}\right)\right]\otimes \mathrm{exp}\left[-i\lambda t\left(a{\sigma }_{+}^{B}+{a}^{+}{\sigma }_{-}^{B}\right)\right]\\ ={U}_{A}\otimes {U}_{B}\end{array}$ (12)

${U}^{+}={U}_{B}^{+}\otimes {U}_{A}^{+}$ (13)

$\rho \left(t\right)=\left(\begin{array}{cccc}{\rho }_{11}& 0& 0& {\rho }_{14}\\ 0& {\rho }_{22}& {\rho }_{23}& 0\\ 0& {\rho }_{32}& {\rho }_{33}& 0\\ {\rho }_{41}& 0& 0& {\rho }_{44}\end{array}\right)$ (14)

$W\left(\tau \right)=Tr\left({\rho }_{B}\left(\tau \right){H}_{B}\right)-Tr\left({\sigma }_{\rho B}{H}_{B}\right)$ (15)

$W\left(\tau \right)={\omega }_{0}\left(2\left({\rho }_{11}+{\rho }_{33}\right)-1\right)\Theta \left(\left({\rho }_{11}+{\rho }_{33}\right)-\frac{1}{2}\right)$ (16)

3. 分析与讨论

(a) (b)

Figure 1. Average photon number $n={10}^{-5}$ : (a) Evolution curve of charger energy (solid line) and battery energy (dotted line); (b) Evolution curves of quantum discord D (solid line) and ergotropy W (dotted line)

(a) (b)

Figure 2. Average photon number $n={10}^{-3}$ : (a) Evolution curve of charger energy (solid line) and battery energy (dotted line); (b) Evolution curves of quantum discord D (solid line) and ergotropy W (dotted line)

(a) (b)

Figure 3. Average photon number n = 0.1: (a) Evolution curve of charger energy (solid line) and battery energy (dotted line); (b) Evolution curves of quantum discord D (solid line) and ergotropy W (dotted line)

(a) (b)

Figure 4. Average photon number $n=5$ : (a) Evolution curve of charger energy (solid line) and battery energy (dotted line); (b) Evolution curves of quantum discord D (solid line) and ergotropy W (dotted line)

4. 结论

NOTES

*通讯作者。

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