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Calculation of thermic and electric properties and valence electron structure for metallic electrodes of Na||Sb-Pb-Sn liquid metal battery

Zhang Jian Wang Xin-Qiao Su Tong Chen Ying Guo Yong-Quan

Yu Juan, Zhang Yan, Wu Yin-Hua, Yang Wen-Hai, Yan Zhi-Hui, Jia Xiao-Jun. Experimental demonstration on quantum coherence evolution of two-mode squeezed state. Acta Phys. Sin., 2023, 72(3): 034202. doi: 10.7498/aps.72.20221923
Citation: Yu Juan, Zhang Yan, Wu Yin-Hua, Yang Wen-Hai, Yan Zhi-Hui, Jia Xiao-Jun. Experimental demonstration on quantum coherence evolution of two-mode squeezed state. Acta Phys. Sin., 2023, 72(3): 034202. doi: 10.7498/aps.72.20221923

Calculation of thermic and electric properties and valence electron structure for metallic electrodes of Na||Sb-Pb-Sn liquid metal battery

Zhang Jian, Wang Xin-Qiao, Su Tong, Chen Ying, Guo Yong-Quan
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  • The valence electron structures and thermal and electric properties of Na||Sb-Pb-Sn liquid metal battery are systematically studies with solid and molecular empirical electron theory (EET). The theoretical studies show that the thermal and electric properties are strongly related to the valence electron structure of electrode. The cathodic alloys Na1–xIAx (IA = K, Rb, Cs) are designed by doping IA group alkali metals (K, Rb, Cs) into Na electrode since the melting points of IA group metals (K, Rb, Cs) are all lower than that of sodium. The theoretical bond lengths and cohesive energy of cathodic alloys Na1–xIAx match the experimental ones well. The theoretical studies show the decreasing tendency of melting point, cohesive energy and electric potential with increasing doping content x in Na1–xIAx alloys, which is due to the modulation of valence electron structure of IA group dopants. According to the analyses of valence structures, the number of lattice electrons decreases with the increasing of the doping content x for the cathodic alloy and causes the melting point, electric potential and cohesive energy to decline. It reveals that the IA group dopant modulates the valence electron structure of cathodic alloy, and induces the electron transformation from lattice electron to covalent electron in s orbital. The anode products such as NaSb3, NaSn, Na15Sn4 and NaPb are formed by transporting Na ions into the anode alloy Sb-Sn-Pb. The calculated bond-lengths and melting points fit the observed ones well for these anode products. Owing to their complex structures with various atomic occupations in unit cell, the thermal property or electric property is not only relative to lattice electron, but also depends on the covalent electron. The sublattice plays an important role in the forming of the four anode products. The lattice electrons are supplied by Na at 4f sites in Na3Sb, Na at 16e and Sn at 32g sites in NaSn, Sn at 16c and Na at 48e sites in Na15Sn4, and Na at 16f and Pb at 32g sites in NaPb, respectively. The open-gate voltage is closely related to the lattice electrons and inversely proportional to the average number of lattice electrons per atom. The open-gate voltage of NaSb3 is the largest among the anode products, however, its averaged number of lattice electron per atom is the least. Since the lattice electron number of NaSn is the largest among the anode products, the open-gate voltage of NaSn is the least. It implies that the lattice electron plays a very important role in Na||Sb-Pb-Sn liquid metal battery, which can modulate the valence electron structures and thermal and electric properties.
      PACS:
      42.50.-p(Quantum optics)
      42.50.Lc(Quantum fluctuations, quantum noise, and quantum jumps)
      42.65.Yj(Optical parametric oscillators and amplifiers)
      Corresponding author: Guo Yong-Quan, yqguo@ncepu.edu.cn
    • Funds: Project supported by the National Key Research and Development Program of China (Grant No. 2018YFB0905600)

    相干的概念最初是用来描述波之间的干涉现象, 态的相干叠加结合可观测量的量子化, 是量子力学区别于经典领域最重要的特征之一. 为了研究量子相干性的作用, 一个基本的问题就是如何更一般地定量分析任意量子态所携带的量子相干性. Baumgratz等[1]根据资源理论提出了一种基于有限维系统的定量描述量子相干性的方法, 至此相干性被看作一种可以被表征量化及操控的量子资源[2,3], 在量子算法[4]、量子信道鉴别[5]、量子密钥分发[6]和量子度量学[7,8]等领域有广泛应用. 目前在不同系统中的量子相干性研究都取得了一定进展, 如在原子的自旋基矢[9]、光子的偏振基矢[10,11]、光子数表象[12]和相干态表象[13]. 且量子相干与其他量子资源如纠缠[14]、光学猫态[12]、非对称性[15]等的关系近年来也引起了广泛关注. 量子态的量子相干性定义为该量子态与非相干态在希尔伯特空间中的最小距离, 度量方式有相对熵度量[1]l1-norm度量[16]、基于距离测量的保真度度量[17], 以及Fisher 信息度量[18], 量子相干性的度量必须满足非负性、单调性、强单调性和凸性条件. 对于基于无限维希尔伯特空间的连续变量量子态相干性的定量分析, 考虑能量不能无限大的约束条件, 量子相干性的度量相较于有限维希尔伯特空间还必须额外满足有限性条件. 2016年张煜然等[19]提出利用相对熵度量无限维系统中的量子相干性, 随后胥建卫 [20]对高斯态的量子相干性进行了量化. 同年, Buono等 [21]采用 Bure距离度量和Hellinger距离度量分别对相干热态、压缩热态及热压缩态的量子相干性进行了量化.

    非经典光场(如压缩态光场和Einstein-Podolsky-Rosen (EPR)纠缠态光场)因其优越的量子特性已被广泛应用于如量子信息网络[22-26]、量子存储[27-29]、量子精密测量[30-35]等方面. 压缩态光场的压缩特性用正交分量的噪声方差描述, 纠缠态光场的纠缠特性用纠缠判据度量[36]. 压缩态光场和EPR纠缠态都具有量子相干性, 量子相干性的研究是沟通压缩和纠缠特性的一个桥梁. 同时, 量子相干性和量子纠缠代表了非经典系统的两个基本特征, 它们都可以在操作资源理论中被描述. 量子相干性反映系统的相干叠加程度, 纠缠是多个关联态的叠加, 反映了两个系统的关联和叠加, 这是相干性与纠缠的区别. 对于纠缠而言, 多体系统中的量子相干性体现了纠缠的本质. Streltsov等[37]于2015年提出任意非零相干态都可以通过双边非相干操作来产生纠缠. Chitambar和Hsieh[38]首次研究了分布式场景中相干和纠缠之间的相互作用, 通过 LIOCC (local incoherent operations and classical communication) 操作研究了纠缠态的形成、蒸馏和局域相干.

    在量子资源实际应用过程中, 难以避免地要将所使用的量子资源与外界环境发生相互作用以及在量子节点间进行传输, 量子信道中存在的损耗将会引起量子资源的退相干, 这意味着量子态的量子特性会逐渐减小甚至消失. 此外, 由于实验制备过程中一些不可避免的损耗和额外噪声的存在会导致量子态的不纯从而影响其相干性. 研究双模压缩态量子相干性在制备和传输过程中的演化对于实际应用具有重要意义. 目前已经有实验证实了压缩、量子纠缠、量子导引和相干性等量子资源在损耗和噪声环境下的退相干现象[39-41], 从实用化角度, 有必要研究高斯量子态的制备过程中的不完美对纠缠特性、压缩特性与量子相干性的影响, 并分析其经过损耗信道的退相干效应.

    目前实验上广泛使用的有效估计量子相干性的方法之一就是态层析, 得到密度矩阵后代入相干性表达式计算出相应的相干性值. 本文从压缩态的实验制备出发, 通过平衡零拍探测重构高斯态协方差矩阵来量化产生装置中损耗与噪声存在导致量子态的不纯对其压缩特性、纠缠特性和量子相干性的影响. 实验演示了双模压缩态光场经损耗信道后量子特性的演化, 实验证明压缩特性、纠缠特性和量子相干性对损耗均是鲁棒的. 特别地, 量子态的压缩特性和纠缠特性在制备过程中热光子数超过一定大小后会消失, 而当压缩和纠缠均已消失时, 量子相干性依然存在. 本研究工作为压缩态、纠缠态光场的量子相干性在量子信息过程中的应用提供了参考.

    在Fock表象中, 量子态ˆρ相干性的相对熵度量可以用下式计算[1]:

    C(ˆρ)=S(ˆρdiag)S(ˆρ) (1)

    其中ˆρ = i,jρi,j|ij|, ˆρdiag = iρi,i|ii|表示对ˆρ的退位相操作, 将ˆρ中含有量子相干性的非对角元剔除仅剩对角元. S表示量子态的von Neumann熵, 可以通过量子态的本征值λi计算, S(ˆρ)=iλilogλi. 当且仅当量子态为纯态时, S(ˆρ)=0. 量子态的相对熵体现的是ˆρˆρdiag之间的距离. 量子力学中光场可以用湮灭算符ˆa表示, 光场的正交振幅算符和正交相位算符用产生湮灭算符表示为ˆx = (ˆa + ˆa)ˆp = i(ˆaˆa). 高斯态ˆρ(ˉx,σ)可以由协方差矩阵σ和位移矢量ˉx很方便刻画[42], 其中ˉx=ˆx=(ˆx1,ˆp1,,ˆxN,ˆpN)T; σ是实对称的正定矩阵, 其矩阵元表示的是相应正交分量之间的起伏, 由σij=Cov(ˆxi,ˆxj)=12ˆxiˆxj+ˆxjˆxiˆxiˆxj给出. 因此, 高斯态的量子相干性可由下式给出:

    C[ˆρ(ˉx,σ)]=S[ˆρ(ˉxth,σth)]S[ˆρ(ˉx,σ)] (2)

    其中, ˆρ(ˉxth,σth)表示与高斯态ˆρ(ˉx,σ)具有相同平均光子数的热态, 其协方差矩阵σth的矩阵元σth 2i1,2i1=σth 2i,2i=12[σ2i1,2i1+σ2i, 2i+(ˉx2i1)2+(ˉx2i)2], 实验中量子态的平移量为0. 高斯态和对应热态的von Neumann熵分别为

    S[ˆρ(ˉx,σ)]=ni=1[(υi+12)log2(υi+12)(υi12)log2(υi12)]S[ˆρ(ˉxth,σth)]=ni=1[(νi+12)log2(νi+12)(νi12)log2(νi12)] (3)

    其中νiυi分别为协方差矩阵σthσ的辛本征值.

    相干态或真空态为最小不确定态, 其正交振幅和相位分量起伏为Δ2ˆx=Δ2ˆp=1, 1被定义为散粒噪声极限 (shot noise limit, SNL). 而压缩态光场是指在满足海森伯不确定性原理的前提下, 某一分量的起伏被抑制降至SNL 之下, 而与其正交的噪声分量会被放大. 对于正交振幅压缩态光场而言, 其协方差矩阵表示为1/2Diag(e2r,e2r), 其中r表示压缩参量. 实际上, 由于量子态的制备过程中一些不可避免的损耗和额外噪声的存在会导致得到的量子态不纯. 尤其对于高压缩度的压缩态而言, 实验制备的压缩态光场往往会出现压缩幅度与反压缩幅度不相等的情况[43]. 把这种压缩分量与反压缩分量不等的压缩态光场称为压缩热态, 引入热光子数nth来描述所制备的压缩态光场的不纯, 单模压缩热态ˆρsqu的协方差矩阵σsqu表示为[21]

    σsqu=(Vs00Vas)=(nth + e2r00nth + e2r). (4)

    其中VsVas分别表示压缩态的正交振幅和正交相位分量起伏, 用分贝表示时, 压缩的相对噪声功率为10lgVs dB, 反压缩的相对噪声功率为10lgVas dB. 连续变量系统中, N模高斯态的纯度仅由全域辛不变量决定[44], 表示为: μ(ˆρ)=1iυi=1detσ, 0. 单模高斯态纯度与正交分量起伏的关系为

    \mu \left( {{{\hat \rho }_{{\text{squ}}}}} \right){\text{ = }}\frac{1}{{\sqrt {\det {{\boldsymbol{\sigma }}_{{\text{squ}}}}} }}{\text{ = }}\frac{1}{{\sqrt {{V_{\text{s}}}{V_{{\text{as}}}}} }}. (5)

    损耗信道中, 信道内的噪声仅来源于真空起伏, 其方差为1. 则经过损耗信道后其协方差矩阵表示为

    \begin{split} {\boldsymbol{\sigma }'_{{\text{squ}}}} &= \left( {\begin{array}{*{20}{c}} {{V'_{\rm s}}}&0 \\ 0&{{V'_{{\text{as}}}}} \end{array}} \right)\\ &=\left( {\begin{array}{*{20}{c}} {\eta {V_{\text{s}}} + (1 - \eta )}&0 \\ 0&{\eta {V_{{\text{as}}}} + (1 - \eta )} \end{array}} \right), \end{split} (6)

    其中 \eta 表示信道的传输效率, 则损耗信道的损耗为 1 - \eta .

    双模压缩态光场的协方差矩阵可以表示为

    {{\boldsymbol{\sigma }}_{{\text{ent}}}} = \left( {\begin{array}{*{20}{c}} {\boldsymbol{{\rm A}}}&{\boldsymbol{C}} \\ {{{\boldsymbol{C}}^{\text{T}}}}&{\boldsymbol{B}} \end{array}} \right){\text{ = }}\left( {\begin{array}{*{20}{c}} \alpha &0&\gamma &0 \\ 0&\alpha &0&{ - \gamma } \\ \gamma &0&\beta &0 \\ 0&{ - \gamma }&0&\beta \end{array}} \right), (7)

    其中 {\boldsymbol{A}} = \alpha {\boldsymbol{I}}, {\boldsymbol{B}} = \beta {\boldsymbol{I}}, {\boldsymbol{C}} = \gamma {\boldsymbol{Z}} , {\boldsymbol{I}} Z为泡利矩阵, {\boldsymbol{I}} = \left( {\begin{array}{*{20}{c}} 1&0 \\ 0&1 \end{array}} \right), {\boldsymbol{Z}} = \left( {\begin{array}{*{20}{c}} 1&0 \\ 0&{ - 1} \end{array}} \right) , \alpha = \beta = ({V_{\text{s}}} + {V_{{\text{as}}}})/2 = {n_{{\text{th}}}}{\text{ + }}\cosh 2 r, \gamma =({V_s} - {V_{{\text{as}}}})/2 = {n_{{\text{th}}}} - \sinh 2 r. 对于双模压缩态而言, 纯度为\mu \left( {{{\hat \rho }_{{\text{ent}}}}} \right)= \dfrac{1}{{\sqrt {\det {{\boldsymbol{\sigma }}_{{\text{ent}}}}} }}= \dfrac{1}{{{V_{\text{s}}}{V_{{\text{as}}}}}}. 判断双模量子纠缠的充分必要条件为PPT (positive partial transposition) 判据, 其值为\sqrt {\dfrac{{\varGamma - \sqrt {{\varGamma ^2} - 4{\text{det}}{{\boldsymbol{\sigma }}_{{\text{ent}}}}} }}{2}}, 其中, \varGamma = \det {\boldsymbol{A}} + \det {\boldsymbol{B}} - 2\det {\boldsymbol{C}}. 当PPT值小于1时, 表明双模量子态是纠缠的. 对于EPR纠缠态的两束光场, 对Alice和Bob端同时进行传输时其协方差矩阵表示为

    {{\boldsymbol{\sigma }}'_{{\text{ent}}}}{\text{ = }}\left( {\begin{array}{*{20}{c}} {{\eta _{\text{A}}}\alpha + 1 - {\eta _{\text{A}}}}&0&{\sqrt {{\eta _{\text{A}}}{\eta _{\text{B}}}} \gamma }&0 \\ 0&{{\eta _{\text{A}}}\alpha + 1 - {\eta _{\text{A}}}}&0&{ - \sqrt {{\eta _{\text{A}}}{\eta _{\text{B}}}} \gamma } \\ {\sqrt {{\eta _{\text{A}}}{\eta _{\text{B}}}} \gamma }&0&{{\eta _{\text{B}}}\beta + 1 - {\eta _{\text{B}}}}&0 \\ 0&{ - \sqrt {{\eta _{\text{A}}}{\eta _{\text{B}}}} \gamma }&0&{{\eta _{\text{B}}}\beta + 1 - {\eta _{\text{B}}}} \end{array}} \right). (8)

    对于波长为1080 nm 的光场而言, 利用KTiOPO4 (KTP) 晶体通过单个非简并光学参量放大器 (nondegenerate optical parametric amplifier, NOPA)即可实现Ⅱ类非临界相位匹配[45], 通过控制NOPA腔后的半波片角度能够实现单模压缩态光场和双模压缩态光场(EPR纠缠态)的切换[46]. 实验装置如图1所示. 所用激光器是连续变量内腔倍频可调谐单频固体激光器, 能够同时输出1080 nm的红外和540 nm的绿光. 两束光场分别经过模式清洁器(MC)后作为NOPA腔的泵浦光场和种子光场. NOPA腔由一个 \alpha -切割的KTP晶体和一个凹面镜组成, 实验中通过调节KTP晶体的温度至相位匹配点附近以满足Ⅱ类非临界相位匹配, 并配合横向移动KTP晶体在腔内的位置, 保证腔长满足信号光、闲置光和泵浦光的三模共振. NOPA腔工作在参量反放大状态时, 当腔后的半波片角度转到0°, 得到一对偏振相互垂直的正交振幅反关联, 正交位相正关联的EPR纠缠态光场, 将半波片角度转为22.5°, 则经由其后棱镜后输出的耦合模分别为正交振幅压缩态光场和正交位相压缩态光场. 实验中分别以正交振幅压缩态光场和EPR纠缠态光场为例进行了量子相干性演化的实验研究. 实验制备的压缩态光场的压缩和反压缩分别为 –4.95 dB和6.53 dB, 纯度为0.83, 其对应的 r = 0.74 , {n_{{\text{th}}}} = 0.096 . 利用半波片和偏振分束器的组合模拟具有不同传输距离的损耗信道, 其中损耗的大小通过调节半波片的角度来改变. 将双模压缩态的两个子模分别发送给Alice和Bob, 对经过损耗信道后的光场进行平衡零拍探测, 将实验数据进行处理重构出量子态的协方差矩阵, 最终得到单模压缩态和双模压缩态在损耗信道的量子相干性演化规律.

    图 1 实验装置示意图, 其中DBS为双色分束镜; MC为模式清洁器; HWP为半波片; HR为高反镜; PBS为偏振分光棱镜; LO为本地振荡光\r\nFig. 1. Schematic diagram of experimental setup. DBS, dichroic beam splitter; MC, mode cleaner; HWP, half-wave plate; HR, high reflection; PBS, polarizing beam splitter; LO, a strong local oscillator beam.
    图 1  实验装置示意图, 其中DBS为双色分束镜; MC为模式清洁器; HWP为半波片; HR为高反镜; PBS为偏振分光棱镜; LO为本地振荡光
    Fig. 1.  Schematic diagram of experimental setup. DBS, dichroic beam splitter; MC, mode cleaner; HWP, half-wave plate; HR, high reflection; PBS, polarizing beam splitter; LO, a strong local oscillator beam.

    首先分析量子态制备过程中热光子数 {n_{{\text{th}}}} 的存在对单模压缩态的压缩特性和量子相干性的影响, 以r = 0.74压缩态为例, 计算结果见图2. 图2(a)中曲线1和2分别表示压缩态的反压缩和压缩分量的相对噪声功率, 曲线3表示SNL. 压缩相对噪声功率随着热光子数的增大而减小, 当 {n_{{\text{th}}}} = 0.77 时, 压缩分量等于SNL, 之后压缩消失. 结合量子相干性计算公式(2)和压缩热态协方差矩阵表达式(4)可计算得到压缩态量子相干性随着热光子数 {n_{{\text{th}}}} 的变化关系, 结果如图2(b)所示. 单模压缩态的量子相干性随着热光子数 {n_{{\text{th}}}} 的增大而减小. 这是由于 {n_{{\text{th}}}} 的增大会导致压缩热态协方差矩阵的对角元难以区分, 特别地, 当 {n_{{\text{th}}}} \to \infty 时, C\left( {\hat \rho } \right) \to 0 , 此时压缩热态变为非相干态.

    图 2 单模压缩态压缩特性和量子相干性随热光子数的变化 (a) 相对噪声功率; (b) 量子相干性\r\nFig. 2. Dependence of squeezing level and quantum coherence of the one-mode squeezed state on the number of thermal photons: (a) Relative noise power; (b) quantum coherence.
    图 2  单模压缩态压缩特性和量子相干性随热光子数的变化 (a) 相对噪声功率; (b) 量子相干性
    Fig. 2.  Dependence of squeezing level and quantum coherence of the one-mode squeezed state on the number of thermal photons: (a) Relative noise power; (b) quantum coherence.

    经过损耗信道后单模压缩态的压缩特性退相干结果如图3(a)所示, 曲线1和2分别表示压缩态光场的反压缩和压缩分量的相对噪声功率, 曲线3表示SNL, 红色圆点和蓝色圆点分别表示压缩和反压缩噪声的实验数据. 在传输效率为1的理想状态时, 对应初始的压缩和反压缩分别为–4.95 dB和6.53 dB. 随着信道传输效率的降低, 压缩度逐渐降低, 直到损耗增大为1时, 压缩特性才完全消失. 图3(b)表示损耗信道中压缩态光场的纯度对量子相干性的影响, 分别取4种不同纯度的压缩态光场计算其量子相干性与损耗信道传输效率的关系. 曲线1, 2, 3, 4分别表示压缩态光场纯度 \mu 为1, 0.83, 0.7, 0.5时量子相干性退相干的理论计算结果, 相应的 {n_{{\text{th}}}} 分别为0, 0.096, 0.22, 0.58. 图中蓝色圆点表示实验制备的压缩态光场在损耗信道中量子相干性的演化结果, 与理论计算相吻合. 结果表明压缩态纯度的降低会引起量子相干性的减小. 压缩态的量子相干性随着损耗的增加而降低, 当且仅当量子态完全被损耗掉后, 量子相干性才变为零. 当传输效率为零, 即损耗等于1时, 压缩态变成了真空态. 因此, 压缩态的压缩特性和量子相干性在损耗信道中是鲁棒的.

    图 3 单模压缩态实验结果 (a) 相对噪声功率随传输效率的变化; (b) 损耗信道中压缩态光场的纯度对量子相干性的影响\r\nFig. 3. Experimental results of the one-mode squeezed state in a lossy channel: (a) Dependence of relative noise power on the transmission efficiency; (b) the influence of purity of squeezed state on quantum coherence in a lossy channel.
    图 3  单模压缩态实验结果 (a) 相对噪声功率随传输效率的变化; (b) 损耗信道中压缩态光场的纯度对量子相干性的影响
    Fig. 3.  Experimental results of the one-mode squeezed state in a lossy channel: (a) Dependence of relative noise power on the transmission efficiency; (b) the influence of purity of squeezed state on quantum coherence in a lossy channel.

    图4(a)图4(b)分别表示r = 0.74的双模压缩态的PPT值和量子相干性随制备系统所引入的热光子数 {n_{{\text{th}}}} 的变化关系. 图4(a)中红线表示双模压缩态的PPT值, 黑虚线表示可分态和纠缠态的边界1, 与单模压缩态相同, 当 {n_{{\text{th}}}} > 0.77 时, PPT值超过1, 此时纠缠态变为可分态, 纠缠特性消失. 双模压缩态量子相干性随热光子数的增大而减小, 并趋近于零, 如图4(b). 通过对比图2图4发现, 当制备过程中热光子数大于0.77时压缩态的压缩特性与纠缠态的纠缠特性均会消失. 而相应的量子相干性随着热光子数的增加而降低, 即使热光子数大于0.77时, 量子相干性依然存在. 这是由于当量子相干性与热光子混合时, 只会让量子相干性所占比例降低, 理论上只有在混合无穷多的热光子数时, 量子相干性所占比例才会趋近于零.

    图 4 双模压缩态实验结果 (a) PPT值随热光子数的变化; (b) 量子相干性随热光子数的变化; (c) 纠缠特性随传输效率的变化; (d) 量子相干性随传输效率的变化\r\nFig. 4. Experimental results of the two-mode squeezed states in lossy channels: (a) Dependence of PPT value on the number of thermal photons; (b) dependence of quantum coherence on the number of thermal photons; (c) dependence of PPT values on the transmission efficiency; (d) decoherence of quantum coherence in the lossy channels.
    图 4  双模压缩态实验结果 (a) PPT值随热光子数的变化; (b) 量子相干性随热光子数的变化; (c) 纠缠特性随传输效率的变化; (d) 量子相干性随传输效率的变化
    Fig. 4.  Experimental results of the two-mode squeezed states in lossy channels: (a) Dependence of PPT value on the number of thermal photons; (b) dependence of quantum coherence on the number of thermal photons; (c) dependence of PPT values on the transmission efficiency; (d) decoherence of quantum coherence in the lossy channels.

    实验制备的双模压缩态的纠缠特性和量子相干性经过损耗信道的退相干结果如图4(c)图4(d)所示, 其中红线表示双模压缩态的单个子模经过损耗信道的传输 ( {\eta _{\text{A}}} = 1, {\text{ }}{\eta _{\text{B}}} = \eta ), 蓝线表示两个子模均经过损耗信道传输的情况( {\eta _{\text{A}}} = {\eta _{\text{B}}} = \eta ), 圆点表示实验测量结果. 当信道传输效率为1时, 对应初始纠缠的PPT值为0.32. 双模压缩态的PPT值随着传输效率的降低而增大, 直到损耗增大为1时, 纠缠特性才完全消失, 变为可分态. EPR纠缠态的量子相干性随着传输效率的减小而降低, 当且仅当纠缠态完全被损耗掉后, 量子相干性才变为零. 显然, 双模压缩态的纠缠特性和量子相干性在损耗信道中是鲁棒的. 对比单模损耗信道与双模损耗信道的结果发现, 对于双模压缩态的纠缠特性和量子相干性, 在单个子模经过损耗信道传输时比两个子模均经过损耗信道时对损耗的容忍度更高. 实验中得到纠缠特性与量子相干性随损耗信道传输效率变化相同的结果. 这是由于量子相干性是反映系统的相干叠加程度, 纠缠是多个关联态的叠加, 反映了多个系统的关联和叠加, 对于纠缠而言, 多体系统中的量子相干性体现了纠缠的本质. 此外, 本实验验证了两种高斯态的量子相干性在损耗信道中传输时的单调性, 由于相干度量的单调性, 在损耗信道这种非相干操作下, 高斯态的量子相干性会降低.

    本文从量子态的实验制备出发, 定量分析了量子态制备过程中热光子数的存在对其压缩特性、纠缠特性和量子相干性的影响, 量化了单模压缩态和双模压缩态经过损耗信道后量子相干性的演化. 实验结果表明, 单模压缩态的压缩特性与双模压缩态的纠缠特性在热光子数超过一定大小后便会消失, 当压缩和纠缠均已消失时, 量子相干性依然存在. 量子态的压缩特性、纠缠特性和量子相干性对损耗均是鲁棒的, 只有在损耗增大为1时, 量子态被完全损耗, 此时压缩特性和纠缠特性均消失, 相应的量子相干性变为零. 对于双模压缩态的纠缠特性和量子相干性, 在单个子模经过损耗信道传输时比两个子模均经过损耗信道时对损耗的容忍度更高.

    最近有研究表明高斯纠缠态可以在单模腔中传输[47], 这为基于高斯量子相干性的潜在应用提供了可能性. 对于量子相干性的退相干, 已经有研究小组实现了量子相干性的单次提纯[48]. 而基于测量的无噪声线性放大技术可以实现对高斯量子相干性的提纯. 我们的研究结果为多组份纠缠态量子相干性在量子信息进程中的应用提供了参考.

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  • 图 1  阴极Na1–xIAx合金的n电子数与熔点的关联图

    Figure 1.  Correlations between various electron numbers and melting points of Na1–xIAx cathode alloys.

    图 2  阴极Na1–xIAx合金的各种电子数与结合能的关联图

    Figure 2.  Correlations between various electrons and cohesive energy of Na1–xIAx cathode alloys.

    图 3  阴极Na1–xIAx合金的各种电子数与电势的关联图

    Figure 3.  Correlations of various electrons and electric potentials for Na1–xIAx anode alloys.

    表 1  Na1–xIAx合金键距计算

    Table 1.  Calculation of bond distances of Na1–xIAx alloy.

    Na1–xIAx {I}_{\alpha } {D}_{\mathrm{uv}}\left({n}_{\alpha }\right) {\bar{D}}_{\mathrm{uv}}\left({n}_{\alpha }\right) {n}_{\rm{A}} { I}_{\alpha } {D}_{\mathrm{uv}}\left({n}_{\alpha }\right) /{ \text{Å} }{\bar{D} }_{\mathrm{uv} }\left({n}_{\alpha }\right)/{ \text{Å} } {n}_{\alpha } |∆D|/{\text{Å} }
    Na83.72963.75020.0516064.30044.32100.008100.0206
    Na0.99K0.0183.73813.75380.0522064.31034.32600.008200.0157
    Na0.99Rb0.0183.74303.75720.0522064.31574.33000.008200.0143
    Na0.99Cs0.0183.74543.76280.0522064.31874.33610.008100.0175
    Na0.98K0.0283.74653.75740.0529064.32024.33100.008230.0108
    Na0.98Rb0.0283.75643.76430.0529064.33104.33890.008200.0079
    Na0.98Cs0.0283.76113.77550.0529064.33704.35130.008170.0143
    Na0.97K0.0383.75503.76100.0534864.33014.33610.008290.0060
    Na0.97Rb0.0383.76973.77130.0535164.34634.34790.008250.0016
    Na0.97Cs0.0383.77693.78810.0535464.35524.36650.008210.0113
    Na0.96K0.0483.76353.76470.0541164.33994.34110.008340.0012
    Na0.96Rb0.0483.78313.77850.0541564.36164.35700.008290.0047
    Na0.96Cs0.0483.79263.80090.0541964.37354.38180.008240.0082
    Na0.95K0.0583.77193.76840.0547464.34984.34630.008400.0036
    Na0.95Rb0.0583.79653.78560.0547964.37694.36610.008340.0109
    Na0.95Cs0.0583.80843.81360.0548464.39184.39710.008270.0052
    DownLoad: CSV

    表 2  Na阴极合金的价电子结构

    Table 2.  Valence electron structures of cathode Na based alloy.

    Na1–xIAxncnsnpnlR(1)
    Na0.46140.46060.00080.53861.4181
    Na0.99K0.010.46680.46600.00080.53321.4217
    Na0.98K0.020.47220.47130.00080.52781.4254
    Na0.97K0.030.47760.47670.00090.52241.4290
    Na0.96K0.040.48300.48210.00090.51701.4327
    Na0.95K0.050.48840.48750.00090.51161.4363
    Na0.99Rb0.010.46680.46600.00080.53321.4235
    Na0.98Rb0.020.47220.47130.00080.52781.4289
    Na0.97Rb0.030.47760.47670.00090.52241.4343
    Na0.96Rb0.040.48300.48210.00090.51701.4397
    Na0.95Rb0.050.48840.48750.00090.51161.4451
    Na0.99Cs0.010.46680.46600.00080.53321.4263
    Na0.98Cs0.020.47220.47130.00080.52781.4345
    Na0.97Cs0.030.47760.47670.00090.52241.4428
    Na0.96Cs0.040.48300.48210.00090.51701.4510
    Na0.95Cs0.050.48840.48750.00090.51161.4592
    DownLoad: CSV

    表 3  阴极Na1–xIAx合金的熔点、结合能与电势

    Table 3.  Melting point, cohesive energy, and electric potentials of cathode Na1–xIAx alloy.

    掺杂量x原子杂阶掺杂杂阶 \bar{T}_{\rm{m}} /K {E}_{\mathrm{c}} /(eV·atom–1) {\bar{E}}_{\mathrm{c}} /(eV·atom–1)\left| { {\Delta E}_{\mathrm{c} } }/{ {E}_{\mathrm{c} } }\right|/{\%}电势/V
    0Na3336.761.1131.1654.670.1482
    0.01Na2K4336.641.1111.1644.770.1481
    0.01Na2Rb4336.451.1101.1634.770.1480
    0.01Na2Cs4336.021.1031.1615.260.1478
    0.02Na2K4336.641.1091.1634.390.1481
    0.02Na2Rb4336.141.1081.1614.780.1478
    0.02Na2Cs4335.291.1101.1584.320.1475
    0.03Na2K4336.601.1031.1625.350.1480
    0.03Na2Rb4335.851.1091.1594.510.1476
    0.03Na2Cs4334.581.1081.1544.150.1471
    0.04Na2K4336.571.1071.1624.970.1479
    0.04Na2Rb4335.571.1031.1574.900.1474
    0.04Na2Cs4333.881.1091.1513.790.1467
    0.05Na2K4336.551.1081.1614.780.1478
    0.05Na2Rb4335.301.1071.1564.430.1472
    0.05Na2Cs4333.201.1081.1473.520.1463
    DownLoad: CSV

    表 4  正极合金的晶体结构

    Table 4.  Crystal structures of anode alloys.

    合金空间群a/\text{Å}b/\text{Å}c/\text{Å}原子占位xyz
    Sb2c0.33330.66660.2500
    Na3SbP63mmc (194)5.3555.3559.496Na12b000.2500
    Na24f0.33330.66660.5830
    NaSnI41/acd (142)10.46010.46017.390Sn32g0.06960.12600.9362
    Na116f0.62580.87580.1250
    Na216e0.872400.2500
    Sn16c0.20830.20830.2083
    Na15Sn4I43d (220)13.14013.14013.140Na112a0.375000.2500
    Na248e0.12700.15480.9670
    Pb32g0.06960.11860.9383
    NaPbI41/acd (142)10.58010.58017.746Na116e0.25000.12500.5000
    Na216f0.12500.37500.6250
    DownLoad: CSV

    表 5  阳极合金的键距

    Table 5.  Bond distances of the anode alloy.

    合金键序成键原子 {I}_{\alpha } {D}_{\mathrm{uv}}\left({n}_{\alpha }\right) /\text{Å} {\bar{D}}_{\mathrm{uv}}\left({n}_{\alpha }\right)/\text{Å} {n}_{\alpha } |ΔD|/\text{Å}
    Na3Sb1Sb-Na263.09753.09100.390550.0065
    2Sb-Na143.16853.16200.194160.0065
    3Na1-Na243.17803.17150.180300.0065
    4Na1-Na163.47693.47040.037380.0065
    5Sb-Na1123.48133.47480.058460.0065
    6Na2-Na1123.48133.47480.056300.0065
    7Na2-Na1124.43104.42450.001470.0065
    8Na2-Na224.75754.75100.000640.0065
    NaSn1Sn-Sn22.97483.02010.426500.0453
    2Sn-Sn42.99253.03780.398490.0453
    3Na1-Sn43.33553.38080.075060.0453
    4Na1-Sn43.35923.40450.068540.0453
    5Na2-Sn43.39743.44270.130640.0453
    6Na2-Sn43.42313.46840.118370.0453
    7Na1-Sn43.48703.53230.041970.0453
    8Na2-Sn23.52253.56780.080830.0453
    9Na2-Sn43.54823.59350.073240.0453
    10Na1-Na243.61483.66010.039850.0453
    11Na1-Na243.66583.71110.032770.0453
    12Na1-Na113.72183.76710.011970.0453
    13Sn-Sn23.74063.78590.022570.0453
    14Sn-Sn24.37804.42330.001960.0453
    15Na1-Na244.49194.53720.001380.0453
    16Na1-Sn44.66744.71270.000450.0453
    17Na2-Na214.70954.75480.001320.0453
    Na15Sn41Sn-Na2243.23783.28540.208500.0476
    2Na2-Na2243.26243.31000.194990.0476
    3Na1-Na2243.34253.39010.110170.0476
    4Na2-Na2123.34683.39440.148300.0476
    5Sn-Na2243.40493.45250.121270.0476
    6Sn-Na2243.41893.46650.115890.0476
    7Na1-Na2243.50263.55020.065550.0476
    8Sn-Na1243.54823.59580.055820.0476
    9Na2-Na2243.81383.86140.032610.047
    10Na2-Na2243.97944.02700.019060.0476
    11Na2-Na2124.17124.21880.010230.0476
    NaPb1Pb-Pb23.14643.14520.334770.0013
    2Pb-Pb43.16183.16060.315560.0013
    3Pb-Na243.36533.36410.198950.0013
    4Pb-Na143.38883.38760.082370.0013
    5Pb-Na243.42153.42030.160350.0013
    6Pb-Na243.48473.48350.125820.0013
    7Pb-Na143.49293.49170.055240.0013
    8Pb-Na143.55493.55370.043540.0013
    9Pb-Na143.61723.61600.034280.0013
    10Pb-Pb23.64183.64060.050010.0013
    11Na1-Na283.69673.69550.034790.0013
    12Na2-Na213.74063.73940.064880.0013
    13Pb-Pb24.40084.39960.002720.0013
    14Na1-Na244.54554.54430.001340.0013
    15Pb-Na244.75134.75010.000970.0013
    DownLoad: CSV

    表 6  阳极产物的价电子结构

    Table 6.  Valence electron structures of anode products

    合金原子杂阶ncnsnpnlR(1)
    Na3SbSb23.00000.56942.430601.4279
    Na141.00000.99820.001801.3070
    Na220.46140.46060.00080.53861.4181
    NaSnSn12.000002.00002.00001.3990
    Na111.00000.99820.001801.3070
    Na240001.00001.5133
    Na15Sn4Sn43.66380.83192.83190.33621.3990
    Na141.00000.99820.001801.3070
    Na230.53500.53400.00100.46501.4029
    NaPbPb22.09620.04812.04811.90381.4300
    Na141.00000.99820.001801.3070
    Na210001.00001.5133
    DownLoad: CSV

    表 7  正极合金的熔点、结合能与电势

    Table 7.  Melting point, cohesive energy, and electric potentials of anode alloy.

    合金Tm/K [35] \bar{T}_{\rm{m}} /K|{\Delta {T}_{\mathrm{m} } }/{ {T}_{\mathrm{m} } }|/%电势/VnβEc/(eV·atom–1)
    Na3Sb11291142.961.21.152040.601.766
    NaSn851813.164.40.734350.602.103
    Na15Sn4681746.169.60.907430.711.318
    NaPb645630.682.20.826360.601.559
    DownLoad: CSV

    表 8  电池的开路电压

    Table 8.  Open gate voltages of the battery.

    Na1–xIAx开路电压/V
    Na3SbNaSnNa15Sn4NaPb
    Na1.00380.58610.75920.6781
    Na0.09K0.011.00390.58620.75930.6782
    Na0.98K0.021.00390.58620.75930.6782
    Na0.97K0.031.00400.58630.75940.6783
    Na0.96K0.041.00410.58640.75950.6784
    Na0.95K0.051.00420.58650.75960.6785
    Na0.99Rb0.011.00400.58630.75940.6783
    Na0.98Rb0.021.00420.58650.75960.6785
    Na0.97Rb0.031.00440.58670.75980.6787
    Na0.96Rb0.041.00460.58690.76000.6789
    Na0.95Rb0.051.00480.58710.76020.6791
    Na0.99Cs0.011.00420.58650.75960.6785
    Na0.98Cs0.021.00450.58680.75990.6788
    Na0.97Cs0.031.00490.58720.76030.6792
    Na0.96Cs0.041.00530.58760.76070.6796
    Na0.95Cs0.051.00570.58800.76110.6800
    nl/atom0.26931.25000.36451.0682
    DownLoad: CSV

    表 A1  IA族元素的乙种杂化表

    Table A1.  B type hybrid table of IA group

    σ1234
    Chσ10.53860.46500
    Ctσ00.46160.53501
    nTσ1111
    nlσ10.53860.46500
    ncσ00.46160.53501
    Rσ(1)H0.37080.32890.32220.2800
    Li1.32601.20891.14400.9860
    Na1.51331.45511.43081.3070
    K1.96281.87941.86011.7820
    Rb2.08702.02702.01751.9570
    Cs2.21402.22602.22792.2400
    注: l, \; m, \;n, \; \tau : 1 0 0 0   l{'}, \; m{'}, \;n{'}, \; \tau {'}: 0.9982 0.0018 0 0
    DownLoad: CSV

    表 A2  VA族元素的甲种杂化表

    Table A2.  A type hybrid table of VA group

    σ1234
    Chσ10.56940.19830
    Ctσ00.43060.80171
    nTσ3 or 53 or 53 or 53 or 5
    nlσ0000
    ncσ3 or 53 or 53 or 53 or 5
    Rσ (1)N0.70000.75170.79730.8200
    P1.09801.11731.13431.1428
    As1.18001.23901.29111.3170
    Sb1.35601.42791.49191.5230
    Bi1.39901.44551.50441.5290
    注: l, \; m, \; n, \; \tau : 1 2 0 1; l{'}, \; m{'}, \; n{'}, \; \tau {'} : 0 3 0 1
    DownLoad: CSV

    表 A3  IVA族元素的甲种杂化表

    Table A3.  A type hybrid table of IVA group

    σ123456
    Chσ10.95020.83200.16810.04810
    Ctσ00.04980.16800.83190.95191
    nTσ444444
    nlσ21.90401.66400.33600.09600
    ncσ22.09602.33603.66403.90404
    Rσ(1)C0.76300.76300.76300.76300.76300.7630
    Si1.17001.17001.17001.17001.17001.1700
    Ge1.22301.22301.22301.22301.22301.2230
    Sn1.39901.39901.39901.39901.39901.3990
    Pb1.43001.43001.43001.43001.43001.4300
    注: l, \; m, \; n, \; \tau ; 2 2 0 0; l{'}, \; m{'}, \; n{'}, \; \tau {'}; 1 3 0 1
    DownLoad: CSV
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Publishing process
  • Received Date:  30 September 2020
  • Accepted Date:  01 December 2020
  • Available Online:  02 April 2021
  • Published Online:  20 April 2021

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