Experimental demonstration on quantum coherence evolution of two-mode squeezed state

Yu Juan; Zhang Yan; Wu Yin-Hua; Yang Wen-Hai; Yan Zhi-Hui; Jia Xiao-Jun

doi:10.7498/aps.72.20221923

Abstract

As one of the most remarkable features of quantum mechanics, quantum coherence is regarded as an important quantum resource in the quantum information processing. The one-mode squeezed state and the two-mode squeezed state (Einstein-Podolsky-Rosen (EPR) entangled states) as the most representative examples of nonclassical states both have quantum coherence. The squeezing property of the squeezed state is described by the variance of quadrature components, and the positive partial transposition (PPT) criterion is used to describe the entanglement of the EPR entangled states. The research of the quantum coherence of Gaussian states is also a bridge between the properties of squeezing and entanglement. It has been shown that the quantum coherence with infinite-dimensional systems can be quantified by relative entropy. One of the widely used effective methods to obtain the value of quantum coherence experimentally is the quantum tomography. The covariance matrices of the quantum states are reconstructed via balanced homodyne detection and then taken into quantum coherence expression to calculate the corresponding value. The main factors affecting quantum coherence are the classical and uncorrelated noise in the actual experimental generation processing and the decoherence effect caused by the coupling between quantum resources and the surrounding environment. And the quantum coherence evolution in the generation and transmission process of the quantum resources is essential for the practical applications. Therefore, we analyze in detail the influences of the impurity of quantum resource on squeezing, entanglement and quantum coherence. The evolutions of quantum coherence of these Gaussian states in the lossy channels are demonstrated experimentally. The quantum coherence is shown to be robust against the loss in the lossy channels, which is similar to the case of squeezing and entanglement. The quantum coherences of the squeezed states and the EPR entangled states are robust against the thermal photons in the actual experimental generation processing, although the squeezing and entanglement of Gaussian states disappear at a certain number of thermal photons. Our research results provide a reference for the practical applications of quantum coherence of the squeezed state and entangled states in the lossy environment.

Keywords:

squeezed state /
Einstein-Podolsky-Rosen entangled states /
purity of quantum state /
quantum coherence

PACS:

42.50.-p	(Quantum optics)
42.50.Lc	(Quantum fluctuations, quantum noise, and quantum jumps)
42.65.Yj	(Optical parametric oscillators and amplifiers)

Authors and contacts

1.
School of Optoelectronic Engineering, Xi’an Technological University, Xi’an 710021, China
2.
State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Opto-Electronics, Shanxi University, Taiyuan 030006, China
3.
Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006, China
4.
China Academy of Space Technology (Xi’an), Xi’an 710000, China

Corresponding author: Yu Juan, yujuan643@126.com

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 62105256, 62122044, 61925503, 11904218, 12147215, 11834010, 62135008, 62001374, 12004276, 12103039), the Natural Science Research Program of the Education Department of Shaanxi Province, China (Grant Nos. 21JK0694, 18JK0386, 21JY016), the Program for the Innovative Talents of Higher Education Institutions of Shanxi, China, the Scientific and Technological Programs of Higher Education Institutions in Shanxi, China (Grant No. 2019L0794), the Program for Sanjin Scholars of Shanxi Province, China, the Fund for Shanxi “1331Project” Key Subjects Construction, China, and the Natural Science Basic Research Program in Shaanxi Province of China (Grant No. 2021JQ-640).

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1. 引　言

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.

2. 量子态相干性定量分析

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

$C\left( {\hat \rho } \right) = S\left( {{{\hat \rho }_{{\text{diag}}}}} \right) - S\left( {\hat \rho } \right) \text{, }$

(1)

其中 $\hat \rho {\text{ = }}\sum\nolimits_{i, j} {{\rho _{i, j}}} \left| i \right\rangle \left\langle j \right|$ , ${\hat \rho _{{\text{diag}}}}{\text{ = }}\sum\nolimits_i {{\rho _{i, i}}} \left| i \right\rangle \left\langle i \right|$ 表示对 $\hat \rho$ 的退位相操作, 将 $\hat \rho$ 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 ${\lambda _i}$ 计算, $S(\hat \rho ) = - {\displaystyle\sum\nolimits_i}{\lambda _i}\log {\lambda _i}$ . 当且仅当量子态为纯态时, $S(\hat \rho ) = 0$ . 量子态的相对熵体现的是 $\hat \rho$ 和 ${\hat \rho _{{\text{diag}}}}$ 之间的距离. 量子力学中光场可以用湮灭算符 $\hat a$ 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 $\hat x{\text{ = }}\left( {{{\hat a}^\dagger }{\text{ + }}\hat a} \right)$ 和 $\hat p{\text{ = i}}\left( {{{\hat a}^\dagger } - \hat a} \right)$ . 高斯态 $\hat \rho \left( {{\boldsymbol{\bar x}}, {\boldsymbol{\sigma }}} \right)$ 可以由协方差矩阵 ${\boldsymbol{\sigma }}$ 和位移矢量 ${\boldsymbol{\bar x}}$ 很方便刻画^[42], 其中 ${\boldsymbol{\bar x}} = \left\langle {{\boldsymbol{\hat x}}} \right\rangle = \left\langle {{{\left( {{{\hat x}_1}, {{\hat p}_1}, \cdots , {{\hat x}_N}, {{\hat p}_N}} \right)}^{\text{T}}}} \right\rangle$ ; ${\boldsymbol{\sigma }}$ 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 ${\sigma _{ij}} = {\text{Cov}}\left( {{{\hat x}_i}, {{\hat x}_j}} \right) = \dfrac{1}{2}\left\langle {{{\hat x}_i}{{\hat x}_j} + {{\hat x}_j}{{\hat x}_i}} \right\rangle - \left\langle {{{\hat x}_i}} \right\rangle \left\langle {{{\hat x}_j}} \right\rangle$ 给出. 因此, 高斯态的量子相干性可由下式给出:

$C\left[\hat{\rho }\left(\bar{{\boldsymbol{x}}},{\boldsymbol{\sigma}} \right)\right]=S\left[\hat{\rho }\left({\bar{{\boldsymbol{x}}}}_{\text{th}},{{\boldsymbol{\sigma}} }_{\text{th}}\right)\right]-S\left[\hat{\rho }(\bar{{\boldsymbol{x}}},{\boldsymbol{\sigma}} )\right]\text{, }$

(2)

其中, $\hat \rho \left( {{{{\boldsymbol{\bar x}}}_{{\text{th}}}}, {{\boldsymbol{\sigma }}_{{\text{th}}}}} \right)$ 表示与高斯态 $\hat \rho ({\boldsymbol{\bar x}}, {\boldsymbol{\sigma }})$ 具有相同平均光子数的热态, 其协方差矩阵 ${{\boldsymbol{\sigma }}_{{\text{th}}}}$ 的矩阵元 ${\sigma _{{\text{th 2}}i{{ - 1, 2}}i{{ - 1}}}} = {\sigma _{{\text{th 2}}i{{, 2}}i}} = \dfrac{1}{2} \Big[ {\sigma _{{{2}}i - 1{{, 2}}i - 1}} + {\sigma _{{\text{2}}i{\text{, 2}}i}} + {{\left( {{{{\boldsymbol{\bar x}}}_{2 i - 1}}} \right)}^2} +{{\left( {{{{\boldsymbol{\bar x}}}_{2 i}}} \right)}^2} \Big]$ , 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

$\begin{aligned} &S\left[\hat{\rho }\left(\bar{{\boldsymbol{x}}},{\boldsymbol{\sigma}} \right)\right]={\displaystyle {\sum }_{i=1}^{n}\left[\left(\frac{{\upsilon }_{i}+1}{2}\right){\rm{log}}_{2}\left(\frac{{\upsilon }_{i}+1}{2}\right)-\left(\frac{{\upsilon }_{i}-1}{2}\right){\rm{log}}_{2}\left(\frac{{\upsilon }_{i}-1}{2}\right)\right]}\text{, }\\ &S\left[\hat{\rho }\left({\bar{{\boldsymbol{x}}}}_{\text{th}},{{\boldsymbol{\sigma}} }_{\text{th}}\right)\right]={\displaystyle {\sum }_{i=1}^{n}\left[\left(\frac{{\nu }_{i}+1}{2}\right){\rm{log}}_{2}\left(\frac{{\nu }_{i}+1}{2}\right)-\left(\frac{{\nu }_{i}-1}{2}\right){\rm{log}}_{2}\left(\frac{{\nu }_{i}-1}{2}\right)\right]}\text{, }\end{aligned}$

(3)

其中 ${\nu _i}$ 和 ${\upsilon _i}$ 分别为协方差矩阵 ${{\boldsymbol{\sigma }}_{{\text{th}}}}$ 和 ${\boldsymbol{\sigma }}$ 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 $\left\langle {{\Delta ^2}\hat x} \right\rangle = \left\langle {{\Delta ^2}\hat p} \right\rangle = 1$ , 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 $1/2{\text{Diag}}({{\text{e}}^{ - 2 r}}, {{\text{e}}^{2 r}})$ , 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 ${n_{{\text{th}}}}$ 来描述所制备的压缩态光场的不纯, 单模压缩热态 ${\hat \rho _{{\text{squ}}}}$ 的协方差矩阵 ${{\boldsymbol{\sigma }}_{{\text{squ}}}}$ 表示为^[21]

${{\boldsymbol{\sigma }}_{{\text{squ}}}} = \left( {\begin{array}{*{20}{c}} {{V_{\text{s}}}}&0 \\ 0&{{V_{{\text{as}}}}} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {{n_{{\text{th}}}}{\text{ + }}{{\text{e}}^{ - 2r}}}&0 \\ 0&{{n_{{\text{th}}}}{\text{ + }}{{\text{e}}^{2r}}} \end{array}} \right).$

(4)

其中 ${V_{\text{s}}}$ 和 ${V_{{\text{as}}}}$ 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 $10\lg {V_{\text{s}}}$ dB, 反压缩的相对噪声功率为 $10\lg {V_{{\text{as}}}}$ 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: $\mu \left( {\hat \rho } \right)=\dfrac{1}{{{\prod _i}{\upsilon _i}}}=\dfrac{1}{{\sqrt {\det {\boldsymbol{\sigma }}} }}$ , $0 \leqslant \mu \leqslant 1$ . 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 ${\boldsymbol{A}} = \alpha {\boldsymbol{I}}, {\boldsymbol{B}} = \beta {\boldsymbol{I}}, {\boldsymbol{C}} = \gamma {\boldsymbol{Z}}$ , ${\boldsymbol{I}}$ 和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)

3. 实验装置及结果分析

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.

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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.

下载: 全尺寸图片幻灯片

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.

下载: 全尺寸图片幻灯片

图4(a)和图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和图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.

下载: 全尺寸图片幻灯片

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)和图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.

4. 结　论

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 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

Figure 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.

DownLoad: Full-Size Img PowerPoint

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

Figure 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.

DownLoad: Full-Size Img PowerPoint

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

Figure 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.

DownLoad: Full-Size Img PowerPoint

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

Figure 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.

DownLoad: Full-Size Img PowerPoint

[1]	Baumgratz T, Cramer M, Plenio M B 2014 Phys. Rev. Lett. 113 140401 Google Scholar
[2]	Li Y C, Lin H Q 2016 Sci. Rep. 6 26365 Google Scholar
[3]	Shi Y H, Shi H L, Wang X H, Hu M L, Liu S Y, Yang W L, Fan H 2020 J. Phys. A 53 085301 Google Scholar
[4]	Hillery M 2016 Phys. Rev. A 93 012111 Google Scholar
[5]	Napoli C, Bromley T R, Cianciaruso M, Piani M, Johnston N, Adesso G 2016 Phys. Rev. Lett. 116 150502 Google Scholar
[6]	Gisin N, Ribordy G, Tittel W, Zbinden H 2002 Rev. Mod. Phys. 74 145 Google Scholar
[7]	Giovannetti V, Lloyd S, Maccone L 2011 Nat. Photonics 5 222 Google Scholar
[8]	Giorda P, Allegra M 2016 J Phys. A-Math. Theor. 51 2 Google Scholar
[9]	Zhang A, Zhang K, Zhou L, ZhangW 2018 Phys. Rev. Lett. 121 073602 Google Scholar
[10]	Yuan Y, Hou Z, Zhao Y Y, Zhong H S, Xiang G Y, Li C F, Guo G C 2018 Opt. Express 26 004470 Google Scholar
[11]	Wu K D, Hou Z, Zhong H S, Yuan Y, Guo G C 2017 Optica 4 454 Google Scholar
[12]	Zhang M, Kang H J, Wang M H, Xu X L, Peng K C 2021 Photonics Res. 9 887 Google Scholar
[13]	Tan K C, Volkoff T, Kwon H, Jeong H 2017 Phys. Rev. Lett. 119 190405 Google Scholar
[14]	Tan K C, Jeong H 2018 Phys. Rev. Lett. 121 220401 Google Scholar
[15]	Lostaglio M, Müller M P 2019 Phys. Rev. Lett. 123 020403 Google Scholar
[16]	林银, 黄明达, 於亚飞, 张智明 2017 物理学报 66 110301 Google Scholar Lin Y, Huang M D, Yu Y F, Zhang Z M 2017 Acta Phys. Sin. 66 110301 Google Scholar
[17]	Braunstein S L, Caves C M 1994 Phys. Rev. Lett. 72 3439 Google Scholar
[18]	Feng X N, Wei L F 2017 Sci. Rep. 7 15492 Google Scholar
[19]	Zhang Y R, Shao L H, Li Y, Fan H 2016 Phys. Rev. A 93 012334 Google Scholar
[20]	Xu J 2016 Phys. Rev. A 93 032111 Google Scholar
[21]	Buono D, Buono G, Petrillo G, Torre G, Zonzo G, Illuminati F 2016 arXiv: 1609.00913
[22]	周瑶瑶, 刘艳红, 闫智辉, 贾晓军 2021 物理学报 70 104203 Google Scholar Zhou Y Y, Liu Y H 2021 Acta Phys. Sin. 70 104203 Google Scholar
[23]	Yan Z H, Qin J, Qin Z Z, Su X L, Jia X J, Xie C D, Peng K C 2021 Fundamental Research 1 43 Google Scholar
[24]	Chou C W, Hume D B, Thorpe M J, Wineland D J, Rosenband T 2011 Phys. Rev. Lett. 106 160801 Google Scholar
[25]	Huo M R, Qin J L, Cheng J L, Yan Z H, Qin Z Z, Su X L, Jia X J, Xie C D, Peng K C 2018 Sci. Adv. 4 eaas9401 Google Scholar
[26]	Liu S S, Lou Y B, Chen Y X, Jing J T 2022 Phys. Rev. Lett. 128 060503 Google Scholar
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[34]	Guo X S, Breum C R, Borregaard J, Izumi S, Larsen M V, Gehring T, Christandl M, Neergaard-Nielsen J S, Andersen U L 2020 Nat. Phys. 16 281 Google Scholar
[35]	Bai S Y, An J H 2021 Phys. Rev. Lett. 127 083602 Google Scholar
[36]	Yan Z H, Wu L, Jia X J, Xie C D, Peng K C 2021 Adv. Quantum Technol. 4 2100071 Google Scholar
[37]	Streltsov A, Singh U, Dhar H S, Bera M N, Adesso G 2015 Phys. Rev. Lett. 115 020403 Google Scholar
[38]	Chitambar E, Hsieh M H 2016 Phys. Rev. Lett. 117 020402 Google Scholar
[39]	Deng X W, Liu Y, Wang M H, Su X L, Peng K C 2021 npj Quantum Inform. 7 65 Google Scholar
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Vol. 72, Issue 3 - 2023-02-05

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