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非局域性是纠缠态所具有的典型量子特性,这种特性无法用经典理论或局域隐变量理论解释,它是量子力学的最本质的特征之一[2]。非局域色散消除是非局域性的表现形式之一。它是指频率纠缠光子对中的信号光s经历的色散可以被闲频光i经历的色散非局域消除掉,只要使两光子经过的色散量大小相等符号相反,无论这两个光子相距有多远,它们之间的符合信号(量子关联性)不会改变。就好像这两个光子没有经过色散介质一样。而换成经典脉冲,经上述色散介质后其符合信号仍然会被展宽。
通过自发参量下转换(spontaneous parametric down conversion, SPDC)过程产生频率纠缠光子对态函数的一般形式可以表示为:
$ |\psi\rangle=\iint \mathrm{d} \omega_{s} \mathrm{~d} \omega_{i} F\left(\omega_{s}, \omega_{i}\right) \hat{a}_{s}^{\dagger}\left(\omega_{s}\right) \hat{a}_{i}^{\dagger}\left(\omega_{i}\right)|0\rangle $
(1) 式中,ω表示光子的频率,下标s, i分别表示下转换产生的信号光和闲频光,F(ωs, ωi)为双光子光谱振幅, $\hat{a}^{\dagger} $为产生算符,|0〉表示真空态。一般F(ωs, ωi)≠F(ωs)F(ωi),即双光子光谱振幅是不可因子化的,因此是频率纠缠的。如图 1所示[3],假设用连续激光抽运BBO晶体产生频率纠缠光子对,则(1)式中的态函数可以简化为:
$ \begin{gathered} |\psi\rangle= \\ \int \mathrm{d} \Omega F(\Omega) \hat{a}_{s}^{\dagger}\left(\omega_{0} / 2+\Omega\right) \hat{a}_{i}^{\dagger}\left(\omega_{0} / 2-\Omega\right)|0\rangle \end{gathered} $
(2) 图 1 非局域色散消除的概念图[3]
式中,Ω=ωs, i-ω0/2为信号(闲频)光子与抽运中心频率ω0一半的偏移量。此态是最大频率纠缠态,且两双光子频率完全反关联。由于在SPDC过程中满足能量守恒,纠缠光子对的频率之和一定等于抽运光频率(是一个固定值)。因此,只要测量其中一个光子的频率,就可以同时精确确定另一个光子的频率。在下面会看到,这一特性决定了纠缠光子对的非局域色散消除效应。
如果让信号光s经过2阶色散系数为β1、长度为z1的色散介质后进入探测器D1得到光子到达时间t1,让闲频光i经过2阶色散系数为β2、长度为z2的色散介质后进入探测器D2得到光子到达时间t2, 则两个探测器的联合探测概率正比于以下的Glauber 2阶关联函数:
$ G^{(2)}\left(t_{1}, t_{2}\right)=\left|\left\langle 0\left|\hat{E}_{2}^{(+)}\left(t_{2}\right) \hat{E}_{1}^{(+)}\left(t_{1}\right)\right| \psi\right\rangle\right|^{2} $
(3) 式中,$ \hat{E}_{1}^{(+)}\left(t_{1}\right)=\int \mathrm{d} \omega_{1} f\left(\omega_{1}\right) \hat{a}_{1}\left(\omega_{1}\right) \exp \left[-\mathrm{i}\left(\omega_{1} t_{1}-\right.\right. \left.\left.k_{1} z_{1}\right)\right]$是光场在探测器D1处的正频分量,$ \hat{\boldsymbol{E}}_{2}^{(+)}\left(t_{2}\right)$的定义类似; 实验中,一般用滤波器对纠缠光谱进行后处理,滤波函数为f(ω)=exp[-(ω-ωf)2/(2σf2)],其中,ωf为滤波的中心频率,σf为滤波器的带宽。假设滤波器的带宽很窄,可以将波数k用泰勒级数展开到2阶(忽略3阶和更高阶项):kj(ω0±Ω)=kj(ω0)±αjΩ+βjΩ2, j=1, 2。其中,α和β分别表示色散介质的1阶和2阶色散系数,它们分别决定着波包的延迟和展宽[3]。经过计算最终得到:
$ G^{(2)}\left(t_{1}-t_{2}\right)=\operatorname{Cexp}\left[-\frac{\left(t_{1}-t_{2}-\tilde{t}\right)^{2}}{2 \sigma_{\mathrm{t}}^{2}}\right] $
(4) 式中,C为常数,$ \tilde{t}=\alpha_{2} z_{2}-\alpha_{1} z_{1}$。实验时通常将探测的光子到达时间信号进行符合处理得到符合计数的分布,以此来反应G(2)的形状。理论上可以得到符合分布的方差数学表达式为[1, 4]:
$ \sigma_{\mathrm{t}}^{2}=\frac{1 / \sigma_{\mathrm{f}}^{4}+\left(\beta_{1} z_{1}+\beta_{2} z_{2}\right)^{2}}{1 / \sigma_{\mathrm{f}}^{2}} $
(5) (5) 式包含了色散系数和的平方,若β1z1=-β2z2, 则信号光s所经历的色散将被闲频光i所经历的色散非局域地消除。如果将上述纠缠光换成经典光源,则符合分布的方差数学表达式[1, 4]变成:
$ \sigma_{\mathrm{c}}^{2}=\frac{1 /\left(2 \sigma_{\mathrm{f}}^{4}\right)+\beta_{1}^{2} z_{1}^{2}+\beta_{2}^{2} z_{2}^{2}}{1 / 2 \sigma_{\mathrm{f}}^{2}} $
(6) (6) 式包含了色散系数平方的和,此时,两束经典光所经历的色散彼此间将无法消除。(5)式和(6)式差异的根本原因来自于频率纠缠光源的纠缠特性。
这一效应已分别在纳秒[3, 5-6]、皮秒[7]和飞秒[8-9]尺度被实验验证。需要说明的是,非局域色散消除必须使用非局域的探测方式,如果探测方式是局域的,则这种效应存在经典类比[10]。因此,非局域的探测方式是验证量子非局域性的必要条件。从这个角度来说,LI等人[7]和MAcLEAN等人[11]利用非局域的探测方法所做的工作才是真正意义上的非局域色散消除。而在HOM干涉中,参与干涉的光子需要在分束器上汇聚,因此其探测方式必然本质上是局域的,因而HOM干涉存在着经典类比[12-14],而非局域色散消除是无法用经典过程来模拟的,不存在经典类比,它是量子力学的独特特征,无法用任何经典理论或基于决定论的定域隐变量理论来解释。
为了定量描述这种效应,2010年,WASAK等人[15]提出了一个判定非局域色散消除的不等式,其表达式可以写为:
$ \left\langle\left(\Delta \tau^{\prime}\right)^{2}\right\rangle \geqslant\left\langle(\Delta \tau)^{2}\right\rangle+\frac{(2 \beta L)^{2}}{\left\langle(\Delta \tau)^{2}\right\rangle} $
(7) 式中,〈(Δτ)2〉和〈(Δτ′)2〉分别表示未加色散和加大小相等正负相反色散后纠缠光子对到达时间差的方差。此时2阶色散系数β1=-β2=β,长度z1=z2=L。其归一化的表达式可以写为:
$W=\frac{\left\langle\left(\Delta \tau^{\prime}\right)^{2}\right\rangle\left\langle(\Delta \tau)^{2}\right\rangle}{\left[\left\langle(\Delta \tau)^{2}\right\rangle\right]^{2}+(2 \beta L)^{2}} \geqslant 1 $
(8) 经典光场满足上述不等式,然而,频率纠缠光场将违背以上不等式。因此,此不等式类似于测试偏振纠缠非局域性所用的Bell不等式,可以作为经典光场和非经典(频率纠缠)光场的判定标准,也可以作为频率纠缠光源非局域性测试的标准。目前,此不等式已经被实验所验证[7, 11]。需要说明的是,类似地,频率正关联纠缠光子对也将违背以上不等式,只是〈(Δτ)2〉和〈(Δτ′)2〉应该改为纠缠光子对到达时间和的方差。
目前,非局域色散消除的研究可以分成两类: 一类是量子纠缠光源的色散消除,即两个光子在经历色散介质之后,时域宽度都被扩展了,但是它们的到达时间关联函数没有被展宽; 另一类是非局域量子干涉仪(Franson干涉仪[16])中的色散消除。
量子色散消除的研究进展
Research progress on quantum dispersion cancellation
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摘要: 非局域色散消除是量子纠缠光源的非经典效应之一,在量子信息科学中有着重要的应用。详细介绍了非局域色散消除的概念、研究意义以及近几年国内外的发展状况。对频率纠缠光源的非局域色散消除、Franson干涉仪中的非局域色散消除、Hong-Ou-Mandel干涉仪中的局域色散消除等3种情况的研究进展进行了对比分析。在此基础上,对量子色散消除的研究前景进行了展望。Abstract: Nonlocal dispersion cancellation is one of the non-classical effects of quantum entangled photon sources, and it has important applications in quantum information science. A detailed introduction to the concept, research significance of nonlocal dispersion cancellation, and the research progress in China and abroad in recent years were reviewed. The following three cases: the nonlocal dispersion cancellation of frequency entangled photon sources, the nonlocal dispersion cancellation in Franson interferometers, and the local dispersion cancellation in the Hong-Ou-Mandel interferometer were then compared and analyzed. On this basis, the prospects for the study of quantum dispersion cancellation were anticipated.
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图 1 非局域色散消除的概念图[3]
图 2 使用纠缠光子的非局域色散消除[3]
图 3 非局域色散消除现象的2维直接表征[11]
图 4 实验得到的符合分布[26]
a—未加色散 b~d—分别给信号光加10km/20km/62km单模光纤,给闲频光加1.245km/2.49km/7.47km色散补偿光纤时的结果
图 5 非最大纠缠态非局域色散消除的最优方案研究[28]
a—实验装置图 b—使用频率反关联双光子的情况 c—使用频率正关联双光子的情况
图 6 3个光子的非局域色散消除[33]
a—3个光子的NDC方案图 b—使用量子纠缠光得到的联合到达时间分布 c—使用经典光脉冲得到的联合到达时间分布
图 7 Franson干涉仪中的非局域色散消除方案[34]
图 9 独立单光子源HOM干涉中的色散消除实验[42]
a—实验装置 b—干涉条纹的干涉度和宽度随色散变化图
图 10 相干光源HOM干涉中的色散消除实验[43]
a—实验原理图 b—实验装置图 c—没有经历色散光纤的HOM干涉条纹 d—经历色散光纤后的HOM干涉条纹
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