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稀土离子在元素周期表中构成了一组特殊的过渡元素。三重电离的稀土离子有一个部分填充的4f壳层,外层填满的5s2和5p6电子壳层将其与外界环境屏蔽,即使稀土离子被掺杂到宿主晶体中,能级屏蔽作用使得晶体仅对自由离子能级起微弱扰动作用。内壳层4f-4f跃迁在光谱上从远红外跨越到真空紫外,部分填充的4fn壳层产生狭窄的谱线[12]。可见,稀土离子的能级结构较特殊且光谱资源丰富。
稀土离子掺入晶体后将产生能级分裂,间隔通常在几百cm-1以上。电磁超精细相互作用不仅提供了附加的结构还提供了长寿命基态自旋相干存储的机遇[12]。5s2和5p6层的屏蔽作用还可以有效减弱晶格与宿主原子对4fn能级的扰动,从而获得较长的光学与自旋相干时间,其中光学相干时间可达毫秒量级,自旋相干时间可达6h[13-14]。稀土离子的相干时间对环境变化较为敏感,低温环境可以较好地抑制声子扰动,提高稀土掺杂固态系综中的光学跃迁与自旋跃迁相干性,施加磁场能使离子产生能级分裂,均匀线宽因此变窄,相干时间得到延长。故在实验中常通过降低环境温度(一般小于4K)和施加磁场来延长相干时间。
在稀土掺杂材料中,存在均匀展宽和非均匀展宽两种主要的展宽机制影响观察光谱。通常认为均匀展宽对晶体中单个离子的影响是相等的。非均匀展宽通常被看作是由于晶体生长、杂质、位错或其它晶格缺陷引起的晶体局部应变造成的静态效应,每个单独的光学中心会在宿主晶体中经受不同的局部环境。这会使单个光学中心的中心频率发生偏移,进而改变跃迁频率的分布[15]。
许多均匀展宽的谱线组合起来,每一条谱线的洛伦兹吸收谱集中在它自己的共振频率上,致使产生更宽的频率分布,通常为高斯分布(集中缺陷引起的应变)或洛伦兹分布(稀释缺陷引起的应变),其宽度称为非均匀展宽Γinh。稀土离子的非均匀展宽从几百兆赫兹到几百吉赫兹等,可实现大带宽存储,还可获得较大的多模容量[16]。许多稀土离子已被应用于固态量子存储,使用不同的稀土离子可以获得不同的工作波长范围。特别的,Er3+在1.5μm附近具有跃迁,因为这种特性,它非常适合用于通信波段的固态量子存储,后面将会看到,通信波段的固态量子存储基本上都在掺Er3+的固态系综中实现。
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稀土掺杂固态量子存储协议可分为光子回波与电磁诱导透明(electromagnetically induced transparency, EIT)[17]两大类。其中光子回波协议又包含受控可逆非均匀展宽协议(controlled reversible inhomogeneous broadening, CRIB)、原子频梳协议(atomic frequency combs, AFC)、静默回波恢复协议(revival of silenced echo, ROSE)等。下面分别对这几类协议进行介绍,特别地,目前大部分固态量子存储均基于光子回波协议实现,本文中将重点介绍该类协议。考虑形成非均匀展宽吸收的原子系综,假设光子入射前系综中所有原子均处于基态|g〉,则光-原子系统的状态可由一个直积态表示:
$ |1\rangle_{\text {in }} \otimes\left|g_{1} \cdots g_{j} \cdots g_{N}\right\rangle $
(1) 如果单光子的光谱与这条展宽线相匹配而被吸收,由于不知道具体是系综中哪一个原子吸收的,则原子系综的激发态可由一个集体激发态表示:
$ \begin{gathered} |\varphi\rangle=\frac{1}{\sqrt{N}} \sum\limits_{j=1}^{N}\left[c_{j} \exp \left(\mathrm{i} 2 \pi \delta_{j} t\right) \exp (\mathrm{i} k z)\right] \times \\ \left|g_{1} \cdots e_{j} \cdots g_{N}\right\rangle \end{gathered} $
(2) 式中, cj代表第j个原子的概率振幅,δj表示第j个原子相对于入射光子载波频率的失谐量,zj表示第j个原子的位置,N为系综原子数,t为从吸收开始计数的时间,k为波数,ej为吸收光子的原子所处的状态,其由基态转变为激发态。一旦吸收发生,由exp(i2πδjt)项可以看出,集体激发态中不同的组分开始积累与其失谐成比例的不同的相位,导致失相的发生。这种失相抑制了系综的集体光发射。因此,如果找到一种方法来消除这种失相,在所有原子经过时间之后相位的变化量为2π的整数倍,这种集体光发射就会发生,即入射光子在原子系综中经历一段时间后被重新发射,携带有入射前编码的量子态信息,这就是存储的基本思想。
基于光子回波的稀土掺杂固态量子存储通常需要先进行吸收线型的制备,如制备原子频率梳的主要手段是通过光抽运将不需要的原子搬运到辅助能级上,而原来基态能级上留下的原子构成所需的梳齿形状,这也称为原子布居反转。制备过程的能级和原子布居如图 1所示。其中|g〉为基态能级,|e〉为激发态能级,|aux〉为辅助能级(下同)。
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该协议的基本重相机制是基于受控可逆非均匀展宽(controlled reversible inhomogeneous broadening,CRIB)。其具体思想如下[18-19]:在透明窗口内产生一条窄吸收线(可以通过光抽运技术来实现),这条最初很窄的吸收线随后被人为加宽(可通过在光传播方向上施加电场梯度来实现); 吸收光子后,原子偶极子将发生相应的失相,原子将处于(2)式给出的状态,即集体激发态。如果在合适的时间改变失谐的符号,即δj→-δj(可通过简单地反转电场极性来实现),各个原子的总相位演化$\int_{0}^{t} \delta_{j} \mathrm{~d} t $将可以回到初始状态。具体的,在失谐发生一段时间t1后(t1从失谐时刻开始计数),改变相位演化的方向,对第j个原子,相位演化可以描述为:
$ \int_{0}^{t_{1}} \delta_{j} \mathrm{~d} t+\int_{t_{1}}^{t}\left(-\delta_{j}\right) \mathrm{d} t $
(3) 失谐反转后(t2从反转时刻开始计数),系统状态可写为:
$ \begin{gathered} |\varphi\rangle=\frac{1}{\sqrt{N}} \sum\limits_{j=1}^{N}\left[c_{j} \exp \left(\mathrm{i} 2 \pi \delta_{j} t_{1}\right) \exp \left(-\mathrm{i} 2 \pi \delta_{j} t_{2}\right) \times\right. \\ \left.\exp \left(\mathrm{i} k z_{j}\right)\right]\left|g_{1} \cdots e_{j} \cdots g_{N}\right\rangle \end{gathered} $
(4) 值得注意的是,t1可在吸收后选择(存储可按需读出)。由(3)式及(4)式可知,当t1=t2=t/2时,(3)式中的积分为0,(4)式中前两个指数项乘积(总相位因子)为1,失相被消除,光子重发射发生。
CRIB协议前向恢复效率为:
$ \eta_{\text {CRII }}=d_{\mathrm{b}}{ }^{2} \exp \left(-d_{\mathrm{b}}\right) \exp \left(-d_{0}\right) \exp \left(-t^{2} \tilde{\gamma}^{2}\right) $
(5) 式中,db代表展宽峰的光学厚度,$ \tilde{\gamma}=2 \pi \gamma$代表初始峰的频谱宽度。第1项给出了光子被吸收和再发射的原始效率。第2项解释了光子被介质重新吸收的可能性。在低效率光抽运的情况下,光在光学厚度为d0的吸收背景下的(重)吸收将会发生,这里也考虑到了这一点。最后一项描述了因初始峰的有限宽度引起的退相干。如果d0=0,使用后向读出可实现100%效率的存储[20]。
前期制备完成后,最简单的CRIB协议可在二能级系统中实现,时刻t1时,反转外场极性,总时间为t=2t1时,重发射发生。为了延长存储时间,可将光学相干转化到自旋跃迁上(需要三能级系统),即在T1时刻施加一个π脉冲,将集体激发态转化到基态自旋能级[21],T2时刻想要读出时,再施加一个π脉冲,自旋相干又转化为光学相干,再次反转外场极性实现读出。
图 2为CRIB协议能级示意图。其中图 2a为简单二能级系统CRIB存储能级示意,图 2b为三能级系统CRIB存储能级示意(将集体激发态转化到基态自旋能级实现长时间存储)。
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原子频率梳(atomic frequency combs,AFC)是在光学厚度(原子密度)-频率(或失谐量)2维坐标系上形成的等间距吸收峰,因形如梳子而得名, 其峰宽为γ(FWHM),峰间距为Δ,单位为Hz,如图 3所示,虚线代表入射光脉冲[22]。
原子频梳协议的具体思想如下:当一个入射光子入射到存储介质,它将会被频梳齿形结构中的原子吸收,原子系综的激发态可用一个集体激发态表示,即(2)式,之后原子因积累不同的失谐而导致失相,与CRIB类似。但与CRIB不同的是,由于原子跃迁频率的离散性和周期性(因AFC是等间距的吸收峰结构),AFC协议的重相是自动发生的,不需要外场干预,这意味着AFC协议会在特定的时间实现集体光发射,但吸收后的重相时刻不能改变。如果要实现按需存储,需要额外地将激发态能级相干转移到基态自旋能级上并返回[22]。
如果AFC的峰宽度相对于峰间隔而言足够窄,(2)式中的失谐量δj可写为离散变量的乘积mjΔ,其中mj为整数。此时原子系综的集体激发态可写为:
$ \begin{aligned} |\varphi\rangle &=\frac{1}{\sqrt{N}} \sum\limits_{j=1}^{N}\left[c_{j} \exp \left(\mathrm{i} 2 \pi m_{j} \Delta t\right) \times\right.\\ &\left.\exp \left(\mathrm{i} k z_{j}\right)\right]\left|g_{1} \cdots e_{j} \cdots g_{N}\right\rangle \end{aligned} $
(6) 当时间为t=n(1/Δ)时,其中n为整数,所有依赖于时间的相位变为2π的整数倍,即所有项都发生重相。此时,输入光以原始编码状态重新发射。理想情况下,重发射后所有的原子均返回基态能级,光-原子系统回到(1)式给出的初始状态。当输入光子在n≥2的重相时刻重新发射时,输出光子称为高阶回波。通常1阶回波(n=1)比高阶回波具有更高的发射概率。因此,通常只考虑1阶回波。AFC协议在前向上的第1个回波效率可由下式给出[22-23]:
$ \eta_{A F C} \approx \frac{d^{2}}{F^{2}} \exp \left(-\frac{d}{F}\right) \exp \left(-d_{0}\right) \exp \left(-\frac{7}{F^{2}}\right) $
(7) 式中, F为AFC频梳精细度,其值为峰间距Δ与吸收峰宽度γ之比。d为峰的光学厚度。注意到此式与(5)式相似,相比于(5)式,AFC协议中吸收峰的光学厚度仅是用有效光学厚度d/F替代。如果将精细度F写成存储时间ts=1/Δ和线宽$ \tilde{\gamma}$的形式,会发现最后一项对应于(5)式中的退相干项。该式已经将吸收背景考虑在内。与CRIB协议类似,在吸收谱制备完成后,最简单的AFC协议可在二能级系统中实现。
基态的原子系统吸收入射光子后,处于集体激发态,由于原子频梳的周期特性,经过时间t=(1/Δ),光子重发射将自发地发生。延长存储时间可通过将集体激发态转化为自旋能级上的自旋波存储来实现(需要三能级系统)。具体的,原子系综吸收光子后,在T1时刻施加一个π脉冲,使得将集体激发态转化到基态自旋能级,此时系统中各个原子的相位演化停止。T2时刻若需读出,则再施加一个π脉冲,此时自旋相干转化为光学相干,系综中各个原子的相位继续演化,在特定时刻可以获得重发射,AFC存储时序如图 4所示。其中虚线脉冲为不施加π脉冲时的回波(一般存储),实线输出脉冲为施加π脉冲后(按需存储)的回波。
图 5为AFC协议能级示意图。图 5a为简单二能级系统AFC存储能级示意,图 5b为三能级系统AFC存储能级示意,将集体激发态转化到基态自旋能级实现长时间存储。注意该图与图 2很相似,但从激发态能级回到基态能级并不需要外场作用。
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如果向非均匀展宽介质中先后输入π/2脉冲和π脉冲,两次输入的时间间隔为t12,则经过相同时间t12后,介质会发射出一个相干辐射脉冲。双脉冲光子回波(two-pulse echo,2PE)协议与此类似。在t=0时刻向介质中输入信号脉冲,在t=t12时刻输入强重相脉冲,则在t=2t12时刻会触发回波脉冲,如图 6所示。
图 6 双脉冲回波示意图[24]
显然,双脉冲回波协议在理论上满足作为存储的条件。但是,许多研究人员指出该协议并不算是一个好的量子存储协议[24-26]。具体来讲,该协议遭受由重相脉冲导致的粒子数反转,为了有效地反转原子相干性的相位,并使它们在稍后的时刻重相,这个脉冲必须同时将原子抽运到光学跃迁的上能级。在增益区工作时,反转介质也会因自发辐射(spontaneous emission, SE)而产生弛豫,这进一步增加了本征噪声,不适合恢复光的初始量子态。此外,由于空间相位匹配的要求,双脉冲回波信号沿着与驱动场相同的方向传播。因此,当回波仅由几个光子组成时,很容易被掩埋在重相脉冲自由感应衰减(free induction decay, FID)的长尾中。
在双脉冲回波协议的基础上,静默回波恢复协议(revival of silenced echo,ROSE)被提出[27], 相比于CRIB和AFC,该协议不需要任何准备步骤。其基本思想如下:首先,一个携带所要存储信息的弱脉冲在时间t1入射到存储介质中,之后一个强重相脉冲在时间t2射入介质,将布洛赫矢量旋转一个角度π。这个π旋转同时逆转了不均匀的相移,并将原子提升到更高的能级。原子相干在时间te=t1+2t12时再次相移,其中tij=tj-ti,并辐射出一个回波信号。由重相脉冲导致的布居反转可产生较大的增益和自发辐射。二者都会影响信息恢复时的保真度,如果在时间t3(t3>te)施加第2个π脉冲,可使原子回到基态。
如图 7所示,在tr=t1+2t23时基态原子发出二次回波,此时避免了增益和自发辐射噪声,理想情况下,通过接收二次回波即可获得所存储的信息。然而这个简单的过程只能恢复存储的部分信息; 另一部分信息在te时已经被一次回波带走。为了避免这种信息丢失,必须想办法消除一次回波。目前有利用斯塔克效应引起的干扰[28-29]或依赖空间相位失配[27]等方法来消除一次回波。
图 7 ROSE协议示意图[27]
通信波段稀土离子掺杂固态量子存储进展
Advances of rare earth ions doped solid-state quantum memory at telecom band
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摘要: 量子互联网是实现多方量子通信、分布式量子计算等量子信息技术的重要基础,量子存储器作为实现互联网的重要部件,对量子信息技术的发展、应用具有举足轻重的作用。如今遍布全球的光纤网络已经是信息传输的有力载体,通信波段的量子存储器因容易嵌入到当前的光纤网络中而备受重视。聚焦于稀土离子掺杂固态体系的通信波段光量子存储,首先介绍稀土离子掺杂固态量子存储的基本原理,包括稀土掺杂材料特性以及存储协议等,然后介绍目前的研究现状,最后简要分析其未来的发展趋势,并对量子互联网的构建做出展望。Abstract: Quantum internet is an important basis for realizing multi-party quantum communication, distributed quantum computing, and other quantum information technologies. Quantum memory, as a significant part for realizing Internet, plays a pivotal role in the development and application of quantum information technology. Nowadays, the global optical fiber network has become a powerful carrier of information transmission, and quantum memory in communication band is highly valued because it is easy to be embedded in the current optical fiber network. Focus on telecom band optical quantum memory with rare earth ions doped solid-state system, the basic principle of rare earth ions doped solid-state quantum memory was firstly introduced, including rare earth doped material properties and memory protocol. the current state of the art was then introduced. Finally, a brief analysis on its future development trend was given, and the prospect for the construction of quantum Internet was made.
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Key words:
- quantum optics /
- telecom band /
- solid-state quantum memory /
- rare earth ions
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图 6 双脉冲回波示意图[24]
图 7 ROSE协议示意图[27]
图 8 CRIB协议存储实验图[33]
图 9 AFC存储实验图[35]
图 10 频率上转换存储实验装置及特性图[36]
图 11 掺铒光纤存储实验装置及测量结果图[37]
图 12 片上存储实验装置及测量结果图[41]
图 13 通信波段多路复用的宽带单光子存储实验装置及测量结果图[44]
图 14 ROSE实验中磁场、光束及其偏振方向[46]
图 15 ROSE回波及其效率[46]
图 16 ROSE效率与光学厚度的函数曲线[46]
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