The quantum internet 量子互联网

In the past two decades, a broad range of fundamental discoveries have been made in the field of quantum information science, from a quantum algorithm that places public-key cryptography at risk to a protocol for the teleportation of quantum states¹. This union of quantum mechanics and information science has allowed great advances in the understanding of the quantum world and in the ability to control coherently individual quantum systems². Unique ways in which quantum systems process and distribute information have been identified, and powerful new perspectives for understanding the complexity and subtleties of quantum dynamical phenomena have emerged.
在过去二十年里,量子信息科学领域取得了一系列基础性的重大发现,从一种可能威胁公钥密码体系的量子算法,到一种用于量子态传态的协议 1。量子力学和信息科学的结合,使我们对量子世界有了更深入的理解,并能更好地控制单个量子系统 2。我们发现了量子系统处理和传输信息的独特方式,并产生了新的视角来认识量子动力学现象的复杂性和微妙性。

In the broad context of quantum information science, quantum networks have an important role, both for the formal analysis and the physical implementation of quantum computing, communication and metrology^2,3,4,5. A notional quantum network based on proposals in refs 4, 6 is shown in Fig. 1a. Quantum information is generated, processed and stored locally in quantum nodes. These nodes are linked by quantum channels, which transport quantum states from site to site with high fidelity and distribute entanglement across the entire network. As an extension of this idea, a 'quantum internet' can be envisaged; with only moderate processing capabilities, such an internet could accomplish tasks that are impossible in the realm of classical physics, including the distribution of 'quantum software'⁷.
量子信息科学的广泛背景中,量子网络在量子计算、通信和测量的正式分析和物理实施中发挥着重要作用。图 1a 展示了基于参考文献 4 和 6 中的提议而构建的一个假想的量子网络。量子信息在量子节点中产生、处理和存储。这些节点通过量子通道相互连接,以高保真度传输量子态并在整个网络中分配纠缠。在这个想法的延伸中,可以设想一个"量子互联网"。即使只有适度的处理能力,这样一个互联网也可以完成在经典物理领域中是不可能的任务,包括"量子软件"的分发。

a, Shown is a notional quantum network composed of quantum nodes for processing and storing quantum states and quantum channels for distributing quantum information. Alternatively, such a network can be viewed as a strongly correlated many-particle system. b, The quantum interface between matter (coloured cube) and light (red curves) is depicted. Coherent interactions in the node are characterized by the rate χ; coupling between the node and photons in the external channel occurs at the rate κ; and parasitic losses occur at the rate γ. c, Quantum state transfer and entanglement distribution from node A to node B is shown in the setting of cavity quantum electrodynamics (QED)⁶. At node A, a pulse of the control field Ω^out_A(t) causes the transformation of atomic state |Ψ〉 into the state of a propagating optical field (that is, into a flying photon). At node B, the pulse Ωⁱⁿ_B(t) is applied to map the state of the flying photon into an atom in the cavity, thereby realizing the transfer of the state |Ψ〉 from node A to node B (ref. 18). d, The distribution of entanglement by using ensembles of a large number of atoms is shown¹³. A single-photon pulse at node A is coherently split into two entangled components that propagate to node B and node C and then are coherently mapped by the control fields Ωⁱⁿ_{B, C}(t) into a state that is entangled between collective excitations in each ensemble at node B and node C. At later times, components of the entangled state can be retrieved from the quantum memories by separate control fields, Ω^out_B,C(t) (ref. 19). H_int(t), interaction hamiltonian; ħ, h/2π (where h is Planck's constant).
显示了一个由量子节点组成的量子网络,用于处理和存储量子态,以及用于分发量子信息的量子通道。此类网络也可视为一个强相关的多粒子系统。 量子节点与光之间的量子接口如图所示。节点中的相干相互作用由速率χ表征;节点与外部通道中的光子的耦合发生于速率κ;而寄生损耗发生于速率γ。 在腔量子电动力学中演示了从节点 A 到节点 B 的量子态传输和纠缠分配。在节点 A,控制场Ωout_A(t)的脉冲导致原子态|Ψ〉转变为传播光场(即飞光子)的状态。在节点 B,脉冲Ωin_B(t)被应用以将飞光子的状态映射到腔中的原子,从而实现了从节点 A 到节点 B 的态|Ψ〉的传输。 利用大量原子组成的集综展示了纠缠的分配。在节点 A,单光子脉冲被相干分裂为两个纠缠分量,传播到节点 B 和节点 C,然后通过控制场Ωin_B,C(t)被相干映射到每个集综中集体激发的纠缠状态。在稍后的时间,可以通过单独的控制场Ωout_B,C(t)从量子存储器中取出纠缠态的分量。

Full size image

Apart from the advantages that might be gained from a particular algorithm, there is an important advantage in using quantum connectivity, as opposed to classical connectivity, between nodes. A network of quantum nodes that is linked by classical channels and comprises k nodes each with n quantum bits (qubits) has a state space of dimension k2ⁿ, whereas a fully quantum network has an exponentially larger state space, 2^kn. Quantum connectivity also provides a potentially powerful means to overcome size-scaling and error-correlation problems that would limit the size of machines for quantum processing⁸. At any stage in the development of quantum technologies, there will be a largest size attainable for the state space of individual quantum processing units, and it will be possible to surpass this size by linking such units together into a fully quantum network.
除了从特定算法中可能获得的优势之外,与经典连接相比,使用量子连接也有一个重要的优势。一个由 k 个节点组成的量子节点网络,每个节点有 n 个量子位(qubits),其状态空间的维度为 k2n,而一个完全量子网络的状态空间则呈指数级增长,达到 2kn。量子连接还可能提供一种强大的方式来克服限制量子处理机器规模的大小缩放和错误相关问题 8。在量子技术发展的任何阶段,个体量子处理单元的状态空间的最大尺度都是有限的,通过将这些单元连接成一个完全量子网络,就可以超越这一限制。

A different perspective of a quantum network is to view the nodes as components of a physical system that interact by way of the quantum channels. In this case, the underlying physical processes used for quantum network protocols are adapted to simulate the evolution of quantum many-body systems⁹. For example, atoms that are localized at separate nodes can have effective spin–spin interactions catalysed by single-photon pulses that travel along the channels between the nodes¹⁰. This 'quantum wiring' of the network allows a wide range for the effective hamiltonian and for the topology of the resultant 'lattice'. Moreover, from this perspective, the extension of entanglement across quantum networks can be related to the classical problem of percolation¹¹.
量子网络的一个不同视角是将节点视为通过量子通道相互作用的物理系统的组件。在这种情况下,用于量子网络协议的基础物理过程已被调整以模拟量子多体系统的演化。例如,局部化在单独节点上的原子可以通过在节点之间传播的单光子脉冲来产生有效的自旋-自旋相互作用。这种网络的"量子布线"为有效哈密顿量和最终"晶格"的拓扑提供了广泛的选择空间。此外,从这个角度来看,跨量子网络的纠缠扩散可以与渗流的古典问题相关联。

These exciting opportunities provide the motivation to examine research related to the physical processes for translating the abstract illustration in Fig. 1a into reality. Such considerations are timely because scientific capabilities are now passing the threshold from a learning phase with individual systems and advancing into a domain of rudimentary functionality for quantum nodes connected by quantum channels.
这些令人兴奋的机会为研究与将图 1a 中的抽象插图转化为现实的物理过程相关的研究提供了动力。这种考虑是及时的,因为科学能力正在从一个个别系统的学习阶段越过阈值,进入量子节点通过量子通道连接的初级功能域。

In this review, I convey some basic principles for the physical implementation of quantum networks, with the aim of stimulating the involvement of a larger community in this endeavour, including in systems-level studies. I focus on current efforts to harness optical processes at the level of single photons and atoms for the transportation of quantum states reliably across complex quantum networks.
在本评论中,我阐述了量子网络物理实现的一些基本原则,目的是激发更广泛社区参与这一努力,包括在系统级研究中。我着重探讨了当前利用单光子和原子光学过程可靠传输量子态跨越复杂量子网络的努力。

Two important research areas are strong coupling of single photons and atoms in the setting of cavity quantum electrodynamics (QED)¹² and quantum information processing with atomic ensembles¹³, for which crucial elements are long-lived quantum memories provided by the atomic system and efficient, quantum interfaces between light and matter. Many other physical systems are also being investigated and are discussed elsewhere (ref. 2 and websites for the Quantum Computation Roadmap (http://qist.lanl.gov/qcomp_map.shtml), the SCALA Integrated Project (http://www.scala-ip.org/public) and Qubit Applications (http://www.qubitapplications.com)).
两个重要的研究领域是在腔量子电动力学环境下单光子和原子的强耦合 12,以及利用原子集合进行量子信息处理 13,其中关键元素是原子系统提供的长寿命量子存储器和高效的光物质量子接口。许多其他物理系统也正在被研究,在其他文献中有讨论(参考文献 2 和量子计算路线图网站(http://qist.lanl.gov/qcomp_map.shtml)、SCALA 综合项目(http://www.scala-ip.org/public)和量子比特应用(http://www.qubitapplications.com)).

A quantum interface between light and matter
光与物质之间的量子接口

The main scientific challenge in the quest to distribute quantum states across a quantum network is to attain coherent control over the interactions of light and matter at the single-photon level. In contrast to atoms and electrons, which have relatively large long-range interactions for their spin and charge degrees of freedom, individual photons typically have interaction cross-sections that are orders of magnitude too small for non-trivial dynamics when coupled to single degrees of freedom for a material system.
在量子网络上分发量子态的主要科学挑战是实现光与物质在单光子水平上的相干控制。与相对较大的长程自旋和电荷相互作用的原子和电子不同,单个光子通常与材料系统的单一自由度耦合时,其相互作用截面小几个数量级,无法产生重要的动力学效应。

The optical physics community began to address this issue in the 1990s, with the development of theoretical protocols for the coherent transfer of quantum states between atoms and photons in the setting of cavity QED^6,14,15. Other important advances have been made in the past decade^2,4, including with atomic ensembles^13,16. The reversible mapping of quantum states between light and matter provides the basis for quantum-optical interconnects and is a fundamental primitive (building block) for quantum networks. Although the original schemes for such interconnects are sensitive to experimental imperfections, a complete set of theoretical protocols has subsequently been developed for the robust distribution of quantum information over quantum networks, including, importantly, the quantum repeater^4,17 and scalable quantum networks with atomic ensembles¹³.
光学物理界在 1990 年代开始解决这个问题,开发了在腔体量子电动力学环境下,在原子和光子之间协调传递量子态的理论协议 6,14,15。过去十年里,包括原子集团 13,16 在内的其他重要进展也已取得 2,4。光与物质量子态的可逆映射为量子光学互联提供了基础,是量子网络的基本原语(构建块)。虽然这种互联的最初方案对实验不完善敏感,但随后已经开发了用于在量子网络上健壮分发量子信息的一整套理论协议,其中包括重要的量子中继器 4,17 和以原子集团为基础的可扩展量子网络 13。

A generic quantum interface between light and matter is depicted in Fig. 1b. This interface is described by the interaction hamiltonian H_int(t), where for typical states 〈H_int(t)〉 ≈ ħ χ(t), with ħ being h/2π (where h is Planck's constant) and χ(t) being the time-dependent coupling strength between the internal material system and the electromagnetic field. Desirable properties for a quantum interface include that χ(t) should be 'user controlled' for the clocking of states to and from the quantum memory (for example, by using an auxiliary laser), that the physical processes used should be robust in the face of imperfections (for example, by using adiabatic transfer) and that mistakes should be efficiently detected and fixed (for example, with quantum error correction). In qualitative terms, the rate κ, which characterizes the bandwidth of the input–output channel, should be large compared with the rate γ, which characterizes parasitic losses, and both of these rates should be small compared with the rate of coherent coupling χ.
图 1b 描述了一种通用的量子界面,它通过相互作用哈密顿量 Hint(t)进行描述,对于典型状态来说,〈Hint(t)〉≈ħχ(t),其中ħ=h/2π(h 为普朗克常数),χ(t)为内部物质系统与电磁场之间的时间依赖耦合强度。量子界面的理想特性包括:χ(t)应该可以由用户控制以对量子存储器中的态进行时钟控制(例如使用辅助激光器);所使用的物理过程应该对缺陷具有稳健性(例如使用绝热转移);错误应该能够有效地检测和修正(例如使用量子纠错)。定性地说,描述输入-输出通道带宽的速率κ应该大于描述寄生损耗的速率γ,而这两个速率都应该小于相干耦合速率χ。

Examples of physical systems for realizing a quantum interface and distributing coherence and entanglement between nodes are shown in Fig. 1c, d. In the first example (Fig. 1c), single atoms are trapped in optical cavities at nodes A and B, which are linked by an optical fibre. External fields control the transfer of the quantum state |Ψ〉 stored in the atom at node A to the atom at node B by way of photons that propagate from node A to node B^6,18. In the second example (Fig. 1d), a single-photon pulse that is generated at node A is coherently split into two components and propagates to nodes B and C, where the entangled photon state is coherently mapped into an entangled state between collective excitations at each of the two nodes^13,19. Subsequent read-out of entanglement from the memories at node B and/or node C as photon pulses is implemented at the 'push of a button'.
图 1c、d 显示了实现量子接口以及在节点之间分配相干性和纠缠的物理系统示例。在第一个例子(图 1c)中,单个原子被困在节点 A 和 B 的光学腔中,这些节点通过光纤相连。外部场控制了从节点 A 到节点 B 的光子传播,从而将存储在节点 A 原子中的量子态|Ψ〉转移到节点 B 的原子。在第二个例子(图 1d)中,在节点 A 产生的单光子脉冲被相干地分裂成两个分量,并传播到节点 B 和 C,在那里,纠缠光子态被相干地映射到每个节点的集体激发态之间的纠缠态。随后,通过按下按钮即可从节点 B 和/或节点 C 的存储器中读出纠缠作为光脉冲。

Cavity QED 腔体量子电动力学

At the forefront of efforts to achieve strong, coherent interactions between light and matter has been the study of cavity QED²⁰. In both the optical^12,21 and the microwave^22,23,24,25 domains, strong coupling of single atoms and photons has been achieved by using electromagnetic resonators of small mode volume (or cavity volume) V_m with quality factors Q ≈ 10⁷–10¹¹. Extensions of cavity QED to other systems²⁶ include quantum dots coupled to micropillars and photonic bandgap cavities²⁷, and Cooper pairs interacting with superconducting resonators (that is, circuit QED; see ref. 28 for a review).
在努力实现光与物质之间强有力、连贯互作用的前沿领域中,腔 QED 的研究一直扮演着重要角色。在光学和微波领域,均已通过使用小模量体(或腔体体积)Vm 、质因数 Q 约为 107-1011 的电磁共振腔器,实现了单个原子和光子的强耦合。腔 QED 所延伸到的其他系统包括:量子点与微柱和光子带隙腔体的耦合,以及 Cooper 对与超导共振腔的互作用(即电路 QED,参见参考文献 28)。

Physical basis of strong coupling
强耦合的物理基础

Depicted in Fig. 2a is a single atom that is located in an optical resonator and for which strong coupling to a photon requires that a single intracavity photon creates a 'large' electric field. Stated more quantitatively, if the coupling frequency of one atom to a single mode of an optical resonator is g (that is, 2g is the one-photon Rabi frequency), then
如图 2a 所示，单个原子位于光学谐振腔内，要求单个腔内光子能产生"大"电场才能实现与光子的强耦合。更定量地说，如果单个原子与光学谐振腔单模的耦合频率为 g（即 2g 为单光子拉比频率），那么

a, Shown is a simple schematic of an atom–cavity system depicting the three governing rates (g, κ, γ) in cavity QED, where g ≈ χ in Fig. 1. Coherent exchange of excitation between the atom and the cavity field proceeds at rate g, as indicated by the dashed arrow for the atom and the green arrows for the cavity field. b, A photograph of two mirror substrates that form the Fabry–Pérot cavity, which is also shown schematically. The cavity length l = 10 µm, waist w₀ = 12 µm transverse to the cavity axis, and finesse F ≈ 5 × 10⁵. The supporting structure allows active servo control of the cavity length to δl ≈ 10−14 m (ref. 12). Scale bar, 3 mm. c, The reduction in the critical photon number n₀ over time is shown for a series of experiments in cavity QED that were carried out by the Caltech Quantum Optics Group. These experiments involved either spherical-mirror Fabry–Pérot cavities (circles) or the whispering-gallery modes of monolithic SiO₂ resonators (squares). The data points shown for 2006 and 2008 are for a microtoroidal SiO₂ resonator^75,76; those for 2009 and 2011 (open squares) are projections for this type of resonator⁷⁷.
如图所示，给出了一个原子-腔室系统的简单示意图，描述了腔量子电动力学中三个主导速率(g、κ、γ)，其中 g≈χ如图 1 所示。原子和腔室场之间激发的相干交换以速率 g 进行，如原子的虚线箭头和腔室场的绿色箭头所示。b 图给出了构成费布里-珀罗腔室的两个镜面基板的照片，以及该腔室的示意图。腔室长度 l=10μm，横截面 w0=12μm，光谱精度 F≈5×105。支撑结构允许通过伺服控制将腔室长度调整到δl≈10-14m(参考文献 12)。尺度栏为 3mm。c 图显示了随时间 cavity QED 实验中关键光子数 n0 的减少,这些实验由加州理工量子光学小组开展,涉及球面镜费布里-珀罗腔室(圆点)或单体 SiO2 谐振腔的横膜波模式(方块)。2006 年和 2008 年的数据点是针对微环形 SiO2 谐振腔 75,76;2009 年和 2011 年的数据点(空心方块)是对该类谐振腔的预测 77。

Full size image

where µ₀ is the transition dipole moment between the relevant atomic states (with transition frequency ω_A), and ω_C ≈ ω_A is the resonant frequency of the cavity field, with polarization vector ε. Experiments in cavity QED explore strong coupling with g ≫ (γ, κ), where γ is the atomic decay rate to modes other than the cavity mode and κ is the decay rate of the cavity mode itself. Expressed in the language of traditional optical physics, the number of photons required to saturate the intracavity atom is n₀ ≈ γ²/g², and the number of atoms required to have an appreciable effect on the intracavity field is N₀ ≈ κγ/g². Strong coupling in cavity QED moves beyond traditional optical physics, for which (n₀, N₀) ≫ 1, to explore a qualitatively new regime with (n₀, N₀) ≪ 1 (ref. 12).
其中μ0 是相关原子态之间的跃迁偶极矩(跃迁频率为ωA),而ωC ≈ ωA 是腔场的共振频率,偏振矢量为ε。腔 QED 实验探索了强耦合 g ≫ (γ, κ),其中γ是原子衰减率到腔模式之外的模式,κ是腔模式本身的衰减率。用传统光学物理的语言表示,使原腔内原子饱和所需的光子数为 n0 ≈ γ2/g2,而使腔内场产生明显影响所需的原子数为 N0 ≈ κγ/g2。腔 QED 中的强耦合超越了传统光学物理,其中(n0, N0) ≫ 1,进入一个定性新的区域(n0, N0) ≪ 1(参考文献 12)。

In the past three decades, a variety of approaches have been used to achieve strong coupling in cavity QED^{12,20,21,22,23,24,25}. In the optical domain, a route to strong coupling is the use of high-finesse optical resonators (F ≈ 10⁵–10⁶) and atomic transitions with a large µ₀ (that is, oscillator strengths near unity). Progress along this path is illustrated in Fig. 2c, with research now far into the domain (n₀, N₀) ≪ 1.
在过去的三十年里,人们采用了各种方法来实现腔道量子电动力学的强耦合 12,20,21,22,23,24,25。在光学领域,实现强耦合的一种途径是使用高品质因数的光学共振腔(F≈105-106)和具有较大μ0(即振子强度接近于 1)的原子跃迁。图 2c 描述了这一发展进程,目前已远远进入(n0,N0)≪1 的领域。

As the cavity volume V_m is reduced to increase g (equation (1)), the requirement for atomic localization becomes more stringent. Not surprisingly, efforts to trap and localize atoms in high-finesse optical cavities in a regime of strong coupling have been central to studies of cavity QED in the past decade, and the initial demonstration was in 1999 (ref. 29). Subsequent advances include extending the time for which an atom is trapped to 10 s (refs 30, 31); see ref. 32 for a review. Quantum control over both internal degrees of freedom (that is, the atomic dipole and the cavity field) and external degrees of freedom (that is, atomic motion) has now been achieved for a strongly coupled atom–cavity system³³. And an exciting prospect is cavity QED with single trapped ions, for which the boundary for strong coupling has been reached³⁴.
随着空腔体积 Vm 的减小以增加 g(方程(1))，对原子局域化的要求变得更加严格。毫不奇怪,在过去十年里,研究腔 QED 的中心是在高优质光腔中捕获和局域化原子,以实现强耦合,其初始演示是在 1999 年(参考文献 29)。后续进展包括将原子被困的时间延长到 10 秒(参考文献 30、31);参见参考文献 32 进行综述。现已实现了对内部自由度(即原子偶极子和腔场)和外部自由度(即原子运动)的量子控制,对于强耦合原子-腔系统 33。一个令人兴奋的前景是利用单个被困离子进行腔 QED,已经达到了强耦合的极限 34。

Coherence and entanglement in cavity QED
腔体量子电动力学中的相干性和纠缠

Applying these advances to quantum networks has allowed single photons to be generated 'on demand' (Box 1). Through strong coupling of the cavity field to an atomic transition, an external control field Ω(t) transfers one photon into the cavity mode and then to free space by way of the cavity output mirror, leading to a single-photon pulse |φ₁(t)〉 as a collimated beam. The temporal structure (both amplitude and phase) of the resultant 'flying photon' |φ₁(t)〉 can be tailored by way of the control field Ω(t) (refs 6, 35), with the spatial structure of the wave packet being set by the cavity mode.
将这些进步应用于量子网络,使得单个光子可 '按需' 产生(框 1)。通过将腔场与原子跃迁强耦合,外部控制场 Ω(t) 将一个光子转移到腔模式,然后通过腔镜输出到自由空间,从而产生单光子脉冲 |φ1(t)〉 作为平行光束。所产生的 '飞行光子' |φ1(t)〉 的时间结构(包括振幅和相位)可通过控制场 Ω(t)来调整(文献 6、35), 而波包的空间结构由腔模式确定。

Several experiments have confirmed the essential aspects of this process for the deterministic generation of single photons^30,34,36. Significantly, in the ideal (adiabatic) limit, the excited state |e〉 of the atom is not populated because of the use of a 'dark state' protocol³⁷. By deterministically generating a bit stream of single-photon pulses from single trapped atoms, these experiments are a first step in the development of quantum networks based on flying photons.
多个实验已经验证了这一过程的关键方面,用于确定性单光子的产生 30,34,36。值得注意的是,在理想(绝热)极限中,由于采用了"黑暗态"协议 37,原子的激发态|e〉没有被占据。通过从单个被困原子确定性地产生单光子脉冲比特流,这些实验是基于飞行光子的量子网络发展的一个重要步骤。

Compared with the generation of single photons by a variety of other systems³⁸, one of the distinguishing aspects of the dark-state protocol (Box 1) is that it should be reversible. That is, a photon that is emitted from a system A should be able to be efficiently transferred to another system B by applying the time-reversed (and suitably delayed) field Ω(t) to system B (Fig. 1c). Such an advance was made¹⁸ by implementing the reversible mapping of a coherent optical field to and from internal states of a single trapped caesium atom. Although this experiment was imperfect, it provides the initial verification of the fundamental primitive on which the protocol for the physical implementation of quantum networks in ref. 6 is based (an important theoretical protocol that has been adapted to many theoretical and experimental settings).
与通过各种其他系统 38 产生单个光子相比,黑暗态协议(框 1)的一个显著特点是它应该是可逆的。也就是说,从系统 A 发射的光子应该能够通过向系统 B 应用时间反转(并适当延迟)的场Ω(t)来有效地转移到另一个系统 B(图 1c)。这种进步是通过实现对单个困住的铯原子的内部态进行可逆光学场映射而实现的 18。尽管这个实验并不完美,但它提供了关于文献 6 中量子网络物理实现协议的基本原理的初步验证(这是一个重要的理论协议,已被改编应用于许多理论和实验环境)。

The adiabatic transfer of quantum states (as described in Box 1, as well as related possibilities^10,35) relies on strong coupling between an atom and a single polarization of the intracavity field. However, by extending the ideas in Box 1 to the two polarization eigenmodes of the cavity for given transverse and longitudinal mode orders, it is possible to generate entanglement between the internal states of the atom and the polarization state of a coherently generated photon^39,40,41. An initial control field Ω₁(t₁) results in entanglement between internal states of the atom b, |b_±〉, and the polarization state of a flying photon |φ^±_field(t₁)〉 that is coherently generated by the coupled atom–cavity system. Applying a second control field Ω₂(t₂) returns the atom to its initial (unentangled) state while generating a second flying photon |ξ^±_field(t₂)〉, thereby leading to entanglement between the polarizations of the fields, φ^±_field and ξ^±_field, emitted at times t₁ and t₂.
量子态的绝热转移(如 Box 1 所述,以及相关可能性 10,35)依赖于原子和单一极化的腔内场之间的强耦合。然而,通过将 Box 1 中的思想扩展到洞腔的两个偏振本征模式,对于给定的横向和纵向模式阶数,可以在原子内态和由耦合原子-腔系统产生的光子极化状态之间产生纠缠 39,40,41。初始控制场Ω1(t1)导致原子内态 b, |b±〉和飞行光子极化态|φ±field(t1)〉之间产生纠缠。施加第二个控制场Ω2(t2)将原子恢复到初始(未纠缠)状态,同时产生第二个飞行光子|ξ±field(t2)〉,从而导致在 t1 和 t2 时刻发出的场的极化φ±field 和ξ±field 之间产生纠缠。

Such a sequence of operations has been applied to single rubidium atoms falling through a high-finesse optical cavity²¹. In this study, entangled photons were generated with a time separation τ = t₂ − t₁ limited by the atomic transit time. Although the atoms arrived randomly into the cavity mode in this case, the protocol itself is intrinsically deterministic. With trapped atoms, it will be possible to generate entangled states at user selected times (t₁, t₂) at the 'push of a button.' Moreover, the scheme is inherently reversible, so the entanglement between atom and field can be used to distribute entanglement to a second atom–cavity system in a network.
这一系列操作已经应用于通过高精度光学腔室下落的单个铷原子 21。在这项研究中,产生了间隔时间τ=t2-t1 的缠结光子,该时间间隔受原子通过时间的限制。尽管在这种情况下原子随机进入腔室模式,但该协议本身是确定性的。对于被捕获的原子,可能在用户选择的时间(t1,t2)生成缠结态,犹如按下按钮一般。此外,该方案本质上是可逆的,因此原子和场之间的缠结可用于将缠结分配给网络中的第二个原子-腔室系统。

In a broader context, important advances have been made in the generation and transfer of quantum states in other physical systems, including quantum dots⁴² and circuits²⁸ coupled to cavities.
在更广泛的背景下，在其他物理系统中量子态的生成和转移方面取得了重要进展，包括与腔室耦合的量子点 42 和电路 28。

With the maturation of experimental capabilities in cavity QED that is now evident, many previously developed theoretical protocols will become possible. These include the sequential generation of entangled multiqubit states⁴³, the teleportation of atomic states from one node to another¹⁵, photonic quantum computation by way of photon–photon interactions at the nodes³⁵ and reversible mapping of quantum states of atomic motion to and from light⁴⁴. Clearly, new technical capabilities beyond conventional (Fabry–Pérot) cavities will be required to facilitate such scientific investigations; several candidate systems are discussed in Box 2.
随着腔体量子电动力学实验能力的成熟,许多先前开发的理论协议将变为可能。这些包括多量子比特纠缠态的连续生成 43、从一个节点到另一个节点的原子态远程传输 15、通过节点处光子-光子相互作用的光子量子计算 35,以及原子运动量子态和光场之间的可逆映射 44。显然,为了促进此类科学研究,将需要超越传统(法布里-珀罗)腔室的新技术能力;Box 2 中讨论了几种候选系统。

Quantum networks with atomic ensembles
利用原子集合体的量子网络

An area of considerable research activity in the quest to distribute coherence and entanglement across quantum networks has been the interaction of light with atomic ensembles that consist of a large collection of identical atoms. For the regime of continuous variables, entanglement has been achieved between two atomic ensembles, each of which consists of ∼10¹² atoms⁴⁵, and the quantum teleportation of light to matter has been demonstrated by mapping coherent optical states to the collective spin states of an atomic memory⁴⁶. Further research of the continuous variables regime is reviewed elsewhere⁴⁷. Here I focus, instead, on the regime of discrete variables, with photons and atomic excitations considered one by one.
在分散量子网络中的相干性和纠缠的过程中,大量的研究集中于光与由大量相同原子组成的原子集合的相互作用。对于连续变量的情况,已经实现了两个原子集合(每个集合约有 10^12 个原子)之间的纠缠,并且通过将相干光学态映射到原子存储器的集体自旋态,实现了从光到物质的量子隧道传输 46。对连续变量情况的进一步研究已在其他地方综述 47。这里我将重点放在离散变量的情况上,即一个一个地考虑光子和原子激发。

Writing and reading collective spin excitations
集体自旋激发的写作和阅读

Research on discrete quantum variables is based on the remarkable theoretical protocol described in ref. 13, in which Luming Duan, Mikhail Lukin, Juan Ignacio Cirac and Peter Zoller presented a realistic scheme for entanglement distribution by way of a quantum-repeater architecture^4,17. Fundamental to this protocol, which is known as the DLCZ protocol, is the generation and retrieval of single 'spin' excitations within an ensemble of a large number of atoms⁴⁸ (Box 3). Together with photoelectric detection of field 1, a laser pulse ('write' pulse) creates a single excitation |1_a〉 that is stored collectively within the atomic ensemble. At a later time, a second laser pulse ('read' pulse) deterministically converts excitation stored within the atomic memory in the state |1_a〉 into a propagating field, denoted field 2.
基于离散量子变量的研究源于参考文献 13 中所描述的出色的理论方案,陆明段、Mikhail Lukin、Juan Ignacio Cirac 和 Peter Zoller 提出了一种利用量子中继架构实现纠缠分发的实际方案 4,17。这一被称为 DLCZ 方案的协议的基础是在大量原子组成的集合体中生成和读取单个'自旋'激发态 48（见方框 3）。通过场 1 的光电探测和一个激光脉冲('写入'脉冲),在原子集合体中产生一个单一的激发态|1a〉,并被集体存储。随后,第二个激光脉冲('读出'脉冲)可以确定性地将存储在原子存储器中的|1a〉态激发转换为一个传播场,即场 2。

The basic processes illustrated in Box 3 can be extended to create an entangled pair of ensembles, L and R (ref. 13; Fig. 3a). The entangled state is generated in a probabilistic but heralded⁴⁹ manner from quantum interference in the measurement process. That is, detection of a photon from one atomic ensemble or the other in an indistinguishable manner results in an entangled state with one collective spin excitation shared coherently between the ensembles. In the ideal case, and to lowest-order probability, a photoelectric detection event at either of the two detectors projects the ensembles into the entangled state |Ψ_L,R〉 = 1//√2 (|0_a〉_L|1_a〉_R ± e^iη₁ |1_a〉_L|0_a〉_R), with the sign (+ or −) set by whether detector 1 or detector 2 records the event. The phase η₁ is determined by the difference between the phase shifts along the two channels, η₁ = β_L - β_R (ref. 49), which must be stable. Any given trial with a 'write' pulse is unlikely to produce a detection event at either detector, and such failed trials require the system to be reinitialized. However, a photoelectric detection event at either detector unambiguously heralds the creation of the entangled state. Limited by the coherence time between the metastable lower atomic states |g〉_i and |s〉_i for all atoms _i = 1, 2, ... , N_a within the ensemble (ref. 50; Box 3), this entangled state is stored in the quantum memory provided by the ensembles and is available 'on demand' for subsequent tasks, such as entanglement connection^13,51.
在框 3 所示的基本过程的基础上,可以延伸创造一对纠缠的集合体 L 和 R(参考 13;图 3a)。纠缠态是通过测量过程中的量子干涉以概率性但被预告的方式产生 49。也就是说,以无法区分的方式从一个原子集合体或另一个集合体中检测到一个光子,会导致一个纠缠态,其中一个集体自旋激发在集合体之间相干共享。在理想情况下,且为最低阶概率,任一探测器检测到光电事件都会将集合体投射到纠缠态|ΨL,R〉=1//√2(|0a〉L|1a〉R±eiη1|1a〉L|0a〉R),其中,标志(+或-)由检测器 1 还是检测器 2 记录该事件决定。相位η1 由两个通道的相位差确定,η1=βL-βR(参考 49),这必须是稳定的。任何一次带有"写入"脉冲的尝试都不太可能在任一探测器上产生检测事件,这种失败的尝试需要重新初始化系统。但是,任一探测器上的光电检测事件无疑预示着纠缠态的创建。受限于集合体内所有原子=1,2,...,Na 之间的稳定态|g〉和|s〉的相干时间(参考 50;框 3),这种纠缠态被存储在集合体提供的量子存储器中,并可根据需要用于后续任务,如纠缠连接 13,51。

A realistic scheme for entanglement distribution by way of a quantum-repeater architecture was proposed by Duan, Lukin, Cirac and Zoller and is known as the DLCZ protocol¹³. a, Measurement-induced entanglement between two atomic ensembles^13,49, L and R, is shown. Synchronized laser pulses incident on the ensembles (denoted write beams, blue arrows) generate small amplitudes for optical fields from spontaneous Raman scattering⁴⁸; these fields are denoted 1_L and 1_R (red arrows). These fields interfere at a 50/50 beam splitter, with outputs directed to two single-photon detectors. A measurement event at either detector (shown for detector 1) projects the ensembles into the entangled state |Ψ_L,R〉 with one quantum of excitation shared remotely between the ensembles. Entanglement is stored in the quantum memory provided by the ensembles and can subsequently be converted to propagating light pulses by a set of 'read' laser pulses (Box 3). b, Experimentally determined components of the density matrix ρ^exp_L,R for entanglement between two atomic ensembles are shown⁵⁰, corresponding to concurrence C = 0.9±0.3, where C = 0 for an unentangled state. The first number in each ket refers to the excitation number for the ensemble L, and the second is for the ensemble R. For comparison, the density matrix ρ^ideal_L,R for the ideal state |ΨL,R〉 is shown, with concurrence C = 1. c, The laboratory set-up is shown for the entanglement of two pairs of atomic ensembles to generate the functional quantum nodes L and R, which are separated by 3 m (ref. 65). Each of the four elongated ovals shows a cylinder of 10⁵ caesium atoms, which forms an atomic ensemble at each site. Entangled states between the upper u and lower l pairs at the L and R nodes, |Ψ^u_L,R〉 |Ψ^l_L,R〉, are generated and stored in an asynchronous manner for each pair (u and l) as is the case in panel a. Atomic excitations for the pairs L_u, L_l and R_u, R_l are subsequently converted to flying photons at each node, with a polarization encoding that results in violation of Bell's inequality⁶⁵. The entire experiment functions under the quantum control of single photon detection events.
一种实现量子中继器架构下量子纠缠分配的现实方案由 Duan、Lukin、Cirac 和 Zoller 提出,称为 DLCZ 协议 13。a,测量诱导的两个原子簇 L 和 R 之间的纠缠 13,49,通过同步激光脉冲(标记为写入光束,蓝色箭头)在原子簇上产生来自自发拉曼散射的弱光场 48,这些光场标记为 1L 和 1R(红色箭头)。这些光场在一个 50/50 光分束器上干涉,输出被导向两个单光子探测器。任一探测器(以探测器 1 为例)上的探测事件将原子簇投射到共享一个量子激发的缠缠态|ΨL,R〉中。该纠缠存储在原子簇提供的量子存储器中,随后可通过一组"读取"激光脉冲转换为传播光脉冲(见附录 3)。b,实验测定的两个原子簇纠缠的密度矩阵分量ρexpL,R50,对应的缠缠度 C=0.9±0.3,其中 C=0 对应无纠缠态。每个 ket 的第一个数字表示簇 L 的激发数,第二个数字表示簇 R 的激发数。为了对比,理想态|ΨL,R〉的密度矩阵ρidealL,R 也给出,其缠缠度 C=1。c,实验室设置展示了两对原子簇纠缠以生成功能量子节点 L 和 R,节点间距离为 3 米 65。每个四个长椭圆代表一个含有 10^5 个铯原子的原子簇。节点 L 和 R 上的上下配对(u 和 l)纠缠态|ΨuL,R〉|ΨL,R〉以异步方式生成和存储,与 a 面板情况类似。 原子激发对于 Lu, L 和 Ru, R 引起的飞行光子在每个节点都会以偏振编码的方式产生,从而违背了贝尔不等式 65。整个实验在单光子探测事件的量子控制下进行。

Full size image

Although the above description is for an ideal case and neglects higher-order terms, the DLCZ protocol is designed to be resilient to important sources of imperfections, including losses in propagation and detection, and detector dark counts. Indeed, the scheme functions with 'built-in entanglement purification'¹³ and enables entanglement to be extended beyond the separation of two ensembles in an efficient and scalable manner. Theoretical extensions^52,53 of the DLCZ protocol have examined related network architectures for optimizing scalability in view of laboratory capabilities (discussed below).
尽管上述描述针对理想情况并忽略了高阶项,但 DLCZ 协议旨在抗御重要的不完善源,包括传播和探测过程中的损耗以及探测器的暗计数。事实上,该方案具有"内置纠缠净化"机制,并能以高效和可扩展的方式延长两个样品间的纠缠。DLCZ 协议的理论扩展已探讨了优化可扩展性的相关网络架构,并考虑了实验室条件(如下所述)。

Coherence and entanglement with atomic ensembles
原子集合体的相干性和纠缠

The initial, enabling, steps in the implementation of the DLCZ protocol were observations of quantum correlations both for single photon pairs^54,55 and for a large number of photons (10³–10⁴) (ref. 56) generated in the collective emission from atomic ensembles. Single photons were generated by the efficient mapping of stored collective atomic excitation to propagating wave packets for field 2 (refs ^{57,58,59,60,61}; Box 3). Conditional read-out efficiencies of 50% in free space⁵⁸ and 84% in a cavity⁶² were realized for state transfer from a single collective 'spin' excitation stored in the atomic ensemble to a single photon for field 2.
实施 DLCZ 协议的初始启用步骤是观察量子相关性,包括单光子对 54,55 和大量光子(103–104 个)56 的集体辐射。单个光子是通过将存储在原子集体中的集体激发有效映射到传播波包以产生场 2 而生成的 57,58,59,60,61(附录 3)。在自由空间中,从单个集体"自旋"激发转移到场 2 中单个光子的状态转移效率达到 50%58,在腔体中达到 84%62。

With these capabilities for coherent control of collective atomic emission, heralded entanglement between ensembles separated by 3 m was achieved in 2005 (ref. 49). More recent work has led to the inference that the concurrence C (ref. 63) of entanglement stored between the two ensembles in Fig. 3 is C = 0.9±0.3 (ref. 50), with the associated density matrix shown in Fig. 3b.
通过对集体原子发射的相干控制,在 2005 年实现了相隔 3 米的集合体之间的预兆纠缠(参考文献 49)。最近的研究表明,在图 3 所示的两个集合体之间存储的纠缠并发度 C(参考文献 63)为 C=0.9±0.3(参考文献 50),相关的密度矩阵如图 3b 所示。

The DLCZ protocol is based on a quantum-repeater architecture involving independent operations on parallel chains of quantum systems¹³, with scalability relying crucially on conditional control of quantum states stored in remote quantum memories⁶⁴. The experiment shown in Fig. 3c took an important step towards this goal by achieving the minimal functionality required for scalable quantum networks⁶⁵.
量子中继器架构的 DLCZ 协议涉及对并行的量子系统链进行独立操作 13,可扩展性关键在于对远程量子存储器中存储的量子态实现条件控制 64。图 3c 所示的实验迈出了朝着可扩展量子网络所需最基本功能的关键一步 65。

Apart from the DLCZ protocol, which involves measurement-induced entanglement, it is also possible to achieve deterministic mapping of quantum states of light into and out of atomic ensembles by using electromagnetically induced transparency^16,66. Pioneering work^67,68 demonstrated the storage and retrieval of classical pulses to and from an atomic ensemble. This work was then extended into the quantum regime of single photons^69,70. Entanglement between two ensembles coupled to a cavity mode was achieved by adiabatic transfer of excitation⁷¹, thereby providing a means for on-demand entanglement. In addition, the reversible mapping of photonic entanglement into and out of pairs of quantum memories has been achieved¹⁹ by an electromagnetically-induced-transparency process, which should assist the distribution of entanglement over quantum networks (Fig. 1d).
除了涉及测量诱导纠缠的 DLCZ 协议外,还可以通过利用电磁诱导透明度来实现光量子态与原子集合体之间的确定性映射 16,66。先驱性的工作 67,68 演示了经典脉冲在原子集合体中的存储和读取。这项工作随后被扩展到单光子的量子领域 69,70。通过激发的绝热传输,实现了与腔模式耦合的两个集合体之间的纠缠 71,从而提供了按需产生纠缠的方法。此外,通过电磁诱导透明度过程,实现了光子纠缠与量子存储器对之间的可逆映射 19,这应该有助于在量子网络上分发纠缠(图 1d)。

Contemporary with this work on heralded and deterministic entanglement, a variety of experiments based on entanglement as a postdiction have been carried out⁷² (that is, for cases in which a physical state is not available for use in a scalable network but which are nonetheless significant). An important advance in this regard is the use of a pair of ensembles for entanglement generation to achieve a posteriori teleportation of light to an atomic memory⁷³.
与这项关于宣告的和决定性的纠缠的工作同时进行,基于纠缠作为预测的各种实验已经被进行 72(即对于某些情况下,物理状态不能用于可扩展的网络,但仍然很重要)。在这方面的一个重要进展是使用一对集合来产生纠缠,以实现光到原子记忆的事后遥测 73。

There has also been considerable effort devoted to the detailed characterization of decoherence for stored atomic excitation and entanglement^50,65,73. Decoherence of entanglement between distinct atomic ensembles has been observed in the decay of the violation of Bell's inequality⁶⁵ and of the fidelity for teleportation⁷³. By measuring concurrence C(t), quantitative characterizations of the relationship between the global evolution of the entangled state and the temporal dynamics of various local correlations were also able to be made⁵⁰.
对于存储原子激发态和纠缠状态的去相干行为,已经进行了大量详细的研究 50,65,73。通过观察贝尔不等式的违背以及远程传输的保真度的衰减,可以测量出不同原子系统之间纠缠态的去相干行为 65,73。通过测量缠结度 C(t)，也可以定量地刻画纠缠态的全局演化与局部相关性时间动力学之间的关系 50。

Extending entanglement for quantum networks
量子网络中的纠缠延伸

The entangled states that have been created so far both in cavity QED and by using the DLCZ protocol are between pairs of systems (known as bipartite entanglement) for which there are definitive procedures for operational verification⁷². The creation of more-general classes of entangled state shared between more than two nodes would be of great interest. However, as researchers progress towards more-complex quantum networks, the issue of entanglement verification becomes increasingly problematic. At present, the theoretical tools and experimental capabilities for characterizing the general states of quantum networks do not exist.
迄今为止,在腔体量子电动力学和 DLCZ 协议中所创造的纠缠态都是在两个系统之间(称为二分纠缠)的,这种纠缠状态有明确的操作验证程序。在多于两个节点之间共享的更一般类型的纠缠态的创造将是非常令人感兴趣的。然而,随着研究人员向更复杂的量子网络发展,纠缠验证问题变得日益棘手。目前,用于描述量子网络一般状态的理论工具和实验能力还不存在。

Perhaps surprisingly, a non-trivial task will be to find out whether a quantum network 'works'. As moderately complex quantum networks are realized in the laboratory, it will become increasingly more difficult to assess the characteristics of a network quantitatively, including whether entanglement extends across the whole network. One strategy, motivated by the underlying physical processes of the network, could be to try to determine the density matrix ρ(t) for the network. However, this approach would fail because of the exponential growth in ρ(t) with the size of a network.
或许令人惊讶的是,确定量子网络是否"工作"这一并非微不足道的任务。随着相对复杂的量子网络在实验室中得到实现,量化地评估网络的特性越来越困难,包括判断纠缠是否遍及整个网络。基于网络的基本物理过程,一种可能的策略是尝试确定网络的密度矩阵ρ(t)。然而,这种方法将失败,因为ρ(t)随网络规模的指数增长。

An alternative strategy could be based on more functional issues of algorithmic capability. An attempt could be made to implement a quantum algorithm for computation or communication to test whether the purported quantum network has greater capabilities than any classical counterpart. This course is, however, problematic because the advantage of a quantum network might only be realized above some threshold in the size of the network. Furthermore, from an experimental perspective, this strategy does not offer much in the way of diagnostics for 'fixing' the network when it fails.
一种替代策略可以基于更多功能性问题的算法能力。可以尝试实施量子算法进行计算或通信,以测试所声称的量子网络是否具有比任何经典对应物更强大的功能。然而,这种做法存在问题,因为量子网络的优势可能只会在网络规模超过某个阈值时才能体现出来。此外,从实验的角度来看,这种策略在网络失败时没有太多有助于"修复"网络的诊断方法。

Another, less obvious, approach might be to adopt more seriously the perspective of a quantum network as a quantum many-body system and to search for more 'physical' characteristics of the network (for example, the scaling behaviour of pair correlation functions and multipartite entanglement). Indeed, an active area of research is the nature of entanglement for systems that undergo quantum phase transitions, and there have been pioneering advances in the study of one-dimensional spin chains⁷⁴.
另一个不太明显的方法可能是更认真地采用量子网络作为量子多体系统的视角,并寻找网络的更多"物理"特性(例如对偶相关函数和多粒子纠缠的缩放行为)。事实上,研究系统在量子相变过程中的纠缠性质是一个活跃的研究领域,在一维自旋链的研究方面也取得了开创性的进展。

Conclusion 结论

Progress has been made towards the development of quantum networks, but the current state of the art is primitive relative to that required for the robust and scalable implementation of sophisticated network protocols, whether over short or long distances. The realization of quantum memories, local quantum processing, quantum repeaters and error-corrected teleportation are ambitious goals. Nevertheless, there is considerable activity directed towards these goals worldwide.
量子网络的发展已取得进展,但与实现复杂网络协议所需的稳健和可扩展性相比,当前的技术水平还很原始,无论是短距离还是长距离。实现量子存储器、本地量子处理、量子中继器和纠错传输是宏伟目标。不过,全球范围内仍有大量致力于实现这些目标的活动。

Here cavity-QED-based networks and networks implemented using the DLCZ protocol were considered separately, but it is clear that quantum networks will evolve as heterogeneous entities. For example, the same protocol that creates the entanglement between the two ensembles shown in Fig. 3a can be used to create an entangled state with one excitation shared between an atom in a cavity and an atomic ensemble. A crucial task will be the development of unambiguous procedures for verifying entanglement, a non-trivial undertaking that has not always been carried out correctly⁷².
这里,基于腔体量子电动力学的网络和使用 DLCZ 协议实现的网络被分别考虑,但很明显量子网络将会演变为异构实体。例如,创造图 3a 中两个原子簇之间纠缠的相同协议可用于创造一个纠缠态,其中一个激发在一个原子和原子簇之间共享。一个关键任务将是制定明确的程序来验证纠缠,这是一个非平凡的任务,并且一直没有被正确地完成 72。

I have used quantum networks as a unifying theme, but the research described here has broader value, including advancing the understanding of quantum dynamical systems and, for the cases considered here, creating new physics from controlled nonlinear interactions of single photons and atoms. These are exciting times in quantum information science as researchers pass from the regime of individual building blocks (for example, a single atom–cavity system) to the realm of complex quantum systems that are assembled block by block from many such units.
我已将量子网络作为统一主题,但此处描述的研究具有更广泛的价值,包括推进对量子动力学系统的理解,以及在此处考虑的情况下,从单个光子和原子的受控非线性相互作用中创造新的物理学。随着研究人员从单个构建块(例如单个原子-腔室系统)的领域进入由许多此类单元组装而成的复杂量子系统的领域,量子信息科学正处于令人兴奋的时期。

Box 1: Mapping quantum states between atoms and photons
盒子 1：在原子和光子之间映射量子态

Reversible transfer of a state between light and a single trapped atom can be achieved through the mappings |b〉|1〉 → |a〉|0〉 and |a〉|0〉 → |b〉|1〉 for the coherent absorption and emission of single photons (in panel A, a and b of the figure, respectively)¹⁸. In this case, |a〉 and |b〉 represent internal states of the atom with long-lived coherence (for example, atomic hyperfine states in the 6S_1/2, F = 3 and F = 4 manifolds of atomic caesium), and |0〉 and |1〉 are Fock states of the photons in the intracavity field with n = 0 and n = 1 excitations, respectively. The transition between |b〉 and |e〉 is strongly coupled to a mode of an optical cavity with interaction energy ħg, where g (in green) is the coherent coupling rate of the atom and the photon. In this simple setting, the interaction hamiltonian for atom and cavity field has a dark state |D〉 (that is, there is no excited state component |e〉)³⁷, as given by
可以通过映射|b〉|1〉→|a〉|0〉和|a〉|0〉→|b〉|1〉来实现光和单个困住的原子之间状态的可逆转移,这对于单个光子的相干吸收和发射(在图 A 的 a 和 b 面板中)很有意义 18。在这种情况下,|a〉和|b〉代表具有长寿命相干性的原子的内部态(例如,原子铯中 6S1/2,F=3 和 F=4 层次的原子超精细态),而|0〉和|1〉则分别是腔内场中光子的 Fock 态,对应于 n=0 和 n=1 激发态。|b〉和|e〉之间的跃迁与一个光腔模式强耦合,相互作用能量为ħg,其中 g(以绿色表示)是原子和光子的相干耦合率。在这个简单的设置中,原子和腔场的相互作用哈密顿量具有一个黑暗态|D〉(即没有激发态分量|e〉)37。

|D〉 = cosθ|a〉|0〉 + sinθ|b〉|1〉, where
|D〉 = cos θ|a〉|0〉 + sin θ|b〉|1〉, 其中

with Ω(t) as a classical control field¹⁴. For Ω(t = 0) = 0, then |D〉 = |a〉|0〉. By contrast, for Ω(t → ∞) >> g, |D〉 ⇒ |b〉|1〉.
以Ω(t)作为经典控制场 14。对于Ω(t = 0) = 0,则|D〉= |a〉|0〉。相比之下,对于Ω(t → ∞) >> g,|D〉⇒|b〉|1〉。

Panel A, a of the figure shows that by adiabatically ramping a control field Ω₁(t) ≫ g from on to off over a time Δt that is slow compared with 1/g, the atomic state is mapped from |b〉 to |a〉 with the accompanying coherent absorption of one intracavity photon. Conversely, in panel A, b of the figure, by turning a control field Ω₂(t) from off to on, the atomic state is mapped from |a〉 to |b〉 with the transfer of one photon into the cavity mode.
图 A, a 部分显示,通过将控制场Ω1(t)≫g 从开到关的绝热变化过程,原子态从|b〉映射到|a〉,并伴有一个腔内光子的相干吸收。相反地,在图 A, b 部分,通过将控制场Ω2(t)从关到开,原子态从|a〉映射到|b〉,并将一个光子转移到腔模式中。

These two processes can be combined to achieve the coherent transfer of the state of a propagating optical field λ(t) = |φ_field(t)〉 into and out of a quantum memory formed by the atomic states |a〉 and |b〉 (ref. 18; figure, panel B). In the ideal case, the mapping is specified by |φ_field(t)〉|b〉 → |0〉(c₁|a〉 + c₀|b〉) ... (storage) ... |0〉(c₁|a〉 + c₀|b〉) → |φ_field(t + τ)〉|b〉, where the field state is taken to be a coherent superposition of zero (c₀) and one (c₁) photon, |φ_field(t)〉 = E(t)[c₀|0〉_field + c₁|1〉_field]. E(t) is the envelope of the field external to the cavity, with ∫|E(t)|²dt = 1; t + τ is a user-selected time (discussed below). Given timing information for the incoming field |φ_field(t)〉, the first step in this process (figure, panel B, a) is accomplished by adiabatically ramping the control field Ω₁(t) from on to off, as in A, a. After this step, the internal states of the atom provide a long-lived quantum memory (figure, panel B, b). At a user-selected later time t + τ, the final step is initiated (figure, panel B, c) by turning Ω₂(t + τ) from off to on (as in A, b), thereby coherently mapping the atomic state c₁|a〉 + c₀|b〉 back to the 'flying' field state β(t) = |φ_field(t + τ)〉.
这两个过程可以结合起来实现光场λ(t) = |φfield(t)〉的状态在原子态|a〉和|b〉形成的量子存储器之间的连贯传输(参考文献 18;图 B 部分)。在理想情况下,映射过程可用公式|φfield(t)〉|b〉→|0〉(c1|a〉+c0|b〉)...(存储)...|0〉(c1|a〉+c0|b〉)→|φfield(t+τ)〉|b〉来表示,其中光场状态被假设为零光子(c0)和一光子(c1)的相干叠加,|φfield(t)〉=E(t)[c0|0〉field+c1|1〉field]。E(t)为场外包络,满足∫|E(t)|2dt=1;t+τ为用户选定的时间(下文讨论)。给定输入光场|φfield(t)〉的时间信息,该过程的第一步(图 B,a)是将控制场Ω1(t)从开到关的缓慢变化,如图 A,a 所示。此后,原子的内部态提供了长寿命的量子存储器(图 B,b)。在用户选定的后续时间 t+τ,通过将Ω2(t+τ)从关到开(如图 A,b 所示)来启动最终步骤(图 B,c),从而将原子态 c1|a〉+c0|b〉连贯地映射回"飞行"光场态β(t)=|φfield(t+τ)〉。

Show less 少一些

Box 2: A new paradigm for cavity QED
箱 2：腔道 QED 的新范式

To build large-scale quantum networks^4,6, many quantum nodes will need to be interconnected over quantum channels. Because conventional (Fabry–Pérot) configurations are ill suited for this purpose, there have been efforts to develop alternative microcavity systems²⁶, both for single atoms^75,76,78 and for atom-like systems (such as nitrogen–vacancy centres in diamond⁷⁹). A quantitative comparison of candidate systems is provided in ref. 77.
要建立大规模的量子网络 4,6,许多量子节点需要通过量子信道互连。由于传统的(法布里-珀罗)构型不适合这一目的,人们已经努力开发替代的微腔系统 26,既适用于单个原子 75,76,78,也适用于原子类系统(如钻石中的氮-空位中心 79)。候选系统的定量比较见文献 77。

A remarkable resonator for this purpose is the microtoroidal cavity that is formed from fused SiO₂ (refs 80,81) (shown in the figure). Such a resonator supports a whispering-gallery mode⁸² circulating around the outer circumference of the toroid (shown in cross-section in grey, in panel a of the figure), with an evanescent field external to the resonator. The intensity of the resonator mode is indicated by the coloured contours. Because of the small mode volume V_m and large quality factor Q, an atom (blue) interacting with the evanescent field of a whispering-gallery mode can be far into the regime of strong coupling, with projected values for the critical photon n₀ and atom N₀ numbers (n₀ ≈ 2 × 10⁻⁵ and N₀ ≈ 10⁻⁶)⁷⁷ that are significantly greater than current¹² and projected⁷⁷ values for cavity QED with Fabry–Pérot cavities (Fig. 2c).
一种用于此目的的显著共振器是由熔融 SiO2 形成的微小环形腔室（参考文献 80、81）（如图所示）。这种共振器支持沿着环形腔室外缘环流的低闷声模式（a 面板中灰色截面所示），并具有外部的暗场。共振器模式的强度由彩色等值线表示。由于模式体积 Vm 小且品质因数 Q 大，与共振器暗场相互作用的原子（蓝色）可以进入强耦合域，临界光子数 n0 和原子数 N0 的预测值（n0 ≈ 2 × 10−5 和 N0 ≈ 10−6）77 明显大于目前 12 和预测 77 的基于 Fabry–Pérot 腔室的腔体量子电动力学值（图 2c）。

Pioneering fabrication techniques^80,81 lend themselves to the integration of many microtoroidal resonators to form optical networks, as illustrated in panel b and c of the figure. Panel b shows a photograph of a silicon chip with a linear array of microtoroidal resonators within an ultrahigh-vacuum apparatus⁷⁶. The toroids appear as small scattering centres on a silicon chip that runs vertically down the centre of the picture. Black arrows indicate a horizontal SiO₂ fibre taper for coupling light to and from one resonator. Scale bar, 2 mm. Panel c is a scanning electron micrograph of an array of microtoroidal resonators (a magnification of the region bounded by the white box in panel b), showing toroids of fused SiO₂ on silicon supports⁸⁰.
开创性的加工技术 80,81 有利于将许多微环形谐振器整合到光学网络中,如图中 b 和 c 面板所示。b 面板显示了一个硅芯片上有线性阵列微环形谐振器的超高真空装置 76 的照片。这些环形体在垂直放置于图像中间的硅芯片上显示为小散射中心。黑色箭头指示水平 SiO2 光纤减肥器,用于将光耦合到一个谐振器及从其取出。比例尺为 2 毫米。c 面板是微环形谐振器阵列的扫描电子显微镜图像(放大了 b 面板白框内的区域),显示了硅支架上熔融的 SiO2 环形体 80。

These resonators have the capability for input–output coupling with small parasitic loss⁸¹ for the configuration shown in panel d (scale bar, 10 µm), which is a micrograph of an individual toroid and fibre taper from panel b⁷⁶. Q = 4 × 10⁸ has been realized at λ = 1,550 nm, and Q ≈ 10⁸ at λ = 850 nm, with good prospects for improvement to Q ≈ 10¹⁰ (ref. 77). For these parameters, the efficiency ε for coupling quantum fields into and out of the resonator could approach ε ≈ 0.99–0.999 while remaining firmly in the regime of strong coupling⁷⁷. Such high efficiency is crucial for the realization of complex quantum networks, including for distributing and processing quantum information^4,6,35 and for investigating the association between quantum many-body systems and quantum networks^9,11.
这些谐振器具有小寄生损耗 81 的输入-输出耦合能力,如面板 d 中所示的配置(比例尺 10 微米),这是面板 b76 中一个单个环形谐振器和光纤锥形耦合器的显微镜图像。在波长 1,550 纳米处实现了 Q=4×10^8,在波长 850 纳米处 Q ≈ 10^8,还有望进一步提高到 Q ≈ 10^10 (参考文献 77)。对于这些参数,将量子场耦合进出谐振器的效率ε可高达ε ≈ 0.99-0.999,同时仍牢固地处于强耦合 77 区域。这种高效率对实现复杂的量子网络至关重要,包括分配和处理量子信息 4,6,35,以及研究量子多体系统与量子网络之间的关联 9,11。

The initial step in this quest to realize a quantum network was the demonstration of strong coupling between individual atoms and the field of a microtoroidal resonator⁷⁵. More recently, non-classical fields have been generated from the interaction of single atoms with a microtoroidal resonator by way of a 'photon turnstile', for which a single atom dynamically regulates the transport of photons one by one through the microtoroidal resonator⁷⁶ (figure, panel d). Only single photons can be transmitted in the forward direction (from right to left in the figure), with excess photons n > 1 dynamically rerouted to the backward direction.
构建量子网络的初始步骤是展示单个原子与微环形共振腔场之间的强耦合 75。最近,通过单个原子与微环形共振腔的交互作用生成了非经典光场,采用了"光子转门"技术 76 (图,面板 d)。只有单个光子可以沿前向方向 (从图中右侧到左侧)传输,多余的光子 n > 1 动态地被重新路由到反向方向。

Show less 少一些

Box 3: Writing and reading single atomic excitations
盒子 3:写入和读取单原子激发

The DLCZ protocol¹³ is based on ensembles of N_a identical atoms (blue) with a Λ-level configuration, as shown in the figure. The metastable lower states |g〉 and |s〉 can be, for example, atomic hyperfine states of the electronic ground level to ensure a long lifetime for coherence. All atoms are initially prepared in state |g〉 with no excitation (figure, panel a), namely |0_a〉 ≡ ${\otimes }_i^{N_a}$ |g〉_i, and a weak off-resonant 'write' pulse is then sent through the ensemble. This results in a small probability of amplitude √p that one of the N_a atoms will be transferred from |g〉 to |s〉 and will emit a photon into the forward-scattered optical mode (designated field 1) with a frequency and/or polarization distinct from the write field.
DLCZ 协议 13 基于具有Λ-电平结构的 Na 个相同原子(蓝色)的集合,如图所示。准稳定的较低态|g〉和|s〉可以是电子基态的超精细态,以确保相干性的长寿命。所有原子最初都被预先准备在|g〉态,没有激发(图 a 面板),即|0a〉≡|g〉,然后通过集合发送一个弱的失谐"写入"脉冲。这会导致 Na 个原子中的一个以概率√p 从|g〉转移到|s〉,并向前向散射的光学模式(被称为场 1)发射一个频率和/或偏振与写入场不同的光子。

For small excitation probability p≪1, in most cases nothing happens as a result of the writing pulse, so the resultant state |φ_a,1〉 for the atomic ensemble and field 1 in the ideal case is given by
对于小激发概率 p≪1 的情况,在大多数情况下,写入脉冲不会产生任何结果,因此理想情况下原子集合和场 1 的结果状态 |φa,1〉 可以表示为

where |n₁〉 is the state of the forward-propagating field 1 with n₁ photons (n₁ = 0 or 1), the phase β is determined by the propagation phases of the write pulse and field 1, and O(p) denotes of order p. The atomic state |1_a〉 in equation (1) (above) is a collective (entangled) state with one excitation shared symmetrically between the N_a atoms (that is, one 'spin flip' from |g〉 to |s〉), where in the ideal case¹³
|n1〉是 n1 个光子(n1=0 或 1)的正向传播场 1 的态,相位β由写入脉冲和场 1 的传播相位决定,O(p)表示阶数 p 的量级。式(1)中的原子态|1a〉是一个集体(纠缠)态,其中一个激发对称地分布在 Na 个原子之间(即从|g〉翻转到|s〉的一个"自旋翻转"),在理想情况下 13

Field 1 is directed to a single-photon detector, where a detection event is recorded with probability p. Such an event for field 1 heralds that a single excitation (or spin flip from |g〉 to |s〉) has been created and stored in the atomic ensemble in the state |1_a〉 with high probability. Higher-order processes with multiple atomic and field 1 excitations are also possible and ideally occur, to lowest order, with probability p².
场 1 指向单光子探测器,检测事件以概率 p 被记录。对于场 1 来说,这种事件预示着一个单一激发(或从|g〉到|s〉的自旋翻转)以很高的概率被创造和存储在原子集合的状态|1a〉中。具有多个原子和场 1 激发的高阶过程也是可能的,且理想情况下以概率 p2 发生。

After a user-defined delay (subject to the finite lifetime of the quantum memory), the collective atomic excitation |1_a〉 can be efficiently converted to a propagating beam (designated field 2) by way of a strong 'read' pulse (figure, panel b), where in the ideal case there is a one-to-one transformation of atomic excitation to field excitation, |1_a〉 to |1₂〉. In the case of resonance with the transition from |s〉 to |e〉, the reading process utilizes the phenomenon of electromagnetically induced transparency^16,66.
在用户定义的延迟(受量子存储器有限寿命的制约)之后,集体原子激发态|1a〉可以通过强"读取"脉冲(图表,面板 b)有效转换为传播光束(指定场 2),在理想情况下,原子激发态到场激发态存在一一对应的转化关系,即|1a〉到|12〉。在与|s〉到|e〉跃迁共振的情况下,读取过程利用了电磁诱导透明 16,66 的现象。

Show less 少一些