INTRODUCTION
Entanglement shared by remote nodes is a fundamental resource for quantum networks1. For example, it is feasible to realize remote quantum communication by entanglement-based approaches2. The long-distance quantum channel established can be adopted to connect quantum processors or quantum sensors in different nodes3,4, striving towards the goal of distributed quantum networks5,6. Quantum networks are composed of a large number of interconnected nodes linked by photon-mediated quantum channels, with each node containing matter qubits for quantum memory and communication qubits supporting long-distance transmission.
Due to the unavoidable transmission losses that scale exponentially with the channel length, the distance between nodes on the ground is currently restricted to the order of one hundred kilometers7,8. To overcome this challenge, a ground-based quantum communication network can be deployed, with the assistance of quantum repeaters9,10. As shown in Fig. 1a, the basic principle of quantum repeaters is to first establish multiple segments of short-distance quantum entanglement between quantum memories and then to progressively perform entanglement swapping to gradually extend the distances. For wide-area coverage, satellites as transfer stations11 have been shown as a feasible solution12. An alternative approach that circumvents the use of long-channel optical fibers is transportable quantum memories13-15 combined with classical transportation methods, as shown in Fig. 1b, whose transmission distance is mainly limited by the decoherence rate of the quantum memories. With satellite-borne transportable quantum memories, the quantum Internet at the global scale could be envisioned16,17.
Fig. 1. Schematic diagrams of quantum repeaters and transportable quantum memories. a, |
Quantum memories can be divided into two types according to their physical systems: the ones based on atomic ensembles or single quantum systems18. For ensemble-based emissive quantum memories, entanglement can be generated between the collective excitation of an atomic ensemble and a scattered photon19, forming atom-photon entanglement, a building block for quantum repeaters and quantum networks. However, this process is intrinsically probabilistic, in which the state fidelity is traded off against the emission probability, resulting in a limited entanglement generation rate. Such systems include gaseous atomic ensembles20,21 and rare-earth-ion-doped crystals22. Alternatively, atomic ensembles could also be used as absorptive quantum memories that support wideband storage and offer striking multiplexing capacity23, and are in principle compatible with deterministic entangled photon sources24,25, thus enabling the efficient generation of deterministic entanglement between photons and quantum memories. However, the efficient interface between quantum memory and external light sources with heterogeneous matter systems remains a significant challenge26-30.
Unlike ensembles, single quantum systems that serve as single emitters can establish entanglement deterministically between the atomic spin and an emitted photon, which is known as spin-photon entanglement. The simple structure greatly reduces the difficulties for building near-term quantum networks. Rapidly developing systems include single trapped atoms and ions31,32, quantum dots33, as well as different color centers or dopants in solids34,35. It is worth mentioning that single quantum systems could also be constructed as absorptive quantum memories with suitable schemes36,37, acting as quantum receiving nodes. The photons generated by single emitters can be reversely and efficiently mapped into another single quantum system that uses the same matter system38, avoiding the bandwidth- and frequency-matching problem with external photon sources based on heterogeneous systems compared to ensemble-based absorptive quantum memories. More importantly, single quantum systems show intrinsic nonlinearity after appropriate cavity-enhanced coupling to the photonic channels, which could be exploited to construct deterministic quantum gates39,40. This method can facilitate the establishment of heralded entanglement across the quantum network nodes41,42 and can even overcome the efficiency limitation of photonic Bell-state measurements43,44, which is of great significance for establishing scalable quantum networks45.
To obtain deterministic interactions between single emitters and photons, efficient light–matter interaction has to be achieved. Considering the interaction between a single-atom system and a beam of light in free space46, the absorption cross section of the atom is $\sigma_{\mathrm{abs}}=3 \lambda^{2} / 2 \pi$, where λ is the optical wavelength of the input photon. The transverse area of the beam is $A=\pi \omega_{0}^{2} / 4$, where ω0 is the mode field diameter of the beam in semiclassical theory. Even if ω0 is focused to reach the order of magnitude of λ, due to finite solid-angle coverage and imperfect mode matching46, it is difficult for a single emitter to interact with photons efficiently in free space. To make quantum memory based on single emitters more efficient and to fulfil the requirement for deterministic interaction, it is necessary to confine the electromagnetic field of photons to enhance the interface between photon and spin. Combining single emitters with an optical resonator has been proven to be an excellent method.
One can obtain the cooperativity of cavity-coupled quantum systems $C=g^{2} / 2 \kappa \gamma $39,45, where g is the coupling constant, κ is the decay rate of the cavity field, and γ is the total dephasing rate of the emitter. γ can be defined via $\gamma=\gamma_{0}+\gamma_{1}+\gamma_{\mathrm{d}} $, where 2γ0 is the spontaneous emission rate of desired optical transitions, 2γ1 is the decay rate over other pathways, including the spontaneous emission rate of other optical transitions and nonradiative decay, and γd is the pure dephasing rate. Note that cooperativity is defined in terms of field decay rates45. When C ≫ 1, the coherent coupling between emitters and photons leads to near-deterministic atom–photon interactions, making it possible to achieve cavity-based quantum gates44,47. Further discussion about deterministic atom–photon interactions can be viewed in ref. 40. With the premise of C ≫ 1, cavity-emitter coupling can be divide into two regimes. When g ≫ κ, γ, the system works in the “strong-coupling” regime. When C ≫ 1 and κ > g ≫ γ, the system works in the “fast-cavity” regime. In the “strong-coupling” regime, photons can be confined in the cavity for a long time, which forms new energy eigenstates and generates the vacuum Rabi splitting phenomenon. For atoms that have large transition dipole moments, such as Rb atoms in vacuum, their g is usually of the same order of magnitude as κ and γ, and thus, such cavity-coupled systems tend to enter the “strong-coupling” regime38. However, for some solid-state emitters, such as rare-earth dopants, due to the small value of the electric dipole, their total decay rate γ0 + γ1 is on the order of Hz48. When solid-state emitters are coupled to a resonator, the value of g is much lower than κ, which makes it difficult for them to work in the “strong-coupling” regime. Whether these systems work in the “fast-cavity” regime or in the regime where C < 1 primarily depends on the value of the pure dephasing rate γd.
The cavity can change the density of the final-state photonic modes of emitters and thus enhance the spontaneous emission rate of the corresponding optical transition, known as the Purcell effect49. This is of great significance for solid-state emitters, which show lower spectral stability than single atoms trapped in vacuum, even when C < 1. In this scenario, the coupled system can also enter the “weak-coupling” regime, or be referred to as the Purcell regime39. Above all, the cavity-enhanced decay rate requires the detection of photons during a short time window, which would increase the spectral purity and the indistinguishability of photons when the emitter linewidth is mainly determined by γd50. In addition, by coupling single emitters to the optical resonator, the emitted photons can be output with a certain cavity mode, thus improving the success probability of coupling and receiving the output photons. This is of great use for emitters embedded in solids, in which the collection efficiency is rather small without a specially designed structure due to the high refractive index of the solid medium51. Furthermore, when the fluorescence spectrum line of the single emitter is wide, optical resonators can be used to realize spectral filtering52.
In principle, large cavity enhancement requires strong confinement of the electric field in both spatial and frequency domains that favors the use of high-quality microcavities with ultralow mode volumes. Among various single quantum systems, solid-state emitters are embedded in a solid matrix that provides stable physical surroundings and thus are more convenient to be integrated with a variety of microcavities. Considering the applications in large-scale quantum networks, single emitters with robust optical interfaces and long-lived spins are crucial requirements. To date, most prominent solid-state systems satisfying this criterion include color centers in semiconductors and rare-earth ions doped in solids. Single emitters usually have optical and spin coherence proprieties that differ greatly when embedded in different host materials and microstructures; thus, choosing appropriate microcavities is essential to obtain high cavity enhancement. For practical applications, more detailed aspects, such as the suitable working wavelength and tunability of cavities, unavoidable damage to the spin environment, and parasitic loss induced in the fabrication process, should be taken into account.
In second section, some single emitters and the properties of cavity-coupled systems are briefly summarized. Then, commonly used micro-optical resonators are summarized and compared with each other, in which emphasize the unique advantages of fiber Fabry-Pérot (F-P) microcavities are emphasized in quantum memory applications. In addition, we introduce the fabrication methods of fiber F-P microcavities and the effects of different parameters on the cavity working performance. In third section, state-of-the-art progress on individual color centers is overviewed and discussed, with a focus on systems that are coupled to fiber F-P microcavities. In fourth section, the advantages of rare-earth dopants in quantum memory are described, explained how a single rare-earth ion can play the role of quantum repeaters and transportable quantum memories, and analyzed the research progress of quantum memory based on single rare-earth ions in fiber F-P microcavities. In fifth section, the contents are summarized in this paper, elaborated the significance of quantum nodes based on single color centers and rare-earth ions in fiber F-P microcavities, and looked forward to their future development direction.
SOLID-STATE SINGLE EMITTERS AND MICRO-OPTICAL RESONATORS
The advantages of combining solid-state single emitters with optical resonators are discussed in first section. This sectionre views these solid-state emitters and micro-optical resonators commonly used to couple these single emitters in the field of cavity quantum electrodynamics (QED).
Solid-state single emitters
Quantum dots, as a kind of semiconductor nanostructure, are characterised by ultra-bright single-photon emission53, entangled light emission54, and ideal optical coherence time55. Various micro-optical resonators have been introduced to increase the photon extraction efficiency of quantum dots to over 50%56,57. The high indistinguishability of emitted single photons58 and entangled photon pairs24,25 allows quantum dots for two-photon interference and furthermore entanglement swapping59 between consecutive photons. Interference between photons emitted by two separate quantum dots was also demonstrated first in 201060 and then in 2022 with near-unity visibility61, enabling the generation of coherent single photons in a scalable manner. The demonstration of spin-photon entanglement62,63 establishes the first step toward implementation of communication qubits consisting of quantum dots. However, due to relatively insufficient spin coherence times, it seems difficult for quantum dots to meet the requirement for memory qubits of long-distance quantum networks33.
Color centers in semiconductors, as crystal defects, may offer suitable optical interfaces and long-lived spins. Different color centers feature different structural symmetries, leading to various optical and spin properties. Although there is a trade-off between operation speed and coherence time64, determined by the coupling strength between spins and the host environment, most color centers stand at a moderate position, so that spin qubits based on color centers usually allow for various quantum applications, including quantum memory65 and nanoscale sensing66. The color centers in diamonds and silicon carbide (SiC) are currently two widely studied types. Color centers in SiC have attracted extensive attention as their fluorescent radiation is close to the telecom band. For color centers in diamonds, nitrogen-vacancy (NV) color centers and group-IV-vacancy color centers, including silicon vacancy (SiV)67-69, germanium vacancy (GeV)68,70, and tin vacancy (SnV),71,72 are two major categories. Among them, NV centers have received considerable attention in the last two decades. They contain optically controllable electron spins for communication qubits that support efficient entanglement with other NV centers, along with long-lived nuclear spins for memory qubits, which is a crucial advantage for quantum network applications73.
As another alternative, rare-earth-ion-doped crystals have become a promising solid-state quantum system in recent years. For triply ionized rare-earth ions in solids, the 4f electron shell is shielded from external disturbance by outer filled 5s and 5p shells74. When the temperature is decreased to below 4 K, the lattice vibration of the crystals is greatly suppressed, and the influence of phonon coupling on optical transitions is almost eliminated. Under this condition, the intra 4f transitions of rare-earth ions can exhibit long optical coherence time up to milliseconds and a narrow fluorescence spectrum with a near-perfect zero-phonon line (ZPL). Rare-earth ions with even numbers of 4f electros, such as Er3+, Yb3+, and Nd3+, are called Kramers ions75. The ground-state spin of Kramers ions has a doublet state, which can be degenerated under external magnetic fields, forming an ideal two-level system. The nuclear spin of Kramers ions may offer coherent time over 1 s under large magnetic field76 where the electronic spin is frozen to the ground state. Compared to Kramers ions, non-Kramers ions with quenched electronic magnetic moments, including Eu3+, Pr3+ and Tm3+, usually exhibit much longer coherence times of nuclear spins13, and thus show an advantage in long-lifetime quantum memory77.
There are also other solid-state host materials, including two-dimensional (2D) materials and carbon nanotubes (CNTs). An overall discussion of solid-state single-photon emitters can be found in refs. 51 and 78. In this work, we will focus on color centers and rare-earth-ion-doped crystals, which can simultaneously meet the requirements of both memory qubits and communication qubits.
Basic principle of cavity-coupled systems
The micro-optical resonator has two parameters directly related to cavity QED: mode volume (V) and quality factor (Q). The mode volume represents the concentration degree of the light field distribution in the cavity79:
$V=\frac{\int_{c \mathrm{ar}} \epsilon(r)|E(r)|^{2} d^{3} r}{\epsilon\left(r_{0}\right) E\left(r_{0}\right)^{2}},$
where ϵ(r) is the relative dielectric constant, E(r) is the electric field strength in the cavity, and r0 is the coordinate where the quantum system in the cavity interacts with the electromagnetic field. Generally, the single emitter is located at the antinode of the standing wave field, where the strength of the electric field is the highest in the cavity. A smaller mode volume means a stronger restriction on the mode of the field inside the cavity and thus a more concentrated distribution of the light field.
Q, which characterises the ability of the resonator to store energy and to restrict the mode of the electromagnetic field in the frequency domain, is defined as
$Q=\frac{\nu}{\delta v},$
where ν is the resonant frequency and δv is the full width at half maxima of the resonance spectrum. With the values of Q and V, the effective Purcell factor49 can be obtained as follows:
$P_{\text {eff }}=\xi \cdot P=\xi \cdot \frac{3 \lambda^{3} Q}{4 \pi^{2} n^{3} V},$
where ξ is the branching ratio of the cavity-resonant transition in free space, that is, the proportion of the desired atomic transition located at the resonance frequency of cavities and n is the refractive index of the intracavity medium. As shown in Eq. (3), Peff can be improved by increasing Q or decreasing V. Meanwhile, the effective Purcell factor can be expressed via cooperativity C as follows:
$P_{\text {eff }}=2 C \cdot \frac{\gamma}{\gamma_{0}}.$
$\frac{\tau_{0}}{\tau_{\mathrm{c}}} \approx \xi \cdot P+1,$
where τ0 is the fluorescence lifetime of emitters in free space and τc is the cavity-enhanced fluorescence lifetime of emitters under Purcell enhancement. This equation provided a good approximation for evaluating of effective cavity enhancement through fluorescence measurement of desired transitions. However, to be more precise, the suppression on the off-resonant spontaneous emissions into free space should be considered82. As the resonant transition is enhanced, the optical branching ratio of emitters changes when coupled to a resonator83. Under these circumstances, the Purcell-enhanced branching ratio of emission into the cavity-resonant transitions, when ignoring the suppression of off-resonant fluorescence, can be obtained via84 the following equation:
$\xi_{\mathrm{c}}=\frac{\xi \cdot(P+1)}{\xi \cdot P+1},$
ξc tends to reach unity under large P even with rather small ξ. Therefore, for a quantum system with optical transitions involving multiple ground-state energy levels, in which rare-earth-ion dopants serve as a typical example, the optical resonator can be utilized to enhance the transitions between specific energy levels and to improve the branching ratio of the required transitions to ensure that the single emitter emits photons in a cycling transition, a prerequisite to complete initialization efficiently and to realize single-shot readout of the spin states84,85.
Commonly used micro-optical resonators
The microring resonator is composed of a set of waveguides, among which at least one waveguide is a closed-loop structure. The total reflection of light waves takes place continuously inside the cavity86. Such a resonator has the functions of excellent frequency selection and multimode coexistence and thus can be used for the design and research of various photonic devices87. Whispering gallery cavities, as a typical kind of microring resonator, provide an ideal optical platform for cavity QED due to their ultrahigh finesse. Whispering gallery cavities with different geometries, including microsphere cavities88, microtoroid cavities89, and microdisk cavities90, have been demonstrated. As an example, the geometry of a microdisk cavity is shown in Fig. 2a. A quality factor up to 1010 can be achieved via a microsphere cavity91. However, it is difficult to construct integrated devices based on such microsphere structures. Microtoroid cavities and microdisk cavities can be more integrated than microsphere cavities92. In spite of high quality factors, the mode volume of whispering gallery cavities is relatively large compared to other micro-optical resonators.
Fig. 2. Types of commonly used microcavities. a, |
A photonic crystal cavity (PCC) is constructed based on an optical waveguide and etched with circular air-hole sequences. According to the working dimension, it can be divided into one-dimensional PCC and 2D PCC. As shown in Fig. 2b, a one-dimensional PCC has a periodic lattice structure in one dimension with periodic Bragg reflectors on both sides. The 2D PCC has a periodic lattice structure in two directions. Both types of PCCs can achieve high Q values, in which one-dimensional PCCs have a relatively easy fabrication process and a smaller mode volume, allowing strong Purcell enhancements, and therefore they are more widely used.
The exact Q value of PCCs is also closely related to the fabrication process of the cavity. There are two main fabrication methods for PCCs: electron beam lithography (EBL) and focused ion beam (FIB). EBL constructs PCCs based on silicon-based materials. In practical applications, such cavities are often placed on the surface of solid-state samples to build solid-state quantum memory. FIB directly conducts ion beam etching on the solid sample itself96, causing damage to the original physical and chemical properties of solid samples. In actual manufacturing, the size instability, position instability, internal surface roughness, and other factors of the round air holes will affect the parameters related to the loss of the microcavity, thereby imposing a limit on the achievable Q value.
Bullseye is a kind of circular structure that confines the electric field in a certain mode and, thus it is featured with high collection efficiency93,97. The geometry of the bullseye is shown in Fig. 2c. Such a planar structure makes it feasible for device integration with other optical components93.
Micropillar cavities, consisting of two distributed Bragg reflectors (DBRs) with a spacer layer inside, are a kind of F-P cavity. Such microcavities allow for dual resonance while maintaining high quality factors94. In addition, micropillar cavities are widely used in quantum dots53, as shown in Fig. 2d, to enhance the collection efficiency of output photons emitted from quantum dots.
Fiber F-P cavities are one kind of open F-P cavity based on a fiber concave mirror, featuring wide tunability and efficient external coupling, as well as ultrahigh quality factors. As shown in Fig. 2e-g, there are three combinations: (i) a fiber concave mirror and a fiber concave mirror; (ii) a fiber concave mirror and a plane mirror; and (iii) a fiber concave mirror and a fiber plane mirror. When building a solid-state quantum memory, the second type of combination is widely used83,98, so that the sample crystal, usually in form of nanoparticles or membranes, can be adsorbed on the plane-mirror end by van der Waals forces without introducing any additional losses. Few interference fringes on the surface of membrane can indicate a successful binding99.
To form a stable fiber F-P microcavity, the mirror distance should be less than the radius of curvature of the concave mirror, which is usually in the order of microns, when coupled to single emitters. When nanoparticles are placed inside the cavity, the losses introduced are mainly due to particle scattering, which are related to the volume and refractive index of the particles100. To minimize scattering losses, it is advisable to reduce the particle size and embed them in a dielectric layer with a suitable refractive index101. When membranes are integrated into fiber F-P microcavities, the situations are more complex as the presence of membranes in cavities changes bare-cavity modes into hybridized modes50,98. Due to different electric field distribution, the losses at end mirrors and membrane-air surface varies for different modes, which mainly depends on the thickness of the membranes. Therefore, the effective losses in the cavity are highly dependent on the precise membrane thickness and should be carefully optimized, taking into account the mirror losses and the roughness of the membranes98. In any case, emitters located at the antinode position of the standing-wave light field can obtain the strongest Purcell enhancement48,102. The confinement of the cavity to the internal light field can be varied by adjusting the position, reflectivity, and other parameters of the mirror at either end.
Both the fiber concave mirror and plane mirror are coated with a high-reflectivity film before coupling the single emitters. Finesse ( $\mathscr{F} $) can be defined according to the transmission or reflection spectrum of the F–P cavity:
$\mathscr{F}=\frac{F S R}{\delta v},$
where FSR = c/2L is the free-spectrum range, which is the frequency interval of adjacent resonant longitudinal modes. The value of $\mathscr{F} $ is correlated to the light field loss inside the fiber F–P microcavity.
$\mathscr{F}=\frac{F S R}{2 \kappa} \approx \frac{2 \pi}{\mathscr{L}},$
Where, $\mathscr{F} $ is the sum of the intracavity loss. The final equivalence is valid when the loss is small enough. Therefore, as long as the cavity length is determined, a smaller cavity loss means a higher finesse and a narrower resonant peak linewidth. Based on the value of finesse, the quality factor of the fiber F–P microcavity can be calculated using the following equation:
$Q=\frac{2 \mathscr{F} L}{\lambda},$
Where, L is the cavity length. Consequently, by reducing the mirror loss, higher $\mathscr{F} $ and Q can be obtained. At present, the Q value of fiber F–P microcavities has already reached an order of 107 35.
$L_{\mathrm{eff}}=\frac{\int_{\mathrm{csv}} \epsilon(z)|E(z)|^{2} d z}{\epsilon\left(z_{0}\right)\left|E\left(z_{0}\right)\right|^{3} / 2},$
Where, z0 is the position coordinate with the highest electromagnetic field strength in the cavity, ϵ(z) is the relative dielectric constant, and E(z) denotes the electric field strength in the cavity. If only the fundamental Gaussian mode is considered, then95
$V=\frac{\pi \omega_{0}^{2}}{4} \cdot \frac{\int_{\operatorname{cav}} \epsilon(z)|E(z)|^{2} d z}{\epsilon\left(z_{0}\right)\left|E\left(z_{0}\right)\right|^{2} / 2}=\frac{\pi \omega_{0}^{2}}{4} L_{\text {eff }},$
where $\omega_{0}$ is the diameter of the beam waist. The mode volume of the fiber F–P microcavity is proportional to the effective cavity length.
The resonant frequency of the resonator must be strictly aligned with the optical transition frequency of the quantum system to achieve resonant operation. Therefore, in actual experiments, it is often necessary to tune the optical resonator. PCC can be tuned within a few nanometres by post-processing diamond etching or adding inert gas during operation81. In contrast, the fiber F-P microcavity can be tuned in a much larger range by changing the distance between two mirrors. In addition, the solid sample is adsorbed directly onto the plane mirror in the fiber F-P microcavity104, maintaining its original physical and chemical properties. As the emitters are normally several micrometers beneath the interface, the coherence time and inhomogeneous broadening of the emitters inside the membrane are comparable to those of emitters in bulk crystals35. Additionally, photons in the fiber F-P microcavity can be output from the fiber end, which can make the collection much more efficient and make it easy to interface with the existing fiber-based quantum network.
The construction and parameters of the fiber F-P microcavity
The core of the fiber F-P microcavity is the fiber concave mirror, whose configuration is shown in Fig. 3b-c. Due to the small cross section of the fiber end, the traditional mirror-polishing method cannot be used to fabricate fiber concave mirrors. In the early years, several methods including wet etching105, gas-bubble method106, transfer technique107, and electrochemical deposition technique108 have been introduced to fabricate fiber concave mirrors. However, by adapting these methods, it is difficult to fabricate fiber mirrors with small and controllable concave surfaces. Utilizing high-speed ion bombardment of samples to achieve etching, FIB can be used to fabricate such concave mirrors109 with arbitrary shape and high-precision positioning. However, FIB etching, featuring high cost and inefficiency, is not suitable for the batch fabrication of fiber concave mirrors. The scanning and non-continuous fabrication process of FIB makes it difficult to achieve the extremely low roughness of a concave surface. As an alternative, the method of laser ablation can also be introduced to fabricate fiber concave mirrors. SiO2 can absorb photons at wavelengths from 9.0 μm to 9.5 μm110. Therefore, the optical fiber with SiO2 as the main component can be ablated by CO2 laser pulses with a wavelength of 9.3 μm111. The typical construction process is shown in Fig. 3a. Using ablation pulses with suitable parameters, a fiber concave mirror with adjustable curvature and excellent surface quality can be produced, whose roughness can be as low as 0.2 nm112. The extremely low roughness loss makes it possible to construct a fiber F-P microcavity with an ultrahigh finesse. In addition, the CO2 laser machining method is convenient in process and low in cost, making it one of the most widely used construction methods for fiber concave mirrors.
Fig. 3. The machining of fiber concave mirrors. a, |
There are four main parameters of the fiber concave mirror113:
(1) Radius of curvature (ROC): the value of ROC directly determines the stability range of the fiber F-P microcavity and thus has an important impact on the mode volume.
(2) Effective diameter of the concave mirror, D: The value of D determines the diffraction loss or clipping loss. In particular, D is a crucial factor when building a millimeter-long fiber F-P microcavity. Due to the Gaussian rather than spherical shape of laser pulses, the maximum of D is limited when using single-laser pulses to fabricate a fiber concave mirror. However, this limitation can be overcome by adapting the multiple-pulse method114.
(3) Deviant distance between the concave mirror center and fiber core, Δd: When the fiber end is selected to output photons, Δd should be as small as possible so that the output efficiency between the cavity and fiber can be higher. The thinner the fiber core is, the more stringent the requirements are.
(4) Mirror ellipticity, e: e = 2(R1 − R2)/(R1 + R2), where R1 and R2 are the ROCs of the long axis and short axis, respectively, obtained by surface fitting. If e is sufficiently large, the fiber F-P microcavity will exhibit polarization-dependent mode splitting95,115,116. A fiber concave mirror with a low mirror ellipticity can be obtained through multi-pulse machining114.
Apart from these parameters, the type of fiber should also be taken into consideration. When using the fiber concave mirror to output photons, the mode-matching efficiency is mainly determined by the overlapping between cavity mode and fiber transmission mode117. For short cavities, it is easy to achieve a high mode-matching efficiency118 using normal single-mode fiber after carefully choosing the cavity parameters. However, for long cavities, in which the mode field diameter of cavity mode is usually large, specially designed integrated mode-matching optics devices119 or fibers with a large mode field diameter114,120 have to be utilized to enhance the efficiency. To date, fiber F-P microcavities have been demonstrated to work with cavity lengths ranging from sub-micrometers118 to a few millimeters114, showing great flexibility for various applications.
After the preparation of fiber concave mirrors, the surfaces of two mirrors that constitute an F-P cavity have to be coated with multilayer dielectric films to obtain high reflectivity111. Usually, this is done by coating a stack of alternating high-index (e.g., Ta2O5) and low-index (e.g., SiO2) thin films with specific thickness on the mirror substrates, forming DBRs. The loss at the fiber concave mirror end mainly includes absorption loss, scattering loss, and clipping loss102. In order to obtain high finesse of the cavities and efficient outcoupling efficiency of emitted photons, the absorption and scattering loss introduced by coated films must be kept as low as possible. However, to ensure the efficient outcoupling, the transmission of the outcoupling mirror should be large enough compared to other losses101.
To this end, in order to enhance light-matter interactions, intracavity losses should be controlled as low as possible, while the transmission of the outcoupling mirror dominates the whole loss.
QUANTUM NODES BASED ON INDIVIDUAL COLOR CENTERS COUPLED WITH FIBER F-P MICROCAVITIES
Great efforts have already been made in coupling color centers, especially NV centers, with fiber F-P microcavities. In this section, we introduce several color centers and take NV centers as an example to illustrate the prospect of coupling color centers with fiber F-P microcavities.
Solid-state color center systems
In addition to near-telecom-band photon emission, color centers in SiC also feature long spin coherence time at room temperature and good prospects for integration121, making them one of the most promising candidates for the upscaling of quantum networks. For color centers in diamonds, the SiV centers and NV centers are the other two prominent systems that have shown enormous experimental progresses recently. The SiV center, as a group-IV defect, is insensitive to external disturbance due to its mirror symmetry122 and thus has a rather small inhomogeneous broadening of emitted photons, showing promising optical properties with good indistinguishability123 but relatively low quantum efficiency124,125. The NV center features outstanding spin coherence properties and stable optical transitions126. However, the inefficiency of the zero-phonon transition limits the application of NV centers for long-distance quantum networks. Fortunately, this limitation can be overcome by introducing optical resonators. A concrete discussion and comparison of different color centers in diamond and SiC can be further found in refs.127,128. In the following part of this section, we will describe advantages and disadvantages of NV centers in free space, demonstrate in which aspect fiber F-P microcavities can improve the properties of NV centers, and analyze further developing routes of such cavity-coupled systems when considering the requirement for long-distance quantum repeater applications.
A large number of research results based on NV centers have emerged since spin-photon entanglement based on NV centers was realized34. The most attractive properties of NV centers in diamond are their robust single-photon emission129 and long spin coherence time even at room temperature. Apart from this, the possibility of mapping its spin state to nearby nuclear spins130 also makes NV centers a suitable emitter for the application of quantum communication. After coupling to a nearby 13C nuclear spin, coherence lifetimes exceeding 1 s131 have been realized. However, the NV center in diamond has a nonzero electric dipole moment, and its luminescence properties are strongly influenced by the environmental stress field and electromagnetic field. For this reason, the fluorescence spectrum of NV centers has a large broadening. Growing a better diamond can reduce the broadening of NV center fluorescence to a certain extent.
Due to coupling with phonons at room temperature, the excited state spin of NV center gets is rapidly mixed, which makes it difficult to generate deterministic spin-photon entanglement. At cryogenic temperatures, coupling between NV centers and phonons is much weaker than that at room temperature132, which makes NV centers more suitable for establishing spin-spin entanglement at different nodes based on entanglement swapping. In 2013, heralded entanglement between two NV centers separated by three meters was realized by Bernien et al.133. The distance between two NV centers was extended to 1.3 km later in 2015126. In this work, robust entanglement between two remote nodes was generated and was used to violate a loophole-free Bell inequality. An entanglement-based quantum network consisting of three spatially separated NV centers was demonstrated73, in which the distributions of multipartite entangled states across the three nodes and entanglement swapping through an intermediate node was established, indicating the great potential of this system with respect to the development of scalable quantum network nodes in the near future.
Despite recent progress, NV centers at cryogenic temperatures are still faced with difficulties in coupling to phonons, which generate phonon sidebands (PSBs) that destroy the indistinguishability of output photons, lead to inefficient connection with photons from other nodes, and ultimately limit the generation rate of remote entanglement. At 4 K, the Debye-Waller factor of the NV center, which weighs the proportion of fluorescence intensity at the ZPL, is merely approximately 0.04134. Considering the collection efficiency, a solid immersion lens was introduced135. However, the theoretical efficiency limit of such a scheme is 50%. An optical resonator can be introduced to enable indistinguishable photons at ZPL to be emitted at a faster speed and simultaneously collected with higher efficiency.
Development of quantum nodes based on individual NV centers coupled with fiber F-P microcavities
As the Stark effects of NV centers caused by charge fluctuation136 can contribute to frequency instability of the optical transition, especially for NV centers close to surfaces136, NV centers coupled with PCCs might not be an ideal choice. One feasible solution is to select other color centers with an inversion symmetry that lacks a linear Stark shift137,138. Another solution is to replace the nanophotonic structure with fiber F-P microcavities, which can accommodate diamond membranes with thicknesses of several micrometres to avoid addressing single NV centers close to surfaces.
For NV centers under strong Purcell enhancements, a shortened fluorescence lifetime and improved output mode can result in not only a high excitation rate but also efficient collection of coherent photons. Correspondingly, the success rate of remote entanglement generation can be improved by approximately three orders of magnitude139, which is of great significance for constructing high-speed quantum nodes based on individual NV centers in diamonds. Moreover, NV centers can be turned into narrow-band emitters140,141 under cavity filtering. To achieve a strong Purcell enhancement, the cavity length is usually reduced as short as possible to obtain a sufficiently small mode volume. However, it should be noted that a longer cavity length, while maintaining a constant finesse, results in a narrower linewidth of the cavity mode and a stronger filtering effect on the ZPL emission spectrum of NV centers. Therefore, when designing a cavity coupled system based on NV centers in fiber F-P microcavities, it is necessary to balance the value of the Purcell factor and the fluorescence linewidth of the emitters by selecting an appropriate cavity length.
Diamond nanoparticles and membranes are often used as carriers of NV centers in fiber F-P microcavities52,102,142. Nanoparticles are small in size, are easy to process, and can be directly attached to a plane mirror, as shown in Fig. 4a. Based on diamond nanoparticles, NV centers were demonstrated to be coupled with a fiber F-P microcavity in 201352. However, due to the broadened emission spectrum in both ZPL and PSBs at room temperature, which is predominantly induced by phonon coupling, only an increase in photon emission into the cavity mode is observed, instead of a shorter emitter's fluorescence lifetime, is observed. Purcell-enhanced photon emission from NV centers in diamond nanoparticles coupled to a fiber F-P microcavity at room temperature was demonstrated in 2016118. In this work, the mode volume of the fiber F-P microcavity was as low as 1 cubic wavelength. The ultralow mode volume is crucial for broadband emitters, such as NV centers, to achieve high Purcell effects due to the limited quality factors of emitters that are estimated from the linewidth of the emission spectrum118. However, the microcavity was coated with thin metal coatings instead of high-reflectivity dielectric coatings, a compromise for ultrasmall mode volume, resulting in a low quality factor. More importantly, further quantum network applications require individual NV centers in diamond nanoparticles operating at cryogenic temperatures.
Fig. 4. Color centers coupled with fiber F-P microcavities. a, |
For nanoparticles, the close proximity of interfaces has a non-negligible influence on the optical and spin coherence properties of NV centers143. Therefore, diamond membranes have become another widely studied carrier of NV centers in the construction of single emitter quantum systems. As shown in Fig. 4b, inside the fiber F-P microcavity, the diamond membrane is attached onto the end of the plane mirror by van der Waals forces. When the NV center is located at the antinode of the standing wave field in the cavity, the intensity of the interaction between light and matter in the cavity reaches its maximum. It is worth mentioning that, as shown in Fig. 4e-f, different membrane thicknesses and air spacings can result in different resonant modes due to hybridized resonances of membrane-air cavities. As the loss caused by scattering at the diamond interface and loss in the mirror are related to the distribution of the electric field102,144,145, the proper mode that favours a higher Purcell factor should be determined according to actual experimental conditions98. Furthermore, the transverse mode mixing and the curvature related cavity loss at the membrane-air interface increase as the ratio between the membrane thickness and the ROC increases146. Therefore, when designing a cavity coupled system, the compromise between improved emitter coherence for thicker and improved cavity performances for thinner membranes should be taken into consideration. For NV centers in a membrane coupled with an open microcavity, which is fabricated on a flat silica substrate by CO2 laser ablation, the Purcell factor at ZPL has been increased to 30142 and can be further improved by increasing Q and decreasing V147. Based on the fiber F-P microcavity, optical addressing of individual NV centers at cryogenic temperatures83, as shown in Fig. 4c, has been demonstrated under a resonant excitation protocol. By tuning the fiber F-P microcavity to the resonant frequency of the ZPL, the fluorescence lifetime is significantly reduced, which can be viewed in Fig. 4d, and an enhancement of the ZPL emission by a factor of four is extracted.
On the basis of these works, several improvements can still be made to improve the Purcell factor towards scalable quantum nodes with high photon emission rates. First, a better coated film and a low roughness membrane are needed to improve the finesse of the fiber F-P microcavity and correspondingly the quality factor. Second, fiber end lapping technology can be utilized to reduce the depth of the fiber concave mirror148, thereby achieving a smaller cavity length and smaller mode volume. In addition, decreasing the vibration in cryogenic systems149 and introducing fast feedback as well as slow feedback150 to obtain a better stability of the cavity length149 can also be reduced the fluorescence lifetime of the color center151,152 and therefore improve the emission rate of the color center. Such improvements can further increase the success rate of quantum protocols such as quantum teleportation153 and remote spin-spin entanglement126 can be further raised.
In addition to NV centers, efforts have been made in the coupling of GeV104,154 and SiV155-158 color centers in nanodiamonds or diamond membranes with fiber F-P microcavities. These color centers typically require cryogenic conditions, even temperatures down to millikelvins159,160, for sufficient ground state coherence time to support remote entanglement distribution. A highly strained device may alleviate the need for ultralow temperature159, 160. For color centers in large-bandgap semiconductors such as SiC, the ability to integrate with mature semiconductor technologies opens a door to the scalable integration of quantum systems and provides new methods for controlling emitters. For example, diodes can be fabricated to modulate the local electrical environment of emitters, achieving broadband Stark tuning of the emission frequency and dramatic narrowing of the optical linewidths161. From the perspective of transmission loss, the coupling between color centers in SiC with near-telecom-band radiation and fiber F-P microcavities becomes a promising platform, with prospects for scaling up quantum networks. Furthermore, as the quantum frequency conversion of single photons from an NV center to the telecom band has already been demonstrated162, for color centers only with optical transitions far from the telecom band, quantum frequency conversion163 becomes a worthwhile choice to extend the distance of entanglement from different quantum nodes.
QUANTUM NODES BASED ON SINGLE RARE-EARTH DOPANTS COUPLED WITH FIBER F-P MICROCAVITIES
In this section, we introduce the advantages of rare-earth-ion-doped crystals in terms of quantum memory, analyze the great potential of constructing quantum memories based on single rare-earth ions, and overview the experimental progress on the coupling of rare-earth ions with optical resonators, especially F-P microcavities.
Quantum memory based on rare-earth-ion-doped crystals
Rare-earth ion ensembles have a large inhomogeneous broadening164, making them suitable for the construction of absorptive quantum memories with large bandwidth and high multimode capacity, for which ensembles of rare-earth ions attracted much attention in early research. To date, a storage efficiency of 69%165, a fidelity of 99%166-168, and a bandwidth of 16 GHz169 have been achieved. In addition, in terms of multimode quantum memory, 24 frequency modes170, 51 spatial modes171, 1250 temporal modes23, and multiple degrees of freedom172 based on rare-earth ion ensembles have been demonstrated.
Among all rare-earth ions, quantum memory based on Er3+ has always been an attractive research topic because its emission band is located around the telecom wavelength. In 2010, storage of weak coherent light pulses at single-photon level based on Er3+:YSO was demonstrated174. Another pioneering work is the storage of entangled photons at the telecom wavelength, which was first demonstrated in an erbium-doped optical fiber175. In 2018, a hyperfine coherence time of Er3+ ions over 1 s was demonstrated76, satisfying the requirement for long-distance quantum repeater applications. More recently, a fiber-integrated quantum memory has been demonstrated based on Er3+ ensembles in an optical waveguide176. With these developments, Er3+-based materials are expected to play an important role in fiber-based quantum networks.
Despite the enormous progress, most quantum memories based on rare-earth ion ensembles suffer from low storage efficiency without cavity enhancement177-179 More importantly, absorptive quantum memories require external high-quality single-photon sources with perfectly matched wavelengths and bandwidths. Although this problem can be circumvented by utilizing the Duan-Lukin-Cirac-Zoller (DLCZ) protocol22, the emission probability has to maintain a low value to avoid multi-photon emission that could deteriorate the fidelity of entanglement. The same problem occurs when using light sources based on spontaneous parametric down-conversion (SPDC) process. Interfacing deterministic entangled-photon sources24,25 with absorptive quantum memories26 may provide an excellent solution. In contrast, quantum memory based on single ions can work without an external photon source and can generate deterministic entanglement between single ions and photons. Especially for a single Er3+ ion, as long as spin-photon entanglement is generated, photons can transmit across the existing fiber network with low transmission loss, reach adjacent nodes, and be utilized to achieve entanglement swapping for entanglement distribution between remote spins, forming the basis of quantum repeaters.
Eu3+, as a non-Kramers ion, has a small magnetic dipole and thus a low decoherence rate. Using the zero-first-order-Zeeman (ZEFOZ) magnetic field and dynamical decoupling, Eu3+ ions in YSO hosts have been proven to have a nuclear spin coherence time of up to 6 h at 2 K13. Such a long coherence time and optical accessibility180 make Eu3+ : YSO an attractive candidate for long lifetime quantum memory, especially transportable quantum memory. Since coherent optical storage for 1 h15 and single-photon-level quantum memory up to 20 ms181 have been demonstrated, the feasibility of transportable quantum memory based on Eu3+ ion has been preliminarily proven. However, due to the spin inhomogeneous broadening, the ZEFOZ magnetic field and dynamical decoupling pulses cannot be perfect for a large ensemble182. The noise introduced by nonuniform dynamical decoupling pulses and magnetic fields can impose a limit on the achievable fidelity of the quantum memory. As single Eu3+ ions lie in the magnetic field with a certain coordinate and do not exhibit inhomogeneous broadening, the decoherence rate and the noise can be quite low when applying the ZEFOZ magnetic field and dynamical decoupling pulses, which provides an attractive solution for the study of transportable quantum memory.
Quantum memory based on single rare-earth ions coupled with F-P microcavities
The 4fn↔4fn transitions of rare-earth ions have rather low oscillator strengths due to their parity-forbidden character by the electric dipole selection rules. This results in an extremely long fluorescence lifetime and thus low emission rates when directly detecting single ions by intra-4f fluorescence183. As an alternative, parity-allowed 4fn↔4fn−15d1 transitions have been adopted for optical detection of Pr3+ and Ce3+ ions due to their much shorter fluorescence lifetime184,185. However, 4fn↔4fn−15d1 transitions are usually located in the ultraviolet range and show strong PSBs in absorption and emission, which is similar to transitions in NV centers in diamond. In order to realize spin-photon interfaces, coherent intra-4f transitions are more appropriate, provided that the excited state lifetimes can be reduced to achieve a reasonable emission rate and, more importantly, to approach the coherence time for the emission of photons with high indistinguishability. From these perspectives, incorporating single rare-earth ions in a crystalline host with a high-quality cavity appears to be a potential approach as the Purcell enhancement could significantly reduce the fluorescence lifetime to satisfy the Fourier-limited condition, thus allowing for quantum swapping between photons emitted from different nodes, which is a key requirement for a scalable quantum network.
Most rare-earth ions lack intrinsic cycling transitions. When the cavity is tuned to resonate with the ground-state optical transition of rare-earth ions, the resonant fluorescence lifetime will be greatly reduced under Purcell enhancement. As long as the Purcell factor is large enough, the optical transition of rare-earth ions can be altered into an approximate cycling transition, which is a prerequisite for single-shot readout of spin qubits84,186.
Based on PCC-coupled rare-earth-ion systems, single-shot readout of single Yb3+ and Er3⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓+ 84,85, optical multiplexing of multiple Er3+ in the frequency domain35,186, Stark tuning of the emission frequency of a single Er3+ 187,188 and nuclear spin-wave quantum register for single Yb3+ electron spins189 have already been accomplished. Under such Purcell enhancement, the fluorescence lifetime of rare-earth dopants can be reduced to the order of μs, greatly reducing the restrictions on the generation of remote entanglement. However, when the rare-earth ion is too close to the surface of the solid sample, the properties of the crystal field will change dramatically, and thus, the decoherence rate will increase190, which increases the emission linewidth of rare-earth ions to an order of several MHz.
Similar to NV centers in diamond, rare-earth ions can also be produced in membranes or nanoparticles to incorporate small-volume fiber F-P microcavities. In 2018, Casabone et al. placed Eu3+ : Y2O3 nanoparticles at the end of a plane mirror and combined them with a fiber concave mirror to form a fiber F-P microcavity with a finesse F of 1000 and a cavity length of 2 μm. As shown in Fig. 5a, by attaching nanoparticles at an antinode point, they achieved an effective Purcell factor of two for an ensemble composed of few Eu3+ ions101. In 2021, the same research group placed Er3+ : YSO nanoparticles in a fiber F-P microcavity, with an effective Purcell factor of 15 for the Er3+ ensemble, as is shown in Fig. 5b, greatly reducing the fluorescence lifetime of the Er3+ ensemble. Besides, by changing the voltage offset of the driving piezo in a few milliseconds, as shown in Fig. 5c and d, dynamic control of the cavity length and, correspondingly, the cavity resonant condition can be achieved173, which will further allow for multiplexing in the frequency domain. Recently, the detection of single Er3+ ions in nanoparticles coupled with a fiber F-P microcavity has been achieved191, opening up the prospect of addressing ions within a nanoscale volume that allows strong dipolar interaction.
Fig. 5. Rare-earth-ion-doped nanoparticals coupled with fiber F-P microcavities. a, |
However, as coupling membranes with fiber F-P microcavities gives consideration to both a large Purcell enhancement factor and the maintenance of optical coherence properties of rare-earth ions, such a coupling protocol has become increasingly appealing in the field of quantum repeaters.
In 2020, Merkel et al. constructed an open F-P microcavity on silica substrates with a bare-cavity quality factor of up to nine million and coupled a 19-μm-thick Er3+ : YSO membrane to this cavity, achieving an effective Purcell factor of 59 for maximally coupled Er3+ dopants. Based on this experimental work, it is observed that the fluorescence lifetime of the Er3+ ensemble decreases so significantly that the ensemble-averaged coherence time might exceed the fluorescence lifetime of Er3+ ions located at the field maximum coordinate48. Then, in 2021, they decreased the total cavity length to 70 μm and achieved optical detection of single Er3+ ions based on a similar cavity-coupled system35. The sketch of this work and the Hanbury Brown-Twiss experiment results can be viewed in Fig. 6a and b, respectively. As the goal is to operate single ions, the concentration of rare-earth dopants needs to be as low as possible, for which a nominally undoped YSO crystal with an actual Er3+ concentration below 0.3 ppm is used in this work. The effective Purcell factor is up to 70 for ions at the cavity-field maximum point, based on which single Er3+ is detected and examined by the Hanbury Brown-Twiss experiment. As shown in Fig. 6c, in this work, for such a single Er3+ emitter, its fluorescence linewidth can be stably below 0.2 MHz for hours, indicating the good frequency stability of rare-earth ions. However, such an optical linewidth still needs to be narrowed to approach the Fourier-limited linewidth for better indistinguishability of emitted photons. One possible approach to further reduce spectral diffusion is to choose host materials with lower nuclear magnetic moments, thereby reducing the magnetic noise caused by spin flip-flops within the hosts. By coupling to a high-Q PCC, Er3+ in CaWO4 has been demonstrated to have a short-term optical linewidth of 150 kHz and a long-term spectral diffusion of 63 kHz, both close to the Fourier-limited linewidth of 21 kHz192. As optical properties of ions can be preserved better in membranes48, coupling similar crystals in the form of membranes to open F-P microcavities with large Purcell factors might be a promising solution toward indistinguishable Er3+-based quantum emitters.
Fig. 6. Rare-earth-ion-doped membranes coupled with fiber F-P microcavities. a, |
Open F-P microcavities fabricated on silica substrates in aforementioned works can be replaced by fiber F-P microcavities. Above all, as fiber F-P microcavity is one of the most miniaturized F-P cavities,193 a smaller cavity length, correspondingly, a smaller mode waist and a smaller mode volume can be achieved. In this way, the branching ratio can be further increased to approach a condition with a better cycling transition. Thus, a more efficient initialization, manipulation, and single-shot readout of spin states can be achieved. In addition, applying a fiber F-P microcavity can help to improve the efficiency of coupling output photons into a fiber-based quantum network. Based on the generation of spin-photon entanglement between telecom photons and emitter-based quantum memories, such as single Er3+, remote entanglement between quantum nodes across fiber links could be established, opening the possibility for the implementation of large-scale quantum networks.
As mentioned earlier, Eu3+ in YSO hosts has great potential for the construction of transportable quantum memory due to its outstanding spin coherence properties. However, the small branching ratio of approximately 1:60101 brings great difficulties for the single-shot readout of single Eu3+ ions. Moreover, the wavelength of the ground-state transition of Eu3+ is near 580 nm, at which the loss of the cavity mirror tends to be greater than that at longer wavelengths, making it more difficult for Eu3+ to achieve a large Purcell factor. For these reasons, the detection and manipulation of single Eu3+ ions requires microcavities with higher finesse, higher quality factor, and smaller mode volume. Moreover, to ensure the efficient output of photons emitted by rare-earth ions, the transmissivity of the output cavity mirror must be far larger than the absorption loss and clipping loss of the mirror surface. Therefore, the level of coating techniques and CO2 laser machining method still need continuous improvement.
Additionally, whether for color centers or rare-earth ions, to achieve stronger Purcell enhancements, a higher finesse, and a smaller mode volume are needed. As both transmission and scattering losses below 1 ppm can be achieved194, a fiber F-P microcavity with a finesse of 106 can be envisioned, with which the cavity Purcell factor can be much higher than that found in current research studies. High-finesse fiber F-P microcavities need to be actively stabilized before they are used to couple the single emitters placed inside. However, the vibration of the commercial cryogenic system is generally strong, for which the precise stabilization of the fiber F-P microcavity will be faced with great difficulties when the finesse of the cavity exceeds 1000152,195. By designing the resonator into an integrated structure and using vibration-isolation connections149,150,195, the relative movement between the plane mirror and the optical fiber concave mirror can be reduced. When the requirements for ultralow temperature are not demanding, the thermal feedback method can be used to compensate for the fluctuation of the cavity length48,152.
CONCLUSION AND OUTLOOK
Constructing quantum nodes based on single emitters shows great significance for the long-distance distribution in quantum entanglement due to its key superiority of the deterministic generation of spin-photon entanglement. In quantum networks, different application scenarios impose different requirements on the technical specifications of quantum memory. Normally, quantum repeater applications require quantum memories with high efficiency and storage times in the order of seconds10, better with large multiplexing capacities196. For transportable quantum memory applications, the storage times have to be in the order of hours to support quantum state transmission over hundreds of kilometers on the ground or global coverage on satellites13-17. Solid-state quantum systems with color centers and rare-earth dopants are revealed to have great optical and spin coherence properties, for which such single emitters could serve as quantum nodes in both quantum repeaters and transportable quantum memory applications, as well as other possible applications in future quantum networks.
As the fluorescence lifetime of color centers generally ranges from approximately 1 ns to tens of nanoseconds184,197, the single-shot readout of individual color centers198 can be achieved even in free space. To date, the entanglement generation rate based on individual color centers can already exceed the decoherence rate through single-photon interference, showing deterministic delivery of remote entanglement across quantum network nodes199. However, to further improve the success rate of entanglement distribution, ZPL photons should be emitted at a higher speed. As analyzed in second section, coupling emitters with fiber F-P microcavities can maintain the spin coherence property of emitters while achieving a strong Purcell enhancement that could greatly enhance the ZPL emission. Furthermore, fiber F-P microcavity makes the coupling between quantum nodes and fiber networks more efficient. However, to extend the scale of a fiber-based quantum network, the transmission loss must be considered. To this end, a color center with a near-telecom-band emission can be explored and studied200,201. As an alternative, high-efficiency quantum frequency conversion can be introduced. This approach is robust and versatile. To date, a total external device efficiency of up to 57% has been demonstrated202.
Different from color centers, as the fluorescence lifetime of rare-earth ions is generally on the order of milliseconds, it is quite difficult to detect single rare-earth dopants through transition within the 4f shell in free space, which makes the cavity-enhanced protocol an inevitable choice. As with quantum repeaters, even though the single Er3+ ion has already been detected via Purcell enhancement, the emission linewidth of single Er3+ ion is still far beyond lifetime-limited35. Solving this problem requires higher Purcell factors to further decrease the fluorescence lifetime. Apart from this, slowing spectral diffusion by diminishing the coupling with the nuclear spin bath can also help to approach nearly lifetime-limited optical linewidths. From this perspective, host crystals with small nuclear magnetic moments might work out, among which CaWO4192 has already been testified to be an outstanding host material. There is still a gap of nearly an order of magnitude between Purcell-enhanced linewidth and lifetime-limited linewidth, even for Er3+ ions in CaWO4192. As Purcell effect cannot be increased boundlessly due to some technical limitations, studying the decoherence mechanisms of single ions in solid matrix and searching for more qualified host materials will be crucial works. Eu3+ ions in YSO hosts have been proven to have a spin coherence time of 6 h, for which Eu3+:YSO has become an important candidate for transportable quantum memory. Because the ZEFOZ magnetic field and dynamical decoupling pulses can be applied more precisely on a single Eu3+ ion than on an Eu3+ ensemble182, constructing quantum memory based on single Eu3+ ions seems to be an excellent choice. However, due to the low branching ratio and the short wavelength of the intra-4f transition of Eu3+ ions, it is difficult to directly detect single Eu3+ ions. Alternatively, it may be feasible to indirectly detect and manipulate the single Eu3+ ion by monitoring the magnetic coupling to other rare-earth ions that exhibit brighter optical emissions, such as Yb3+ or Er3⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓+203, which is also an on-going work in our group.
In any case, the spectral stability of the emitters is one of the most important indexes, especially for scaling up of the quantum network. The resonator, as an indispensable part of solid-state quantum systems, needs to be improved to support a much higher Purcell factor. For fiber F-P microcavities with a membrane placed inside, the mode volume is limited by the thickness of the membrane. However, to maintain the coherence property and the spectral stability of single emitters inside the membrane, the membrane cannot be infinitely thin. To this end, the technology of stabilizing fiber F-P microcavities under cryogenic conditions and the micromachining technique of membranes will play an important role in the future development of such cavity-coupled systems.
At present, the primary focus of research is to enhance the performance of individual emitters within fiber F-P microcavities. For practical applications, substantial efforts should be directed toward the miniaturization and integration of these devices. Various potential approaches are worth exploring, with one obvious solution involving the incorporation of arrays of fiber mirrors combined with a thin membrane or an array of nanoparticles to create multiple microcavities. Nevertheless, this approach is resource intensive and presents challenges in scaling up for large-scale integration. A more promising alternative is to harness the inherent tunability of fiber F-P microcavities by utilizing high-precision three-dimensional positioning of fiber mirrors with dynamical and active control. This approach has the potential to achieve both spatial and frequency multiplexing of numerous addressable emitters173,191, albeit requiring extensive technical exploration. If successful, the integration of high-density emitters within a single device could significantly advance the development of high-speed quantum networks.
MISCELLANEA
Acknowledgments Z.-Q. Z. acknowledges the support from the Youth Innovation Promotion Association CAS. X. L. acknowledges the support from the Xiaomi Young Talents Program.
Declaration of Competing Interest The authors declare no competing interests.
Funding This work is supported by the Innovation Program for Quantum Science and Technology (No. 2021ZD0301200), the National Natural Science Foundation of China (Nos. 12222411, 11821404 and 12204459), and Anhui Provincial Natural Science Foundation (No. 2108085QA26).
