Tian Zhong

Philippe Goldner

Emerging rare-earth doped material platforms for quantum nanophotonics

Keywords:

Rare-earth dopants are arguably one of the most studied optical impurities in solids, with applications spanning from laser optoelectronics, biosensing, lighting to displays. Nevertheless harnessing rare-earth dopants' extraordinary coherence properties for quantum information technologies is a relatively new endeavor, and has been rapidly advancing in recent years. Leveraging the state-of-theart photonic technologies, on-chip rare-earth quantum devices functioning as quantum memories, single photon sources and transducers have emerged, often with performances unrivaled by other solid-state quantum technologies. These existing quantum devices, however, nearly exclusively rely on macroscopic bulk materials as substrates, which fundamentally limits the future scalability and potential functionalities of such quantum systems. Thus the development of new platforms beyond single crystal bulk materials has become a priority. In this review article, we summarize the latest progress towards nanoscale, low-dimensional rare-earth doped materials for enabling next generation rare-earth quantum devices. Different platforms with variety of synthesis methods are surveyed.

Their key metrics measured to date are presented and compared. Special attention is placed on the connection between the topology of each platform to its target applications. Lastly, an outlook for near term prospects of these platforms are given, with a hope to spur broader interests in rare-earth doped materials as a promising candidate for quantum information technologies.

INTRODUCTION

Solid-state quantum nanophotonics has emerged as one of the most active fields in quantum engineering. In particular, optically active atomic or atom-like defects in solids that simultaneously possess long-lived spin coherence and stable optical transitions play key roles in the development of quantum networks, quantum sensing and quantum information processing. Among numerous candidates, rare-earth doped materials have attracted significant interests in recent years, and is rapidly advancing. The emergent field of rare-earth nanophotonics inherits the rich history of rare-earth spectroscopy, and leverages the cutting-edge nanophotonic technologies to enable a new class of devices that are highly coherent, efficient and scalable. At the heart of this development is the fundamental science and engineering of rare-earth doped materials with wide variety of chemical compositions, physical behaviors, topologies and synthesis methods. The rich library of material properties offers nearly unlimited opportunities for quantum device engineering. At the same time, crucial device metrics also dictate the best combination of material topologies and growth technique to achieve those specifications.

Only recently the connection between material growths and quantum device engineering has been made, and the importance of material design and synthesis has been recognized. After decades of arduous efforts in perfecting bulk, single crystal hosts, newer platforms based on rareearth doped nanocrystals, polycrystalline ceramics and films tailored for quantum technologies have been gain-ing momentum and begin to reap promising results. In this article, we summarize some of the most recent developments in emerging material platforms beyond bulk crystals for building next generation rare-earth quantum photonic devices. We attempt to establish a common ground for material chemists, spectroscopists and device engineers in this field by highlightening the relationship between material metrics and device functionalities. The hope is to spur further research in fundamental material science to understand, control and design better materials to meet the needs of fast-growing rare-earth quantum technologies. It should be pointed out that we do not intend to give a thorough review on rare-earth doped bulk crystals, nor the spectroscopic techniques used to characterize them. A number of works on those topics are already available to the interested reader [START_REF] Macfarlane | High-resolution laser spectroscopy of rare-earth doped insulators: a personal perspective[END_REF][START_REF] Goldner | Rare Earth-Doped Crystals for Quantum Information Processing[END_REF][START_REF] Thiel | Rare-earth-doped materials for applications in quantum information storage and signal processing[END_REF][START_REF] Sun | Recent progress in developing new rare earth materials for hole burning and coherent transient applications[END_REF][START_REF] Kunkel | Recent Advances in Rare Earth Doped Inorganic Crystalline Materials for Quantum Information Processing[END_REF].

This article is organized as follows: Section 2 gives a review of rare-earth quantum photonic device developed to date and highlight the trend in underlying material platforms. Section 3 surveys in more details the progress in several emerging material platforms. In Section 4, we give an outlook for future material development and elaborate several key metrics need to be accomplished in order to push forward the field of rare-earth nanophotonics. such as yttrium aluminum garnet (Y 3 Al 5 O 12 or YAG), Y 2 SiO 5 (YSO), Y 2 O 3 , or YVO 4 (YVO). The 4f shell of rare-earth ions in crystals is partially occupied and is shielded from the interaction with the crystalline environment by 5s and 5p outer shells. The 4f electrons are highly localized and the 4f-4f intra-shell transitions are parity forbidden in free space, but become weakly allowed in the presence of crystal field, which result in sharp optical transitions with high quantum efficiency. Sharp optical transitions (equivalent quality factor over 10 11 ) at cryogenic temperatures are correlated with long optical T 1 lifetimes (hundreds of µs to ms) but also weak oscillator strengths. Electronic Zeeman and hyperfine transitions in the radio-frequency or microwave regimes are abundant. When doped into host with small nuclear magnetic moments such as YSO, record long spin coherence times 6 hours in Eu:YSO [START_REF] Zhong | Optically addressable nuclear spins in a solid with a six-hour coherence time[END_REF] and 1.3 seconds in Er:YSO [START_REF] Rančić | Coherence time of over a second in a telecomcompatible quantum memory storage material[END_REF] have been achieved. Such long coherence times combined with a rich selection of spin levels (both electronic and nuclear) and optical addressability make rare-earth ions attractive candidates for qubits as a building block in quantum information science. Furthermore, optical transitions exhibit inhomogeneous broadenings that are typically 10 5-6 times the homogeneous linewidths, indicating an excellent prospect for dense spectral multiplexing of qubits or memories to enable large data processing and storage bandwidths [START_REF] Usmani | Mapping multiple photonic qubits into and out of one solid-state atomic ensemble[END_REF][START_REF] Sinclair | Spectral multiplexing for scalable quantum photonics using an atomic frequency comb quantum memory and feed-forward control[END_REF].

One of the most successful application of rare-earth doped crystals in quantum technology is optical quantum memories. These memories which faithfully store a qubit encoded on a single photon have performances unmatched by any other solid-state system. For reviews on rare-earth quantum memories, readers are referred to ref. [START_REF] Tittel | Photon-echo quantum memory in solid state systems[END_REF][START_REF] Lvovsky | Optical quantum memory[END_REF]. Landmark ensemble experiments include storage of light up to a minute using electromagnetically induced transparency (EIT) in praseodymium doped yttrium orthosilicate (Pr 3+ :YSO) [START_REF] Longdell | Stopped light with storage times greater than one second using electromagnetically induced transparency in a solid[END_REF][START_REF] Heinze | Stopped light and image storage by electromagnetically induced transparency up to the regime of one minute[END_REF], quantum storage of photonic entanglement in neodymium (Nd) doped YSO (Nd 3+ :YSO) [START_REF] Clausen | Quantum storage of photonic entanglement in a crystal[END_REF][START_REF] Bussières | Quantum teleportation from a telecomwavelength photon to a solid-state quantum memory[END_REF] and thuliumdoped lithium niobate waveguides [START_REF] Saglamyurek | Broadband waveguide quantum memory for entangled photons[END_REF], optical memory at the single photon level in Nd doped yttrium orthovanadate (Nd:YVO) [START_REF] De Riedmatten | A solid-state light-matter interface at the single-photon level[END_REF][START_REF] Zhou | Realization of reliable solid-state quantum memory for photonic polarization qubit[END_REF][START_REF] Zhong | Nanophotonic rare-earth quantum memory with optically controlled retrieval[END_REF] and quantum level spin storage in Eu 3+ :YSO [START_REF] Jobez | Coherent Spin Control at the Quantum Level in an Ensemble-Based Optical Memory[END_REF], conditional phase shifts between two ensembles of europium (Eu) in YSO [START_REF] Longdell | Demonstration of conditional quantum phase shift between ions in a solid[END_REF], up to 69% efficient quantum memory for light in Pr 3+ :YSO [START_REF] Hedges | Efficient quantum memory for light[END_REF][START_REF] Afzelius | Demonstration of atomic frequency comb memory for light with spin-wave storage[END_REF][START_REF] Sabooni | Efficient quantum memory using a weakly absorbing sample[END_REF], and a quantum optical connection between a cold atomic gas and Pr 3+ in YSO [START_REF] Maring | Photonic quantum state transfer between a cold atomic gas and a crystal[END_REF]. These ensemble-based memories provide natural interfaces between optical photons and spins at radio-frequency or microwave regimes, which can also be harnessed to realize quantum transducers interconnecting distant superconducting quantum circuits. Hybrid quantum systems based on rare-earth ensembles see the opportunities of microwave quantum storage [START_REF] Wolfowicz | Coherent storage of microwave excitations in rare earth nuclear spins[END_REF] and microwave-to-optical conversion [START_REF] Brien | Interfacing superconducting qubits and telecom photons via a rare-earth-doped crystal[END_REF][START_REF] Williamson | Magneto-optic modulator with unit quantum efficiency[END_REF]. To date, high cooperativity coupling of rare-earth spins to a superconducting resonator has been demonstrated [START_REF] Probst | Anisotropic rare-earth spin ensemble strongly coupled to a superconducting resonator[END_REF][START_REF] Dold | High-cooperativity coupling of a rareearth spin ensemble to a superconducting resonator using yttrium orthosilicate as a substrate[END_REF]. Unit efficiency microwave to optical conversion via magneto-optic coupling has been proposed [START_REF] Williamson | Magneto-optic modulator with unit quantum efficiency[END_REF], and its experimental implementation is being pursued [START_REF] Fernandez-Gonzalvo | Coherent frequency up-conversion of microwaves to the optical telecommunications band in an er:yso crystal[END_REF][START_REF] Fernandez-Gonzalvo | Cavity enhanced raman heterodyne spectroscopy in er:yso for microwave to optical signal conversion[END_REF][START_REF] Kindem | Characterization of 171 Yb 3+ : yvo 4 for photonic quantum technologies[END_REF].

Ultimately, the most powerful rare-earth quantum technologies will be devices operating on invidividual ions [START_REF] Brien | Nondestructive photon detection using a single rare-earth ion coupled to a photonic cavity[END_REF]. To that end, optical addressing of single ions is coming into focus of this research. Such a task, however, is not easy. The progress has been hindered by long optical lifetimes of rare-earth ions and resultant faint photoluminescence. So far, only a few experiments have succeeded in isolating individual praseodymium [START_REF] Kolesov | Optical detection of a single rareearth ion in a crystal[END_REF][START_REF] Utikal | Spectroscopic detection and state preparation of a single praseodymium ion in a crystal[END_REF][START_REF] Nakamura | Spectroscopy of single pr 3+ ion in laf3 crystal at 1.5 k[END_REF], cerium [START_REF] Kolesov | Mapping spin coherence of a single rare-earth ion in a crystal onto a single photon polarization state[END_REF][START_REF] Siyushev | Coherent properties of single rare-earth spin qubits[END_REF][START_REF] Xia | All-optical preparation of coherent dark states of a single rare earth ion spin in a crystal[END_REF], neodymium [START_REF] Zhong | Optically addressing single rare-earth ions in a nanophotonic cavity[END_REF] and erbium [START_REF] Yin | Optical addressing of an individual erbium ion in silicon[END_REF][START_REF] Dibos | Atomic source of single photons in the telecom band[END_REF] ions. It is worth noting that majority of these experiments were not probing ions via their highly coherent but weak 4f-4f optical transitions. Taking Pr and Ce as examples, bright 4f-5d transitions from high lying electronic levels were exploited to enhance the detection sensitivity of individual ions. To directly access the 4f-4f transitions of single ions, the emission rates of individual emitter need to be enhanced. Recent works by [START_REF] Zhong | Optically addressing single rare-earth ions in a nanophotonic cavity[END_REF][START_REF] Dibos | Atomic source of single photons in the telecom band[END_REF] have been successful in that by coupling ions to nanophotonic cavities with sub-wavelength scale dimensions (more discussions on cavities in the next paragraph). Both experiments achieved Purcell enhancement of ions' emission on the order of hundreds, allowing direct optical detection of individual ions with sufficient signal-noise performance using state-of-the-art superconducting nanowire detector technology. These results point at a viable approach to efficiently detect and coherently control individual ions in an on-chip nanophotonic platform.

The weak oscillator strengths of rare-earth emitters require significant enhancement of their atomic absorption cross-sections in order to strongly interact with light at the quantum level. One effective way to achieve such enhancement is to modify the local photonic density of states using nanophotonic structures including waveguides and cavities. Rare-earth doped photonic waveguides have been realized by ion-indiffusion in Tm or Er doped LiNbO 3 [START_REF] Saglamyurek | Broadband waveguide quantum memory for entangled photons[END_REF][START_REF] Sinclair | Properties of a rare-earth-ion-doped waveguide at sub-kelvin temperatures for quantum signal processing[END_REF][START_REF] Askarani | Storage and reemission of heralded telecommunication-wavelength photons using a crystal waveguide[END_REF], laser written channels in Pr:YSO crystals [START_REF] Corrielli | Integrated optical memory based on laser-written waveguides[END_REF], and high-index membranes deposited or transferred on top of doped oxide substrates [START_REF] Dibos | Atomic source of single photons in the telecom band[END_REF][START_REF] Marzban | Observation of photon echoes from evanescently coupled rare-earth ions in a planar waveguide[END_REF][START_REF] Miyazono | Coupling erbium dopants in yttrium orthosilicate to silicon photonic resonators and waveguides[END_REF][START_REF] Ding | Multidimensional purcell effect in an ytterbium-doped ring resonator[END_REF]. These waveguides provide transverse confinement of the optical field interacting with dopants, thus enhancing the coupling strengths. In the cases of ion-indiffused and laser written devices, the rare-earth emitters align with the peak field intensity in the waveguide. Whereas in systems based on high-index materials placed on top of doped crystals, the optical field evanescently couples to the dopants near the material interfaces. All of these designs have demonstrated enhanced lightmatter interactions, and preserved optical coherence of the dopants comparable to bulk crystal hosts. The long coherence despite the fabrication processes highlights the remarkable robustness of rare-earth dopant as a quantum resource. The waveguide scheme has also been ex-tended to cavities by forming a micro-ring [START_REF] Miyazono | Coupling erbium dopants in yttrium orthosilicate to silicon photonic resonators and waveguides[END_REF][START_REF] Ding | Multidimensional purcell effect in an ytterbium-doped ring resonator[END_REF] or patterning photonic crystal lattices [START_REF] Dibos | Atomic source of single photons in the telecom band[END_REF] along the waveguide. For cavity-based system, an important figure of merit is the Q/V ratio where Q is the quality factor and V is the normalized mode volume of the cavity in a unit of (λ/n) 3 [START_REF] Mcauslan | Strong-coupling cavity qed using rare-earth-metal-ion dopants in mono-lithic resonators: What you can do with a weak oscillator[END_REF]. Focused ion beam-milled photonic crystal nanobeam resonator achieving Q/V on the order of 10 4 have been fabricated [START_REF] Zhong | Nanophotonic coherent light-matter interfaces based on rare-earth doped crystals[END_REF][START_REF] Zhong | High quality factor nanophotonic resonators in bulk rare-earth doped crystals[END_REF], which enabled first nanophotonic quantum memories [START_REF] Zhong | Nanophotonic rare-earth quantum memory with optically controlled retrieval[END_REF][START_REF] Zhong | Interfacing broadband photonic qubits to a cavityprotected rare-earth ensemble[END_REF] based on mesoscopic ensemble of dopants and coherent optical addressing of individual ions. The photonic crystal cavities were proven effective to strongly enhance the spontaneous emission rate of rare-earth ions via the Purcell effect [START_REF] Zhong | Optically addressing single rare-earth ions in a nanophotonic cavity[END_REF][START_REF] Dibos | Atomic source of single photons in the telecom band[END_REF]. When the enhancement is sufficient large compared to other optical dephasing processes of the emitter, radiatively-limited single photon emission from individual ions becomes feasible and has been demonstrated recently [START_REF] Zhong | Optically addressing single rare-earth ions in a nanophotonic cavity[END_REF]. Besides waveguides and cavities, other techniques for controlling the rare-earth emission rates were also proposed. One example is a hybrid scheme in which the rare-earths are coupled to highly optically absorbing materials such as graphene via near field interactions [START_REF] Tielrooij | Electrical control of optical emitter relaxation pathways enabled by graphene[END_REF]. FIG. 1: Evolving material platforms for rare-earth quantum devices from macroscopic bulk crystals (Reprinted from [START_REF] Tittel | Photon-echo quantum memory in solid state systems[END_REF]) to low-dimensional materials including doped nanocrystals (Reprinted from [START_REF] Zhang | Transparent Er3+-Doped Y2O3 Ceramics with Long Optical Coherence Lifetime[END_REF]) and sub-micron thick films.

RECENT PROGRESS TOWARDS LOW-DIMENSIONAL RARE-EARTH DOPED MATERIALS

Majority of rare-earth quantum photonic devices demonstrated to date have been based on bulk, single crystal doped materials. These macroscopic bulk matrices, while having superb crystalline qualities, remains difficult to grow and lacks the essential prospect for scalable fabrication desired by future quantum technologies. Perhaps most scalable platforms for on-chip device integration are submicron thick membranes, on which large arrays of devices can be lithographically patterned and etched. High quality thin membranes of rare-earth doped materials have not been commonly available due to their complicated chemical compositions and difficult growth processes. Most promising synthesis methods so far are atomic layer deposition and molecular beam epitaxy, the latest development of which will be discussed in the next section. Another class of doped materials promising for nanophotonic applications is nanocrystal and polycrystalline ceramics. These powder form materials, once believed less promising for quantum applications, have gain significant tractions due to recent reports of long coherence even though the size of these nanocrystals is as small as 60nm [START_REF] Perrot | Narrow Optical Homogeneous Linewidths in Rare Earth Doped Nanocrystals[END_REF]. The rare-earth doped nanocrystals are promising systems for single ion detection and control since they minimize background signal and can allow for higher Rabi frequencies in microscopy setups [START_REF] Utikal | Spectroscopic detection and state preparation of a single praseodymium ion in a crystal[END_REF][START_REF] Perrot | Narrow Optical Homogeneous Linewidths in Rare Earth Doped Nanocrystals[END_REF][START_REF] Kolesov | Optical detection of a single rare-earth ion in a crystal[END_REF]. They can also be inserted into high quality factor micro-cavities to enhance light-atom interactions, increasing emission rates through the Purcell effect and collection efficiencies [START_REF] Casabone | Cavity-enhanced spectroscopy of a few-ion ensemble in Eu3+:Y2O3[END_REF]. Perhaps the biggest advantage of nanocrystals is their relatively easy synthesis methods. Compared to single crystal bulk materials or epitaxial films, many nanocrystals can be grown with commercially available raw materials and in-house processing capabilities. Once this technology is matured to show consistent, robust coherence, the rare-earth doped nanocrystals can find widespread adoption as practical quantum technologies for sensing and communications.

Low dimensional host matrix

Nanocrystals and polycrystalline ceramic materials

A first approach to obtain these materials is to use chemical methods, the so-called bottom-up strategy. This has been first used to produce Eu 3+ :Y 2 O 3 micro-size particles with 60 nm crystalline domains by a solvothermal method [START_REF] Perrot | Narrow Optical Homogeneous Linewidths in Rare Earth Doped Nanocrystals[END_REF][START_REF] Yang | Y2O3: Eu3+Microspheres: Solvothermal Synthesis and Luminescence Properties[END_REF]. These particles, as most of those studied so far by high-resolution and coherent spectroscopy, are polycrystalline, i.e. each particle is made of several crystallites. Optical properties were determined on samples in the form of a powder. Photon echoes, well known for providing an accurate measurement of homogeneous linewidths in transparent materials [START_REF] Abella | Photon echoes[END_REF][START_REF] Meltzer | Line Broadening Mechanisms and Their Measurement[END_REF], can also be observed in powders that strongly scatter light as first demonstrated by Beaudoux et al. [START_REF] Beaudoux | Emission of photon echoes in a strongly scattering medium[END_REF] and recently theoretically analyzed [START_REF] Pierrat | Photon echoes in strongly scattering media: A diagrammatic approach[END_REF]. Application of this technique to the above particles led to a homogeneous linewidth for Eu 3+ 7 F 0 → 5 D 0 transition at 581 nm of 86 kHz at 1.3 K [START_REF] Perrot | Narrow Optical Homogeneous Linewidths in Rare Earth Doped Nanocrystals[END_REF]. This was about one to two orders of magnitude lower than previous results obtained on Eu:Y 2 O 3 nanocrystals ( [START_REF] Meltzer | Electron-phonon interactions in insulating nanoparticles: Eu2O3[END_REF]). It is however much larger than values obtained in the best single crystals (300 Hz [START_REF] Thiel | Cone in Quantum Light Matter Interaction in Solid State Systems[END_REF]) or transparent ceramics (3.5 kHz) [START_REF] Kunkel | Dephasing mechanisms of optical transitions in rare-earth-doped transparent ceramics[END_REF], although it shoud be noted that considerable dispersion of linewidths is found in single crystals grown by different methods [START_REF] Flinn | Sampledependent optical dephasing in bulk crystalline samples of Y2O3:Eu 3+[END_REF].

Light scattering by particles introduces losses that are very detrimental to cavities quality factor, and should be as small as possible. The particle diameter d is a major factor to consider since the scattering cross-section varies as d 6 [START_REF] Casabone | Cavity-enhanced spectroscopy of a few-ion ensemble in Eu3+:Y2O3[END_REF]. Moreover, to be able to use isolated particles, aggregation should be avoided. This was made possible using homogeneous precipitation, a technique that lead to low particle size dispersion and spherical shapes [START_REF] Sohn | Synthesis and characterization of near-monodisperse yttria particles by homogeneous precipitation method[END_REF].

To obtain Y 2 O 3 cubic phase, high temperature annealing of the precursor hydrocarbonates particles is required. It was shown that treatment at 1200 • C reduced inhomogeneous linewidths down to ≈10 GHz for 0.5% Eu 3+ doping, a value comparable to bulk ones [START_REF] De Oliveira Lima | Influence of defects on sub-Å optical linewidths in Eu3+:Y2O3 particles[END_REF]. This was observed for particle and crystallite sizes in the ranges 150-540 nm and 90-120 nm respectively, suggesting that the decrease in linewidths was due to a decrease in defects, such as oxygen vacancies, in the volume of the nanocrystals. On particles of 400 nm diameter (130 nm crystallite diameter), an extensive study of homogeneous linewidths Γ h as a function of temperature, magnetic field and time was performed by 2 and 3-pulse photon echoes and spectral holeburning [START_REF] Bartholomew | Optical Line Width Broadening Mechanisms at the 10 kHz Level in Eu3+:Y2O3 Nanoparticles[END_REF]. A surprising result was that magnetic fields up to 2.5 T had no effect on Γ h , suggesting that dephasing from magnetic defects or impurities was not the main contribution to Γ h . Indeed, it was concluded that for temperatures < 12 K, Γ h was due to interactions with two-level systems (TLS) and a temperature independent dephasing. While TLS effect has been previously identified in Y 2 O 3 bulk crystals and nanoparticles [START_REF] Perrot | Narrow Optical Homogeneous Linewidths in Rare Earth Doped Nanocrystals[END_REF][START_REF] Flinn | Sampledependent optical dephasing in bulk crystalline samples of Y2O3:Eu 3+[END_REF], it was not the case for the latter process. Based on a simple model, it was suggested that it can be due to electric field noise created by surface charges, a mechanism already observed in NV -centers in diamond for example [START_REF] Kim | Decoherence of Near-Surface Nitrogen-Vacancy Centers Due to Electric Field Noise[END_REF]. Importantly, no sign of interaction between Eu 3+ and a modified phonon density of states, related to the crystallite sizes [START_REF] Meltzer | Electron-phonon interactions in insulating nanoparticles: Eu2O3[END_REF], was observed. This indicates that there is no intrinsic size-related contribution, which would be very difficult to overcome, to the homogeneous linewidth of the studied particles. The narrowest homogeneous linewidths reported were 45 kHz at 1.3 K with a broadening of ≈100 kHz over 1s [START_REF] Bartholomew | Optical Line Width Broadening Mechanisms at the 10 kHz Level in Eu3+:Y2O3 Nanoparticles[END_REF]. Narrow linewidths were also demonstrated in smaller particles. Because of the required high temperature annealing that promote sintering, it can be challenging to directly obtain small particles that stay non-aggregated. An alternative approach was proposed in which large particles are reduced in size by wet chemical etching in a weak acid. Starting from dispersed and well crystallized particles of 450 nm diameter, particles down to 150 nm were obtained showing homogeneous linewidths of 34 kHz at 1.4 K, the smallest value reported so far for a nano-material [START_REF] Liu | Controlled size reduction of rare earth doped nanoparticles for optical quantum technologies[END_REF].

The limited broadening induced by etching (≈ 10 kHz), while the particle size is reduced by 65% but the crystallite size remains unchanged, suggests that the latter is the relevant parameter for dephasing processes (Fig. 2). This was further supported by showing that etching proceeds by opening grain boundaries between crystallites in the particles. Single crystalline particles can therefore be produced, which may be useful when, for instance, a magnetic field should be applied in a specific direction or well defined Rabi frequencies are needed.

To assert the potential of rare-earth doped nanoparticles as quantum memories or processors, it is necessary to measure spin coherence lifetimes T 2 since qubits are usually defined within the energy level structure of ground state spins [START_REF] Walther | High-fidelity readout scheme for rare-earth solid-state quantum computing[END_REF]. This was performed on 400 nm Eu 3+ :Y 2 O 3 particles using an all-optical technique for spin coherence manipulation and spin echo detection [START_REF] Hartmann | Photon, spin, and Raman echoes[END_REF][START_REF] Ham | Spin coherence excitation and rephasing with optically shelved atoms[END_REF][START_REF] Guillot-Noël | Hyperfine structure and hyperfine coherent properties of praseodymium in single-crystalline La2(WO4)3 by hole-burning and photon-echo techniques[END_REF]. Powders cooled to 5 K showed nuclear spin echo signals for 151 Eu 3+ with signal to noise ratio of about 10, proving that the technique actually works in a scattering medium, and sufficient for T 2 measurements [START_REF] Serrano | All-optical control of long-lived nuclear spins in rare-earth doped nanoparticles[END_REF]. Spin coherence lifetimes of 1.3 ms were recorded, increasing to about 3 ms under a 9 mT magnetic field (Fig. 3). These values, one order of magnitude lower than in transparent ceramics [START_REF] Karlsson | Nuclear spin coherence properties of 151Eu3+ and 153Eu3+ in a Y2O3 transparent ceramic[END_REF] but still the longest reported for any nanoparticle with optically addressable spins, provide valuable insight into dephasing mechanisms. First, the strong dependance of T 2 for very weak magnetic field suggests that the main dephasing is due to magnetic dipole-dipole interaction with electron spins carried by defects or impurities. Indeed, when the Zeeman interactions become larger than the dipole-dipole interaction, the effect of the latter is reduced. This magnetic noise also affect the optical transition, which can be expected to broaden to a similar level than the spin transition, i.e. ≈250 Hz, based on the excited and ground state gyromagnetic ratios in other hosts [START_REF] Longdell | Characterization of the hyperfine interaction in europiumdoped yttrium orthosilicate and europium chloride hexahydrate[END_REF][START_REF] Ahlefeldt | Minimizing Zeeman sensitivity on optical and hyperfine transitions in EuCl3 6H2O to extend coherence times[END_REF][START_REF] Cruzeiro | Characterization of the hyperfine interaction of the excited 5 D0 state of Eu 3+ :Y2SiO5[END_REF]. This is much too small to account for the optical linewidth of 45 kHz and supports the hypothesis of electric field perturbations as the main optical dephasing process. This is also in agree- ment with Stark coefficient of the optical transition [START_REF] Macfarlane | Optical Stark spectroscopy of solids[END_REF] being 4 to 5 orders of magnitude larger than the spin one [START_REF] Macfarlane | Optical Measurement of the Effect of Electric Fields on the Nuclear Spin Coherence of Rare-Earth Ions in Solids[END_REF]. Coherent spin control was further demonstrated by an all-optical dynamical decoupling (DD) technique that resulted in T 2 up to 8 ms, limited by disorder and light scattering in the powders that prevent pulses with accurate area to be applied [START_REF] Serrano | All-optical control of long-lived nuclear spins in rare-earth doped nanoparticles[END_REF]. This highlights the interest of using single particles with single crystalline structure for optimal decoherence control. It was also shown that spin state phase is well preserved in echo and DD sequences, which is a requirement for quantum storage. Apart from experiment on powders, spectroscopy of single Eu 3+ :Y 2 O 3 nanoparticles has been studied using a fiber micro-cavity [START_REF] Casabone | Cavity-enhanced spectroscopy of a few-ion ensemble in Eu3+:Y2O3[END_REF]. In this setup, the particles are placed on a flat mirror which faces a concave profile lasermachined in an optical fiber. It was found that Eu 3+ in 90 nm particles had inhomogeneous linewidths identical to what had been measured on powders, and that the cavity allowed detecting about 10 ions taking advantage of the Purcell effect.

Similar results have also been reported in Pr 3+ :Y 2 O 3 nanoparticles [D. Serrano et al., in preparation], obtained by homogeneous precipitation, for optical and spin transitions. Compared to Eu 3+ , Pr 3+ has generally stronger optical transitions and more favorable branching ratios that could be beneficial for coupling to a cavity. Hyperfine splittings are however smaller [START_REF] Goldner | Rare Earth-Doped Crystals for Quantum Information Processing[END_REF], which limits the minimal pulse duration that can be applied to a specific transition between hyperfine levels and therefore operation fidelity. Homogeneous linewidths of between 100 and 300 kHz were observed for the 3 H 4 → 1 D 2 transition at 619 nm and particles between 400 and 150 nm (crystallite size 120 and 80 nm). Interestingly, spin coherence lifetimes reached nearly 1 ms, longer than reported so far in bulk Pr 3+ doped crystals, like YSO [START_REF] Fraval | Method of Extending Hyperfine Coherence Times in Pr3+:Y2SiO5[END_REF][START_REF] Goldner | Long coherence lifetime and electromagnetically induced transparency in a highly-spinconcentrated solid[END_REF]. This could be explained by lower gyromagnetic ratios for Pr 3+ in Y 2 O 3 according to calculations based on crystal field modeling of electronic levels [START_REF] Guillot-Noël | Calculation and analysis of hyperfine and quadrupole interactions in praseodymium-doped La2(WO4)3[END_REF][START_REF] Lovrić | Hyperfine characterization and spin coherence lifetime extension in Pr 3+ :La2(WO4)3[END_REF].

Pioneering experiments on high resolution spectroscopy of rare earth doped nanomaterials used a topdown approach, laser ablation, to produce Y 2 O 3 nanocrystals with a monoclinic phase [START_REF] Meltzer | Electron-phonon interactions in insulating nanoparticles: Eu2O3[END_REF][START_REF] Meltzer | Evidence for longrange interactions between rare-earth impurity ions in nanocrystals embedded in amorphous matrices with the two-level systems of the matrix[END_REF]. Spectral hole burning was used to probe Eu 3+ homogeneous linewidths, which were found in the several MHz range for crystals of 10s of nm, likely to be highly agglomerated. Interactions with modified phonon density of states were proposed to explain the broadening.

Other top-down approaches have also been investigated, in which a single crystal is ground to nanoparticles. In Pr 3+ :YSO, optical homogeneous linewidths of a few MHz for single ions were reported for particles between 100s of nm and a few microns, as well as spectral diffusion of 10 MHz over minutes for the larger particles [START_REF] Utikal | Spectroscopic detection and state preparation of a single praseodymium ion in a crystal[END_REF][START_REF] Eichhammer | Spectroscopic detection of single Pr3+ ions on the 3H4 1D2 transition[END_REF]. Strain induced by grinding and associated TLS could be the source of this significant broadening compared to the 80 kHz linewidth in the bulk crystal. Such mechanisms were also put forward in studies on Tm 3+ doped crystals ground to micron size powders and probed by spectral hole burning [START_REF] Lutz | Effects of fabrication methods on spin relaxation and crystallite quality in Tm-doped powders studied using spectral hole burning[END_REF]. The main effect of strain was observed on spectral hole lifetimes that strongly shortened under milling (several hours down to 10s of minutes) and could be partially recovered with high-temperature annealing. Homogeneous linewidths, that can be estimated from hole widths, followed the same trends although the already broad values of several MHz for bulk crystals do not allow studying very narrow linewidths. Milled, micron-size powders of Er 3+ :LiNbO 3 were also studied by spectral hole burning and photon echoes, and showed that even low energy milling impacts kHz range homogeneous linewidths [START_REF] Lutz | Effects of mechanical processing and annealing on optical coherence properties of Er3+:LiNbO3 powders[END_REF]. For powders with 1-100 µm crystals, annealing was able to recover close to bulk properties.

Another class of polycrystalline materials are transparent ceramics, which are usually obtained by sintering cubic phase nanoparticles under high temperature and high isostatic pressure (HIP). This results in the growth of micron size crystals with random orientation and a strong decrease of the material porosity, ultimately leading to negligible scattering losses [START_REF] Ikesue | Synthesis and Performance of Advanced Ceramic Lasers[END_REF]. Thanks to their high flexibility in composition, shape, and ability to build composite materials, transparent ceramics are thus an alternative to single crystals in a variety of photonic applications like lasers or scintillators, and could also allow obtaining new high performance materials in quantum technologies [START_REF] Ikesue | Ceramic laser materials[END_REF][START_REF] Liu | Towards Bright and Fast Lu3Al5O12:Ce,Mg Optical Ceramics Scintillators[END_REF][START_REF] Ferrier | Narrow inhomogeneous and homogeneous optical linewidths in a rare earth doped transparent ceramic[END_REF]. Eu 3+ in Y 2 O 3 transparent ceramic properties have been investigated in details, as a function of doping concentration, post-synthesis treatment and co-doping with sintering aids. Optical transition inhomogeneous linewidths down to 9 GHz were observed for 1000 ppm Eu 3+ doping [START_REF] Ferrier | Narrow inhomogeneous and homogeneous optical linewidths in a rare earth doped transparent ceramic[END_REF][START_REF] Kunkel | Rareearth doped transparent ceramics for spectral filtering and quantum information processing[END_REF], comparable to bulk values, while co-doping with Zr 4+ (5000 ppm), a common sintering aid, increased the linewidth to 100 GHz [START_REF] Kunkel | High-Resolution Optical Line Width Measurements as a Material Characterization Tool[END_REF]. Homogeneous linewidths were also studied as a function of temperature, magnetic field and delay in 3 pulse experiments to reveal spectral diffusion processes [START_REF] Kunkel | Dephasing mechanisms of optical transitions in rare-earth-doped transparent ceramics[END_REF]. This revealed complex behaviors, that could be in some cases analyzed by taking into account the dependence of perturbation correlation times on temperature and magnetic field. The narrowest homogeneous lines were 3.2 kHz for ceramics post-annealed under air and are due to several contributions, including TLS and possibly magnetic perturbations. Dephasing was more pronounced in samples without post-annealing, underlining the role of oxygen in controlling defects concentration and nature. This was further investigated by comparing high resolution spectroscopy with photoluminescence, electron paramagnetic resonance and thermo-luminescence, showing that defects related to oxygen vacancies, like F centers, seem to be correlated to optical linewidth [START_REF] Kunkel | High-Resolution Optical Line Width Measurements as a Material Characterization Tool[END_REF]. Nuclear spin properties of 151 Eu 3+ and 153 Eu 3+ were also measured by the all-optical technique described above and showed remarkably long coherence lifetimes, up to 16 ms under a 3 mT magnetic field [START_REF] Karlsson | Nuclear spin coherence properties of 151Eu3+ and 153Eu3+ in a Y2O3 transparent ceramic[END_REF], close to values measured in Eu 3+ :Y 2 SiO 5 single crystal [START_REF] Alexander | Measurement of the ground-state hyperfine coherence time of 151 Eu 3+ :Y2SiO5[END_REF][START_REF] Arcangeli | Spectroscopy and coherence lifetime extension of hyperfine transitions in 151 Eu 3+ :Y2SiO5[END_REF]. As in the nanoparticles, spin transitions are much less sensitive to perturbations than optical transitions, pointing to a possible role of electric noise.

Very narrow inhomogeneous lines, as low as 442 MHz, have also be observed in Er 3+ in Y 2 O 3 transparent ceramics. This indicates the very high crystalline quality of the samples. Homogeneous linewidths measured at 2.5 K and under 0.65 T magnetic field vary between 10 and 50 kHz, depending on Er 3+ concentration. In the case of paramagnetic ions, interaction between ground state spins themselves can cause strong dephasing, which can explain the observed dependence on Er 3+ concentration [START_REF] Böttger | Optical decoherence and spectral diffusion at 1.5µm in Er 3+ :Y2SiO5 versus magnetic field, temperature, and Er 3+ concentration[END_REF]. At low Er 3+ concentration, temperature dependence of Γ h under magnetic field suggests that TLS contribution dominates the homogeneous linewidth. This suggests a low concentration in defects or impurities carrying electron spins, that could induce significant broadening by coupling to Er 3+ electron spin.

Rare-earth doped films

Atomic layer deposition (ALD) is a common technique for obtaining Y 2 O 3 films with well controlled thickness and good crystalline quality [START_REF] Miikkulainen | Crystallinity of inorganic films grown by atomic layer deposition: Overview and general trends[END_REF]. A study of Eu 3+ and Er 3+ doped ALD films in the context of applications in quantum technologies was reported [START_REF] Scarafagio | Ultrathin Eu-and Er-Doped Y2O3 Films with Optimized Optical Properties for Quantum Technologies[END_REF]. It showed that optimized growth followed by high temperature annealing resulted in Eu 3+ luminescence spectra corresponding mainly to the Y 2 O 3 cubic phase. In a 100 nm thick sample doped with 5% Eu 3+ and deposited on (100) silicon wafers, 5 D 0 level lifetimes reached 0.8 ms, close to bulk values. The 7 F 0 → 5 D 0 inhomogeneous linewidth was about 200 GHz, a factor of 2 larger than expected from studies on transparent ceramics [START_REF] Kunkel | Rareearth doped transparent ceramics for spectral filtering and quantum information processing[END_REF]. This could be due to stress in the film induced by differences in thermal expansion coefficients between silicon and Y 2 O 3 , which would also explain the shift of the line with the annealing temperature. It may also stem from the presence of defects at grain boundaries since grain size in such polycrystalline thin films is of the order of 30 nm. Ultra-thin films, which are of interest for dopant localization and hybrid devices [START_REF] Tielrooij | Electrical control of optical emitter relaxation pathways enabled by graphene[END_REF], have also been investigated [START_REF] Scarafagio | Ultrathin Eu-and Er-Doped Y2O3 Films with Optimized Optical Properties for Quantum Technologies[END_REF]. In films as thin as 7 nm, Er 3+ luminescence at 1.5 µm showed decay times of about 1.5 ms, a long value for Er 3+ nanomaterials, although still shorter than the 7.7 ms of bulk Er 3+ :Y 2 O 3 . Further studies on coherent properties are needed to assert the potential of these systems for quantum technologies, which may require specific setups such as waveguides to deal with film/substrate geometries and the low number of active ions in very thin layers [START_REF] Marzban | Observation of Photon Echoes From Evanescently Coupled Rare-Earth Ions in a Planar Waveguide[END_REF].

The simple chemical composition and stable cubic phase of Y 2 O 3 makes it possible to grow wafer-scale, epitaxial thin films. Indeed, single crystal, epitaxial Y 2 O 3 have been synthesized using molecular beam epitaxy (MBE) on silicon wafers for high-k dielectric gate in MOSFET [START_REF] Guha | Atomic beam deposition of lanthanum-and yttrium-based oxide thin films for gate dielectrics[END_REF][START_REF] Guha | Latticematched, epitaxial, silicon-insulating lanthanum yttrium oxide heterostructures[END_REF]. Recently, this process has been improved to achieve exceptional qualities of Y 2 O 3 films doped with Er 3+ [START_REF] Singh | Characterization of epitaxially grown erbium doped y2o3 for quantum optics applications[END_REF]. Controls of doping concentrations in the range of 1 to 100 parts per millions can be achieved. The thickness of the film up to 1.5 µm is also controllable with a precision down to a monolayer of atoms. The growth occurs via a two-dimensional layer-by-layer growth mode, resulting in smooth, uniform films, which offers several distinct advantages: first, the epitaxial film has significantly less optical loss and suggests better coherence for dopants; second, it allows atomic precision placement of rare-earth dopants via delta doping; third, the wafer-size film allows scalable top-down device patterning using standard lithography and etching techniques. The photoluminescence of Er 3+ was measured, which clearly showed two distinct transitions corresponding to two crystal symmetry sites: C2 site at 1532 nm and C3i at 1542 nm. The wavelengths of the peaks agree with that in bulk Er 3+ :Y 2 O 3 [START_REF] Reinemer | Optical characterization of perturned sites in rare earth doped oxide crystals[END_REF], confirming the correct Y site occupancies. Optical lifetimes and ground-state electron paramagnetic resonance were measured [START_REF] Singh | Characterization of epitaxially grown erbium doped y2o3 for quantum optics applications[END_REF]. The photoluminescence decay gives a T 1 of 8.7 ms for C2 site, in excellent agreement with bulk Er 3+ :Y 2 O 3 measurement [START_REF] Reinemer | Optical characterization of perturned sites in rare earth doped oxide crystals[END_REF]. Optical and spin coherence are next experiments to fully characterize this epitaxial film material.

Hybrid rare-earth materials

Other hybrid architectures have been considered, based on thin films or nanoparticles. A first example is the electrical control of coupling between rare-earth ions and a graphene layer [START_REF] Tielrooij | Electrical control of optical emitter relaxation pathways enabled by graphene[END_REF]. Depending on the Fermi level in graphene, rare-earth can decay through photon emission, non-radiatively by creation of electron-hole pairs in the graphene layer or by excitation of graphene plasmons. This process however is only efficient at distances below 10 nm and is therefore maximized for very thin rare-earth doped films. This effect was observed in a Er 3+ :Y 2 O 3 thin film of about 60 nm thickness obtained by the metal organic decomposition method on a Si substrate [START_REF] Andriamiadamanana | Erbium-doped yttria thin films prepared by metal organic decomposition for up-conversion[END_REF]. A graphene layer was placed on top of the film and covered by a transparent solid polymer electrolyte to allow tuning of graphene Fermi level by a gate voltage. Comparing Er 3+ emission below or outside the graphene sheet as a function of the voltage applied to the device, it was possible to identify a regime of low emission in which Er 3+ is quenched by the graphene, a stronger emission when electron hole pair creation becomes reduced and finally a low emission again when Er 3+ is able to excite plasmon in the graphene layer [START_REF] Tielrooij | Electrical control of optical emitter relaxation pathways enabled by graphene[END_REF]. This demonstrates a unique capability of electrically controlling the local density of optical states of an emitter. The on-demand coupling of emitters with plasmons opens promising possibilities for quantum optoelectronics, which could take advantage of the coherence properties of rare-earth transitions. This will require large couplings that could be obtained with thinner films, provided that rare-earth properties can be preserved (see above, [START_REF] Scarafagio | Ultrathin Eu-and Er-Doped Y2O3 Films with Optimized Optical Properties for Quantum Technologies[END_REF]).

A second example is the incorporation of rare-earth ions in diamond, one of the major material for quantum technologies. Such a system could combine the excellent properties of diamond such as high purity and crystalline quality with the narrow lines of rare-earth ions. More-over, diamond can host a variety of defects with strong emissions and/or carrying spins with long coherence lifetimes that could also interact with rare-earths in hybrid quantum systems. Incorporating rare-earth ions, which have a large ionic radius, into the diamond lattice is however difficult. One approach is to chemically self-assemble rare-earth complexes on bulk or nano-diamonds before growing a new diamond layer by CVD [START_REF] Magyar | Synthesis of luminescent europium defectsin diamond[END_REF][START_REF] Vanpoucke | Can europium atoms form luminescent centres in diamond A combined theoreticalexperimental study[END_REF]. Another one consist in implanting rare-earth into bulk or nano diamond [START_REF] Cajzl | Erbium ion implantation into diamond -measurement and modelling of the crystal structure[END_REF][START_REF] Cajzl | Co-implantation of Er and Yb ions into single-crystalline and nano-crystalline diamond[END_REF]. Rare-earth ions luminescence has been observed with these different methods, with long lifetimes in the case of Eu 3+ ions [START_REF] Magyar | Synthesis of luminescent europium defectsin diamond[END_REF]. The lack of well resolved crystal field structure in the spectra however prevents a detailed analysis of the rare-earth local environment and comparison with simulations [START_REF] Magyar | Synthesis of luminescent europium defectsin diamond[END_REF][START_REF] Vanpoucke | Can europium atoms form luminescent centres in diamond A combined theoreticalexperimental study[END_REF][START_REF] Cajzl | Erbium ion implantation into diamond -measurement and modelling of the crystal structure[END_REF], which could unambiguously confirm their incorporation into diamond, still under debate [START_REF] Vanpoucke | Can europium atoms form luminescent centres in diamond A combined theoreticalexperimental study[END_REF]. High-resolution or coherent spectroscopy has not yet been reported on these systems.

OUTLOOK

Future advances in rare-earth nanophotonics hinges on breakthroughs in material synthesis, control and engineering. A set of material properties forms the key metrics for guiding this material development and optimization. For nanocrystals, the particle size should be small enough for precise localization of the emitter with respective to the optical field it couples to, while large enough to suppress unwanted decoherence due to proximity to particle surfaces. For thin films, well controlled thickness and surface roughness are important for high quality photonic devices. The dopant spectroscopic properties, including narrow optical and spin inhomogeneous linewidths, optical lifetimes, homogeneous linewidth, long spin T 1 and T 2 , will continue to be the qualifying parameters that drive the growth optimization process. Some of these parameters are less restrictive than others for specific applications. For instance, for single ion devices, inhomogeneous broadening is of secondary concern as long as individual ion exhibits longlived coherence.

One decisively enabling capabilities for rare-earth nanophotonics is the deterministic control of doping concentration and placement in host matrices. The former is notoriously difficult in bulk, yttrium-based doped crystals, because the background doping of all sorts of rareearth elements is at ∼ppm level in the raw yttrium material. This limitation might be overcome with less common growth techniques such as floating zone growth or molecular beam epitaxy in which elements of rare-earths and yttrium can be better separated by their disparate vapor pressures or other physical properties. In terms of dopant placement, ALD and MBE offers powerful deltadoping options to precisely control the distance of dopant to material interfaces, and in principle could result in a mono layer thickness of the doped material. The mastery of 2D film growth technique will further spur new ways to control dopant placement in 3D, for instance, by combining delta-doping with aperture-masked doping at prescribed locations on the substrate. With nanocrystals containing one or a few rare-earth ions, its placement can be controlled by functionalizing the substrate surface and chemically docking the nanocrystal to the desired location.

Once the above metrics and capabilities are realized, they open up powerful tools to nanoscopically control and tailor interactions of rare-earth ions with other quantum degrees of freedoms. With regard to nanophotonics, such controls will enable unprecedented enhancement of light-matter interaction at single rare-earth ion level, and potentially unlock new coupling modalities such as rare-earth-phonon, rare-earth-electric field interactions at varying length and energy scales. Together, these new rare-earth materials and accompanying processing techniques present an exciting prospect of engineering highly coherent quantum systems with on demand functionality and performance.

FIG. 2 :

 2 FIG.2: Optical echo decays in Eu 3+ :Y2O3 nanoparticles before and after chemical etching. Despite the size reduction from 450 nm to 150 nm, only limited broadening (≈ 10 kHz) is observed. Inset: 200 nm etched particles. Reproduced from Ref.[START_REF] Liu | Controlled size reduction of rare earth doped nanoparticles for optical quantum technologies[END_REF] with permission from the Royal Society of Chemistry.

FIG. 3 :

 3 FIG. 3:Nuclear spin homogeneous linewidth of 151 Eu 3+ :Y2O3 (±1/2 → ±3/2 ground state transition) in Y2O3 nanoparticles as a function of external magnetic field. Reproduced from Ref.[START_REF] Serrano | All-optical control of long-lived nuclear spins in rare-earth doped nanoparticles[END_REF], CC-BY-4.0.

3 FIG. 4 :

 34 FIG.4: Photoluminescence of Er 3+ in an epitaxial Y2O3 film grown on silicon using molecular beam epitaxy. Two optical transitions confirm the correct crystal symmetry site occupancies. The inset shows a transmission electron microscope image of the Y2O3 lattice on silicon.

We would like to acknowledge D. Serrano and A. Tallaire for their comments on the manuscript. This project has received funding from the National Science Foundation EAGER award No 1843044, European Union's Horizon 2020 research and innovation programme under grant agreement No 712721 (NanOQTech).


M Businger

A Tiranov

† K T Kaczmarek

S Welinski

‡ Z Zhang

A Ferrier

P Goldner

M Afzelius
email: mikael.afzelius@unige.ch

Optical spin-wave storage in a solid-state hybridized electron-nuclear spin ensemble

Solid-state impurity spins with optical control are currently investigated for quantum networks and repeaters. Among these, rare-earth-ion doped crystals are promising as quantum memories for light, with potentially long storage time, high multimode capacity, and high bandwidth. However, with spins there is often a tradeoff between bandwidth, which favors electronic spin, and memory time, which favors nuclear spins. Here, we present optical storage experiments using highly hybridized electron-nuclear hyperfine states in 171 Yb 3+ :Y2SiO5, where the hybridization can potentially offer both long storage time and high bandwidth. We reach a storage time of 1.2 ms and an optical storage bandwidth of 10 MHz that is currently only limited by the Rabi frequency of the optical control pulses. The memory efficiency in this proof-of-principle demonstration was about 3%. The experiment constitutes the first optical storage using spin states in any rare-earth ion with electronic spin. These results pave the way for rare-earth based quantum memories with high bandwidth, long storage time and high multimode capacity, a key resource for quantum repeaters.

Solid-state impurity spins play an increasingly important role in quantum information technologies, with applications in communication, computing and sensing [1][2][3]. In quantum communication, solid-state spins that can be interfaced with photons are promising candidates for nodes in quantum networks and quantum repeaters. In such systems, optical transitions are used to convert quantum information between internal spin states and optical photons, where spins store and possibly process quantum information within the solid [3,4].

There is a current interest in using both the electronic and nuclear degrees of freedom of solid-state spin systems [3,5], with the goal of simultaneously achieving efficient manipulation and long-duration storage. Electronic spins couple strongly to external fields, making them ideal for high-bandwidth operations and highly sensitive sensors. The weaker coupling of the nuclear spins shields them from the environment, allowing longduration quantum storage. However, finding spin systems that simultaneously possess good optical, electronic and nuclear spin properties is challenging. A prominent example is the nitrogen vacancy (NV) center in diamond, where the electron spin of the NV can be coupled to photons [6], while the hyperfine interaction with neighbouring 13 C nuclear spin provides a long-duration memory [5,7]. Similar hybrid electron-nuclear systems are investigated using 31 P phosphor donors in silicon [8,9] and quantum dots [10].

Rare-earth-ion (RE) doped crystals represent another prominent example of solid-state impurities with excellent optical [11] and spin properties [12]. RE crystals have emerged as strong candidates for ensemble-based quantum memories (QMs) [13][14][15][16], and more recently for single ion/spin [17][18][19][20] quantum information processing. For ensemble-based QMs, thus far only RE nuclear spin systems have been used for storing optical pulses using spin states [21][22][23][24], based on the nuclear quadrupole states of either Pr 3+ or Eu 3+ ions. While these non-Kramers RE ions with quenched electronic spin provide excellent memory times, the purely nuclear states limit the memory bandwidth to < 10 MHz. RE ions with non-zero electronic spins, so-called Kramers ions, could potentially provide a solution to the bandwidth limit, provided that long coherence times can be engineered in such electronic spin systems.

Recently, it has been shown that such long spin coherence times can be found in some Kramers RE ions with electron spin S [25][26][27][28] and non zero nuclear spin I by exploiting the hyperfine coupling S • A • I, where A is the hyperfine tensor. In Refs [27,28] it was particularly shown that at zero applied magnetic field, an anisotropic hyperfine interaction leads to strong mixing of S and I, resulting in highly hybridized electron-nuclear states with zero first-order Zeeman (ZEFOZ) effect [29]. Using this feature we showed simultaneous long optical and spin coherence times of T o 2 = 180 µs and T s 2 = 1.5 ms, respectively, in 171 Yb 3+ :Y 2 SiO 5 [28]. We also showed that, while the hybridized states are insensitive at first order to slowly fluctuating magnetic DC fields (at the ZEFOZ point), the magnetic AC transition moment between the hybridized states remains electronic, resulting in high Rabi frequency and fast operations. These results are promising for broadband and long-duration optical quantum memories, but thus far there has been no demonstration of optical storage using spin states in any Kramers RE ion system.

In this Letter we demonstrate an optical memory using the hybridized electron-nuclear states at zero field in 171 Yb 3+ :Y 2 SiO 5 , based on the atomic frequency comb (AFC) memory scheme [30]. We reach a spin storage time of 1.2 ms and a bandwidth of 10 MHz, which is only limited by the current optical Rabi frequency. In addition, we show efficient optical AFC echoes with delays one order of magnitude longer than previously achieved in any Kramers system, which we attribute to reduced superhyperfine coupling in the zero-field ZEFOZ point.

The AFC memory is based on a ∆-periodic structure of highly absorbing peaks within an inhomogeneously broadened optical transition [30]. An input pulse then produces an optical AFC echo, with a delay of 1/∆. The AFC echo process allows high temporal multimode storage, provided that 1/∆ is much longer than the input pulse duration [31]. To achieve on-demand read out, an optical control pulse can be applied before the AFC echo, thereby converting the optical coherence into a spin coherence. This spin-wave memory [24,[32][33][34] is read out by applying another control pulse after a time T S , which results in an output pulse with a storage time of T M = 1/∆ + T S . The spin-wave storage time can be extended to the spin coherence time T s 2 by applying a spin echo sequence [34].

The memory puts specific demands on the atomic system. It requires an excited state coupled to two spin states, a so-called Λ system, where the optical memory bandwidth is ultimately limited by the spin-state energy split [30]. Efficient AFC echoes and long 1/∆ delays require an optically deep and high resolution comb, which in turn requires a long optical coherence time and efficient optical pumping. Finally, long memory lifetime requires a long T s 2 and efficient spin manipulation through microwave (MW) pulses.

The optical transition we use in 171 Yb 3+ :Y 2 SiO 5 connects the lowest crystal field levels in the electronic 2 F 7/2 ground and 2 F 5/2 excited states, at 978.854 nm (in vacuum) for site II in Y 2 SiO 5 [35]. The highly anisotropic hyperfine tensors splits both levels into four non-degenerate hyperfine states at zero applied magnetic field [36], as shown in FIG. 1a. Many different combinations of transitions could be used as the Λ system for optical storage, with corresponding microwave frequencies from 529 to 3026 MHz. Here we focus on the particular Λ system formed by the ν 1 and ν 2 transitions, see FIG. 1a, with a spin transition at ν MW = 655 MHz, which has the required optical and spin coherence times [28]. A crucial first step towards optical spin-wave storage is then to demonstrate efficient optical pumping and optical AFC echoes.

Optical pumping of all hyperfine states requires addressing transitions involving all four hyperfine states. Our setup (FIG. 1d) is based on two lasers (master and slave), where the master is locked to a high-finesse cavity at the frequency of the ν 1 = 306263.0 GHz transition. The slave is locked to the master laser with an offset ν 2 = ν 1 + 0.6547 GHz using an optical phase lock loop (PLL). Additionally the slave laser addresses the ν 3 = ν 2 + 6.2594 GHz and ν 4 = ν 2 + 7.0762 GHz transitions by phase modulation (FIG. 1a andd). The Y 2 SiO 5 crystal is doped with 5 ppm of 171 Yb 3+ , with an isotopic purity of ≈ 95%. The optical inhomogeneous broadening is 1.3 GHz, hence there is clear overlap between optical transitions (FIG. 1b). This issue can be solved by standard pumping sequences for RE crystals, refereed to as class cleaning [37,38]. By class cleaning for 400 ms with all ν 1 -ν 4 frequencies and then state initializing into state |4 g using frequencies ν 2 -ν 4 , we obtain the absorption spectrum in FIG. 1b. The strong optical absorption on the ν 1 transition and the deep holes on the ν 2 -ν 4 transitions are evidence of efficient state initialization.

A measurement of the lifetime of the absorption structure showed that it rethermalizes with two different exponential time constants (FIG. 1c); a faster 36(6) ms decay and a longer 390(55) ms decay. The double decay is due the different relaxation rates between hyperfine states, as also observed in 145 Nd 3+ [39]. The population relaxation lifetimes are significantly longer than in the 10 ppm doped 171 Yb 3+ :Y 2 SiO 5 sample used in Ref. [28], which indicates that the population relaxation is due to flip-flops between ions in different hyperfine states [39][40][41]. A more detailed investigation of optical pumping and hyperfine flip-flop relaxation processes in this material will be presented elsewhere [42].

In a first set of storage experiments the optical AFC echo was studied (FIG. 2), without applying the optical control pulses. The AFC was created on the ν 1 transition, which has the strongest absorption in the Λ system. In particular we measured the AFC echo efficiency η AF C as a function of the AFC delay 1/∆, which depends strongly on the contrast and shape of the comb. The ideal shape of the comb teeth is squarish, with an optimal comb finesse given only by the maximum optical depth [43]. To create such combs we use the optical pumping method presented in Ref. [31], which was specifically designed to create optimal AFCs over a large frequency bandwidth. The AFC bandwidth was set to 20 MHz, and the 60 ms long comb preparation sequence directly followed the state initialization and class cleaning sequences.

An example of an optical AFC echo is shown in FIG. 2a, and the echo efficiencies are plotted in FIG. 2b. At the shortest delay of 1/∆ = 1 µs, the AFC echo reaches a combined storage and retrieval efficiency of η AF C = 24%. The associated comb is shown in FIG. 2c, and it features a high contrast, squarish shape that is close to optimal. The theoretical efficiency for an optimal comb for an optical depth of d = 4 is 32% [43]. But the experimental comb at 1/∆ = 1 µs has a background optical depth of about d 0 ≈ 0.3, which reduces the efficiency to 0.32 exp(-d 0 ) = 24% [44], consistent with the experimental data. Higher efficiencies should be achieved by improving the optical pumping, which requires a lower 171 Yb 3+ concentration to reduce the flip-flop rate [40], and using optical cavities [45,46].

The decay of the AFC echo as a function of 1/∆ is due to a reduction in contrast and a small deviation in shape from the optimal square one, as exemplified for 1/∆ = 5 µs in FIG. 2c. The decay curve can be fitted using the formula exp -4/(∆ • T 2 ) , where ideally the T 2 is the optical coherence time [31]. However, shorter T 2 are typically obtained due to technical noise such as laser coherence time limitations and cryostat vibrations. The data in FIG. 2b shows a double exponential decay with T 2 = 15 and 165 µs, respectively, while the optical coherence time in this material we measured to be as long as 600 µs with photon echoes. We believe the laser spectrum to be the main limitation to the observed decay constants.

The timescale of the AFC delays shown in FIG. 2b is up to two orders of magnitude longer than previously achieved delays (0.1 -1 µs) in RE ions with electronic spin degrees of freedom, such as in Nd 3+ [14,47] or Er 3+ [48] doped crystals. Those short decays have been explained [47,48] by invoking superhyperfine interaction between the RE electronic spin and the nuclear spin of Y 3+ ions in the host, which causes spectral nuclear spinflip sidebands [49] and effectively enlarges the homoge-neous linewidth of the RE ion. The efficient AFC echoes at long delays suggest that superhyperfine interaction is strongly suppressed at the zero-field ZEFOZ point. The delays are similar to those achieved in the purely nuclear RE spin systems Eu 3+ [31] and Pr 3+ [50], which yet again highlights the interest of the hybridization of the electronic and nuclear spins at zero field in 171 Yb 3+ :Y 2 SiO 5 .

We now turn to the optical spin-wave storage experiments, which in addition require coherent manipulation of the optical and microwave transitions, see FIG. 3. Optically one requires an efficient population transfer of the control pulse on the ν 2 transition, see FIG. 1a, which has a dipole moment about 3 times weaker than the ν 1 transition (see Supplemental materials [START_REF]See Supplemental Material at URL for additional experimental data, a[END_REF]). On the ν 1 transition we achieved an optical Rabi frequency of Ω O = 2.0 MHz, see FIG. 3a, which gives an estimated Rabi frequency of Ω O = 0.6 MHz on ν 2 . The resulting π-pulse duration of 0.8 µs implies an efficient population transfer over less than its Fourier limited bandwidth of 1.2 MHz, clearly insufficient for the 10 MHz bandwidth of the 100 ns input pulse. To increase the bandwidth over which efficient population transfer can be achieved, one can employ longer frequency-chirped adiabatic pulses [START_REF] Minář | [END_REF]. In this case we employ hyperbolic-square-hyperbolic (HSH) pulses [53] of duration 5 µs, chirped over 10 MHz, which resulted in a measured population transfer efficiency of about 90% per HSH pulse. To reach larger memory bandwidths would require higher control pulse Rabi frequencies (see Supplemental materials [START_REF]See Supplemental Material at URL for additional experimental data, a[END_REF]).

On the microwave transition ν MW , the stored spin coherence induced by the optical control pulse will dephase with the inverse of the inhomogeneous spin linewidth 1/Γ MW . We measured a spin linewidth of Γ MW = 0.7 MHz (see Supplemental materials [START_REF]See Supplemental Material at URL for additional experimental data, a[END_REF]), which practically makes it impossible to read out the memory given the duration of the optical control pulse. But the spin coherence can be rephased with a spin echo sequence [34,54], in this case a pair of MW π pulses greatly extends the storage time. Using a simple coil wrapped around the crystal we reached a Rabi frequency of Ω MW = 0.65 MHz, see FIG. 3b. We note that with respect to pure nuclear non-Kramers systems, the spin linewidth is significantly larger. But this is more than compensated for by the electronic spin transition moment, as reflected by the large Rabi frequency. Using adiabatic MW pulses with a duration of 10 µs and a chirp bandwidth of 3 MHz, we readily reach an estimated transfer efficiency of >95%.

The final AFC spin-wave storage experiment data is shown in FIG. 3d, as a function of the total storage time T M = 1/∆ + T S . Memory output pulses were detectable beyond 1 ms and the memory lifetime is consistent with previous measurements of the spin coherence time in 171 Yb 3+ :Y 2 SiO 5 [28]. It should be emphasized that we reach spin-wave storage times presently only achieved in Eu 3+ :Y 2 SiO 5 crystals using nuclear states. For the shortest spin storage time of T S = 100 µs, the total memory efficiency was η M = 3.3%. While the AFC echo efficiency could be understood by the optical depth, see FIG. 2, the total memory efficiency falls short of our predictions by a factor of 4, given the optical and MW control pulse efficiencies given above (see Supplemental materials [START_REF]See Supplemental Material at URL for additional experimental data, a[END_REF]). Possibly these were overestimated and/or their phase coherence was not sufficient.

To summarize, in this Letter we have demonstrated storage of optical pulses using the electronic-nuclear hyperfine states in 171 Yb 3+ :Y 2 SiO 5 , which constitutes the first demonstration of spin-wave storage in any RE ion with electronic spin. Moreover, the AFC echo delay (which is related to temporal multimode capacity) and the total spin-wave storage time reach similar performance as in the pure nuclear 151 Eu 3+ :Y 2 SiO 5 system, but with 5 times larger optical bandwidths [55].

To conclude, we briefly discuss current limitations and future prospects of the memory. The memory bandwidth of 10 MHz is currently limited by the 0.6 MHz Rabi frequency of the control pulses. Simulations show (see Supplemental materials [START_REF]See Supplemental Material at URL for additional experimental data, a[END_REF]) that a 2 MHz Rabi frequency could increase the memory bandwidth to 100 MHz. Other Λ systems could reach such Rabi frequencies using different polarization modes. To efficiently excite mi-crowave transitions at > 1 GHz in these Λ systems, one can use lumped-element MW cavities [56,57]. The optical Rabi frequency can also be greatly increased by using laser-written waveguides [50,58] and could potentially allow memory bandwidths in the range of 100 MHz to 1 GHz. This would facilitate interfacing with quantum photon pair sources [15,59], and possibly allow interfacing with quantum dot single photon sources [60][61][62]. There is also the prospect of greatly increasing the AFC multimode capacity, which would increase the rate of a quantum repeater [63]. By increasing the bandwidth and by achieving AFC echo delays only limited by the long optical coherence time in 171 Yb 3+ :Y 2 SiO 5 , the temporal multimode memory capacity could potentially reach 1000 modes. An important future step will be to demonstrate storage of weak coherent states, a strong requirement for a quantum memory. This will crucially depend on the efficiency with which one can optically pump ions out of the storage state, to decrease unwanted photon emission noise due to the control pulses. Recent work has shown that optical pumping of hyperfine states can be particularly efficient [39].

Our sample is a 171 Yb 3+ :Y 2 SiO 5 crystal with 5 ppm doping concentration and with an Yb 3+ isotope purity of 95%. The inhomogeneous broadening on the optical 2 F 7/2 (0) ←→ 2 F 5/2 (0) transition was measured to be 1.3 GHz which is noticeably larger than 0.56 GHz measured in the 10 ppm doped sample [1]. We attribute this to the imperfections during the growing process of the crystal. The crystal was grown using the Czochralski method and cut along orthogonal polarization extinction axes of the Y 2 SiO 5 crystal usually denoted as D 1 , D 2 and b, where b is the crystallographic C 2h crystal symmetry axis. The dimensions along D 1 × D 2 × b are 3.5 × 4.5 × 12 mm, respectively. The crystal was polished for light propagating along the b axis. The polarization of the light was along the D 2 axis to maximize the absorption for the ν 1 transition.

Population relaxation

The population relaxation was probed using the optical spectral hole burning technique [2]. As mentioned in the main text, several decay constants were observed which can be interpreted as relaxation processes involving various hyperfine transitions [3]. The longest relaxation time was measured to be about 400 ms, while the shortest is 40 ms long. From the decay of the sidehole and the antihole structure we can attribute the fast decay to the relaxation of the |4 g ↔ |3 g and the |1 g ↔ |2 g transitions coming from the spin cross-relaxation. The long decay can be attributed to the relaxation of the |4 g ↔ |2 g and the |1 g ↔ |3 g transitions.

Spin Rabi frequency

To address the spin |4 g ↔ |3 g transition we use an unmatched inductive copper coil wrapped around the crystal with 6 turns aligned with the b axis of the crystal. To drive the transition the Agilent ESG signal generator is used together with a 4 W microwave amplifier. Using the impedance-unmatched coil with an inductance of ≈ 200 nH an induced oscillating magnetic field at 655 MHz on the order of 10 -20 µT was achieved.

The coherent operation on the spin |4 g ↔ |3 g transition was tested using the optical probe pulse synchronised with a microwave square pulse (FIG. 1). By looking at the absorption of the probe signal a Rabi oscillation with up to Ω MW = 0.65 MHz frequency was measured. This is in accordance with the estimated microwave power cou- pled to the unmatched coil, taking into account the magnetic dipole moment of the transition gµ B with g = 6.06 and µ B /h = 13.99 GHz/T for a microwave excitation along the b axis.

To generate efficient π-rotations in the microwave domain we used sech/tanh adiabatic-shape chirped pulses. This allows one to achieve efficient population inversion over the entire spin inhomogeneous linewidth of the spin ensemble. For this we use the IQ-modulation channels of the Agilent ESG signal generator which is controlled by an arbitrary function generator (Tektronix 3022C).

The efficiency of the π-pulse was estimated by probing the population using the optical probe pulse and measuring the population change before and after several pairs of π pulses. An efficiency of up to 97(1)% was deduced from this measurement. We attribute this imperfection to the spatial inhomogeneity of the Rabi frequency induced by the copper coil along the crystal.

Spin rephasing

The inhomogeneous broadening of the hyperfine transition was previously estimated using Raman heterodyne scattering to be about Γ MW = 730 kHz. This value is an order of magnitude bigger than for the hyperfine transitions of non-Kramers ions. This leads to a much faster dephasing of the induced spin coherence, which happens in ≈ 500 ns. Such a fast dephasing imposes a strong constraint on the separation between the two control field pulses to observe the three-level AFC echo after transferring from the spin state.

This issue can be solved by applying a pair of π pulses on the microwave transition in order to rephase the induced spin coherence (see FIG. 5c). The distance between the pulses τ has to match the timing between the control field pulses T S such that τ = T S /2. To probe the dephasing process one can measure the efficiency of the rephased signal as a function of the delay between the π pulses η M (τ ). If we assume a Gaussian profile for the spin inhomogeneous broadening, the decay of the stored echo is given by

η M (τ ) = A exp - π 2 Γ 2 MW (T S -2τ ) 2 2 ln 2 , ( 1
)
where A is the constant including the efficiency of the The results are depicted in FIG. 3 together with the fit by Eq. ( 1). The extracted FWHM of the inhomogeneous broadening of the spin transition of Γ MW = 0.73(4) MHz is in accordance with the previous estimation.

Optical branching ratio

The optical branching ratio TABLE I was measured for a sample with a 10 ppm concentration. The light was polarized along the D 2 axis. The spectral hole burning technique was used to extract hole/antihole amplitudes for every optical transition. For this the spectral hole was prepared at ν 1 laser frequency, while the probe frequency was scanned over a wide range (10 GHz) to measure the whole optical absorption spectra. The hole/antihole structure was further extracted for the measured spectra.

The optical inhomogeneous broadening for that sample is 0.56 GHz, which results in a low overlap between different spectral classes. In this case the amplitudes for most of the holes/antiholes represent the absorption coefficient for certain optical transitions. This way, the optical branching ratios for the given ground states connected to different excited states were extracted. Further normalization was used to reconstruct the whole 4 × 4 table (TABLE I).

Optical control pulse bandwidth

Using ≈ 300 mW power for the optical control field we measured the Rabi oscillation of ≈ 2 MHz for the |4 g ↔ |1 e optical transition (FIG. 2). This value is limited by the low oscillator strength of the transition and, in principle, can be enhanced by an external optical cavity or a strong light confinement using waveguide structures. An order of magnitude increase of the Rabi frequency has been demonstrated previously using those methods [4,5].

By using the optical branching ratio table (TABLE I)

we estimate the Rabi frequency of |3 g ↔ |1 e to be ≈ 0.6 MHz. Hence, this is the maximum Rabi frequency we currently can achieve with the optical control pulse. To increase the bandwidth over which one can achieve efficient population (or coherence) transfer one can employ frequency-chirped adiabatic pulses, as discussed in detail in Ref. [6]. In this experiment we use hyperbolicsquare-hyperbolic (HSH) pulses [7], which were specifically designed to achieve efficient transfer over a large bandwidth with respect to the available Rabi frequency. Using a simple Bloch calculation we simulated the population transfer as a function of detuning, for a given Rabi frequency. The input parameters were the HSH pulse duration T , Rabi frequency Ω O , and the total frequency chirp bandwidth Γ of the HSH pulse. Note that T refers to the central part of the HSH pulse, which is the period during which the Rabi frequency is constant. The smoothly rising and falling edges of the HSH pulse are much shorter for the pulses used during this work, and they have negligible effect on the resulting transfer efficiency.

The average efficiency η t of the HSH pulse is then calculated by averaging the population transfer over the chirp bandwidth Γ. In the adiabatic limit, the HSH pulse performs equally well over this entire bandwidth, hence it has a top-hat profile as a function of detuning [7]. However, if the pulse is not adiabatic the transfer function will have a smoother Gaussian shape, and the average efficiency will be lower than the peak efficiency at zero detuning, see for instance [8].

In FIG. 4, average transfer efficiency η t is plotted as a function of Γ, for Rabi frequencies ranging from Ω O /(2π)=0.6 MHz to Ω O /(2π)=2 MHz. The pulse duration was set to T = 5µs. For lower chirp bandwidths, the behavior is complex with the appearance of a maximum efficiency. We attribute this to the effect of non-adiabaticity discussed above. In the regime of large chirp bandwidth, however, the efficiency drops as expected when the chirp bandwidth is increased. It is also clear that for our Rabi frequency of Ω O /(2π)=0.6 MHz, the maximum efficiency is reached for about Γ/(2π)=10 MHz, and increasing the bandwidth to Γ/(2π)=20 MHz significantly reduces the efficiency. Note that the total AFC spin-wave memory efficiency scales as η 2 t . The HSH pulse is an extension of the well-known adiabatic population inversion pulse called hyperbolic secant (SECH) pulse, for which one can analytically solve the Bloch equations [8]. The HSH pulse is a time strectched SECH pulse that has larger effective pulse area for the same pulse duration. As a consequence it has a higher population transfer efficiency for finite pulse duration and Rabi frequency, with respect to the basic SECH pulse. But as it is derived from a SECH pulse, the analytical results of SECH pulses still apply. In Ref. [6] the average population transfer efficiency was calculated for a SECH pulse, for a given set of pulse parameters. Our simulations show that the HSH pulse has the same scaling as a function of these parameters, with the theoretical efficiency

η HSH t ≈ 1 -exp -0.5 πT Ω 2 O Γ . (2)
The formula was simplified under the approximation that Γ >> Ω O . The formula has the same functional form as for SECH pulses, but we had to fit the pre-factor of 0.5 appearing in the exponential to obtain exact agreement with the Bloch simulations. The theoretical efficiencies calculated using Eq. ( 2) are shown as dashed lines in FIG. 4. The quadratic dependence on the Rabi frequency Ω O explains the strong influence of the Rabi frequency on the achievable bandwidth.

Two-level AFC

By varying the frequency separation of the comb teeth ∆ one can probe the quality of the prepared comb structure. For this the two-level AFC echo experiment was performed for various storage times (see FIG. 2b in the main text). Due to the finite width of the created structure the intensity of the echo reemission will degrade for longer 1/∆ storage times with characteristic decay time T * 2 characterizing the spectral comb

η AFC = η opt AFC exp - 4 T * 2 ∆ , (3)
where η opt AFC is the maximum efficiency for a given optical depth of the comb d, optimal finesse F and background absorption d 0 given by

η opt AFC = d F 2 sinc 2 π F e -d/F -d0 . ( 4
)
The optimal finesse is F = π/arctan(2π/d) and the ultimate limit for the decay time T * 2 is given by the coherence lifetime of the optical transition T o 2 . The highest efficiency for a given optical depth d is obtained for square peaks, where the intrinsic dephasing from the absorption peak shape is given by the second term in the Eq. ( 4). In this scenario the echo efficiency is bounded by 54% due to reabsorption (the last term in the Eq. ( 4)). This limit can be overcome by forcing the reemission in the backward direction or by placing the crystal inside an impedance-matched cavity to reach a theoretical limit of 100% [5,9].

The effective decay T * 2 extracted from the experiment contains two decay constants of ≈ 16 µs and ≈ 160 µs (see FIG. 2b). These values are still much lower than the measured optical coherence time T o 2 of the optical transition of 600 µs. The effects of the spectral resolution and the laser linewidth are the most probable limitations.

Experimental setup

Our laser setup is based on two external cavity laser diodes at 978.8 nm (Toptica DLpro and DL100) (see FIG. 5). The first (master ) laser is stabilized using a custom Fabry-Perot cavity with finesse approaching ≈ 5000 by the Pound-Drever-Hall technique at ν 1 ≈ 306263.0 GHz frequency. The cavity is placed inside a custom vacuum system and is passively stabilized. This laser is used for the preparation of the atomic frequency comb structure at the |4 g ↔ |1 e transition and the creation of the input pulse (see FIG. 5a). This method of locking provides the laser linewidth of ≈ 1 kHz comparable with the homogeneous linewidth of the optical transition Γ h = (πT o 2 ) -1 , which is estimated to be ≈ 2 kHz. The second (slave) laser is stabilized with a custom optical phase locked loop setup using the external signal generator at ν 2 ≈ ν 1 +0.6547 GHz optical frequency with respect to the master laser [10]. The tapered amplifier (Toptica Boosta) is used to reach 1.2 W optical power level utilized for the control mode. Part of this power is going through the fibre-based Eospace phase modulator to create two additional frequencies ν 3 = ν 2 +6.2594 GHz and ν 4 = ν 2 + 7.0762 GHz to address |2 g ↔ |3 e and |1 g ↔ |4 e optical transitions, respectively. These frequencies are used to polarize the spin ensemble into the |4 g ground state before the preparation of the AFC structure (see FIG. 5b).

Acousto-optic modulators (AOM) are used to create the necessary light pulse sequence for the storage experiment. It consists of class cleaning (400 ms), population polarization (300 ms), AFC preparation (60 ms) The pulse sequence for the optical spin-wave storage experiment. The polarization step takes 300 ms, while the comb is prepared in 60 ms. A waiting time of 4 ms is used between the end of the preparation and the start of the three level AFC sequence. TS is the time between the control pulses, and 1/∆ is the AFC storage time. The total storage time is TM = TS + 1/∆. The separation between the microwave (MW) pulses is TS/2. and the storage sequence (see FIG. 5b). The RF signals used to drive the AOMs are generated using an arbitrary waveform generator (Tabor WW1281A), arbitrary function generators (Tektronix 3102C) combined with custom AOM drivers. We use two different optical paths to interact with the sample to spatially separate the input and control modes, which prevents the noise from the strong control mode polluting the echo signal (see FIG. 5c).

After each AOM the light is coupled into a single-mode polarization maintaining fibre and then out-coupled on a separate optical table, where the closed cycle cryogenic cooler (Oxford, PT-405) with an optical access is located (see FIG. 5b). Our 171 Yb 3+ :Y 2 SiO 5 crystal is attached to the cold head using a custom copper support. The support is based on copper braids and suppresses vibrations coming from the cryogenic cold head. This allowed us to run all the experiments without detrimental effects of mechanical vibrations.

The 125 mm lens is used to focus both modes on to the crystal, resulting in a beam width of around 90 µm for the input mode and 150 µm for the control mode. The maximum power in the control mode before the cryostat window is measured to be 300 mW. The maximum power available in the input mode is 5 mW. An additional 125 mm lens is placed after the crystal for spa-tial mode matching. Two right angle mirrors are used to reflect the input mode symmetrically around the control mode to pass a second time through the crystal, as well an additional mirror is placed to back reflect the input mode. This results in quadruple-passage through the crystal to increase the overall absorption. The optical depth of ≈ 4 -5 at the |4 g ↔ |1 e transition was achieved after the polarization sequence.

The output mode is separated using the free-space Faraday rotator (FR) together with a polarization beam splitter (PBS) and a half-wave λ/2 plate. The polarization in the quadruple path is set by the half-wave λ/2 plate parallel to D 2 axis of the crystal to maximize the absorption (see FIG. 5b). The separate half-wave λ/2 plate is used for the control mode to match the polarization of the input mode. The signal is detected using the Thorlabs photodiode with switchable gain corresponding to a bandwidth of 4 MHz.

FIG. 1 .

 1 FIG. 1. (color online) Experimental setup. (a) Energy level diagram of the optical 2 F 7/2 (0) ←→ 2 F 5/2 (0) transition for site II of 171 Yb 3+ :Y2SiO5 crystal at zero magnetic field. The Λ system used for optical storage uses transition ν1 for the input/output pulses and transition ν2 for the control pulses, and νMW for the MW pulses. Transitions ν3 and ν4 are added for the different optical pumping steps. (b) The absorption spectrum after performing class cleaning and state initialization to the |4 g ground state over a 40 MHz bandwidth. Inset: zoom on the ν1 and ν2 transitions of the Λ system. (c) Optical depth of the anti-hole at ν1 frequency and the hole at ν2 frequency, as a function of delay after the state initialization. (d) Experimental setup (see text for details). FP -Fabry-Perot cavity, PDH -Pound-Drever-Hall module, PM -phase modulator, PLL -phase locked loop, λ/2 -half-wave plate, PBS -polarization beamsplitter, , FR -Faraday rotator, ×2 -microwave frequency doubler, SW -microwave switch.

FIG. 2 .

 2 FIG. 2. (color online) Optical AFC echoes. (a) Example of an optical AFC echo for a delay of 1/∆ = 5 µs, with an efficiency of ηAF C = 15%. The input pulse duration was 100 ns. (b) AFC echo efficiencies as a function of delay 1/∆ (see text for fit model). (c) Two examples of measured AFCs for 1/∆ = 1 µs and 1/∆ = 5 µs, in terms of optical depth d as a function of frequency detuning on the transition ν1.

  FIG. 3. (color online) AFC spin-wave storage results. (a,b) Measured Rabi oscillations on the optical and MW transitions (see Supplemental materials [51]). (c) Pulse and timing sequence of the AFC spin-wave storage experiment, including input/output pulses (transition ν1), control pulses (transition ν2) and MW pulses (transition νMW). Note that for a perfect control pulse the AFC echo at 1/∆ is completely suppressed. (d) The intensity of optical output pulse as a function of the total memory storage time TM, with 1/∆ = 7 µs. The data was fitted to the function exp(-2TS/T s 2 ), resulting in a spin coherence time of T s 2 = 1.2(2) ms. Inset: Example of output pulse trace for TM = 107 µs.

FIG. 1 .

 1 FIG. 1. (color online) Coherent driving of the spin transition. (a) The oscillation of the transmitted optical signal synchronised with the 10 µs microwave pulse. The measurement is done for various microwave power levels at the input of the microwave 33 dB amplifier. (b) Temporal trace measured for -3 dBm input power. The measured oscillation corresponds to a Rabi frequency of Ωmw = 0.65 MHz. (c) Measured Rabi frequency for various input power levels of the microwave pulse. The square root √ Pmw dependence of the measured frequency confirms the coherent nature of the observed spin manipulation.

FIG. 2 .

 2 FIG. 2. (color online) Coherent driving of the optical transition. (a) Optical transmission for |4 g ↔ |1 e measured right after the strong optical control pulse. The transmission is measured by scanning the frequency of the probe pulse for different durations of the control pulse. Clear oscillations are visible due to the coherent driving of the optical transition. The faster oscillations are visible for detuned spectral components due to the square temporal shape of the optical pulse and a faster Rabi frequency for off-resonant excitations. (b) Lower figure shows the transmission of the probe pulse for the central part around zero detuning as a function of the pulse length. The oscillation corresponds to a Rabi frequency of ΩO = 2 MHz and was measured for the maximum optical power of ≈ 300 mW before the cryostat.

  FIG. 3. (color online) Spin inhomogeneous linewidth measurement. (a) The intensity of spin-wave storage signal measured for different temporal positions of the second optical transfer TS for fixed temporal separation between two microwave pulses τ . The delay from the optimal value of ∆TS = TS -2τ leads to a fast dephasing in the spin transition due to the finite inhomogeneous broadening. By fitting to the Eq. (1) we extract the inhomogeneous linewidth of ΓMW = 0.73(4) MHz. This value fits well with measurements done with Raman heterodyne signal. (b) The pulse sequence used for this experiment with timing indications.

FIG. 4 .

 4 FIG. 4. (color online) Theoretical optical control pulse efficiency. Solid lines show the simulated average transfer efficiency ηt as a function of the HSH bandwidth Γ/(2π), for Rabi frequencies ΩO/(2π) = 0.6, 1.0, 1.5, 2.0 MHz. Dashed lines show the corresponding efficiency calculated using Eq. (2).

ν 1 , ν 2 ,

 12 FIG. 5. (color online) Experimental scheme. (a) Energy level diagram for optical 2 F 7/2 (0) ←→ 2 F 5/2 (0) transition for site II of 171 Yb 3+ :Y2SiO5 crystal at zero magnetic field. The |4 g ↔ |1 e (ν1 frequency) transition if used for a storage while the |3 g ↔ |1 e (ν2 frequency) transition is used for the optical transfer into the spin state and back. |1 g ↔ |4 e and |2 g ↔ |3 e (ν3 and ν4) transitions are used for spin ensemble polarization. (b) Experimental setup (see text for details). FP -Fabry-Perot cavity, PDH -Pound-Drever-Hall module, AOM -acousto-optic modulator, PM -phase modulator, PLL -phase locked loop, λ/2 -half-wave plate, PBS -polarization beamsplitter, ×2 -microwave frequency doubler, FR -Faraday rotator, SWmicrowave switch. (c)The pulse sequence for the optical spin-wave storage experiment. The polarization step takes 300 ms, while the comb is prepared in 60 ms. A waiting time of 4 ms is used between the end of the preparation and the start of the three level AFC sequence. TS is the time between the control pulses, and 1/∆ is the AFC storage time. The total storage time is TM = TS + 1/∆. The separation between the microwave (MW) pulses is TS/2.

TABLE I .

 I Measured relative optical oscillator strengths for 171 Yb 3+ :Y2SiO5 optical 2 F 7/2 (0) ←→ 2 F 5/2 (0) transition for crystallographic site II measured for light polarized along the D2 axis using the spectral holeburning technique. The lambda system used for the spin-wave storage experiment is indicated by bold. AFC and the finite efficiency of the transfer η t and microwave η MW pulses.

		|1 e	|2 e	|3 e	|4 e
	1| g	0.15	0.06	0.08	0.71
	2| g	0.06	0.19	0.71	0.04
	3| g	0.07	0.71	0.16	0.06
	4| g	0.72	0.04	0.05	0.19
	two-level AFC η

† These authors contributed equally to this work.; Present address: The Niels Bohr Institute, University of Copenagen, DK-2100 Copenhagen , Denmark

ACKNOWLEDGEMENTS

We acknowledge funding from FNS Research Project No 172590, EUs H2020 programme under the Marie Sk lodowska-Curie project QCALL (GA 675662), IMTO Cancer AVIESAN (Cancer Plan, C16027HS, MALT).