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The ability to coherently control and read out qubits with long coherence times in a scalable system is a crucial requirement for any quantum processor. Nuclear spins in the solid state have shown great promise as long-lived qubits. Control and readout of individual nuclear spin qubit registers has made major progress in the recent years using individual electron spin ancilla addressed either electrically or optically. Here, we present a new platform for quantum information processing, consisting of $^{183}$W nuclear spin qubits adjacent to an Er$^{3+}$ impurity in a CaWO$_4$ crystal, interfaced via a superconducting resonator and detected using a microwave photon counter at 10mK. We study two nuclear spin qubits with $T_2^*$ of $0.8(2)~$s and $1.2(3)~$s, $T_2$ of $3.4(4)~$s and $4.4(6)~$ s, respectively. We demonstrate single-shot quantum non-demolition readout of each nuclear spin qubit using the Er$^{3+}$ spin as an ancilla. We introduce a new scheme for all-microwave single- and two-qubit gates, based on stimulated Raman driving of the coupled electron-nuclear spin system. We realize single- and two-qubit gates on a timescale of a few milliseconds, and prepare a decoherence-protected Bell state with 88% fidelity and $T_2^*$ of $1.7(2)~$s. Our results are a proof-of-principle demonstrating the potential of solid-state nuclear spin qubits as a promising platform for quantum information processing. With the potential to scale to tens or hundreds of qubits, this platform has prospects for the development of scalable quantum processors with long-lived qubits., Comment: 14 pages, 4 main figures, 7 supplementary figures

The integration of coherent emitters into low-loss photonic circuits is a key technology for quantum networking. In this context, nanophotonic silicon devices implanted with erbium are a promising hardware platform that combines advanced wafer-scale nanofabrication technology with coherent emission in the minimal-loss band of optical fibers. Recent studies have reported two distinct sites in the silicon lattice in which erbium can be reproducibly integrated with particularly promising properties. Here, for an in-depth analysis of these sites, resonant fluorescence spectroscopy is performed on a nanophotonic waveguide in magnetic fields applied along different orientations. In this way, the site symmetry is determined, the spin Hamiltonian is reconstructed and a partial fit of the crystal field Hamiltonian is performed. The obtained quantitative description of the magnetic interaction allows the optimization of Zeeman splittings, optical branching ratios or microwave driving to the needs of future experiments. Beyond that, the derived site symmetry constrains the location of the erbium dopant in the silicon unit cell. This is a key step towards a detailed microscopic understanding of the erbium sites, which may help to improve the integration yield, thus paving the way to efficient nanophotonic quantum memories based on the Er:Si platform.

Rare-earth-ion (REI) ensembles in crystals have remarkable optical and spin properties characterized by narrow homogeneous linewidths relative to the inhomogeneous ensemble broadening. This makes it possible to precisely tailor the ensemble spectral density and therefore the absorption profile by applying narrow-linewidth radiation to transfer population into auxiliary levels, a process broadly known as spectral hole burning (SHB). REI-doped crystals find applications in information processing, both classical (pattern recognition, filtering, spectral analysis) and quantum (photon storage), all protocols requiring suitable ensemble preparation by SHB as a first step. In Er$^{3+}$-doped materials, the longest reported hole lifetime is one minute, and longer lifetimes are desirable. Here, we report SHB and accumulated echo measurements in a scheelite crystal of CaWO$_4$ by pumping the electron spin transition of Er$^{3+}$ ions at microwave frequencies and millikelvin temperatures, with nuclear spin states of neighboring $^{183}$W atoms serving as the auxiliary levels. The lifetime of the holes and accumulated echoes rises steeply as the sample temperature is decreased, exceeding a month at 10 mK. Our results demonstrate that millikelvin temperatures can be beneficial for signal processing applications requiring long spectral hole lifetimes.

Rare-earth ion doped crystals are state-of-the-art platforms for processing quantum information, particularly thanks to their excellent optical and spin coherence properties at cryogenic temperatures. Experimental observations have shown that the application of a static magnetic bias field significantly improves the coherence times in the rare-earth ions ensemble, but only a few studies have focused on its the dependency as a function of both magnetic field direction and amplitude. This is especially true for magnetic field amplitudes under the mT, and for low magnetic dipole moment ions. In this paper, we investigate the relationship between the magnetic field parameters and the decoherence caused by magnetic dipole-dipole coupling with the nearest neighbors nuclear spins in the crystal. The primary non-Kramers rare-earth ions investigated here are europium and praseodymium, but we also extend our study to the ytterbium Kramers ion due to its low magnetic dipole in the mT range. We perform theoretical investigations and simulations of the energy structure and coherence time evolution and identify good correspondences between experimental and simulated spin echo data. This work allows us to pinpoint the most relevant decoherence mechanisms in the considered magnetic field regime, and to predict favorable magnetic configurations., Comment: 20 pages, 14 figures

We use an erbium doped CaWO$_4$ crystal as a resonant transducer between the RF and optical domains at 12 GHz and 1532 nm respectively. We employ a RF resonator to enhance the spin coupling but keep a single-pass (non-resonant) optical setup. The overall efficiency is low but we carefully characterize the transduction process and show that the performance can be described by two different metrics that we define and distinguish: the electro-optics and the quantum efficiencies. We reach an electro-optics efficiency of -84 dB for 15.7 dBm RF power. The corresponding quantum efficiency is -142 dB for 0.4 dBm optical power. We develop the Schr\"odinger-Maxwell formalism, well-known to describe light-matter interactions in atomic systems, in order to model the conversion process. We explicitly make the connection with the cavity quantum electrodynamics (cavity QED) approach that are generally used to describe quantum transduction., Comment: 40 pages, 10 figures

It was recently shown that the optical excitation of rare-earth ions produces a local change of the host matrix shape, attributed to a change of the rare-earth ion's electronic orbital geometry. In this work we investigate the consequences of this piezo-orbital backaction and show from a macroscopic model how it yields a disregarded ion-ion interaction mediated by mechanical strain. This interaction scales as $1/r^3$, similarly to the other archetypal ion-ion interactions, namely electric and magnetic dipole-dipole interactions. We quantitatively assess and compare the magnitude of these three interactions from the angle of the instantaneous spectral diffusion mechanism, and reexamine the scientific literature in a range of rare-earth doped systems in the light of this generally underestimated contribution., Comment: 15 pages, 3 figures, 3 tables

Electron spin resonance (ESR) spectroscopy is the method of choice for characterizing paramagnetic impurities, with applications ranging from chemistry to quantum computing, but it gives access only to ensemble-averaged quantities due to its limited signal-to-noise ratio. Single-electron-spin sensitivity has however been reached using spin-dependent photoluminescence, transport measurements, and scanning-probe techniques. These methods are system-specific or sensitive only in a small detection volume, so that practical single spin detection remains an open challenge. Here, we demonstrate single electron magnetic resonance by spin fluorescence detection, using a microwave photon counter at cryogenic temperatures. We detect individual paramagnetic erbium ions in a scheelite crystal coupled to a high-quality factor planar superconducting resonator to enhance their radiative decay rate, with a signal-to-noise ratio of 1.9 in one second integration time. The fluorescence signal shows anti-bunching, proving that it comes from individual emitters. Coherence times up to 3 ms are measured, limited by the spin radiative lifetime. The method has the potential to apply to arbitrary paramagnetic species with long enough non-radiative relaxation time, and allows single-spin detection in a volume as large as the resonator magnetic mode volume ( 10 um^3 in the present experiment), orders of magnitude larger than other single-spin detection techniques. As such, it may find applications in magnetic resonance and quantum computing.

The coherent interaction of solid-state spins with both optical and microwave fields provides a platform for a range of quantum technologies, such as quantum sensing, microwave-to-optical quantum transduction and optical quantum memories. Rare-earth ions with electronic spins are interesting in this context, but it is challenging to simultaneously and efficiently drive both optical and microwave transitions over a long crystal. In this work, we use a loop-gap microwave resonator to coherently drive optical and microwave clock transitions in $^{171}$Yb$^{3+}$:Y$_2$SiO$_5$, at close to zero external magnetic field. The low magnetic field regime is particularly interesting for interfacing these spin transitions with superconducting circuits. We achieve a Rabi frequency of 0.56 MHz at 2.497 GHz, over a 1-cm long crystal. Furthermore, we provide new insights into the spin dephasing mechanism at very low fields, showing that superhyperfine-induced collapse of the Hahn echo signal plays an important role at low fields. Our calculations and measurements reveal that the effective magnetic moment can be manipulated in $^{171}$Yb$^{3+}$:Y$_2$SiO$_5$, allowing to suppress the superhyperfine interaction at the clock transition. At a doping concentration of 2 ppm and a temperature of $3.4$ K, we achieve the longest spin coherence time of $10.0 \pm 0.4 ~\text{ms}$ reported in $^{171}$Yb$^{3+}$:Y$_2$SiO$_5$., Comment: 8 pages, 4 figures, SI in ancillary files

Counting the microwave photons emitted by an ensemble of electron spins when they relax radiatively has recently been proposed as a sensitive method for electron paramagnetic resonance (EPR) spectroscopy, enabled by the development of operational Single Microwave Photon Detectors (SMPD) at millikelvin temperature. Here, we report the detection of spin echoes in the spin fluorescence signal. The echo manifests itself as a coherent modulation of the number of photons spontaneously emitted after a $\pi/2_X - \tau - \pi_Y - \tau - \pi/2_\Phi $ sequence, dependent on the relative phase $\Phi$. We demonstrate experimentally this detection method using an ensemble of $\mathrm{Er}^{3+}$ ion spins in a scheelite crystal of $\mathrm{CaWO}_4$. We use fluorescence-detected echoes to measure the erbium spin coherence time, as well as the echo envelope modulation due to the coupling to the $^{183}\mathrm{W}$ nuclear spins surrounding each ion. We finally compare the signal-to-noise ratio of inductively-detected and fluorescence-detected echoes, and show that it is larger with the fluorescence method.

Erbium-doped crystals offer a versatile platform for hybrid quantum devices because they combine magnetically-sensitive electron-spin transitions with telecom-wavelength optical transitions. At the high doping concentrations necessary for many quantum applications, however, strong magnetic interactions of the electron-spin bath lead to excess spectral diffusion and rapid decoherence. Here we lithographically fabricate a 4.4 GHz superconducting planar micro-resonator on a $\text{CaWO}_{4}$ crystal doped with Er ions at a concentration of twenty parts per million relative to Ca. Using the microwave resonator, we characterize the spectral diffusion processes that limit the electron-spin coherence of Er ions at millikelvin temperatures by applying 2- and 3-pulse echo sequences. The coherence time shows a strong temperature dependence, reaching 1.3 ms at 23 mK for an electron-spin transition of $^{167}\text{Er}$., Comment: 10 pages, 5 figures

Cryogenics is a pivotal aspect in the development of quantum technologies. Closed-cycle devices have recently emerged as an environmentally friendly and low-maintenance alternative to liquid helium cryostats. Yet the larger level of vibrations in dry cryocoolers forbids their use in most sensitive applications. In a recent work, we have proposed an inertial, broadband, contactless sensor based on the piezospectroscopic effect, ie the natural sensitivity of optical lines to strain exhibited by impurities in solids. This sensor builds on the exceptional spectroscopic properties of rare earth ions and operates below 4K, where spectral hole burning considerably enhances the sensitivity. In this paper, we investigate the fundamental and technical limitations of this vibration sensor by comparing a rigid sample attachment to cold stage of a pulse-tube cryocooler and a custom-designed exchange gas chamber for acoustic isolation., Comment: 7 pages, 5 figures

We investigate a system composed of an ensemble of room temperature rare-earth ions embedded in a bulk crystal, intrinsically coupled to internal strain via their sensitivity to the surrounding crystal field. We evidence the generation of a mechanical response under resonant atomic excitation. We find this motion to be the sum of two fundamental, resonant optomechanical backaction processes: a conservative, piezo-orbital mechanism, resulting from the modification of the crystal field associated with the promotion of the ions to their excited state, and a dissipative, non-radiative photothermal process related to the phonons generated throughout the atomic population relaxation. Our work opens new research avenues in hybrid optomechanics, and highlights new interactions that may be key for understanding the dephasing dynamics of ultra-coherent rare-earth ions., Comment: Main text: 6 pages, 4 figures. Supplementary Material: 7 pages, 6 figures

Erbium ions doped into crystals have unique properties for quantum information processing, because of their optical transition at 1.5 $\mu$m and of the large magnetic moment of their effective spin-1/2 electronic ground state. Most applications of erbium require however long electron spin coherence times, and this has so far been missing. Here, by selecting a host matrix with a low nuclear-spin density (CaWO$_4$) and by quenching the spectral diffusion due to residual paramagnetic impurities at millikelvin temperatures, we obtain an Er$^{3+}$ electron spin coherence time of 23 ms. This is the longest electron spin coherence time measured in a material with a natural abundance of nuclear spins and on a magnetically-sensitive transition. Our results establish Er$^{3+}$:CaWO$_4$ as a leading platform for quantum networks.

We study the possibility to enhance the phase space density of non-interacting particles submitted to a classical laser field without spontaneous emission. We clearly state that, when no spontaneous emission is present, a quantum description of the atomic motion is more reliable than semi-classical description which can lead to large errors especially if no care is taken to smooth structures smaller than the Heisenberg uncertainty principle. Whatever the definition of position - momentum phase space density, its gain is severely bounded especially when started from a thermal sample. More precisely, the maximum phase space density, can only be improved by a factor M for M-level atoms. This bound comes from a transfer between the external and internal degrees of freedom. To circumvent this limit, one can use non-coherent light fields, informational feedback cooling schemes, involve collectives states between fields and atoms, or allow a single spontaneous emission even, Comment: 3 figures, 4 pages

We review a series of quantum memory protocols designed to store the quantum information carried by light into atomic ensembles. In particular, we show how a simple semiclassical formalism allows to gain insight into various memory protocols and to highlight strong analogies between them. These analogies naturally lead to a classification of light storage protocols into two categories, namely photon echo and slow-light memories. We focus on the storage and retrieval dynamics as a key step to map the optical information into the atomic excitation. We finally review various criteria adapted for both continuous variables and photon-counting measurement techniques to certify the quantum nature of these memory protocols.

In Er$^{3+}$:Y$_2$SiO$_5$, we demonstrate the selective optical addressing of the $^{89}$Y$^{3+}$ nuclear spins through their superhyperfine coupling with the Er$^{3+}$ electronic spins possessing large Land\'e $g$-factors. We experimentally probe the electron-nuclear spin mixing with photon echo techniques and validate our model. The site-selective optical addressing of the Y$^{3+}$ nuclear spins is designed by adjusting the magnetic field strength and orientation. This constitutes an important step towards the realization of long-lived solid-state qubits optically addressed by telecom photons., Comment: 5 pages, 4 figures, supplementary material (3 pages)

We demonstrate a solid state spin-wave optical memory based on stopped light in a spectral hole. A long lived narrow spectral hole is created by optical pumping in the inhomogeneous absorption profile of a Pr$^{3+}$:Y$_2$SiO$_5$ crystal. Optical pulses sent through the spectral hole experience a strong reduction of their group velocity and are spatially compressed in the crystal. A short Raman pulse transfers the optical excitation to the spin state before the light pulse exits the crystal, effectively stopping the light. After a controllable delay, a second Raman pulse is sent, which leads to the emission of the stored photons. We reach storage and retrieval efficiencies for bright pulses of up to $39\,\%$ in a $5 \,\mathrm{mm}$-long crystal. We also show that our device works at the single photon level by storing and retrieving $3\,\mathrm{\mu s}$-long weak coherent pulses with efficiencies up to $31\,\%$, demonstrating the most efficient spin-wave solid state optical memory at the single-photon level so far. We reach an unconditional noise level of $(9\pm1)\times 10^{-3}$ photons per pulse in a detection window of $4\,\mathrm{\mu s}$ leading to a signal-to-noise ratio of $33 \pm 4$ for an average input photon number of 1, making our device promising for long-lived storage of non-classical light., Comment: 5 pages, 4 figures

We show that the AC-Stark shift (light-shift) is a powerful and versatile tool to control the emission of a photon-echo in the context of optical storage. As a proof-of-principle, we demonstrate that the photon-echo efficiency can be fully modulated by applying light-shift control pulses in an erbium doped solid. The control of the echo emission is attributed to the spatial gradient induced by the light-shift beam.

We study the bandwidth and multiplexing capacity of an erbium-doped optical memory for quantum storage purposes. We concentrate on the protocol ROSE (Revival of a Silenced Echo) because it has the largest potential multiplexing capacity. Our analysis is applicable to other protocols that involve strong optical excitation. We show that the memory performance is limited by instantaneous spectral diffusion and we describe how this effect can be minimised to achieve optimal performance.

Interlaced Spin Grating is a scheme for the preparation of spectro-spatial periodic absorption gratings in a inhomogeneously broadened absorption profile. It relies on the optical pumping of atoms in a nearby long-lived ground state sublevel. The scheme takes advantage of the sublevel proximity to build large contrast gratings with unlimited bandwidth and preserved average optical depth. It is particularly suited to Tm-doped crystals in the context of classical and quantum signal processing. In this paper, we study the optical pumping dynamics at play in an Interlaced Spin Grating and describe the corresponding absorption profile shape in an optically thick atomic ensemble. We show that, in Tm:YAG, the diffraction efficiency of such a grating can reach 18.3% in the small angle, and 11.6% in the large angle configuration when the excitation is made of simple pulse pairs, considerably outperforming conventional gratings., Comment: 11 pages, 13 figures in Physical Review A, 2015

We implement the ROSE protocol in an erbium doped solid, compatible with the telecom range. The ROSE scheme is an adaptation of the standard 2-pulse photon echo to make it suitable for a quantum memory. We observe an efficiency of 40% in a forward direction by using specific orientations of the light polarizations, magnetic field and crystal axes.

I theoretically study the behavior of strong pulses exciting emitters inside a cavity. The ensemble is supposed to be inhomogeneously broadened and the cavity matched finding application in quantum storage of optical or RF photons. My analysis is based on energy and pulse area conservation rules predicting important distortions for specific areas. It is well supported by numerical simulations. I propose a qualitative interpretation in terms of slow-light. The analogy with the free space situation is remarkable.

Based on the similarity of paraxial diffraction and dispersion mathematical descriptions, the temporal imaging of optical pulses combines linear dispersive filters and quadratic phase modulations operating as time lenses. We consider programming a dispersive filter near atomic resonance in rare earth ion doped crystals, which leads to unprecedented high values of dispersive power. This filter is used in an approximate imaging scheme, combining a single time lens and a single dispersive section and operating as a time reversing device, with potential applications in radio-frequency signal processing. This scheme is closely related to three-pulse photon echo with chirped pulses but the connection with temporal imaging and dispersive filtering emphasizes new features., Comment: 21 pages, 11 figures

Coherence rephasing is an essential step in quantum storage protocols that use echo-based strategies. We present a thorough analysis on how two adiabatic rapid passages (ARP) are able to rephase atomic coherences in an inhomogeneously broadened ensemble. We consider both the cases of optical and spin coherences, rephased by optical or radio-frequency (rf) ARPs, respectively. We show how a rephasing sequence consisting of two ARPs in a double-echo scheme is equivalent to the identity operator (any state can be recovered), as long as certain conditions are fulfilled. Our mathematical treatment of the ARPs leads to a very simple geometrical interpretation within the Bloch sphere that permits a visual comprehension of the rephasing process. We also identify the conditions that ensure the rephasing, finding that the phase of the optical or rf ARP fields plays a key role in the capability of the sequence to preserve the phase of the superposition state. This settles a difference between optical and rf ARPs, since field phase control is not readily guaranteed in the former case. We also provide a quantitative comparison between $\pi$-pulse and ARP rephasing efficiencies, showing the superiority of the latter. We experimentally verify the conclusions of our analysis through rf ARP rephasing sequencies performed on the rare-earth ion-doped crystal Tm$^{3+}$:YAG, of interest in quantum memories., Comment: 24 pages, 7 figures

In low concentration Tm$^{3+}$:YAG, we observe efficient adiabatic rapid passage (ARP) of thulium nuclear spin over flipping times much longer than $T_2$. Efficient ARP with long flipping time has been observed in monoatomic solids for decades and has been analyzed in terms of spin temperature and of the thermodynamic equilibrium of a coupled spin ensemble. In low-concentration impurity-doped crystals the spin temperature concept may be questioned. A single spin model should be preferred since the impurity ions are weakly coupled together but interact with the numerous off-resonant matrix ions that originate the spin-spin relaxation. The experiment takes place in the context of quantum information investigation, involving impurity-doped crystals, spin hyperpolarization by optical pumping, and optical detection of the spin evolution., Comment: 6 pages, 4 figures

The three-pulse photon echo is a well-known technique to store intense light pulses in an inhomogeneously broadened atomic ensemble. This protocol is attractive because it is relatively simple and it is well suited for the storage of multiple temporal modes. Furthermore, it offers very long storage times, greater than the phase relaxation time. Here, we consider the three-pulse photon echo in both two- and three-level systems as a potential technique for the storage of light at the single-photon level. By explicit calculations, we show that the ratio between the echo signal corresponding to a single-photon input and the noise is smaller than one. This severely limits the achievable fidelity of the quantum state storage, making the three-pulse photon echo unsuitable for single-photon quantum memory., Comment: 6 pages, 4 figures

We report on light slowing down in a rare earth ion doped crystal by persistent spectral hole burning. The absence of motion of the active ions, the large inhomogeneous broadening, the small homogeneous width and the long lifetime of the hyperfine shelving states make this material convenient for the burning of narrow persistent spectral holes. Since the hole can be burnt long before the arrival of the input signal, there is no need for a strong coupling field, illuminating the sample simultaneously with the input signal, in contrast with procedures such as Electromagnetically Induced Transparency or Coherent Population Oscillations. Analyzing the slowing down process, we point out the role played by off resonance atoms where most of the incoming information is carried over while the pulse is confined within the sample., Comment: 14 pages, 6 figures

We consider in this paper a two-pulse photon echo sequence as a potential quantum light storage protocol. It is widely believed that a two-pulse scheme should lead to very low efficiency and is then not relevant for this specific application. We show experimentally by using a Tm${}^{3+}$:YAG crystal that such a protocol is on contrary very efficient and even too efficient to be considered as a good quantum storage protocol. Our experimental work allows us to point out on one side the real limitations of this scheme and on the other side its benefits which can be a source of inspiration to conceive more promising procedures with rare-earth ion doped crystals.

A thulium-doped crystal is experimentally shown to be an excellent candidate for broadband quantum storage in a solid-state medium. For the first time, nuclear spin coherence is optically excited, detected and characterized in such a crystal. The lifetime of the spin coherence -- the potential storage entity -- is measured by means of Raman echo to be about 300 $\mu$s over a wide range of ground state splittings. This flexibility, attractive for broadband operation, and well fitted to existing quantum sources, results from the simple hyperfine structure, contrasting with Pr- and Eu- doped crystals.