Gershon Kurizki
email: gershon.kurizki@weizmann.ac.il.

Patrice Bertet

Yuimaru Kubo

Klaus Mølmer

David Petrosyan

Peter Rabl

Jörg Schmiedmayer

Quantum technologies with hybrid systems

Keywords: hybrid quantum systems, quantum technologies, quantum

An extensively pursued current direction of research in physics aims at the development of practical technologies that exploit the effects of quantum mechanics. As part of this ongoing effort, devices for quantum information processing, secure communication, and high-precision sensing are being implemented with diverse systems, ranging from photons, atoms, and spins to mesoscopic superconducting and nanomechanical structures. Their physical properties make some of these systems better suited than others for specific tasks; thus, photons are well suited for transmitting quantum information, weakly interacting spins can serve as long-lived quantum memories, and superconducting elements can rapidly process information encoded in their quantum states. A central goal of the envisaged quantum technologies is to develop devices that can simultaneously perform several of these tasks, namely, reliably store, process, and transmit quantum information. Hybrid quantum systems composed of different physical components with complementary functionalities may provide precisely such multitasking capabilities. This article reviews some of the driving theoretical ideas and first experimental realizations of hybrid quantum systems and the opportunities and challenges they present and offers a glance at the near-and long-term perspectives of this fascinating and rapidly expanding field.

During the last several decades, quantum physics has evolved from being primarily the conceptual framework for the description of microscopic phenomena to providing inspiration for new technological applications. A range of ideas for quantum information processing (1) and secure communication (2, 3), quantum enhanced sensing (4-8), and the simulation of complex dynamics [START_REF] Buluta | Quantum simulators[END_REF][START_REF] Cirac | Goals and opportunities in quantum simulation[END_REF][START_REF] Bloch | Quantum simulations with ultracold quantum gases[END_REF][START_REF] Blatt | Quantum simulations with trapped ions[END_REF][START_REF] Aspuru-Guzik | Photonic quantum simulators[END_REF][START_REF] Houck | On-chip quantum simulation with superconducting circuits[END_REF] has given rise to expectations that society may before long benefit from such quantum technologies. These developments are driven by our rapidly evolving abilities to experimentally manipulate and control quantum dynamics in diverse systems, ranging from single photons (2, 13), atoms and ions [START_REF] Bloch | Quantum simulations with ultracold quantum gases[END_REF][START_REF] Blatt | Quantum simulations with trapped ions[END_REF], and individual electron and nuclear spins [START_REF] Morton | Hybrid solid state qubits: The powerful role of electron spins[END_REF][START_REF] Maurer | Room-temperature quantum bit memory exceeding one second[END_REF][START_REF] Pla | High-fidelity readout and control of a nuclear spin qubit in silicon[END_REF], to mesoscopic superconducting [START_REF] Houck | On-chip quantum simulation with superconducting circuits[END_REF][START_REF] Schoelkopf | Wiring up quantum systems[END_REF] and nanomechanical devices [START_REF] Poot | Mechanical systems in the quantum regime[END_REF][START_REF] Aspelmeyer | Cavity optomechanics[END_REF]. As a rule, each of these systems can execute one or a few specific tasks, but no single system can be universally suitable for all envisioned applications. Thus, photons are best suited for transmitting quantum information, weakly interacting spins may serve as long-lived quantum memories, and the dynamics of electronic states of atoms or electric charges in semiconductors and superconducting elements may realize rapid processing of information encoded in their quantum states. The implementation of devices that can simultaneously perform several or all of these tasks, e.g., reliably store, process, and transmit quantum states, calls for a new paradigm: that of hybrid quantum systems (HQSs) [START_REF] Morton | Hybrid solid state qubits: The powerful role of electron spins[END_REF][START_REF] Wallquist | Hybrid quantum devices and quantum engineering[END_REF][START_REF] Xiang | Hybrid quantum circuits: Superconducting circuits interacting with other quantum systems[END_REF][START_REF] Daniilidis | Quantum interfaces between atomic and solid-state systems[END_REF][START_REF] Treutlein | Hybrid mechanical systems. Cavity Optomechanics[END_REF]. HQSs attain their multitasking capabilities by combining different physical components with complementary functionalities.

Many of the early ideas for HQSs emerged from the field of quantum information processing and communication (QIPC) and were, to a large extent, inspired by the development of QIPC architectures in which superconducting qubits are coupled to highquality microwave resonators [START_REF] Schoelkopf | Wiring up quantum systems[END_REF]25). Superconducting qubits are very-well-controlled quantum systems [START_REF] Devoret | Superconducting circuits for quantum information: An outlook[END_REF][START_REF] You | Atomic physics and quantum optics using superconducting circuits[END_REF], but in contrast to atoms, they suffer from comparatively short coherence times and do not couple coherently to optical photons. A microwave resonator, such as, for example, a lumped-element LC-circuit or coplanar waveguide (CPW) resonator, can serve as an interface between superconducting qubits and also between superconducting qubits and other quantum systems with longer coherence times and optical transitions [START_REF] Schoelkopf | Wiring up quantum systems[END_REF][START_REF] Xiang | Hybrid quantum circuits: Superconducting circuits interacting with other quantum systems[END_REF][START_REF] Daniilidis | Quantum interfaces between atomic and solid-state systems[END_REF][START_REF] Bensky | Controlling quantum information processing in hybrid systems on chips[END_REF]. It has thus been proposed to couple superconducting qubits, via a "microwave quantum bus," to ions [START_REF] Tian | Interfacing quantum-optical and solid-state qubits[END_REF], atoms [START_REF] Sørensen | Capacitive coupling of atomic systems to mesoscopic conductors[END_REF][START_REF] Verdú | Strong magnetic coupling of an ultracold gas to a superconducting waveguide cavity[END_REF][START_REF] Petrosyan | Reversible state transfer between superconducting qubits and atomic ensembles[END_REF], polar molecules [START_REF] Rabl | Hybrid quantum processors: Molecular ensembles as quantum memory for solid state circuits[END_REF], electrons confined above a liquid helium surface [START_REF] Schuster | Proposal for manipulating and detecting spin and orbital States of trapped electrons on helium using cavity quantum electrodynamics[END_REF], and spin-doped crystals [START_REF] Morton | Hybrid solid state qubits: The powerful role of electron spins[END_REF][START_REF] Imamo Glu | Cavity QED based on collective magnetic dipole coupling: Spin ensembles as hybrid two-level systems[END_REF][START_REF] Wesenberg | Quantum computing with an electron spin ensemble[END_REF][START_REF] Marcos | Coupling nitrogen-vacancy centers in diamond to superconducting flux qubits[END_REF]. With the recent advances in the control of micro-and nanomechanical systems [START_REF] Poot | Mechanical systems in the quantum regime[END_REF][START_REF] Aspelmeyer | Cavity optomechanics[END_REF], the use of a mechanical quantum bus has also been identified as an alternative promising route for interfacing and communicating between various quantum systems [START_REF] Treutlein | Hybrid mechanical systems. Cavity Optomechanics[END_REF][START_REF] Tian | Optoelectromechanical transducer: Reversible conversion between microwave and optical photons[END_REF]. Here one exploits the ability of functionalized resonators to respond sensitively to weak electric, magnetic, or optical signals while still being sufficiently well isolated from the environment to enable coherent transmission of quantum states.

Nearly a decade after the initial proposals, the HQS idea has developed into a rapidly growing interdisciplinary field of research. The basic working principles of HQS have by now been demonstrated in several experiments, and a growing number of theoretical and experimental research activities is presently devoted to further exploration of this approach. It is thus timely to assess the current status of the field and highlight some of its most promising short-and long-term perspectives. In what follows, this is done by reviewing the overarching goals and focusing on a few examples of HQS representing some of the most active research directions in this field. References are provided where the reader may find further information on this field.

General Concepts and Experimental Implementations of HQSs

A necessary prerequisite for realizing a functional HQS is the ability to communicate quantum states and properties between its different components with high fidelity. For two physical systems A and B, this requires an interaction Hamiltonian H AB , which either conditions the evolution of one system on the state of the other or drives, in a correlated fashion, transitions in the two systems. In most of the examples discussed below, we will encounter interaction Hamiltonians of the form

H AB ' Zg eff ða + b + b + aÞ; [1]
where a ða + Þ and b ðb + Þ are de-excitation (excitation) operators, which, in a generic sense, cause transitions between states within systems A and B, respectively. The product a + b thus indicates a swap process in which an excitation in one system is accompanied by a de-excitation in the other. With an appropriate identification of the operators, Eq. 1 represents interacting systems such as quantum optical fields coupled by a beam splitter, atoms interacting resonantly with a cavity field, and naturally occurring spins interacting via their magnetic moments. If systems A and B have very different physical properties, it may be difficult to identify appropriate degrees of freedom that experience interactions of the form of Eq. 1.

One obstacle may arise from the effective coupling strength g eff being weak due to inadequate spatial (or impedance) matching between the systems. The couplings between microscopic systems, such as the spin-orbit and spin-spin interactions responsible for fine and hyperfine structure in atoms and molecules, are relatively strong because the electrons and nuclei are confined within Ångström distances. Alternatively, mesoscopic superconducting qubits may strongly couple to each other because of their large electric dipoles, associated with the spatial extent of the region traversed by the sustained electric currents. However, when a single atom, ion, or electron is placed near a micrometer-or millimeter-sized superconducting system, the coupling between the two is several orders of magnitude weaker. Another common challenge is rooted in the difference of the energy scales in the systems that we intend to couple. Even in the presence of a strong interaction, the swap process described by H AB will not take place if it does not conserve energy, meaning that the excitation energies of the two subsystems are very dissimilar. Much of the research in the field of HQS is devoted to overcoming these obstacles.

In Fig. 1, we illustrate various candidate components of HQSs, characterized by their typical Bohr excitation frequencies (vertical axis) and their coherence times (horizontal axis). The location of each system on the horizontal axis of the figure identifies the tasks that are best delegated to that component of a HQS: e.g., spins are useful for storage, whereas superconducting qubits may be more practical for rapid processing of quantum states. The coherence time T 2 , i.e., the time over which quantum superposition states survive, determines the minimal coupling strength required for a HQS component to function with sufficiently high fidelity: The (effective) coupling rate g eff between systems A and B must be large enough to allow quantum state transfer between them within the shortest coherence time of the two, g eff T 2 min 1.

The arrows connecting different components of HQS in Fig. 1 are labeled by approximate values of g eff that can be realistically achieved with present-day technology. Some of the larger coupling strengths in Fig. 1 seem to contradict our observation concerning the weak coupling between very different physical systems. This contradiction is resolved by noting that the coupling of a mesoscopic system via light or microwave fields to ensembles rather than to single atoms or spin dopants (see below). The red and blue arrows in the figure indicate the single-system and ensemble coupling strengths, respectively. Fig. 1 also shows various examples for the coupling of systems with strongly dissimilar excitation energies (dashed lines). In such cases, the coupling mechanism involves an external source or sink, such as a (classical) laser or microwave field, which bridges the energy mismatch to make the processes described by Eq. 1 resonant when they are accompanied by absorption or stimulated emission of photons. This coupling mechanism applies to the well-known processes of laser-assisted optical Raman transitions in atoms and molecules. In optomechanics [START_REF] Aspelmeyer | Cavity optomechanics[END_REF], parametric coupling via an applied control field is used to bridge the energy difference between mechanical vibrational modes and optical photons and to enhance the interaction strength g eff .

We next present in more detail the specifics of different systems sketched in Fig. 1 and describe some of the ideas for their hybridization. Spin-Ensemble Quantum Memories for Superconducting Qubits. The simplest superconducting qubit is an electrical LC circuit (resonator) in which the inductance is replaced by a nonlinear Josephson junction.

The excitation spectrum of this resonator then becomes anharmonic at the singlequantum level. It can thus behave as an effective two-level system [START_REF] Devoret | Superconducting circuits for quantum information: An outlook[END_REF][START_REF] You | Atomic physics and quantum optics using superconducting circuits[END_REF]. Due to their macroscopic size, typically between 100 μm and 1 mm, superconducting qubits possess a large electric dipole moment, and hence they couple strongly to the microwave field of a CPW resonator [START_REF] Schoelkopf | Wiring up quantum systems[END_REF]. This strong coupling allows efficient qubit readout, as well as rapid (100-ns duration) exchange of photons with the resonator, which can mediate high-fidelity quantum gates between different qubits (25). The strong interactions of superconducting qubits with the environment, however, lead to modest coherence times, of the order of 10-100 μs at best [START_REF] Paik | Observation of high coherence in Josephson junction qubits measured in a three-dimensional circuit QED architecture[END_REF][START_REF] Barends | Coherent Josephson qubit suitable for scalable quantum integrated circuits[END_REF].

On the other hand, due to the small magnetic moments, electronic or nuclear spins interact only weakly with their environment, and even at room temperature, spins embedded in a solid-state matrix can exhibit coherence time reaching seconds for electron spins [START_REF] Bar-Gill | Solid-state electronic spin coherence time approaching one second[END_REF] and many minutes, up to hours, for nuclear spins [START_REF] Saeedi | Room-temperature quantum bit storage exceeding 39 minutes using ionized donors in silicon-28[END_REF]. At the same time, spins can be densely packed because their mutual influence is weak. Thus, spin ensembles provide a natural storage medium for quantum states.

The prospect of combining fast superconducting processing qubits with long-lived spin quantum memories has led to some of the first HQS proposals [START_REF] Imamo Glu | Cavity QED based on collective magnetic dipole coupling: Spin ensembles as hybrid two-level systems[END_REF][START_REF] Wesenberg | Quantum computing with an electron spin ensemble[END_REF], in which spins and qubits are coupled to the same CPW microwave cavity, and storage of a qubit state is mediated by a cavity photon that is absorbed by the spins. The magnetic coupling strength between a single spin and a microwave photon is only about g ∼ 10 Hz in typical CPW resonators, which is too small for a photon to be absorbed before it leaks out of the cavity. It has therefore been proposed to use a large ensemble of N spins that couple to a photon ffiffiffiffi N p times stronger than a single spin. Indeed, the resonant coupling of an ensemble of N spins to a single cavity mode is described by the Tavis-Cummings model with the Hamiltonian

H = Zg X N n=1 À c ↑ n ↓ + c + ↓ n ↑ Á = Zg ffiffiffiffi N p ðc S + + c + S Þ; [2]
where c (c + ) denotes the annihilation (creation) operator of the cavity field, and the operator

S + = ð1= ffiffiffiffi N p Þ P ↑ n ↓ creates a
collective spin excitation distributed over the entire ensemble. Eq. 2 clearly reveals the collective enhancement factor g eff = g ffiffiffiffi N p of the effective coupling rate between the cavity mode and the spin ensemble, which, for N = 10 11 -10 12 , can exceed both the spin decoherence rate 1=T 2 and typical cavity damping rates κ ≈ 10 5 s -1 . Following these early proposals, the strong collective coupling regime has been observed in a number of systems, P1 centers in diamond [START_REF] Schuster | High-cooperativity coupling of electron-spin ensembles to superconducting cavities[END_REF][START_REF] Ranjan | Probing dynamics of an electron-spin ensemble via a superconducting resonator[END_REF], nitrogen-vacancy (NV) centers in diamond [START_REF] Kubo | Strong coupling of a spin ensemble to a superconducting resonator[END_REF][START_REF] Amsüss | Cavity QED with magnetically coupled collective spin states[END_REF], and erbium ions in YSiO 2 [START_REF] Probst | Anisotropic rare-earth spin ensemble strongly coupled to a superconducting resonator[END_REF]. Much larger collective coupling constants can be reached if the ensemble of noninteracting spins is replaced by exchange-coupled electron spins in insulating ferromagnets such as yttrium iron garnet, providing a density of electron spins up to four orders of magnitude larger [START_REF] Huebl | High cooperativity in coupled microwave resonator ferrimagnetic insulator hybrids[END_REF][START_REF] Tabuchi | Hybridizing ferromagnetic magnons and microwave photons in the quantum limit[END_REF][START_REF] Zhang | Strongly coupled magnons and cavity microwave photons[END_REF][START_REF] Goryachev | High-cooperativity cavity QED with magnons at microwave frequencies[END_REF]. Although the shorter lifetime of the magnonic elementary excitations in these systems (1 μs at best) makes them unsuitable for quantum memory applications, these novel hybrid systems open up new possibilities in the areas of quantum magnonics and quantum transducers.

Recent experiments have demonstrated the possibility of quantum-state transfer from a superconducting qubit to a spin ensemble [START_REF] Zhu | Coherent coupling of a superconducting flux qubit to an electron spin ensemble in diamond[END_REF][START_REF] Kubo | Hybrid quantum circuit with a superconducting qubit coupled to a spin ensemble[END_REF][START_REF] Saito | Towards realizing a quantum memory for a superconducting qubit: Storage and retrieval of quantum states[END_REF][START_REF] Tabuchi | Coherent coupling between ferromagnetic magnon and superconducting qubit[END_REF]. The experiment in ref. [START_REF] Kubo | Hybrid quantum circuit with a superconducting qubit coupled to a spin ensemble[END_REF] is schematically depicted in Fig. 2A: a superconducting qubit is electrically coupled to a superconducting quantum interference device (SQUID)-based CPW resonator, which in turn is magnetically coupled to an NVcenter ensemble. An arbitrary superposition αjgi + βjei of the qubit ground ðjgiÞ and excited ðjeiÞ states is first transferred to the corresponding superposition of the CPW bus (resonator) energy states αj0i + βj1i, with j0i and j1i denoting the 0-and 1-photon Fock states. This state is then mapped onto a superposition αj↓i + βS + j↓i of the spin-ensemble ground state j↓i, corresponding to all spins in the ground state, and the collective, symmetric, single-excitation state

S + j↓i = ðj↑ 1 ↓ 2 ::↓ N i+ j↓ 1 ↑ 2 ::↓ N i+ . . . +j↓ 1 ↓ 2 ::↑ N iÞ= ffiffiffiffi N p
. Typical results are shown in Fig. 2B for the initial qubit state jψi = ðjgi + jeiÞ=√2 undergoing periodic storage and retrieval cycles in this proof-of-principle demonstration of a spin-ensemble quantum memory.

The retrieval signal in Fig. 2B is seen to decay in a few hundred nanoseconds, which is many orders of magnitude faster than coherence times reported for individual NV centers in diamond. This rapid decay stems from the inhomogeneous broadening of the spin ensemble, namely, from each spin having a slightly different resonance frequency ω i in its specific local environment (formed by other electronic spin impurities, nuclear spins, local strain, etc.). The spread in frequency of the spin ensemble causes the collective single-excitation state S + j↓i to evolve during time τ into the state

jψðτÞi = À j↑ 1 ↓ 2 ::↓ N ie iϕ 1 + j↓ 1 ↑ 2 ::↓ N ie iϕ 2 + . . . + j↓ 1 ↓ 2 ::↑ N ie iϕ N Á ffiffiffiffi N p ; [3]
with ϕ i = -ω i τ. The accumulation of different phases implies that the terms in the superposition of Eq. 3 no longer interfere constructively and the stored state cannot be retrieved as a cavity photon after storage time exceeding T* 2 ≈ 1=Δω, which is determined by the (inhomogeneous) linewidth Δω of the spin ensemble [START_REF] Kubo | Hybrid quantum circuit with a superconducting qubit coupled to a spin ensemble[END_REF].

The extension of storage time of the solidstate spin-ensemble quantum memories beyond the inhomogeneous linewidth limit is a subject of ongoing research. One remedy to the problem can be to use stronger couplings, g eff Δω, to shift the frequency of the hybridized spin wave-cavity excitation out of the continuum of spin transition frequencies. This approach has indeed been demonstrated to lead to a significantly reduced decay of collective excitations in such systems [START_REF] Putz | Protecting a spin ensemble against decoherence in the strong-coupling regime of cavity QED[END_REF].

Another potentially viable approach involves a prior optimal spectral filtering of the spin ensemble [START_REF] Bensky | Optimizing inhomogeneous spin ensembles for quantum memory[END_REF], which leaves an appropriate subset of the spins to serve as a high-fidelity memory. Finally, because individual spins in the inhomogeneously broadened ensemble evolve unitarily, each having its own transition frequency, the precession direction of all of the spins can be reversed by applying wellknown refocusing techniques, such as Hahn echo sequences used in magnetic resonance experiments. This refocusing can enable the retrieval of the stored quantum state as a spin-echo long after the ensemble state would have lost its phase coherence due to inhomogeneous broadening [START_REF] Wu | Storage of multiple coherent microwave excitations in an electron spin ensemble[END_REF][START_REF] Afzelius | Proposal for a coherent quantum memory for propagating microwave photons[END_REF][START_REF] Julsgaard | Quantum memory for microwave photons in an inhomogeneously broadened spin ensemble[END_REF][START_REF] Grezes | Multimode storage and retrieval of microwave fields in a spin ensemble[END_REF].

On the other hand, inhomogeneous broadening and its effective refocusing by π pulses offer the possibility of using the spin ensemble for multimode storage. Because the photonic excitation stored in the ensemble at time t is dephased during time T 2 *, another photon can be transferred to a collective spinexcitation mode at time t + τ, if τ > T 2 *. By exploiting the vanishing overlap between any two such storage modes, this transfer process can be repeated several times [START_REF] Wu | Storage of multiple coherent microwave excitations in an electron spin ensemble[END_REF][START_REF] Afzelius | Proposal for a coherent quantum memory for propagating microwave photons[END_REF][START_REF] Julsgaard | Quantum memory for microwave photons in an inhomogeneously broadened spin ensemble[END_REF][START_REF] Grezes | Multimode storage and retrieval of microwave fields in a spin ensemble[END_REF]. The number of photon pulses stored within the homogeneous lifetime T 2 (T 2 *) of individual spins is limited by the time-bandwidth product T 2 /T 2 *, allowing the storage of up to a few hundred photonic qubits in an ensemble of NV centers [START_REF] Julsgaard | Quantum memory for microwave photons in an inhomogeneously broadened spin ensemble[END_REF]. Encouraging experimental results have already been obtained for the sequential storage of several weak microwave pulses and their retrieval tens of microseconds later [START_REF] Grezes | Multimode storage and retrieval of microwave fields in a spin ensemble[END_REF]. Multimode storage has also been demonstrated in a phosphorus-doped silicon crystal [START_REF] Wu | Storage of multiple coherent microwave excitations in an electron spin ensemble[END_REF]. In this experiment, the strong hyperfine interaction between each individual electron and its parent ion was further exploited to convert a sequence of collective electron-spin excitations into nuclear excitations, which were then stored and subsequently transferred back to the electron spins and the microwave field after seconds of storage time.

These initial results and ideas indicate the feasibility of a practical spin-ensemble quantum memory, capable of simultaneously storing the states of hundreds of superconducting qubits, for many seconds, and, potentially, for hours. Such memories would constitute a prime example of the benefits of the HQS approach.

Atomic Ensembles as Memories and

Optical Interfaces. In addition to information processing and storage, quantum states may be used for secure transmission of data. Distribution of data within larger QIPC architectures and practical quantum cryptography can only be achieved using optical photons propagating in free space or optical fibers (2, 3). The implementation of coherent interfaces between quantum memories, processing qubits, and "flying" optical qubits is thus of general importance in quantum information science.

Isolated atoms have optical (Raman) transitions with excellent properties for coherent absorption and emission, as well as storage of photons. Therefore, the coupling of ensembles of cold trapped atoms [START_REF] Verdú | Strong magnetic coupling of an ultracold gas to a superconducting waveguide cavity[END_REF][START_REF] Petrosyan | Reversible state transfer between superconducting qubits and atomic ensembles[END_REF] and molecules [START_REF] Rabl | Hybrid quantum processors: Molecular ensembles as quantum memory for solid state circuits[END_REF] to CPW resonators was among the initial proposals for HQSs capable of providing both long storage times and efficient optical interfaces. Compared with solid-state spin ensembles, cooling and trapping atoms in the vicinity of superconducting CPW, however, introduce severe experimental complications, and the realistic number of trapped particles of about N = 10 5 -10 6 is much smaller than what is achievable with spin-doped crystals and hence the coupling to atomic spin ensembles is generally weaker [START_REF] Verdú | Strong magnetic coupling of an ultracold gas to a superconducting waveguide cavity[END_REF].

To compensate for the reduced coupling, strong electric-dipole transitions between rotational states in polar molecules [START_REF] Rabl | Hybrid quantum processors: Molecular ensembles as quantum memory for solid state circuits[END_REF] or between highly excited Rydberg states of atoms [START_REF] Petrosyan | Reversible state transfer between superconducting qubits and atomic ensembles[END_REF][START_REF] Petrosyan | Quantum information processing with single photons and atomic ensembles in microwave coplanar waveguide resonators[END_REF]) can be used. In particular, the transitions in the range of tens of gigahertz between circular Rydberg states with huge dipole moments have been used [START_REF] Raimond | Manipulating quantum entanglement with atoms and photons in a cavity[END_REF] to strongly couple Rydberg atoms to single microwave photons in 3D resonators. Preliminary findings show that similar coupling can also be realized in on-chip CPW cavities [START_REF] Hogan | Driving Rydberg-Rydberg transitions from a coplanar microwave waveguide[END_REF][START_REF] Hermann-Avigliano | Long coherence times for Rydberg qubits on a superconducting atom chip[END_REF]. Ref. 32 details a scheme where a cloud of atoms, initially prepared in the ground state jgi, is coupled to a collective Rydberg excitation state via a two-photon process involving an optical pump field and a single photon of a CPW resonator. Overall, this scheme realizes an interaction Hamiltonian of the generic form given by Eq. 1

H = Zg ffiffiffiffi N p ðc R + + c + R Þ; [4]
where the operator R + = ð1= ffiffiffiffi N p Þ P r n g creates a symmetric (collective) excitation of the Rydberg state jri, and g = Ω gi η ir =δ is the effective two photon coupling rate. This rate is proportional to the Rabi frequency Ω gi of the optical pump field between the ground state jgi and an intermediate Rydberg state jii and the Rabi frequency η ir of a single microwave photon interacting with the Rydberg states jii and jri, and is inversely proportional to the detuning δ Ω gi ; η ir from the intermediate state. With a strong enough pump field Ω gi and a very large dipole matrix element (η ir ∼ n 2 ), such as obtained for the transition between neighboring Rydberg states with high principal quantum number n ≅ 70, the necessary effective coupling strength of g eff = g ffiffiffiffi N p J 1 MHz can already be reached with a reasonable number of N = 10 6 atoms.

Once the state transfer between the microwave photon and the atoms is complete, additional optical transitions can be used to transfer the collective Rydberg excitation to a long-lived spin excitation in the groundstate hyperfine manifold and back or map it onto a propagating photon mode. In free space, stimulated Raman techniques, such as electromagnetically induced transparency [START_REF] Fleischhauer | Electromagnetically induced transparency: Optics in coherent media[END_REF], enable a coherent and reversible conversion of collective spin excitations into photons in well-defined spatiotemporal modes. This process requires, however, a large optical depth of the medium, OD = σρl > 1, where σ is the single-atom absorption cross section, ρ is the atom density, and l is the length of the medium [START_REF] Gorshkov | Universal approach to optimal photon storage in atomic media[END_REF]. By placing the atomic medium in an optical resonator with high finesse F, the transfer efficiency can be further increased, reaching the optimal value of C=ð1 + CÞ [START_REF] Gorshkov | Photon storage in Λ-type optically dense atomic media. I. Cavity model[END_REF]. Here C = F × OD is the optical cooperativity that represents the key figure of merit for the optical interface. In view of the low atomic densities in such setups, the integration of atomic traps with on-chip photonic structures [START_REF] Hung | Trapped atoms in one-dimensional photonic crystals[END_REF][START_REF] Tiecke | Nanophotonic quantum phase switch with a single atom[END_REF] is a promising experimental approach toward achieving hybrid interfaces with C 1.

Trapping and cooling of atoms in chipbased traps is a well-established technique, which is, however, technically demanding at cryogenic temperatures and close to a superconducting surface. Nevertheless, magnetic trapping and even cooling of a cloud of atoms down to quantum degeneracy [i.e., to Bose-Einstein condensation (BEC)] above a superconducting chip has been demonstrated [START_REF] Nirrengarten | Realization of a superconducting atom chip[END_REF][START_REF] Bernon | Manipulation and coherence of ultra-cold atoms on a superconducting atom chip[END_REF]. New ways of using superconducting resonators and circuits directly for trapping are currently being explored [START_REF] Bothner | Inductively coupled superconducting half wavelength resonators as persistent current traps for ultracold atoms[END_REF]. Other experimental approaches to the realization of atomic HQSs are being pursued [START_REF] Hafezi | Atomic interface between microwave and optical photons[END_REF], where atoms are optically trapped in the evanescent field of a tapered fiber [START_REF] Vetsch | Optical interface created by laser-cooled atoms trapped in the evanescent field surrounding an optical nanofiber[END_REF]. This proximity of the atoms to the fiber would enable highfidelity conversion of atomic quantum states into propagating photons in the fiber while minimizing the perturbation of the superconducting circuit because of the localization of the trapping and Raman laser beams.

In parallel to these experimental efforts on atomic microwave-to-optics interfaces, other approaches using optical and microwave transitions in spin-doped crystals are being developed [START_REF] O'brien | Interfacing superconducting qubits and telecom photons via a rareearth-doped crystal[END_REF][START_REF] Williamson | Magneto-optic modulator with unit quantum efficiency[END_REF]. The goal here is to achieve a maximal overlap between the spin excitations created by the respective optical and microwave modes while minimizing the absorption of optical photons on the superconductor. Using for this purpose larger 3D microwave resonators, instead of planar CPW, may be beneficial, because both the optical and microwave modes can have their maxima at the center of such resonators, away from the superconducting walls [START_REF] Williamson | Magneto-optic modulator with unit quantum efficiency[END_REF]. The reduced magnetic or optical coupling to a single spin is compensated in this approach by the larger number of particles that can be enclosed within the microwave mode volume [START_REF] Chiorescu | Magnetic strong coupling in a spin-photon system and transition to classical regime[END_REF][START_REF] Abe | Electron spin ensemble strongly coupled to a three-dimensional microwave cavity[END_REF][START_REF] Probst | Three-dimensional cavity quantum electrodynamics with a rare-earth spin ensemble[END_REF].

Mechanical Quantum Transducers. As an alternative to optical and microwave photons, quantum information can be converted and transmitted via quantized mechanical vibrations of opto-and nano-mechanical systems [START_REF] Poot | Mechanical systems in the quantum regime[END_REF][START_REF] Aspelmeyer | Cavity optomechanics[END_REF]. High-Q micro-and nano-mechanical resonators, such as tiny cantilevers or suspended membranes, respond very sensitively to applied forces, which makes them suitable for diverse applications relying on the detection of weak signals. When applied at the level of single quanta, the same principle of force sensing-for example, the conversion of a weak magnetic force into a detectable electric or optical signal (81, 82)-opens up new possibilities for mechanically interfacing quantum systems of different types [START_REF] Treutlein | Hybrid mechanical systems. Cavity Optomechanics[END_REF]. The recently demonstrated cooling of mechanical vibrational modes close to the quantum ground state [START_REF] Connell | Quantum ground state and singlephonon control of a mechanical resonator[END_REF][START_REF] Teufel | Sideband cooling of micromechanical motion to the quantum ground state[END_REF][START_REF] Chan | Laser cooling of a nanomechanical oscillator into its quantum ground state[END_REF], and the current experimental efforts to couple mechanical resonators to superconducting circuits [START_REF] Connell | Quantum ground state and singlephonon control of a mechanical resonator[END_REF][START_REF] Lahaye | Nanomechanical measurements of a superconducting qubit[END_REF][START_REF] Pirkkalainen | Hybrid circuit cavity quantum electrodynamics with a micromechanical resonator[END_REF], atoms [START_REF] Hunger | Resonant coupling of a Bose-Einstein condensate to a micromechanical oscillator[END_REF], and spins [START_REF] Arcizet | A single nitrogen-vacancy defect coupled to a nanomechanical oscillator[END_REF][START_REF] Kolkowitz | Coherent sensing of a mechanical resonator with a single-spin qubit[END_REF] are important initial steps in this direction.

The basic ideas and potential applications of mechanical quantum transducers are illustrated in Fig. 3. The setup shown in Fig. 3A depicts a mechanical spin transducer, where localized electronic spin qubits are coupled to the quantized motion of a magnetized vibrating tip. In the presence of strong magnetic field gradients, the motion of the tip modulates the Zeeman splitting of the spin eigenstates below the tip and results in a spin-resonator interaction of the form ( 91)

H int = Zλðb + b + Þσ z : [5]
Here, b is the annihilation operator for the mechanical oscillator mode, σ z is the Pauli operator for the spin, and λ is the coupling strength per phonon. Unlike Eq. 1, H int does not describe a resonant exchange of excitations. Instead, it represents a spindependent force, which evolves an initial spin superposition state into an equivalent superposition of displaced mechanical states.

If the resonator is electrically charged, the weak magnetic moments of the spins are effectively amplified via this process into large electric dipoles, enabling, for example, strong electric interactions between separated spin qubits. Similar principles underlie the realization of various other mechanical hybrid systems in which spins or superconducting qubits are mechanically coupled with each other or interfaced with photons (92), trapped ions [START_REF] Tian | Coupled ion-nanomechanical systems[END_REF][START_REF] Hensinger | Ion trap transducers for quantum electromechanical oscillators[END_REF], or atomic systems [START_REF] Hunger | Resonant coupling of a Bose-Einstein condensate to a micromechanical oscillator[END_REF][START_REF] Hammerer | Strong coupling of a mechanical oscillator and a single atom[END_REF][START_REF] Gao | Coupling Rydberg atoms to superconducting qubits via nanomechanical resonator[END_REF][START_REF] Bariani | Single-atom quantum control of macroscopic mechanical oscillators[END_REF].

First experiments-still in the classical regime-have shown that micromechanical oscillators can be magnetically coupled to hyperfine states of cold atoms [START_REF] Hunger | Resonant coupling of a Bose-Einstein condensate to a micromechanical oscillator[END_REF] or individual impurity spins [START_REF] Arcizet | A single nitrogen-vacancy defect coupled to a nanomechanical oscillator[END_REF][START_REF] Kolkowitz | Coherent sensing of a mechanical resonator with a single-spin qubit[END_REF]. For a coherent coupling of two or multiple qubits via mechanical channels, it is necessary to reach the regime of strong (mechanical) cooperativity, C m = λ 2 T 2 T m > 1 (91). Here T 2 is the qubit coherence time and T -1 m = k B T=ðZQÞ is the characteristic mechanical decoherence rate, where Q is the quality factor of the resonator mode, T is the support temperature, and k B is Boltzmann's constant. Simple estimates show that for spin qubits, the condition C m > 1 can be realistically achieved using state-of-the-art mechanical resonators with Q ∼ 10 5 -10 6 and working at T ≤ 1 K temperatures [START_REF] Kolkowitz | Coherent sensing of a mechanical resonator with a single-spin qubit[END_REF][START_REF] Rabl | A quantum spin transducer based on nano electro-mechancial resonator arrays[END_REF]. For superconducting qubits, the electrostatic interaction with nano-mechanical resonators can be significantly stronger [START_REF] Armour | Quantum Dynamics of a Cooper-Pair Box Coupled to a Micromechanical Resonator[END_REF]. Initial experiments accessing the full strong-coupling regime λ > T -1 2 ; T -1 m have already been performed [START_REF] Connell | Quantum ground state and singlephonon control of a mechanical resonator[END_REF][START_REF] Pirkkalainen | Hybrid circuit cavity quantum electrodynamics with a micromechanical resonator[END_REF].

Nano-mechanical systems are of particular interest for the development of a universal opto-mechanical (OM) transducer for coherently interfacing optical and microwave photons (38, 99-101) (Fig. 3 C andD). In OM systems, the frequency ω c of an optical cavity mode is modulated by the motion of a mechanical resonator with mechanical frequency ω m . Common examples of OM systems include Fabry-Perot cavities with a movable end mirror or with a semitransparent membrane placed inside the cavity [START_REF] Aspelmeyer | Cavity optomechanics[END_REF]. Nano-photonic systems (102, 103) and photonic bandgap structures (99, 104) also exhibit similar interactions. The OM system is described by the Hamiltonian

H = Zω c c + c + Zω m b + b + Zg 0 c + c ðb + b + Þ; [6]
where c is the annihilation operator for the optical mode. The first two terms in this equation represent the unperturbed energies of the optical and mechanical modes, respectively, and the third term describes the radiation pressure coupling. The coupling constant g 0 is the optical frequency shift per vibrational quantum, which is typically very small. However, the OM interaction can be parametrically amplified by driving the cavity with a strong laser of frequency ω d . In this case, the total field inside the cavity is c = [α(t) +δc]exp(-iω d t), where δc represents the quantum fluctuations around a large classical field amplitude α(t). By choosing the resonance condition ω c = ω d + ω m , the dominant OM coupling term then becomes analogous to a beam-splitter interaction (20)

H ΟΜ ' ZGðtÞðδc + b + b + δcÞ; [7]
where, similar to an anti-Stokes scattering process, low-frequency mechanical excitations are up-converted by the driving field into optical signal photons and vice versa.

The effective coupling G(t)= α(t)g 0 is then enhanced and controlled by the external driving field.

In the microwave domain, an analogous coupling arises when mechanical oscillations modulate the capacitance of a superconducting CPW resonator or LC circuit [START_REF] Teufel | Sideband cooling of micromechanical motion to the quantum ground state[END_REF]105). Here too, the coupling is amplified and controlled by a strong microwave driving field that bridges the frequency difference between the circuit and mechanical resonances. It is important that due to the underlying parametric interaction, the effective photon-phonon interface in Eq. 5 does not rely on the absolute frequency of the optical or microwave mode, which in both cases is compensated by the frequency of the external driving field (Fig. 3D). Therefore, by using a single mechanical membrane as both a mirror and a capacitor, an effective optics-to-microwave interface is achieved, whereby microwave photons are converted to phonons and successively to (flying) optical photons.

Various designs for the experimental implementation of coherent microwaveto-optics transducers are currently being explored. An efficient OM conversion between optical and microwave signals has already been demonstrated in both room temperature (106, 107) and cryogenic (108) environments. Despite many obstacles that currently still hinder a fully coherent operation of such experiments at a single-photon level, the prospects for OM quantum interfaces between solid-state, atomic, and optical systems are very promising.

Outlook: Quo Vadis, Quantum Hybridium?

Hybrid quantum systems are still a long way from implementing general quantum information processing and communication tasks with the fidelities needed for practical applications. The integration of very different physical components presents technological and scientific challenges that are absent when controlling each component individually. However, the experimental realizations of HQSs described above show that these obstacles may be overcome. Protocols with controlled time-dependent couplings are currently being investigated to optimize quantum state transfer speed and fidelity. The shape of the coupling fields can be tailored to the temporal response of the noisy (decohering) environments (109), and quantum measurements and feedback may also be applied (110), in an effort to reduce or eliminate the effect of dissipation and decoherence at HQS interfaces. There is little doubt that continued progress along these directions will enable hybrid, multitasking quantum technologies of increasing sophistication.

To provide an overview of tasks for which HQSs may be used in the near-and moredistant future, we indicate in Fig. 4 the infidelities tolerated by various potential technological applications of quantum effects. Obviously, the ultimate goal of realizing a large-scale quantum computer including algorithmic quantum error correction is beyond current capabilities. At the other, more modest, end of the scale are communication and sensing, which may function, albeit at a lower rate, even if the coupling process has a low fidelity or is heralded with low success probability. Here, HQSs are expected to have a great impact rather soon. For example, the unprecedented level of control and low-noise amplification at microwave frequencies achieved in superconducting circuits has already been used to detect an electron-spin resonance at the level of few excitations (111). Similarly, OM transducers are being explored for low-noise optical detection of weak radio frequency signals (107). More generally, the anticipated ability of HQSs to transfer highly nonclassical (entangled) states between different physical platforms may extend quantum-enhanced sensing schemes to systems where such a high level of quantum control is a priori not available. The mapping of squeezed microwave or optical fields onto spin ensembles for magnetometry and the preparation of nanomechanical sensors in highly sensitive quantum superposition states are possible applications along these lines.

Turning from practical applications to more fundamentally oriented research, the scaling-up of HQSs may offer new possibilities to simulate and study complex phenomena in quantum many-body systems. Especially, the combination of different systems with optimized coherent and tailored (engineered) dissipative properties may be used to investigate issues related to nonequilibrium phase transitions in open quantum many-body systems [START_REF] Cirac | Goals and opportunities in quantum simulation[END_REF](112)(113)(114). Such systems are currently difficult to realize in the laboratory, but analyses of dissipative Dicketype lattice models with HQS arrays of spin ensembles and superconducting circuits (115-118) show the potential for achieving scalable systems with sufficiently large interaction strengths.

Overall, the HQS approach concurs with the long-term vision of a "quantum information era," in which quantum information is processed, stored, and transmitted in a modular and platform-versatile manner. The fascinating prospects of this research will keep scientists occupied for some time and are likely to stimulate many ideas and motivate experts from research and engineering areas not even mentioned in this review to confront the challenges of HQSs.

Fig. 1 .

 1 Fig. 1. HQS overview. The diagram shows a selection of physical systems that represent components of HQS with different functionalities. The individual systems are positioned in the diagram according to their characteristic excitation frequencies (vertical axis) and coherence times (horizontal axis). The arrows indicate possible coupling mechanisms and the corresponding coupling strengths g eff that can be realistically achieved. The red and the blue arrows represent the coupling between single systems and the coupling to and between ensembles, respectively. Couplings represented by dashed lines are assisted by additional classical laser or microwave fields to bridge the apparent mismatch of the excitation energies. See text for more detail.

Fig. 2 .

 2 Fig. 2. Spin-ensemble quantum memory. (A) Photograph (Upper) and schematic drawing (Lower) of the hybrid quantum circuit realized in ref. 53. A transmon qubit (red) is coupled to an ensemble of NV-center electron spins (pink) via a frequency-tunable quantum bus resonator (orange). (B) Swap oscillations of a coherent superposition of quantum states, (jg〉 +je〉)/√2 initially prepared in the qubit, showing cycles of storage (dashed arrows) in and retrieval (solid arrows) from the spin ensemble.

Fig. 3 .

 3 Fig. 3. Mechanical quantum transducers. (A) A magnetized mechanical resonator is coupled to a localized electronic spin qubit and converts small spin-induced displacements into electric signals. Thereby spin qubits can be "wired up" electrically or coupled to other charged quantum systems. (B) Illustration of an OM interface between a superconducting qubit and optical (flying) photons. Here the mechanical system, represented by a semitransparent membrane, simultaneously acts as a capacitor and an optical reflector. (C) Experimental setup used in ref. 108 to implement an optomechanical microwave-to-optics interface via simultaneous coupling of a partially metallized membrane to an optical cavity and an LC circuit. (D) A signal photon of frequency ∼ ω op c enters the optical cavity and is down-converted via the driven, parametric OM interaction into a phonon of frequency ω m . Then, via an equivalent process, this mechanical excitation is up-converted again into a microwave photon of frequency ∼ ω mw c in the LC circuit. Via this mechanism and its reverse, quantum information encoded in microwave excitations of a superconducting qubit or LC resonator can be coherently transferred into optical signals for long-distance quantum communication.

Fig.

  Fig. Potential applications that can be performed by a HQS with achievable infidelity 1-F.

| www.pnas.org/cgi/doi/10.1073/pnas.1419326112 Kurizki et al.

| www.pnas.org/cgi/doi/10.1073/pnas.1419326112 Kurizki et al.

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5(4):222-229.

| www.pnas.org/cgi/doi/10.1073/pnas.1419326112 Kurizki et al.

ACKNOWLEDGMENTS. We would like to thank the many colleagues and collaborators for their contributions to the field of hybrid quantum systems and to our own earlier work. This article was initiated upon successful completion of the European Commission project MIDAS (Macroscopic Interference Devices for Atoms and Solids). We acknowledge support from ISF, United States-Israel Binational Science Foundation and Alternative Energy Research Initiative (G.K.), the Villum Foundation (K.M.), the Humboldt Foundation (D.P.), European Projects SIQS (P.R. and J.S.) and SCALEQIT (Scalable Quantum Information with Transmons) (P.B., Y.K.), the Austrian Science Fund through SFB FOQUS (P.R., J.S.), the START Grant Y 591-N16 (P.R.), the Agence Nationale de la Recherche through the European Coordinated Research on Longterm Challenges in Information and Communication Sciences & Technologies ERA-Net (CHIST-ERA) program QINVC (Quantum Information with NV Centers) (P.B., Y.K.), and JSPS (Japan Society for the Promotion of Science) (Y.K.).


W L Ma

M Stern

A Bienfait

C Grezes

B Julsgaard

Y Kubo

K Nakamura

J Isoya

S Onoda

T Ohshima

V Jacques

D Vion

D Esteve

R B Liu

K Mølmer

P Bertet

Storage and retrieval of microwave fields at the single-photon level in a spin ensemble

Keywords: number(s): 03.67.Lx, 42.50.Ct, 42.50.Pq, 85.25.Cp

published or not. The documents may come

Superconducting qubits are attractive candidates for solidstate implementations of quantum information processing, but suffer from coherence times shorter than ∼100 μs [1][2][3]. To circumvent this issue, it has been proposed to use ensembles of spins in semiconductors [4][5][6][7][8][9] as a multimode quantum memory, able to store multiple qubit states over longer periods of time, and to retrieve them on demand [START_REF] Kurizki | Proc. Natl. Acad. Sci. USA[END_REF]. Inspired by research on optical quantum memories [START_REF] Damon | [END_REF][12][13], realistic protocols have been proposed recently [14,15]. The state of a superconducting qubit is first converted into the state of a microwave photon, propagating or trapped in a resonator. This photon is then resonantly and collectively absorbed by the spin ensemble, resulting in a transverse magnetization which, due to the spread of resonance frequency within the ensemble, decays in a time T * 2 called the free-induction decay time. The write step is later followed by the need to read the stored quantum state. Both protocols [14,15] propose to apply sequences of π pulses to the spins, combined with dynamical tuning of the resonator frequency [16,17] and quality factor [18,19] in order to trigger the rephasing of the spins, resulting in the emission of an echo at a chosen time that faithfully reproduces the initial quantum state.

Whereas the transfer of a qubit state into a spin ensemble has been demonstrated experimentally [20][21][22][23], implementing the read step remains the major obstacle to an operational microwave quantum memory. An intermediate goal consists in storing a classical microwave pulse with an ultralow power corresponding to an average of one photon in the resonator and to retrieve it as an echo after a refocusing pulse, as was achieved at optical frequencies [24,25]. First results in this direction were obtained using ensembles of negatively charged nitrogen-vacancy (NV) color centers in diamond [26,27] and of rare-earth ions in a Y 2 SiO 5 crystal [28]. The NV's electronic spin is a spin triplet (S = 1) well suited for a quantum memory because of its long coherence times in pure crystals [29,30] and the possibility of repumping it into its ground state m S = 0 by optical irradiation at 532 nm [31] (see Fig. 1). In [27], successive low-power microwave pulses were stored in an NV ensemble, and retrieved later as a series of echoes after a refocusing microwave pulse was applied. A key aspect in this experiment was an active reset of the NVs to increase the repetition rate of successive experimental sequences to obtain sufficient statistics; this was achieved by applying optical pumping laser pulses injected through an optical fiber introduced in the cryostat. The echo efficiency, defined as the ratio of the echo energy and the stored pulse energy, was, however, not sufficient in [27] to observe an echo below 100 photons on average in the resonator.

Here, using a sample with a longer coherence time and an improved optical pumping scheme, we increase the echo efficiency and the storage time by one order of magnitude. This allows us to observe an echo with an initial pulse power corresponding to on average only one photon in the resonator. The diamond single crystal was synthesized by the temperature gradient method at high pressure and high temperature using 99.97% 12 C-enriched pyrolytic carbon prepared from 12 Cenriched methane as a carbon source [32], resulting in a nominal 300 ppm concentration of 13 C nuclei. The original 1.4 ppm concentration of substitutional nitrogen impurities (so-called P1 centers) was partially converted into NV centers by 2 MeV electron irradiation at room temperature followed by annealing for 2 h at 1000 • C, yielding a final [NV -] = 0.4 ppm and [P1] = 0.6 ppm. A scheme of the experimental setup is shown in Fig. 1(a). The diamond is glued onto the inductance of a superconducting planar lumped-element LC resonator patterned in a niobium thin film on a silicon substrate. Microwave pulses can be sent to the resonator input; the reflected signal is amplified at 4 K and its amplitude and phase are measured at room temperature by a homodyne detection setup. For resetting the spins, laser pulses at 532 nm can be sent onto the sample through direct optical access in the dilution cryostat. The resonator reflection coefficient S 11 (ω), measured with a network analyzer, is shown in Fig. 2(a), yielding the resonance frequency ω r /2π = 2.915 GHz and quality factor Q = 650, fixed by the coupling to the measurement line through the input capacitor. NV centers are detected by their absorption of the microwave whenever their transition frequency matches ω r . The energy levels of the NV centers are schematically shown in Fig. 1(c). The electronic spin is coupled by hyperfine interaction to the spin-triplet (I = 1) nuclear spin of the 14 full width at half maximum (FWHM) linewidth is measured for each line of the triplet, much narrower than in previous work [20,27], due to a lower P1 center concentration and to the isotopic enrichment in 12 C.

For optical pumping of the NV centers, the laser beam is focused to a 0.6 mm diameter at the sample level, with a power of 0.28 mW. In this geometry, it is straightforward to optimize the laser beam position on the sample in order to minimize the amount of power needed to reset the spins in their ground state. The efficiency of the optical pumping is measured as explained in [27]. The experimental sequence includes an initial strong microwave pulse that results in spin saturation, followed by a laser pulse of varying duration T L . After a 300 μs delay necessary for relaxation of the quasiparticles generated in the superconducting thin film and in the silicon substrate, the reflected amplitude of a few-photon microwave pulse reveals the spin polarization. The extracted polarization level p(T L ) is shown in Fig. 2(c). It changes only slightly The spins are first reset in their ground state by a laser pulse of power 280 μW and duration 0.2 s. A first 1 μs microwave pulse θ at ω r , of power -71 dBm, induces a transverse magnetization which decays within T * 2 . A 1-μs-long microwave refocusing pulse (R) of power -20 dBm is applied at τ = 50 μs, which rephases the spins at 2τ . The microwave amplitude (blue curve) shows both the reflection of the two microwave pulses driving the spins (with their amplitude trimmed by saturation of our detection chain, as indicated in red), as well as the echo emitted at 2τ upon rephasing of the spins. (b) Measured decay of the echo amplitude A as a function of 2τ (open circles). Calculated decay due to a bath of 213 ppm of 13 C (dashed orange curve), 0.6 ppm of P1 centers causing spectral diffusion (dotted green line), 0.2 ppm of NV centers causing instantaneous diffusion (dash-dotted blue line), and the combination of the three contributions (solid red line). The theory curves have been scaled in amplitude according to the data. An exponential fit (black solid line) yields a coherence time T 2 = 84 μs.

above T L = 1 s, indicating that the maximum NV polarization possible with irradiation at 532 nm (p max = 90% according to [33]) is reached. Compared to earlier work [27] where the laser position could not be optimized, the maximum polarization can be reached with ∼20 times lower pulse energy. This makes it possible to perform the experiments at a faster repetition rate (0.2 Hz) and at a cryostat temperature T = 100 mK instead of 400 mK [27], thus closely approaching the condition kT ω r required in order for thermal noise to be completely suppressed.

High-power Hahn echoes are measured at B NV = 1.74 mT, using microwave pulses at ω = ω r according to the sequence shown in Fig. 3(a). The sequence starts with a laser pulse of duration T L = 0.2 s, resulting in a spin polarization p = 0.72 p max = 0.65. At t = 0, a first pulse generates a transverse magnetization in the ensemble, followed by a refocusing microwave pulse at t = τ which induces rephasing of the spins at 2τ and emission of a spin echo into the measuring line, as seen in Fig. 3(a). Note that due to spatial inhomogeneity of the microwave field generated by the planar inductance, it is not possible to apply a well-defined Rabi angle to all the spins, which results in a reduced echo visibility. The echo amplitude is measured as a function of the delay 2τ between the first pulse and the echo, and is found to decay approximately exponentially with a time constant T 2 = 84 μs [see Fig. 3(b)]. Decoherence occurs due to dipolar interactions with the bath of paramagnetic species present in the sample ( 13 C nuclei, P1 centers, and NV centers), whose dynamical evolution causes a randomization of the phase acquired by NV centers during the two halves of the spin-echo sequence. The 13 C nuclei bath precesses at the Larmor frequency γ n B NV = 2π × 130 kHz (γ n being the 13 C gyromagnetic ratio), giving rise to a characteristic oscillation pattern [34][35][36] in the spin-echo amplitude, visible in the data of Fig. 3(b). The dynamics due to flip-flop events within the P1 center bath is responsible for a decoherence process known as spectral diffusion [37]. Finally, the bath consisting of NV centers at frequency ω r (only half of the total NV concentration) unavoidably undergoes spin flips due to the refocusing pulse itself, which constitutes an efficient decoherence process called instantaneous diffusion [38]. The various contributions of each bath were calculated using the cluster-correlation expansion method [39,40] Overall, dipolar interactions between NV centers appear to be the dominant source of decoherence in our experiment.

Since the echo efficiency was limited by the finite spin coherence time in earlier work [27], a significant improvement is expected with this new sample. The echo efficiency is first measured by performing a Hahn echo sequence with a lowpower microwave pulse. The experimental sequence, shown in Fig. 4, starts with a 0.2 s repumping laser pulse, followed after 300 μs by a microwave pulse populating the resonator with on average 60 photons, and τ = 50 μs later, by a refocusing pulse. A spin echo is detected at t = 2τ . The efficiency, defined as the energy recovered during the echo divided by the absorbed energy, reaches E = 0.3%.

The whole spin-echo sequence was simulated by numerical integration of the coupled differential equations obtained after discretizing the resonance frequency and coupling constant distribution in the spin ensemble (see [14]). A 130 kHz linewidth of each hyperfine peak and a collective coupling constant g ens /2π = 410 kHz (when all spins are polarized), are compatible with the experimentally determined parameters and yield quantitative agreement with the shape of both the absorbed microwave pulse and the spin-echo amplitude. The discrepancy noted in [27] is absent in the present experiment, probably because decoherence is negligible during the driven evolution. The finite T 2 and the imperfect π pulse due to the spread in Rabi frequencies are the main factors limiting E, while the finite cooperativity C = 0.22 [14] limits both the absorption and the echo emission out of the cavity.

Owing to the larger value of E, it becomes possible to reach the level where a spin echo can be observed for an initial pulse populating the resonator with only a single microwave photon on average. This is shown in Fig. 4(b). Note that a shorter repumping time of 0.1 s (with the same laser power) was used, in order to enable a larger number 5 × 10 5 of repetitions of the experiment. The shorter repumping step yields a lower spin polarization p = 0.56 as shown in Fig. 2 and a lower cooperativity of 0.19, which results in a correspondingly lower echo efficiency than in Fig. 4(a). These results are again quantitatively reproduced by the simulations with the same parameters as mentioned earlier, using the experimentally determined repumping efficiency.

The coherence times demonstrated in this experiment match those requested in a realistic quantum memory protocol [14], which suggests that a first implementation is within reach. The remaining challenges are the improvement of the refocusing pulse using adiabatic passage as demonstrated recently [41], and the integration of dynamical tuning of the resonator frequency and quality factor with more elaborate spin-echo sequences. The latter is needed in particular to silence the echo emission [START_REF] Damon | [END_REF]12] in-between the two π pulses in the course of the read step of the protocol. The microwave currents needed to drive the spins during the refocusing pulses are much stronger than typical Josephson junction critical currents. This precludes the use of integrated superconducting quantum interference devices as in [20,42], while a combination of coupled linear and tunable resonators [43] may be employed as tuning elements in the resonator.

In conclusion we report the measurement of a spin echo with an initial microwave pulse at the single-photon level. The figures of merit reached are sufficient to envision first implementations of a spin-ensemble multimode quantum memory for superconducting qubits.

FIG. 1 .

 1 FIG. 1. (Color online) (a) Experimental setup and principle of the experiment. An ensemble of ∼10 10 spins is inductively coupled to a planar superconducting LC resonator of frequency ω r (with a collective coupling constant g ens ), cooled at 10 mK. The resonator is measured in reflection through an input coupling capacitance. Microwave pulses are produced by mixing a continuous microwave source with dc pulses generated by an arbitrary wave form generator (AWG). They drive the spins via the microwave current induced in the resonator inductance. The reflected microwave signal (including the emitted echo) is amplified at low temperature and demodulated at room temperature, yielding its amplitude A(t) and phase φ(t). (b) The spins are nitrogen-vacancy color centers in diamond, which consist of a nitrogen impurity next to a vacancy of the diamond lattice. A dc magnetic field B NV applied parallel to the chip along the [110] crystalline axis so that only NV centers whose axis are nonorthogonal to the field (shown in blue in the figure) are Zeeman shifted and contribute to the signal. Laser pulses can be sent onto the diamond via a direct optical access to the cryostat mixing chamber. (c) NV centers energy levels in a weak magnetic field. The electronic ground state is a spin triplet S = 1, with a zero-field splitting D/2π = 2.88 GHz, coupled by hyperfine interaction to the 14 N nuclear spin triplet I = 1. This splits each of the |m S = 0 → |m S = +1 transitions into a triplet of lines.

  N atom, resulting in a splitting of the |m S = 0 → |m S = +1 transition into three resonances separated by 2.2 MHz corresponding to the three different m I states of the 14 N. A dc magnetic field B NV is applied parallel to the chip, along the [110] crystalline axis of the diamond. Out of the four possible orientations of NV centers along [111] crystalline axes, two are perpendicular to the field and are therefore not Zeeman shifted, so that they do not contribute to the signal. The remaining two families are brought into resonance with ω r at B NV ∼ 1.8 mT, as shown in Fig. 2(b) where the hyperfine triplet is clearly seen as dips in |S 11 (ω)| when they cross the resonance. A ∼200 kHz

FIG. 2 .

 2 FIG. 2. (Color online) (a) Phase φ of the resonator reflection coefficient S 11 (ω) measured with a vector network analyzer (blue dots), yielding ω r /2π = 2.915 GHz and Q = 650 (red line is a fit to the data). (b) S 11 (ω) as a function of B NV around 1.8 mT, showing the NV centers as a triplet of absorption dips. The inset shows |S 11 |(B NV ) at ω/2π = 2.915 GHz; blue dots are data, and red line is a fit to a sum of three Lorentzians with linewidth 0.012 mT. (c) Measured spin polarization for a laser pulse of power P L = 280 μW as a function of its duration T L , renormalized to its maximal value. Dashed red and black lines indicate the laser pulse durations T L = 0.2 and 0.1 s used in the experiments shown in Figs. 3 and 4, corresponding to relative polarizations p/p max = 0.72 and 0.62.

FIG. 3 .

 3 FIG. 3. (Color online) (a)Hahn echo sequence. The spins are first reset in their ground state by a laser pulse of power 280 μW and duration 0.2 s. A first 1 μs microwave pulse θ at ω r , of power -71 dBm, induces a transverse magnetization which decays within T * 2 . A 1-μs-long microwave refocusing pulse (R) of power -20 dBm is applied at τ = 50 μs, which rephases the spins at 2τ . The microwave amplitude (blue curve) shows both the reflection of the two microwave pulses driving the spins (with their amplitude trimmed by saturation of our detection chain, as indicated in red), as well as the echo emitted at 2τ upon rephasing of the spins. (b) Measured decay of the echo amplitude A as a function of 2τ (open circles). Calculated decay due to a bath of 213 ppm of 13 C (dashed orange curve), 0.6 ppm of P1 centers causing spectral diffusion (dotted green line), 0.2 ppm of NV centers causing instantaneous diffusion (dash-dotted blue line), and the combination of the three contributions (solid red line). The theory curves have been scaled in amplitude according to the data. An exponential fit (black solid line) yields a coherence time T 2 = 84 μs.

  , with concentrations [P1] = 0.6 ppm, [NV -] = 0.4/2 = 0.2 ppm, and [ 13 C] = 213 ppm, compatible with the sample parameters. Good agreement with the data is obtained [see Fig. 3(b)].

FIG. 4 .

 4 FIG. 4. (Color online) Spin echo at the few-photon level. The experimental sequence is the same as in Fig.3, but with an initial microwave pulse having a power corresponding to (a) 60 and (b) one photon inside the resonator. The signal was averaged over 3 × 10 4 (a) and 5 × 10 5 (b) sequences, with a repetition rate of 5 Hz (a) and 10 Hz (b), limited by the laser pulse duration. Blue solid lines are experimental data, and red dashed-dotted lines are the results of simulations as explained in the text. The signal-to-noise ratio, defined as the ratio between signal intensity integrated over the echo duration and the corresponding noise intensity, is 4 for the 60-photon pulse (a) and 1.5 in the one-photon pulse (b).

We acknowledge technical support from P. Sénat, D. Duet, J.-C. Tack, P. Pari, and P. Forget, as well as useful discussions within the Quantronics Goup and with A. Dréau, J.-F. Roch, T. Chanelière, and J. Morton. We acknowledge the support of the French National Research Agency (ANR) with the QINVC project from CHISTERA program, the European project SCALEQIT, the C'Nano IdF project QUANTRO-CRYO, JST, and JSPS KAKENHI (Grant No. 26246001). Y.K. acknowledges support from the JSPS, and B.J. and K.M. from the Villum Foundation.