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Thesis Rare-earth quantum memories for single photons and entanglement USMANI, Imam Abstract The ability of storing and retrieving quantum states of light is an important experimental challenge in quantum information science. A powerful quantum memory for light is required in a quantum repeater, which would allow long distance (>500km) quantum communications. To be implemented in such application, the quantum memory must allow on-demand readout, with high fidelity and efficiency, and a long storage time. Additionally a multimode capacity (for temporal or spatial modes) would allow multiplexing. Our approaches focus on rare-earth doped crystals, i.e. solid state quantum memory. I present, in this work, our contributions for a solid-state quantum storage, with good performances in every criteria. In particular, I present the preservation of quantum entanglement during the storage, which paves the way for the implementation of quantum memories in quantum repeaters. Reference USMANI, Imam. Rare-earth quantum memories for single photons and entanglement. Thèse de doctorat : Univ. Genève, 2013, no. Sc. 4544 URN : urn:nbn:ch:unige-276020 DOI : 10.13097/archive-ouverte/unige:27602 Available at: http://archive-ouverte.unige.ch/unige:27602 Disclaimer: layout of this document may differ from the published version. UNIVERSITÉ DE GENÈVE Groupe de Physique Appliquée - Optique FACULTÉ DES SCIENCES Prof. N. Gisin Rare-earth quantum memories for single photons and entanglement THÈSE présentée à la Faculté des Sciences de l'Université de Genève pour obtenir le grade de Docteur ès Sciences, mention Physique par Imam Azim Usmani du Grand-Saconnex (GE) Thèse N◦ 4544 GENÈVE 2013 utïvrRsrr 3r çENûvr rrc*lrf rgs scltlrtcgt Doctorqfès sciences Men|îon physîgue Thèse aeA/lonsieurImam Azim US/VIANI i n ti tu l é e: " RoreEorthQuqnlum Memoriesfor single-photons qnd " Enlonglemenf Lo Foculté des sciences,sur le préovis de Messieurs N. GlSlN,professeur ordinoireet d ire c t e urde t h è se( s e c t i o nd e p h ysi q u eG , r ou p ed e p hy s iq u eo p p l i q u é e ) ,M . A F ZE L IU S , , r o u p e d e p h y si q u eo p pl i q u é e ) ,M o d om e N .T I M ON E y , doc t eu r ( Sec t i o nd e p h y siq u eG doc t eu re ( S e ct i o n de physique, Groupe de p h ys i q u e o p p l i qu é e ) , Messieurs Ph.GRANGIER,professeur(lnstitut d'Optique Groduoie School, porisTech, Poloiseou,Fronce),et H. de RIEDMATTEN, professeur (TheInstituteof PhotonicSciences, Porc Mediterronide lo Tecnologio,Costelldefels, Borcelono,Espoho),ouloriseI'impression de lo présentethèse,sonsexprimerd'opinionsurlespropositions quiy sonténoncées. Gen è v e , l e 3 o vr i l 2 0 ' 1 3 Thèse - 4544, Jeon-Morc TRISCONE N. B . Lo thèse doit porter lo déclorotion précédenfe et remplir les conditions énumérées dons les "lnformotionsrelotives oux thèses de doctorot à I' U ni ver si té de Ge nè ve". Résumé de la Thèse L'information quantique a, depuis environ une vingtaine d'années, apporté une vision supplémentaire à la mécanique quantique et permis d'en entrevoir des applications très prometteuses. Elle se base sur des concepts nouveaux par rapport à la mécanique classique, comme la superposition d'états quantiques ou l'intrication. Parmi les possibles champs d'applications, on retrouve les ordinateurs quantiques, qui permettraient de résoudre certains problème beaucoup plus rapidement qu'un ordinateur classique, ainsi que les communications quantiques. Celles-ci incluent par exemple la cryptographie quantique qui illustre très bien comment on peut proter des particularités de la mécanique quantique pour une application concrète. En eet, le secret d'une clé d'encryption est garanti par les lois de la physique quantique. En général, l'information est encodée sous forme de qubits. Ceux-ci, à l'instar d'un bit, peuvent prendre les valeurs 0 ou 1, mais aussi toute superposition de ces deux valeurs. La valeur d'un qubit peut ainsi être représentée par une coordonnée sur une sphère de rayon unité et contient donc plus d'information qu'un bit classique. Pour des réalisations expérimentales, il faut pouvoir générer des états quantiques, mais aussi être capable de les modier, les mesurer et les transporter. Les photons sont idéalement adaptés pour encoder des qubits et peuvent en particulier être aisément transportés par bre optique. Toutefois, il peut être nécessaire de les stocker pour un temps donné, c'est à dire de transférer leur état quantique (de manière réversible) dans un système stationnaire comme des atomes. Ceci requière comme outil une mémoire quantique pour les photons. Il faut remarquer que, contrairement à une mémoire classique, un état quantique stocké reste inconnu. En eet, la mesure d'un système physique modierait irrémédiablement son état quantique, et il ne serait de toute façon pas possible de connaître de manière déterministe son état. Une mémoire quantique doit évidemment préserver l'état d'un photon, et en particulier maintenir l'intrication éventuel qu'il aurait avec un autre photon. La réalisation d'une mémoire quantique performante promettrait de nouvelles applications, comme par exemple les répéteurs quantiques. Ceux-ci amèneraient la possibilité d'étendre la cryptographie quantique sur de longues distances ou de créer des réseaux quantiques. Les mémoires, placées dans les n÷uds du réseaux auraient un rôle de synchronisation entre les diérents liens. La recherche pour la réalisation d'une mémoire quantique est un domaine très actif iii et demande de trouver un système stationnaire fortement couplé à la lumière et capable de maintenir susamment longtemps un état quantique. Dans le groupe de physique appliquée (GAP) à Genève, la recherche se concentre sur les cristaux dopés aux terres rares, refroidis à des température cryogénique (≈3 K). Ceci permet d'utiliser un grand nombre d'atomes qui sont piégés naturellement par le cristal. Ces systèmes on été employés avec succès dans le domaine des lasers, mais la recherche pour le stockage d'état quantique est beaucoup plus récente. Mon travail, dans le cadre de cette thèse, a été de chercher à réaliser une mémoire quantique pour photons, qui pourrait être utilisée dans des future répéteurs quantiques. Tout d'abord, nous avons eectué des mesures spectroscopiques dans un candidat potentiel pour une mémoire quantique, un cristal dopé au néodyme : Nd3+:Y2SiO5. Cela a permis de trouver une conguration pour implémenter un protocole de mémoire quantique, le peigne en fréquence atomique. Nous avons alors réalisé diverses expériences pour montrer le potentiel d'une mémoire quantique dans un cristal dopé au terres rares. D'une part, nous avons cherché à maximiser l'ecacité de stockage dans le Nd3+:Y2SiO5, ainsi que dans un autre cristal (Eu3+:Y2SiO5) à l'aide d'une cavité optique. Puis, nous avons démontré le stockage de plusieurs qubits dans un seul ensemble d'atomes, et montré une capacité jusqu'à 64 modes temporels de la mémoire. Aussi, nous avons accompli le stockage de lumière dans une onde de spin, ce qui a permis d'allonger signicativement le temps de stockage. A chaque fois, nous avons cherché à réaliser ces mesures avec des états cohérents avec, en moyenne, environ un photon par impulsion, pour démontrer le caractère quantique du stockage. Cependant, nous avons pu démontrer des tests plus forts, en stockant des vrai photons uniques, pour la première fois dans une mémoire à état solide. Ceci a permis de montrer que le stockage préservait également l'intrication d'un photon. Nous avons également pu réaliser la création annoncée d'intrication entre deux mémoires diérentes, séparé par 1.3 cm. Ces expériences ont contribué à montrer qu'il est possible de réaliser une mémoire dans les cristaux dopés aux terres rares, en atteignant tous les critères possibles pour un répéteur quantique. Dans le futur, il faudra trouver un système qui combinera, au moins, toutes les performances atteintes jusqu'ici dans les diérentes expériences. iv Contents Résumé de la Thèse 1 Introduction 2 Theory iii 1 5 2.1 Medium for quantum storage . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Quantum storage protocol . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Spectral hole burning . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3 Realization of a quantum memory 3.1 3.2 3.3 3.4 3.5 Spectroscopy of a rare earth doped crystal Storage eciency . . . . . . . . . . . . . . Multimode quantum storage . . . . . . . . On-demand storage . . . . . . . . . . . . . Testing a QM with single photons . . . . . 4 Discussions and Outlook Bibliography Acknowledgements A Additional spectroscopy List of Publications B Published articles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 13 25 31 31 35 39 47 57 59 65 67 Demonstration of atomic frequency comb memory for light with spin-wave storage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 Towards an ecient atomic frequency comb quantum memory . . . . . . . . 72 Mapping multiple photonic qubits into and out of one solid-state atomic ensemble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 v Quantum storage of photonic entanglement in a crystal . . . . . . . . . . Heralded quantum entanglement between two crystals . . . . . . . . . . . Atomic frequency comb memory with spin-wave storage in Eu3+:Y2SiO5 Single-photon-level optical storage in solid-state spin-wave memory . . . vi . . . . . . . . 92 101 114 122 Chapter 1 Introduction A memory for classical information is a very old and basic concept. It takes a large variety of forms, from a simple piece of paper to a hard drive. Nowadays, it is possible to massively produce devices with impressively large capacities and fast access times. Improving their performances or nding new systems is still an active eld, but the concept of a classical memory remains the same. However, quantum mechanics has changed our fundamental description of physics. It introduces new concepts such as a quantum state, quantum superposition or entanglement. Based on that, fascinating applications have been developed. For example, quantum cryptography [1] uses quantum key distribution (QKD) where security is guaranteed through the laws of quantum physics. To progress experimentally though, the eld of quantum information needs also new tools, such as a quantum memory (QM). Such a device must have the ability to store and faithfully release a quantum state. In particular, one would need an on-demand readout, that is to say the output can be released whenever it is needed. Quantum storage must face some intrinsic concepts of quantum mechanics which differentiates it from a classical memory. For example, we know a measure of a quantum system will aect its quantum state in an irreversible way. Because of this, an initially unknown state cannot be determined with 100% probability and, more generally, only limited amount of information can be extracted from a nite quantum ensemble [2]. This particularity is actually used against a potential eavesdropper in QKD. However, because of this, a quantum memory cannot be based on a measure and write-down strategy. Here, I focus on optical quantum memories [3]. Photons are indeed ideal carriers of a quantum states and can travel through large distances. The diculty, though, is to store them in a stationary device for a certain amount of time. A quantum memory would be an elementary tool in many quantum optics experiments, as are today optical bres, single photon detectors or photon sources. Moreover, there are several potential applications in quantum information as I will describe here. In linear optics quantum computation (LOQC) [4, 5], a QM is a unitary gate, which has the 1 role of synchronization between dierent computing channels. Also quantum memory could be a useful component of a photon source. For example, using spontaneous parametric down conversion (SPDC) one can herald the presence of a single photon [6, 7]. It is however emitted at random time which makes it unsuitable for some applications. This heralded photon could be stored in a QM and released when needed, thus realizing an on-demand source of single photons[8]. Our main interest, though, is the implementation of quantum memories in quantum communications (QC) [9]. I mentioned for example quantum key distribution. Since the original idea of Benett and Brassard [10], it has been realized experimentally, in real-eld conditions and even commercially systems are available. While in 1992 the rst experimental proof was demonstrated over a short link of 30 cm[11], the distance has been impressively increased. Using polarization based QKD in free space a distance of 144 km has been reached [12], while time-bin qubits have been used for QKD over 250 km in optical bres [13]. One should note that a protocol can be secure even if it is not based on true single photons. In particular, an important alternative approach based on continuous variables uses coherent and squeezed states [14, 15]. The distance of communication is however limited to few hundreds of km. Indeed, the signal sent through an optical bre undergoes losses increasing exponentially with the communication distance. For example, even with ultra-low loss optical bres (-0.16 dB/km), a photon has a probability of 10−16 to travel through 1000 km. In classical communications, this problem is solved with ampliers. It is however not a solution in QC, since an arbitrary quantum state cannot be perfectly copied deterministically (non-cloning theorem [16]). Fortunately, the idea of a quantum repeater was proposed [17, 18, 19] to distribute entanglement over large distances. The principle is to split a long communication distance into shorter links (g.1.1). Using entangled photon pair sources, one attempts to share entanglement in each link. It is a probabilistic process, and the losses in a link must be low enough, so that this probability is reasonable. Because this does not happen necessarily at the same time in all links, this entanglement must be maintained in quantum memories placed at each border of a link to allow synchronization. Afterwards, Bell measurements [20, 21] in intermediate stations will allow to swap the entanglement between QMs separated by the complete communication distance. In this context, we can consider more generally a quantum network [22] where nodes generate, process and store quantum information, while photons transport quantum states from site to site and distribute entanglement over the entire network. Additionally, i would like to point out that a quantum repeater could even be used in an optical interferometric telescope which has the potential to image extra-solar planets [23]. Now, to test if a QM is suitable for those dierent applications, I will use a variety of criteria [3] that I summarize here. First, the most important for a quantum memory is to re-emit the most faithfully a quantum that was stored. One can calculate the delity of the output to the input state and, ultimately, it should be above a threshold 2 a) Entanglement creation QM QM QM A B C QM ... QM D ... W QM QM QM X Y Z b) First entanglement swapping QM A QM ... QM QM ... W Z D c) Last entanglement swapping QM QM A Z Figure 1.1: Quantum repeater scheme. for possible error correction, for any given input state. In practice, it very convenient to dene some more specic criteria, such as the eciency η which is the ratio between the energies of the output and input states. Ideally, it should be the closest to one. However, some classical optical storage schemes can easily reach 100% (and even more due to light amplication processes) even though they are not suitable for a quantum memory [24, 25]. Therefore, it is necessary to complete this criterion with a measure of delity, which can be conditioned on the re-emission of an output. Indeed, when working with single photon detectors, it is possible to remove the vacuum component of the state by post-selecting on the detections. This is not the case with homodyne and heterodyne measurements used in continuous variable techniques, since there is always a detection. Also, a measure of the noise level helps to determine if a QM can potentially store true single photons with a good signal-to-noise ratio. Indeed, if there is no input, the probability that the QM emits a noise photon must be close to zero. Additionally, a quantum memory should ideally have the capacity of storing many dierent qubits at the same time. This multimode capacity is quantied by the number of modes it can accept and depends strongly of the storage scheme [26]. The storage time is another criterion and it should be long enough to perform a particular task. While classical memories have no real limit and can always be copied if they undergo physical damage, a quantum memory must usually face decoherence that grows with time. Finally, we note that an optical quantum memory usually works for a specic wavelength and frequency bandwidth. This must be taken into account for potential applications. For example, a quantum memory working at telecom wavelengths would be a clear advantage in quantum communications [27]. Research in quantum storage is an active eld and many realizations have already been made in various systems. Compact overviews can be found in ref.[3, 28, 29, 18]. Nevertheless, no QMs have yet been implemented in a real application. Our longterm motivation is therefore to realize a QM so that it will full all the requirements to achieve this, in particular to implement it in a future quantum repeater. The 3 realization of a QM requires a stationary medium that interacts strongly with light and is capable of maintaining a quantum state, for example an atomic ensemble. Secondly, we need to use a quantum memory protocol so that the user can map a quantum state into the medium and release it when it is needed. This can be achieved using electro-magnetically induced transparency (EIT), photon echoes or Raman transitions. Finally, a QM needs to be characterized. This can be done by using some criteria, or by storing some quantum states and measuring the delity of the storage. In the next chapter, I will discuss the choice of a suitable medium for quantum storage and, in particular, the properties of rare earth doped crystals which are used in this thesis. Also, I will describe quantum storage protocols, specically the atomic frequency comb (AFC) protocol based on photon echoes. In chapter 3, I will discuss the experimental realization of a QM. My work in this thesis was guided by the various performance criteria a QM needs for the implementation in a quantum repeater. In addition, we were able to test our QMs with the storage of true single photons and a measure of entanglement between two QMs. 4 Chapter 2 Theory 2.1 Medium for quantum storage We discuss now the system in which a QM can be implemented. The question of nding the ideal system is still open, as it is the case for example in quantum computing. For QC applications we would like to work with light at optical or nearinfrared wavelength, say from 400 to 1600 nm. One can use individual systems such as trapped ions [30] or atoms in high nesse cavities [31]. Another approach is the use of atomic ensembles [28] featuring a strong light matter interaction due to a large optical thickness [28]. Quantum storage was rst demonstrated in vapours of rubidium or cesium[32, 33, 34]. A usual diculty in such system is the decoherence due to atomic collisions. To overcome this, one solution is to cool and trap the atoms, using magnetooptical trap (MOT) [34] or optical lattices [35]. The drawback is a higher complexity of the experiment. Rare earth doped crystals Promising alternatives are solid-state atomic ensembles. Specically, rare earth ion doped crystals [29, 36] have already been widely studied in the context of laser applications and storage of strong laser pulses. The ions are impurities with a doping level from 10 to 1000 parts-per-millions (ppm) and since they are naturally trapped in the host crystal, the ensemble is sometimes described as a frozen gas. Because of this natural trapping, the crystal needs only to be cooled down with commercial cryostat and the setup is not in principle that complex. The crystals are often the same as the ones used in lasers, only the doping concentration is usually smaller. They are of good optical quality and it is useful to cut it along an optical axis, so that the light does not necessarily undergo birefringence. Rare earth elements have the particularity that for most of them, the 4f electronic shell is incomplete and optical transitions occur in it (see g. 2.1). We note that for free ions, such transitions would be forbidden by 5 selection rules, but they become here weakly allowed because the crystal eld changes the wave functions of the electrons. The 4f shell is closer to the nucleus than some (complete) outer shell (5s, 5p and 6s) which induces a screening. This isolates the 4f shell from the environment and it leads to very long coherence time. Indeed, the optical coherence time is usually in the 1 µs to 1 ms regime below 4 K. We note that the level structure will depend on whether it is a Kramers ion (odd number of electron in 4f) or a non-Kramers ion (even number of electrons). Note that we here consider trivalent rare earth ions (RE3+). For a Kramers ion, the ground state is a doublet whose degeneracy can be lifted with an external magnetic eld. These Zeeman states can be easily separated by tenths of GHz, and are used in quantum storage protocols. For a non-Kramers ions, though, the angular momentum degeneracy is completely lifted by the crystal eld. Therefore, if we need more than one ground state, we can use ions with a nuclear spin, which will induce a hyperne structure. These electronic and nuclear spin levels also have impressively long coherence times at cryogenic conditions, from 1 ms to 1 s. Another point is that environment variations in the crystal induce an inhomogeneous broadening Γinh in the optical transition, typically between 100 MHz and 10 GHz. We note that this is much broader than the homogeneous linewidth (γh =1kHz-1MHz) (g. 2.2). As a consequence, the dierent resonance transitions of an ion may not be distinguished in a broadened spectrum and this leads to dierent classes of atoms. However, this may help for the realization of a QM with a large bandwidth and, because of a high Γγ ratio, a high multimode capacity. This is the basis of the atomic frequency comb memory. inh h 2.2 Quantum storage protocol Atomic frequency comb To realize a quantum memory, one needs to apply a protocol that allows the reversible mapping of light. For example, using electro-magnetically induced transparency (EIT) [37] one can slow and stop light for a certain time. Alternatively, in a Duan-Lukin-Cirac-Zoller (DLCZ) scheme [17], through an optical excitation and the detection of a Stokes photon, the creation of a single excitation in an atomic ensemble is heralded. This single excitation can be released, on-demand with a π pulse, through the emission of an anti-Stokes photon. This type of quantum memory, sometimes described as a photon pistol can be used to herald entanglement between remote atomic ensembles [17]. The two schemes have been combined in a single experiment, both in a warm vapour [32] or in a cold gas [34]. However, in rare earth doped crystals, the large inhomogeneous broadening induces a fast dephasing, much faster than the coherence time, which suppresses all coherent emissions of the atoms. Fortunately, photon echo protocols, which have already been used for optical data storage, allow to compensate 6 Figure 2.1: An example of energy levels of rare earth elements. 7 Figure 2.2: Illustration of a inhomogeneous broadening in a rare earth doped crystal. Γinh is typically between 100 MHz and 10 GHz, which is much larger than the homogeneous linewidth (γh =1kHz-1MHz) the inhomogeneous broadening. We illustrate here the principle of such protocol with the state of the atomic ensemble after the absorption of a photon: |Ψi = N X j=1 cj eiδj t e−ikzj |g1 . . . ej . . . gN i (2.1) N is the number of atoms in the ensemble, cj a factor that depends on the frequency and position of the atom j, δj its frequency detuning and zj its spatial position while k is the wave number of the light eld. This state represents one excitation delocalized among the N atoms. As it can be predicted from Maxwell equations, the polarization of the atomic ensemble induces the emission of an electric eld. However, because of the inhomogeneous broadening, δj has a large spread which will lead to a fast dephasing. Therefore, after a short time (about the input pulse duration) the emission is incoherent, that is to say its intensity is proportional to N, and in all directions. The principle of any photon echo protocol is to act such as the term e−iδ t is the same for all j at a particular time. This leads to constructive interference, thus coherent emission. It will take form of an emitted pulse (a photon echo), with intensity proportional to N2 and in a specic direction determined by the spatial phase imprinted in the atoms e−ikz . Because N is very large, the incoherent emission is completely negligible compared to the photon echo. Some protocols are dened as classical, such as a two-pulse photon echo (2PE). Because of intrinsic noise created by the optical π pulse, it can never work as quantum memory for single photons[24], even though it can reach eciencies of 100% (or more). Protocols for quantum storage have been proposed in the last ten years, such as controlled and reversible inhomogeneous broadening (CRIB) [38, 39, 40] or atomic frequency combs (AFC) [41]. Our work here focuses on this last. It has the advantage, as we will see later, of being highly multimode. Also, compared to CRIB, it is not necessary to apply an external electric eld gradient. The principle of it, is to tailor a periodic function in the inhomogeneous absorption prole (g. 2.3a). More j j 8 (a) Atomic density lds l fie Output mode tro Con Input m ode e s D g g aux Atomic detuning d Intensity (b) Input mode Control fields 2p / D - T0 Ts Output mode Time T0 Figure 2.3: (a) schematic level structure of the atomic ensemble, with a periodic absorption prole for AFC storage (b) An input is absorbed by the comb, transferred into a spin state for a time Ts , and re-emitted when the atoms are in resonance. precisely, it consists in a series of peaks of specic shapes separated by a detuning 2π∆, which composes the atomic frequency comb. This periodicity induces that at the time t=1/∆ after absorption of the wave packet, all atoms are in phase which forces the emission of the echo (g. 2.3b). This intuitive explanation is conrmed in a detailed analysis [41] and it is shown that the protocol works for any input state that is much weaker than a π pulse. We note that from the point of view of Bonarota et al.[42], AFC is more similar to EIT because it is based on a dispersion (caused by the variations in the absorption prole), while CRIB is an absorbing storage protocol (the absorption prole is at). It is very useful to simulate or calculate numerically the eect of an AFC for various absorption prole that cannot be solved analytically. For this purpose, one can use a Maxwell-Bloch simulator or, alternatively, a general formula has been derived [43] which assumes only that the susceptibility function is periodic. Here, we used a numerical method, working for any absorption prole, that we describe here. The linear part of the susceptibility χ for an ensemble of two-level atoms can be written as a perturbation solution[44]: N X cj (2.2) χ(ω) ∝ (ω − ω) − iγ j j=1 h ω is the angular frequency of the light eld, ωj is the resonant frequency of atom j and γh is the homogeneous linewidth. Hence, the complex wave number can be calculated 9 and depends strongly on ω: p k(ω) = ω/c 1 + χ(ω) The absorption coecient is given by: α = 2k 00 where k00 is the imaginary part of k. If necessary, we can adjust the amplitude in Eq.2.2 to obtain the desired absorption coecient. Now, we suppose the input eld can be written as: Z Ein (t) = Ẽ(ω)e−iωt dω Through the medium of length L, each components propagates and acquire a phase k(ω)L. Hence the output eld can be calculated: Eout (t) = Z Ẽ(ω)ei(k(ω)L−ωt) dω For various atomic distributions, we calculated numerically Eout and the dierent resulting parameters, such as the storage eciency, which were in good accordance with Maxwell-Bloch simulations and analytical results. Moreover, we were able to study some subtle eects arising, for example when the AFC is not innite. We described here a two-level AFC protocol, for which the storage time (1/∆)must be chosen in advance, during the AFC preparation. Moreover, it is limited by the coherence time of the optical transition. In the goal of achieving a quantum memory with on-demand readout and long storage time, we must use a 3-level AFC scheme [41] which includes the transfer to an additional ground level |si (a spin state for example). After an input has been absorbed, we apply a π pulse (control eld) on the |si − |ei transition which moves the coherence to the |gi−|si transition (see g. 2.3). Assuming there is no spin inhomogeneous broadening, the phase evolution in Eq.2.1 will stop. After a time Ts, we apply a second π pulse, which moves back the coherence to the |gi−|ei transition and the phases evolve again. This leads to the re-emission of an echo at the time Ts + 1/∆ (g.2.3b). Therefore, a complete AFC protocol would allow to realize an on-demand quantum memory in an inhomogeneously broadened ensemble. Recently, a DLCZ scheme using an AFC in an inhomogeneous ensemble has been proposed [45]. This would allow to implement a photon pairs source in the medium. It requires the same techniques as the implementation for a complete AFC scheme. Therefore the path to achieve both protocols are the same. 2.3 Spectral hole burning We briey describe the process of spectral hole burning [46][47], which can be used to tailor a specic structure in the absorption prole, or to do spectroscopic 10 (a) 1 2 Absorption Class: (b) Angular frequency Figure 2.4: Spectral hole burning in a three-level system. measurements in a material. We rst consider a probe eld propagating in a medium of two-level atoms. If the light is in resonance with the atoms, the light intensity decreases exponentially with the length L of the medium: I(L) = e−αL I(0) is the intensity of the light at position l and α is the absorption coecient. We often use the optical depth d = αL in the context of quantum storage. The absorption coecient can easily be measured by the use of a probe eld transmitted through the medium. The important point here is that it depends on the population in the ground (Ng ) and excited state (Ne): I(l) α = (Ng − Ne ) σ. where σ is the cross section. The population can be changed via optical pumping which leads to a decrease of absorption (a spectral hole ) or an increase of it (a spectral antihole ). To illustrate this process, we consider a simple case of an atomic ensemble with two ground states and one excited state (g.2.4a). The inhomogeneous broadening is larger than the split ∆Eg in the ground state. The temperature makes that the two ground states are equally populated at equilibrium. We consider the relaxation time of the ground states (Tz ) is longer than the excited state lifetime (T1). A pumping beam at an angular frequency ωpump is shined into the medium and, because of the inhomogeneous broadening, it is resonant with 2 dierent types of atoms (g.2.4a). After the optical pumping, the population in the ground states of the 2 classes of atoms has changed, which modies the absorption spectrum (g.2.4b) by creating spectral holes and anti-holes. 11 Many spectroscopic measurements can be done using this technique. The decay of a spectral hole after the pumping process allows to determinate the relaxation time of the ground states TZ . The positions of the holes and anti-holes are functions of the energy splits, thus one can measure the g tensor by applying various magnetic elds. Alternatively, with a more elaborate pumping beam, it is possible to tailor any structure in the absorption prole, such as an AFC. A crucial parameter is the eciency of the optical pumping, that is to say what fraction of atoms will be moved from one ground state to another (the degree of spin polarization). This may indeed eect the AFC storage eciency. A simple rate equation model of a 3-level system allows to determine the ratio between the populations in the ground states [48]. In a steady state, after a long optical pumping, it depends of the ratio between TZ and T1: TZ ρ2 =1+2 , ρ1 T1 where ρ1(ρ2) is the population fraction in the initial (nal) ground state. For this reason, we need the ground state relaxation time to be long compared to the excited state lifetime. We note that we have not taken into account the branching ratio between the dierent transitions. In the case the dierent transitions have similar strengths, the formula is a good approximation. Another important feature is the ability to burn narrow holes, since it determines the duration of an AFC storage in the optical transition. While, ultimately, it is limited by the homogeneous linewidth of the atoms[49], other processes may broaden it. First, the laser linewidth must be narrow enough to attain this limit. Also, uctuations of the resonance frequency of an atom during the hole burning, that is to say spectral diusion, will broaden a spectral hole. Finally, to achieve ecient optical pumping, the pumping beam must have a high intensity and long duration, which induces power broadening[49]. We conclude by noting that, in many rare earth doped crystals, the level structure is richer than in our simple model. There may be a lot of dierent transitions possible, which leads to more complicate hole burning spectra but the principles introduced here remain the same. 12 Chapter 3 Realization of a quantum memory The realization of an optical quantum memory is a dicult and challenging task. We want not only to realize a proof-of-principle of quantum storage, we would also like to realize a quantum memory whose performances would allow to use it some day in a quantum repeater [17, 18, 19]. I will present here the dierent aspects on which I worked for such a goal. First, one needs to nd a suitable system to implement a quantum memory. In Geneva, our research focuses on rare earth doped crystals. Even though we believe strongly that these are promising systems [29], we need to nd the most suitable medium for our interests. I will rst present our spectroscopic measurements in a neodymium doped ortho-silicate crystal, Nd3+:Y2SiO5 with the goal of using it for quantum storage. In the following sections, I will discuss the various criteria of a QM: the storage eciency, multimode capacity and storage time. Finally, I will present tests of QMs: storage of single photons and heralded entanglement between two QMs. 3.1 Spectroscopy of a rare earth doped crystal Introduction Motivation We believe rare earth doped crystals have a high potential for the realization of a quantum memory [29, 36]. Of course, some of them are more promising than others, because of their dierent properties. They have been widely studied in the context of laser applications and optical data storage, but research for quantum storage is more recent. Therefore, spectroscopy of rare earth doped crystals is an active eld, with the goal of nding the ideal system with the best properties for a quantum memory. It is important to understand that it is not possible to build ourselves the ideal system by xing the dierent parameters, such as the resonant frequency of the atoms or their coherence time. In fact, these are xed by the laws of physics - or Nature. What we 13 can do, is to chose the host crystal and the dopant (with a particular concentration). Then, we can play with just a few settings of the environment such as magnetic eld or temperature (the lowest possible in principle). This doesn't mean that doing spectroscopy consists of testing randomly a large number of materials - though it can lead to new discoveries. One should understand and model the dierent processes and interactions in these systems. With the help of dierent spectroscopic studies published in the past, it should be possible to nd potential candidates for the realization of a QM and study them. Requirements We specify now the required properties of a medium for the implementation of a quantum memory, in particular for an AFC protocol. As depicted in g.2.3 a resonant transition at an optical frequency is necessary to absorb an input eld. We want for this transition a high oscillator strength (for a strong light-matter coupling) and long coherence time. To tailor an AFC in the absorption prole, an auxiliary state |auxi is needed to transfer some atoms through spectral hole burning. The lifetime of this auxiliary state must be long (compared to the excited state lifetime) so that this process is ecient. Also, an ecient AFC has a narrow structure. Hence, it is necessary to burn very narrow spectral holes, which depends on the homogeneous linewidth and other processes such as spectral diusion. Finally an additional ground state, usually a spin state |si is required to include a spin-wave storage and on-demand readout. For a long storage time, |si should have a long coherence time with respect to |gi. Choice of Nd :Y SiO 3+ 2 5 We are interested here in nding a rare earth doped crystal for a 2-level AFC protocol that would be highly multimode and ecient enough to store true single photons. Kramers ions are suited for this, since a magnetic eld allows to use exactly two Zeeman states as |gi and |auxi. At the time this thesis was started (2008), Er3+:Y2SiO5 was a very promising material, since very long optical coherence times and low spectral diusions were measured at low temperature and high magnetic eld [50]. However, because of a bad branching ratio and short Zeeman state lifetime (compared to the excited state lifetime) [51], the optical pumping was not ecient [48]. We note a realization of a CRIB protocol was nevertheless demonstrated [52], for the rst time at the single photon level and at a telecom wavelength. The rst mapping of light at the single photon level in a solid-state (independently of the protocol) was achieved in 2008 using Nd3+:YVO4 [53]. An advantage of this material was a high absorption of α = 40 cm−1 . However, the storage time and the eciency were quite low. These limitations were due to inecient optical pumping and strong superhyperne interactions between Neodymium and Vanadium ions which aected the quality of the AFC prole. 14 This interaction was also observed and problematic in Er3+:LiNbO3. With the knowledge of these previous results, we chose to investigate Nd3+:Y2SiO5. Indeed, there is no abundant elements with a strong nuclear magnetic moment (such as niobium or vanadium) in the host crystal that could cause a strong superhyperne interactions. Also, the optical pumping is potentially very ecient in neodymium compared to erbium because of the short excited state lifetime (≈ 300µs for Nd, ≈10 ms for Er). It is however necessary to nd a magnetic eld conguration in which the Zeeman state lifetime is long compared to that. Setup We did a series of measurement in Nd3+:Y2SiO5crystals, cooled to 3 K, to nd a conguration suited for an ecient AFC. We measured the absorption prole and its inhomogeneous broadening, the optical coherence time, the Zeeman state lifetime and the minimal width of a spectral hole. In appendix, we present a partial measure of the g tensor in the ground and excited state. The samples had a doping concentration of 35 ppm and were of various lengths, from 1 to 10 mm. The light was propagating along the crystallographic axis b and its polarization was in the plane dened by D1 and D2 which are the eigen axes of the index of refraction. Magnetic elds have also been applied in the D1-D2 plane with dierent intensities and directions. At rst, we used permanent magnets outside the cryostat. With such a conguration, the direction of the magnetic eld could easily be varied by changing the magnets position, but the intensity of the eld was limited to 30 mT. To increase this, we placed permanent magnets inside the cryostat, which allowed to apply a eld of 300 mT. However, we could test only a few directions, since it was more complicated to change the position of the magnets. Finally, by using another cryostat, a superconducting magnet was available, which could generate a variable eld up to 2 T. For technical reasons, this cryostat was not used for AFC storage, but only for spectroscopy measurements. Absorption and inhomogeneous broadening of the optical transition As expected [54], Nd3+:Y2SiO5has a resonance at 883.2350(6) nm corresponding to the 4I -4F transition of the site 1 in Y2SiO5. With a probe eld, we could measure the absorption coecient as a function of laser frequency and observe an inhomogeneous broadening (g.3.1). It has symmetric distribution, close to a Gaussian with a full width at half maximum (FWHM) of 4.6 GHz. The absorption is maximal when the polarization is parallel to D1 (α = 3.43 cm−1) and minimal when its parallel to D2 (α = 1.32 cm−1). From additional measurements[55], it appears that the eigen axes of refraction index coincide with the eigen axes of absorption. 9 2 3 2 15 Inhomogeneous Broadening α [cm-1] 3 polarization along D1 polarization along D2 2 1 0 -7 -6 -5 -4 -3 -2 -1 0 1 Detuning [GHz] 2 3 4 5 6 7 Figure 3.1: Absorption spectrum of Nd3+:Y2SiO5. The FWHM of the inhomogeneous broad- ening is 4.6 GHz. Also, the eigen axes of absorption coincide with the eigen axes of refraction index D1 and D2 . The absorption coecient is α = 3.43 cm−1 and α = 1.32 cm−1 , along D1 and D2 respectively. The absorption coecient will determine the storage eciency through the optical depth (d = αL). For example, with a 2 cm long crystal, the eciency should reach 41% for an optimized AFC storage [56] in the forward direction. However, in some cases the absorption can be reduced, if for instance, we apply a strong magnetic eld or if we want to use both polarizations. This could be overcome by increasing the doping concentration, multiple passes in the crystal or using an impedance-matched cavity [57]. Zeeman state lifetime Because Nd3+ is a Kramers ion, each level has a remaining degeneracy that can be lifted with an external magnetic eld (a partial study of the g tensor can be found in the appendix). In particular, this creates two Zeeman states (Ms = ±1/2) in the ground level. They can be used as |gi and |auxi (2.3) for a two-level AFC protocol. For an ecient optical pumping, thus ecient AFC storage, the Zeeman state lifetime (TZ ) must be long compared to the excited state lifetime (T1) as we have seen in section 2.3. From ref.[54] and our own measurements, we know the excited state lifetime is about 300 µs. Therefore, we need TZ to be at least tenths of ms. We will present in this section measures of TZ for various magnetic elds. Although the population relaxation occurs in radio frequencies, we can measure TZ with an optical laser, through the dynamic of 16 14 12 TZ [ms] 10 8 6 4 2 0 -90 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 90 Angle of B with D2 Figure 3.2: The Zeeman lifetime has been measured for dierent orientations of the magnetic eld (B≈ 20 mT). a spectral hole. With an ecient optical pumping, a majority of atoms are transferred from an initial to a nal ground state, which creates a spectral hole at the frequency of the pump (g.2.4). However, because of population relaxation, the spectral hole is subject to an exponential decay at a rate 1/TZ . Therefore, we measured the dynamic of a spectral hole after the optical pumping, and the obtained decay time is TZ . We note that if the optical pumping is inecient (as in Er3+:Y2SiO5for example), the excited state is still populated at the end of the hole burning. Because of this, the spectral hole dynamic would include a decay at a rate 1/T1. To our knowledge, there are no previous measurements of TZ in Nd3+:Y2SiO5. We started with a moderate magnetic eld (≈ 20 mT) by the use of permanent magnets placed outside the cryostat. We measured TZ for various angles θ between the D2 axis and the magnetic eld. We note that θ and θ+180are equivalent, since it corresponds to an inversion of the magnetic eld. There is however an ambiguity on θ. Indeed, although the D1 and D2 axes can be well identied, their directions are unknown without a crystallographic study of the sample. For this reason, an angle of +θ could be taken as −θ in another experiment. We observed a Zeeman state lifetime between 5 and 14 ms (g.3.2). The longest lifetime arises for an angle of about −40, while at an angle of 90the spectral hole was to small to measure its dynamic. Interestingly, D1 and D2 do not appear to be axes of symmetry as is the case for the g tensor (g. A.4 in appendix). We must say we don't have a good explanation for this angular dependence of TZ . To search for longer Zeeman lifetimes, we increased the magnetic eld (up to 2 T) 17 for two particular angles, θ = −30and θ=90(g.3.3a). For both angles, TZ starts to increase with the magnetic eld, reaches a maximum, and nishes to decrease. In particular, we obtained a Zeeman lifetime of 150 ms for a magnetic eld of 370 mT at an angle of -30with the D2 axis. Comparing the two angles, we note the maximum of TZ does not arise for the same magnetic eld. However, this may be explained by the fact that the g factor of the ground state is dierent for both angles. Actually, while the optimal magnetic elds are dierent for the two angles, the ground state splits are almost the same (≈ 12GHz) when TZ reaches a maximum (g.3.3b). This is probably not a coincidence, but we are yet far of having a full theoretical understanding of it. Many processes have been identied to explain the spin relaxation rate [50]. The one-phonon direct process, involving the absorption or emission of a resonant phonon, increases with the magnetic eld. The two-phonon Raman and two-phonon resonant Orbach processes depend on the temperature but not on the magnetic eld. A process that should decrease with the magnetic eld is the ip-op rate (exchange of spin with a neighbour ion). At high magnetic eld, we may neglect the ip-op, and the spin relaxation rate (R = 1/TZ ) can be written as[50]: 5 R(∆E) = αD · (∆E) coth ∆E 2kB T gµB B 2kB T + αRO (T ), , (3.1) where αD is an anisotropic constant, ∆E the split in the ground state, kB the Boltzmann constant, and αRO (T ) is the relaxation rate due to the Raman and Orbach processes. In g.3.4 , we use this formula to t our data for ∆E >12 GHz. The good agreement between the data and the tted curve indicates that the direct process must be the cause for the increase of spin relaxation rate with the magnetic eld. However, it is more dicult to explain the spin relaxation at low magnetic elds. Indeed, the ip op rate should be[50]: 2 Rf f = αf f sech where αf f is an anisotropic constant and µB the Bohr magneton. If we t the formula to the data, g is huge (around 50), much larger than the observed g factor of the Zeeman state (which is always less than 4 g.A.4). Therefore, it appears we don't have a good explanation of the spin relaxation rate process at low magnetic eld. We also would like to study TZ if we increase the temperature. This can help us to understand the underlying process of spin relaxation. For example, the Raman process has a dependency of T 9 and it may dominate the spin relaxation rate above a certain temperature. Also, for practical reasons, we would like to know at which temperature must be the crystal to operate well as a quantum memory. We measured spectral hole decays for various temperatures (g.3.5) at a particular angle (300 mT with θ=-30). It appears that the spin relaxation time is roughly constant between 3 and 4 K and it 18 Figure 3.3: Zeeman state lifetime for various magnetic elds. top: For both angles of B with D2 , we see the same behaviour, an increase of lifetime followed by a decrease, but the maximum arise for a dierent B. However, we can plot(bottom) the lifetime as a function of the split in the ground state, and we see that the maximum arise exactly for the same split. 19 Spin relaxation rate (Hz) 2 10 1 10 0 5 10 15 20 25 30 35 40 45 Zeeman states split (GHz) Figure 3.4: Spin relaxation rate (1/TZ ) for an angle of B with D2 of -30(blue dots) and 90(green squares). A t using eq.3.1 is in good agreement with experimental data for large splits. increases above this temperature. We note we observed spectral holes even above 6 K, however with a fast decay. Finally, we investigate the eciency of the optical pumping for the same particular magnetic eld (which will be used for AFC storage). The laser frequency was scanned during the optical pumping, which created a large spectral pit (g.3.6). After this, we measured the remaining optical depth in the centre of the pit (d0) and compared to the initial optical depth (d). The ratio d0/d was about 4%. This is a signicant improvement if we compare to Er3+:Y2SiO5, where the optical pumping was very inefcient. We will see in section 3.2 how it does aect the storage eciency. In a simple 3-level model (sec.2.3), the ratio d0/d should be equal to the ratio between the excited state lifetime and the Zeeman state lifetime (if T 1 TZ ). Here, T1=300µs and for this magnetic eld TZ =120 ms, so that we have T1/TZ =0.25% which does not agree with the measure of d0/d. We note that the branching ratio is close to 0.5 (measures in the appendix), so that it shouldn't aect much the optical pumping eciency. We do not have yet a good explanation for this disagreement. One possibility is that T1 is longer than estimated. Indeed, the average time for an ions to decay from the excited state to the ground state may be much longer than 300µs if it is trapped in a meta-stable level. 20 population fraction in initial ground state 1 0.8 0.6 3.2 K 3.7 K 4.2 K 4. 8K 5.4 K 0.4 0.2 0 0 100 200 300 waiting time after optical pumping [ms] 400 Figure 3.5: Spectral hole decay for various temperatures. Under 4 K, the spin relaxation rate is constant. 3.5 3 optical depth 2.5 2 1.5 1 0.5 0 -20 -10 0 10 20 30 40 50 Detuning [MHz[ Figure 3.6: Pit of absorption measured shortly after optical pumping. Even if the burning time has been long enough, there is a residual absorption in the pit of around 4% of the initial value. 21 B[mT] 20 77 300 20 300 θ[] -30 -30 -30 5 30 T2[µs] 4.9±0.2 15±9 93±15 6.3±0.3 60±3 x 2 1 1 2 1 Table 3.1: Coherence time for dierent magnetic eld congurations. The value for x is not a tting parameter. It was xed to 1 if the decay appeared to be exponential, and xed to 2 if it was not the case. Coherence time of the optical transition Upon absorption of a wave packet, an atomic ensemble is in a particular entangled state where the excitation is delocalized among all the ions (eq.2.1). The coherence between the ground and excited states is essential, since it allows a collective interference and coherent emission of a photon echo. However, because of the interaction with environment, this coherence undergoes an exponential decay with a time constant T2, the coherence time. This denes the homogeneous linewidth γh = πT1 and a spectral hole cannot be narrower than 2γh [49]. In the context of AFC storage, this will aect the storage eciency when the optical storage time approaches T2. Indeed, it would require to tailor, in the absorption prole, narrow structures of widths approaching γh. We present here measures of T2 by the use of two-pulse photon echoes (2PE) [58] for various magnetic elds. The principle is to send two short but strong pulses into the medium, separated by a time t12. A coherence is created between the ground and excited states, which results in an emission of an echo in the same spatial mode at a time t12 after the second pulse. The exponential decay of the echo intensity with storage time allows to measure directly the coherence time. The echo intensity can be described in a empirical form proposed by Mims[59]: Iecho = I0e−(4t /T ) where x ∈ [1; 2] is a phenomenological constant resulting from spectral diusion. The obtained values of T2 for various magnetic elds are given in table 3.1. The highest measured value is T2=93µs for a magnetic eld of 300 mT with θ = −30. This is relatively high, and approaches the excited state lifetime T1 =300µs1 . Also, for a high magnetic eld, the decay is almost purely exponential (x=1) which implies low spectral diusion. With this result, it appears that it should be possible to burn very narrow spectral holes, since the corresponding homogeneous linewidth is γh =3.4 kHz. However, with a more precise measure, we observed a small oscillation in the echo intensity with a period of about 1.5 µs. This is probably due to an additional level structure and we will see in the next section how it does aect the spectral hole 2 12 1T 1 2 x has been measured through uorescence detection and stimulated photon echoes 22 90 80 70 T2 [μs] 60 50 40 30 20 10 0 2.5 3 3.5 4 4.5 5 Temperature [K] 5.5 6 6.5 Figure 3.7: Optical coherence time with increasing temperature. burning. Finally, we measured T2 for higher temperatures, with B=300 mT and θ = −30(g.3.7). Not surprisingly, the coherence time is shorter for higher temperatures. There is however still a strong photon echo at 6 K. Narrow spectral hole burning For ecient and long AFC storage, it is necessary to tailor narrow structures in the absorption prole. In particular, it is necessary to burn very narrow spectral holes. Their width is ultimately limited by the the homogeneous linewidth which has been measured here through 2PE spectroscopy. However, after a certain time, a spectral hole may be broadened by spectral diusion. 2PE is not much sensitive to this eect, it will only lead to a non-exponential decay of an echo. Therefore, a reliable method to measure spectral diusion is simply to perform spectral hole burning spectroscopy. However, to avoid power broadening, the optical pumping power must be small, so that only a small fraction of population is transferred. Also, the laser linewidth should be narrow enough, so that a spectral hole width is not limited by the coherence of the laser2. We present here narrow spectral hole burning for a magnetic eld of 20 mT and 300 mT at an angle of -30with D2 (g. 3.8). This last conguration was chosen because it is close to an optimum in term of optical pumping eciency and for practical 2 We note that SHB may be used to measure the linewidth of a laser if the atoms linewidth is known 23 optical depth 3 2 1 0 -4'000 B=10mT B=300mT -2'000 0 detuning [kHz] 2'000 4'000 Figure 3.8: Spectral holes for two dierent magnetic elds at an angle of -30with D2. The hole widths are 240 kHz. For B=300 mT, we observe side holes at ±640 kHz that are probably due to a superhyperne interaction with yttrium ions. reasons (it is dicult to get a higher B with permanent magnets). In both case, the measured hole width is about 240 kHz. We note this is larger than the homogeneous linewidth: γh=3.4kHz for 300 mT and γh=65 kHz for 20 mT. The broadening for the low magnetic eld is probably due to spectral diusion, as it was expected from a non-exponential 2PE decay. We note that this broadening is quite moderate, since it is usually dicult to attain the limit of the homogeneous linewidth. For the high magnetic eld though, we believe there may be here some power broadening, since we observed narrower spectral holes. This is however negligible compared to side holes that appears at ±650 kHz. This makes it problematic to tailor an AFC for long storage times. Indeed, not only it adds an unwanted structure, but it appears that atoms between the central and the side holes are also aected by optical pumping. For this reason, we can consider the eective linewidth for this magnetic eld is much larger than 240 kHz. The side holes also induced an oscillation on the 2PE decay. This is the result of a quantum beat between dierent transitions. We believe these transitions occur from a superhyperne interaction with yttrium ions in the crystal. Indeed the positions of the side holes depend on the external magnetic eld. For B=77 mT, they were at 24 kHz. If we assume a linear dependency, it appears to be about 2 MHz/T. This is compatible with the gyromagnetic ratio of yttrium ions (Y=2.1 MHz/T). We note that the magnetic moment of yttrium is small compared to the one of vanadium, which explains why the superhyperne interaction is here less strong than in Nd3+:YVO4. Nevertheless, yttrium is an abundant element with a magnetic moment and it appears superhyperne interaction is inevitable. At 300 mT, the split of 650 kHz makes it problematic, and the question remains open whether a stronger magnetic eld would suppress this interaction or not. ±194 Conclusion on Nd3+ :Y2 SiO5 spectroscopy In this work, we presented various spectroscopic measures on Nd3+:Y2SiO5 to nd a suitable conguration for AFC storage. We rst measured the inhomogeneous broadening together with the absorption coecient for two axes of polarization. For various magnetic elds and temperatures, we performed spectral hole burning, which allowed to measure Zeeman state lifetimes. Additionally, we measured the optical pumping eciency and the width of a spectral hole. This was completed with measures of coherence time through 2PE. Measures of g tensors are presented in the appendix. We note that we did not test all possible magnetic congurations, and we do no have yet a full theoretical understanding of the various mechanisms in the material. Nevertheless, setting a magnetic eld at an angle θ=-30is, for now, the most promising for AFC storage. With a low magnetic eld, (≈20 mT), the spectral diusion is reasonable, so that it is possible to obtain a hole linewidth of 240 kHz. However, the optical pumping is inecient, because of a short Zeeman states lifetime. On the contrary, the optical pumping is ecient for a strong magnetic eld (300-400 mT). There is however a strong superhyperne interaction with yttrium ions, that forbids to tailor narrow structures for an AFC. We conclude that such a conguration is promising to implement a twolevel AFC storage. It is potentially multimode (sec.3.3) if we create large combs and it may also be ecient (sec.3.2and sec.3.5) if we restrain it to short storage times. 3.2 Storage eciency Motivation The storage eciency η of a quantum memory is dened as the ratio between the energies of the input and output states. We want to maximize it, since it is a crucial parameter in any applications. For example, in the case of a quantum repeater, one usually assumes an eciency of 90% to maintain a high entanglement distribution rate. Additionally, a high eciency would allow us to realize elaborate experiments, such as storage of true single photons where the diculty is to distinguish a weak signal from the noise. Before the present work (2008), demonstrations of a quantum 25 storage protocol in solid-state materials were quite inecient. In our group, storage of weak coherent states was achieved in Nd3+:YVO4[53] and Er3+:Y2SiO5[52] with storage eciency of 0.5% and 0.2%, respectively. Calculation of eciency For ecient storage, we need a high optical depth of the medium. This allows a strong light-matter coupling, thus a high probability of absorption and emission. Additionally, in a photon echo quantum memory, all atoms need to be in phase at the time of the echo. Therefore the rephasing eciency needs to be high. This is usually achieved by tailoring narrow structures in the absorption prole. As an example, we give the formulas of storage eciency in the case of an AFC constituted of Gaussian peaks (see g.2.3). This was obtained by solving Maxwell-Bloch equations in the original AFC paper [41]. If the emission is in the forward mode, we have: 7 ˜ ηf = d˜2 e−d e− F 2 , where F is the nesse of the AFC: F = ∆/γ with ∆ as the separation between the Gaussian peaks and γ their FWHM. d˜ is the eective optical depth and can be expressed as d˜ ≈ d/F where d is the amplitude of a peak (in other words, the maximal optical depth). We note that because of the re-absorbing factor e−d˜, the eciency is limited to ηmax =4e−2 ≈54%. If the emission can be done in the backward direction, (by the use of anti-propagating control elds for example) it is possible to reach unit eciency: ˜ 7 ηb = (1 − e−d )2 e− F 2 Ideally, we would need highly absorbing (d˜) and narrow peaks (F ). However, for a given medium, d is xed. Therefore, to increase d˜, it is necessary to decrease the nesse. This would however aect the rephasing eciency (e− ). This implies that there is a trade-o, for a given d, between high absorption and ecient rephasing. The optimal nesse can be calculated analytically in this case, but also empirically if the eciency is more dicult to predict. Originally, the eciency was calculated for Gaussian peaks. However, depending on the AFC preparation, the peaks have sometimes a Lorentzian shape [60, 43]. An analytical formula for such a case has been given by Chanelière et al.[43] and it was noted that the optimal eciency is worse if the peaks are Lorentzian rather than Gaussian. This brings the question of the optimal shape of the peaks to maximize the storage eciency. To answer this, we may use a formula given by Bonarota et al.[56], which assumes only that the absorption of the medium is a periodic in P function −iω/∆ frequency. If the absorption is decomposed in a Fourier series: α(ω) = n αne , the eciency can be formulated as: ηf (L) = |α−1 L|2 e−α L (3.2) 7 F2 0 26 0.6 0.35 0.3 optimized efficiency ηf Optimized efficiency ηf 0.5 0.4 0.3 Peak shape: Square Super Gaussian Gaussian Triangle Lorentzian 0.2 0.1 0 0.25 0.2 0.15 0.1 0.05 0 5 10 Maximal optical depth d 15 20 (a) 0 0 5 10 15 20 25 maximal optical depth d 30 35 40 (b) Figure 3.9: (a) The storage eciency is calculated for various shapes of an AFC using eq.3.2. It appears that square peaks are optimal.(b) If we introduce an absorbing background proportional to the maximal optical depth (here d0 = 0.04 · d), the eciency cannot reach 54%. Instead, we observe here an optimum for d=8 of 33% and the absorbing background dominates for higher optical depths. Using this general formula, we calculated (numerically) the optimal eciency for various peak shapes as a function of the maximal optical depth (g.3.9(a)). It appears that the square shape is the most ecient among the tested possibilities and it is expected it would be more ecient than any shape [56]. In principle this could be proven analytically using formula 3.2, but an intuitive explanation can also justify it. Indeed, the rephasing of an AFC is the most ecient if the atoms are the closest to the centre of a peak. Since the optical depth at any frequency is limited by d, a square shape is the solution that saturates this constraint. We note that tailoring such a square comb is, in a way, the most dicult solution, since it would require an innite coherence to tailor vertical edges. Therefore, the homogeneous linewidth should be small compared to the width of a peak. We mention also that, during the AFC preparation, the optical pumping is not necessarily perfect. A fraction of atoms remains in their initial state, which results in an absorbing background d0. This aects the storage eciency by a factor e−d [53]. In principle, d0 is proportional to d, assuming the preparation is not more dicult at high absorption. For this reason, there is a trade-o between a high absorption in the AFC and a background absorption, and this results in an optimal optical depth of the sample(g.3.9(b)). 0 27 AFC in Nd :Y SiO 3+ 2 5 We present here AFC storage in a Nd3+:Y2SiO5crystal. From the knowledge of our spectroscopic measures, we applied a magnetic eld of 300 mT at an angle of −30with respect D2 to obtain a long Zeeman lifetime. Because the split in the ground state is larger than the inhomogeneous broadening, the absorption is divided by 2 compared to the situation without a magnetic eld. Using a 1 cm long crystal and a doublepass conguration, we obtain a maximal optical depth of about 3. An AFC is tailored (g.3.10(a)) by the use of a non-coherent preparation. It consists of scanning the laser frequency and modulating its intensity, so that it permits a frequency-selective optical pumping. If the pre-programmed storage time is short (less than 100 ns), the width of a spectral hole is narrow enough to create squarish and non-overlapping peaks. We tested the AFC by storing a classical pulse during 33 ns and observed an eciency of 17% (3.10(b)). We note that we are here at the limit of the bandwidth of the photo-diode and of the acousto-optic modulator (AOM) which creates the input pulse. For this reason it was not possible to measure here the storage eciency for shorter storage times, but we mention we reached an eciency of 21% for a storage time of 25 ns in the single photon regime [61]. To increase more the eciency, we would need a higher optical depth. However, because of the absorbing background, it would not be possible to reach more than 33%(g.3.9(b)). This would occur for an optical depth of 8 and a comb nesse of 4.6, and we estimate this would bring many technical diculties. Therefore, to increase signicantly the storage eciency, we would probably need another material. For example, we reached η=35% in a Pr3+:Y2SiO5crystal in collaboration with Lund university [60] and Hedges et al.. demonstrated a storage eciency of 69%, by the use of a gradient-echo memory [62]. Nonetheless, we demonstrated here a signicant improvement of storage eciency, and it will help us to realize more elaborate experiments. AFC in Eu :Y SiO with an impedance-matched cavity 3+ 2 5 In this thesis, we worked also on the storage of light in a Eu3+:Y2SiO5crystal, which is a non-Kramers ion. In the perspective of ecient storage, it has a very good potential: the optical pumping is extremely ecient because of an ultra long spin states lifetime [63] and it would be possible to tailor very narrow structures because of a very narrow homogeneous linewidth [63, 64, 65]. However, our main motivation comes from the fact that it is a very promising system to implement a QM with an on-demand readout and a highly multimode capacity [66]. The splits in the ground states are large compared to Pr3+:Y2SiO5[67], but the oscillator strength is much lower. Unfortunately, this cannot be compensated with a higher doping concentration, since it increases a lot the inhomogenous broadening and only a little the absorption coecient [63]. Increasing the crystal length is also not a satisfying solution, since it would bring many 28 1 3.5 0.9 3 0.8 input 0.7 relative intensity optical depth 2.5 2 1.5 0.6 0.5 0.4 transmission (23%) 0.3 1 echo (17%) 0.2 0.5 0.1 0 0 20 40 60 Detuning [MHz] 80 0 -40 100 (a) -30 -20 -10 0 10 20 30 40 time [ns] 50 60 70 80 90 100 (b) Figure 3.10: (a) AFC with periodicity of 30 MHz in a Nd3+:Y2SiO5crystal. (b) It is tested by storing a classical pulse for 33 ns and we obtain an eciency of 17%. technical diculties. An alternative has been proposed [57, 68] to place the crystal into an asymmetric cavity (Gires-Tournoi interferometer) to reach perfect absorption (g.3.11) . If the QM has an optical depth d˜, the impedance-matching conditions are reached when the reectivity is R1 = e−2d˜ for the rst mirror and R2 = 1 for the second one. Indeed, similarly to a standard Fabry-Perot cavity, in the backward direction, the eld emitted by the cavity interferes with the eld reected by the rst mirror. In resonance, this interference is destructive, so that the intensity is zero at the output of the cavity and the QM has absorbed 100% of the light. Moreover, an echo will be emitted at the expected storage time, and the eciency can now reach 100% [57]. With such a scheme, it is therefore possible to attain high eciencies, even with a weakly absorbing medium. If the spatial interference is perfect, η is only limited by the rephasing eciency (narrow peaks) and the re-absorption in a single pass. We worked here to realize a cavity-enhanced storage in a Eu3+:Y2SiO5crystal, by the use of mirrors placed outside the cryostat 3.12(a). Because the cavity is sensitive to any environment perturbations (vibrations, sounds, temperature uctuations...), it needs to be actively stabilized using a piezo-electric on a mirror and a probe light to obtain an error signal. The advantage of such setup relies on the possibility to add another crossed-beam that would not be aected by the mirrors (g.3.12(a)). Such beam can be used to prepare the AFC, scan the absorption prole and, most importantly, to implement a control eld to allow spin-wave storage. We could observe the cavity modes and a very high absorption (g. 3.12(b)), but we were not yet able to implement an AFC storage. We need to overcome few technical details, but we believe it is only a matter of time to be able realize a cavity-enhanced AFC storage. 29 Figure 3.11: Schematic setup of a QM in an impedance-matched cavity. The light emitted by the cavity interferes with the reection on the rst mirror. This allows to reach 100% storage eciency, even for a weakly aborbing medium. 1.1 stabilisation light 1 0.9 probe light photo-diode 0.8 PID box mirror R=70% amplifier reflected intensity preparation light (optical pumping) 0.7 empty cavity 0.6 0.5 0.4 0.3 cryostat cavity containing an attenuator 0.2 0.1 miror R=100% 0 laser frequency piezo-electric (a) (b) Figure 3.12: (a) Setup to realize cavity-enhanced storage in a Eu3+:Y2SiO5crystal. (b)By scanning the laser frequency, we observe resonances in the reected light by the cavity. The QM is simulated here with a variable attenuator. Thanks to the impedance matching conditions, it is possible to absorb about 90% of the light. It is here limited by the quality of the spatial interference between all the modes of the cavity. If the light is out of resonance or if the cavity is empty, all the light is rejected. 30 3.3 Multimode quantum storage Classical memories have today the capacity to store a huge amount of information. In analogy, we expect a QM could have a multimode capacity, that is to say, it would store more than one qubit at the same time. The number of modes (N ) it can store is a crucial criterion depending for many applications. For example, it can speedup the entanglement generation in some quantum repeater architectures [69]. Of course, the largest N would be the best, but we note that a value of N=100 is often assumed to compare quantum repeater architectures [18]. In general, one can consider a spatial, frequency or temporal multiplexing. The last two possibilities appear to be the most adapted for the coupling to single mode bres. We will here consider temporal modes. For almost any storage protocol (EIT, CRIB...), an increase of the number of modes requires an increase of optical depth [26]. For this reason, before this work, quantum storage demonstrations were usually not multimode. However, in an AFC protocol, the number of temporal modes is independent of the optical depth [41, 26] and it is considered intrinsically multimode. Indeed, N is proportional to the storage time and to the bandwidth of a QM: the time-bandwidth product. For an AFC, the bandwidth can be increased by increasing the number of peaks without changing their separation ∆ (g.3.13). Alternatively, on can increase the storage time by increasing the density of peaks. Both solution does not require a higher optical depth. Therefore, to realize a highly multimode QM, we need to create an AFC with a large number of peaks. We note that rare earth doped crystals are well adapted to this, since the inhomogeneous broadening is usually large compared to the homogeneous linewidth. Here [70], we demonstrated the multimode potential of a quantum memory based on AFC by storing up to 64 dierent time modes in a Nd3+:Y2SiO5 crystal (g.3.14). We had to limit the storage time to roughly 2µs because of material limits (sec.3.1). The comb bandwidth (100 MHz), however, could have been increased to few GHz with a more elaborate optical pumping. We used weak coherent states with less than a photon per pulse, and the detection was performed with a silicon single photon counter. This demonstrates, in comparison to previous multimode optical data storage[71], the potential to store true quantum states. Moreover, we also showed the preservation of coherence by performing interference between consecutive pulses with various phase dierences (g.3.15). For more details, we refer to the published paper [70] (appendix B, p.79) 3.4 On-demand storage Up to here, I presented experiments with a preprogrammed storage time with a 2level AFC. Although it could be rapidly tuned, the storage time could not be changed after an input had been absorbed in the QM. An on-demand readout is however nec31 (a) (b) 3.0 2.5 Efficiency [%] Optical depth 6 2.0 1.5 5 4 3 2 1.0 1 0.5 -50 -25 0 25 0 50 5 Optical Detuning [MHz] 10 15 20 25 30 Input pulse duration [ns] Figure 3.13: (a) By adding absorbing peaks to an initial AFC (green line), we increase its Normalized counts bandwidth and keep the optical depth (dashed black line) (b) This allows to store shorter pulses with (at least) the same eciency, thus to increase the number of temporal modes. The same principle could be applied to increase the storage time. 1.0 Input modes Output modes x50 0.8 0.6 0.4 0.2 0.0 0.0 0.4 0.8 1.2 1.6 2.0 2.4 Time (ms) Figure 3.14: Because of a large time-bandwidth product, it is possible to send 64 dierent short pulses into the QM before the light is re-emitted. Each full pulse have an average number of photons n̄ < 1. 32 (b) 140 700 120 600 100 500 80 400 Counts Counts (a) 60 300 40 200 20 100 0 0 Noise level 0 200 400 600 800 1000 1200 0 1 Time [ns] 2 3 4 Phase [rad] Figure 3.15: (a) A series of pulses with various phases is stored in a medium with two AFCs with dierent periodicities. This induces, at the output, the interference between consecutive pulses. Thanks to the multimode capacity, it is possible to visualize a complete interference fringe in one measurement. (b) The raw visibility (red curve) of this interference is 78±3%. Substracting the detector dark count, the net visibility is 86±3%.) essary in most of the applications, in particular for a quantum repeater. Additionally, for a 2-level AFC, the storage time is limited (in the best case) to the optical coherence time. However, quantum repeaters require long storage times. Indeed, the memory storage time must be much larger than the time for a photon to travel through an elementary link. This can go from 10 ms to 1 s [18]. The solution is to add 2 control pulses to include a spin-wave storage [41]. This allows to re-emit a photon whenever it is needed, and the storage duration is now limited by the spin coherence time which can be extremely long. Spin-wave storage has already be implemented in solid-state memories in an EIT protocol, and for classical light [72, 73]. The goal, here, was to realize it with an AFC. Spin-wave storage in Pr :Y SiO and in Eu :Y SiO 3+ 2 3+ 5 2 5 To realize a complete AFC protocol, we need a system with at least three long-lived ground states and one optical excited state. Excellent candidates are non-Kramers ions with a hyperne structure. For example, Pr3+:Y2SiO5 and Eu3+:Y2SiO5 have exactly 3 ground states at zero magnetic elds with very long coherence times. Pr3+:Y2SiO5 has a high oscillator strength, which allowed an ecient 2-level AFC storage [60]. Therefore, we realized the rst complete AFC protocol in this system, in collaboration with Lund university [74] (g.3.16). It allowed us to store bright pulses up to 20µs. Later, we were able to implement also spin-wave storage in a Eu3+:Y2SiO5crystal [75]. More details 33 Normalized intensity Normalized intensity 14 12 10 8 16 12 8 4 0 0 6 4 1 2 3 Time (ms) 4 Detector gate off x10 2 Control pulses 0 0 2 4 6 Time (ms) 8 10 12 Figure 3.16: Spin-wave storage in a Pr3+:Y2SiO5crystal. The inset shows a 2-level AFC storage. When the two control pulses are added, the storage time is extended can be found in the published papers [74, 75] (appendix B p.67 and p.114). Storage of weak coherent states The future challenge is to implement spin-wave storage with true single photons as it would be in a quantum repeater. However, all the previous experiments, were realized with bright classical pulses. Therefore, the rst step is to store weak coherent states. This is however much more dicult than with a 2-level AFC, because of the very bright control pulses: they may induce dierent kind of noise, such as free induction decay, uorescence, but also after-pulses and detector blinding. In this perspective, Eu3+:Y2SiO5 is more adapted to Pr3+:Y2SiO5, because of the larger level spacings in the ground states. Indeed, the frequency separation between the control eld and the echo is larger, which makes the ltering more feasible. We note also, that in an AFC protocol, unlike EIT, the control elds is switched o during the photon echo. Nonetheless, we had to use a ltering cavity, temporal gating and dierent spatial mode to obtain a satisfying signal-to-noise ratio at the single photon level. Very recently, we were able to store weak coherent states with an average of 2.5±0.6 photon per pulses (g.3.17). We shown also a relatively low noise of 7.1 ± 2.3 · 10−3 photons per mode. These results are not yet published, but an article is under a review process [76] (appendix B p.122). Spin refocusing In all the experiment I presented here, the storage times (roughly tens of µs) were much shorter than the spin coherence times (15.5 ms in Eu3+:Y2SiO5 [77]). This is attributed to inhomogeneous spin dephasing. Indeed, the spin transitions have a inhomogeneous broadening of usually tenth of kHz. This can however be compensated 34 Figure 3.17: First demonstration of a spin-wave storage in a solid-state memory with weak coherent states. Here the average number photon per pulses is 2.5±0.6. with spin echo techniques. An impressive demonstration was realized by Longdell et al.. to stop light for more than 1 s in a Pr3+:Y2SiO5crystal with an EIT protocol. We note that a dynamical decoupling sequence allowed to extend even the spin coherence time. To achieve this, the time between the refocusing pulses must be much shorter than the coherence time. We are now working on implementing spin refocusing in our Eu3+:Y2SiO5 quantum memory. We were able to observe a clear extension of the storage time (g.3.18) to roughly 10 ms. This work is however under progress and we need to understand why we observe no echo for certain storage times. 3.5 Testing a QM with single photons A QM must have the ability to store faithfully a quantum state, which implies it must also preserve quantum entanglement. In previous experiments [52, 53], weak coherent states were stored. This demonstrated that it is possible to obtain a satisfying signal-to-noise ratio, even at the single photon level, but the preservation of a quantum state or entanglement was not shown. We want here to test a QM by the use of a single photon source. Additionally, this would be a step forward in the realization of quantum repeater, which includes always the generation of quantum states. We note that before 2011, the storage of true single photons was never realized in a solid-state device. Single photon sources Single photons can be produced by an external source or by the QM itself (as in a DLCZ scheme [17, 45]). We note that, for previous experiments in rubidium or cesium vapours [34, 32], single photons were emitted by a separate atomic ensemble. This 35 Figure 3.18: Spin-wave storage of classical pulses including a spin refocusing technique. The storage time has been increase to almost 10 ms, but we don't understand yet large variations of the echo intensity. implied that the photons had a wavelength and bandwidth perfectly adapted to the QM. However, a QM based on AFC in a rare-earth crystal have potentially a large bandwidth. Therefore, it opens up the possibility to use other kind of sources that produces large bandwidth photons. For example, SPDC in a non-linear medium [6, 7] is widely used in quantum optics experiments to create correlated photon pairs with a high rate. Moreover, such a source is part of certain quantum repeater schemes [69, 18]. Here we used a PPKTP waveguide pumped by a 532 nm light, which is converted into a idler eld at a telecom wavelength (1338 nm) and into a signal eld at the wavelength of the QM (883 nm). Both modes are ltered with cavities to match the AFC bandwidth (120 MHz). More details and characterization of this source can be found in the supplementary information of ref.[78] (Appendix B p.101). Single photon storage We used this source to demonstrate the rst storage of true single photons in a solid-state device [61]. The signal photon is sent to a Nd3+:Y2SiO5 crystal to be stored for up to 200 ns and the idler photons was coupled to a 50 m long telecom bre. A histogram of the coincidence detections is shown in g.3.19 and we measured a cross (2) correlation well above the classical limit of gsi =2. Additionally, we could demonstrate that the storage process preserved the energy-time entanglement between the signal and idler photons. By the use of a Franson interferometer [79] (g.3.20), we performed a Bell experiment to violate a Clauser-Horne-Shimony-Holt (CHSH) inequality[80]. We found the Bell parameter (S = 2.64 ± 0.23) to be by three standard deviation above the classical limit (S=2). For more details about these results, I refer to the published paper [61] (appendix B p.92). We note that Saglamyurek et al. have performed a 36 a with AFC b Classical regime Figure 3.19: (a) The histogram of coincidence detections shows that the signal photon was stored for various durations in the QM (b) The cross-correlation is maintain well above the classical limit for any storage time very similar experiments with a thulium-doped lithium niobate waveguide, which was published in the same issue[81]. Heralded entanglement between two QMs A quantum repeater is composed of distant and entangled quantum memories. Therefore we estimate an important test is to perform heralded entanglement between two separated QMs, at least for a short spatial separation. Also, such experiment has never been demonstrated for solid-state devices before we started this work. Here, we used our photon pairs source to herald a single-photon entanglement and store it into two Nd3+:Y2SiO5 crystals separated by 1.3 cm [78] (g.3.21(a)). After the re-emission of the light eld, a tomography was performed on the photonic state. This allowed to calculate the concurrence, which is a measure of entanglement. It goes from 0 for a separable state to 1 for a maximally entangled state. We found a lower-bound on the concurrence that was positive, which revealed the entanglement of the QMs (g.3.21(b)). Such a setup was inspired by previous works in cold gases [82, 83, 84]. One can found more details in the published paper [78] (appendix B p.101). We mention a similar work has been published shortly before this, demonstrating the entanglement of diamonds at room temperature [85]. 37 a b Path Combinations: short-short long-long long-short short-long Start Long Arm SSPD 50% Short Arm Stop 50% Si APD Source of Entangled Photons 883 nm 1338 nm 50 % Long Storage Time % 50 : Faraday Mirror Short Storage Time Figure 3.20: (a) Franson-type setup. It allows the interference between the |shorti|shorti and |longi|longi modes. (b) The good visibilities of this interference (78±4% and 84±4) is sucient to violate a Bell inequality m 1c Heralding detector MB B Filtering 133 88 m 5 Memory preparation MA Detector 1 PBS PPKTP waveguide cm BS DM n 32 A FR m 3n 8nm 1.3 Switch Detector 2 (a) (b) Figure 3.21: (a) An SPDC source heralds the presence of a single photon which is sent into a beamsplitter. This creates a single photon entangled state, delocalized between to spatial modes. This state is transferred to two quantum memories that become entangled and keep the excitation for 50 ns. To measure the amount of entanglement, the excitation is transferred back to a light eld, and a tomography is performed with two single photon detectors.(b) This allowed to calculate a lower bound on the concurrence. A positive value revealed the presence of entanglement. 38 Chapter 4 Discussions and Outlook In this thesis, I worked on dierent aspects of a quantum memory with the prospect of implementing it in a quantum repeater. First, our spectroscopic measurements in a Nd3+:Y2SiO5 crystal allowed to nd a system for a promising implementation of an AFC protocol. Also, we worked on the important properties of a quantum memory: the storage eciency, the multimode capacity and the storage time (with an on-demand readout). Finally, we tested our QMs, through the storage of single photons, preservation of time-bin entanglement and heralding of entanglement between two crystals. Spectroscopy Through our spectroscopic measures in Nd3+:Y2SiO5, we found interesting magnetic eld congurations for a two-level AFC. It allowed us to realize many breakthroughs for quantum storage in a solid-state device [61, 70, 78, 55], and we hope more will come. Unfortunately, with a low magnetic eld, the optical pumping was not ecient. For this reason, the experiments were done with a high magnetic eld. However, a superhyperne interaction with yttrium ions limited the storage time to less than 2µs. Also, even for a high magnetic eld, the optical pumping is not perfect and about 4% of atoms remain in the spectral hole where the AFC is created. Therefore, it should be interesting to nd another conguration with a longer Zeeman lifetime and less aecting superhyperne interactions. To nd this, we would need to do a more systematic spectroscopy for various magnetic elds in the D1-D2 plane. A better theoretical understanding of the spin relaxation process as a function of the magnetic eld would also help for nding such conguration. Also, we observed some very long lived spectral holes (∼1 s) which probably results from the hyperne structure in 143 Nd and 145Nd isotopes. This would promise very ecient optical pumping and an additional ground state that could be used for a complete AFC scheme. However, the high value of the nuclear spin (7/2), together with the 1/2 electronic spin, generates 16 39 dierent levels in the ground and excited states, thus 256 transitions! Because of this, a hole burning spectrum is extremely dicult to read, in particular if the two isotopes are present in a naturally doped crystal. Therefore, it would be interesting to work with a isotopically pure sample, to measure the eciency of the optical pumping and maybe nd a Λ system suitable for a 3-level AFC storage. I also believe that it is necessary to search for other possible systems, that is to say ions and host crystals. For example, Nd3+:YLiF4 isotopically pure in 7Li presents a very small inhomogeneous broadening of 45 MHz [86]. Not only could it lead to high absorption, but it also allows to resolve the dierent optical and hyperne transitions without the need for spectral hole burning. With only one class of atoms, it would be much easier to nd a Λ system. It would be interesting to continue this work by doing coherence measurements and spectral hole burning in this system. I would also like to mention Er3+:Y2SiO5 which has the great advantage of having its resonance transitions at telecom wavelength. Previous work [50] demonstrated impressively long coherence times and low spectral diusion at very low temperature and strong magnetic eld. Unfortunately, further experiments in Geneva [51] showed this was counterbalanced by inecient optical pumping due to a short Zeeman lifetime compared to the excited state lifetime. For this reason, there is no more research, to our knowledge, for quantum storage in this material. However, we know that the Zeeman lifetime can be strongly increased with a moderate magnetic eld at a particular angle, as we have seen in Nd3+:Y2SiO5. Hence, there is still a potential in Er3+:Y2SiO5, but we would need to search more systematically for long Zeeman lifetime for various magnetic eld conguration. Another idea is the use of stimulated emission to increase the eciency of optical pumping. It was already demonstrated in Er3+:Y2SiO5[48] and it could be interesting to implement in any other materials, including Nd3+:Y2SiO5. At present, the most promising materials for a spin-wave storage are probably nonKramers ions, in particular Pr3+:Y2SiO5 and Eu3+:Y2SiO5 because of extremely long coherence times, very weak spectral diusion and very ecient optical pumping. Also, at zero magnetic eld, they have exactly the 3 ground states required for a complete AFC protocol. We note however that they have a narrower bandwidth and a wavelength less accessible to common diode lasers. We nally note that the spin coherence time can be dramatically increased by operating at a magnetic eld where a transition has a ZEro First Order Zeeman (ZEFOZ) shift [87]. This has been applied successfully to Pr3+:Y2SiO5[88] but there are many other candidates [87]. However, one must take into account that by applying such eld, the degeneracy of the spin states is usually lifted, which complicates the level structure and makes the spectral tailoring (e.g. AFC preparation) more dicult. 40 Storage eciency In this thesis, one of our goals was to increase the eciency of quantum storage, since it helps to realize more sophisticated experiments, and it is also a crucial parameter in a future quantum repeater. As we described in sec.3.2, it depends on the quality of the atomic frequency comb, which should have highly absorbing peaks with square shapes and the lowest possible absorbing background (d0). These parameters, in turn, depend on the material used, which should have a high initial absorption, narrow homogeneous linewidth and long spin/Zeeman lifetimes. We demonstrated a storage eciency in Nd3+:Y2SiO5 of 20% which allowed us to realize important breakthroughs for quantum storage in a solid-state memory[61, 78]. It was possible to tailor square peaks, but only for short storage times (<100 ns) because of the eective homogeneous linewidth in this material. In principle, a higher initial absorption, for example with higher doping concentration, would permit to increase the eciency. However, because of the absorbing background, it would be limited, in the ideal case, to 33% (in a backward emission), assuming that the preparation is not more dicult at high optical depths. This could be improved by nding a better magnetic eld conguration as we discussed in the previous section. Recently, our researches focused on Eu3+:Y2SiO5, which has a good potential for a QM with spin-wave storage for single photons with a high multimode capacity. The extremely narrow homogeneous linewidth of 122 Hz [64] is promising for long storage time in the optical transition. Our experiments, though, were restricted by the laser linewidth (≈30 kHz), limiting our ability to create squarish peaks. The priority for future experiments in this material should be to realize a much more stable laser, for example by using an ultra low expansion (ULE) [89, 90] cavity in vacuum. Also, Eu3+:Y2SiO5 has a relatively weak oscillator strength, making it dicult to obtain high absorption. One solution is to increase the doping concentration. But this will also increase the optical inhomogeneous broadening, thus the absorption would not much higher. Another solution is to place the crystal in an asymmetric cavity (Gires-Tournoi interferometer) which may allow 100% absorption using an impedance matching condition [57, 68]. Then, a weakly absorbing medium could allow very ecient storage. Our preliminary results for such experiments indicate absorption of more than 90%, however without performing AFC storage. To achieve this, the crucial points are alignment of the cavity (spatial mode matching), its length stabilization and reduction of intra-cavity losses. The conguration of Sabooni et al.[91, 92], using reection coating on the crystal surfaces, is more robust and does not require active stabilization. This led to a cavity enhanced storage with 58% eciency. However, our setup (g.3.12(a)), with mirrors outside the cryostat, allows to use a crossed beam, which overlaps with the probe eld in the region of the cryostat, but which is not reected by the cavity mirrors. Hence, a control eld, even out of resonance with the cavity, could be sent through this crossed beam and allow spin-wave storage. Therefore, our plan in this 41 project is to rst demonstrate a cavity enhanced storage for short storage time. Then, with a narrower laser linewidth, we hope to increase the storage time and implement spin-wave storage. In general, cavity-enhanced storage will be useful to compensate a weakly absorbing medium. Therefore it opens up quantum storage for more materials, even for those with a weak oscillator strength. Also, for a two-level AFC, it allows to reach 100% eciency [57] while a forward echo is limited to 54% [41]. Meanwhile, the impact of an impedance-matched cavity on other storage protocols remains unknown and could be investigated. While a cavity-enhanced CRIB would certainly be similar, a DLCZ scheme in a cavity would need a more detailed analysis, since the Stokes and anti-Stokes have dierent frequencies and can have dierent spatial directions. Also, an interesting eect has been recently observed [91]. The strong dispersion in a hole-burning medium (slow light in a spectral hole) induces a drastic reduction of the cavity mode spacing and bandwidth. This may be a complication for quantum storage in term of achievable bandwidth, and thus, this eect should be more deeply studied. Multimode storage The multimode capacity of a QM is an important characteristic that allows the speed up of entanglement distribution rate in some quantum repeater architectures [69]. AFC is intrinsically multimode in the time domain, since, unlike other protocols, the number N of temporal modes is independent of the available optical depth[26, 41]. N can be increased via a large AFC bandwidth and long two-level AFC storage time. Also, rare earth doped materials, because of a large inhomogeneous broadening compared to the homogeneous linewidth, are well adapted to multimode storage. Here we demonstrated the reversible mapping of 64 temporal modes at the single photon level in a Nd3+:Y2SiO5crystal. Through the interference of consecutive modes, the preservation of coherence was shown. This proof-of-principle experiment demonstrated that future QM could be highly multimode if based on AFC. The number of modes can clearly be much higher [56, 71], but usually a value of 100 modes is typically assumed for a quantum repeater to compete with direct transmission [18]. Therefore, the next challenge seems to store roughly this number of modes in more sophisticated experiments. For example, we demonstrated true single photon storage in Nd3+:Y2SiO5 [61]. However only few temporal modes were stored, because the experiment required high eciency, thus short storage time. This could be relatively easily compensated by increasing the AFC bandwidth to 1 GHz or more, which would allow to show a multimode storage with true single photons in a solid. On the other hand, non-Kramers ions are promising systems for spin-wave storage. In this case, though, a large AFC bandwidth is dicult to reach, because the separations between the hyperne states are small and we would need extremely high Rabi frequency to create, for the control eld, a π pulse with a large bandwidth. Therefore, the solution would be to increase 42 the storage time for the two-level AFC echo, to several tenths of µs. We believe this to be within reach in Eu3+:Y2SiO5, if we had a laser with narrow enough linewidth. We note however we would be, at a certain point, limited by a temperature-dependent spectral diusion [90]. On-demand readout & storage time For any applications we mentioned in the introduction, a quantum memory must include an on-demand readout. Also, the storage time must be long enough to achieve a particular task. For example, quantum repeaters require 10 ms to 1 s depending of the architecture [18]. This can be done with a complete AFC protocol, which includes a spin-wave storage by the use of control elds. We realized this in a Pr3+:Y2SiO5 crystal, in collaboration with Lund University, which was the rst realization of an AFC with spin-wave storage in a solid-state [74]. We could later reproduce this in a Eu3+:Y2SiO5 crystal which has very promising properties for a QM adapted to a quantum repeater [75]. Using dierent ltering stages to suppress the noise induced by the control elds, we observed a noise of only (7.1 ± 2.3) · 10−3 photons per mode, which allowed to store weak coherent pulses with an average of 2.5 ± 0.6 photons [76]. The phase preservation was demonstrated through interferences between two consecutive modes. The noise is in principle low enough to store true single photons from an external source. However, the low storage eciency (≈ 0.4%) decreases the signal-to-noise ratio, which explains why we could not store coherent pulses containing less than 2.5 photons in average. Therefore, it is necessary to increase the storage eciency in a manner that would not increase the noise. This could be done with an impedance matched cavity or a laser with longer coherence time. Also, this system could be used to implement a photon pairs source based on a DLCZ scheme [45]. We estimate that it should be feasible in a short term period in our system with a few ameliorations. The total storage time was limited by the spin inhomogeneous broadening which induced dephasing in the spin state. This can be overcome with spin refocusing using RF pulses [93, 94]. A sequence of pulses resonant with the spin transition allows to compensate the inhomogeneous broadening and even extend the spin coherence time using dynamical decoupling [93, 73]. Our current eort to implement spin refocusing in Eu3+:Y2SiO5 indicates a total storage time of roughly 10 ms, which already represents an increase of 3 orders of magnitude. We hope we will be able to soon combine this with spin-wave storage of weak coherent pulses. We could then measure if the noise level would increase with the number of RF pulses [95]. I would nally like to mention that the storage time could be increased even more dramatically with dynamical decoupling sequence combine to a ZEFOZ transition. Indeed, Fraval et al.. [93] have extend the coherence time in Pr3+:Y2SiO5 from 82 ms to 30 s with such a sequence. In Eu3+:Y2SiO5 though, it has been suggested that the coherence time (without dynamical decoupling) could reach hundreds of second at a ZEFOZ transition [96]. Therefore, one 43 could extrapolate an extended coherence time of several hours! And this would still be shorter than the spin lifetime that has been estimated to reach 20 days [63]. Entanglement in Quantum memories To characterize a quantum memory, a direct method is to store quantum states, in our case single photons, and measure the storage delity. This also goes in the direction of realizing a complete quantum repeater, which not only requires quantum memories, but also single photon sources, single photon detectors, large interferometers and Bell measurements. Here we used a photon pair source based on SPDC in a non-linear crystal [61]. To match the QM bandwidth, the idler and signal photons were ltered with an optical cavity and etalons. The idler eld, at telecom wavelength, was sent through a 50 m long optical bre while the signal eld was stored in a Nd3+:Y2SiO5 crystal and re-emitted using an AFC protocol. The measured cross correlations between the idler and signal photons demonstrated the quantum character of the source. Moreover, using a Franson interferometer, we could perform a Bell experiment and demonstrate, through the violation of a CHSH inequality, that the energy-time entanglement between the idler and signal eld was preserved during the storage. In another experiment [78], we were able to herald and measure entanglement between two QMs by mapping a single photon entangled state into two crystals. The QMs were separated by 1.3 cm and the distance could not be increased in that setup, since additional optical losses would decrease the amount of entanglement. If we would like to herald entanglement between two QMs separated by a large distance, i.e. an elementary link of a quantum repeater, a solution would be to duplicate the photon pair source and use the scheme of Simon et al.. [69]. In that case, though, the distance would not be very large, since the storage time in Nd3+:Y2SiO5 didn't exceed 100 ns (although we could probably push it to 1 µs). It would be more interesting to use a Eu3+:Y2SiO5 crystal, since we already implemented a spin-wave storage in it. However this material requires photons with a narrow bandwith (≈1-5 MHz) which would be dicult to produce with SPDC, in particular for this wavelength (580 nm). It would probably be necessary to use a doubly resonant Optical Parametric Oscillator (OPO) [97] so that additional ltering would not be necessary. We believe a more promising solution is to use a DLCZ scheme as we discussed in the previous section, so that the photons are directly produced by the atomic ensemble. However, photons at 580 nm would undergo large losses in an optical bre(≈10 dB/km), thus it would be necessary to coherently down convert them to a telecom wavelength [98]. Such a scheme, although experimentally challenging, would allow to herald entanglement between two QMs separated by a large distance. 44 Outlook To sum up, I would argue that from the results presented here and from other related experiments realized in the community, solid-state devices have a strong potential for the realization of a quantum memory adapted to a quantum repeater. Indeed, dierent experiments have achieved high performances in the various criteria such as the storage eciency, storage time, noise level and multimode capacity. These numbers have however not yet been achieved in the same system. Which leads to the question whether there is a material in which we can combine all the required performances. We believe that Eu3+:Y2SiO5 has the potential to do it, and our future work will focus on it. This will however also require the development of a highly coherent laser, the use of an impedance matched cavity and reliable spin refocusing techniques. We believe also that, in parallel, it is still necessary to search for alternative systems and study the potential of various rare earth doped crystals. We are now optimistic that in a few years a prototype of a quantum repeater (or at least an elementary link of it) based on solid-state QMs could be realized. However, it is still a huge step to realize a quantum repeater that could compete with direct transmission. For example, we will have to take into account all the important optical losses arising in a more complex setup. Surely, it will take time until a quantum repeater would be commercialised as is today QKD, but progress may be faster than we think. 45 46 Bibliography [1] N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden. Quantum cryptography. Rev. Mod. Phys., 74(1):145195, Mar 2002. [2] S. Massar and S. Popescu. 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Dynamic decoherence control of a solid-state nuclear-quadrupole qubit. Phys. Rev. Lett., 95(3):0305064, July 2005. [94] E. L. Hahn. Spin echoes. Phys. Rev., 80(4):580594, November 1950. [95] Khabat Heshami, Nicolas Sangouard, Ji °íMinár, Hugues de Riedmatten, and Christoph Simon. Precision requirements for spin-echo-based quantum memories. Phys. Rev. A, 83:032315, Mar 2011. [96] J. J. Longdell, A. L. Alexander, and M. J. Sellars. Characterization of the hyperne interaction in europium-doped yttrium orthosilicate and europium chloride hexahydrate. Phys. Rev. B, 74(19):195101, Nov 2006. [97] Enrico Pomarico, Bruno Sanguinetti, Nicolas Gisin, Robert Thew, Hugo Zbinden, Gerhard Schreiber, Abu Thomas, and Wolfgang Sohler. Waveguide-based opo source of entangled photon pairs. New Journal of Physics, 11(11):113042, 2009. [98] Noe Curtz, Rob Thew, Christoph Simon, Nicolas Gisin, and Hugo Zbinden. Coherent frequency-down-conversion interface for quantum repeaters. Opt. Express, 18(21):2209922104, October 2010. [99] R. Marino. Propriétés magnétiques et optiques de monocristaux dopés terres rares pour l'information quantique. PhD thesis, Université Lille 1 & Chimie Paris Tech, 2011. 55 56 Acknowledgements Cette thèse a été un long travail et, heureusement, je n'ai pas été seul pour l'accomplir. J'aimerai en proter pour remercier tout les gens qui m'ont aidé dans mon travail, mais aussi ceux qui m'ont soutenu ou tout simplement accompagné pour que je passe 5 très belles années! Avant tout, je crois que j'aimerais remercier tout le groupe du GAP-optique dans son ensemble. En eet, l'ambiance y était excellente, et elle le sera toujours j'en suis sûr. Ceci m'a permis de passer de très bon moments et il a été très facile de collaborer avec tout le monde. Nicolas Gisin porte sûrement la responsabilité de cette atmosphère positive, et je le remercie de m'avoir accueilli dans ce groupe. J'ai beaucoup apprécié d'avoir été supervisé par Mikael Afelius. Il a su me guider tout en écoutant toutes les idées que je pouvais proposer. En plus, nous avons passé de très bon moments au pub et dans les gradins de la patinoire! Hugues a également été un excellent un superviseur pour mon début de thèse, et je suis très content qu'il ait été dans mon jury. Merci encore à Nuala qui a su être une très bonne partenaire de labo et membre du jury. Merci également à Philippe Grangier qui a fait le déplacement, j'ai été très honoré de sa présence. Encore au début de la thèse, j'ai eu le plaisir d'aller à Lund pour une collaboration très fructueuse. J'aimerais remercier Stefan Kröll de nous avoir accueilli, et il a été très agréable de travailler avec Atia. Je remercie tout mes collègues de labo: Björn, Christoph, Félix, Pierre et Nuala. Cela s'est passé merveilleusement bien, et surtout nous avons pu être ami en dehors du labo et partager plein d'activités. L'ambiance dans le bureau a aussi été parfaite (même si ça détournait du travail..), alors merci à Cyril, Tommaso, Pierre, Emmanuel, Alexey, Nino, Björn et Cyril B. J'ai aussi beaucoup aimé les activités annexes au bureau, cartes et brassage! Alors merci encore à Anthony, Tomy et ce bon vieux Pavel. En fait, je remercie simplement tous les membres du GAP, même si votre nom n'est pas là! Merci aussi à tous les footeux avec qui j'ai passé d'excellents jeudi midi et même quelques matchs du tournoi universitaire. Je suis content d'avoir succédé à Noé et Emmanuel pour l'organisation du foot, et j'essaierai de revenir le plus souvent possible. J'ai aussi beaucoup apprécié la disponibilité et l'aimabilité de toutes les secrétaires, Isabelle, Laurence et Natalie. J'ai aussi beaucoup été aidé par tous les techniciens j'ai 57 aimé discuté avec vous tous, Jean-Daniel, Claudio, Olivier, Raphael et Mathieu. Une petite dédicace encore à mes amis physiciens du master/bachelor. J'ai passé d'immenses bon moments avec vous tous, et je suis heureux qu'on se voit toujours régulièrement. Un grand merci également à tous ceux qui étaient à la soutenance, j'ai vraiment été très content que vous soyez là. Mais je remercie bien sûr aussi tous mes amis qui n'étaient pas présents à ce moment, mais avec qui simplement je passe du bon temps. Et j'aimerais nir par le plus important, un immense merci à tous les membres de ma famille, sans qui je ne serais pas qui je suis. 58 Appendix A Additional spectroscopy g tensor of Nd Nd3+:Y2SiO5is a Kramers doublets, hence the ground and the excited states behave like a spin 12 system. The Hamiltonian of the system is : → − → − H = µB B gµν S → − where→ µB = 14GHz/T is the Bohr magneton, B the magnetic eld, g the Landé tensor and −S a column vector with the three Pauli matrix. This 2 dimensions Hamiltonian will give two eigen states separated by ∆E. Here we choose an external magnetic eld B in the D1-D2 plane, at an angle θ with D2. Then, we have a Landé factor depending of θ : ∆E = g(θ)µB B We will label ∆Eg/e and gg/e for the ground/excited state. We want here to measure gg/e for dierent θ. The rst technique to measure g is to do spectral hole burning. By burning a spectral hole at a central frequency, the absorption spectrum will present holes and anti-holes of absorption, because atoms have been moved to dierent states (see g. A.1). The position of this holes and anti-holes depends of ∆Eg /e and can be moved by a variable magnetic eld. Since their position is proportional to ge, gg and gg ±ge, we could deduce the Landé factor from a linear t. For B k D2, we nd gg (θ = 0) = 2.91 and ge (θ = 0) = 0.16. Then, by changing only the magnetic eld orientation, we could deduce ge for all angle θ by measuring the side-holes position (g. A.2). We note that we could work with relatively low magnetic eld (<100 mT), because magnetic dipole is high for electronic orbitals. There is another method to measure the Landé factor without hole burning. If we apply a strong magnetic eld, the splitting in the ground state and excited state 59 Hole burning spectrum 400 350 250 200 200 hole/anti-hole position [MHz] α [m-1] 300 150 100 50 0 150 hole anti-hole 1 anti-hole 2 anti-hole 3 -150 0 5 10 15 Magnetic field [mT] -100 -50 20 25 0 Detuning [MHz] 50 100 Figure A.1: Absorption spectrum after hole burning with Bk D2. Because the Nd ions have a split in the ground and excited state, there is a hole at central frequency, 2 side holes at ±∆Ee and 6 anti-holes at ±∆Eg and ±∆ Eg ±Ee . Inset: position of a side-hole and three anti-holes in function of magnetic eld. Using a linear t, we can deduce the Landé factor in the ground and excited state: gg (θ = 0) = 2.91 and ge (θ = 0) = 0.16 60 0.9 0.8 0.7 0.6 ge 0.5 0.4 0.3 0.2 0.1 0 -90 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 90 angle of B with D2 Figure A.2: Measure of the Landé factor in the excited state by measuring the position of side holes for dierent orientation of magnetic eld. D1 and D2 are close of being axes of symmetry. become larger than the inhomogeneous broadening. Then, by looking at the absorption spectrum(g. A.3), we can directly measure the splits for dierent magnetic elds and calculate g using a linear t. The results are given in tab. A.1 As a useful result, we have now the values of ge for all directions of B in the D1-D2 plane. This is important to know the potential bandwidth of quantum memory in this material. We can now predict the necessary magnetic eld for a given orientation to have a minimal bandwidth. In any case, we note that the splits are large compare to what we can have in non-Kramers doublets. The choice of the magnetic eld will however depend of other important parameter such as coherence and lifetime in the optical and spin transition. θ 0 90 30 150 gg ge 2.86 ±0.16 1.43 ± 0.05 0.78 ± 0.07 2.7±0.1 2.38±0.05 0.42±0.07 Table A.1: value of the Landé factor for dierent angle θ between magnetic eld and D2 using the absorption spectrum with a variable magnetic eld. The value for ge are consistent with the measures using spectral holes in g.A.2. 61 2 α [cm-1] 1.5 1 0.5 0 Detuning Figure A.3: Absorption spectrum for a magnetic eld of 0.95 T parallel to D1. The splits of the ground and excited being larger than the inhomogeneous broadening, we can distinguish the 4 transitions. Then, we have the values for gg for a few angle. This can be compared to the work of [99] where the whole tensor has been measured (g. A.4). The splits have probably an inuence in the Zeeman state lifetime as we will see in the next section. Also, we see that a strong magnetic eld divide the absorption by a factor 2 or 4 when the splits become larger than the inhomogeneous broadening. In principle, this eect could be compensated by an additional optical pumping. Additionally we can estimate the branching ratio between the dierent transitions using g. A.3. Indeed, we see the absorption is weaker for the optical transitions that don't conserve the spin. We normalize the absorption, so that we get the coecient βi. When both transitions are equally strong, then β1 = β2 = 0.5. We got an estimation for two magnetic conguration in tab. A.2. For an angle of 150 with D2, β1=0.64, which is encouraging for ecient optical pumping. For comparison, the ideal value would be 0.5, but the branching ratio in Er3+:Y2SiO5is β ≈ 0.95, which aected badly the optical pumping. 62 4 gg 3 2 1 -90 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 Angle of B with D2 30 40 50 60 70 80 90 Figure A.4: Landé factor for a megnetic eld in the D1-D2 plane. We compare our results (dots) with the work from [99] (line). It seems compatible within the errors. θ 90 150 β1 0.76 ± 0.01 0.64 ± 0.01 Table A.2: Branching ratio for the strongest transition (conserving the spin) for two angles between B and D2 . 63 64 List of Publications [1] Mikael Afzelius, Imam Usmani, Atia Amari, Björn Lauritzen, Andreas Walther, Christoph Simon, Nicolas Sangouard, Ji°í Miná°, Hugues de Riedmatten, Nicolas Gisin, and Stefan Kröll. Demonstration of atomic frequency comb memory for light with spin-wave storage. Phys. Rev. Lett., 104(4):040503, January 2010. [2] A. Amari, A. Walther, M. Sabooni, M. Huang, S. Kroll, M. Afzelius, I. Usmani, B. Lauritzen, N. Sangouard, H. de Riedmatten, and N. Gisin. Towards an ecient atomic frequency comb quantum memory. Journal of Luminescence, 130(9, Sp. Iss. SI):15791585, SEP 2010. [3] Imam Usmani, Mikael Afzelius, Hugues de Riedmatten, and Nicolas Gisin. Mapping multiple photonic qubits into and out of one solid-state atomic ensemble. Nat Commun, 1:12, April 2010. [4] Christoph Clausen, Imam Usmani, Felix Bussieres, Nicolas Sangouard, Mikael Afzelius, Hugues de Riedmatten, and Nicolas Gisin. Quantum storage of photonic entanglement in a crystal. Nature, 469(7331):508511, January 2011. [5] Imam Usmani, Christoph Clausen, Felix Bussieres, Nicolas Sangouard, Mikael Afzelius, and Nicolas Gisin. Heralded quantum entanglement between two crystals. Nat Photon, 6(4):234237, April 2012. [6] N Timoney, B Lauritzen, I Usmani, M Afzelius, and N Gisin. Atomic frequency comb memory with spin-wave storage in 153 eu 3 + :y 2 sio 5. Journal of Physics B: Atomic, Molecular and Optical Physics, 45(12):124001, 2012. [7] N Timoney, I Usmani, P Jobez, M Afzelius, and N Gisin. Single-photon-level optical storage in a solid-state spin-wave memory, 2013. 65 66 Appendix B Published articles 67 PRL 104, 040503 (2010) PHYSICAL REVIEW LETTERS week ending 29 JANUARY 2010 Demonstration of Atomic Frequency Comb Memory for Light with Spin-Wave Storage Mikael Afzelius,1,* Imam Usmani,1 Atia Amari,2 Björn Lauritzen,1 Andreas Walther,2 Christoph Simon,1 Nicolas Sangouard,1 Jiřı́ Minář,1 Hugues de Riedmatten,1 Nicolas Gisin,1 and Stefan Kröll2 1 Group of Applied Physics, University of Geneva, CH-1211 Geneva 4, Switzerland 2 Department of Physics, Lund University, Box 118, SE-22100 Lund, Sweden (Received 14 August 2009; published 27 January 2010) We present a light-storage experiment in a praseodymium-doped crystal where the light is mapped onto an inhomogeneously broadened optical transition shaped into an atomic frequency comb. After absorption of the light, the optical excitation is converted into a spin-wave excitation by a control pulse. A second control pulse reads the memory (on-demand) by reconverting the spin-wave excitation to an optical one, where the comb structure causes a photon-echo-type rephasing of the dipole moments and directional retrieval of the light. This combination of photon-echo and spin-wave storage allows us to store submicrosecond (450 ns) pulses for up to 20 s. The scheme has a high potential for storing multiple temporal modes in the single-photon regime, which is an important resource for future long-distance quantum communication based on quantum repeaters. DOI: 10.1103/PhysRevLett.104.040503 PACS numbers: 03.67.Hk, 42.50.Gy, 42.50.Md A quantum memory (QM) for photons is a light-matter interface that can achieve a coherent and reversible transfer of quantum information between a light field and a material system [1]. A QM should enable efficient, high-fidelity storage of nonclassical states of light, which is a key resource for future quantum networks, particularly in quantum repeaters [2–6]. In order to achieve reasonable entanglement distribution rates, it has been shown that some type of multiplexing is required [4,5], using for instance independent frequency, spatial or temporal modes (multimode QM). Several types of light-matter interactions have been proposed for building a QM, for instance electromagnetically induced transparency [7–10], Raman interactions [11–14], or photon-echo techniques [15–21]. Photonecho techniques in rare-earth-ion doped crystals have an especially high multimode capacity for storing classical light [22]. Classical photon echoes are not useful, however, for single-photon storage due to inherent noise problems [23]. The photon-echo QM based on controlled reversible inhomogeneous broadening [15–18] is free of these noise problems. But this technique has a lower time-multiplexing capacity than classical photon echoes, for a given optical depth, due to loss of storage efficiency as the controlled frequency bandwidth is increased [19,24]. Some of us recently proposed a photon-echo type QM based on an atomic frequency comb (AFC) [19] that has a storage efficiency independent of the bandwidth, allowing optimal use of the inhomogeneous broadening of rare-earth-doped crystals. An AFC memory has the potential for providing multimode storage capacity [19,24] crucial to quantum repeaters. The few reported AFC experiments [20,21] have been investigating the physics of the optical AFC echo, where the memory storage time is predetermined by the periodicity of the comb. For quantum repeaters it is crucial to be able to choose the time of the memory 0031-9007=10=104(4)=040503(4) readout (on-demand readout). Here we present the first light-storage experiment where an AFC is used in combination with reversible transfer of the excitation to a spin state [19], resulting in on-demand readout and storage times longer than 20 s. The underlying idea of the AFC QM is to shape an inhomogeneously broadened optical transition jgi ! jei into a periodic series of narrow and highly absorbing peaks with periodicity , see Fig. 1. A photon with a bandwidth that is matched to the width of the AFC structure is then stored as an optical excitation delocalized over the peaks, which we can write as j c i ¼ PN ij t ikzj e jg1 ej gN i where N is the number j¼1 cj e of atoms in the AFC, j is the detuning of atom j with respect to the laser frequency, zj is the position, k is the wave number of the light field, and the amplitudes cj depend on the frequency and on the spatial position of the particular atom j. The terms in this large superposition state accumulate different phases due to the inhomogeneous distribution of atomic resonance frequencies, resulting in a loss of the initially strong collective coupling to the light mode. But the periodic AFC peak separation leads to a rephasing of the terms after a time 1=, which restores the strong collective coupling, leading to a photon-echo type reemission [25], the AFC echo. The narrow and highly absorbing peaks can theoretically absorb all the light and completely emit the energy in the AFC echo [19]. A large number of peaks leads to a high multimode capacity [19,24]. In the original proposal [19] on-demand readout and longer storage times rely on reversible transfer of the optical excitation to a long-lived spin state by strong control pulses (see Fig. 1). If we imagine a perfect pulse applied at time T 0 , each term in the superposition state becomes jg1 sj gN i; thus, we have a single collec- 040503-1 Ó 2010 The American Physical Society PRL 104, 040503 (2010) (b) (a) g 5/2 2.0 e 1.5 e 3/2 Output mode Control fields Input mode 0.5 2.0 0.0 0.4 0.5 0.6 0.7 0.8 1.5 s g 3/2 s 5/2 aux e 1.0 0.5 Absorption depth d 1.0 1/2 1/2 0.0 (c) Intensity 0 2 4 6 8 10 Frequency (MHz) Input mode Output mode T week ending 29 JANUARY 2010 PHYSICAL REVIEW LETTERS Ts T Time FIG. 1 (color online). (a) The experiment was performed on the 3 H4 ! 1 D2 transition in Pr3þ . The ground and excited state manifolds both have three hyperfine levels denoted MI ¼ 1=2, 3=2, 5=2. The three-level lambda system was formed by the levels labeled jgi, jsi and jei, following the notation in [19]. (b) Experimental absorption spectrum showing the AFC on the jgi $ jei transition created within a 18 MHz wide transmission hole using the spectral holeburning sequence described in the text. Here the comb consists of 9 peaks with spacing ¼ 250 kHz. The holeburning sequence also empties the jsi level, whereas jauxi is used for population storage. Note that the second AFC in the center of the spectrum is due to the weaker transition from the ground state 1=2 to the excited state 5=2. (c) The pulse sequence showing the input pulse to store, the two control fields for the back-and-forth transfer to the spin state, and the retrieved output pulse. tive spin-wave excitation. If we assume that the spin transition is homogeneously broadened, then each term is 0 frozen with the phase term eij T due to the time spent in the excited state [19]. In practice, inhomogeneous spin broadening adds other phase factors and reduces the collective spin wave, as will be discussed below. After a time Ts in the spin state, another control pulse transfers the excitation back to the excited state and the AFC evolution resumes, leading to a reemission after a total storage time T 0 þ Ts þ T 00 where T 0 þ T 00 ¼ 1=. The storage material used in this experiment is a praseodymium-doped Y2 SiO5 crystal (Pr3þ concentration of 0.05%) with an optical transition at 606 nm. The optical homogeneous linewidth at cryogenic temperature is around 1 kHz, whereas the inhomogeneous broadening is about 5 GHz [26]. The ground and excited states both have a hyperfine manifold consisting of three closely spaced levels [Fig. 1(a)], assuming no applied magnetic field. Three ground-state levels are necessary for the experiment, see below, which was our main motivation for choosing Pr3þ . The different hyperfine transitions are usually hidden within the large inhomogeneous broadening. By spectral hole burning techniques one can, however, isolate a subensemble of atoms whose different hyperfine transitions can be unambiguously excited. This distillation technique, which we will summarize here, has been the subject of several papers [27–29]. A laser beam whose frequency is swept pumps atoms from ground levels jgi and jsi to the auxiliary storage level jauxi; see Fig. 1, which creates a wide transmission window within the inhomogeneous profile. In the next step a narrow absorption peak is created in the hole by coherently transferring back atoms, within a narrow frequency range, from jauxi to jgi [28]. In this experiment we extended this method by transferring back atoms at different frequencies to create a frequency comb. In Fig. 1(b) we show the absorption spectrum recorded after the preparation sequence. The AFC created on the jgi $ jei transition is clearly visible. The jsi level is used for the spin-wave storage, which means that the control pulses will be applied on the jsi $ jei transition displaced by 10.2 MHz with respect to the jgi $ jei transition. Note that the absorption spectrum in Fig. 1(b) is shown only for visualization purposes, since the fast frequency scan method [29] we used leads to distortions for absorption depths d * 2. The experimental setup is shown in Fig. 2. The control pulses were both counterpropagating with respect to the input pulse. By phase matching condition the output pulse then copropagates with the input signal [19]. Using this configuration we could reduce noise due to off-resonant free-induction decay emission produced by the strong control pulses. In a preliminary experiment we investigate the AFC echo on the jgi $ jei transition without applying the control pulses, see inset in Fig. 3. This allows us to optimize the relevant comb parameters in order to obtain a strong echo. The input pulse duration was set to 450 ns so that the bandwidth is entirely contained in the 2 MHz wide AFC. The efficiency of this AFC echo, which we define as the Control Input Pr:Y2 SiO5 15% Output 85% BS Input/Output PD FIG. 2 (color online). The experimental setup. The spectral hole burning and storage pulse sequences were created using a frequency-stabilized laser and acousto-optic modulators similar to the setup in [28] (not shown). A beam splitter (BS) split the light into a strong and a weak beam, whose two modes were overlapped in the crystal cooled to 2.1 K. The detected output pulse propagating to the right originated from the combination of a weak input pulse incident on the crystal from the left, and two strong control pulses incident from the right. Note that the signal disappeared when the control pulses were blocked directly after the BS. An AOM was used to direct only the transmitted input pulse and the output pulse onto the photodiode (PD), effectively working as a detector gate. 040503-2 PRL 104, 040503 (2010) week ending 29 JANUARY 2010 PHYSICAL REVIEW LETTERS ratio of the AFC echo area to the input pulse area, depend on the shape of the AFC [19]. Two critical parameters are the peak absorption depth d and the finesse defined as F ¼ = where is the full-width at half maximum of a peak. For instance, a high finesse leads to low decoherence during the storage time 1=, but also to a lower effective absorption d=F of the input pulse [19]. The peak absorption d could be controlled by the power of the laser beam creating the peaks in the transmission hole, but a high power also had an impact on the finesse by causing power broadening of the peaks. The peak width was also limited by laser frequency stability, resulting in typical widths of 100 kHz. For a periodicity of ¼ 1 MHz, the optimized efficiency was about 15%. The delay of 1 s was not sufficiently long, however, for applying the control pulses before the emission of the AFC echo. We therefore set the periodicity to ¼ 250 kHz giving us 4 s to apply the control pulses. The closer spacing of the peaks lowered the finesse of the comb, thus lowering the efficiency to 5% (see inset in Fig. 3). This efficiency is in reasonable agreement with numerical simulations using the experimentally estimated peak absorption depth and finesse of the comb. In Fig. 3 we show the main result where two control pulses are applied on jsi $ jei to transfer the excitation to the jsi hyperfine level. The retrieved pulse is clearly observed above the noise level. This realizes a true storage of the input pulses, with on-demand readout. Thus, the control pulses provide a mechanism for momentarily interrupting the predetermined AFC evolution [19]. We tested this mechanism in detail by varying the time at which the first control pulse was applied T 0 ¼ ð1:17; 1:63; 2:23Þ s [cf. Fig. 1(c) for notation]. This resulted in different measured durations T 00 ¼ ð2:84; 2:41; 1:85Þ s. The total time spent in the excited state jei, however, is constant (within the measurement error), T 0 þ T 00 ¼ ð4:01; 4:04; 4:08Þ s, corresponding to the expected 1=. In Fig. 4 we show storage experiments where the spinwave storage time Ts is varied. The output signal is clearly visible up to 20 s of total storage time (1= þ Ts ). The exponential decay of the output signal as a function of Ts can be attributed to inhomogeneous spin dephasing, corresponding to an inhomogeneous broadening of 26 kHz consistent with previous measurements [30]. We point out that this can be compensated for by spin echo techniques. With such techniques Longdell et al. [31] stopped light during >1 sec in a Pr3þ :Y2 SiO5 crystal using electromagnetically induced transparency. We now discuss the total storage efficiency . Clearly is bounded by the AFC echo efficiency e ¼ 4%–5% for 1= ¼ 4 s (cf. Fig. 3 inset). This can be improved by increasing the finesse and optical depth (see discussion above). The spin-wave storage further decreases the efficiency. By extrapolating the experimentally measured to the limit Ts ! 0 we find that ¼ 0:5%–1%, which is a value independent of spin dephasing. The effect of the control pulses can now be understood by a simple model. We assume that one pulse has a single-atom jsi ! jei transfer efficiency T , which is constant as a function of detuning. Then the application of the two control pulses reduce the efficiency to ¼ e 2T . Numerical simulations using 3-level Maxwell-Bloch equations show that this simple model is correct if the control pulses do not induce any additional phase decoherence. From the experimental values given above we thus find T 0:30–0:45. To better understand this value we have performed numerical calculations. The control pulses were two identical complex hyperbolic secant pulses, which can achieve Intensity (arb. units) 1.0 Normalized intensity 14 12 10 8 16 Intensity (arb. units) Normalized intensity 5 12 8 4 0 0 6 1 2 3 Time ( s) 4 4 x10 2 Control pulses 2 4 6 8 10 3 0.6 0.4 0.2 0.0 2 0 5 10 15 20 25 30 Duration of spin storage TS ( s) 1 0 0 0 4 0.8 12 -2 Time ( s) FIG. 3 (color online). Storage in the spin state using two control pulses. Shown are (from left) the partially transmitted input pulse, control pulses (strongly attenuated by the closed optical gate), and the output pulse (magnified by a factor 10 for clarity). Here 1= ¼ 4 s and Ts ¼ 7:6 s, resulting in a total storage time 1= þ Ts ¼ 11:6 s. Inset: The AFC echo observed at 1= ¼ 4 s when the control pulses are not applied. Note that the vertical scales have been normalized to 100 with respect to the input pulse before the crystal; thus, these yield (in percent) the efficiency for the echo and transmission coefficient for the input pulse. 0 10 12 14 16 Time ( s) 18 20 FIG. 4 (color online). Experimental traces for spin-storage times Ts ¼ 5:6, 7.6, 10.6, and 15:6 s. All other parameters are the same as those in Fig. 3. The input pulses are superimposed to the left (truncated) and the output pulses for different Ts are seen to the right. The leakage of the control pulses through the optical gate is not shown. For clarity there is also a break in the horizontal scale. The decay of the signal (see inset) as a function of Ts is due to inhomogeneous spin dephasing. The solid curve is a fitted Gaussian function corresponding to 26 kHz (full-width at half maximum) spin broadening. 040503-3 PRL 104, 040503 (2010) Input modes 1.0 Intensity (arb. units) PHYSICAL REVIEW LETTERS 0.8 0.6 Control pulses 0.4 Output modes 0.2 0.0 0 2 4 6 8 10 12 14 16 Time ( s) FIG. 5 (color online). Storage of two temporal input modes. The normalized input modes shown here (dashed line) are recorded before the crystal using a reference detector. The AFC was prepared with a periodicity of ¼ 200 kHz, corresponding to 5 s storage time on the optical transition. The spinstorage time was set to Ts ¼ 7:6 s, resulting in a total storage time of 1= þ Ts ¼ 12:6 s. efficient, broadband transfer of population [28] and coherence [29]. Our pulses had duration 600 ns and Rabi frequency 1:2 MHz (close to the maximal value in this experiment), the frequency chirp being 2 MHz. These values were found by empirically optimizing the size of the output pulse. Using the numerical model we find T ¼ 0:75, also averaged over the bandwidth of the AFC. Based on this we would expect a total efficiency ¼ 5% 0:752 ¼ 2:8% using the simplified model above, which is significantly higher than what we observe. The most probable explanation for the discrepancy is imperfect spatial mode overlap between the counterpropagating input and control pulses (see Fig. 2). A larger control beam would make a more spatially uniform Rabi frequency and would facilitate mode alignment. Note that the theoretical T can be improved by increasing the Rabi frequency (by a factor of 2) and adapting the duration and chirp of the pulse in order to achieve an efficient (T 95%) transfer. We finally show an example of storage of two temporal modes; see Fig. 5. Note that both modes are stored with the same efficiency, which is a particular feature of the AFC memory due to the fact that each mode spend the same total time (1= þ Ts ) in the memory. The number of modes we could store was mainly limited by the number of peaks Np that could be created in the AFC, since the number of input modes one can store is proportional to Np [19]. Np could be increased by making narrower peaks and/or a wider AFC. The width is currently limited by the separation of the hyperfine transitions (cf. Fig. 1), which could be increased via the nuclear Zeeman effect. The most significant improvement can be made by creating narrower peaks, which in principle can approach the homogeneous linewidth of about 1 kHz. A multimode storage capacity in the range of tens of modes appears feasible. Other rareearth-ion-doped crystals have even higher multimode potential, for instance Eu3þ :Y2 SiO5 as discussed in Ref. [19]. In conclusion, we have demonstrated the first lightstorage experiment combining an atomic frequency comb week ending 29 JANUARY 2010 and spin-wave storage. Using this method we stored optical submicrosecond (450 ns) pulses for up to 20 s as a spinwave in Pr3þ :Y2 SiO5 . This optical bandwidth is more than 1 order of magnitude higher than previous stopped-light experiments demonstrated in rare-earth crystals [31]. The spin-storage time could be greatly extended by spin echo techniques [31]. The authors acknowledge useful discussions with Pavel Sekatski. The work was supported by the Swiss NCCR Quantum Photonics, the Swedish Research Council, the Knut and Alice Wallenberg Foundation, the Crafoord Foundation, the ERC Advanced Grant QORE, the Lund Laser Center, and the EC projects Qubit Applications (QAP) and FP7 Grant No. 228334. *[email protected] [1] K. Hammerer, A. S. Sørensen, and E. S. Polzik, arXiv:0807.3358. [2] H.-J. Briegel et al., Phys. Rev. Lett. 81, 5932 (1998). [3] L.-M. Duan et al., Nature (London) 414, 413 (2001). [4] C. Simon et al., Phys. Rev. Lett. 98, 190503 (2007). [5] O. A. Collins et al., Phys. Rev. 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Kröll a, M. Afzelius b, I. Usmani b, B. Lauritzen b, N. Sangouard b, H. de Riedmatten b, N. Gisin b a b Department of Physics, Lund University, P.O. Box 118, SE-22100 Lund, Sweden Group of Applied Physics, University of Geneva, CH-1211 Geneva 4, Switzerland a r t i c l e in fo abstract Available online 1 February 2010 We present an efficient photon-echo experiment based on atomic frequency combs [Phys. Rev. A 79 (2009) 052329]. Echoes containing an energy of up to 35% of that of the input pulse are observed in a Pr3 + -doped Y2SiO5 crystal. This material allows for the precise spectral holeburning needed to make a sharp and highly absorbing comb structure. We compare our results with a simple theoretical model with satisfactory agreement. Our results show that atomic frequency combs has the potential for highefficiency storage of single photons as required in future long-distance communication based on quantum repeaters. & 2010 Elsevier B.V. All rights reserved. Keywords: Quantum repeater Quantum memory Atomic frequency comb Storage efficiency 1. Introduction The distribution of entanglement over long distances is a critical capability for future long-distance quantum communication (e.g. quantum cryptography) and more generally for quantum networks. This can be achieved via so-called quantum repeaters [1–4], which can overcome the exponential transmission losses in optical fiber networks. Quantum memories (QM) for photons [5– 9] are key components in quantum repeaters, because the distribution of entanglement using photons is of probabilistic nature due to the transmission losses over long quantum channels. QMs enables storage of entanglement in one repeater segment until entanglement has also been established in the adjacent sections. For quantum repeaters a QM should be able to store single-photon states with high conditional fidelity F and with high storage and retrieval efficiency, Z [4]. Further, it has recently been shown that in order to reach useful entanglement distribution rates in a repeater, QMs with multiplexing capacity (multimode QM) are necessary [3,10]. Significant progress have been achieved lately using atomic ensembles for manipulating the propagation and quantum state of an optical field, see Hammerer et al. [11] for a recent review. Storage of single photons using electromagnetically induced transparency (EIT) has been demonstrated with warm [7] and cold vapors [6,8] of alkali atoms. Storage of light at the single photon level has been demonstrated also in rare-earth-ion-doped crystals (REIC) [9]. REICs are characterized by large optical inhomogeneous broadening which enables storage and recall of Corresponding author. Tel.:+ 46 462229625. E-mail address: [email protected] (A. Amari). 0022-2313/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jlumin.2010.01.012 coherent information by manipulating and controlling the inhomogeneous dephasing using echo techniques. Although traditional photon echoes cannot be used in the single-photon regime due to spontaneous-emission noise induced by the p-pulse [12], photon echo techniques avoiding this noise have been proposed; controlled reversible inhomogeneous broadening (CRIB) [13–17] and more recently atomic frequency combs (AFC) [18]. The AFC protocol may offer a breakthrough for the practical construction of quantum repeaters capable of achieving sufficient entanglement distribution rate, since the number of modes that can be stored in an AFC QM is independent of the memory material absorption depth. Since the proposal of the AFC scheme, storage of light pulses at the single photon level (so called weak coherent states) has been shown in Nd3 + :YVO4 [9]. The fidelity of the storage was measured by storing a time-bin qubit and performing an interference measurement on the recalled qubit. The resulting interference fringe visibility was V= 95%, which corresponds to a fidelity F= (1 +V)/2 [19] of 97.5%. This shows that light at the single photon level can be stored and retrieved without introducing noise, and future experiments are likely to improve the fidelity further. The combined storage and retrieval efficiency, however, was only 0.5% in that experiment. A more recent experiment in Tm3 + :YAG [20] showed improved efficiency of 9%, also with weak coherent states. Finally, storage of 64 weak coherent states encoded in different temporal modes has been achieved in Nd3 + :Y2SiO5 [21], underlying the high multimode capacity of the AFC scheme. In view of these encouraging results in terms of fidelity and multimode storage, it is clear that increasing the efficiency is of great importance, particularly in the perspective future long-distance quantum repeaters where QM efficiencies of 90% are necessary with the architectures known today [4]. ARTICLE IN PRESS 1580 A. Amari et al. / Journal of Luminescence 130 (2010) 1579–1585 Here we report a photon-echo experiment based on an AFC in a Pr -doped Y2SiO5 crystal. We measure echoes containing an energy of up to 35% of that of the input pulse, which is the highest AFC echo efficiencies measured so far. This shows that AFC-based schemes can be used for efficient light storage. This improvement is possible because of a good control of the procedure that creates the atomic frequency comb, via optical pumping techniques, and because of a storage medium with high optical depth. Combs with peaks of widths 100–300 kHz with peak absorption depths approaching 10 were created inside a transparent region created by optical pumping techniques in a part of the inhomogeneous profile in a Pr3 þ : Y2 SiO5 crystal. We also examine parameters related to the experimental optimization of the efficiency and compare to a theoretical model in order to understand how to further improve the efficiency of storage and retrieval from a memory using the AFC scheme. The paper is organized in the following way. In Section 2 we give an overview of the theory of AFC. In Section 3 the experimental setup is described. Preparation of a narrow periodic series of absorption peaks is discussed in Section 4. In Section 5 experimental results of AFC echoes are presented and compared with the theoretical model. Conclusions are given in Section 6. 3+ pffiffiffiffiffiffiffiffiffiffiffiffi 2 2 the decay (dephasing) is given by et g~ =2 where g~ ¼ g= 8 ln 2. For the first echo emission at t ¼ 2p=D this dephasing factor becomes 2 2 eð1=F Þðp =4ln2Þ (note that the factor applies to the field amplitude), where F ¼ D=g. From this observation it follows that a highfinesse grating strongly reduces the intrinsic dephasing. In general the dephasing factor (for the field amplitude) is given by the Fourier-transform of one peak in the comb. Obtaining a high efficiency echo also requires a strong interaction between the ensemble of atoms and the field, which can be achieved by a high absorption depth, d. It is shown in Ref. [18] that the comb absorbs uniformly over the photon bandwidth, under the assumption that gp 4 D. The effective absorption d~ depth depend on the exact shape of the peaks in the comb, but in general it decreases with increasing F for a given peak amplitude, since the total number of atoms decreases. For Gaussian peaks one finds that d~ d=F, and the fraction of the input light that is transmitted through the AFC is given by [18] ~ T ¼ ed ; ð1Þ while the absorption is simply given by 1 T. For an AFC consisting of peaks with Gaussian line shape the resulting echo efficiency is given by (see Ref. [18] for the derivation) 2 ~ 1 p2 Z ¼ d~ ed eF2 2 ln 2 ; 2. Theory of AFC We consider an ensemble of atoms with a transition jgSjeS having a narrow homogeneous linewidth gh , but a large inhomogeneous broadening Gin b gh . There are thus many addressable spectral channels within the optical transition. We also assume that there is at least one more meta-stable ground state, jauxS, having a long population lifetime. This allows a highresolution spectral shaping of the jgSjeS transition by spectral hole burning, where jauxS is used as population storage reservoir. These properties are often found in rare-earth-ion-doped crystals [22,23], which are considered here. The detailed experimental procedure for precise spectral shaping depend on the particular system. In Section 4 we discuss the procedure for Praseodymium doped Y2SiO5 crystals. We assume that the inhomogeneously broadened transition has been shaped into a periodic series of narrow peaks, called an atomic frequency comb, see Fig. 3. We further assume that the light pulse to be stored has a spectral bandwidth, gp , larger than the periodicity in the comb (gp 4 D), but smaller than the total comb structure. The interaction between the input pulse and a ground-state population grating versus frequency generally results in a photon echo emission after a time 1=D, which is used in accumulated or spectrally programmed photon echoes [24–29]. The echo emission arises from the evolution of the atomic coherence induced by the input pulse, which periodically rephases due to the periodicity in the atomic population grating. In typical echo experiments only a small fraction of the input pulse is re-emitted in the echo and the storage time is not variable since it is set by the predetermined grating periodicity D. This is not useful for quantum repeaters where efficiencies close to 100% and on-demand read-out of the quantum memory is necessary [3]. Solutions to these issues were, however, recently proposed in Ref. [18]. In Ref. [18] it is shown theoretically that a comb-shaped grating consisting of sharp and strongly absorbing peaks could generate a very efficient echo. This can be understood in terms of the Fourier-transform of the grating function, which governs the evolution of the atomic coherence. The periodicity in frequency results in a periodic time evolution, with an overall decay given by the width of the peak in the comb. For a series of well-separated Gaussian peaks, with full width at half maximum (FWHM) g, ð2Þ where qualitatively the first factor can be understood as the coherent response of the sample, the second factor the re-absorption of the echo and the last factor the previously mentioned dephasing. For a high finesse, F, and high peak absorption d, the efficiency tends to a maximum of 54% for an effective absorption depth d~ ¼ 2, limited by re-absorption of the echo. Higher efficiency can be achieved using three-level storage and counter-propagating fields [18] (see below). In this work we show experimental efficiencies up to 35%, which is significantly higher than previous AFC experiments. This improvement results from our ability to make high finesse, high absorbing comb structures. We also note that a solution to the predetermined storage time was proposed in Ref. [18] (see above). It is based on coherent transfer of the excited state amplitude to a long-lived ground state coherence, for instance a spin coherence, before the appearance of the echo. The memory can be read-out by transferring back the amplitude to the excited state, after a time determined by the user. This aspect of the proposal was recently demonstrated experimentally [30]. A three-level system and counterpropagating control pulses allows for a spatial reversal of the propagation of the echo (so called backward recall). In the absence of dephasing backward recall can reach 100% efficiency by cancelation of the re-absorption, as discussed in Refs. [13,15,18,31,32]. 3. Experiment The measurements were performed on the site 1 transition D2 3H4 at 605.977 nm in a Pr3 þ : Y2 SiO5 crystal immersed in liquid helium at a temperature close to 2.1 K. The sample was 20 10 10 mm3 and had a Pr3 + concentration of 0.05% which gives an absorption depth in the range 60 od o80 [33] at the center of the inhomogeneous profile. The high absorption was critical in order to obtain highly absorbing peaks (see Section 4). A ring dye laser (Coherent699-21) using Rhodamine 6G pumped by Nd:YVO4 laser (Coherent Verdi) is used to give 600 mW output power at l ¼ 605:977 nm. The laser is stabilized against a spectral hole in a second Pr3 þ : Y2 SiO5 crystal, yielding a coherence time 4 100 ms and a frequency drift o1 kHz=s [34]. In order to create the desired pulse shapes and to eliminate beam 1 ARTICLE IN PRESS A. Amari et al. / Journal of Luminescence 130 (2010) 1579–1585 movement accompanying frequency shifts, the laser light was passed twice through a 200 MHz acousto-optic modulator (AOM) with a bandwidth of 100 MHz. A 1 GS/s arbitrary waveform generator (Tektronix AWG520) controlled the AOM, allowing direct control of the light pulse amplitude, phase, and frequency. After the AOM, the light passed through a single mode optical fiber to clean up the spatial mode. A beam sampler removed a small percentage of the light before the cryostat to be used as a reference beam. The rest of the beam passed through a l=2 plate, such that the light polarization could be aligned along the transition dipole moment direction to give maximum absorption. The beam was then focused to a 100 mm radius at the center of the sample, which gave Rabi frequencies of maximum 2 MHz for the strongest transitions. The spectral structures were measured by scanning the light frequency across the spectral structure and recording the intensities of both the transmitted and the reference beams [35]. The signals from the detectors were divided to reduce the effect of laser amplitude fluctuations. The intensity of the probe pulses were chosen such that they did not affect the created spectral structures during the readout process. The scan rate was also set such that it had negligible effect on the resolution of the recorded spectra (see discussion below). 1581 electromagnetically induced transparency (EIT) [36], slow light [33] or quantum computing [37,38]. Since the precise control of the absorption structure is of particular concern for this paper, we will here make a detailed description of how to create a test platform for AFC experiments and essentially the same technique can be used also for the other listed experiments mentioned above. 4.1. Pit creation The inhomogeneous 1D2 3H4 absorption line in Pr3 þ : Y2 SiO5 is about 5 GHz and the homogeneous line width of a single Pr ion is about a kHz at temperatures below 5 K. A chirped laser pulse applied somewhere within the inhomogeneous profile will create a spectral hole through optical pumping. The maximum width of the spectral hole burnt by such a scan is given by the specific level structure of the Pr3 þ ion [39] (see Fig. 2). After relaxation from excited states the ion has to be in one of the three hyperfine ground levels, which have a total separation of 27.5 MHz, but the maximum spectral hole width is reduced by the hyperfine splitting of the excited state levels of 9.4 MHz, yielding a final spectral hole interval of 18.1 MHz. The scanned pulse will thus create a simple, wide spectral hole, henceforth called a pit, and is shown in Fig. 1a. When light pulses are applied to perform operations inside the pit, they also have a probability to interact with the tails of the absorption profile of the ions outside the pit, and in particular with the ions immediately outside, forming the walls of the pit. Generally one would like to avoid such interactions and the simple pit in Fig. 1a then is not optimal. Fortunately, it is possible to shuffle many ions in the walls of the pit further away from the pit. This is illustrated in the top right part of Fig. 1. In this figure, a class of ions having the j0S-je1S transition at some specific frequency just inside the pit and the transitions from the other two ground state levels outside the pit, is displayed. For this ion class, the simple burning pulse only targets the j0S-state, so only this ground state will be emptied. However, it is clear that these 4. Preparation of AFC Creating the atomic frequency comb structure, with good control of all necessary parameters, such as peak height, width, separation and number of peaks, can be challenging. Especially considering that the frequency comb structure also preferably should be well separated in frequency from all other absorbing atoms in the material. However, in rare-earth-metal-ion-doped crystals the inhomogeneous absorption profile can indeed be efficiently manipulated, providing the flexibility needed to meet all those requirements. This flexibility is useful not only for AFC, but also for many other similar experiments, such as Absorption Absorption Laser scan 10.2 MHz 1.2 1 0.8 0.6 0.4 0.2 0 -10 Absorption(αL) Absorption(αL) <18 MHz -5 0 5 Frequency (MHz) 10 0 2.5 17.3 MHz aux 1 2 1.5 1 0.5 0 -20 -10 0 10 20 30 Frequency (MHz) Fig. 1. (color online) In (a) a simple spectral pit created only by scanning a pulse across a specific interval o 18 MHz is shown (upper part shows a schematic view and lower part shows actual experimental data). In (b) a more optimal pit is shown, where additional burning pulses on different frequency intervals have iteratively been applied to spectrally remove ions as far from the spectral pit as possible (see text). For an exact pulse sequence, see Appendix A. Note that the frequency scale in the two experimental figures is different. ARTICLE IN PRESS 1582 A. Amari et al. / Journal of Luminescence 130 (2010) 1579–1585 ions do not have to be in the j1S state, in fact, it would be better if they could be further shifted outwards so that all end up in the jauxS state. This can be done by additional burning pulses at the j1S-je1S transitions. This will cause some ions to go to the jauxS-state but also cause some of the ions to fall back down into the pit, and thus, to get the optimal effect, one would have to iterate between the pulses burning at the center of the pit and the ones burning outside it to improve the walls. Similar techniques can be used on the lower frequency side of pit to obtain the final optimal pit, as shown in the lower part of Fig. 1b. Sechyp 2 Absorption Sechyp 1 12 |0 > 4.3. Peak creation After a suitable pit has been created, a narrow selection of ions is coherently burnt back into the pit. This narrow ensemble of ions now forms an absorption peak spectrally clearly separated from all other ions. The pulses used for this transfer are two complex hyperbolic secant pulses (sechyp for short). The first one targets the j5=2gS-j5=2eS transition, of ions having their j1=2gS-j1=2eS transition at frequency zero (0) MHz. The second pulse is applied immediately after the first, before the excited ions can decay spontaneously, on the j5=2eS-j1=2gS transition. One could imagine taking other routes to burn back a peak, but this particular route is advantageous because the first pulse targets the strongest transition, which means the exciting pulse power, and thus power broadening effects, can be kept at a minimum. The deexcitation pulse on the other hand then targets a weak transition, but since this transition is inside the pit and spectrally far away from other ions, the power can here be increased without any power broadening effects. Fig. 2 briefly illustrates how to create such a peak structure, and also displays an experimentally created version. Creating a full atomic frequency comb from this situation, is now the relatively simple matter of adding additional coherent burn-back pulses, with appropriate frequency offsets, creating the additional peaks. The shape of the peaks, as well as width and absorption height, is determined by the burn-back pulses. Changing the spectral shape of these pulses will change the shape of the peaks, and increasing the pulse power will cause more ions to be transferred, which results in higher absorption peaks (as long as there is enough ions available in the crystal at that frequency). Absorption (αL) 5/2e 3/2e 1/2e 1/2g→1/2e 0.6 |0 > |1 > |aux > |aux > |1 > 4.2. Experimental implementation The exact sequence of pit burning pulses differ depending on the exact level structure of the ion used. Table 1 in Appendix A lists the explicit pulses used to create an optimal pit in Pr3 þ : Y2 SiO5 in this work, and the order in which we have applied those pulses is also listed in Appendix A. When working in other materials, essentially the same sequencing can be used, but of course, the actual frequencies have to be changed to match the transitions of the ion in question. The different optical pumping BurnPitX pulses listed in Table 1 (see Appendix A) are repeated and iterated, as explained in the previous section, in order to create good shallow walls while maintaining no atoms inside the pit. The repetition sequence given at the end of the Appendix A is somewhat arbitrary. A higher number of repetitions reflects the fact that the primarily target transition has relatively low oscillator strength. The exact numbers can be changed a bit up or down without significant effect on the result. There is a 1 ms waiting time after every single BurnPitX pulse (see Appendix A), to give excited ions time to decay back to the ground state before the next pulse arrives. The excited state lifetime is T1 ¼ 164 ms [39]. |e > |e > 1/2g→3/2e 0.4 0.2 1/2g→5/2e 0 -2 0 2 4 6 8 10 Frequency(MHz) 12 1/2g 14 16 |0 > |1 > |aux > Fig. 2. (Color online) Top part shows the sequence and position of the pulses that burns back a narrow ensemble of ions into an empty spectral pit. The lower (b) part shows an experimental version of such a created peak structure. The difference in height of the three different transition from the j0S state, comes from the fact that these transitions have different oscillator strength. This yields a good control over all the essential parameters of the AFC. 4.4. Comb structure measurement One of the goals of this work was to compare the observed AFC echo efficiency with the one predicted by the theoretical model discussed in Section 2. This requires a precise measurement of the AFC structure in order to determine the shape, width and height of the peaks. To do this the laser was slowly swept in frequency across the created AFC structure and the transmitted light intensity was detected after the sample. From this transmission profile of the structure, the absorption spectrum can be calculated. The intensity of the scan pulse was chosen such that it did not affect the created spectral structure. For most comb measurements we chose to scan the laser over one peak only and lowered the scan rate to a minimum. This is to reduce the effect of the scan rate on the measurement resolution [40,35]. By varying the rate we confirmed that the measured width was indeed independent of the scan rate. The comb was created on the j1=2gS-j1=2eS transition in order to maximize the absorption. It is, however, very challenging to measure absorptions above d= 3–4. In order to circumvent this problem we instead measured the comb structure on the weaker j1=2gS-j5=2eS transition, cf. Fig. 2. The ratio of these two transitions is known from previous work [41], thus the optical depth of the j1=2gS-j1=2eS transition is readily inferred from the measured absorption spectrum. 5. Results and discussion The input pulse is stored on the j1=2gS-j1=2eS transition, which is the transition for ions in state j1=2gS with the highest oscillator strength. This results in a comb with high optical depth d. The bandwidth of the AFC is limited by the frequency separation between the excited states and in the present case this is about 4.6 MHz, as set by the j1=2eS and j3=2eS separation ARTICLE IN PRESS A. Amari et al. / Journal of Luminescence 130 (2010) 1579–1585 (cf. Fig. 2). Fig. 3 shows a comb containing four peaks, and where the width of each peak is about g ¼ 150 kHz. The separation between the peaks was set to D ¼ 1:2 MHz. The pulse to be stored has a Gaussian shape with duration 200 ns, resulting in a frequency power spectrum with FWHM 2 MHz (see Fig. 3). A high-efficiency echo is shown in Fig. 4. The emitted echo is observed after 800 ns, as expected from the comb periodicity ts ¼ 1=D ¼ 800 ns. To be able to calculate the efficiency of the echo, first a reference input pulse is sent through the empty pit (no AFC prepared). This pulse is thus completely transmitted through the sample. The AFC is then prepared inside the pit and an identical pulse (the storage pulse) is sent in. This pulse is partially absorbed in the medium and produces an echo. The ratio between the area of the echo and the area of the reference pulse gives the storage efficiency. A small part of the reference pulse as well as the storage pulse is split off before they enter the cryostat so unintentional input power differences between the reference and storage pulses can be compensated for. For the data shown in Fig. 4 we measured an efficiency of 35%. To our knowledge this is the highest AFC echo efficiency observed up to date. 1.2 MHz As discussed in Section 2 the efficiency depends strongly on the shape of the AFC. Our comb structure measurements show peaks that have near Gaussian shapes. This facilitates the theoretical modeling since we can use the simple model discussed in Section 2 and we need only then to measure two parameters; the peak absorption d and peak width g (the finesse is calculated from the relationship F ¼ D=g). In order to make a quantitative comparison with the model, we varied the peak absorption d and measured the resulting input pulse transmission and echo efficiency. This was done by increasing the power of the back burning pulses used in the peak creation (see Section 4.3). For each power setting we also measured the comb structure to find d and F. The width of the peaks could be varied by changing the chirp width of the sechyp pulses used for peak creation. We did measurements as a function of d for two settings of the chirp; 200 and 300 kHz. The measured peak widths for these two settings were 175 and 245 kHz, respectively, corresponding to F= 6.9 and 4.9. With increasing back burning power (hence increasing d) the peaks were slightly broadened due to power broadening, but the observed increase was only 10–15% for the data considered here. The widths given above are averages over all back burning powers (hence d values). In Fig. 5 we show measured transmission coefficients of the input pulse and the efficiencies of the echo for the two data sets. The data are plotted as a function of the measured peak 6 70 2 0 -3000 -2000 -1000 0 1000 2000 3000 45 F=5 F=5 F=7 F=4 F=4 60 40 35 30 50 25 40 20 30 15 20 10 Frequency (kHz) 10 5 0 0 1 2 3 4 5 6 7 8 1.0 80 F=5 70 F=4 F=5 0.6 0.4 0.2 F=4 60 25 40 F=3 30 10 5 1 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Time (us) Fig. 4. (Color online) The dashed line shows the input pulse that is completely transmitted through the empty pit (no absorption). The solid line shows the partially transmitted input pulse and the subsequent echo emission with the AFC created in the pit (cf. Fig. 3). The echo efficiency is 35% of the input pulse (see text for details). 20 15 20 0 -0.4 35 30 50 10 0.0 45 40 F=3 0.8 9 Absorption depth d Transmission (%) Normalised intensity (arb. units) Fig. 3. (Color online) From the inhomogeneous absorption profile, four peaks with ions, all absorbing on the j1=2gS-j1=2eS transition are created. The peak width (FWHM) is g ¼ 150 kHz and they are separated by D ¼ 1:2 MHz. The input pulse has a Gaussian power spectrum with FWHM= 2 MHz. Efficiency (%) 4 F=7 Efficiency (%) 80 150 kHz Transmission (%) Absorption depth 8 1583 2 3 4 5 6 7 8 9 0 10 Absorption depth d Fig. 5. (Color online) Measured transmission of the input pulse (open circles and left axis) and echo efficiency (closed circles and right axis) as a function of the measured optical depth for two different experimental values of the finesse, (a) F= 6.9 and (b) F = 4.9. The dashed lines are theoretically calculated transmission coefficients and solid lines are calculated efficiencies for (a) F = 4,5 and 7, and for (b) F = 3,4 and 5 (see text for details). ARTICLE IN PRESS 1584 A. Amari et al. / Journal of Luminescence 130 (2010) 1579–1585 absorption d as extracted from the AFC spectra. Theoretical transmission and efficiency curves are also shown. These were calculated using the experimental d values, for different values of finesse. It is observed that the transmission coefficient is very sensitive to the finesse, whereas the efficiency is less sensitive up to d =4 5. In general the best agreement for the transmission is obtained for a finesse lower than the one measured from the comb spectra (see above). We can also see that the best-fit finesse is lower for the 300 kHz data set, than for the 200 kHz, which is to be expected. The echo efficiency shows a reasonably good agreement with all three values of the finesse up to d= 4 5. But for d Z 5 the discrepancy between the experimental and theoretical values becomes significant for F= 4 and 5, while for F= 3 it is still satisfactory. The general trend, for both transmission and efficiency, is that our data fit better with a finesse lower than the measured one. This comparison is, however, made within the theoretical framework of Section 2 where a comb of Gaussian peaks was assumed. Both the transmission and echo efficiency are strongly dependent on the actual peak shape [18,20]. For instance, the same comparison with a Lorentzian model [20] yields very different best-fit values for the finesse. Although much effort was devoted to the precise measurement of the comb structure, it is still conceivable that the actual peak shape deviate from a pure Gaussian shape. Particularly since the sechyp pulses used for peak creation have power spectra with more super-Gaussian shape [35]. Another source of error could be imperfections in the peak creation pulses, where the high power needed to obtain high optical depth might generate an increase in the absorption background due to offresonant excitation. Such an additional absorption will reduce both the experimental transmission and efficiency [9] compared to the theoretical model in Section 2 where such an absorption background is neglected. This would particularly affect the high d range, where indeed we observe a larger discrepancy. Appendix A The optical pumping pulses used for creating the pit structure in Fig. 1b are presented in Table 1. Table 1 List of pulses used for the pit burning sequence, with start and end frequencies. Pulse nstart ðMHzÞ nend ðMHzÞ Orel BurnPit1 BurnPit2 BurnPit3 BurnPit4 BurnPit5 BurnPit6 BurnPit7 BurnPit8 BurnPit9 BurnPit10 + 31.85 + 23.85 + 15.95 + 23.85 16.85 8.85 + 15.95 +7.35 1.10 +7.65 + 24.15 + 16.15 + 7.65 + 16.15 9.15 1.15 + 7.65 1.10 + 7.35 + 15.95 3=2g -1=2e 3=2g -5=2e 3=2g -5=2e 3=2g -5=2e 5=2g -5=2e 5=2g -1=2e 3=2g -5=2e 3=2g -5=2e 5=2g -1=2e 5=2g -1=2e This set of pulses will create the pit in Fig. 1b with zero absorption (no absorbing ions) from 1.2 up to 16.2 MHz. The frequency scale is defined by denoting the j1=2gS-j1=2eS transition, for an arbitrarily selected ion class, as zero MHz. The column Orel lists the primary target transition of the scan for the purpose of knowing what light intensity to choose in order to match the Rabi frequency to the relative oscillator strength. Note that pulse numbers 2 and 4 are the same, as is pulses 3 and 7. These pulses are then repeated in an iterative sequence in the following manner (1) (2) (3) (4) Repeat Repeat Repeat Repeat 60 30 20 30 times: times: times: times: BurnPit5, BurnPit6. BurnPit1-4, BurnPit6-10. BurnPit1-4, BurnPit6. BurnPit7-10. Finally yielding the pit in Fig. 1b. 6. Conclusions We have in detail described optical pumping and preparation procedures for creating AFC structures in Pr3 þ : Y2 SiO5 . We were able to make comb structures yielding 4 30% AFC echo efficiency, which are the most efficient AFC echoes observed up to date. We believe that further progress will be possible, by carefully optimizing the comb parameters. It should thus be possible to approach the theoretical limit of 54% used in the present forward-propagation configuration. In order to make a significant further progress, recall in the backward direction would be necessary, in which case 100% efficiency is theoretically possible in the absence of dephasing. In this work we also compared the experimentally observed efficiencies to a theoretical model. Considerable care has been put into determining the line shape and line width of the generated AFC structure with good precision in order to be able to theoretically model the experimental efficiencies. Still, at high optical densities the finesse required to theoretically reproduce the experimental echo efficiency are lower than those measured experimentally. Nevertheless, the present results indeed show that high efficiency QMs can be created using the AFC technique. Acknowledgments This work was supported by the Swedish Research Council, the Knut and Alice Wallenberg Foundation, the Swiss NCCR Quantum Photonics, the European Commission through the integrated project QAP, and the ERC Advanced Grant QORE. References [1] H.-J. Briegel, W. Dür, J.I. Cirac, P. Zoller, Phys. Rev. Lett. 81 (1998) 5932. [2] L.M. Duan, M.D. Lukin, J.I. Cirac, P. Zoller, Nature 414 (2001) 413. [3] C. Simon, H. de Riedmatten, M. Afzelius, N. Sangouard, Z. Zbinden, N. Gisin, Phys. Rev. Lett. 98 (2007) 190503. [4] N. Sangouard, C. Simon, H. de Riedmatten, N. Gisin, Quantum repeaters based on atomic ensembles and linear optics, arXiv:0906.2699. [5] B. Julsgaard, J. Sherson, J.I. Cirac, J. Fiurasek, E.S. Polzik, Nature 432 (2004) 482. [6] T. Chanelie re, D.N. Matsukevich, S.D. Jenkins, S.-Y. Lan, T.B. Kennedy, Nature 438 (2005) 833. [7] M.D. Eisaman, A. Andre, F. Massou, M. Fleischhauer, A.S. Zibrov, M.D. Lukin, Nature 67 (2005) 452. [8] K.S. Choi, H. Deng, J. Laurat, H.J. 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Lejay, J.L. Le Gouët, T. Chanelie re, L. Rippe, A. Amari, A. Walther, S. Kröll, Phys. Rev. A 79 (2009) 033809. [37] J.J. Longdell, M.J. Sellars, Phys. Rev. A 69 (2004) 032307. [38] L. Rippe, B. Julsgaard, A. Walther, Y. Ying, S. Kröll, Phys. Rev. A 77 (2008) 022307. [39] R.W. Equall, R.L. Cone, R.M. Macfarlane, Phys. Rev. B 52 (1995) 3963. [40] T. Chang, M.Z. Tian, R.K. Mohan, C. Renner, K.D. Merkel, W.R. Babbitt, Opt. Lett. 30 (2005) 1129. [41] M. Nilsson, L. Rippe, R. Klieber, D. Suter, S. Kröll, Phys. Rev. B 70 (2004) 214116. ARTICLE Received 18 Feb 2010 | Accepted 5 Mar 2010 | Published 12 Apr 2010 DOI: 10.1038/ncomms1010 Mapping multiple photonic qubits into and out of one solid-state atomic ensemble Imam Usmani1, Mikael Afzelius1, Hugues de Riedmatten1 & Nicolas Gisin1 The future challenge of quantum communication is scalable quantum networks, which require coherent and reversible mapping of photonic qubits onto atomic systems (quantum memories). A crucial requirement for realistic networks is the ability to efficiently store multiple qubits in one quantum memory. In this study, we show a coherent and reversible mapping of 64 optical modes at the single-photon level in the time domain onto one solid-state ensemble of rare-earth ions. Our light–matter interface is based on a high-bandwidth (100 MHz) atomic frequency comb, with a predetermined storage time of ⲏ1 μs. We can then encode many qubits in short ( < 10 ns) temporal modes (time-bin qubits). We show the good coherence of mapping by simultaneously storing and analysing multiple time-bin qubits. 1 Group of Applied Physics, University of Geneva, CH-1211 Geneva 4, Switzerland. Correspondence and requests for materials should be addressed to H.d.R. (email: [email protected]) or to M.A. (email: [email protected]). NATURE COMMUNICATIONS | 1:12 | DOI: 10.1038/ncomms1010 | www.nature.com/naturecommunications 1 ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms1010 aux | W 〉 = ∑ cnei 2pd nt e −ikzn | g en g 〉 0.100 2.0 1.5 γ d 1.0 0.5 d0 0.0 –20 –10 0 (1) n Normalized counts Optical depth o ech AFC oton t ph Inpu g Results The light–matter interface. An AFC is based on a periodic modulation (with periodicity Δ) of the absorption profile of an inhomogeneously broadened optical transition |g〉→|e〉 (see Fig. 1). The modulation should ideally consist of sharp teeth (with full-width at half-maximum γ ) having a high peak absorption depth d (see Fig. 1b). Such a modulation can be created by optical pumping techniques (see Experiment section and Methods). This requires an atomic ensemble with a static inhomogeneous broadening and many independently addressable spectral channels. This can be found in RE-doped solids in which inhomogeneous broadening is of the order of 1–10 GHz and the homogeneous linewidth is of the order of 1–100 kHz when cooled to < 4 K. When a weak photonic coherent state |α〉L with n– < 1 is absorbed by the atoms in the comb, the state of the atoms can be written as |α〉A = |G〉 + α|W〉 + O(α2). Here, |G〉 = |g1…gN〉 represents the ground atomic state and Δ 2.5 e generally d modes can encode a qudit. Time-bin qubits are widely used in fibre-based quantum communication1,2 because of their resilience against polarization decoherence in fibres. Temporal multimode storage is difficult, however, because of the scaling of the number of stored modes Nm as a function of optical depth d of the storage medium21,27. For EIT and Raman interactions, Nm scales as √d, making it very difficult to store many modes27. Recently, we proposed21 a multimode storage scheme based on atomic frequency combs (AFC) with a high intrinsic temporal multimode capacity21,27. Using this method, we recently showed22 that a weak coherent state |α〉L with mean photon number n– = |α|2 < 1 can be coherently and reversibly mapped onto a YVO4 crystal doped with neodymium ions. Later experiments28–31 in other RE-doped materials have improved the overall storage efficiency (35%) and storage time (20 μs). Yet, in these experiments, only a maximum of four modes have actually been stored at the single-photon level; thus, the predicted21,27 high multimode capacity has yet to be shown experimentally. In this study, we show reversible mapping of 64 temporal modes containing weak coherent states at the singlephoton level onto one atomic ensemble in a single spatial mode using an AFC-based light–matter interface. 10 Optical detuning (MHz) 20 Transmitted photons Efficiency (%) Q uantum communication1 offers the possibility of secure transmission of messages using quantum key distribution2 and teleportation of unknown quantum states3. Quantum communication relies on the creation, manipulation and transmission of qubits in photonic channels. Photons have proven to be robust carriers of quantum information. Yet, the transmission of photons through a fibre link, for instance, is inherently a lossy process. This leads to a probabilistic nature of the outcome of experiments. In large-scale quantum networks4, the possibility of synchronizing independent and probabilistic quantum channels is required for scalability5,6. A quantum memory enables this by momentarily holding a photon and then releasing it when another part of the network is ready. To reach reasonable rates in a realistic network, it is necessary to use multiplexing7, which demands quantum memories capable of storing many single photons in different modes. A quantum memory requires a coherent medium with strong coupling to a light mode. Strong and coherent interactions can be found in ensembles of atoms8, for instance, alkali atoms or rare-earth (RE) ions doped into crystals. The latter are attractive for quantum storage applications, as they provide solid-state systems with a large number of stationary atoms having excellent coherence properties. Optical coherence times of up to milliseconds9 and spin coherence times greater than seconds10 have been shown at low temperature (ⱗ4 K). A quantum memory also requires a scheme for achieving efficient and reversible mapping of the photonic qubit onto the atomic ensemble. Techniques investigated include stopped light based on electromagnetically induced transparency (EIT)11–13, Raman interactions14–17 or photon-echo-based schemes18–22. Much progress has been made in terms of quantum memory efficiency13,23 and storage time24,25. Storage of multiple qubits is challenging, however, because it necessitates a quantum memory that can store many optical modes into which qubits can be encoded. A mode can be defined in time, space or frequency. Lan et al.26 recently showed that a multiplexed quantum memory can be achieved by storing several single excitations in separate spatial modes, each functioning as an independent memory. Time multiplexing, as used in classical communication, has the great advantage of requiring only a single spatial mode7,21,27, hence a single quantum memory. Moreover, each pair of temporal modes can be used to encode different time-bin qubits1, or more 0.075 0.050 6 5 4 3 2 1 0 0.0 0.5 1.0 1.5 2.0 Storage time (μs) Emitted photons 0.025 2.5 0.000 0.0 0.2 0.4 0.6 0.8 Time (μs) 1.0 Figure 1 | AFC storage scheme. (a) Simplified level scheme of the Nd ions doped into Y2SiO5. We use the optical transition at 883 nm between the 4I9/2 ground state and 4F3/2 excited state. The former is split into two Zeeman levels by a 0.3 T magnetic field (|g〉 and |aux〉). The experiment is conducted on |g〉 − |e〉, where the absorption profile is shaped into an AFC by optically pumping atoms into |aux〉. The basic idea is to send in a pulse sequence on |g〉 − |e〉 that has a periodic spectral density (due its Fourier spectrum). Some of the excited atoms have a certain probability to spontaneously de-excite to |aux〉. The atoms left behind in |g〉 form the grating (see panel (b)). To build up a deep grating, the sequence is repeated many times (up to the timescale of the population relaxation between |g〉 and |aux〉). More details on the preparation can be found in the Experimental setup and Methods section. (b) An example of a generated comb with periodicity Δ = 10 MHz. The relevant AFC parameters defined in the text are indicated. (c) Mapping of weak coherent states with n– = 0.5 (in a single temporal mode) onto the Nd-doped crystal. Shown are two different experiments with Δ = 10 MHz (dashed line offset vertically) and 1 MHz (solid line). The photons that are transmitted without being absorbed are detected at t = 0, whereas the absorbed and reemitted photons are detected around t = 1/Δ. The vertical scale has been normalized such that it yields efficiency. Inset: The overall write and read efficiency as a function of 1/Δ. 2 NATURE COMMUNICATIONS | 1:12 | DOI: 10.1038/ncomms1010 | www.nature.com/naturecommunications ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms1010 represents one induced optical excitation delocalized over all the N atoms in the comb. In equation (1), zn is the position of atom n, k is the wave number of the single-mode light field, δn is the detuning of the atom with respect to light frequency and the amplitude cn depends on the frequency and spatial position of the particular atom n. The initial (at t = 0) collective strong coupling between the light mode and atoms is rapidly lost because of inhomogeneous dephasing caused by exp(i2πδnt) phase factors. If we assume that the peaks are narrow compared with the periodicity (that is, a high comb finesse F = Δ/γ), then δn≈mnΔ (where mn is an integer) and the W state will rephase after a preprogrammed time T = 1/Δ. The rephased collective state W will cause a strong emission in the forward direction (as defined by the absorbed light). This type of photon-echo emission is also observed in accumulated or spectrally programmed photon echoes32–35, which inspired our proposal. Spectral atomic gratings have also been proposed36 and shown37 for coherent optical delay of streams of strong classical pulses. The interest in spectral gratings was recently renewed in the context of quantum memories, when it was realized how to achieve a much more efficient spectral grating than previously possible. In fact, a 100% efficient echo process is theoretically possible in a backward emission configuration21. This is possible because of the highly absorbing and sharp peaks in the AFC structure. In practice, the finite finesse of the comb still needs to be accounted for, which causes a partial loss of the collective state. However, in ref. 21 we show theoretically that F = 10 induces a negligible loss, which in combination with a high optical depth d makes the AFC scheme very efficient. High-efficiency mapping using high-finesse combs have recently been shown experimentally28,30,31. These experiments and the present study store light for a predetermined time given by 1/Δ. We thus emphasize that we also proposed21 and experimentally showed29 a way to achieve on-demand readout by combining AFC with spin-wave storage. On-demand readout is a crucial resource for applications in quantum networks to render different quantum channels independent. The multimode property of an AFC memory can easily be understood qualitatively. For a periodicity Δ and Np peaks, its total bandwidth is of the order of ~NpΔ, indicating that a pulse of duration τ~1/(NpΔ) can be stored. The multimode capacity stems from the fact that the grating can absorb a train of weak pulses before the first pulse is reemitted after T = 1/Δ (see Fig. 1c). This simple calculation results in a multimode capacity Nm∝T/τ∝Np. Thus, a comb with many peaks, Np, allows us to create a highly multimode memory in the temporal domain. In this context, RE-doped solids are particularly interesting because of their high spectral channel density. Experiment. In this study, we work with a neodymium-doped Y2SiO5 crystal, cooled to 3 K, having a transition wavelength at 883 nm with good coherence properties (see Methods for spectro- scopic information). This wavelength is convenient as we can work with a diode laser and silicon-based single-photon counters having low noise (300 Hz) and high efficiency (32%). The comb is prepared on the |g〉 − |e〉 transition by frequencyselective pumping of atoms into an auxiliary state |aux〉 (see Fig. 1). There are different techniques for achieving this. For instance, by creating a large spectral hole and then transferring back atoms from an auxiliary state to create a comb, as used in ref. 29. Here, we use a technique similar to that employed in ref. 22, in which a series of pulses separated by a time, T, pump atoms from |g〉 to |aux〉 (through |e〉) with a power spectrum having a periodicity 1/T = Δ. This technique is also frequently used in accumulated photon-echo techniques32,37. Here, each pulse sequence consisted of three pulses in which the central pulse is π-dephased (see Fig. 3c). This sequence has a power spectrum with ‘holes’. A Fourier analysis shows that the width of the holes in the power spectrum decreases when the number of pulses in the sequence increases, resulting in a higher comb finesse. In this experiment, three pulses were enough to reach the optimal comb finesse (F≈3) to achieve the maximal efficiency for our optical depth. We refer to the Methods section for more details on the preparation sequence. The experimental sequence is divided into two parts: the preparation of AFC and the storage of weak pulses. To increase the efficiency of optical pumping during the preparation, and thus the depth of the comb, the pulse sequence was repeated 2,000 times, with a delay of 16 μs between each sequence. This is followed by a delay of 5 ms (≈17T1) to avoid fluorescence noise from atoms left in the excited state. During the storage sequence, 1,000 independent trials are performed at a repetition rate of 200 kHz. The entire preparation and storage sequence is then repeated with a rate of 5 Hz. An overview of the experimental setup is given in Fig. 2. In Fig. 1c, we show storage experiments with predetermined storage times of T = 100 ns and 1 μs, for a single temporal mode. The overall in–out mapping efficiencies, defined as the ratio of the output pulse counts to the input pulse counts, are ~6 and ~1%, respectively (see inset of Fig. 1c). In the Methods section, we present a theoretical analysis of the efficiency performance. The efficiency for single-mode storage is currently lower than that achieved in bestperformance single-mode memories, for example, those given in refs 13, 16, 17, 20, 28, 30, 38. However, as explained later, our interface compares very favourably with these experiments in terms of potential multimode storage efficiency. Multimode storage results. The main goal of this study is to show high multimode storage. Following the discussion above, we should maximize the number of peaks in the comb. This can be carried out by increasing the density of peaks in a given spectral region (that is, increasing the storage time T) or by changing the width of the AFC (that is, increasing the bandwidth). Here, we fix the storage time Figure 2 | Experimental setup. Description of the optical setup used for multiqubit storage experiments. Indicated are the polarization beam splitter (PBS), λ/2 plates, polarization controllers (PC), Faraday rotator (FR), double-pass acousto-optic modulators (AOM), the fast electro-optic amplitude modulator (EOM), neutral density (ND) filters, mechanical choppers (MC) and the silicon single-photon counter (APD). See Methods for more details. NATURE COMMUNICATIONS | 1:12 | DOI: 10.1038/ncomms1010 | www.nature.com/naturecommunications 3 ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms1010 6 2.5 Efficiency (%) Optical depth 3.0 2.0 1.5 1.0 5 4 3 2 1 0.5 0 –50 –25 0 25 50 Optical detuning (MHz) E0 5 10 15 20 25 Input pulse duration (ns) 30 1 μs Time 16 μs f1 =–40 MHz f2 = –20 MHz f3=0 MHz f4=20 MHz f5=40 MHz Normalized counts Figure 3 | Increasing spectral bandwidth. (a) Experimental combs created using preparation sequences with either single (solid line) or five (dashed line) simultaneous pump frequencies. The frequency-shifted sequences allow us to enlarge the frequency range over which the optical pumping is efficient, thereby creating a wide 100 MHz comb. (b) Efficiency as a function of the duration (full-width at half-maximum, FWHM) of the input pulse for a single(circles) and five (squares)-frequency preparation. As the duration decreases, the bandwidth of the input pulse increases. The decrease in efficiency for short pulses is due to bandwidth mismatch for large bandwidths when using a single-preparation frequency. This experiment clearly illustrates the gain in bandwidth in the extended preparation sequence for which only a small decrease in efficiency is observed. (c) Pulse sequence for atomic frequency comb preparation (see text). To increase the bandwidth, pulses are repeated with shifted frequencies f = 0, ±20 and ±40 MHz. This pulse sequence was used for most of our experiments. Here, it creates a comb of 100 MHz bandwidth and a periodicity of 1 MHz. The total sequence takes 16 μs. 1.0 0.8 0.6 0.4 0.2 0.0 0.0 Input modes 0.4 Output modes ×50 0.8 1.2 1.6 Time (μs) 2.0 2.4 Figure 4 | Storage of 64 temporal modes. The input (left part) is a random sequence of full and empty time bins, in which the mean photon number in the full ones is n– ⱗ 1. The output (right part) clearly preserves the amplitude information to an excellent degree. The predetermined storage time was T = 1.32 μs; the duration of the input, 1.28 μs; the mode separation, 20 ns; and the mode duration (full-width at half-maximum, FWHM), 5 ns. The output has been multiplied with a factor of 50 for clarity. The average storage and retrieval efficiency was 1.3%. Other examples of multimode storage, for example, with all time bins filled, can be found in the Supplementary Information. to T = 1.3 μs, by which we reach an efficiency of ⲏ1%, and concentrate our efforts on increasing the bandwidth. The spectral width of the grating is essentially given by the width of the power spectrum of the preparation sequence, which, using the pulse sequence described above, only results in a width of about 20–30 MHz. We can, however, substantially increase the total width by inserting more pulses in the preparation sequence, which are shifted in frequency (see Fig. 3c). We thus optically pump atoms over a much larger frequency range. Note that the frequency shift should be a multiple of Δ to form a grating without discontinuities. In this way, we managed to extend the bandwidth of the interface to 100 MHz, as shown in Fig. 3a, without significantly affecting the AFC echo efficiency. This is illustrated in Fig. 3b, in which we show storage efficiency as a function of the duration of the input pulse when the preparation sequence contains a single or five frequencies. The maximum bandwidth allows us to map short, ⱗ5 ns pulses into memory. 4 In addition to the present motivation for multimode storage, a large bandwidth is equally interesting for interfacing a memory with non-classical single-photon or photon-pair sources. These usually have a large intrinsic bandwidth that requires extensive filtering for matching bandwidths. In this case, our extended bandwidth (×5) would require a corresponding factor of less filtering. We show the high multimode capacity of our interface by storing 64 temporal modes during a predetermined time of 1.3 μs (see Fig. 4), with an overall efficiency of 1.3%. This capacity is more than an order of magnitude higher than that previously achieved for multiplexing a quantum memory in a single spatial mode16,22. As shown, we can store a random sequence of weak coherent states. Storage of random trains of single-photon states has been proposed for multiplexing long-distance quantum communication systems on the basis of the so-called quantum repeaters5,6. The maximum rate of communication would then be proportional to the number of modes that can be stored7. Our experiment clearly shows the gain NATURE COMMUNICATIONS | 1:12 | DOI: 10.1038/ncomms1010 | www.nature.com/naturecommunications ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms1010 Figure 5 | Multimode coherence measurement. (a) The output signal (solid line) generated by the double-AFC scheme, which causes an interference between consecutive modes. The input sequence (not shown) is a series of weak coherent states (n̄ ≈1.8) of equal amplitude ck2, in which the relative phase difference between consecutive modes φk + 1 − φk ranges from − π/6 to 8π/6 with a step of π/6. This allows us to capture a complete interference fringe in one measurement. It also clearly shows the preservation of coherence over the complete multimode output. By changing the detuning of the centre of the second AFC with respect to the carrier frequency of the light, we can impose an additional relative π-phase21 on the corresponding output (see Methods). This shifts the interference fringe with half a period (dashed line). (b) The corresponding net interference visibilities are 86±3% (open circles) and 85±3% (filled squares), with detector noise (dashed line) subtracted. The uncorrected raw visibilities were 78±3% and 76±2%. Error bars represent statistical uncertainties of photon counts (±1 s.d.). Discussion For temporal multimode storage, the efficiency of our interface would outperform the current EIT and Raman-based quantum memories in homogeneously broadened media, although impressive efficiencies have been achieved for single-mode storage13,23. This is because of the poor scaling of efficiency as a function of the number of modes for a given optical depth27. It would also compare favourably with the recent few modes storage experiment16 using gradient echo memory, another echo-based storage scheme, also because of the scaling of mode capacity for a given optical depth (Nm~d)7,27. Nevertheless, an increase in storage efficiency and on-demand readout are necessary for applications in quantum communication. The next grand challenge is to combine multimode storage, high efficiency30 and on-demand readout29 in one experiment. The immediate efforts will most probably be devoted to praseodymiumand europium-doped Y2SiO5 crystals, in which the crystal-field Input Output 0.25 0.20 Normalized counts that can be made using an AFC-based quantum memory. It thus opens up a route towards achieving efficient quantum communication using quantum repeaters. It is now possible to use consecutive temporal modes |k〉 and |k + 1〉 to encode time-bin qubits ck | k 〉 + ck +1ei(fk +1 −fk ) | k + 1〉 , in which case, a good coherence between modes is crucial. The coherence can be measured by preparing superposition states and performing projective measurements using an interferometric setup. Projective measurements on time-bin qubits are usually performed using an imbalanced Mach–Zehnder interferometer, in which consecutive time bins interfere1. We can perform the same task with our light–matter interface using a double-AFC scheme (with Δ1 and Δ2) as shown in ref. 22. In short, the difference in delay 1/Δ1 − 1/Δ2 is the delay in an imbalanced Mach–Zehnder interferometer. The technique used for preparing a double-AFC structure is explained in Methods. As shown in Fig. 5, we observe excellent coherence over all modes with an average visibility of V = 86±3%, corresponding to a conditional qubit fidelity of F = (1 + V)/2≈93%. To further illustrate our ability to store multimode light states, we create a light pulse with a random amplitude modulation. As shown in Fig. 6, we can faithfully store this kind of light pulse. The possibility of storing weak arbitrary light states using photon-echo-based schemes was pointed out already by Kraus et al. (ref. 19). We believe that this work, in which complex phase and amplitude information are reversible and coherently mapped onto one atomic ensemble, is the first experimental realization showing these properties at the single-photon level. 0.15 0.10 0.05 0.00 0.2 0.4 0.6 0.8 1.0 Time (μs) Figure 6 | Mapping of a randomly amplitude-modulated pulse. As seen, the overlap between the normalized input (dashed line) and output (solid line) pulses is excellent. The total average number of photons in the 1 μs long input pulse is n– ≈4. ground state has the necessary number of spin levels (three levels) for implementing the on-demand readout. The recent achievements in praseodymium-doped Y2SiO5 crystals are very encouraging29,30, although the bandwidth was limited to a few MHz because of the hyperfine level splitting. Europium-doped Y2SiO5 has the potential of offering higher bandwidths (up to 70 MHz) and narrower comb peaks, which result in a higher multimode capacity21. To exploit the high-bandwidth results reported in this work, using neodymiumdoped crystals, one needs to find a third spin level with a long spin coherence lifetime. An interesting path forward is to investigate neodymium isotopes with a hyperfine structure (143Nd and 145Nd)39. Recent results on a similar system40, 167Er3 + :CaWO4, show coherence times approaching 100 μs for hyperfine transitions. Clearly, this path requires extensive spectroscopic studies to optimize the spin population and coherence lifetimes. However, it is very interesting, as it opens up several material candidates (for example, doped with erbium41 and neodymium) for quantum memory applications. To summarize, we have shown the reversible mapping of up to 64 optical temporal modes at the single-photon level onto one solid-state atomic ensemble. We have shown that the quantum coherence of the stored modes is preserved to a high extent. The NATURE COMMUNICATIONS | 1:12 | DOI: 10.1038/ncomms1010 | www.nature.com/naturecommunications 5 ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms1010 different modes can then be used to encode multiple time-bin photonic qubits. Alternatively, they could also be considered as high-dimensional qudit states. This opens up possibilities to store higher dimensional quantum states such as entangled qudits encoded in time-bin bases42. Our experiment opens the way to multiqubit quantum memories, which are a crucial requirement for realistic quantum networks. Methods Sample. The sample is a 10 mm-long neodymium-doped yttrium orthosilicate crystal (Nd3 + :Y2SiO5) with a low Nd3 + concentration of 30 p.p.m. The inhomogeneous broadening of the 4I9/2–4F3/2 absorption line is around 6 GHz and the optical depth is 1.5 for this sample (in single-pass configuration), when the polarization of the light is along the D1 crystallographic axis. We measured an excited state lifetime of T1 = 300 μs using fluorescence spectroscopy and stimulated photon echoes. With conventional photon echoes (two-pulse), we measured a homogeneous linewidth of 3.5 kHz (T2 = 90 μs) for a magnetic field of 300 mT and temperature of 3 K. Each level is a Kramer’s doublet that splits into two spin states in a magnetic field. For the field orientation (at 30° relative to the D2 axis) used in this experiment, we measured g-factors of gg = 2.6 and ge = 0.5. In a 300 mT magnetic field, the ground and excited states were thus separated by 10.9 and 2.1 GHz. We measured a ground state Zeeman population relaxation lifetime of around T1Z = 100 ms by spectral hole burning. In the spectral hole-burning measurements, we also observed a superhyperfine interaction of neodymium ions with yttrium. This causes additional spectral side holes at around 640 kHz (for the present magnetic field), thus the effective homogeneous linewidth is around 1 MHz. This was our main limitation for the efficiency of our light–matter interface, as it affected our ability to create a good comb for longer storage times (1/Δ≈1 μs). Experimental setup. We present more details on the optical setup shown in Fig. 2. We used a diode laser that is actively frequency stabilized ( < 100 kHz), using a spectral hole as reference. The output was split into two beams using a polarization beam splitter. Each beam could be amplitude, frequency and phase modulated using double-pass acousto-optic modulators. One beam was used for creating preparation pulses and the other one for creating the weak pulses to be stored (strongly attenuated using neutral density filters). In the weak path, an additional electro-optic amplitude modulator was used to create short-input pulses for the multimode storage experiments. The paths were mode overlapped using a fibrecoupled beam combiner. The light was sent through the crystal, again in free space, in a double-pass setup to double the optical depth to d = 3. Using a Faraday rotator and a λ/2 plate before the crystal, we could separate the input and output modes on the polarization beam splitter, while keeping a constant linear polarization optimized for maximum absorption in the crystal. Output light was collected with a multimode fibre and detected by a silicon single-photon counter. Two synchronized mechanical choppers blocked the detector during the preparation sequence and blocked the preparation beam during the storage sequence. The transmission between the input of the cryostat and the detector was typically between 25 and 30%. Comb preparation. We now explain in more detail the preparation sequence allowing us to create the desired comb. The goal is to optically pump atoms from |g〉 to |aux〉 in a frequency-selective manner (see Fig. 1a), wherein the atoms left in |g〉 will form the comb. This can be achieved by two pulses of duration τ separated by time T, as done in ref. 22, which has a power spectrum of width ~1/τ with a sinusoidal modulation of periodicity 1/T. To create a sharper comb structure having higher finesse, it is useful to have a wide power spectrum with sharp ‘holes’ instead of the sinusoidal modulation above. This can be achieved by increasing the number of pulses in the sequence in which the central pulse is π-dephased and has a field amplitude corresponding to the sum of the amplitudes of the side pulses (see Fig. 3c). In frequency space, the short and intense central pulse interferes destructively with the periodic spectrum of the side pulses, creating a wide power spectrum with the desired holes. From this simple Fourier argument, it is clear that the width of these holes in the spectrum is proportional to the number of pulses. In our study, the optimal finesse is close to 3 for the optical depth of our material, in which case, three pulses were enough to achieve this finesse. For the interference experiment, shown in Fig. 5, one temporal input mode must be split into two possible temporal output modes. For this purpose, we can create two superimposed combs, with a different periodicity Δi. An incoming photon then has an equal probability of being absorbed by either of the combs and will consequently be reemitted as a coherent superposition of the two corresponding output modes. To create two combs with periodicity Δ1 and Δ2, we use the sequence shown in Supplementary Fig. S3. A sequence with N = 3 pulses separated by τ1 is superposed with another sequence of N = 3 pulses separated by τ2, with the central pulse of each sequence being centred at t = 0. Thus, the total number of pulses will be 2N − 1 = 5, and the pulse area of the central one should be equal to the sum of the others in order to achieve the appropriate interference effect in the power spectrum. 6 It is also possible to impose a phase change φ on the output mode related to a particular comb. This is achieved by frequency shifting the corresponding comb by a fraction of the comb spacing Δ, as discussed in ref. 21. To do so using our preparation method, we must add a phase to each pulse preparing the corresponding comb (see Supplementary Fig. S3). If we label the pulses with k = ±1, ±2, ±3… (excluding the central pulse), then the phase of each pulse should be kφ. It is straightforward to show by Fourier analysis that the corresponding power spectrum will see a frequency shift in the position of the holes, which in turn will cause a shift in the position of the created comb. Storage efficiency analysis. Efficiency can be calculated theoretically using the 2 formula21,22 h ≈ (d / F )2 e−d / F e−7 / F e−d0 . The different terms can be given a qualitative understanding. The first term represents collective coupling, the second the reabsorption of the reemitted light, the third is an intrinsic dephasing factor due to finesse and the last term is a loss due to an absorption background d0. For the comb with Δ = 10 MHz, we measure d≈1.7, F≈2.7 and d0≈0.5 (see Fig. 1b), resulting in a theoretical efficiency of η ≈ 5% in close agreement with the experiment (see Fig. 1c). The major limiting factor here is d0 (caused by an imperfect preparation of the comb) and then the optical depth of the comb d (the finesse being close to optimum for this d, see ref. 21). The decrease in efficiency for longer storage time periods (see inset of Fig. 1c) is principally due to an increase in background absorption d0 and an accompanying decrease in peak absorption d. 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Tailoring photonic entanglement in high-dimensional Hilbert spaces. Phys. Rev. A 69, 050304 (2004). Acknowledgments We acknowledge financial support from the Swiss NCCR Quantum Photonics, the EC projects Qubit Applications (QAP) and the ERC Advanced Grant (QORE). We also acknowledge useful discussions with Christoph Simon and Nicolas Sangouard. Author contributions I.U. built the experimental setup, carried out the experiment and analysed the data, which were supervised by M.A. and H.d.R. All authors contributed to the conception and analysis of the experiment and to the writing of the paper. Additional information Supplementary Information accompanies this paper on www.nature.com/ naturecommunications. Competing financial interests: The authors declare no competing financial interests. Reprints and permission information is available online at http://npg.nature.com/ reprintsandpermissions/ How to cite this article: Usmani, I. et al. Mapping multiple photonic qubits into and out of one solid-state atomic ensemble. Nat. Commun. 1:12 doi: 10.1038/ncomms1010 (2010). NATURE COMMUNICATIONS | 1:12 | DOI: 10.1038/ncomms1010 | www.nature.com/naturecommunications 7 Supplementary Information Mapping multiple photonic qubits onto one solid-state atomic ensemble Imam Usmani,Mikael Afzelius,Hugues de Riedmatten and Nicolas Gisin Group of Applied Physics, University of Geneva, CH-1211 Geneva 4, Switzerland (Dated: March 5, 2010) In this supplementary material we provide a figure (Fig. S1) of the comb structure for 1 µs storage time. We also show additional results (Fig. S2) of multimode storage experiments and an additional figure (Fig. S3) describing the preparation sequence for double-AFC experiments. 2 Optical depth 2.0 1.5 1.0 0.5 -50 -25 0 25 Optical Detuning [MHz] Figure S 1: Experimental atomic frequency combs. We here show experimental combs with 1MHz peak separation (corresponding to 1 µs storage time) for 20 MHz (green line) and 100 MHz (black line) bandwidths, respectively. The bandwidth enlargement technique used in this work provides a larger bandwidth without deteriorating other comb properties such as optical depth d or finesse F . The AFC echo efficiency is thus unaffected (cf. Figure 3 in the article). However, when compared to combs with larger peak separation (eg. shorter storage time) as shown in Fig. 1b, the peak height d decreases, and the absorption background d0 increases. This explains the lower storage efficiency for 1 µs storage time, as shown in Fig. 1c (see Methods for a discussion on storage efficiency modeling). As explained in the paper, the number of modes is proportional to the number of peaks in the comb. Here we create more than 100 peaks which allows us to store 64 modes with negligible mode overlap. Note that the variation in optical depth as a function of frequency is caused by laser intensity variations during the measurement. normalized counts 3 1.0 Input mode Output mode x50 0.8 0.6 0.4 0.2 0.0 0.0 0.5 1.0 1.5 2.0 2.5 2.0 2.5 normalized counts time [ s] 1.0 Input mode Output mode x50 0.8 0.6 0.4 0.2 0.0 0.0 0.5 1.0 1.5 time [ s] Figure S2: Storage of 64 optical modes. Show are two more examples of storage of 64 temporal modes, with different input sequences as compared to Fig. 4 in the article. In the upper trace, all the modes are full, which allows us in principle to store 32 time-bin qubits. An efficiency of 1.4% was measured, with a mean photon number per input pulse of n̄ ≈ 0.5. In the lower trace, we modulated the amplitude of the input pulses. An efficiency of 1.6% was measured, with a mean photon number in the most intense input pulses of n̄ ≈ 0.9 4 Figure S 3: Double-AFC preparation technique. Pulse sequence for the preparation of two atomic frequency combs, as required for the double read out in the interference experiments. In this example, the two AFCs have a periodicity 1/T1 and 1/T2 . The phase change ±φ on the pulses creating the 1/T2 comb will cause a corresponding phase change φ of the AFC echo. See Methods section in article for details. NEWS & VIEWS RESEaRch a mere first — albeit promising — step on a fresh path in the emerging field of cognitiveenhancement research. These outstanding questions notwithstanding, the current study2 has yet another interesting twist. The genes encoding IGF-II and IGF-IIR are known to be imprinted (they display a dichotomous effect on growth in most tissues studied8). The IGF-II gene — like many other genes for which only the paternal copy is expressed — favours growth. Conversely, the IGF-IIR gene — like most maternally expressed genes — suppresses growth. Chen and co-workers’ data are not only the first to attribute a clear-cut function to these genes in the brain, but also to show that, at least in the rat hippocampus, their protein products cooperate to enhance memory. But, given that the spatial and developmental patterns of IGF-II and IGF-IIR expression in the brain are highly complex9, more work is needed to decipher their exact regulation. One process by which the expression of maternal and paternal copies of IGF-II and IGF-IIR is regulated is DNA methylation8 — an epigenetic mechanism that is dynamically altered after learning10. So it would be interesting to know whether learning per se can also trigger epigenetic changes and so the subsequent alterations in the expression of IGF-II and IGF-IIR. Moreover, ‘natural cognition enhancers’ such as environmental enrichment, which are known to act through epigenetic mechanisms11, might also epigenetically regulate IGF-II and IGF-IIR. If so, for those in search of a memory boost, natural enhancers would be an attractive alternative to purely pharmacological agents. ■ QUAN TU M I NF ORM ATI O N Entanglement on ice The ability to store entangled photons in a solid-state memory, and to retrieve them while preserving the entanglement, is a required step on the way to practical quantum communication. This step has now been taken. See Letters p.508 & p.512 JEVON LONGdELL C ommunication over long distances needs repeaters — for classical communication these are devices that receive input data and retransmit them. In an important move towards long-distance communication of quantum information, two groups1,2 (pages 508 and 512 of this issue) have demonstrated the basic building block of a solid-state quantum repeater. The authors have managed to put entangled photons ‘on ice’. That is, they were able to show that they could keep a pair of photons entangled, even after storing and retrieving one of the photons in a cryogenically cooled crystal. Two quantum-mechanical systems are entangled when it is impossible to properly describe the quantum-mechanical state of one of them in isolation. The concept was introduced by Einstein, Podolsky and Rosen in their famous 1935 paper3. Quantum mechanics predicts completely random, but perfectly correlated, results for some measurements of the two entangled systems. It also states that the results of these measurements are not determined until one of the systems is measured. However, the fact that these correlations still existed when the systems were too far apart for any signalling between them to be possible, led Einstein and colleagues3 to conclude that quantum mechanics was not a complete description of entangled systems. They argued that the measurement results must have been determined by some ‘hidden variable’ when the entangled pair was created, and not at the time of measurement. Since 1935, entangled states have been produced, and the strange predictions that quantum mechanics makes about them have been validated. Furthermore, Bell showed4 that all hidden-variable theories will give predictions that, in certain circumstances, differ from those of quantum mechanics. It was therefore possible to do experiments that supported quantum mechanics over hiddenvariable theories. Today, entanglement is at the heart of quantum information. The challenge in achieving secure, long-distance communication of quantum information is entangling systems that are spatially far apart. In many ways, photons are ideal carriers of quantum information: they are easy to manipulate with optics, can be sent long distances either through optical fibres or in free space, and good single-photon detectors are also available. There are downsides, however. Unless speeding through a vacuum, photons have very short lives. This problem is exacerbated by the fact that most methods for creating, and many methods for manipulating, entangled photons are probabilistic. And this is why the ability to store entangled photons in a memory and then recall them, while preserving the entanglement, which has now been demonstrated by Clausen et al.1 and Saglamyurek et al.2, is so important. It enables much more sophisticated operations Johannes Gräff and Li-Huei Tsai are at the Picower Institute for Learning and Memory, Department of Brain and Cognitive Sciences, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA. e-mails: [email protected]; [email protected] Maher, B. Nature 452, 674–675 (2008). Chen, D. Y. et al. Nature 469, 491–497 (2011). Russo, V. C. et al. Endocr. Rev. 26, 916–943 (2005). Sonntag, W. E. et al. Neuroscience 88, 269–279 (1999). 5. Dudai, Y. Annu. Rev. Psychol. 55, 51–86 (2004). 6. Milekic, M. H. & Alberini, C. M. Neuron 36, 521–525 (2002). 7. Feinberg, A. P. Nature 447, 433–440 (2007). 8. Reik, W. & Walter, J. Nature Rev. Genet. 2, 21–32 (2001). 9. Gregg, C. et al. Science 329, 643–648 (2010). 10. Day, J. J. & Sweatt, J. D. Nature Neurosci. 13, 1319–1323 (2010). 11. Fischer, A., Sananbenesi, F., Wang, X., Dobbin, M. & Tsai, L.-H. Nature 447, 178–182 (2007). 1. 2. 3. 4. through synchronization of these probabilistic processes. An example of these more sophisticated operations is long-distance quantum communication. Entanglement, and therefore secure communication, protected by quantum mechanics5, can be achieved over moderate distances (tens of kilometres) by simple propagation of entangled pairs of photons. But at larger distances, photon loss quickly becomes a significant problem, preventing the distribution of entanglement. One way to solve this problem is to break the link between the two parties into a small number of sub-links joined together with quantum repeaters6. Entangled photons would then be transmitted between the repeaters. The memories in the repeaters can store entangled photon pairs associated with successful transmission along part of the link until other parts of the link are also successful. In this way, quantum repeaters prevent the effective bit rates for transferring information between the two parties from dropping off exponentially (or worse) with distance. This is because, with repeaters, entanglement can be achieved without one photon having to travel along the whole link, which for long links can be very unlikely. When implementing quantum repeaters, it has been the memory aspect that has been the most troublesome. But now Clausen et al.1 and Saglamyurek et al.2 have given us proof-ofprinciple demonstrations. They have produced entangled photons by means of spontaneous parametric downconversion. In this process, an entangled pair of two lower-energy photons is created from a higher-energy photon in an optically nonlinear crystal. The wavelength of the ‘pump’ lasers that produce the higherenergy photon was chosen by both groups1,2 such that one member of the entangled photon pair had a suitable wavelength for being stored in a quantum memory based on a cryogenic crystal doped with rare-earth ions; and, as required by quantum repeaters, the wavelength 2 7 JA N UA Ry 2 0 1 1 | VO L 4 6 9 | N AT U R E | 4 7 5 © 2011 Macmillan Publishers Limited. All rights reserved RESEaRch NEWS & VIEWS of the other member of the pair was suitable for long-distance transmission in optical fibre. Mapping of photonic entanglement into and out of a quantum memory has been demonstrated already with trapped-atom systems7. However, the new work1,2 is the first to achieve it using a solid-state memory. The use of the solid state offers certain practical advantages. For example, Saglamyurek and colleagues2 formed the memory in an optical waveguide, which could enable integrated devices to be built. More significantly though, the authors’ approach1,2 to quantum memories using cryogenic rare-earth-ion-doped crystals is rapidly developing and has already surpassed the storage bandwidths2, capacities8, efficiencies9 and storage times10 of other approaches. Such cryogenic rare-earth-ion-doped systems have already been studied for classical optical signal processing because of these systems’ large ratio of inhomogeneous to homogeneous broadening11 for their optical absorption lines. That is, the optical absorption linewidth of each dopant atom is very narrow, whereas the linewidth of the ensemble of dopants can be very large. This makes them very suitable systems for photon echoes, which is where the dopants emit a pulse of light (echo) in response to earlier applied light pulses. In particular, photon echoes allow signal processing with simultaneous large bandwidth, determined by ensemble linewidth, and high resolution, determined by singledopant linewidth. The rapid advances of rareearth quantum memories have been, in large part, due to the development of photon-echo techniques, which are suitable for preserving quantum states of light. Although the storage of entanglement in a solid is a significant step, the efficiencies and storage times in the entanglement-storage experiments of Clausen et al.1 and Saglamyurek et al.2 need to be improved; they are currently inferior to those that can be achieved in a small spool of optical fibre. And whereas good efficiency, storage time and bandwidth have all been demonstrated by others, in separate demonstrations, the next challenge awaiting researchers is to achieve all these performance metrics for the same memory. This should CI RCA dI AN RHY TH M S Redox redux Oscillations in gene transcription that occur in response to biological daily clocks coordinate the physiological workings of living organisms. But turnover in cellular energy may be sufficient to make the clock tick. See Article p.498 & Letter p.554 J O S E P H B A S S & J O S E P H S . TA K A H A S H I L ast spring, a visitor at the biennial meeting of the Society for Research on Biological Rhythms in Florida approached the geneticist Sydney Brenner inquiring as to what it was that scientists studying circadian rhythms actually do. With a glimmer in his eye, Brenner responded that the meeting concerned “those things that only happen once each day”. Indeed, all forms of life undergo circadian (roughly 24-hour) fluctuations in energy availability that are tied to alternating cycles of light and darkness. Biological clocks organize such internal energetic cycles through transcription–translation feedback loops. But two papers1,2 in this issue show that, in both humans and green algae, rhythmic cycles in the activity of peroxiredoxin enzymes can occur independently of transcription. Biological circadian oscillators have long been recognized as a self-sustained phenomenon, their 24-hour length being both invariant over a wide range of temperatures and responsive to light. Early indications that genes underlie the clocks came3 from the isolation of mutant fruitflies carrying altered, and yet heritable, circadian rhythms. This and subsequent work4,5 established that endogenous molecular clocks consist of a transcription– translation feedback loop that oscillates every 24 hours in cyanobacteria, plants, fungi and animals. Although the specific clock genes are not evolutionarily conserved across distinct phyla, their architecture is similar. The forward limb of the clock involves a set of transcriptional activators that induce the transcription of a set of repressors. The latter comprise the negative limb, which feeds back to inhibit the forward limb. This cycle repeats itself every 24 hours (Fig. 1). Energetic cycles are one type of physiological process that shows transcription-dependent circadian periodicity6,7; such cycles include the alternating oxygenic and nitrogen-fixing phases of photosynthesis, and the glycolytic and oxidative cycles in eukaryotes (organisms with nucleated cells). The idea that biochemical flux per se may couple circadian and energetic cycles was first suggested by McKnight and colleagues8, who showed that varying the redox state of the metabolic cofactor NAD(P) affects the activity of two clock proteins, and it gained further support from subsequent studies9–14. 4 7 6 | N AT U R E | VO L 4 6 9 | 2 7 JA N UA Ry 2 0 1 1 © 2011 Macmillan Publishers Limited. All rights reserved open up new capabilities and technologies that will stretch quantum mechanics in a way that we have not yet been able to. Who knows, it might break. ■ Jevon Longdell is at the Jack Dodd Centre for Quantum Technology, Department of Physics, University of Otago, Dunedin 9054, New Zealand. e-mail: [email protected] 1. Clausen, C. et al. Nature 469, 508–511 (2011). 2. Saglamyurek, E. et al. Nature 469, 512–515 (2011). 3. Einstein, A., Podolsky, B. & Rosen, N. Phys. Rev. 47, 777–780 (1935). 4. Bell, J. S. Rev. Mod. Phys. 38, 447–452 (1966). 5. Ekert, A. K. Phys. Rev. Lett. 67, 661–663 (1991). 6. Briegel, H.-J., Dür, W., Cirac, J. I. & Zoller, P. Phys. Rev. Lett. 81, 5932–5935 (1998). 7. Choi, K. S., Deng, H., Laurat, J. & Kimble, H. J. Nature 452, 67–71 (2008). 8. Usmani, I., Afzelius, M., de Riedmatten, H. & Gisin, N. Nature Commun. 1, 12 (2010). 9. Hedges, M. P., Longdell, J. J., Li, Y. & Sellars, M. J. Nature 465, 1052–1056 (2010). 10. Longdell, J. J., Fraval, E., Sellars, M. J. & Manson, N. B. Phys. Rev. Lett. 95, 063601 (2005). 11. Barber, Z. W. et al. J. Lumin. 130, 1614–1618 (2010). But exactly how transcriptional and nontranscriptional cycles may be interrelated was still not fully understood. To address this relationship, O’Neill and Reddy1 (page 498) examined the rhythmic properties of human red blood cells (RBCs). In their mature form, these cells lack both a nucleus and most other organelles, including energy-producing mitochondria. They function mainly as oxygen shuttles, utilizing the protein haemoglobin as the delivery vehicle. Some of the most abundant proteins in mature RBCs are the evolutionarily conserved enzymes of the peroxiredoxin family, which can inactivate reactive oxygen species (ROS). Class-2 peroxiredoxins contain a cysteine amino-acid residue in their active site that undergoes oxidation when ROS accumulate. This results in the enzyme’s transition from a monomeric to a dimeric state. Excess ROS accumulation induces the formation of even higher-order oligomers. Peroxiredoxin function is essential for RBC survival, as defects in the expression or activity of these enzymes lead to the breakdown of the cells. A previous survey15 searching for proteins that show circadian rhythms of expression in liver identified peroxiredoxins. In their study, O’Neill and Reddy1 monitored the monomer– dimer transition of these proteins in RBCs from three humans. They observed two main circadian features in these enucleated cells. First, the oligomerization pattern was selfsustained over several cycles within an approximate 24-hour period and was not affected by temperature. Second, peroxiredoxin oxidation cycles were synchronized in response to temperature cycles, a property called entrainment that is a hallmark of circadian oscillators. LETTER doi:10.1038/nature09662 Quantum storage of photonic entanglement in a crystal Christoph Clausen1*, Imam Usmani1*, Félix Bussières1, Nicolas Sangouard1, Mikael Afzelius1, Hugues de Riedmatten1,2,3 & Nicolas Gisin1 intensity correlations between the two photons still exist after storage and retrieval. We then show, through a violation of a Bell inequality, that the storage process creates a light–matter entangled state. In addition, these results represent the first successful mapping of energy–time entangled photons onto a quantum memory. Our experiment consists of a coherent solid-state quantum memory and a source of entangled photons. A schematic of the experiment is shown in Fig. 1. The source is based on non-degenerate SPDC in a nonlinear waveguide pumped by continuous wave light at 532 nm. This yields energy–time entangled photons with the signal photon at Quantum memory e 3n 88 883-nm laser m AFC preparation AOM g Fibre optic switch aux PBS FR Nd:Y2SiO5 Etalon 532-nm laser Etalon Chopper 88 3 nm PPKTP waveguide 8 3 ,3 nm Grating 1 % Source of entangled photons FBG Si APD 10% Coincidence logic 50 m SSPD 90 Entanglement is the fundamental characteristic of quantum physics— much experimental effort is devoted to harnessing it between various physical systems. In particular, entanglement between light and material systems is interesting owing to their anticipated respective roles as ‘flying’ and stationary qubits in quantum information technologies (such as quantum repeaters1–3 and quantum networks4). Here we report the demonstration of entanglement between a photon at a telecommunication wavelength (1,338 nm) and a single collective atomic excitation stored in a crystal. One photon from an energy– time entangled pair5 is mapped onto the crystal and then released into a well-defined spatial mode after a predetermined storage time. The other (telecommunication wavelength) photon is sent directly through a 50-metre fibre link to an analyser. Successful storage of entanglement in the crystal is proved by a violation of the Clauser– Horne–Shimony–Holt inequality6 by almost three standard deviations (S 5 2.64 6 0.23). These results represent an important step towards quantum communication technologies based on solid-state devices. In particular, our resources pave the way for building multiplexed quantum repeaters7 for long-distance quantum networks. Although single atoms8,9 and cold atomic gases10–15 are currently some of the most advanced light–matter quantum interfaces, there is a strong motivation to control light–matter entanglement with more practical systems, such as solid-state devices16. Solid-state quantum memories for photons can be implemented with cryogenically cooled crystals doped with rare-earth-metal ions17, which have impressive coherence properties at temperatures below 4 K. They have the advantage of simple implementation because rare-earth-metal-doped crystals are widely produced for solid-state lasers, and closed-cycle cryogenic coolers are commercially available. Important progress has been made over the last years in the context of light storage into solid-state memories, including long storage times18, high efficiency19 and storage of light at the single photon level with high coherence and negligible noise19–23. Yet these experiments were realized with classical bright pulses or weak coherent states of light. Although this is sufficient to characterize the performance of the memory, and even to infer the quantum characteristics of the device19,20, it is not sufficient for the implementation of more sophisticated experiments involving entanglement, as required for most applications in quantum information science. For this purpose, it is necessary to store non-classical light, in particular individual photons that are part of an entangled state (generated, for example, through spontaneous parametric downconversion, SPDC), similar to previous demonstrations using electromagnetically induced transparency in cold atomic gases14,15. In addition, for quantum communication applications, the other part of the entangled state should be a photon at telecommunication wavelength in order to minimize loss during transmission in optical fibres. In this Letter, we report on an experiment in which a photon from an entangled pair is stored in a quantum memory based on a rare-earthmetal-doped crystal. More specifically, we show that non-classical InGaAs APD Filtering cavity Figure 1 | Experimental set-up. The experimental set-up can be divided into three parts: the Nd:Y2SiO5 crystal serving as quantum memory, the laser system for the preparation of the AFC in the crystal, and the source of entangled photons with associated spectral filtering. During the experiment we periodically switch between 15 ms of AFC preparation and frequency stabilization and a 15-ms measurement phase, in which single photons are stored. During the preparation, the comb structure is prepared by frequencyselective optical pumping of atoms from the ground state | gæ to the auxiliary state | auxæ using light from an 883-nm diode laser in combination with an acousto-optic modulator (AOM). The fibre optic switch is in the upper position, and the silicon avalanche photodiode (Si APD) is protected from the bright light by a chopper. During the measurement phase, the positions of switch and chopper are reversed. Now, photon pairs are generated in the periodically poled potassium titanyl phosphate (PPKTP) waveguide via SPDC. The two photons in a pair are spatially separated by a diffraction grating and then strongly filtered by two etalons, a cavity and a fibre Bragg grating (FBG). Photons at 883 nm are sent through the crystal in a double-pass configuration to increase the absorption probability, and are afterwards detected by the Si avalanche photodiode. Photons at 1,338 nm are directed towards a superconducting single photon detector (SSPD) located in another laboratory 50 m away. All relevant quantities are extracted from the coincidence statistics of the two detectors. Details of the frequency stabilization and the filtering system are given in the Methods. PBS, polarizing beam splitter; FR, Faraday rotator. 1 Group of Applied Physics, University of Geneva, CH-1211 Geneva 4, Switzerland. 2ICFO—Institut de Ciencies Fotoniques, Mediterranean Technology Park, 08860 Castelldefels, Barcelona, Spain. 3ICREA— Institució Catalana de Recerca i Estudis Avançats, 08015 Barcelona, Spain. *These authors contributed equally to this work. 5 0 8 | N AT U R E | VO L 4 6 9 | 2 7 J A N U A RY 2 0 1 1 ©2011 Macmillan Publishers Limited. All rights reserved LETTER RESEARCH the memory wavelength of 883 nm, and the idler photon at the telecom wavelength of 1,338 nm. Both photons initially have a spectral width of approximately 1.5 THz, a factor of 104 larger than the 120-MHz bandwidth of the memory. Hence, strong filtering is crucial14 to achieve signal-to-noise ratios sufficiently large to reveal the presence of entanglement during storage. After filtering, the signal photon is sent to the memory, and the idler photon is coupled into a fibre leading to a detector in another laboratory 50 m away. Owing to the low loss at telecommunication wavelengths, this distance could, in principle, be extended to several kilometres without significantly affecting the results presented here. The quantum memory is a 1-cm-long Y2SiO5 crystal, impuritydoped with neodymium ions having a resonance at 883 nm with good coherence properties23. It is based on a photon-echo-type interaction using an atomic frequency comb (AFC) (see ref. 24 and Supplementary Information). In an AFC, the absorption profile of the atomic ensemble is shaped into a comb-like structure by optical pumping. A photon is then, with some efficiency, absorbed and re-emitted into a well-defined spatial mode due to a collective rephasing of the atoms in the comb structure. The time of re-emission depends on the period of the comb and is predetermined. We have previously shown that this kind of memory can store multiple temporal modes23 and is therefore perfectly suited for storing energy–time entangled photons. For the work presented here, we have significantly improved the storage efficiency to obtain sufficiently large signal-to-noise ratios. Indeed, using a new optical pumping scheme for the preparation of the AFC (see Supplementary Information), the efficiency was increased by a factor of three for storage times below 200 ns, now reaching values up to 21% (see results below). In a first experiment we verified that the non-classical nature of the intensity correlations between the signal (883 nm) and idler (1,338 nm) modes is preserved after the storage and retrieval process. If we assume ð2Þ second-order auto-correlations of signal and idler gx (where x 5 ‘s’ ð2Þ for signal or ‘i’ for idler) satisfying 1ƒgx ƒ2, then non-classicality is ð2Þ proved by measuring a cross-correlation gsi ~Psi =Ps Pi greater than 2 (see ref. 25). Here, Ps (or Pi) is the probability of detecting a signal (or idler) photon, and Psi the probability of a coincidence detection (see Methods). ð2Þ We first measured gsi as a function of the pump power of the source, as shown in Fig. 2a. We find an optimum around a pump power ð2Þ ð2 Þ of 3 mW, where gsi <115 without the AFC memory, and gsi < 30 after a 25-ns storage, thus proving the quantum character of the storage (note that all results presented in this Letter are without any subtraction of background noise). The reduction in the cross-correlation with a b 35 140 the storage is due to limited efficiency (21%), which effectively increases the contribution of accidental coincidences stemming from dark counts and multiple pair emissions. Next, we measured the memory efficiency and the cross-correlation for different storage times, as shown in Fig. 2b and c. We now turn our attention towards a particular kind of quantum correlation, namely entanglement. By performing a two-photon quantum interference experiment, we show that the entanglement of the photon pair is preserved when the signal photon is stored in the crystal. Photon pairs generated by our source are energy–time entangled, that is, the two photons in a pair are created simultaneously to ensure energy conservation, but the pair-creation time is uncertain to within the coherence time of the pump laser. We wish to reveal the presence of this entanglement using a Franson-type set-up5. As detailed in the Supplementary Information, the correlations can be interpreted as stemming from local measurements performed on a post-selected time-bin entangled state: p1ffiffi2 ðjEs Ei izjLs Li iÞ, where the early and late time bins jEs,iæ and jLs,iæ are separated by a time of 25 ns set by the analysing interferometer (see Fig. 3a). In our experiment, however, the state of the signal photon is stored as a collective atomic excitation in the quantum memory before the measurement. Moreover, using a double AFC scheme20,23, the memory is used not only to store the entangled photon, but also to analyse it as part of the measurement. More precisely, the incident time-bins jEsæ and jLsæ are mapped to distinct AFC modes jEQMæ and jLQMæ, respectively (where subscript QM denotes quantum memory). Storage of the entangled signal photon then creates a light–matter entangled state: 1 pffiffiffi ðjEQM Ei izjLQM Li iÞ ð1Þ 2 The predetermined storage times of jEQMæ and jLQMæ are 75 ns and 50 ns, respectively. After absorption, both AFCs coherently re-emit the stored excitation into the same well-defined temporal and spatial mode with a relative phase Dws. This re-emission, followed by detection, constitutes the measurement of the state of the memory. The idler photon is measured using a fibre-based time-bin qubit analyser with a 25-ns delay and a relative phase Dwi between the short and long arms. The coincidence detection probability is given by: Psi !1zV cosðDws zDwi Þ where V is the visibility of interference. Figure 3b shows the measured coincidence rate as a function of Dws for two values of Dwi. The raw visibilities are V 5 (84 6 4)% and (78 6 4)%. Storage time 200 c 50 25 20 80 60 15 40 10 20 0 5 15 10 Pump power (mW) 20 5 100 ns 66 ns 100 50 ns 33 ns 50 40 25 ns 0 0 50 Figure 2 | Non-classical correlations and storage efficiency. a, Crossð2Þ correlation gsi as a function of the pump power incident on the wave guide. Data points shown were taken with an AFC memory storage time of 25 ns (brown square symbols), and for comparison, with the crystal prepared with a 120-MHzwide transmission window, that is, without AFC (blue circle symbols). The size of the coincidence window is 10 ns. b, Coincidence histograms for different predetermined storage times, vertically offset for clarity. For comparison, the lowest histogram was taken without AFC. The pump power was 3 mW. c, Cross- 100 150 Delay (ns) 200 Efficiency (%) 133 ns 150 gsi(2) with AFC gsi(2) without AFC 25 100 gsi(2) with AFC 30 Coincidences in one hour 200 ns 120 ð2Þ 250 30 20 15 10 5 0 0 20 50 100 150 200 10 0 0 50 100 150 Storage time (ns) 200 ð2Þ correlation gsi as a function of storage time with 10-ns coincidence window, extracted from b. For storage times up to 200 ns the correlations stay well above ð2Þ the classical limit of gsi ~2 (dashed line). The inset shows the storage efficiency for the same range of storage times. With increasing storage times, limiting factors in the storage medium degrade the comb shape and reduce the efficiency and cross-correlation (see Supplementary Information). However, the latter stays well above the classical limit for storage times up to 200 ns. Error bars show 61 standard deviation (s.d.). 2 7 J A N U A RY 2 0 1 1 | VO L 4 6 9 | N AT U R E | 5 0 9 ©2011 Macmillan Publishers Limited. All rights reserved RESEARCH LETTER Start SSPD 50% Faraday Δφi mirrors 50% Idler Coincidences in two hours 18 b Coincidence window for post-selection 16 14 80 12 10 8 Stop 6 4 2 0 –40 Source of entangled photons –20 0 20 40 Tstop – Tstart (ns) 60 80 100 Si APD Signal Coincidences in two hours a 60 40 20 EQM (75 ns) Es Ls Δφs 0 LQM (50 ns) Storage –240 –180 –120 Measurement –60 0 Δφs (°) 60 120 180 240 Figure 3 | Storage of photonic entanglement in a crystal. a, Franson-type set-up used to reveal the entanglement. A qubit analyser consisting of an unbalanced, fibre-based Michelson interferometer with 25-ns delay and relative phase Dwi is inserted before the SSPD used to detect the idler photon (see also Fig. 1). The signal photon is stored in the crystal, yielding a light– matter entangled state. The state of the memory is measured through reemission and detection of the photon in the time-window at zero-time delay (central peak) of the coincidence histogram (inset). This post-selects measurement on the entangled state of equation (1). The relative phase Dws can be reliably set to any desired value (see the Supplementary information). b, Number of coincidences in the central peak in two hours as a function of the relative phase Dws for two values of Dwi. The pump power was 5 mW, and the size of the coincidence window 10 ns. The solid and dashed lines result from fits to equation (2) and respectively give visibilities of V 5 (78 6 4)% and (84 6 4)%. The visibilities are mainly limited by the level of accidental coincidences (cross symbols). The fit also gives a difference between the two values of Dwi of 75u 6 10u. These values closely match settings necessary for a maximal violation of the CHSH inequality. Error bars are 61 s.d. Quantum entanglement can be revealed by a violation of the Clauser– Horne–Shimony–Holt (CHSH) inequality6. The possibility of violating this inequality, that is, of finding a CHSH parameter Sp .ffiffiffi2, can be inferred indirectly from a visibility larger than 1 2<70:7%. Nevertheless, we performed the measurements necessary for a direct violation of the inequality and obtained S 5 2.64 6 0.23. This proves the presence of entanglement between the idler photon and the matter qubit in the crystal, provided the effect of the memory on single photons is appropriately described as storage followed by measurement (see Supplementary Information). This description is correct within the theory of AFC memories24, which is supported by experiments storing weak coherent states of light11,21–23. Note also that we do not claim any demonstration of nonlocal correlations. Indeed, besides the usual locality and detection loopholes, here the measurement setting has to be chosen before the photonic qubit is mapped onto the crystal. This could have been avoided by adding an interferometer after the memory, the latter being used for storage only. We did not do so because we think that it is elegant and simple to use the memory also as a small quantum processor that performs the measurement. A particularly intriguing situation arises when post-selecting on the case where only jEsæ is stored in the crystal for 25 ns using a single AFC scheme, while jLsæ is directly transmitted. Indeed, the imbalance between the storage efficiency and the transmission probability offers a well-suited qubit analyser for a violation of the CHSH inequality using bases lying in the x–z plane of the Bloch sphere. We performed such a measurement and observed S 5 2.62 6 0.15 (see Supplementary Information). This implies that the initial photon–photon entangled state is mapped onto a state of the form: pffiffiffiffiffiffiffiffi gabs jEQM Ei izjLs Li i ð3Þ using optical fibres3. To achieve this long-term goal, several future advances are required. The user must be able to trigger the re-emission of the memory, whereas in our experiment the duration of the storage is pre-determined. We have proposed24 and demonstrated26 a method for achieving on-demand re-emission using so-called spin-wave storage. This has the additional benefit of allowing longer storage times owing to the more robust spin coherence. Another crucial aspect is the efficiency, which is directly linked to the optical depth of the material24. It can be increased by using longer crystals19 or optical cavities27,28. The creation of entanglement between a single photon and a macroscopic object—in this case a single collective atomic excitation delocalized over a 1-cm-long crystal—is fascinating in itself. Beyond its fundamental interest, we believe that our demonstration of storage of entanglement in a crystal represents an important step towards quantum repeaters based on solid-state quantum memories. We note that, parallel to this work, Saglamyurek et al. have demonstrated storage and retrieval of an entangled photon using a thuliumdoped lithium niobate waveguide29. where gabs is the absorption efficiency. This is an entangled state between a telecommunication-wavelength qubit and a light–matter hybrid qubit. We note that this kind of hybrid qubit is the key ingredient of an efficient quantum repeater protocol based on atomic ensembles and linear optics3. This work is part of the effort towards implementing a quantum repeater, which could provide a solution to the distance limit (due to intrinsic loss) for entanglement distribution and quantum cryptography METHODS SUMMARY Spectral filtering and detection. The bandwidth of the photon pairs is reduced by a factor of 104 in several steps. Pump, signal and idler photons are spatially separated by a diffraction grating (see Fig. 1). In combination with coupling into single-mode fibres, this reduces the bandwidth to tens of gigahertz. A subsequent passage through a Fabry–Perot cavity reduces the bandwidth of the idler photon to 45 MHz (corresponding to a coherence time of about 4 ns), and a fibre Bragg grating blocks all but one of the longitudinal cavity modes. The signal photon is filtered by two etalons with a linewidth of 600 MHz each, and different free spectral ranges. The detector efficiency is 8% for the idler photon with 10-Hz dark counts, and 30% with 100-Hz dark counts for the signal photon. Frequency stabilization. We must ensure, for the whole duration of a measurement, that the central frequency of the optical filtering system at 1,338 nm and of the AFC at 883 nm both satisfy the energy conservation of the SPDC process. To do this, a small fraction of the light at 883 nm is overlapped with the light of the 532-nm laser that pumps the PPKTP waveguide. This leads to the creation of light at 1,338 nm by difference frequency generation (DFG). Using this DFG signal, the frequency of the 532-nm light is adjusted such that the detection rate on a separate InGaAs avalanche photodiode (see Fig. 1) stays constant, which means that the 1,338-nm DFG light is in resonance with the filtering cavity. Long-term stability of the 883-nm laser itself is achieved by continuously referencing it to a Fabry–Perot cavity. 5 1 0 | N AT U R E | VO L 4 6 9 | 2 7 J A N U A RY 2 0 1 1 ©2011 Macmillan Publishers Limited. All rights reserved LETTER RESEARCH Full Methods and any associated references are available in the online version of the paper at www.nature.com/nature. Received 24 August; accepted 9 November 2010. Published online 12 January 2011. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 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Efficient light storage in a crystal using an atomic frequency comb. N. J. Phys. 12, 023025 (2010). 22. Sabooni, M. et al. Storage and recall of weak coherent optical pulses with an efficiency of 25%. Phys. Rev. Lett. 105, 060501 (2010). 23. Usmani, I., Afzelius, M., de Riedmatten, H. & Gisin, N. Mapping multiple photonic qubits into and out of one solid-state atomic ensemble. Nature Commun. 1, 12 (2010). 24. Afzelius, M., Simon, C., de Riedmatten, H. & Gisin, N. Multimode quantum memory based on atomic frequency combs. Phys. Rev. A 79, 052329 (2009). 25. Kuzmich, A. et al. Generation of nonclassical photon pairs for scalable quantum communication with atomic ensembles. Nature 423, 731–734 (2003). 26. Afzelius, M. et al. Demonstration of atomic frequency comb memory for light with spin-wave storage. Phys. Rev. Lett. 104, 040503 (2010). 27. Afzelius, M. & Simon, C. Impedance-matched cavity quantum memory. Phys. Rev. A 82, 022310 (2010). 28. Moiseev, S. A., Andrianov, S. N. & Gubaidullin, F. F. Efficient multimode quantum memory based on photon echo in an optimal QED cavity. Phys. Rev. A 82, 022311 (2010). 29. Saglamyurek, E. et al. Broadband waveguide quantum memory for entangled photons. Nature doi:10.1038/nature09719 (this issue). Supplementary Information is linked to the online version of the paper at www.nature.com/nature. Acknowledgements We thank R. Locher for help during the early stages of the experiment. We are grateful to A. Beveratos and W. Tittel for lending us avalanche photodiodes. This work was supported by the Swiss NCCR Quantum Photonics, the Science and Technology Cooperation Program Switzerland–Russia, as well as by the European projects QuRep and ERC-Qore. F.B. was supported in part by FQRNT. Author Contributions All authors contributed extensively to the work presented in this paper. Author Information Reprints and permissions information is available at www.nature.com/reprints. The authors declare no competing financial interests. Readers are welcome to comment on the online version of this article at www.nature.com/nature. Correspondence and requests for materials should be addressed to M.A. ([email protected]). 2 7 J A N U A RY 2 0 1 1 | VO L 4 6 9 | N AT U R E | 5 1 1 ©2011 Macmillan Publishers Limited. All rights reserved RESEARCH LETTER METHODS Spectral filtering and detection. The narrowband filtering of the SPDC photons consists of several steps (see Fig. 1). First, a diffraction grating spatially separates the pump, signal and idler photons and, in combination with coupling into singlemode fibres, reduces the bandwidth of the photons at 883 nm (or 1,338 nm) to 90 GHz (or 60 GHz). Photons at 1,338 nm are then coupled through a Fabry–Perot cavity with linewidth 45 MHz and free spectral range of 23.9 GHz. Subsequently, a fibre Bragg grating with 16 GHz bandwidth ensures that only a single longitudinal cavity mode remains. Filtering one of the photons in the pair is the same as filtering the photon pair as a whole, because energy conservation guarantees that photons measured in coincidence have the same bandwidth. However, uncorrelated photons would then contribute significantly to the accidental coincidences. Therefore, complementary filtering at 883 nm was necessary. To do this, we used one solid and one air-spaced etalon, both with bandwidths around 600 MHz. Different free spectral ranges of 42 and 50 GHz eliminate uncorrelated longitudinal modes. Additionally, outside the 120-MHz bandwidth of the AFC, the absorption of the crystal with an inhomogeneous linewidth of 6 GHz provides a final filtering step. We used detectors with 30% detection efficiency and approximately 100 Hz dark counts at 883 nm, and detectors with 8% detection efficiency and approximately 10 Hz at 1,338 nm. Together with a transmission of the filtering system for the signal (or idler) photon of 45% (or 14%), and 4% (or 14%) for the remainder of the optical set-up, we reached an overall detection efficiency of 0.5% (or 0.15%) (see also Supplementary Information). These numbers could, in principle, be significantly improved through optimized optical alignment, the use of antireflection-coated elements, and so on. Frequency stabilization. In the experiment, coincidence rates are typically a few per minute. With accumulation times thus reaching several hours, a high degree of frequency stability of the lasers and filtering elements is indispensable. In particular, frequency drifts of the AFC preparation laser with respect to the pump laser of the SPDC source have to be eliminated. Otherwise, the photon-pair frequencies v883 1 v1338 5 v532 imposed by energy conservation in the SPDC would not simultaneously match the centre of the AFC and that of the filtering system at 1,338 nm. Drifts were eliminated using the following method. First, the long-term stability of the 883-nm laser was dramatically increased by locking it to a temperature-stabilized Fabry–Perot cavity. Second, during the 15-ms preparation cycle, we injected a fraction of the 883-nm light into the waveguide. The frequency of the light created at 1,338 nm via difference frequency generation (DFG) was tuned by controlling the frequency of the pump laser at 532 nm. Using a side-of-fringe technique, we could then lock the frequency of the DFG signal to the transmission peak of the filtering cavity. As a result, long-term frequency deviations between the centre of the AFC structure and the filtered photon pairs were reduced to about 1 MHz over several hours. For measurements involving the unbalanced Michelson interferometer for the idler photon, the phase of the interferometer was also stabilized using the highly coherent DFG light. Photon correlations in SPDC. Neglecting the exact frequency dependence, the state of the photons created in the SPDC process is described by pffiffiffi j0s , 0i iz pj1s , 1i izOð pÞ, where the subscript ‘s’ (or ‘i’) indicates the signal (or idler) mode at 883 nm (or 1,338 nm). Here, the pair creation probability p is assumed to be small and proportional to the pump power. In such a state, the signal and idler modes individually exhibit the statistics of a classical thermal field, ð2Þ that is, their auto-correlations are gx ~2 for x 5 s or i. We stress, however, that the criterion for non-classicality of the cross-correlation that we used, namely ð2Þ ð2Þ gsi ~Psi =Ps Pi w2, requires only that 1ƒgx ƒ2, which is always fulfilled by non-degenerate photon pairs created through SPDC. In practice, Psi (or PsPi) is determined by the number of coincidences in a certain time window centred on (or away from) the coincidence peak. For low pump powers, the measured crosscorrelation is usually limited by detector dark counts, and at high pump powers it is reduced by the contribution of multiple pairs, that is, higher-order terms in p. ©2011 Macmillan Publishers Limited. All rights reserved SUPPLEMENTARY INFORMATION doi:10.1038/nature09662 Supplementary Information: Quantum Storage of Photonic Entanglement in a Crystal Christoph Clausen, Imam Usmani, Félix Bussières, Nicolas Sangouard, Mikael Afzelius, Hugues de Riedmatten, and Nicolas Gisin Group of Applied Physics, University of Geneva, CH-1211 Geneva 4, Switzerland (Dated: November 4, 2010) ATOMIC FREQUENCY COMB The atomic frequency comb (AFC) memory is a photon-echo based scheme, where the absorption profile of the crystal is shaped into a comb-like structure using optical pumping, thus making it possible to take full advantage of the high atomic density in a doped crystal, despite of inhomogeneous broadening [1]. When a photon enters the crystal, with a spectral bandwidth covering a large part of the AFC spectrum, it is absorbed, provided that the optical depth is sufficient. After the absorption, the photon is stored in a single atomic excitation delocalized over all the atoms, corresponding to a collective Dicke type state, N j cj e−ikzj ei2πδj t |g1 · · · ej · · · gN , (1) where zj is the position of atom j (note that it is sufficient to consider only the forward propagating spatial mode), k is the wave-number of the light field, δj the detuning of the atom with respect to the laser frequency and the amplitudes cj depend on the frequency and on the spatial position of the particular atom j, which can be in the ground state |gj or the excited state |ej . In the AFC the distribution of atomic detunings δj is periodic with period ∆. After a time ts = 1/∆, the components of the state (1) are in phase and lead to the reemission of a photon in a well defined spatial mode k with high probability. Here we used as storage medium a 1 cm long Y2 SiO5 crystal doped with Nd3+ ions to a concentration of 30 ppm, and cooled to 3 K. This crystal has already shown a high multimode capacity [2] with storage and retrieval efficiency between 6% (for ts = 100 ns) and 1% (ts = 1.5 µs). In the work presented here we have improved the storage efficiency in the range ts = 25 to 200 ns by creating comb peaks with an approximate square shape. This has been shown to be the optimal shape in terms of storage efficiency [3]. Note also that the AFC structure is prepared by using an incoherent optical pumping technique consisting of simultaneously sweeping the laser frequency and modulating its intensity, as also used in [4]. The light used for preparation and storage is focussed onto the crystal, with a beam waist diameter of 40 µm. As a result we observe an efficiency of 21% for ts = 25 ns and 12% for ts = 100 ns. The decay of the efficiency with longer storage time FIG. S1. Experimental absorption spectrum of an AFC prepared by an incoherent optical pumping technique. The peaks have a width larger than the effective linewidth of the material, so it is possible to use the whole available optical depth and to have almost squareshaped peaks, which optimizes the rephasing. The dashed lines delimit the 120 MHz bandwidth of the AFC. (thus for closer comb spacing ∆) is due to limitations in the coherence properties of the material, which deteriorates the comb shape for closely spaced peaks. We refer to Ref. [2] for a more detailed discussion on this optical pumping issue. For the shortest storage time of 25 ns, however, this problem was almost negligible. Hence, the obtained efficiency is mainly limited by the absorption depth of the crystal. An example of a close to optimal comb for ts = 25 ns is shown in Fig. S1. Using the measured comb as an input to a numerical Maxwell-Bloch simulation gave an efficiency close to the measured one. CHSH TEST AS WITNESS FOR LIGHT-MATTER ENTANGLEMENT Here, we discuss the way we create and measure the entanglement between a photon at telecommunication wavelength and a collective atomic excitation stored in a crystal. Our experiment starts with a source of photon pairs (signal-idler) entangled in energy and time [5]. To reveal such an entanglement, the usual way consists of using a w w w . n a t u r e . c o m / NATURE | 1 RESEARCH SUPPLEMENTARY INFORMATION 2 Franson setup (see Fig. S2a) involving two unbalanced Mach-Zehnder interferometers, one for each of the signal and idler photons. By choosing appropriately the arm length of the interferometers to avoid single photon interferences, but such that the delay is shorter than the coherence time of the pump, the uncertainty in the creation time leads to a quantum interference between the two path combinations short-short and long-long. The non-local character of this interference can be revealed through the violation of a Bell inequality by properly choosing the phases and the beamsplitters of the interferometers [6]. Since our experiment presents strong similarities with the Franson experiment, let us analyze in detail the violation of the CHSH inequality [7]. Instead of energytime entanglement, consider for simplicity time-bin entanglement (obtained by post-selection in the Franson setup) involving two particular modes, the early E and late L time bins separated by the delay of the interferometers [8], 1 √ (|Es Ei + |Ls Li ). 2 (2) It is instructive to describe the interferometers in the Franson setup as a tool allowing (i) to transfer the entanglement between temporal modes to an entanglement of spatial modes and (ii) to measure these modes in the appropriate bases. Let us detail these two points separately. (i) First, the signal mode is sent on a 1 × 2 optical switch such that the entanglement (2) becomes 1 √ (|longs , Ei + |shorts , Li ) 2 (3) where long and short describe two spatial modes corresponding to the long and short arms of the interferometer. Note that, in practice, the optical switch is usually replaced by a 50/50 beamsplitter followed by a postselection, as in Fig. S2a. (ii) Once the time-bin encoding is mapped into a spatial encoding, the signal mode is measured, e.g. on the equator of the Bloch sphere, by choosing the relative phase of the short and long modes to be either ∆φ1s = π/4 or ∆φ2s = 3π/4 and then by combining the short and long modes on a 50/50 beamsplitter. This corresponds to the measurement bases 1 1 X1 = √ (σx + σy ), X2 = √ (−σx + σy ). 2 2 where σx and σy are Pauli matrices (see Fig. S3a). If the same transformation and a similar measurement (with either ∆φ1i = π/2 or ∆φ2i = 0) are performed on the idler photon, corresponding to the measurement bases (Fig. S3a) Y1 = σy , Y2 = σx , 2 | WWW . NATURE . COM / NATURE a) spatial ps bs bs bs bs energy time measurement measurement b) ps measurement crystal - photon ps bs bs memory energy time measurement FIG. S2. a) The Franson setup. The energy-time entanglement carried by a photon pair is first mapped onto a spatial entanglement. A projective measurement is then performed using a phase shifter (ps), a beamsplitter (bs) and a single-photon detector (half-circle). b) Our experiment. A crystal, with two AFCs, replaces one of the interferometers. The energy-time entanglement is mapped onto crystal-photon entanglement, i.e. an entanglement involving two collective atomic excitation modes and two spatial photonic modes. the CHSH correlators are given by 1 Ē(X1 , Y1 ) = Ē(X2 , Y1 ) = − √ 2 1 −Ē(X1 , Y2 ) = Ē(X2 , Y2 ) = − √ 2 leading to a maximal CHSH parameter of S = |Ē(X1 , Y1 ) + Ē(X2 , Y1 ) − Ē(X1 , Y2 ) + Ē(X2 , Y2 )| √ = 2 2, which violates the local upper bound of S = 2. We now analyze our experiment when using the double AFC scheme [2], where one of the Mach-Zehnder interferometers is replaced by a light-matter interface with two atomic frequency combs (Fig. S2b), which differ by their periodicity. This interface is equivalent to two distinct atomic ensembles capable of storing, with close-toequal efficiency, a temporal mode for either τ1 = 50 ns or τ2 = 75 ns. Consider the state (2) as starting point. By sending the signal mode to the crystal and by focusing on memory outputs resulting either from a long storage of an early mode or from a late photon stored for a short time, we post-select the following light-matter entangled state, 1 √ (|EQM Ei + |LQM Li ) . 2 (4) where EQM (LQM ) denotes a single collective atomic excitation (see Eq. (1)) for the comb associated with the SUPPLEMENTARY INFORMATION RESEARCH 3 τ2 (τ1 ) storage time. This is the heart of our result, the entanglement between a photon and a single atomic excitation delocalized over a 1 cm long crystal. To measure this entanglement, we perform a projection onto 1 √ (LQM | + e−i∆φs EQM |) 2 a b (5) by sending the photon emitted coherently by the two combs to the detector. This requires that both AFCs can reemit the stored excitation with close-to-equal probability. This emission merely plays the role of the beamsplitter measurement in the Franson setup. To perform the CHSH test previously described the relative phase ∆φs is reliably set to either ∆φ1s or ∆φ2s . This is done by shifting the comb patterns with respect to the central frequency of the signal photon [1]. This phase is very stable over the storage time, resulting in a direct measurement of the matter qubit with high fidelity. Note that, on the idler side, the fiber interferometer is actively stabilized to maintain a desired relative phase. In the experiment, the projection (5) is achieved by post-selecting events in the central coincidence peak (see Fig. 3a in the main text). In the figure, the left sidepeak corresponds to the detection of |Es that was stored in |LQM , and the right sidepeak to the detection of |Ls that was stored in |EQM . Since we had only one detector on each side, we had to add a π-phase shift every time we wanted to access the second outcome of the chosen basis. For this reason, we had to make 4 measurements per correlator, i.e. 16 for the whole Bell test. For a pump power of 3 mW we obtained Ē(X1 , Y1 ) = 0.68 ± 0.12, Ē(X2 , Y1 ) = 0.79 ± 0.10, Ē(X1 , Y2 ) = −0.60 ± 0.10, Ē(X2 , Y2 ) = 0.57 ± 0.14, leading to S = 2.64 ± 0.23. Note that the pump power used for this measurement is lower than the 5 mW used for the visibility measurements (Fig. 3b in the main text), resulting in a higher signal-to-noise ratio (c.f. Fig. 2a in the main text). Here, the uncertainties are standard deviations related to the poissonian statistics of the coincidence events. This is a clear violation of the CHSH inequality that proves, regardless of experimental noise, that the idler photon was entangled with a collective excitation in the crystal. FIG. S3. Measurement bases for the violation of the CHSH inequality. Arrows indicate signal (idler) bases X1 and X2 (Y1 and Y2 ). (a) For the violation of the CHSH inequality, all four bases lie on the equator of the Bloch sphere with the appropriate angles between them. (b) The hybrid qubit represents a state with unequal weights between the basis states |EQM and |Ls , and optimal violation can only be obtained on a plane including the poles of the Bloch sphere. spanned by the states |EQM , |Ls , associated to the signal mode. Contrary to the treatment above, we only use a single AFC. In the post-selection we consider only those cases where the early signal mode is stored in the AFC mode |EQM for a time of exactly τ = 25 ns. The detection of this mode, which we will hereafter refer to as echo, is thus made indistinguishable from the detection of the late mode |Ls , which is directly transmitted through the memory. Taking the echo efficiency ηecho and the transmission probability ηtrans into account, this corresponds to a projection onto the vector cos θ/2Ls | + e−i∆φs sin θ/2EQM |, (6) ηtrans ηecho with cos θ/2 = ηtrans +ηecho , and sin θ/2 = ηtrans +ηecho . The phase factor is controlled via the AFC structure [1] and is chosen to be either ∆φ1s = 0 or ∆φ2s = π. This corresponds to measuring X1 = sin θ σx + cos θ σz , X2 = − sin θ σx + cos θ σz . Fifty metres away, the idler photon, at telecommunication wavelength, is projected either onto the z-direction (corresponding to the operator Y1 = σz ) by measuring the time of arrival at the detector or onto the x-direction (Y2 = σx ) using a Michelson interferometer. One then finds for the correlators, CHSH INEQUALITY WITH A HYBRID QUBIT Ē(X1 , Y1 ) = Ē(X2 , Y1 ) = cos θ, Ē(X1 , Y2 ) = −Ē(X2 , Y2 ) = sin θ. The violation of the CHSH inequality using the hybrid qubit is done in analogue to the violation described above. However, we now consider the entanglement between the idler photon and a hybrid light-matter qubit, so that the CHSH polynomial S = 2 cos θ√+ 2 sin θ is √ −ηecho maximized to 2 2 for cos θ = ηηtrans = 22 , i.e. for a trans +ηecho w w w . n a t u r e . c o m / NATURE | 3 RESEARCH SUPPLEMENTARY INFORMATION 4 ratio between the echo and transmitted pulses 1 ηecho , ≈ ηtrans 5.8 (7) which corresponds to the bases indicated in Fig. S3b. In the experiment, the AFC structure is modified to approximate this requirement (we measured ηtrans ≈ 0.36, ηecho ≈ 0.05, giving a ratio of 1/7.2). Under the assumption that the marginals are the same for the idler photon, independent of the result of the measurement on the signal mode, we measured Ē(X1 , Y1 ) = 0.68 ± 0.05 Ē(X2 , Y1 ) = 0.71 ± 0.06 the contributions of the two terms in Eq. (8), thereby yielding optimal CHSH violation from a non-maximally entangled state. SPECTRAL FILTERING OF PHOTON PAIRS Here we give an overview of bandwidth and efficiency for the elements of the optical setup, where the efficiency for optical elements equals the peak transferred intensity normalized to incoming intensity. The free spectral ranges (FSR) of etalons and cavity, as well as the dark count rate (DC) of the detectors, are given in parenthesis. Wavelength Element Ē(X1 , Y2 ) = 0.63 ± 0.09 883 nm Ē(X2 , Y2 ) = −0.60 ± 0.09 leading to S = 2.62 ± 0.15, a violation of the CHSH inequality by more than 4 standard deviations. This clearly shows that when the early mode was stored in our crystal, the two qubit state involving the hybrid qubit and the idler qubit was entangled. Taking a closer look, it might at first be surprising that the violation of the Bell inequality can be maximal, since directly after the absorption the system is described by the post-selected, non-maximally entangled state (neglecting noise) √ 1 (α|EQM Ei + |Ls Li ) , 1 + α2 90 GHz 600 MHz 600 MHz 120 MHz – 70% 80% 80% – 4% – 120 MHz 30% 0.5% Grating Filter Cavity (FSR 24 GHz) Fiber Bragg Grating Fiber coupling, circulator, polarization controller, fiber beam splitter and mirrors Detector (DC 10 Hz) Total 60 GHz 45 MHz 16 GHz – 90% 30% 50% 14% – 45 MHz 8% 0.15% (8) where α is related to the absorption efficiency ηabs of the √ quantum memory by α = ηabs . (The absorption efficiency corresponds to the absorption by the comb peaks and does not include the absorption by residual atoms whose resonance frequencies fall between the peaks, so that ηtrans = 1 − ηabs .) The explanation is that the memory based measurement is a generalized measure2 ment. More precisely, assuming that ηecho = ηabs η [9], the measurement consists of two generalized measurements {X1 , X2 } made from {Πs+1 , Πs−1 , 1 − Πs+1 − Πs−1 }, i.e. projections onto the non-orthogonal vectors √ √ Πs+1 : ηtrans Ls | + e−i∆φs ηηabs EQM |, ηtrans √ EQM |, Πs−1 : e−i(∆φs +π) ηηabs Ls | + ηabs where φs is controlled from the AFC structure to be either φs,1 = 0◦ or φs,2 = 180◦ for X1 and X2 respectively. Under the fair sampling assumption where inconclusive results are discarded, one can only take successful projections on Πs+1 and Πs−1 into account. If one assigns the value +1 (−1) to a conclusive projection into Πs+1 (Πs−1 ), one finds random marginals and a Bell violation √ of 2 2 provided that the condition (7) is fulfilled. This is analogue to the distillation of entanglement reported in ref. [10] where a non unitary filtering process equalizes 4 | WWW . NATURE . COM / NATURE 1338 nm Bandwidth Efficiency Grating Solid etalon (FSR 42 GHz) Air-spaced etalon (FSR 50 GHz) AFC Fiber coupling, polarization controller, fiber optic switch, mirrors, lenses and windows Detector (DC 100 Hz) Total [1] Afzelius, M., Simon, C., de Riedmatten, H. & Gisin, N. Multimode quantum memory based on atomic frequency combs. Phys. Rev. A 79, 052329 (2009). [2] Usmani, I., Afzelius, M., de Riedmatten, H. & Gisin, N. Mapping multiple photonic qubits into and out of one solid-state atomic ensemble. Nat Commun 1, 1–7 (2010). [3] Bonarota, M., Ruggiero, J., Le Gouët, J.-L. & Chanelière, T. Efficiency optimization for atomic frequency comb storage. Phys. Rev. A 81, 033803 (2010). [4] Lauritzen, B. et al. Telecommunication-wavelength solidstate memory at the single photon level. Phys. Rev. Lett. 104, 080502 (2010). [5] Franson, J. D. Bell inequality for position and time. Phys. Rev. Lett. 62, 2205–2208 (1989). [6] Tittel, W., Brendel, J., Zbinden, H. & Gisin, N. Violation of bell inequalities by photons more than 10 km apart. Phys. Rev. Lett. 81, 3563–3566 (1998). [7] Clauser, J. F., Horne, M. A., Shimony, A. & Holt, R. A. Proposed experiment to test local hidden-variable theories. Phys. Rev. Lett. 23, 880–884 (1969). [8] Brendel, J., Gisin, N., Tittel, W. & Zbinden, H. Pulsed energy-time entangled twin-photon source for quantum communication. Phys. Rev. Lett. 82, 2594–2597 (1999). [9] η describes decoherence in the memory due to the finite finesse of the comb. Note that this equation can be used in the limit where re-absorption is negligible (see Ref. [1]), which is the case in our experiment. [10] Kwiat, P. G., Barraza-Lopez, S., Stefanov, A. & Gisin, N. Experimental entanglement distillation and ‘hidden’ non-locality. Nature 409, 1014–1017 (2001). news & views light sources in optoelectronic circuits for communications, opens up completely new avenues of research in nanoscale sensing or simply gives us a better understanding of how light and matter interact. ❒ Rupert F. Oulton is at the Department of Physics at Imperial College London, The Blackett Laboratory, Experimental Solid State Physics, Imperial College London, Prince Consort Road, London SW7 2AZ, UK e-mail: [email protected] References 1. 2. 3. 4. Oulton, R. F. Mater. Today 15, 592–600 (January/February 2012). Berini, P. & De Leon, I. Nature Photon. 6, 16–24 (2012). Khurgin, J. B. & Sun, G. Appl. Phys. Lett. 100, 011105 (2012). Wang, F. & Shen, Y. R. Phys. Rev. Lett. 97, 206806 (2006). 5. Johnson, P. B. & Christy, R. Phys. Rev. B 6, 4370–4379 (1972). 6. Ma, R.‑M. et al. Nature Mater. 10, 110–113 (2010). 7. Noginov, M. A. et al. Nature 460, 1110–1112 (2009). 8. Hill, M. T. et al. Opt. Express 17, 11107–11112 (2009). 9. Genov, D. A. et al. Phys. Rev. B 83, 245312 (2011). 10.Gontijo, I. et al. Phys. Rev. B 60, 11564–11567 (1999). 11.Ma, R.‑M. et al. Laser Photon. Rev. http://dx.doi.org/10.1002/ lpor.201100040 (2012). 12.Ding, K. et al. Phys. Rev. B 85, 041301 (2012) QUANTUM OPTICS Linking crystals with a single photon Linking distant quantum memories with light has been a goal of the quantum information community for many years. A team at the University of Geneva has now demonstrated that memories made from rare-earth-ion-doped crystals can be connected using a single photon. Steven Olmschenk T he ability to transmit quantum states over large distances is a goal that is being hotly pursued by researchers across the globe. Although distances as great as 144 km have been successfully spanned by single photons1, connections over arbitrary distances will probably require the use of quantum memories (capable of storing quantum states) to allow repeaters to overcome the detrimental losses associated with long transmission distances. Although simple amplification of a quantum signal using a conventional optical amplifier is ruled out by the no-cloning theorem2, which prohibits copying an unknown quantum state, quantum memories make it possible to break the link up into several shorter segments. The deployment of quantum memories may not only enable long-distance quantum communication for ultrasecure information transfer, but might also be used to perform distributed quantum information processing (or quantum cloud computing), allowing access to computational problems that are inaccessible through classical approaches. For these reasons, significant effort is currently being directed towards linking quantum memories with photons to establish quantum connections between distant quantum memories. Entanglement is widely viewed as an essential resource for creating such a quantum connection in both quantum communication and quantuminformation processing. A particularly useful type of entanglement-creating operation is one that is ‘heralded’, whereby a ‘heralding event’ (such as the detection of a photon) signals the creation of entanglement between the quantum memories. Heraldedentanglement operations between quantum memories have been realized in a number of a b Figure 1 | Classical and quantum shell games. a, In a classical shell game, a single pea is hidden under one of two shells. b, In a quantum shell game, the pea is under both shells simultaneously in the form of a superposition — this is analogous to the form of entanglement used in the work of Usmani et al. physical implementations, including atomic ensembles3 and trapped atomic ions4. Now, writing in Nature Photonics, Usmani et al. experimentally demonstrate how a heralded single photon can create entanglement between two rare-earth-ion-doped crystals separated on the centimetre scale5. One of the advantages of this system is that established fabrication techniques for these solid-state devices may assist scaling for future advanced operations in quantum information. The basic form of entanglement employed by Usmani et al. is a bit like a quantum version of a shell game (Fig. 1). In a classic shell game, a single ‘pea’ is placed under one of multiple shells and the player guesses where the pea resides. The quantum version of this game would involve a single pea that is in a superposition of being under all shells simultaneously. Once a measurement is made, the quantum superposition collapses and the pea is found under one of the shells. The entangled state produced by the experiment presented in this issue of Nature Photonics is analogous to this situation, where a single photon replaces the pea, and crystals replace the shells. Usmani et al. began by producing a pair of photons through the standard quantumoptics technique of spontaneous parametric downconversion. Because the generation of this photon pair is probabilistic, detecting one of the two photons heralds the presence of the other, thus giving researchers a guarantee that they are sending just one photon into their experiment. The latter of these photons is the ‘pea’ in the quantum shell game. Usmani et al. sent this photon to a beamsplitter and directed the two output paths towards two separate crystals (Fig. 2). The photon is absorbed by one of the crystals, where it excites one of the many optically active rare-earth ions confined therein. However, the inherent lack of information regarding which path the photon travelled — and thus by which crystal it was absorbed — results in the creation of an entangled state between the two crystals. One of the two crystals ought to contain an excitation, but the excitation is in an entangled superposition state between the two crystals — this is analogous to the pea being under both shells simultaneously. Usmani et al. were able to store this superposition state for up to 33 ns. Because directly measuring the excitation of the crystal is a significant experimental challenge, Usmani et al. verified the entanglement by using an echo technique to write the stored entangled state back to a photon. Correlation measurements between the two paths of this re-emitted photon established a lower bound for the quality of NATURE PHOTONICS | VOL 6 | APRIL 2012 | www.nature.com/naturephotonics © 2012 Macmillan Publishers Limited. All rights reserved. 221 news & views the entanglement between the two crystals. Using these measurements, Usmani et al. were able to demonstrate that their single heralded photon indeed created an entangled state between the two crystals. Of course, there are plenty of challenges that must be overcome before this system can be used to establish a large quantum network. One of the biggest challenges — one faced by all such probabilistic entangling schemes demonstrated so far — will be increasing the overall efficiency of the system, which limits the rate of entanglement generation and thus the rate of information transfer. Another necessary improvement is increasing the distance between the crystals — Usmani et al. employed a spacing of just 1.3 cm to avoid the use of multiple cryostats. Other issues include improving the quality of the entanglement and lengthening the storage duration of the quantum state. However, researchers have already demonstrated impressive progress towards tackling many of these issues, including demonstrations of long-lived coherences in crystals6 for potentially storing received information and proposed architectures to increase the distance between crystal quantum memories7. Moreover, the intriguing possibility of fabricating structures Crystals Beamsplitter Heralded photon Mirror Figure 2 | Creating entanglement between two quantum memories. A heralded single photon (the ‘pea’) first passes through a beamsplitter. The outputs of the beamsplitter are directed to the two crystals (the ‘shells’), where the photon is absorbed to produce a single excitation. The lack of information regarding the path of the photon creates an entangled state between the two crystals. directly into the crystals to improve the overall efficiency of the system adds to the appeal of these solid-state devices. Combining such advances with the spatial and temporal multiplexing abilities of these crystals, which might also be used to improve the entanglement rate, will make this system a strong contender for the scalable technology needed to implement large quantum networks. ❒ Steven Olmschenk is at the Joint Quantum Institute, a research partnership between the University of Maryland and the National Institute of Standards and Technology, College Park, Maryland 20742, USA. e-mail: [email protected] References 1. Ursin, R. et al. Nature Phys. 3, 481–486 (2007). 2. Wootters, W. K. & Zurek, W. H. Nature 299, 802–803 (1982). 3. Sangouard, N., Simon, C., de Riedmatten, H. & Gisin, N. Rev. Mod. Phys. 83, 33–80 (2011). 4. Duan, L.‑M. & Monroe, C. Rev. Mod. Phys. 82, 1209–1224 (2010). 5. Usmani, I. et al. Nature Photon. 6, 234–237 (2012). 6. Fraval, E., Sellars, M. J. & Longdell, J. J. Phys. Rev. Lett. 95, 030506 (2005). 7. Simon, C. et al. Phys. Rev. Lett. 98, 190503 (2007). VIEW FROM... SPIE PHOTONICS WEST 2012 Photons, neurons and wellbeing Techniques for the targeted optical stimulation of neurons may offer new ways to tackle medical problems such as heart defects, epilepsy, Parkinson’s, blindness and hearing loss. David Pile U sing light to control biological processes is a relatively new joint research direction of the biological and physical sciences. Yet there is already a great deal of motivation for developing such technologies, as made clear from the “Photons and Neurons” sessions at SPIE Photonics West 2012, held in San Francisco, USA, on 21–26 January 2012. Over a billion people worldwide suffer from brain disorders such as stroke, depression, migraine, epilepsy, Parkinson’s, chronic pain and blindness. According to Edward Boydon, an expert on brain disorders from the Massachusetts Institute of Technology Media Lab, few of these disorders are effectively treatable by drugs or neurosurgical procedures. Furthermore, the treatments that do exist are often partial or present undesirable side effects. Part of the problem 222 is the complexity of the dense neural circuits in the brain. “Ideally we would be able to hone in on the precise circuits that can best contribute to the remedy of disease, and then use those circuits as drug targets, or as targets for neurosurgeons to implant electrodes to reduce symptoms,” Boydon explained. “To do this, we invented a new technology that allows us to control specific cells embedded within dense neural circuits with light. We do this by exploiting the photosynthetic and photosensory proteins found in many biological species, which convert light into electrical energy.” Installing photosensory cells in a particular region of the brain modifies the neuronal cells to respond to light while leaving the surrounding cells unaffected. Boydon’s team deliver the genes that encode for these proteins to targeted neurons by employing gene therapy viral vectors currently used for gene therapy trials in humans. They deliver the light by inserting compact optical probes such as optical fibres attached to small LEDs or lasers into the brain. Boydon and colleagues are now working on three-dimensional (3D) lightdelivery devices — arrays of waveguides — to achieve improved control over 3D neural circuits. Light delivery methods in the field of optogenetics (the stimulation of nerve cells with light) was one of the key discussion points at the meeting. Partrick Degenaar from Newcastle University in the UK explained that the rise of optogenetics has many exciting applications to neuroprosthesis. Future developments will require researchers to find ways of efficiently delivering light at the required intensity and depth — particularly because biological tissue is a strong scatterer NATURE PHOTONICS | VOL 6 | APRIL 2012 | www.nature.com/naturephotonics © 2012 Macmillan Publishers Limited. All rights reserved. LETTERS PUBLISHED ONLINE: 4 MARCH 2012 | DOI: 10.1038/NPHOTON.2012.34 Heralded quantum entanglement between two crystals Imam Usmani, Christoph Clausen, Félix Bussières, Nicolas Sangouard, Mikael Afzelius* and Nicolas Gisin Quantum networks must have the crucial ability to entangle quantum nodes1. A prominent example is the quantum repeater2–4, which allows the distance barrier of direct transmission of single photons to be overcome, provided remote quantum memories can be entangled in a heralded fashion. Here, we report the observation of heralded entanglement between two ensembles of rare-earth ions doped into separate crystals. A heralded single photon is sent through a 50/50 beamsplitter, creating a single-photon entangled state delocalized between two spatial modes. The quantum state of each mode is subsequently mapped onto a crystal, leading to an entangled state consisting of a single collective excitation delocalized between two crystals. This entanglement is revealed by mapping it back to optical modes and by estimating the concurrence of the retrieved light state5. Our results highlight the potential of crystals doped with rare-earth ions for entangled quantum nodes and bring quantum networks based on solid-state resources one step closer. Quantum entanglement challenges our intuition about physical reality. At the same time, it is an essential ingredient in quantum communication6, quantum precision measurements7 and quantum computing8,9. In quantum communication, photons are naturally used as carriers of entanglement using either freespace or optical-fibre transmission. However, even with ultralowloss telecommunication optical fibre, the transmission probability decreases exponentially with distance, limiting the achievable communication distance to a few hundred kilometres10. A potential solution is to use quantum repeaters2 based on linear optics and quantum memories3,4 with which the entanglement distribution time scales polynomially with the transmission distance, provided entanglement between quantum memories in remote locations can be heralded. In this context, one can more generally consider prospective quantum networks1 where nodes generate, process and store quantum information, while photons transport quantum states from site to site and distribute entanglement over the entire network. An essential step towards the implementation of such potential technologies is to create entanglement between two quantum memories in a heralded manner3,4. Experimental observation of heralded entanglement between two independent atomic systems for quantum networks has been achieved using cold gas ensembles involving either two distinct ensembles5 or two spatial modes in the same ensemble11–14. Complete elementary links of a quantum repeater (based on heralded entanglement) have been implemented in cold gas systems15,16 and with two trapped ions17. For the realization of scalable quantum repeaters, solid-state devices are technologically appealing18. In this context, important results have already been obtained by using crystals doped with rare-earth ions (REs) as quantum memories. These results have demonstrated light storage times greater than one second19, a storage efficiency of 69% (ref. 20) and quantum storage of 64 independent optical modes in one crystal21 (see also ref. 22). Recent achievements23,24 include the storage, in a single RE-doped crystal, of photonic time-bin entanglement generated through spontaneous parametric downconversion (SPDC). Heralded entanglement between two RE-doped crystals using an SPDC source, a common resource in quantum optics, represents an important step towards the implementation of quantum repeater architectures based on solid-state devices25. Here, we present the observation of heralded quantum entanglement between two neodymium ensembles doped in two yttrium orthosilicate crystals (Nd3þ:Y2SiO5) separated by 1.3 cm (Fig. 1 and Methods). A nonlinear optical waveguide is pumped to produce photon pairs by means of SPDC. The resulting idler (1,338 nm) and signal (883 nm) photons are strongly filtered to match the working bandwidth of the crystals, yielding a coherence time of 7 ns. In the limit where the probability of creating a single pair is much smaller than one, the detection of an idler photon heralds the presence of a single signal photon. By sending the latter through a balanced beamsplitter, one heralds, √ neglecting optical losses, a single-photon entangled state26 1/ 2(|1lA |0lB + |0lA |1lB ) between the two spatial output modes A and B. In each of the modes a crystal acts as quantum memory, denoted MA or MB. Upon absorption of the single photon27, the detection of an idler photon heralds the creation of a single collective √ excitation delo calized between the two crystals, written as 1/ 2(|WlA |0lB + |0lA |WlB ). Here, |WlA(or |WlB) is the Dicke-like state created by the absorption of a single photon in MA (or MB). To determine the presence of entanglement between the memories, we use a photon-echo technique based on an atomic frequency comb27–31 that reconverts the collective excitation into optical modes A and B after a preprogrammed storage time of 33 ns (with a total efficiency of 15%). The resulting fields can then be probed using single-photon detectors to reveal heralded entanglement between the memories. As the entanglement cannot increase through local operations on the optical modes A and B, the entanglement of the retrieved light fields provides a lower bound for entanglement between the two memories. The photonic state retrieved from the memories is described by a density matrix r, including loss and noise, expressed in the Fock state basis. To reveal entanglement in this basis, a tomographic approach based on single-photon detectors can be used5,13,14. Specifically, from the values of the heralded probabilities pmn of detecting m [ {0,1} photons in mode A and n [ {0,1} in mode B, combined with the magnitude of the coherence between these modes, a lower bound on the concurrence C of the detected fields is obtained by √ C ≥ max(0, V( p01 + p10 ) − 2 p00 p11 ) (1) Group of Applied Physics, University of Geneva, CH-1211 Geneva 4, Switzerland. * e-mail: [email protected] 234 NATURE PHOTONICS | VOL 6 | APRIL 2012 | www.nature.com/naturephotonics © 2012 Macmillan Publishers Limited. All rights reserved. NATURE PHOTONICS LETTERS DOI: 10.1038/NPHOTON.2012.34 1 cm 1.3 c Heralding detector MB B Filtering 1,33 8n m 88 3 A m MA FR nm BS DM 5 Detector 2 PBS m n 32 PPKTP waveguide Memory preparation Detector 1 Switch Figure 1 | Experimental set-up. Quantum memories MA and MB are implemented using neodymium ions doped into yttrium ortho-silicate crystals (Nd3þ:Y2SiO5) separated by 1.3 cm and cooled to 3 K using a cryostat (see ref. 23 for details). The total efficiency of each memory (used in a double-pass configuration) is 15%. A fibre-optic switch is used to alternate between a 15-ms-long preparation of the two neodymium ensembles as atomic frequency combs on the 4I9/2 4F3/2 transition, followed by attempts at entanglement creation for another 15 ms. The preparation includes a 4 ms waiting time to avoid fluorescence from atoms left in the excited state. For entanglement creation, continuous-wave light at 532 nm is coupled into a periodically poled (PP) KTP waveguide, leading to the production of pairs of photons at wavelengths of 883 nm and 1,338 nm through SPDC. Photons from each pair are separated on a dichroic mirror (DM) and frequency filtered to below the 120 MHz bandwidth of the quantum memories. Detection of an idler photon at 1,338 nm (using a low-noise superconducting single photon detector38) heralds the presence of a signal photon at 883 nm. The signal photon now traverses the switch, a polarizing beamsplitter (PBS) and a Faraday rotator (FR), before a 50/50 beamsplitter (BS) creates single-photon entanglement between spatial modes A and B. This entanglement is, upon absorption, mapped onto crystals MA and MB. After a preprogrammed storage time of 33 ns, the photons are re-emitted and pass through the beamsplitter again. Depending from which output mode of the beamsplitter they emerge, they either reach detector 2 (silicon-based single-photon detector) or are rotated in polarization by the Faraday rotator and reflected by the polarizing beamsplitter towards detector 1. (concurrence is a measure of entanglement, ranging from 0 for a separable state to 1 for a maximally entangled state). The term V is the interference visibility obtained by recombining optical modes A and B on a 50/50 beamsplitter and is directly proportional to the coherence between the retrieved fields in modes A and B. To obtain a large concurrence, one should maximize V (the coherence) and p10 þ p01 (the probability to detect the heralded photon), and minimize p00 and p11 (the probabilities of detecting separable states |0lA|0lB and |1lA|1lB stemming from a lost signal photon and from two signal photons, respectively). To estimate V, p00 , p10 , p01 and p11 , we used the set-up of Fig. 1 (see Methods for details). For p11 , in particular, we used two different methods, as described in the following. In the first method, we use a direct measurement of threefold coincidences, that is, involving all three detectors (see Supplementary Information). With a pump power of 16 mW, we obtained C (MLE) ¼ 6.3+3.8 × 1025 using a maximum likelihood estimation (MLE) of the threefold coincidence probability, and C (CE) ¼ 3.9+3.8 × 1025 using a more conservative estimation (CE). Both estimations yield a concurrence that is greater than 0 by at least one standard deviation, which is consistent with the presence of entanglement between the two crystals. This measurement required 166 h, a period in which two threefold coincidences were observed. The prohibitively long integration time of this method prevented us from attempting it with lower pump powers (that is, for a lower probability of creating more than one pair). Hence, to study how the concurrence changes with pump power, we used a second method based on twofold coincidences, which we now describe. In the second method, p11 is estimated using the fact that our SPDC source produces a state that is very close to a two-mode-squeezed state (TMSS) (see Methods and Supplementary Information). We therefore assume that the measured zero-time cross-correlation g s,i is consistent with a TMSS and can be written as g s,i = 1 + 1/p, where p and p 2 are interpreted as the probabilities of creating one and two photon pairs, respectively (where p ≪ 1). In practice, g s,i is taken as the average value of gAs,i and gBs,i, where gAs,i (or gBs,i) is obtained by blocking mode B (or mode A). From the relationship between p and p11 , we find p11 = 4p10 p01 g s,i − 1 (2) In the Methods and Supplementary Information, we present additional measurements that support our assumption about our source, and show that it leads to a lower bound on the concurrence. We performed a series of measurements for several values of the pump power, which is proportional to p provided p ≪ 1. Figure 2a shows the interference of the delocalized single photon retrieved from the memories. The visibility does not depend on the pump power, and has an average value of 96.9+1.5%. Figure 2b shows the measured g s,i , from which it can be seen that reducing the pump power increases the cross-correlation, as expected. Using additional measurements, we estimated the transmission loss, memory efficiency, dark count probability and pair creation probability of our set-up. These values were then used in a theoretical model (shaded region in Fig. 2b) that is in excellent agreement with the measured values of g s,i , providing additional evidence that the estimated value of p11 yields a lower bound on the concurrence (see Supplementary Information). Figure 2c shows the lower bound on the concurrence for all pump powers, calculated using equations (1) and (2). The concurrence decreases with pump power because p11 increases, but all other terms in equation (1) depend on photon loss only and hence are approximately constant. Nevertheless, the concurrence remains larger than zero for all used pump powers, consistent with heralded entanglement between the atomic ensembles inside the two crystals. The results of the measurement of the concurrence based on threefold coincidences are plotted in Fig. 2c and agree, within uncertainty, with the results of the method based on measurement of the cross-correlation. The observed values of the concurrence are lower bounds on the amount of entanglement of the detected fields and are almost entirely determined by optical loss. Factoring out the detector inefficiency and interferometer loss yields a lower NATURE PHOTONICS | VOL 6 | APRIL 2012 | www.nature.com/naturephotonics © 2012 Macmillan Publishers Limited. All rights reserved. 235 LETTERS NATURE PHOTONICS a 1.0 0.9 Coincidences Visibility 800 0.8 0.7 0.6 600 400 200 0 0.5 0.25 0.30 0.35 0.40 Phase (a.u.) 0.45 0.4 b 0 2 4 6 8 10 Pump power (mW) 12 14 16 0 2 4 6 8 10 Pump power (mW) 12 14 16 0 2 4 6 8 10 Pump power (mW) 12 14 16 30 25 g—s, i 20 15 10 5 0 c 1.6 Concurrence (10−4) 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 Figure 2 | Results. a, Visibility as a function of pump power. Visibility is approximately constant with an average of 96.5+1.2% (green shaded region). Inset: visibility curves at 16 mW measured with detectors 1 (blue filled circles) and 2 (red open squares) (15 min acquisition time per point; error bars are smaller than the symbols). The different amplitudes result from non-uniform loss after recombination of modes A and B on the beamsplitter. The visibilities agree, within uncertainty, to the fits. b, Zerotime cross-correlation gs,i as a function of pump power. The decreasing values agree well with a theoretical model (shaded region; see Supplementary Information). c, Lower bound on the concurrence estimated using the cross-correlation measurement as a function of pump power (blue filled circles), calculated using equations (1) and (2), and the mean visibility of a. The concurrence decreases with pump power, as expected, but remains positive up to 16 mW. The shaded region corresponds to the model in b. All measured values of p10 , p01 and p11 are given in the Supplementary Information. All values are based on raw counts (that is, without subtraction of dark counts and accidental coincidences) and were obtained with coincidence detection windows of 10 ns. Uncertainties are obtained assuming Poissonian detection statistics. The lower bound on the concurrence at 16 mW obtained from measured threefold coincidences using either the MLE (green open diamond) or the CE (green filled diamond) are also shown (horizontally offset for clarity). 236 DOI: 10.1038/NPHOTON.2012.34 bound of 0.01 for the concurrence of the field retrieved just after the crystals (see Methods). The high interference visibility indicates that the stored entanglement does not significantly decohere during the 33 ns, and the coherent nature of the storage endures for longer times, as shown previously using storage of weak coherent states21,28. Increasing the storage time does, however, lower the cross-correlation function g s,i , as measured in ref. 23, and this should consequently lower the concurrence of the stored entanglement (this is essentially due to the decrease of the memory efficiency with increasing storage time, as shown in the Supplementary Information and in ref. 23). With Nd3þ:Y2SiO5 one could, in principle, store entanglement for up to 1 ms (ref. 21). A promising approach to go beyond these limits, and to allow on-demand retrieval of the stored entanglement, is to implement spin-wave storage27 to increase the storage times (as demonstrated recently29 in Pr3þ:Y2SiO5), and to place the crystal inside an impedance-matched cavity to increase the efficiency32,33. Such improvements are necessary for the development of a quantum repeater based on solid-state quantum memories, and are the subject of current research. In conclusion, we have reported an experimental observation of heralded quantum entanglement between two separate solid-state quantum memories. We emphasize that although the entangled state involves only one excitation, the observed entanglement shows that the stored excitation is coherently delocalized among all the neodymium ions in resonance with the photon, meaning 1 × 1010 ions in each crystal. Our results demonstrate that RE ensembles, naturally trapped in crystals, have the potential to form compact, stable and coherent quantum network nodes. Moreover, single-photon entanglement is a simple form of entanglement that can be used for teleportation34 and entanglement swapping operations3 and can also be purified using linear optical elements35. It is also a critical resource in several proposals for quantum repeaters3,25,36 as it is less sensitive to transmission loss and detector inefficiencies4. This, however, is counterbalanced by the extra requirement of phase stabilization over long distances, and the duplication of the resources for post-selecting entanglement shared between two photons, which is needed to implement qubits measurements4. Our experimental approach is based on solid-state devices, the key components being the PPKTP chip (the photon source) with an integrated waveguide and the crystal memory. We believe that this approach opens up possibilities for the integration of components, such as frequency filtering directly on the chip37 or waveguide quantum memories24. The prospect of combining solidstate photon sources and quantum memories is therefore attractive for practical future quantum networks. One important challenge in this context is to create entanglement between two remote solids in a heralded way using two distant sources of photon pairs and a central station performing a single-photon Bell state measurement25, that is, the realization of an elementary link for quantum repeaters. Methods Experimental set-up and concurrence estimation. To reveal entanglement in the Fock state basis, one cannot resort to violating a Bell inequality given solely inefficient and noisy single-photon detectors. Instead, quantum state tomography using single-photon detectors was developed5,13,14. Implicit to this method are the assumptions that (i) the creation of more than two pairs is negligible and (ii) the off-diagonal elements of r with different number of photons vanish (this is valid because no local oscillator providing an aphase reference was used; ref. 5). To estimate V, p00 , p10 and p01 , we used the set-up of Fig. 1 in the following way. First, the visibility V was measured by allowing the re-emitted delocalized photon to interfere with itself using a balanced Michelson interferometer for which the phase was actively stabilized. We then blocked spatial mode B to estimate p10 by summing the number of detections on detectors 1 and 2, conditioned on a heralding signal. This is justified, as the probability of creating two photon pairs is much smaller than the probability of creating a single one. Probability p01 is estimated similarly by blocking mode A instead of B. Alternatively, we could also estimate the sum p10 þ p01 directly by randomizing the phase of the interferometer, with both arms unblocked (further clarifications and justifications on these measurements are NATURE PHOTONICS | VOL 6 | APRIL 2012 | www.nature.com/naturephotonics © 2012 Macmillan Publishers Limited. All rights reserved. NATURE PHOTONICS LETTERS DOI: 10.1038/NPHOTON.2012.34 given in the Supplementary Information). Then, p00 is estimated through normalization of the total probability, p00 þ p10 þ p01 ≈ 1, which is justified by the fact that p11 ≪ p10 þ p01 ≪ p00. The estimation of p11 using the cross-correlation, which is motivated by the results of ref. 13, assumes that all the observed detections stem from a TMSS. In the Supplementary Information, we show that this assumption is conservative and that it yields a lower bound for the concurrence. Using the cross-correlation to estimate p11 requires only the measurement of twofold coincidences, rather than threefold coincidences as for the other method. For our specific set-up, this resulted in a reduction of the measurement time by a factor of 106 to achieve a similar statistical confidence on the concurrence. Moreover, this method requires no physical modifications to the optical circuit to measure the different components of the retrieved fields, which simplifies its implementation. In the Supplementary Information, we present additional measurement that provide evidence that our source produces a state that is very close to an ideal TMSS, and we summarize the results here. We measured the second-order autocorrelation of the signal (2) (or idler) mode without storage to be g(2) s,s (0) ¼ 1.81(2) (or gi,i (0) ¼ 1.86(9)), which is very close to the maximal value of 2 associated with the thermal photon statistics (the lower observed value can be entirely attributed to the finite temporal resolution of our detectors). We also measured the zero-time second-order autocorrelation function of the heralded signal photon just before storage and (2) obtained gs,s|i (0) ¼ 0.061(4) for a pump power of 8 mW, which is consistent with p ≪ 1. The observed values of the concurrence are lower bounds on the amount of entanglement of the detected fields and are almost entirely determined by optical loss. To see this, we first note that when both the multi-pair creation probability and the total transmission probability h of thesignal photon are small, the concurrence is approximately given by C ≈ h(V − 2/ g s,i − 1). At 8 mW of pump power, we measured gs,i ≈ 10, p00 ¼ 0.9997831(71) and p11 ¼ 5.18(40) × 1029, and hence C ≈ 0.3h. With h ≈ p10 þ p01 ¼ 2.2 × 1024, we see that C ≈ 6.6 × 1025 ≪ 0.3. Using this, an estimate of the concurrence of the fields retrieved just after the crystals is obtained by noticing that the transmission h is the product of the probability to find the signal photon inside the fibre per heralding signal (20%), the memory efficiency (15%), the transmission in the interferometer (2.4%) and detector efficiency (30%). Factoring out the detector inefficiency and the interferometer loss yields a lower bound of 0.01 for the concurrence of the field retrieved just after the crystals. All results were derived from raw counts, that is, without the subtraction of dark counts and accidental coincidences, and were obtained with coincidence detection windows of 10 ns. We note that there is no measurable temporal overlap between stored and transmitted photons, and that the transmitted photons do not contribute to the entanglement (see Supplementary Information for details, where histograms showing the temporal profiles of detected photons are shown). Received 24 August 2011; accepted 27 January 2012; published online 4 March 2012 References 1. Kimble, H. J. The quantum internet. Nature 453, 1023–1030 (2008). 2. Briegel, H.-J., Dür, W., Cirac, J. I. & Zoller, P. Quantum repeaters: the role of imperfect local operations in quantum communication. Phys. Rev. Lett. 81, 5932–5935 (1998). 3. Duan, L.-M., Lukin, M. D., Cirac, J. I. & Zoller, P. Long-distance quantum communication with atomic ensembles and linear optics. Nature 414, 413–418 (2001). 4. Sangouard, N., Simon, C., de Riedmatten, H. & Gisin, N. Quantum repeaters based on atomic ensembles and linear optics. Rev. Mod. Phys. 83, 33–80 (2011). 5. Chou, C. W. et al. 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This work was supported by the Swiss National Centres of Competence in Research (NCCR) project ‘Quantum Science Technology (QSIT)’, the Science and Technology Cooperation Program Switzerland– Russia, the European Union FP7 project 247743 ‘Quantum repeaters for long distance fibre-based quantum communication (QUREP)’ and the European Research Council Advanced Grant ‘Quantum correlations (QORE)’. F.B. was supported in part by le Fond Québécois de la Recherche sur la Nature et les Technologies. Author contributions All authors conceived the experiment. I.U., C.C. and F.B. performed the measurements. I.U., C.C., F.B., N.S. and M.A. analysed the data. All authors contributed to the writing of the manuscript. I.U., C.C. and F.B. contributed equally to this work. Additional information The authors declare no competing financial interests. Supplementary information accompanies this paper at www.nature.com/naturephotonics. Reprints and permission information is available online at http://www.nature.com/reprints. Correspondence and requests for materials should be addressed to M.A. NATURE PHOTONICS | VOL 6 | APRIL 2012 | www.nature.com/naturephotonics © 2012 Macmillan Publishers Limited. All rights reserved. 237 SUPPLEMENTARY INFORMATION doi: 10.1038/nphoton.2012.34 Supplementary information for “Heralded quantum entanglement between two crystals”quantum entanglement between two Supplementary information for “Heralded crystals” Imam Usmani, Christoph Clausen, Félix Bussières, Nicolas Sangouard, Mikael Afzelius, and Nicolas Gisin Group of Applied Physics, University of Geneva, Chemin de Pinchat 22, CH-1211 Geneva 4, Switzerland Group of Applied Physics, University of Geneva, (Dated: January 26, 2012) Chemin de Pinchat 22, CH-1211 Geneva 4, Switzerland (Dated: January 26, 2012) Imam Usmani, Christoph Clausen, Félix Bussières, Nicolas Sangouard, Mikael Afzelius, and Nicolas Gisin I. OVERVIEW I. OVERVIEW This appendix provides details on the method we used to This estimate the threefold probability p11used usappendix providescoincidence details on the method we ing either the cross-correlation g or by a direct meas,i to estimate the threefold coincidence probability p11 ussurement thecross-correlation threefold coincidence probability. Specifing eitherofthe gs,i or by a direct meaically, we present in section II a model describing our surement of the threefold coincidence probability. Specifsetup composed of the source of photon pairs and ically, we present in section II a model describing the our memories, and includes the effect of darkpairs counts. setup composed of the source of photon and We the provide evidence that the the method on the measurememories, and includes effectbased of dark counts. We ment of evidence gs,i to estimate yields based a lower on the provide that thep11 method onbound the measureconcurrence at detection and, consequently, on the ment of gs,i to estimate p11 yields a lower bound onconthe currence of the excitation stored inonthe concurrence at delocalized detection and, consequently, thequancontum memories. Finally, in section III, stored we give on currence of the delocalized excitation in details the quanour method to estimate p using threefold coincidences, 11 tum memories. Finally, in section III, we give details on we the uncertainties calculated and we ourdiscuss methodhow to estimate p11 using are threefold coincidences, show the results. we discuss how the uncertainties are calculated and we show the results. II. A. CONCURRENCE USING THE CROSS-CORRELATION II. CONCURRENCE USING THE CROSS-CORRELATION Characterization of the source of photon pairs A. Characterization of the source of photon pairs Let us begin by a description of the source of photonLet pairs we used our experiment. Photonof pairs us begin by ain description of the source phoare produced by spontaneous parametric downconversion ton pairs we used in our experiment. Photon pairs (SPDC), yielding a two-modeparametric squeezed state [1]: are produced by spontaneous downconversion (SPDC), yielding a +∞ two-mode squeezed state [1]: cosh−2 r +∞ tanh2n r |ni ns ni ns |. cosh−2 rn=0 tanh2n r |ni ns ni ns |. (A1) (A1) n=0 Here, ni and ns correspond to the number of photons in the idler andn signal modes (and are both equal to n). Here, ni and s correspond to the number of photons The coherences between terms(and withare different number of in the idler and signal modes both equal to n). pairs are lost due to the lack of a shared phase reference. The coherences between terms with different number of The r can be lack written αPpump ,reference. where α pairsparameter are lost due to the of a as shared P phase is the downconversion efficiency and is the con-α pump The parameter r can be written as αPpump , where tinuous wave pump power. It is convenient to consider is the downconversion efficiency and Ppump is the con2 the parameter p = tanh r, such the probability to tinuous wave pump power. It isnthat convenient to consider 2(1 − p)p , which corresponds to create n pairs is given by the parameter p = tanh r, such that the probability to acreate thermal photon number When the pump n pairs is given by (1distribution. − p)pn , which corresponds to power is small such that r, p 1, then the probabilities a thermal photon number distribution. When the pump of creating one or twothat pairsr,are, a very approxpower is small such p with 1, then thegood probabilities 2 imation, given by p and p , respectively. In this situation of creating one or two pairs are, with a very good approxwe have given by p and p2 , respectively. In this situation imation, we have (A2) p = tanh r2 ≈ r2 = αPpump (p 1). p = tanh r2 ≈ r2 = αPpump nature photonics | www.nature.com/naturephotonics (p 1). (A2) The signal and idler modes of the two-mode squeezed state, individually, also exhibit therThe when signalconsidered and idler modes of the two-mode squeezed mal statistics. Hence, a measurement of their zero-time state, when considered individually, also exhibit ther(2) (0)zero-time should second-order functionof gtheir mal statistics.auto-correlation Hence, a measurement yield a value of 2 [1]. In the limit whereg (2) p (0) 1, this should second-order auto-correlation function can be tested directly using a Hanbury Brown & Twiss yield a value of 2 [1]. In the limit where p 1, this setup obtained by using inserting a 50/50Brown beam &splitter can be[2]tested directly a Hanbury Twiss after the frequency filter of the signal field and,splitter simisetup [2] obtained by inserting a 50/50 beam larly, of the idler field. Note that the thermal statistics after the frequency filter of the signal field and, simiregime experimentally accessible herethermal becausestatistics the colarly, ofisthe idler field. Note that the herence time of the filtered signal and idler fields (0.9 regime is experimentally accessible here because theand co7herence ns, corresponding to 350signal and 43 MHz retime of the filtered and idlerlinewidths, fields (0.9 and spectively) are larger to than temporal of the 7 ns, corresponding 350the and 43 MHzresolution linewidths, refree-running single photon detectors used for detection spectively) are larger than the temporal resolution of the (less than 350single ps for photon both thedetectors signal and idler free-running used fordetectors). detection Hence, the detection system approximately selects a sin(less than 350 ps for both the signal and idler detectors). gle temporal mode of the SPDC process. Hence, the detection system approximately selects a singleThe temporal modeauto-correlation of the SPDC process. measured of the idler field, (2) (τ ), measured is shown on Fig. A1-a. Atofτ the = 0,idler we have gi,iThe auto-correlation field, (2) (2) ggi,i (0) = 1.86(9) with a 320-ps temporal binning. The (τ ), is shown on Fig. A1-a. At τ = 0, we have i,i (2) points are compared against a theoretical curve obdata gi,i (0) = 1.86(9) with a 320-ps temporal binning. The tained assuming a 350 ps detection and acurve 43 MHz data points are compared against a jitter theoretical obwide spectrum (full width at half maximum). The tained assuming a 350 ps detection jitter and a 43agreeMHz ment is good, despite the large observed fluctuations. In wide spectrum (full width at half maximum). The agreeparticular, the fluctuations give the impression that the ment is good, despite the large observed fluctuations. In data points the around the maximum on average lower particular, fluctuations give theare impression that the than the theoretical curve. These fluctuations are bedata points around the maximum are on average lower lieved to be statistical, but they also be are caused than the theoretical curve. Thesecould fluctuations beby an underestimation of the temporal resolution. The lieved to be statistical, but they could also be caused (2) autocorrelation of the signal gs,s (τresolution. ), is shownThe on by an underestimation of thefield, temporal (2)g (2) (0) = 1.81(2), Fig. A1-b and has maximum value gofs,s s,s), is shown on autocorrelation ofathe signal field, (τ (2) in with the theoretical ob= 1.81(2), Fig.excellent A1-b andagreement has a maximum value of gs,s (0) curve tained assuming a 350 MHz wide (full with at half maxin excellent agreement with the theoretical curve obimum) spectrum.a In cases, tained assuming 350both MHz widethe (fullvalue withat atmaximum half maxis slightly lower than 2; this is attributed entirely to the imum) spectrum. In both cases, the value at maximum finite temporal resolution of our detection. Furthermore, is slightly lower than 2; this is attributed entirely to the we note that theresolution value at maximum is lowerFurthermore, than in the finite temporal of our detection. idler case simply because the temporal spectrum we note that the value at maximum is lower thanisinnarthe rower, hence accentuating thetemporal impact of the finite residler case simply because the spectrum is narolution on thisaccentuating reduction. Overall, the ofmeasured rower, hence the impact the finiteautorescorrelations are consistent with the assumption thatautoour olution on this reduction. Overall, the measured source produces a state of the Eq. (A1). that our correlations are consistent withform the of assumption source produces a state of the form of source Eq. (A1). Another important parameter of the is the zerotime auto-correlation of the signal conditioned thezerodeAnother important parameter of the source isonthe (2) (0). This quantifies the tection of an idler photon, g time auto-correlation of the signal s,s|i conditioned on the desub-poissonnian nature of theg (2) heralded signal field, hence tection of an idler photon, s,s|i (0). This quantifies the its single-photon character [1]. For a two-mode squeezed sub-poissonnian nature of the heralded signal field, hence its single-photon character [1]. For a two-mode squeezed1 © 2012 Macmillan Publishers Limited. All rights reserved. supplementary information a doi: 10.1038/nphoton.2012.34 2 2.0 Memory g (2) i,i (τ ) 1.8 Pair source 1.6 not reemitted ( : absorbed) Decoherence 1.4 Loss Not absorbed 1.2 1.0 5 10 0 Delay τ (ns) 5 −20 −15 −10 −5 b 0 15 20 2.0 g (2) s,s (τ ) 1.8 1.6 1.4 1.2 1.0 −10 −5 10 FIG. A1. Measurements of second-order auto-correlation functions of a, the idler photon (1338 nm), and b, the signal photon (883 nm) before the crystals. The values at zero delay in a and b are close to the ideal value of 2. The red lines are theoretical curves based on the spectra of the optical filtering elements and the jitter of the detection system. Data was accumulated for 72 hours with a pump power of 16 mW in a, and for 12 hours at 8 mW in b. Error bars are due to Poisson counting statistics. (2) state, the relation between gs,s|i (0) and the parameter FIG. A2. Model of our experimental setup. The source produces idler-signal pairs described by a two-mode squeezed state. The memory is represented by three beamsplitters (to account for the absorption, the decoherence and the reemission respectively). Each temporal mode after the memory ds comes either from a signal photon emitted at an early time, then stored and later retrieved (mode se ), or from a photon produced at a late time and directly transmitted through the memory (mode s ). To take both contributions into account, a virtual source creating idler-signal pairs in a late time bin has been introduced. The correlations between the modes ds and di are characterized through the measurement of the second-order cross-correlation function. later retrieved; or from a photon created at a later time and not absorbed by the medium. Hence, we need to consider two temporal modes. The storage process is a linear operation, mapping the light field onto an atomic state. In our model, it can be represented by a beamsplitter with a transmission corresponding to the absorption probability, c.f. Fig. A2. Another beamsplitter is introduced to account for the decoherence, inherent to the storage process [3]. Finally, a last beamsplitter accounts for the reemission of the photon. It is identical to the first one and combines the two temporal modes into a single temporal mode ds . The output of the last beamsplitter is detected to access the cross-correlation function 2 p = tanh r is (2) gs,s|i (0) ḡs,i = 2 =2− ≈ 4p for p 1. (1 + p)2 (A3) For p 1, this conditional zero-time auto-correlation of the signal photon can be measured using a Hanbury (2) Brown & Twiss setup. We measured gs,s|i (0) = 0.061(4) with a pump power of 8 mW. This shows the very good single photon character of the heralded signal field, and hence that p 1. B. Modelling the source and the quantum memory Let us continue with the description of our model by focusing on the quantum memory. For now we will consider only the case of a single memory, which corresponds to the situation with one arm of the interferometer blocked. Due to non-unit absorption efficiency, each signal photon detection after the memory stems either from a photon created at an early time, stored in the medium, and 2 d†i di d†s ds d†i di d†s ds (A4) between detection on the early idler mode di and signal mode ds . Note that the overall transmission (in intensity) of the three beamsplitters corresponds to the efficiency of the storage and retrieval processes ηecho , while the reflection of the last beamsplitter is the transmission of the atomic ensemble ηtrans . Using the Heisenberg picture d†s ds = 0|U † d†s U U † ds U |0 where √ ηecho cosh r se + sinh r i†e √ + ηtrans cosh r s + sinh r i† U † ds U = leading to d†s ds = (ηtrans + ηecho ) sinh2 r. nature photonics | www.nature.com/naturephotonics © 2012 Macmillan Publishers Limited. All rights reserved. 3 supplementary information doi: 10.1038/nphoton.2012.34 Furthermore, we have leading to TABLE A1. Conditional probabilities p01 , p10 , p11 and the ratio ηtrans /ηecho for various pump powers. U † di U = cosh r ie + sinh r s†e d†i di = sinh2 r. Following similar lines, we find d†i di d†s ds = sinh2 r sinh2 r (ηtrans + ηecho ) + cosh2 r ηecho . Therefore, the cross-correlation function between the idler-signal modes is given by 1+ 1 tanh2 r 1 + ηtrans ηecho . C. Note that the effect of the memories is to add the term (1 + ηηtrans ), which decreases the cross-correlation. One echo can further take the detector noise into account by adding an additional source of noise with a Poissonian photon distribution. We finally find gs,i = 1 + 2 tanh r 1 + 1 ηtrans ηecho + Pump power 1 mW 2 mW 3 mW 4 mW 8 mW 13 mW 16 mW Mean ηdark pc (A5) where ηdark is the probability to get a dark count within the detection window and pc is the conditional probability of detecting the signal photon retrieved from the memory. We recall that for small pump powers, tanh2 r ≈ αPpump . To support the model presented above, we performed additional measurements to determine the values of α, ηtrans /ηecho , and pc . First, we estimated α = 2.71(8) × 10−3 pairs/mW (per 10 ns within a 43 MHz wide spectral window) from a measurement of the cross-correlation as function of pump power with one crystal prepared as a transmission window (no storage) while the path of other crystal was blocked. Then, ηtrans /ηecho was measured from the ratio of the probabilities of detecting the heralded signal photon when it is either transmitted through the memory (no storage), or stored and retrieved from the memory. This measurement was performed twice – once for each arm of the interferometer (with the other arm blocked) – and the outcomes were averaged. The conditional probability pc was taken as the sum of probabilities p10 and p01 of detecting the heralded signal photon retrieved from the memories (see main text and Table A1). Finally, we measured a dark count probability of ηdark = 2 × 10−6 per 10 ns detection window for the signal mode detector. Idler photons were detected using a superconducting nanowire single photon detector with a dark count rate of 20 Hz. This low dark count rate is neglected in our calculations. Using these parameters, we compared the measured cross-correlation ḡs,i with values of gs,i predicted by Eq. (A5). The comparison is shown on Fig. 2B in the main text. We find excellent agreement within the statistical uncertainties, thereby supporting our model for the source and the memory. p01 (10−4 ) p10 (10−4 ) p11 (10−9 ) 1.04(14) 1.193(75) 0.952(72) 1.105(72) 1.185(51) 1.247(56) 1.146(47) 1.123(30) 0.82(12) 0.809(63) 0.878(70) 0.902(66) 0.984(50) 1.131(52) 1.175(48) 0.957(27) 1.33(30) 1.63(19) 1.61(20) 2.82(31) 5.18(40) 8.79(66) 9.56(64) ηtrans ηecho 2.84(33) 3.03(17) 2.59(17) 3.35(19) 3.13(12) 2.86(11) 2.748(93) 2.936(69) Lower bound on the concurrence from the cross-correlation We now provide evidence, based on our model, that the method we used to estimate the threefold coincidence probability from the measurement of the crosscorrelation leads to an underestimation of the concurrence. For this, we consider a hypothetical situation where photons are detected directly in modes A and B, that is, before the modes are recombined on the beamsplitter (see Fig. 1 of the main text). Hence, singlephoton interference and two-photon bunching do not arise. The overall transmission for each memory is given by p01 and p10 , including detection (we recall that in our experiment, p10 is the probability per heralding signal to detect a photon at detector 1 or 2 when the arm of crystal B is blocked, and similarly for p01 ). We then compare the threefold coincidence probability pth 11 predicted by our model with the one obtained from the method based on the measured cross-correlation ḡs,i . Let us first concentrate on pth 11 . Using our model for the source and the memory, we can predict the threefold coincidence probability should be given by ηtrans ηdark th p11 = 4p10 p01 αPpump 1 + + . (A6) 2ηecho pc In the derivation of this formula we considered the probability to detect an idler photon to be very small (it is of order 10−3 per 10 ns). Furthermore, only contributions of single and double pairs were included. This approximation is well supported since the single pair creation probability p is of order 10−3 per 10 ns. The above expression should be compared with the threefold coincidence probability estimated using ḡs,i = 1 + 1/p, i.e. assuming detections stem from a two-mode squeezed state. Combining this method with our model (Eq. A5) yields p11 = 4p10 p01 ḡs,i − 1 = 4p10 p01 αPpump ηtrans 1+ ηecho ηdark + . pc One can see that the ratio ηtrans /ηecho enters in p11 with- 3 nature photonics | www.nature.com/naturephotonics © 2012 Macmillan Publishers Limited. All rights reserved. supplementary information doi: 10.1038/nphoton.2012.34 4 out the prefactor of one-half, such that p11 > pth 11 . From Eq. (1) of the main text, we directly see that an overestimation of p11 leads to a lower bound on the concurrence at detection, and hence on the concurrence of the entanglement stored in the memories. 300 250 200 ḡsi = 14.91(76) 350 ḡsi = 1.44(23) Note that both expressions for and p11 are obtained by assuming that the double pair creation probability is p2 , consistent with a thermal statistics of a two-mode squeezed state. One can naturally ask if our lower bound is still valid when the photon number distribution is not exactly thermal, but possibly in between thermal and Poisson statistics. The Poissonnian case is obtained when the temporal resolution is much larger than the coherence time of the photons, yielding a highly multimode case and second-order auto-correlations of the signal and idler modes equal to 1 [4, 5]. A temporal resolution that is of the same order as the coherence time of the photons therefore yields a double pair creation probability equal to ap2 , where 1/2 < a < 1 [4], and hence an autocorrelation function that is approximately given by 2a, comprised between 1 and 2, as observed here. In this case, the expression 4p10 p01 in both pth 11 and p11 above should be replaced with 4ap10 p01 < 4p10 p01 . Therefore, we see directly that assuming that we have thermal statistics (a = 1), even if this is not exactly the case as in this experiment, can only overestimate p11 , and hence underestimate the concurrence. 400 Counts in 2h pth 11 450 150 100 50 0 −100 −50 0 50 Time difference (ns) 100 FIG. A3. Coincidence histogram of the measurements taken with a pump power of 4 mW. The x-axis corresponds to the time difference between the detection of a signal photon and an idler photon. The photons that are transmitted through the memory (without storage) give rise to the leftmost peak at a zero time difference. The rightmost peak at 33 ns stems from stored and retrieved photons. The light-blue region marks the coincidence window used for the calculation of the crosscorrelation gs,i and of the concurrence. For comparison, the value of gs,i with a coincidence window positioned half-way between the two peaks is shown (light-red). III. CONCURRENCE USING A DIRECT MEASUREMENT OF THE THREEFOLD COINCIDENCE PROBABILITY A. D. Coincidence window selection Figure A3 shows the histogram of the time difference between the detection of a signal photon (at either detector 1 or 2) and an idler photon and with the arm of memory B blocked, added to the similar histogram obtained with the arm of memory A blocked and B opened. We see two clear peaks corresponding to signal photons that are not absorbed by the crystal, and those that are stored and reemitted 33 ns later. Also shown in the figure is the 10 ns coincidence window, centered on the second peak, used in the calculation of the concurrence. The peaks are well separated, and have no measurable overlap at 33 ns. Hence, the transmitted photons did not contribute the observed entanglement. In fact, shifting the coincidence window to a delay of half of the storage time reduces the cross-correlation to gs,i ≈ 1.4 (instead of ≈ 15 at 33 ns), yielding a concurrence of 0, and hence no entanglement. Hence, photons detected around 17 ns do not show entanglement. 4 Method We now describe how we estimate the probability p11 from the measurement of threefold coincidences. For this, we use the experimental setup shown on Fig. A4; this is the same setup shown on Fig. 1 of the main text. Unlike our estimation based on the measurement of the crosscorrelation (section II), this direct method makes use of the experimental setup in its entirety, that is, with recombination of the modes A and B on the beamsplitter. Furthermore, it does not require assumptions on the statistics of the source of photon pairs and on how the memories operate. Let us consider the two-photon part, denoted ρ2 , of the complete density matrix describing the state of the retrieved fields just after the crystals. If the relative phase between the spatial modes is randomized, then the coherences are zero and we can write ρ2 = q11 |1111| + q20 |2020| + q02 |0202|. (A7) Here, q11 is the probability per heralding signal that the fields retrieved from crystals A and B each contain one photon, q20 (q02 ) the probability per heralding signal that nature photonics | www.nature.com/naturephotonics © 2012 Macmillan Publishers Limited. All rights reserved. doi: 10.1038/nphoton.2012.34 supplementary information5 FIG. A4. Experimental setup used to estimate p̄11 directly. This setup is the same of Fig. 1 of the main text. The relative phase between the arms of the interferometer was randomized by letting the interferometer (see Fig. A4) drift by itself over the total measurement period of more than 100 hours. PBS: polarizing beamsplitter; FR: Faraday rotator; BS: beamsplitter; MA and MB are crystal memories A and B. The detection efficiency from the output modes of the beamsplitter to detectors 1 and 2 and η1 and η2 , respectively. The amplitude transmission and reflection coefficients of modes a and b right after the crystals, through the beamsplitter, are shown. the fields retrieved from crystals A and B each contain 2 and 0 (0 and 2) photons, respectively. Estimation of p11 can be performed by measuring the q11 probability in Eq. A7, as in Ref. [6]. We show here that one can also estimate p11 by measuring q20 and q02 . We first derive useful relations between the occupation probabilities. We use the following definitions: PH is the probability to herald the presence of a signal photon (i.e. the probability to detect an idler photon durA B ing the detection time window); ηecho (ηecho ) is the storage and retrieval efficiency of crystal A (B); P2 is the probability that the source of photon pair emits 2 pairs of photons during the detection time window; R and T are the intensity reflection and transmission coefficients of the beamsplitter when photons are incident from the left hand side, satisfying R + T ≤ 1 and R, T > 0 (see Fig. A4). Using these definitions, and the linear nature of the optical storage process, we have 1 A B 2RT ηecho ηecho P2 , PH 1 2 A 2 = T (ηecho ) P2 , PH 1 2 B 2 = R (ηecho ) P2 , PH 0 < αt2 , αr2 , βt2 , βr2 ≤ 1/2, and we note that R = βr2 and T = αt2 . Using these notations, we can show that the probability to get a coincidence at detectors 1 and 2 per heralding signal is given by p̄11 = (a11 q11 + a20 q20 + a02 q02 )η1 η2 , where η1 (η2 ) is the transmission from output mode a (b) – after the beamsplitter – to detector 1 (detector 2), including the detector efficiency. We also have 2 β2 a11 = βr2 αr − t , αr a20 = 2αt2 αr2 , a02 = 2βt2 βr2 . We characterized our beamsplitter and obtained αt2 = T = 0.479, αr2 = 0.422, βt2 = 0.482 and βr2 = R = 0.409. These values yield q11 = q20 q02 a11 = 0.0028, a20 = 0.394, a02 = 0.404. with which we can write q20 and q02 as a function of q11 : q20 = q11 (R/2T ), q02 = q11 (T /2R). (A8) We use these expressions later. We now consider recombination at the beamsplitter. We describe the effect of the beamsplitter on mode operators a and b as follows: a † → α t a † + α r b† + α l l † , b † → β r a † + β t b† + β l l † , where mode l represents loss. We set all coefficients to be real. Energy conservation yields αr2 + αt2 + αl2 = 1 (and similarly for the coefficients of mode b). Also, unitarity imposes that αr αt + βr βt = 0. Finally, we suppose that (A9) (A10) We immediately notice that a11 can be neglected with respect to a20 and a02 . This essentially means that whenever a photon is retrieved from each crystal, they bunch with almost certainty and do not significantly contribute to threefold coincidences. It also requires indistinguishability of the retrieved photons, which is consistent with the large single photon interference visibility observed (96.9%). Combining Eq. A9 and Eq. A10, and writing a20 ≈ a02 ≈ 0.4, we get p̄11 ≈ 0.4 η1 η2 (q20 + q02 ). (A11) Finally, using Eq. A8, we get T R + q11 q20 + q02 = 2T 2R = 1.012 q11 ≈ q11 5 nature photonics | www.nature.com/naturephotonics © 2012 Macmillan Publishers Limited. All rights reserved. supplementary information and thus p̄11 ≈ 0.4 η1 η2 q11 . (A12) We now have to determine how to use Eq. A12 in the calculation of the concurrence. This method should give a result consistent with the way we measured p10 + p01 , which was done as follows. We let the phase of the interferometer drift to randomize it, and measured the probability to get a detection at detector 1 or 2 per heralding signal. This is equivalent to, first, measure crystal A (with the path of crystal B blocked) using an effective detector of efficiency ηA = αt2 η1 + αr2 η2 , and then measure crystal B (with the path of crystal A blocked) using an effective detector of efficiency ηB = βr2 η1 + βt2 η2 , and then add the results together. Hence, if both crystals were measured simultaneously and separately, the threefold coincidence probability per heralding signal would be q11 ηA ηB . (A13) We can now find the relation between Eq. A12 and A13. For this, we use the fact that, in our experiment, η2 was larger than η1 , which is apparent from the visibility plot in the main text (inset of Fig. 2-A). Hence, by setting η1 = η2 /2, we get q11 ηA ηB ≈ 0.454 η22 q11 . Therefore, we immediately see that if we multiply Eq. A12 by 2.27, we get Eq. A13. In summary, to estimate the probability p11 used in the formula of the concurrence, namely √ C ≥ max(0, V (p10 + p01 ) − 2 p00 p11 ), (A14) we multiply the measured p̄11 by 2.27. This multiplicative factor was obtained using approximations, and hence has some uncertainty associated to it. Nevertheless, as explained in the next subsection, this uncertainty can be neglected in front of the large statistical uncertainty of p̄11 . B. Statistical uncertainty Here we provide details on how the statistical uncertainty on p̄11 was calculated in the method based on the measurement of threefold coincidences (subsection III A). We denote NH the number of heralding signals during the measurement period T , P̄11 the stochastic variable representing the probability to register a threefold coincidence per heralding signal, and p̄11 our estimation of P̄11 obtained from the measurement. Since p̄11 1 and NH 1, the distribution of the number of coincidences between detectors 1 and 2 per heralding signal (i.e. threefold coincidences) is accurately described by a Poisson distribution. Specifically, the probability to get n coincidences given NH heralding signals is given by P (n|p̄11 ) = e−NH p̄11 6 n (NH p̄11 ) . n! (A15) 6 doi: 10.1038/nphoton.2012.34 The probability P (n|p̄11 ) is the likelihood function of p̄11 given n. Hence, P̄11 can be estimated by maximizing this likelihood, which yields √ n n (MLE) = ± , p̄11 NH NH where MLE stands for “maximum likelihood estimation”. This method yields an appropriate estimate of P̄11 , unless one measures n = 0, in which case one would conclude that p̄11 = 0 ± 0. A more conservative estimate can be obtained in the following way. Using Bayes’ law, we can write the probability density function Ω(p̄11 |n) of getting P̄11 = p̄11 given n measured threefold coincidences: Ω(p̄11 |n) = 1 0 P (n|p̄11 ) P (n|p̄11 ) dp̄11 . The denominator of Eq. A15 can be calculated: 1 n n k NH 1 −NH p̄11 (NH p̄11 ) −NH dp̄11 = 1−e e n! NH k! 0 k=0 1 . ≈ NH The last equality holds because n NH . Therefore: Ω(p̄11 |n) = NH P (n|p̄11 ). (A16) Using Eq. A16, we can calculate the expected value of P̄11 and its second moment, respectively denoted P̄11 2 and P̄11 : 1 n+1 , p̄11 Ω(p̄11 |n) dp̄11 ≈ P̄11 = NH 0 1 (n + 1)(n + 2) 2 P̄11 = p̄211 Ω(p̄11 |n) dp̄11 ≈ . 2 NH 0 The standard deviation of P̄11 is therefore √ n+1 2 2 . P̄11 − P̄11 = NH Hence, given n measured threefold coincidences and NH heralding signals, this conservative method yields √ n+1 n+1 (CE) ± , p̄11 = NH NH where CE stands for “conservative estimation”. In our results, we estimate the concurrence with both the maximum likelihood and the conservative estimations of P̄11 . C. Experimental results The estimation of p00 , p10 + p01 and p11 proceeded as follows. For a time T = 166 hours, we let the interferometer drift to randomize the phase. During that period of nature photonics | www.nature.com/naturephotonics © 2012 Macmillan Publishers Limited. All rights reserved. supplementary information7 doi: 10.1038/nphoton.2012.34 time, we recorded the total number of heralded twofold coincidences between the heralding detector and detector 1 (N1|H ), and between the heralding detector and detector 2 (N2|H ), and calculated N2 = N1|H + N2|H . We also recorded the total number of heralded threefold coincidences, denoted N12|H , between all three detectors. We used a coincidence time window of 10 ns; this is the same value we used in the estimation of the concurrence based on the cross-correlation. We calculated p10 + p01 and the CE: (CE) p11 = 2.27 N12|H + 1 ± NH (MLE) p11 (MLE) p11 = 2.27 N12|H ± NH . = 2.9 ± 2.1 × 10−9 and (CE) C (MLE) ≥ 6.3 ± 3.8 × 10−5 p11 For p11 , we used both the MLE, yielding N12|H + 1 NH Finally, we estimated p00 = 1 − p10 − p01 − p11 . With a 10 ns coincidence window, we obtained p10 + p01 = 1.7777(34) × 10−4 , V = 96.9 ± 1.5%, and observed two threefold coincidences (N12|H = 2). This yields √ N2 N2 = ± . NH NH = 3.9 ± 2.2 × 10−9 C (CE) ≥ 3.9 ± 3.8 × 10−5 N12|H NH , [1] L. Mandel and E. Wolf, Optical Coherence And Quantum Optics (Cambridge University Press, Cambridge, 1995). [2] R. H. Brown and R. Q. Twiss, Nature 178, 1046 (1956). [3] M. Afzelius, C. Simon, H. de Riedmatten, and N. Gisin, Phys. Rev. A 79, 052329 (2009). [4] H. D. Riedmatten, V. Scarani, I. Marcikic, A. Acin, Both the MLE and CE yield a lower bound for the concurrence that is greater than 0 by at least one standard deviation. W. Tittel, H. Zbinden, and N. Gisin, J. Mod. Opt. 51, 1637 (2004). [5] P. Sekatski, N. Sangouard, F. Bussières, C. Clausen, N. Gisin, and H. Zbinden, Preprint: arXiv:1109.0194 (2011). [6] C. W. Chou, H. de Riedmatten, D. Felinto, S. V. Polyakov, S. J. van Enk, and H. J. Kimble, Nature 438, 828 (2005). 7 nature photonics | www.nature.com/naturephotonics © 2012 Macmillan Publishers Limited. All rights reserved. IOP PUBLISHING JOURNAL OF PHYSICS B: ATOMIC, MOLECULAR AND OPTICAL PHYSICS doi:10.1088/0953-4075/45/12/124001 J. Phys. B: At. Mol. Opt. Phys. 45 (2012) 124001 (8pp) Atomic frequency comb memory with spin-wave storage in 153Eu3+:Y2SiO5 N Timoney, B Lauritzen, I Usmani, M Afzelius and N Gisin Group of Applied Physics, University of Geneva, CH-1211 Geneva 4, Switzerland E-mail: [email protected] Received 16 December 2011, in final form 8 February 2012 Published 8 June 2012 Online at stacks.iop.org/JPhysB/45/124001 Abstract 153 Eu3+ :Y2 SiO5 is a very attractive candidate for a long-lived, multimode quantum memory due to the long spin coherence time (∼15 ms), the relatively large hyperfine splitting (100 MHz) and the narrow optical homogeneous linewidth (∼100 Hz). Here we show an atomic frequency comb memory with spin-wave storage in a promising material 153 Eu3+ :Y2 SiO5 , reaching storage times slightly beyond 10 μs. We analyse the efficiency of the storage process and discuss ways of improving it. We also measure the inhomogeneous spin linewidth of 153 Eu3+ :Y2 SiO5 , which we find to be 69 ± 3 kHz. These results represent a further step towards realizing a long-lived, multimode solid-state quantum memory. (Some figures may appear in colour only in the online journal) 1. Introduction echo is referred to as a two-level echo in the remainder of this paper. The efficiency of the echo in the forward direction, assuming a comb which can be described by a sum of Gaussian functions, is given by [14] 2 d 2 η≈ e−7/F e−d/F e−d0 , (1) F where d (= αL, α is the absorption coefficient and L the sample length) is the optical depth and F is the comb finesse. More specifically, F = /γ , where is as before and γ is the comb tooth full-width at half-maximum (FWHM). Here we also include an additional loss factor (the last factor) due to an absorbing background d0 , which often occurs due to imperfect preparation of the comb. Note that although we consider Gaussian-shaped teeth here, which fit well with our experimental data, other shapes have been considered elsewhere [15, 16]. Important results have been obtained using such a two-level AFC scheme, where a heralded single photon was stored in a crystal [17, 18] or heralded entanglement was generated between two crystals [19]. Two-level echo efficiencies using an AFC scheme of 15–25% are seen in many experiments [13, 16, 17, 20, 21]. In these papers and the work presented here, the inhomogeneously broadened medium is an inorganic crystal weakly doped with rare-earth ions. Such crystals placed in commercially available cryostats cooled to less than 4 K have impressive coherence properties on optical and spin transitions [9, 22]. We refer to [8] for a comprehensive review of these materials in terms of quantum memories. In Quantum communication [1] provides resources and capabilities, such as quantum key distribution [2], that are not possible to obtain using classical communication. A major challenge to quantum communication, however, is to overcome the inherent losses of quantum channels, e.g. optical fibres. A solution to the problem of long-distance quantum communication is the quantum repeater [3–6], which in principle can work over arbitrary distances. To implement a quantum repeater, quantum memories and effective delay lines of variable duration are required. Atom-based memories are attractive candidates; indeed much research has been done in recent years to implement them [7, 8]. In addition to faithfully reproducing the input mode and storing for potentially long times [9, 10], memories should also have a multimode capacity [11, 12] and an on-demand readout [13] to realize a good quantum memory for quantum repeaters. The quantum memory based on an atomic frequency comb (AFC) [14] has the potential of achieving these ambitious goals in one memory. An AFC memory is one which is based on an inhomogeneously broadened medium which has been tailored to contain a frequency comb of narrow peaks (teeth) of atomic population. The frequency spacing between the narrow peaks () dictates the storage time of the memory: coherent reemission of the input mode, an echo, is seen in the same forward direction a time 1/ after the input mode. Such an 0953-4075/12/124001+08$33.00 1 © 2012 IOP Publishing Ltd Printed in the UK & the USA J. Phys. B: At. Mol. Opt. Phys. 45 (2012) 124001 N Timoney et al the memory in an auxiliary state. In terms of rare-earthdoped crystals, praseodymium and europium are interesting candidates. Their nuclear spin of I = 5/2, results in three hyperfine levels at zero applied magnetic field [9]. A full AFC scheme was demonstrated for the first time in praseodymium [13]. The nature of the material used in these results places a limit of the number of modes to tens of modes. This limit is due to the hyperfine splitting of the material (approximately 10 MHz) and the optical homogeneous width of the material—1 kHz. Europium on the other hand has a larger hyperfine splitting (approximately 100 MHz) and an optical homogeneous linewidth of the order of 100 Hz [22]; with such a material it should be possible to create a multimode memory with at least an order of magnitude more modes [14]. Also, larger frequency separation of the input and control frequencies will be useful for spectral filtering, a likely requirement for future single-photon storage. The drawback of europium-doped materials is the low oscillator strength, which results in the low Rabi frequency of the control fields and the low optical depth. Both these factors are serious limitations in our present experiment, as we will describe. We note, however, that a material with a small optical depth does not necessarily have a poor efficiency, as equation (1) would imply. By placing the crystal in an impedance-matched cavity, it should be possible to achieve high efficiencies despite a low optical depth [25, 26]. e s g Figure 1. An illustration indicating the time order of an AFC involving spin-wave storage. The input mode is in resonance with the |g → |e transition; the control pulses are applied on the |e → |s transition. The time 1/ is defined by the periodic frequency separation of the teeth of the AFC. this particular work, we use 153 Eu3+ :Y2 SiO5 , which we believe has the potential of fulfilling the requirements of a quantum memory for quantum repeaters [5, 14]. The scheme described above, however, is not the complete AFC scheme as proposed in [14]. The conversion of the optical excitation to a spin excitation is missing. This requires the presence of another ground state. For a complete scheme, the input mode is followed by a control pulse which transfers the optical coherence between |g and |e to a spin coherence between |g and |s. The time line and an illustration of the relevant levels is shown figure 1. This control pulse ‘stops the clock’ of the predefined memory time of 1/. A second control pulse ‘restarts the clock’ by reverting to the optical coherence between |g and |e. The time between the control pulses is not predefined, such that the application of the second control pulse allows an on-demand readout of the memory. The explanation above is simplified since it does not take into account spin dephasing due to inhomogeneous broadening, which leads to a decay in storage efficiency as a function of spin storage time. This dephasing can, however, be compensated for by using spin-echo techniques, allowing long storage times only limited by the spin coherence time. An interesting aspect of the AFC scheme is that it is possible to implement a multimode memory in the time domain where the multimode capacity is not dependent on increasing optical depth [14, 23]. Rather the limit on the number of temporal modes which can be stored depends only on the number of peaks in the AFC. This is restricted, in turn, by the width of the AFC, generally limited by the hyperfine transition spacing and the smallest comb tooth width which can be obtained. Another restriction, of a technical nature, is imposed by the frequency bandwidth of the control pulses, which must spectrally cover the entire AFC spectrum. To transfer a large bandwidth, the control pulses would most likely take the form of chirped-shaped pulses as discussed in [24]. To implement a full AFC scheme, the atomic element must have three levels in the ground state, so that the atoms not required for the comb can be spectrally separated from 2. Europium Europium-doped Y2 SiO5 has been the subject of several spectroscopic studies, measuring for instance absorption coefficients, inhomogeneous and homogeneous broadenings and hyperfine level spacings [22, 27–29]. Until recently, however, the ordering of the hyperfine levels and the transition branching ratios between these were unknown in 153 Eu3+ :Y2 SiO5 . We thus conducted spectroscopic investigations of an isotopically enriched 153 Eu3+ :Y2 SiO5 crystal [30] to determine a suitable -system in which to perform a complete AFC scheme. In this work, we use one of the potential -systems identified in our previous work, shown in figure 2. We use the stronger ±|3/2g → ±|3/2e transition, with a greater optical depth, for the input mode, while the control fields are applied on the weaker ±|5/2g → ±|3/2e transition. It should be noted that from [30] three other potential -systems could be identified, which could all work as well or even better than the selected one (i.e. have larger oscillator strengths). This particular configuration was chosen for a first proof-of-principle demonstration due to available frequency shifts by our frequency shifters. Other configurations will be tested in future work. Our crystal has a 153 Eu3+ doping level of 100 ppm, an inhomogeneous broadening of 700 MHz and an absorption coefficient of 1.2 cm−1 on the 7 F0 → 5 D0 transition. We note that larger optical depths are possible with increased doping [27]. 2 J. Phys. B: At. Mol. Opt. Phys. 45 (2012) 124001 ±5/2e 5 N Timoney et al figure 3. The losses due to these AOMs are compensated for with the amplifier after the 3 × 3 coupler; such recuperation could not be obtained at 580 nm. Once the light is produced at 580 nm, it passes through an AOM (AOM:M) which determines the intensity and duration of the light in the crystal. AOM:M is in double pass, which allows small frequency scans of at least 15 MHz to be made uniformly in intensity. In fact more than one spatial mode is required for the spin-wave storage measurements performed in this paper. This is due to the low amplitude of the observed echo signal, compared to the strong control pulse. In addition, we observed that the control pulse causes free induction decay in its spatial mode, presumably due to off-resonant excitation, which will act as a background noise on the weak echo. To avoid this noise, we use two spatial modes: one for the comb preparation and the control modes, and another for the input pulse. In order to have fast independent control over each of the spatial modes, we use two AOMs, one for each mode. The zero order from the first pass of AOM:M is thus diverted to another AOM (AOM:I) which operates at the same frequency. AOM:I is also in a double-pass configuration. The resulting light from AOM:I and AOM:M crosses the europium crystal. Before the cryostat, there can be up to 53 mW of light from AOM:M, depending on the radio frequency amplitude applied to AOM:M. This light is focussed in the crystal with a beam waist of 60 μm. The light from AOM:I is significantly weaker; it can only reach a maximum power of 3 mW. The light hitting the detector can be attenuated using AOM:D in single pass after the cryostat. The light at 580 nm is frequency stabilized using a second continuous wave SFG source. This source is not shown in figure 3. Light from both diode lasers is taken before the amplifiers, combined on a dichroic mirror and sent through a second PPKTP waveguide. A cavity with a free spectral range of 1 GHz and finesse of 600 is used in a Pound–Drever– Hall configuration. The cavity mirrors are separated by an invar spacer and the structure is temperature stabilized. The correction for the error signal is applied to the 930 nm laser. The frequency stabilization removed slow frequency drifts and narrowed the laser linewidth to slightly less than 50 kHz as measured by spectral hole burning. 260 MHz D0 ±3/2e 194 MHz 580 nm ±1/2e 7 ±1/2 g F0 ±3/2 g ±5/2g 90 MHz 119 MHz Figure 2. Hyperfine structure of the 7 F0 → 5 D0 transition in 153 Eu3+ :Y2 SiO5 . The blue and green arrows indicate the -system used for this work, where ±|3/2g → ±|3/2e acts as the |g → |e transition for the input mode, and ±|5/2g → ±|3/2e the |e → |s transition for the control fields. Although a -system containing two ground states is all that is required for the complete AFC scheme, a third ground state (an auxiliary state) is required to store the atoms which are not part of the AFC. For the -system used in this paper, the auxiliary state is ±|1/2g. The ±|1/2g → ±|5/2e transition is used in the comb preparation stage, as explained in the text. 3. Experimental description The 7 F0 → 5 D0 transition requires a light source at 580 nm. This wavelength is not covered by diode lasers at present, so we have chosen to generate it using sum frequency generation (SFG) of two wavelengths: 1540 nm and 930 nm. The nonlinear medium used is a PPKTP waveguide, which produces 110 mW of light at 580 nm at the output where there is 1.5 W of light at 1540 nm and 400 mW of light at 930 nm at the input. To reduce the losses of the light produced at 580 nm, the three frequencies required for the measurements shown in this paper are produced using acousto optical modulators (AOMs) on the 1540 nm light. This light is recombined in a 3 × 3 coupler; the polarization of each AOM can be controlled before it is combined, thus adjusting the polarization required for the waveguide. The polarization control is suppressed in 930 nm T.Amp AOM 0 1540 nm Er Amp AOM Er Amp AOM:I 1 Europium Crystal 1 AOM:M PPKTP AOM:D AOM Figure 3. A very basic illustration of the experiment. The sum frequency process in the PPKTP waveguide generates the light at 580 nm from the 1540 and 930 nm diode lasers. The light from both lasers is amplified; a tapered amplifier is used for the light at 930 nm and two erbium amplifiers for the light at 1540 nm. The AOMs at 1540 nm select the atomic transition used. Two AOMs (AOM:M and AOM:I) at 580 nm before the europium crystal determine the amplitude and the duration of light in either the control or the input mode. These two modes overlap in the europium crystal. Further explanation of the experiment can be found in the text. 3 J. Phys. B: At. Mol. Opt. Phys. 45 (2012) 124001 N Timoney et al 1 10 0.7 Input mode Two level echo (η = 1.5 %) 0.6 0 10 Intensity, au Optical depth 0.5 0.4 0.3 0.2 −1 10 −2 10 0.1 0 −3 −0.5 0 Frequency, MHz 10 0.5 −1 0 1 2 Times, μs 3 4 5 Figure 4. (a) A sample comb created with a periodicity of 0.5 MHz. The blue solid line shows the measured comb. This comb is obtained by measuring the absorption of ±|3/2g → ±|3/2e with a small scan of <2 MHz over 200 μs. The comb shown is an average of 20 measurements. The green-dashed line shows a Gaussian comb with γ = 165 kHz (corresponding to F = 3.03), d0 = 0.04 and d = 0.54. (b) The corresponding echo, where the efficiency is measured to be 1.5%. It was necessary to change the detector gain for the two measurements shown, which results in more noise on the background of the input mode. structure of ions. A fraction of the excited ions in ±|5/2e will decay to |3/2g, thus forming the desired spectral comb of ions on ±|3/2g → ±|3/2e . Since only a fraction of the ions decay to |3/2g, it is necessary to repeat the comb preparation many times. Ions that fall down in |5/2g must also be removed by optical pumping, forcing most of these ions into the desired |3/2g state. In the experiments shown here, 80 repetitions are used. More details on all the steps used in the comb preparation can be found in [30]. A sample comb is shown in figure 4(a) where the frequency separation of the comb is = 0.5 MHz (this corresponds to a two-level AFC storage time of 2 μs). The comb measurement is performed by measuring the absorption of ±|3/2g → ±|3/2e . The maximum optical depth that we could obtain on this transition is d = 0.8, higher than the maximum peak height shown in figure 4(a), where the maximum value is d = 0.54. Peaks with higher absorption are obtainable with more power in the pulse stream; however, this results in broader peaks. Broader peaks reduce the finesse (F) of equation (1), thus lowering the maximum echo efficiency possible. The green-dotted line shown in figure 4(a) represents a Gaussian comb in frequency (ν) of the form The europium crystal itself is housed in a pulse tube cooler, where the cold finger has a temperature of 2.8 K. We observe spectral broadening of the teeth in the comb when the cryostat cooler is switched on compared to when it is switched off. The spectral broadening was very shot-to-shot dependent, indicating that it depends on where the measurement is performed within the pulse tube cycle (with a period of 700 ms). It is likely that this spectral broadening is caused by crystal movement in the same direction as the beam propagation, induced by the vibrations from the cryostat compressor or rotary valve. Although the exact physical mechanism leading to this broadening remains unclear, we believe it to be due to phase noise in the atom–light interaction induced by the modulation of the laser–crystal distance, leading to an effective laser line broadening. By triggering the experimental sequence on the vibrations in the cooling tubes using a piezo, the observed jitter on the width of a comb tooth is reduced by a factor of 3, from 150 to 50 kHz. We conclude this section by noting that the technical spectral broadening due to laser linewidth and cooler vibrations adds up to about 100 kHz. j=+4 4. Comb preparation and two-level echo efficiency n(ν) = d j=−4 The frequency comb required for an AFC quantum memory can be prepared using a stream of pulses where the inverse of the time separation gives the frequency separation () of the comb produced [11, 16]. Spectral tailoring techniques similar to those used in [13] and [30] are necessary before the comb creation to isolate an atomic system such as that described in figure 2. Note that the spectral tailoring is performed over a certain frequency range, in our case over roughly 10 MHz. The spectral tailoring includes a spin polarization process, which prepares the atoms in one of the ground states, in this case in the auxiliary state ±|1/2g (see figure 2). The comb is then prepared by sending a stream of 15 pulses on the ±|1/2g → ±|5/2e transition, exciting a spectral comb 2 − (ν− j) 2 e 2γ̄ + d0 , (2) √ where the peak width is given by γ̄ = γ / 8 ln 2 (all other parameters are defined as before). The finesse of each comb is obtained by measuring the width of the peaks created. The minimum width of a peak in the comb is limited by the laser linewidth and the frequency noise induced by vibrations from the cryostat. Additional contributions include possible power broadening from the peak preparation pulses and the inhomogeneous spin linewidth of the material. The echo which can be seen using the comb in figure 4(a) is shown in figure 4(b). The efficiency of the emitted two-level echo is obtained by comparing the area of the input mode when there is no absorption on the input mode transition (the blue 4 N Timoney et al 2.5 5 2 4 1.5 3 1 2 0.5 1 0 1.5 2 2.5 3 3.5 Storage time (μs) 4 4.5 length between the laser source and the crystal; this could be done using an interferometric setup. In addition, due to our preparation method, the inhomogeneous spin linewidth is also likely to contribute towards the minimum linewidth. This issue can, however, easily be resolved by changing the preparation method. The maximum efficiency obtained at 2 μs storage time is limited by the low optical depth of our material. Higher efficiencies can certainly be reached using a crystal with higher doping concentration. Indeed a peak optical depth three times higher has been obtained [27], which ideally would result in a tenfold increase in the efficiency (see equation (1)). Another interesting solution would be to place a cavity around the crystal [25, 26]. Both methods might be required in order to approach unit efficiency. We conclude this section by noting that whilst not dictating an absolute limit, these efficiencies do represent a serious limitation in our present system, which is relevant to the following results of spin-wave storage in the complete AFC scheme. Finesse Echo efficiency % J. Phys. B: At. Mol. Opt. Phys. 45 (2012) 124001 0 5 Figure 5. The decay in the echo efficiency of a two-level AFC echo. The squares show the efficiencies using a comb preparation described in the text for a range of AFC storage times (1/). The finesse expected from γ = 165 kHz is plotted using a solid red line (right-hand y axis). 5. Spin-wave storage solid line) to the area of echo signal (the green solid line). The efficiency measured here is 1.5%, which is higher than that expected (1.2%) using equation (1) and the values of d and d0 obtained in figure 4(b) (cf figure 5). We estimate an error of ±5% on the efficiencies shown due to the nonlinear reaction of changing the gain on the detector. Additionally, the optical depth measured in the comb measurement of figure 4 is likely to be too low. First, the optical depth has been reduced by the input pulse which was not suppressed for this measurement. Second, the rate of the scan, 2 MHz in 200 μs, implies a scan resolution of roughly 100 kHz, large enough to detrimentally affect the values extracted for d and d0 . Using the same preparation method, the time separation of the comb preparation pulses and thus the frequency separation of the comb teeth () was varied. The two-level echo efficiency for a storage time of up to 5 μs was measured and the results are plotted in figure 5. A comb has been measured for each efficiency shown in figure 5. For longer storage times, we do not deem this comb measurement to be accurate. The effects of our scan resolution, the vibrating cryostat and the laser linewidth are hard to separate from the actual comb. Instead we merely extract a value of γ from these plots of 165 kHz, and show the measured finesse on the right-hand axis. The trend of decreasing finesse accounts for the trend of decreasing efficiency. The optimum storage time in our system is currently 2 μs. But these results do not represent a fundamental limit to the maximum storage time or two-level efficiency in this material. The storage time is clearly limited by the AFC tooth width (γ ). The minimum width obtained was limited by our laser linewidth and the effective linewidth broadening probably due to crystal movement induced by the cryostat cooler. In order to improve the storage time, we should improve the laser frequency stabilization and reduce the presumed effect of crystal movement. The latter can be done by employing a low-vibration pulse tube cooler or by stabilizing the path In order to perform spin-wave storage, we spatially separate the input mode from the control mode using a cross beam configuration, as already discussed in the experimental description section (see also figure 3). While reducing the free induction decay noise caused by the control fields [13], the crossed beam configuration also has a negative effect; it reduces the efficiency of the two-level echo. A sample twolevel echo is shown in figure 6. This echo is measured without the control pulses, but in a crossed mode configuration where the comb is prepared using AOM:M. The efficiency of the twolevel echo is measured to be 0.24% at 1/ = 4 μs. This should be compared to the single spatial mode measurement, where the efficiency was measured to be 0.6% (see figure 5). The two-level echo efficiency is thus reduced by almost a factor of 3. This is attributed to imperfect overlap of the two beams. In the introduction, we talked about using chirped pulses for our control pulses. Such pulses would allow us to efficiently transfer a large bandwidth, potentially all of the teeth in the AFC spectrum. But these pulses are generally of longer duration as compared to a π -pulse [24]. There is however a limit on the time which we can use to perform the control pulses. The duration and shape of the input pulse and one control pulse cannot be longer than the predefined two-level echo time 1/. In addition, the duration of the input pulse defines the bandwidth which the control pulse must transfer. In our current experimental setup, the Rabi frequency of the control pulse transition, ±|5/2g → ±|3/2e , is estimated to be of the order of 300 2π kHz. To transfer a bandwidth of 300 kHz, using a π -pulse, 1.7 μs are necessary at this Rabi frequency. If we set 1/ = 2 μs to obtain the highest two-level echo efficiency, the input pulse would have to be shorter than 0.3 μs. The spectral bandwidth of such an input pulse would be >3 MHz, the majority of which will not be transferred by a 1.7 μs control pulse. This illustrates that a compromise must be made between the efficiency of the two-level echo and the 5 J. Phys. B: At. Mol. Opt. Phys. 45 (2012) 124001 N Timoney et al 2 10 Input mode Two level echo (η = 0.24 %) Three level echo (η = 0.02%) Control pulses 1 γ Intensity au Intensity, au 10 0.03 0 10 IS 0.025 = 69 ±3 kHz 0.02 0.015 −1 10 0.01 −2 10 0 −3 10 −2 0 2 4 6 Times, μs 8 10 2 4 6 8 Ts (μs) 10 12 14 Figure 7. Inhomogeneous spin linewidth measurement. Each point represents the maximum of the Gaussian function which has been fitted to each trace. Some traces and fits are shown; the rest are suppressed for clarity. The formula used to fit the data is described in the text and yields an inhomogeneous linewidth of 69 ± 3 kHz. 12 Figure 6. An input mode (blue), its two-level echo with no control pulses (green) and a spin stored echo, a three-level echo, when the control pulses are applied (red). Compared to the two-level echoes shown in figure 5, the input mode is not in the same spatial mode as the preparation of the AFC. This is to reduce free induction decay background noise on the three-level echo. The input mode and the preparation beams overlap in the crystal. The control fields are in the same mode as that of the preparation. Imperfect overlap of the two modes accounts for the efficiency of the two-level echo dropping by almost a factor of 3 compared to figure 5. Changing the gain on the detector distorts the pulse shape of the echo. All of the measurements represent an average of 20 traces. where Ts defines the spin storage time, A is a constant and γIS is the inhomogeneous spin linewidth. An echo of increasing spin storage time is shown in figure 7. The complete memory time is given by Ts + 1/. Each recorded echo is fitted with a Gaussian function, the maximum of which is used to fit equation (3). We obtain an inhomogeneous spin linewidth of 69 ± 3 kHz. Some echo traces and their fits are shown, but most have been suppressed for the sake of clarity. Armed with the knowledge of the inhomogeneous spin linewidth, we can estimate the efficiency of the transfer pulses used. For the example shown in figure 6, the measured efficiency can be extrapolated to Ts = 0, resulting in an estimated 0.04% efficiency without spin dephasing. Hence, the estimated three-level echo at Ts = 0 is 16% of the twolevel echo (0.04%/0.24% = 0.16). Following the simple model discussed in [13], we can then calculate the transfer efficiency per pulse, which we find to be 40%. It should be noted that since the bandwidth of the control pulses is not much larger than the input pulse bandwidth, we should take this value as an effective average over the bandwidth. It is also difficult to estimate the effect of imperfect beam overlap on the efficiency. However, considering that we observe a strong decrease in the two-level echo due to insufficient overlap, it is likely that the overlap has a non-negligible effect on the estimated transfer pulse efficiency. We emphasize that it is possible to refocus the spin coherence using radio frequency pulses (spin echo), which would allow us to store for durations of the order of the spin coherence time. This was measured to be 15.5 ms for the 151 Eu3+ isotope in Y2 SiO5 at zero magnetic field [31]. One can expect a similar value for 153 Eu3+ . As a future perspective, one can also think of applying a magnetic field to cancel the firstorder Zeeman effect, creating a memory which for europium could be of the order of many seconds [32]. This is similar to the work performed in praseodymium [9]. The excellent spin coherence properties are a clear strength of europium-doped materials. efficiency of the transfer. We use the final echo efficiency as an indicator of the best compromise. The input pulse had approximately a Gaussian shape, while the identical control pulses were square shaped. We set the two-level storage time to 1/ = 4μs. The durations of the input pulse and the control pulses were optimized by looking at the efficiency of the three-level echo. The highest efficiency was reached with an input pulse with a full-width at halfmaximum of 1.3 μs and control pulses of duration 1.55 μs. The resulting echo with spin-wave storage is shown in figure 6, where the spin storage time is roughly 5 μs, leading to a total storage time of 9 μs. We checked that the echo is not present if we remove the first control pulse or if we remove the input mode, as expected. In figure 6, we also show the associated two-level echo (no control pulses applied) obtained with an identical comb structure. Note that AOM:D is used to gate the detector around the measured echoes, as the input mode would saturate the detector at the gain required to see the echoes. Maintaining this 1/ = 4 μs AFC, it is possible to vary the time between the control pulses (Ts ). This is the on-demand storage time of the full AFC scheme. For a rare-earth-doped crystal, we expect to see a decay given by the inhomogeneous spin linewidth of the material [13]. The storage time of the complete AFC scheme as described in the introduction of this paper is limited by this linewidth. If we assume a Gaussian distribution for the spin broadening, the decay of the stored echo is given by −Ts2 γ 2 π 2 IS echo height = A e 2ln2 , (3) 6 J. Phys. B: At. Mol. Opt. Phys. 45 (2012) 124001 N Timoney et al 6. Conclusion, discussion and outlook Acknowledgments Here we show for the first time an AFC memory with spin-wave storage in a europium crystal. Europium is an interesting material for quantum memories thanks to its long spin coherence time and its potential for multimode storage. The experiments we report here represent a first step in this direction. Yet, several important aspects of the memory must be significantly improved. First, the efficiencies which we report are very low, indeed europiums’ handicap lies in its low optical depth. There exist promising proposals to increase the optical depth by placing the crystal in a cavity thus increasing the efficiency of a twolevel echo. The measurements shown in this paper do not represent a fundamental limit of the efficiency nor of the storage time of a two-level echo. The latter has an impact on the multimode capacity of the memory. The storage time could be improved by changing the comb preparation method to one such as that found in [11], which would remove the effect of the inhomogeneous linewidth from the minimum peak width obtainable. Indeed peaks as narrow as 1 kHz can be found in the recent publication [33]. Further technical improvements on the laser linewidth or a cryostat with less vibration would also improve the storage time shown in this paper, allowing for multimode storage and the use of more efficient, chirped control pulses. Another technical difficulty is as follows: the relatively small dipole moment of this transition means that large amounts of power are required to transfer a relatively small bandwidth. To increase the bandwidth of the transfer, significantly larger amounts of power are required. Recent developments of powerful lasers for sodium-based systems at 589 nm [34] can also be applied to developing narrowband, powerful and compact lasers at 580 nm. In the spin-wave storage experiments presented here, we could store for up to about 10 μs, limited by inhomogeneous spin dephasing. From these measurements, we could estimate the spin linewidth, which we found to be 69 ± 3 kHz. The inhomogeneous spin linewidth is not a restriction, however, on the maximum storage time of the medium. The storage time can be increased using spin refocussing pulses such as [9]. An important milestone for quantum memories based on rare-earth-doped crystals would be to store an optical pulse on the single-photon level as a spin-wave excitation, a milestone that has not yet been reached in any rare-earthbased memory. The low overall efficiencies obtained in these experiments currently make this a very challenging experiment in 153 Eu3+ :Y2 SiO5 . The possible improvements that we have detailed, however, should make it possible in the near future. In this context, we recently proposed a method of generating a photon pair source with variable delay [35], using the same resources used for spin-wave storage-based AFC quantum memory. The overall efficiency of this source also has a less strong dependence on optical depth and control pulse transfer efficiency. An advantage of 153 Eu3+ :Y2 SiO5 in this context is the large hyperfine splitting, which facilitates spectral filtering as compared to praseodymium-doped Y2 SiO5 . We would like to thank Claudio Barreiro for technical assistance and N Sangouard and H de Riedmatten for useful discussions. We are also grateful to Y Sun, R L Cone and R M Macfarlane for kindly lending us the 153 Eu3+ doped Y2 SiO5 crystal. 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Express 17 14687–93 Sekatski P, Sangouard N, Gisin N, de Riedmatten H and Afzelius M 2011 Photon-pair source with controllable delay based on shaped inhomogeneous broadening of rare-earth-metal-doped solids Phys. Rev. A 83 053840 Single-photon-level optical storage in a solid-state spin-wave memory N. Timoney, I. Usmani, P. Jobez, M. Afzelius,∗ and N. Gisin arXiv:1301.6924v1 [quant-ph] 29 Jan 2013 Group of Applied Physics, University of Geneva, CH-1211 Geneva 4, Switzerland A long-lived quantum memory is a firm requirement for implementing a quantum repeater scheme. Recent progress in solid-state rare-earth-ion-doped systems justifies their status as very strong candidates for such systems. Nonetheless an optical memory based on spin-wave storage at the singlephoton-level has not been shown in such a system to date, which is crucial for achieving the long storage times required for quantum repeaters. In this letter we show that it is possible to execute a complete atomic frequency comb (AFC) scheme, including spin-wave storage, with weak coherent pulses of n̄ = 2.5 ± 0.6 photons per pulse. We discuss in detail the experimental steps required to obtain this result and demonstrate the coherence of a stored time-bin pulse. We show a noise level of (7.1 ± 2.3) · 10−3 photons per mode during storage, this relatively low-noise level paves the way for future quantum optics experiments using spin-waves in rare-earth-doped crystals. Quantum communication if rigorously executed provides us with a provably secure method of communication [1]. However, inherently lossy channels limit the distance over which the communication can be performed, which today is roughly 250 km [2, 3]. A quantum repeater which can in principle allow quantum communication over longer distances [4–6], provided that the required quantum memories are developed. Prime candidates for quantum memories are atomic systems, which are capable of maintaining the coherence of stored excitations for long times. Atomic systems that are currently investigated range from individual quantum systems[7, 8], lasercooled atomic gases [9, 10], room-temperature atomic vapours [11–13], to rare-earth-ion-doped crystals [14, 15]. Crystals doped with rare-earth-ion impurities have attractive coherence properties when cooled < 4K, in particular hyperfine states can have coherence times which can approach seconds [16]. This has provided a strong motivation for developing quantum memories using such systems. Following the first storage experiment at the single-photon level [17], a succession of experiments demonstrated storage of single photons [18, 19], generation of light-matter [14, 15] and matter-matter entanglement using crystals [20]. The quantum memory performances have also been strongly developed, particularly in terms of storage efficiency [21, 22], multimode capacity [23, 24] and polarization qubit storage [19, 25, 26]. These experiments were performed for short storage times (in the 10 ns to few µs regime) using an optical coherence, rather than exploiting long spin coherence times. Spin storage experiments require strong optical control fields to convert the initial optical coherence to a spin coherence. Photon noise is induced by such an operation, which has been nonetheless shown to work for alkali atomic systems [7–10, 12, 13]. In rare-earth-iondoped solids the task is complicated since there is less spectral separation between the weak signal field and the optical control field (roughly 100 times less). Scattering from the control field is thus more likely, as it propagates through a dense solid-state crystal. Two quantum memory schemes were specifically pro- posed for solid-state ensembles; the controlled and reversible inhomogeneous broadening (CRIB) memory (see [27] and references therein) and the atomic frequency comb (AFC) memory [28]. The AFC has a particularly high multimode capacity, which is the ability to store trains of single photon pulses [28, 29]. This is crucial for speeding-up quantum repeater protocols [5]. The AFC scheme is based on an echo induced by a regular spectral grating of periodicity ∆, in the absorption profile of an atomic ensemble. An AFC echo is emitted a time defined by 1/∆, unless the optical coherence is transferred (written) to a spin coherence before the time 1/∆ has elapsed. Reversing the transfer retrieves an optical pulse (referred to as an AFC spin-wave echo). AFC memories which only use the optical coherence are delay lines unless combined with spin-wave storage [28], which allow for on-demand read out and significantly longer storage times. Only a few AFC spin-wave storage experiments have been reported, all involving storage of bright classical pulses [30–32]. Here we demonstrate storage of an optical pulse containing a few photons on average, using an AFC memory combined with spin-wave storage in a europium doped Y2 SiO5 crystal. We apply a strategy of filtering in space, time and frequency in order to reduce unwanted emission from the crystal at the moment the weak pulse is recovered from the crystal. To quantify the degree of noise we measure the unconditional noise floor [12],which is the probability for the memory to produce a noise photon when the memory is read. We report that the unconditional noise floor can be reduced to (7.1 ± 2.3) · 10−3 by our filtering strategy, which is low enough to allow for a range of quantum information schemes that require manipulation of spin coherence. Using the ability of the AFC memory to store multiple time bins, we also store and analyse a time-bin pulse with higher photon numbers, showing the high coherence of our quantum memory. Europium is a promising candidate for quantum memories due to its fine coherence properties at T < 6K [33– 35], which ultimately could lead to an extremely long- 2 (a) (b) FIG. 1. (a)The atomic level scheme of the optical transition 7 F0 →5 D0 in 151 Eu3+ :Y2 SiO5 . (b) A schematic of the experimental setup around the memory, the rest of the experiment has been suppressed for simplicity. The control and preparation beam is in single pass (wide labelled arrow). The input mode (thin dashed line) is in double pass, with the help of a Faraday Rotator (FR) and a polarizing beamsplitter (PBS). On return from the crystal the input mode passes through a Fabry-Perot(FP) cavity (bandwidth of 7.5 MHz). A classical detector (Sd) and 10 µW of horizontally polarized light (thin dotted line) is used to actively and intermittently stabilize the cavity to the frequency of the input mode. An accousto optical modulator (AOM) in double pass acts as a detector gate. lived [36] and multimode memory [28]. In this work we use the optical 7 F0 →5 D0 transition at 580 nm. The crystal is isotopically pure 151 Eu3+ :Y2 SiO5 (100ppm). At a temperature of around 3 K we measure an overall absorption coefficient of α = 1.5 cm−1 and an optical inhomogeneous linewidth of 500 MHz. The relevant energy diagram is shown in Fig. 1a. Our input and control fields excite two optical-hyperfine transitions separated by 35.4 MHz. The schematic of the experimental set up (Fig. 1b) shows only the optics around the cryostat containing the 151 Eu3+ :Y2 SiO5 crystal of length L=1 cm. The storage mode crosses the control and preparation mode through the crystal. Given the measured angular separation of the beams before the cryostat we estimate a spatial mode overlap of 95 %. A double-pass configuration was implemented on the storage mode to increase the optical depth [23], while the control mode was in single pass. The laser and the acousto-optic modulators (AOMs) used for spectral control are not shown in Fig. 1b. The laser at 580 nm is a commercially available system based on an amplified diode laser at 1160 nm and a frequency doubling stage. Before the cryostat the intense control pulses had peak powers of up to 300 mW. The diode laser is stabilized to have a spectral linewidth of approximately 30 kHz. The AFC comb structures are created with frequency selective optical pumping techniques, which are now wellestablished techniques for spectral shaping of inhomoge- neously broadened transitions, see for instance [30, 37]. A particular feature of our preparation sequence is that we first pump all ions into the ±|1/2ig state, and then create the comb-structure by removing atoms from this state. This has the benefit of reducing the effect of the inhomogeneous spin linewidth, which could otherwise limit the minimum tooth width in the comb structure [31]. The maximum optical depth we can achieve on the input transition is αL = 2.4, in double-pass configuration. We first characterize our memory using bright input pulses of many photons and detecting the pulses with a linear photodiode. We observe AFC echo efficiencies of more than 5% for 1/∆ = 6 µs, and AFC spin-wave echo efficiencies of 1% for spin-wave storage time TS of 18 µs. The reduction in efficiency is mostly due to imperfect control pulses. We estimate the transfer efficiency per control pulse to be 0.49. By measuring the decay of the spin-wave echo as a function of TS , we estimate the inhomogeneous spin linewidth to be 8 kHz. This measurement will be further detailed in a future publication. The 8 kHz linewidth is surprisingly low, a factor of 8 less than for the 153 Eu3+ :Y2 SiO5 (100 ppm) sample we previously used [31]. This results in a spin-wave memory time of about 50 µs, defined by the point where the efficiency is reduced to exp(−1), the longest so far obtained in an AFC memory. By applying spin refocussing techniques we can expect to increase it further, up to the spin coherence time of 15 ms [35]. AFC spin-wave storage for weak coherent pulses with average photon numbers between n̄ = 2.5 ± 0.6 and n̄ = 11.2 ± 0.6 are shown in Fig. 2. The input pulse is 2 µs long, the memory parameters are 1/∆=6 µs and TS =21 µs, leading to a total storage time of 27 µs. The duration and shape of the control pulses were optimized for the highest signal-to-noise ratio (SNR), see discussion below. These measurements are performed, as all of the measurements shown in this letter, without the cryostat switched on to reduce the effect of vibrations on the comb structure [31]. There are two principal mechanisms which are responsible for the noise created by the bright control pulses. One is scattering of the laser light itself from optical surfaces. Another is emission from the atoms which have been excited by the pulses, this includes incoherent fluorescence, coherent free-induction-decay (FID) type emission and an unexpected off resonantly excited echo. Spatial separation of the input and control modes is used to shield the single photon counting detector from scattered light, but this did not lead to sufficient suppression. A double-pass AOM (shown in Fig. 1b) is used as a detector gate in time, exploiting the temporal separation between the control fields and the emitted spin-wave echo, providing a suppression of roughly 106 . This proved sufficient to prevent detector blinding or significant afterpulsing. The emission noise is, however, also present in the tem- 3 (b) Input Input 100 C1 x 30 50 Counts Counts 150 0 (c) 150 C2 10 20 Time ( µs) 30 250 100 0 25 C1 50 x 30 Output 0 500 Counts 200 (d) C2 Output 0 0 10 20 Time ( µs) 30 30 Time ( µs) 35 5 15 10 SNR (a) 5 0 0 n̄ 10 FIG. 2. Storage of a weak coherent pulse with (a) n̄ = 2.5 ± 0.6 and (b) n̄ = 11.2 ± 0.6. The input mode recorded with no comb, the position of the control fields (C1 and C2), and finally a magnified (x 30) signal of the AFC spin-wave echo (blue curve), the associated noise without an input pulse (green curve) and the detector dark counts (red curve) are shown. Note that the total measurement time differs between the data sets in (a) and (b). (c) The same echo data of (a) with the temporally separated off-resonant echo (OREO). (d) SNR for different n̄. Shown is also a fitted linear slope fixed to 1 at n̄ = 0 by definition. poral mode of the output mode. A diffraction grating and a Fabry Perot (FP) cavity are used to spectrally filter this noise. The FP cavity is necessary in particular to reduce noise originating from FID. Sharp spectral features about the control transition, by products of the spectral tailoring required to prepare the AFC, cause the FID. We could reduce this noise by altering our preparation sequence, to increase the transparency window around the control field transition. The frequency of this noise is close to that of the control field, a fact which we observe by changing the frequency at which we lock our FP cavity. In addition to the fluorescence and FID noise, we also observed an unexpected noise source at the input frequency. We believe its presence to be due to off-resonant excitation from the control fields, we thus call this an off-resonant echo (OREO) (see Fig. 2c). The OREO is observed a time 1/∆ after C2, supporting this hypothesis. We also observe a strengthening of the echo if C1 is present. We explain this by supposing that the offresonant excitation of C1 is combined with transfer to the spin state. C2 then reads out the excitation in the same manner which it does the single-photon-level input pulse. The observed 1/∆ dependence of the OREO, also fits to this explanation. Although the OREO is considerably larger than the AFC spin-wave echo which we are seeking to retrieve, the two echoes occur in temporally separated modes (see Figs 2a and c). We could reduce the impact of temporal mode leakage by carefully tuning the shape of the control fields, which is consistent with an off-resonant excitation mechanism. Note that since the FID and the OREO are coherent processes, the corresponding emission should only be strong in the control mode. Scattering inside the crystal does however introduce significant cross talk between the spatial modes. The temporal shape of the remaining noise we observe in Fig. 2a is indicative of FID noise. This gives us rea- son to believe that a more efficient filtering system would permit us to increase the power in the control fields thus increasing their efficiency. The remaining noise, in this particular measurement, amounts to (5.1 ± 1.3) · 10−3 photons per mode emitted at the crystal. The SNR up to n̄ = 11.2 ± 0.6 is shown in Fig. 2d. These measurements were taken on a range of different days for the same experimental parameters. The SNR follows a linear dependence within the experimental errors, see fitted linear slope in Fig. 2d. Measurements carried out for higher average photon numbers (not shown) confirmed this behaviour. The final memory efficiency in the photon counting experiment was significantly lower than for the bright pulse storage. The optimization of the duration and shape of the control pulses led to a lower transfer efficiency. Furthermore, a photon counting experiment requires time averaging, for example, the measurement for n̄ = 2.5±0.6 was taken over the course of three hours. This challenges the stability of the experiment, in particular, laser fluctuations create reduced quality combs, which negatively affect the AFC echo efficiency. Averaging over all the measurements shown in Fig. 2d, we obtain a global memory efficiency of (3.8 ± 1.5) · 10−3 and an unconditional noise floor of (7.1 ± 2.3) · 10−3 . Finally we show the coherence of the AFC spin-wave echo. To do this we store a time-bin pulse in the memory where we vary the phase of one of the time bins. We then self interfere the time-bin pulse using a temporal beamsplitter and examine the interference curve. The visibility of the curve gives a measure of coherence preservation in the memory. For the measurement shown in this letter, the temporal beamsplitter comes in the form of the control pulses. The scheme is pictorially shown in Fig. 3a. To store and analyse the time-bin pulse, we need clean temporal separation between the retrieved pulses and 4 (a) (b) (c) 300 100 V fit 80 = 0.87 ±0.06 Counts Sum counts 400 200 100 0 0 60 40 20 0.5 1 1.5 Phase shift, π 2 0 26 28 30 Time, µs 32 FIG. 3. (a) The method used to measure the coherence of the AFC spin-wave echoes. The single write operation of Fig. 2a,b (C1) is replaced by a double write operation(W1, W2). If the temporal separation (T) of the input mode is equal to that of the double write operation , the first echo of the second write operation and second echo of the first operation interfere. (b) The visibility curve for two pulses with n̄ = 176 ± 8 . (c) The signal of a constructive and destructive case. The thick dashed lines show the temporal window which was used to obtain the interference curve. The detector gate has cut some of the first echo and the OREO is not shown in this temporal slice. enough time to see the triple pulse structure shown in Fig. 3c after the final control field. To do this we extend the AFC time from 6 to 8 µs, and reduce the pulse width of the input pulses and the entire pulse length of the control pulses. These measures further reduce the efficiency with which we can store in the memory to ηs = (6.3±0.1)·10−4 for each mode, including the reduction in storage efficiency due to the double write operation. TS was set to 21 µs in this experiment, yielding a total memory time of about 29 µs. A visibility curve for n̄ = 176±8 is shown in figure 3b, where we measure V = 0.87 ±0.06. We suspect that laser phase noise contributes negatively to our visibility curve. A simple calculation shows that frequency noise with σf = 25 kHz reduces the baseline to V = 0.95. Together with the noise level this accounts for the visibility we measure. For n̄ = 51 ± 3 we observe a further drop in visibility to V = 0.71 ±0.1. This is due to the increasingly important role of noise in determining the minimum of the visibility curve. We note that with higher storage efficiency, it should be possible to obtain high visibilities for lower photon numbers. The unconditional noise floor achieved in our experiment should in principle allow us to store a single-photonlevel optical pulse with high SNR. The limited SNR obtained at a few photons is entirely given by the low overall memory efficiency. Future experiments should therefore aim at increasing the efficiency, while we consider the filtering to be sufficient for quantum applications. The memory inefficiency is due to two major factors; 1) insufficient optical depth and 2) insufficient control field transfer efficiency. To increase the optical depth we will implement an impedance-matched cavity around the memory, as proposed in [38, 39]. Indeed recent results using such an impedance-matched cavity have shown an optical AFC efficiency of 58% using a crystal with optical depth comparable to ours if in a single-pass configuration [22]. 2) The control field transfer efficiency can most easily be improved by using longer adiabatic transfer pulses [40]. Such long temporal windows can be created by increasing the AFC echo time (1/∆). Using the narrowest measured optical homogeneous linewidth measured with europium of 122 Hz [33], 1/∆ times of around 1 ms are in principle possible. Furthermore increasing 1/∆ will also allow us to exploit the multimode capability of the AFC scheme. Currently, however, our laser linewidth represents serious technical obstacle in increasing 1/∆ towards this limit, beyond the shown 6-8 µs. We also note that employing long adiabatic control fields should reduce off resonant excitation, further decreasing the remaining noise. To conclude, we have demonstrated the first optical storage as a spin-wave in a solid-state memory, in the regime of a few photons per input pulse. This was made possible by a strategy of extensive filtering and by carefully shaping the temporal envelope of the strong control pulses. The final unconditional noise floor of (7.1 ± 2.3) · 10−3 is low enough to allow for quantum schemes using spin-wave storage and manipulation, such as the generation of quantum-correlated spin-wave and photonic excitations using variant of the DLCZ [41] approach adapted to the solid-state [42–44]. These schemes will, in turn, allow for generation of entanglement between light and matter and entanglement of solid state remote quantum memories, a basic building block for quantum repeaters. We would like to thank C. Barreiro for technical assistance, and N. Sangouard and P. Goldner for useful discussions. We gratefully acknowledge R. Cone and R.M. 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