Rare-earth quantum memories for single photons and entanglement

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Thesis

Reference

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.

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.

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UNIVERSITÉ DE GENÈVE FACULTÉ DES SCIENCES

Groupe de Physique Appliquée - Optique 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

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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 mé- canique classique, comme la superposition d'états quantiques ou l'intrication. Parmi les possibles champs d'applications, on retrouve les ordinateurs quantiques, qui per- mettraient 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 particu- larités 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 expéri- mentales, 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 irrémédi- ablement 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 amèn- eraient 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

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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.

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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 dif- ferentiates 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

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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

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QM QM QM QM QM QM QM QM

QM QM

QM QM QM QM

A Z

a) Entanglement creation

b) First entanglement swapping

B C D ... W X Y

D

A Z

...

W

c) Last entanglement swapping

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 eciencyη 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 long- term 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

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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.

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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 near- infrared 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 magneto- optical 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

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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 Γ_inh in the optical transition, typically between 100 MHz and 10 GHz. We note that this is much broader than the homogeneous linewidth (γ_h =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 ^Γ_γ^inh_h ratio, a high multimode capacity. This is the basis of the atomic frequency comb memory.

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 aπ 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

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Figure 2.1: An example of energy levels of rare earth elements.

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Figure 2.2: Illustration of a inhomogeneous broadening in a rare earth doped crystal. Γ_inh 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= XN

j=1

cje^iδ^j^te^−ikz^j|g1. . . ej. . . gNi (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,δ_j 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, δ_j 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^j^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^j. 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 opticalπ 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

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e

g

Atomicdensity aux

Atomic detuningd Inputmode

Outputmode Co

ntro lfie

lds

s

D g (a)

Intensity

Time Input

mode

Output Control fields mode

/ 0

2p D-T Ts T0

(b)

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 detuning2π∆, 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 aπpulse. 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 susceptibilityχfor an ensemble of two-level atoms can be written as a perturbation solution[44]:

χ(ω)∝ XN

j=1

c_j

(ω_j−ω)−iγ_h (2.2)

ω is the angular frequency of the light eld,ω_j is the resonant frequency of atom j and γ_h is the homogeneous linewidth. Hence, the complex wave number can be calculated

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and depends strongly on ω:

k(ω) = ω/cp

1 +χ(ω) The absorption coecient is given by:

α= 2k⁰⁰

wherek⁰⁰ is the imaginary part ofk. 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:

E_in(t) = Z

E(ω)e˜ ^−iωtdω

Through the medium of length L, each components propagates and acquire a phase k(ω)L. Hence the output eld can be calculated:

E_out(t) = Z

E(ω)e˜ i(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−|sitransition (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−|eitransition and the phases evolve again. This leads to the re-emission of an echo at the time T_s+ 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

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Class: 1 2

(a) (b)

Angular frequency

Absorption

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^−αLI(0)

I(l) is the intensity of the light at position l and α is the absorption coecient. We often use the optical depthd=αLin 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 (N_g) and excited state (N_e):

α= (N_g−N_e)σ.

whereσis 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 anti- hole). 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 ∆E_g 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 (T_z) is longer than the excited state lifetime (T₁). A pumping beam at an angular frequency ω_pump 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.

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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 T_Z. 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 T_Z and T₁:

ρ₂

ρ₁ = 1 + 2T_Z 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.

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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

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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 Nd³⁺:Y2SiO5

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 Zee- man 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 lim- itations were due to inecient optical pumping and strong superhyperne interactions between Neodymium and Vanadium ions which aected the quality of the AFC prole.

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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 er- bium 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 tran- sition

As expected [54], Nd³⁺:Y2SiO5has a resonance at 883.2350(6) nm corresponding to the⁴I⁹₂-⁴F³₂ 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 = 1.32cm⁻¹). From additional measurements[55], it appears that the eigen axes of refraction index coincide with the eigen axes of absorption.

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Inhomogeneous Broadening

α [cm-1]

0 1 2 3

Detuning [GHz]

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

polarization along D1

polarization along D2

Figure 3.1: Absorption spectrum of Nd³⁺:Y2SiO5. The FWHM of the inhomogeneous broadening 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 (M_s = ±1/2) in the ground level. They can be used as|giand|auxi(2.3) for a two-level AFC protocol. For an ecient optical pumping, thus ecient AFC storage, the Zeeman state lifetime (T_Z) 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 300µs. Therefore, we needT_Zto be at least tenths of ms. We will present in this section measures of T_Z for various magnetic elds. Although the population relaxation occurs in radio frequencies, we can measureT_Z with an optical laser, through the dynamic of

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TZ [ms]

0 2 4 6 8 10 12 14

Angle of B with D2

-90 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 90

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 rate1/T_Z. Therefore, we measured the dynamic of a spectral hole after the optical pumping, and the obtained decay time isTZ. 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 T_Z in Nd³⁺:Y2SiO5. We started with a moderate magnetic eld (≈ 20 mT) by the use of permanent magnets placed outside the cryostat. We measuredT_Z for various anglesθ between the D₂ axis and the magnetic eld. We note thatθ andθ+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 T_Z.

To search for longer Zeeman lifetimes, we increased the magnetic eld (up to 2 T)

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for two particular angles, θ = −30and θ=90(g.3.3a). For both angles, T_Z 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 T_Z 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 T_Z 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/T_Z) can be written as[50]:

R(∆E) =α_D ·(∆E)⁵coth

∆E 2k_BT

+α_RO(T), (3.1)

where α_D is an anisotropic constant, ∆E the split in the ground state, k_B the Boltz- mann 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]:

R_{f f} =α_{f f}sech²

gµ_BB 2kBT

,

whereα_{f f} is an anisotropic constant andµ_B 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 T_Z 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 T⁹ 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 withθ=-30).

It appears that the spin relaxation time is roughly constant between 3 and 4 K and it

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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.

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0 5 10 15 20 25 30 35 40 45 10¹

10²

Zeeman states split (GHz)

Spin relaxation rate (Hz)

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 inef- cient. We will see in section 3.2 how it does aect the storage eciency. In a simple 3-level model (sec.2.3), the ratiod₀/d should be equal to the ratio between the excited state lifetime and the Zeeman state lifetime (ifT1T_Z). Here,T₁=300µs and for this magnetic eld T_Z=120 ms, so that we have T₁/T_Z=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.

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population fraction in initial ground state 0 0.2 0.4 0.6 0.8 1

waiting time after optical pumping [ms]

0 100 200 300 400

3.2 K 3.7 K 4.2 K 4. 8K 5.4 K

Figure 3.5: Spectral hole decay for various temperatures. Under 4 K, the spin relaxation rate is constant.

optical depth

0 0.5 1 1.5 2 2.5 3 3.5

Detuning [MHz[

-20 -10 0 10 20 30 40 50

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.

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B[mT] θ[] T2[µs] x 20 -30 4.9±0.2 2

77 -30 15±9 1

300 -30 93±15 1

20 5 6.3±0.3 2

300 30 60±3 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

2 and a spectral hole cannot be narrower than 2γ_h [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 approachingγh. 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 t12after 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]: I_echo = I₀e^−(4t¹²^/T²⁾^x where x ∈ [1; 2] is a phenomenological constant resulting from spectral diusion. The obtained values ofT2

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 γ_h=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

1T1has been measured through uorescence detection and stimulated photon echoes

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T2 [μs]

0 10 20 30 40 50 60 70 80 90

Temperature [K]

2.5 3 3.5 4 4.5 5 5.5 6 6.5

Figure 3.7: Optical coherence time with increasing temperature.

burning.

Finally, we measuredT₂for higher temperatures, with B=300 mT andθ =−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

2We note that SHB may be used to measure the linewidth of a laser if the atoms linewidth is known

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optical depth

0 1 2 3

detuning [kHz]

-4'000 -2'000 0 2'000 4'000

B=10mT B=300mT

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: γ_h=3.4kHz for 300 mT and γ_h=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

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±194 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.

Conclusion on Nd

³⁺

:Y

2

SiO

5

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 two- level 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

Rare-earth quantum memories for single photons and entanglement

Thesis

Reference