Solid-state light-matter interfaces on the quantum test bench

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Thesis

Reference

Solid-state light-matter interfaces on the quantum test bench

CLAUSEN, Christoph

Abstract

Solid-state light-matter interfaces based on crystals doped with rare-earth ions have shown great progress in recent years, heading towards quantum technology applications. In this thesis we present three experiments testing the suitability of an atomic frequency comb light-matter interface, implemented in an Nd:YSO crystal, for the storage of quantum information. A dedicated source of entangled photons has been developed, and the storage and retrieval of quantum correlations and entanglement been studied. We show the successful transfer of photonic entanglement to the crystal, and the generation of a matter-matter entangled state between two crystals. These are two of the main steps in a quantum repeater protocol. We also demonstrate faithful storage and retrieval of quantum information encoded in the polarization of heralded single photons. These experiments prove the strong potential of solid-state light-matter interfaces based on atomic frequency combs for quantum communication.

CLAUSEN, Christoph. Solid-state light-matter interfaces on the quantum test bench. Thèse de doctorat : Univ. Genève, 2013, no. Sc. 4545

URN : urn:nbn:ch:unige-277063

DOI : 10.13097/archive-ouverte/unige:27706

Available at:

http://archive-ouverte.unige.ch/unige:27706

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 Professeur Nicolas Gisin

solid-state light-matter

interfaces on the quantum test bench

thèse

présentée à la Faculté des Sciences de l’Université de Genève pour obtenir le grade de Docteur ès sciences, mention physique

par

christoph clausen

de Flensburg (Allemagne)

Thèse N^o 4545

Genève Mars 2013

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A B S T R A C T

The understanding of quantum physics has evolved to the point of realization that certain phenomena, and in particular entanglement, offer unique resources with great potential for technological applications. As an example, quantum key distribution allows two distant parties to exchange cryptographic keys whose security is in principle guaranteed by the laws of quantum mechanics. Quantum key distribution belongs to the broader field of quantum communication, for which one of the pri- mary goals is the long-distance distribution of entanglement. Once in place, the entanglement could be used for remote quantum computation or to transfer quantum information with the help of teleportation.

Current quantum communication channels are subject to a kind of distance barrier caused by the finite transparency of optical fibers:

The probability that the individual photons carrying the entanglement reach the other end of the channel decreases exponentially with distance, which is thus limited to the order of100km. Possibilities for the implementation of a quantum repeater, with which this distance barrier could be breached, are currently being researched, including the development of suitable photon sources and quantum memories.

A quantum memory is a special kind of light-matter interface – a device that can transfer information from one to the other – that can accept and preserve quantum information. To this category belong the rare-earth-ion doped crystals prepared as atomic frequency combs, which have recently been developed. These particular interfaces have an inherent capacity for extensive temporal multiplexing, which makes it ideal for quantum repeaters.

In this thesis we present a series of experiments that test the compat- ibility of the atomic frequency comb light-matter interface with quantum information. The device under test is an yttrium-orthosilicate crystal doped with neodymium ions. For this purpose, a dedicated source of entangled photons has been developed, and the storage and retrieval of quantum correlations and entanglement been studied. We show the successful transfer of photonic entanglement to the crystal, and the generation of a matter-matter entangled state between two crystals. These are two of the main steps in a quantum repeater protocol. In a third experiment we demonstrate the faithful mapping of quantum information encoded in the polarization of a heralded single photon onto the crystal and back. Together, these experiments prove the strong potential of solid-state light-matter interfaces based on atomic frequency combs for quantum communication applications.

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R É S U M É

La compréhension de la physique quantique a évoluée jusqu’au point où il a été realisé que certains phénomènes, et particulièrement l’intrication, offrent des ressources uniques avec un potentiel énorme pour des applications techniques. La distribution quantique de clés, par exemple, permet à deux interlocuteurs distants d’échanger une clé de chiffre- ment, dont la confidentialité est en principe garantie par les lois de la mécanique quantique. La distribution quantique de clés fait partie du champ de la communication quantique, où un des buts primaires est la distribution d’intrication à grande distance. Une fois établie, l’intrication pourrait servir aux calculs quantiques répartis ou au transfert de l’information quantique par téléportation.

Les canaux de la communication quantique actuels sont soumis à une barrière de distance à cause de la transparence finie des fibres optiques : la probabilité qu’un photon individuel porteur d’intrication arrive à l’autre côté decroît exponentiellement avec la distance, qui est donc limitée à environ 100km. Les possibilités de construire un répé- teur quantique, qui permet de surmonter cette barrière, font présen- tement l’objet de plusieurs travaux de recherche, ce qui nécessite le développement de sources de photons et mémoires quantiques.

Une mémoire quantique est une interface entre lumière et matière – un dispositif permettant de transferer d’information de l’une à l’autre – qui accepte et préserve l’information quantique. Dans cette catégorie se trouve le cristaux dopés aux terres-rares préparés comme peigne de fréquence atomique, une technique récemment développée. Cette technique possède une grande capacité de multiplexage temporel, une propriété idéale pour le répéteur quantique.

Dans cette thèse nous présentons une série d’expériences qui testent la compatibilité d’une interface lumière-matière basée sur le peigne de fréquence atomique avec l’information quantique. L’échantillon étudié est un cristal d’orthosilicate d’yttrium dopé avec des ions de néodyme.

Pour les besoins de la cause une source de photons intriqués a été dé- veloppée, et nous avons investigué le stockage ainsi que la lecture des corrélations quantiques et de l’intrication. Nous montrons le transfert d’intrication au cristal et la création d’un état intriqué matière-matière entre deux cristaux – ce qui correspond à la réalisation de deux élé- ments fundamentaux d’un répéteur quantique. Dans une troisième ex- périence nous avons démontré le stockage fidèle de l’information quantique encodé dans la polarisation d’un photon unique annoncé. La to- talité des expérience prouve le fort potentiel des interfaces lumière- matière basées sur les cristaux dopés aux terres-rares en combinaison avec la technique de peigne de fréquence atomique dans la régime de la communication quantique.

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C O N T E N T S

1 introduction 2

2 quantum repeaters based on multimode quantum mem-

ories 6

2 . 1 The quantum repeater . . . 7

2 . 2 A quantum repeater based on atomic ensembles . . . 8

2 . 2 . 1 Entanglement generation . . . 9

2 . 2 . 2 Entangling two atomic ensembles . . . 10

2 . 2 . 3 Readout and entanglement swapping . . . 11

2 . 3 Photon-pair sources and multimode memories . . . 11

2 . 4 Conclusion . . . 12

3 a multimode, solid-state light-matter interface 14 3 . 1 Introduction . . . 14

3 . 2 The atomic frequency comb . . . 16

3 . 2 . 1 Complete protocol with on-demand readout . . . 17

3 . 2 . 2 Efficiency . . . 18

3 . 2 . 3 Multimode capacity . . . 20

3 . 3 Important experimental results for the AFC . . . 21

3 . 4 The neodymium quantum memory . . . 22

3 . 4 . 1 Preparation and properties of the AFC . . . 23

4 an afc-compatible photon-pair source 26 4 . 1 Introduction . . . 26

4 . 2 Spontaneous parametric down-conversion . . . 27

4 . 2 . 1 The two-mode squeezed state . . . 28

4 . 2 . 2 Some aspects of multimode SPDC . . . 29

4 . 3 Implementation . . . 31

4 . 3 . 1 The waveguide . . . 32

4 . 3 . 2 Filtering . . . 32

4 . 3 . 3 Stabilization . . . 34

4 . 3 . 4 Second generation source . . . 35

4 . 3 . 5 Characterization . . . 36

5 exposing the quantum memory to quantum light 40 5 . 1 Quantum storage of photonic entanglement in a crystal 40 5 . 1 . 1 Non-classical intensity correlations . . . 41

5 . 1 . 2 Light-matter entanglement . . . 41

5 . 2 Heralded quantum entanglement between two crystals . 43 5 . 3 Storage of polarization qubits . . . 45

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contents 1

5 . 4 Conclusion . . . 46 6 photon number counting with quantum memories 48

7 conclusion and outlook 50

7 . 1 Outlook . . . 51 7 . 1 . 1 Towards the implementation of an elementary link 51 7 . 1 . 2 Better quantum memories . . . 52 7 . 1 . 3 Long-term perspectives . . . 53

Acknowledgements 56

Bibliography 58

Published articles 66

Quantum storage of photonic entanglement in a crystal . . . . A-1 Heralded quantum entanglement between two crystals . . . . B-1 Quantum storage of heralded polarization qubits in birefrin-

gent and anisotropically absorbing materials . . . C-1 Detector imperfections in photon-pair source characterization D-1 Analysis of a photon number resolving detector based on flu-

orescence readout of an ion Coulomb crystal quantum memory inside an optical cavity . . . E-1

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1 I N T R O D U C T I O N

In the past few decades, quantum mechanics has left the regime of pure science and entered the realm of technology and applications.

This is the essence of what some people call thesecond quantum revo- lution(Dowling and Milburn,2003). Thefirstquantum revolution was started by the introduction of the wave-particle duality, a concept which soon enabled physicists to explain the structure of matter, e.g. the pe- riodic table, to a large extent. These new insights led to a series of important technological developments, most notably the semiconductors which now pervade our everyday life. However, quantum mechanics itself retained its mostly descriptive nature: it had not yet been realized that quantum phenoma like superposition and entanglement are resources that can be exploited for technological innovation.

The second quantum revolution was triggered by a combination of several factors. It had soon been realized that the constant push towards smaller devices inevitably leads to a point where the laws of quantum mechanics become important for device production and func- tionality. Yet, the transition to quantum mechanical devices was not just an unavoidable evil looming somewhere in the future; it was discov- ered that quantum entanglement offers a new kind of computational resource with great promise for applications, in particular in information processing and metrology. The proposal of the first quantum key distribution protocol (Bennett and Brassard,1984), and the quantum algorithm for factoring large numbers (Shor,1994) revealed that quantum technology can offer solutions to problems which were previously unsolvable or intractable. The actual implementations of these kinds of proposals require robust quantum systems with specific properties, which are not necessarily easy to obtain or isolate in nature. Luckily, technology has advanced sufficiently to allow for the creation or en- gineering of new quantum systems and quantum states, and in many areas of physics large efforts are now devoted in this endeavour.

Among the new quantum technologies, quantum communication plays a key role. The idea of being able to communicate over channels whose security is based on the laws of physics, quickly fell into fruit- ful ground, and commercial implementations of quantum key distribution systems have recently become available. In a more general setting,

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introduction 3

quantum communication investigates the possibilities for establishing data channels between quantum computation units. The length scales over which such a communication can take place may vary by several orders of magnitude. An implementation, for example, of a quantum computer has to fulfill two seemingly contradictory conditions (Ladd et al.,2010). On the one hand, the individual quantum bits need to be very well protected from the environment to avoid errors. On the other hand, the efficient manipulation of the quantum bits requires a strong sensitivity to some part of the environment. One could now imagine a quantum processor on a chip using different species of quantum bits for these two purposes and exchange the information between them by means of quantum communication. However, the implementation of a quantum computer with even only a small number of quantum bits faces extreme challenges. One way to obtain scalability is the use of an architecture for distributed computing, similar to the internet (Kimble, 2008). Indeed, it is not unlikely that only a few selected institutions will possess the resources to build and maintain such quantum computers, whose computing power could then be enhanced by connecting them with quantum links.

For long-distance quantum communication it would be highly advan- tageous to keep using the existing networks of optical fibers. However, the direct transmission of quantum information is bound to fail for long distances because of the inherent absorption and decoherence in the fiber. Even the remarkable fibers employed for telecommunication to- day cause an attenuation by one order of magnitude for every50km.

The classical solution of straightforward amplification is forbidden for quantum states (Wootters and Zurek,1982), but Briegel et al.(1998) proposed a scheme for aquantum repeaterthat efficiently creates entanglement between a pair of particles separated by an arbitrary distance.

The entangled pair can subsequently be used to transfer quantum information by means of quantum teleportation (Bennett, Brassard, et al.,1993).

The quantum repeater is based on a divide-and-conquer strategy. Ba- sically, the communication distance is broken down into a series of short elementary links at whose ends entanglement can be established via direct transmission. The distance spanned by the entangled pair is then increased link by link via entanglement swapping operations.

For this to be efficient, the preparation of entanglement in the elementary links needs to be asynchronous. Synchronization between adjacent links is achieved by the use ofquantum memories, a device that faith- fully stores quantum bits and releases them on demand.

In the context of optical communication, a quantum memory is typically realized by coherent absorption and reemission of single photons. For a few years, hot or cold atomic vapors were the only systems that provided sufficient control and coherence. Solid-state systems in the form of rare-earth ion doped crystals later entered the field with

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4 introduction

promises of easier handling and longer storage times (Lvovsky, Sanders, and Tittel,2009). Specialized protocols were developed to deal with and even take advantage of the relatively large inhomogeneous broadening, which is characteristic for these systems. One of the newest of these protocols, and arguably among the most promising for quantum repeaters thanks to the ability of temporal multiplexing, is the atomic frequency comb, introduced by Afzelius, Simon,et al.(2009) from the University of Geneva. While its capabilities as an efficient light-matter interface were quickly confirmed by experiments, its suitability for the storage and retrieval of quantum states was yet to be investigated.

During the course of this thesis a series of proof-of-principle experiments have been developed and performed to confirm the suitability of solid-state light-matter interfaces, based on the atomic frequency comb protocol, for quantum communication purposes. In particular, a source of quantum light compatible with the requirements of quantum repeaters has been developed. The source was then combined with the quantum memory to demonstrate the following capabilities, which are at the heart of a possible implementation of quantum repeaters:

• preservation of quantum correlations, and in particular quantum entanglement during the storage,

• creation of heralded entanglement between two quantum memories,

• and the storage of polarization qubits despite birefringence and anisotropic absorption.

The thesis is organized as follows. Chapter 2 revises the original idea of the quantum repeater, and a seminal protocol by Duan et al.

(2001) based on the use of atomic ensembles. In Chapter 3 we discuss quantum memories in general and present the atomic frequency comb protocol, its characteristics and the specific implementation used for the experiments of this thesis. Chapter 4introduces the source of entangled photons that has been developed following the compatibil- ity requirements of the quantum memory, and the quantum nature of the generated light is discussed. The following chapter contains the main results of this thesis, obtained through experiments where the photon source and the quantum memory were combined. In chapter6 we present a technique for number-resolved single-photon detection based on quantum-memory techniques, and discuss how it can be implemented. A conclusion and outlook are given in Chapter7.

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2 Q U A N T U M R E P E A T E R S B A S E D O N M U L T I M O D E Q U A N T U M M E M O R I E S

Light is the fastest possible carrier of information and has by now established itself as the medium of choice for telecommunication purposes.

A large range of high-end devices has been developed for that purpose, including light sources, optical fibers and detectors. Light continues to be an excellent information carrier when working at the single photon level. Additionally, single photons are only weakly or not at all affected by many typical noise sources, they have several degrees of freedom suitable for encoding quantum information, and they are easily manip- ulated (O’Brien,2007).

The ability to transfer quantum information over long distances is attractive for many reasons of both fundamental and applied nature.

One could, for instance, wonder if there is a limit on the distance that can be spanned by quantum correlations such as entanglement. On the applied side research is mostly spurred by security considerations. On the one hand, the invention of the quantum computer threatens the security of widely used algorithms for public-key cryptography. On the other hand, quantum information processing itself offers the ideal solution, namely quantum key distribution, which by now has been made commercially available. But also outside the regime of quantum information processing advantages can be gained by making use of quantum communication, such as a greatly enhanced resolution of interferomet- ric telescopes (Gottesman, Jennewein, and Croke,2012).

The main obstacle for long-distance quantum communication is photon loss, which limits the distance over which photons can be transmit- ted directly. Quantum repeaters offer a possible solution to this prob- lem. In the remainder of this chapter the basic idea of the quantum repeater is reviewed, followed by a discussion of the seminal protocol proposed by Duan et al.(2001). Finally, it is shown how one can significantly increase the communication rates by introducing multimode quantum memories and spontaneous down-conversion sources (Simon et al.,2007).

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2 . 1 the quantum repeater 7

QM QM QM QM QM QM QM QM

A C₁ C₂ C₃

QM QM

C₄ B

Elementary link QM QM QM

A C₂

QM QM

C₄ B

Entanglement swapping

QM

A B

Entanglement purification Entanglement creation

QM

A B

QM

A B

×M

×1 QM

QM

QM QM

Figure 2.1:The quantum repeater. Intermediate stationsC_iare introduced between partiesAandB and entanglement created in the resulting elementary links by direct transmission. The distance of entanglement is increased by entanglement swapping at the intermediate stations, until the whole distance is bridged. Errors lead to the reduction of entanglement fidelity, indicated by the color of the links. Via entanglement purification a high fidelity entangled pair can be distilled from many low-fidelity pairs.

2 . 1 the quantum repeater

In classical communication, long distances are bridged with the help of repeaters. These intermediate stations receive the signals and retrans- mit fresh copies. The copying of general quantum states is not possible (Wootters and Zurek,1982), but Briegelet al.(1998) showed that the repeater concept can nevertheless be used to distribute an entangled pair between distant locations. In their proposal, whose essence is illustrated in Fig. 2.1, the communication channel is divided into elemen- tary linksof length L₀ which is sufficiently short to permit the creation of entanglement between its endpoints via direct transmission. This is done simultaneously, but independently, in all the elementary links.

Once two neighbouring links each contain entangled pairs of sufficient quality, the two pairs can be transformed into a single pair by teleportation of entanglement (Bennett, Brassard, et al.,1993; ˙Zukowski et al.,1993), also calledentanglement swapping. The new pair now spans twice the distance. Using these larger pairs one can again perform entanglement swapping to extend the entanglement even further until, eventually, an entangled pair is obtained that spans the whole distance.

The quality of an entangled pair, represented by a density matrix_ρ, is typically quantified by its resemblance to a maximally entangled state

|ΨM E〉via thefidelity, defined as

F(ρ) =^p_〈ΨM E|ρ|ΨM E〉. (2.1)

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8 quantum repeaters based on multimode quantum memories

Imperfections during the entanglement creation and entanglement swapping operations lead to a reduction of the entanglement fidelity as the spanned distance increases. If the fidelity risks to drop below a certain threshold value, the process can be repeated several times for the same distance, and multiple entangled pairs combined to form a single entangled pair of higher fidelity. In the original proposal, this is calledpurificationof entanglement. Here, the specific threshold fidelity depends on the intended purpose of the entanglement pair. As an example, the security of entanglement-based quantum key distribution (Ekert,1991) is guaranteed by the violation of a Bell’s inequality. As-

suming that the entanglement degradation can be modelled by white noise, that is ρ = V

ΨM E

+ (1−V) 1/⁴, the violation of the Clauser-Horne-Shimony-Holt inequality (Clauseret al.,1969) requires V >1/^p2⇔F>0.88.

Important is here that even with purification the time and resources required to establish the final entangled pair depend only polynomially on the total distance, instead of exponentially, which is the case for direct transmission. Two key points that make this possible are

• the elementary links can be prepared independently from one another, and

• once entanglement is created in the elementary links or in subsequent steps of the protocol, it can be maintained until used. For the endpoints AandB of the entire link, the entanglement must be maintained until the very end of the protocol.

The second point is equivalent to maintaining the total quantum state of two separate objects, one in each end of the elementary link, for an extended amount of time. In other words, each end of the link requires aquantum memory.

2 . 2 a quantum repeater based on atomic ensembles

Three years after the invention of the quantum repeater, a concrete implementation was proposed by Duanet al.(2001), now often referred to as the DLCZ protocol. The protocol is based on ensembles of atoms as stationary systems and linear optics for entanglement generation and swapping, both of which are simple to realize. The implementation consists of the following steps

1. locally generating entanglement between a collective atomic excitation and a photon,

2. a Bell-state measurement on two such photonic modes to create entanglement between two remote atomic excitations, and

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2 . 2 a quantum repeater based on atomic ensembles 9

~k_w

~k_r

~kS

~kAS

|g〉

|e〉

|s〉 write

|g〉

|e〉

|s〉 read

EnsembleL Bell-state EnsembleR

measurement a_L

b_L

a_R b_R

Figure 2.2:Elementary link for a quantum repeater based in the DLCZ protocol. Two distant atomic ensembles are exposed to write-pulse which induce spontaneous Raman transitions with some low probability. The Stokes modes are coupled into optical fibers and sent to a central station. Upon detection of a single photon, the passage through the beam splitter has erased the information of which ensemble the photon came from, leading to entanglement between the two ensembles. Entanglement swapping can be performed by sending a read-pulse to convert the stored excitations into anti- Stokes photons, which are then subjected to the same kind of Bell- state measurement.

3. converting atomic excitations into photons to perform entanglement swapping between neighbouring links.

In the following these individual steps are described in detail.

2 . 2 . 1 Entanglement generation

The entanglement generation is based on spontaneous Raman scatter- ing in an atomic ensemble, see Fig. 2.2. The ensemble consists of N identical atoms with ground state_|g〉, a metastable state_|s〉and an excited state _|e〉. The state _|e〉 is higher in energy than both _|g〉 and_|s〉 and connected to both via optical dipole transitions. After preparation of the system such that all the atoms are in the ground state it is exposed to an off-resonant laser pulse on the_|g〉 ↔ |e〉transistion. With a certain probability p1 this induces a Raman transition of one of the atoms into state_|s〉with the simultaneous emission of a Stokes photon. If there is no means to tell which of the atoms has undergone the transition, the atomic state is now described by the collective excitation,

S^†|vac_〉= _p¹ N

XN

j=1

c_jeⁱ⁽^~^k^w⁻^~^k^S⁾^~^x^j|g₁g₂. . .s_j. . .g_N〉, (2.2) where the vacuum state_|vac_〉denotes the state without any excitations.

The weights c_j depend on the coupling between the j’th atom at posi-

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tion~x_jand the light field. Each term carries a phase that is proportional to the difference between the wave-vectors of the write (^~k_w) and Stokes fields (^~k_S). Denoting the mode of the Stokes photon bya, the combined state of light and matter is entangled and of the form

|φ0〉=1+^ppS^†a^†

|vac_〉+O(p). (2.3)

Here,O(p)represents terms that correspond to multiple spontaneous Raman transitions into the same Stokes mode, which happen with a probability of at mostp².

2 . 2 . 2 Entangling two atomic ensembles

When two distant atomic ensembles L and R are prepared simultaneously, one obtains a product state|φ0〉L⊗ |φ0〉Rof the combined system.

However, entanglement between the two atomic ensembles can subsequently be created by entanglement swapping, triggered by a joint measurement on the photonic modesa_Landa_R. This is most easily done by combining the two modes on a 50/50 beam splitter with detectors at the outputs. Assuming that at most one photon has been emitted by the two ensembles, the detection of a single photon after the beam splitter corresponds to a projection onto the states

b^†_L|vac_〉= _p¹ 2

a^†_L+e^iφa^†_R

|vac_〉= _p¹ 2

|1〉L|0〉R+e^iφ|0〉L|1〉R

b_R^†|vac_〉= _p¹ 2

a^†_L−e^iφa^†_R

|vac_〉= _p¹ 2

|1〉L|0〉R−e^iφ|0〉L|1〉R

(2.4)

whereφis a relative phase accumulated during the creation and propagation of the photons. The detection leaves the atomic ensembles in a maximally entangled state with a single delocalized excitation,

|ψ_atoms〉= _p¹ 2

|1〉L|0〉R±eⁱ^φ|0〉L|1〉R

, (2.5)

where the kets now represent atomic modes.

Two important points need to be made. First, the creation of the state

|ψatoms〉is probabilistic and happens with probability p·η, where_η is the product of the channel transmission and the detection efficiency. By far the most of the trials will be without success, because one will want to keep p1 to avoid multiple excitations. However, once a photon is detected it is certain that the atomic ensembles are entangled as in Eq. (2.5). The detection event thus heralds the creation of entanglement. Second, the entanglement is encoded in long-lived atomic states and can be preserved until the neighbouring links are ready to proceed to the next step of the protocol. Hence, the atomic ensembles them- selves serve as a quantum memory, although the quantum state to be stored has been created directlyinsidethe memory.

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2 . 3 photon-pair sources and multimode memories 11

2 . 2 . 3 Readout and entanglement swapping

We have just seen how to perform a joint measurement on two photonic modes with the help of a beam splitter. The same procedure can be used to connect neighbouring links in order to extend the distance spanned by the entangled pair. To do this, the atomic excitation needs to be reliably read out, that is, converted into a photon. This is the step where the collective enhancement of the atoms in the ensemble becomes important. A laser pulse resonant with the_|s〉 ↔ |e〉transition and wave-vector^~k_rbrings the collective excitation in Eq. (2.2) from_|s〉 to _|e〉, adding to each term the corresponding phase of the read pulse.

Now, each of these terms can decay back to the collective ground state by emitting an anti-Stokes photon with wave-vector^~k_AS, and the ampli- tude of this process is proportional to

N

X

j=1

c_jeⁱ⁽^~^k^w⁻^~^k^S⁺^~^k^r⁻^~^k^AS⁾^~^x^j, (2.6) assuming that the atoms do not move between the write and read pulses. We see that the direction of emission of the anti-Stokes photon is subject to a phase matching condition. Hence, when N is large, there will be strong constructive interference for emitting the photon in the direction^~k_AS=^~k_w+^~k_r−~k_S, making efficient collection possible. As before, the detection of a single anti-Stokes photon heralds the success of the entanglement swapping.

2 . 3 photon-pair sources and multimode mem- ories

Since its proposal, the DLCZ protocol has been studied extensively, problems have been identified, and improved protocols proposed (San- gouardet al.,2011). The work in this thesis has been inspired by possible solutions to two problems in particular:

• Suitable transitions in atomic ensembles are typically found in the visible to near-infrared spectral regions from 600nm to 900nm.

These wavelengths are inconvenient for long-distance transmission through optical fibers: the loss for every kilometer of fiber is at least 10 times higher than in the so-called telecom bands around1300nm and1500nm.

• In the elementary link, only a single attempt of entanglement creation can be made at a time, because the memories need to be emptied before the next trial. It turns out that this puts a strong limit on the rate of entanglement generation, given by the time that it takes for the signals to travel between the atomic ensemble

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and the central measurement station. Typical distances are here around50km, which corresponds to a single trial every500µs.

The first point could be addressed by methods of optical frequency conversion, e.g. by four-wave mixing in atomic ensembles (Radnaevet al., 2010), or difference-frequency generation in non-linear crystals (Curtz et al.,2010). However, achieving efficient conversion without adding noise remains experimentally challenging.

The second point calls for some kind of multiplexing. Imagine, for instance, using a whole series of atomic ensembles instead of just a single one. The write pulses are sent to the individual ensembles in sequence. If the resulting Stokes photons are coupled into the same fiber using some kind of multi-port switch, the time of detection at the central station reveals which one of the ensembles contains the excitations, and this ensemble is used for the remainder of the protocol. In such a scenario, the speed-up is proportional to the number of multiplexed atomic ensembles.

A repeater protocol that provides solutions to both points at the same time has been proposed by Simonet al. (2007)¹. In the DLCZ protocol, a photon is created simultaneously with an excitation inside the quantum memory. Simonet al.suggest that the same situation can be reached by first generating a pair of photons, and then storing one of them in the memory. The separation of the photon creation from the storage procedure gives more flexibility in terms of the frequencies of the generated photons and the implementation of the quantum memory. In particular, the two photons in a pair can have widely different frequencies, where one is adapted to the quantum memory and the other in a telecom band. Additionally, one can now choose to use a multimode quantum memory which for a single preparation can store and retrieve many temporal modes, equivalent to the collection of atomic ensembles mentioned above.

2 . 4 conclusion

Long-distance quantum communication could open up interesting possibilities for fundamental research and applications of quantum technologies. By using a quantum repeater it is in principle possible to create remote entangled pairs at a rate which is polynomial in the distance of communication. For sufficiently large distances this will beat the direct transmission, where the rate drops exponentially with the distance.

Duanet al. (2001) identified the key steps of the quantum repeater protocol and showed that they can be implemented using spontaneous Raman transitions in atomic ensembles. Among the requirements are

1 For a different approach to parallelization and multiplexing, see also Collins et al.

(2007)

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2 . 4 conclusion 13

• the creation of entanglement between a “flying” and a stationary quantum system, i.e. between a photon and an atomic ensemble,

• the preservation of a stationary quantum state for the duration that is required to complete the entire repeater protocol,

• the controlled and coherent conversion of the collective atomic excitations into photons at a desired time (i.e. on-demand), and

• the heralded creation of entanglement between two quantum memories.

While the proposed implementation should work as expected, it is required that the photons are at wavelengths suitable for long-distance transmission through optical fibers. Additionally, the entanglement creation rate is limited by the time that it takes for the signal to travel between the different stations.

A variation of the DLCZ protocol has been proposed by Simonet al.

(2007), where the stationary quantum systems are replaced by quantum memories which can absorb and emit single photons. This way, a significant speed-up can be obtained by using quantum memories that support temporal multiplexing. Such memories are the subject of the next chapter. These memories are then combined with photon-pair sources, where one photon is created at a telecom wavelength and the other compatible with the memory. An implementation of such a photon source shall be discussed in chapter4.

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3 A M U L T I M O D E , S O L I D - S T A T E L I G H T- M A T T E R I N T E R F A C E

3 . 1 introduction

It has been argued in the previous chapter that a quantum memory is an essential building block of a quantum repeater. To that end, a quantum memory is a device that performs a delayed identity transfor- mation, i.e. it leaves the input quantum state unchanged and releases it after a variable delay. Please note that this is not equivalent to what is sometimes called a delay-line, which might have a variable length, but once the signal enters the delay line, the time at which it comes out can no longer be changed. For quantum repeaters and most other purposes, the input state is first stored, and later retrievedon demand.

The most obvious purpose of a quantum memory is the synchronization of otherwise independent parts of a quantum device. Along the same line it can make probabilistic processes semi-deterministic. Con- sider, for example, a probabilistic source of photon pairs. The detection of one of the photons heralds the presence of the other, which is stored in a quantum memory. The quantum memory is now, assuming unit efficiency, a deterministic single-photon source with the drawback that it needs to be recharged after every emission. Recently, the range of possible applications for quantum memories has been extended to include photon number resolving detectors (see chapter6), and the selective delay of optical sidebands with respect to the carrier for ultrasound detection (McAuslan, Taylor, and Longdell,2012).

In the context of long-distance quantum communication, a quantum memory is a device that can store a quantum state of light at the single photon level for a variable amount of time. The light is significantly slowed down or even stopped inside the medium. This can only be achieved by converting the photonic quantum state into a quantum state of matter, which hence needs to fulfill the following requirements:

• strong coupling between light and matter to allow for the efficient and coherent absorption and reemission of photons,

• a means of coherent manipulation of the quantum state of the matter system for on-demand readout, and

14

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3 . 1 introduction 15

• long coherence times to achieve a high-fidelity readout even for long storage times.

A matter systems that directly meets these requirements is an ensemble of free atoms in a vapor cell. In particular for the alkali atoms, the energy-level structure has been well-understood for a long time, they have optical transitions at convenient wavelengths and long-lived hyperfine ground states that can be used for long-term storage. Strong coupling between light and a single atom typically requires sophisti- cated techniques involving high-finesse optical cavities. For atomic ensembles the coupling strength is proportional to the square root of the number of atomsN, andcollectivestrong coupling can be achieved by making N sufficiently large. In terms of the resonant optical depth d = αL ∝ N, where α is the attenuation coefficient and L the length of the medium, the strong coupling condition is given byd 1(Ham- merer, Sørensen, and Polzik,2010).

Implementations of optical quantum memories can roughly be divided into two categories (Lvovsky, Sanders, and Tittel, 2009; Ham- merer, Sørensen, and Polzik,2010). The first category is that ofphoton echoesin inhomogeneously broadened media. The photon is absorbed and creates a collective atomic excitation. The reemission is induced by letting the individual atomic dipoles rephase at a later time. Protocols that belong to this kind of quantum memory are the coherent reversible inhomogeneous broadening (CRIB) (Moiseev and Kröll,2001; Krauset al.,2006) and the atomic frequency comb (AFC) (Afzelius, Simon,et al., 2009). Adiabatic protocols form the second category. Here, the photons are converted into collective excitations and back by sufficiently slowly varying a control field. Prominent protocols in this category are electro- magnetically induced transparency (EIT) and Raman¹ type quantum memories.

The first photon echo was observed as early as in 1964 (Kurnit, Abella, and Hartmann, 1964) and the storage of light pulses extensively studied in the following decades. However, serious efforts towards quantum memories did not start before the experiment by Hau et al. (1999), who used the anomalous dispersion due to EIT in an ultracold sodium gas to reduce the group velocity of a light pulse to 17m/s, corresponding to a delay of more than10µs. A short time later it was noted that a quantum memory could be realized by extending this technique to completely stop the light pulse in the medium (Fleis- chhauer and Lukin, 2000), and first experimental realizations quickly followed (Liuet al.,2001).

Since the first EIT-based implementations, remarkable results have been achieved for quantum memories based on ensembles of free atoms.

These include combined storage and retrieval efficiencies of87% after

1 Although the Raman-process could seem to be too fast for being adiabatic, the far- off-resonant nature gives the same adiabaticity criterion as for EIT (Gorshkovet al., 2007).

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16 a multimode, solid-state light-matter interface

7µs in a hot Rubidium vapor using a photon-echo technique (Hos- seini et al.,2011), 73% retrieval efficiency and a storage time of a few milliseconds in a DLCZ-type memory (Bao et al.,2012), as well as coherence times of a few hundreds of milliseconds (Schnorrberger et al., 2009). The main decoherence mechanisms in these systems are motional diffusion for hot vapors and fluctuations in the optical and magnetic fields used for trapping of the cold gases. These effects can be avoided by switching to naturally trapped ensembles like im- purity dopant ions in crystals. Although this approach so far requires cryogenic temperatures, the absence of the need for laser cooling significantly reduces experimental complexity, and the solid-state aspect could facilitate the development of compact integrated solutions in the future. Suitable crystals are readily available from the laser industry, and in particular ions of the rare-earth elements display a series of promising properties. The optical transitions within the 4f shell² are shielded from the crystal environment by the larger orbits of the filled 5s and 5p shells. This is evidenced by long optical coherence times which can reach the order of 1ms (Sun et al.,2002). On spin transitions, decoherence times on the order of 100ms have been reported, and been extended to beyond30s by applying dynamical decoupling techniques (Fraval, Sellars, and Longdell,2005).

Even though the 4f-transitions in many aspects behave very much like optical transitions in free atoms, they can be very different in a few aspects. The embedding in the crystal lattice breaks many of the symmetries that are at the origin of the rather strict selection rules taught in atomic physics classes. As a rule of thumb, all transitions are weakly allowed with excited state lifetimes from100µs to10ms. Addi- tionally, the transitions are inhomogeneously broadened. Typical inhomogeneous linewidths are on the order of GHz, which is often larger than the hyperfine splittings in the system. As a consequence, optical spectroscopy is often tricky at best, and optical pumping schemes used for state preparation in free atoms cannot necessarily be applied.

3 . 2 the atomic frequency comb

The techniques of CRIB and EIT for the implementation of quantum memories are also applicable to rare-earth-ion doped crystals. However, the inhomogeneous broadening requires using spectral hole-burning to isolate a class of atoms with a spectrally narrow absorption line.

The immediate drawback is that most of the atoms are excluded from the interaction with the light field and the effective optical depth is strongly reduced, in particular when the bandwidth of the light field is

2 For free atoms, transitions between shells with the same total orbital angular momen- tuml are forbidden. Inside the crystal,l is no longer a good quantum number, and admixtures of states with different parity make such transitions possible.

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3 . 2 the atomic frequency comb 17

large. The atomic frequency comb (Afzelius, Simon,et al.,2009) is a photon-echo type protocol which takes explicit advantage of the inhomogeneous broadening. To understand the intuition behind the protocol, consider the collective state ensemble of an ensemble of atoms that has absorbed a photon on the inhomogeneously broadened transition

|g〉 ↔ |e〉,

|Ψ_atoms(t)_〉= _p¹ N

XN

j=1

c_jeⁱ⁽^~^k^j^~^x^j^−δ^j^t⁾|g₁g₂. . .e_j. . .g_N〉. (3.1) This state is similar to the atomic state of Eq. (2.3) after detection of the Stokes photon, except that it includes an additional phase that depends on the detuningδ_j between the photon and the resonance of the respective atom. The broad and continuous distribution of the _δ_j’s in an inhomogeneous ensemble leads to rapid dephasing of the individual terms. In the CRIB protocol the collective reemission of the photon at a later time is achieved by actively controlling the detunings such that the dephasing is reversed and eventually undone. The AFC exploits the natural time evolution of the atomic state by shaping the distribution of the_δ_j’s beforehand. In particular, a distribution of detunings corresponding to a set of narrow lines with spacing ∆will lead to a photon echo at time t_e=2π/∆, where the temporal phases of all the terms in Eq. (3.1) become a multiple of2π.

3 . 2 . 1 Complete protocol with on-demand readout

The atomic frequency comb described above implements a quantum memory with a storage time that is fixed at the time of the spectral preparation of the memory. The complete protocol that allows on- demand readout is illustrated in Fig.3.1. The implementation requires an atomic system which besides a Λ-type level structure contains another metastable state _|aux_〉, typically a third hyperfine ground state.

The individual steps of the protocol including so-called spin-wave storage are

1. The inhomogeneous absorption spectrum in the transition_|g〉 ↔

|e〉 is shaped into an AFC by optically pumping the unwanted atoms into the state_|aux_〉.

2. The resonant input photon is absorbed by the comb such that the atoms are put into the collective state specified in Eq. (3.1).

3. Before the photon is reemitted the single collective excitation is transferred to _|s〉 by applying suitable optical control pulses.

If we assume that the inhomogeneous broadening between the ground states is negligible, the phase evolution is stopped while the atoms are in this state. Additionally, coherence times are much longer, allowing for longer storage.

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a

b

|g〉

|aux〉

|e〉

|s〉

Frequency

Density Density

Preparation Storage and retrieval

|g〉

|aux〉

|e〉

|s〉

Frequency input

control output

preparation

Time

Intensity

TS

2π∆ −τ τ

input output

Figure 3.1:The AFC including spin-wave storage. a First, a comb structure is prepared on the|g〉 ↔ |e〉transition by optical pumping. The input pulse is stored by collective absorption of the atoms in the comb, and the excitation transferred to the empty state|s〉with a control pulse. The retrieval is triggered by a second control pulse that transfers the excitation back to|e〉.bThe corresponding tim- ing sequence.

4. After the desired storage time T_s, the collective excitation is put back into _|e〉with the help of another control pulse. The phase evolution continues, and the photon is reemitted at a total time t =T_s+2π/∆after the absorption.

3 . 2 . 2 Efficiency

As mentioned earlier, it is the optical depth of the medium that quan- tifies the strength of the coupling between the light and the atomic ensemble. It is hence one of the important parameters for determining the efficiency of a quantum memory protocol. A series of other parameters that also need to be considered are shown in Fig. 3.2a. Their role can be intuitively understood by noticing that it is the collective enhancement due to the rephasing of the atoms in the different comb lines which determines the efficiency of the photon echo. It is clear that if the frequency ranges between the comb lines cannot be emptied completely, the remaining atoms will give rise to a background optical depthd₀, but not contribute to the rephasing. Similarly, any finite widthγof the comb lines will lead to imperfect rephasing and reduce the echo efficiency. The effective optical depth d_eff =d/F is the ratio between the optical depth and the finesse F = ∆/γof the comb. For a given optical depth, a lower finesse corresponds to a larger number of absorbing atoms but leads to less perfect rephasing, and the optimal efficiency is a compromise between the two. Additionally, the photon

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3 . 2 the atomic frequency comb 19

Frequency

Opticaldepth

d₀ γ d

∆

0.0 0.1 0.2 0.3 0.4 0.5 0.6

Efficiency Square

Gaussian Lorentzian

0 5 10 15 20 25 30

Optical depth 05

1015 2025 Optimal finesse

a b

Figure 3.2:Parameters and theoretical performance of the AFC. a The AFC profile can be defined by the comb spacing∆, the linewidthγ, the optical depthdof the comb lines, and the background absorption d₀. bFor a given optical depth andd₀ =0, the efficiency of the photon echo can be optimized via the line shape and the finesse F =∆/γof the comb. A square shape is always optimal, and the smoother the line shape, the higher the optimal finesse.

echo risks being reabsorbed on the way out of the crystal when the optical depth is too high.

An analytical expression for the efficiency of the protocol can be obtained by solving the Maxwell-Bloch equations for the propagation of the light pulse through the medium (Afzelius, Simon,et al.,2009;

Bonarota, Ruggiero,et al.,2010). Here, we shall only consider the situation of forward reemission without spin-wave storage. It is then help- ful to consider a frequency-dependent optical depthd(ω)with perfect periodicity and infinite width. For a period of2π/T, the optical depth can be expressed as the Fourier series,

d(ω) = X∞ n=0

d˜_ne^inωT, (3.2)

where the Fourier coefficients are calculated in the usual way, d˜_n= ^T

2π Z _π/T

−π/T

d(ω)e^−inωTdω. (3.3)

It can then be shown that the efficiency for the photon echo after a storage time T is given by the first two Fourier coefficients (Bonarota, Ruggiero,et al.,2010),

η=d^˜₁

2e⁻^d^˜⁰. (3.4)

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Here, the first term characterizes the strength of the collective absorption and reemission (_d˜₁ _∼ _d_eff), and the second term the attenuation of the light pulse during the propagation through the medium (_d˜₀ _∼ d_eff+d₀). For a given optical depth one can use these formulas to find 1. the shape of the comb that optimizes the efficiency. This amounts

to finding the function that optimizes the ratio_d˜₁/ ˜d₀, and 2. the finesse of the comb that optimizes the efficiency for a given

shape.

It turns out that the optimal comb shape is square, for which the efficiency and optimal finesse are given by

η=d_eff² e⁻^d^eff sinc²(^π

F), F_opt= ^π arctan²^π

d

. (3.5)

The two functions are plotted in Fig.3.2b and compared to the optimal performance for Gaussian and Lorentzian combs. Due to the possibility of reabsorption, the efficiency saturates at54%. An efficiency approach- ing unity can, however, be achieved by using spin-wave storage, where the control fields are applied such that the echo is emitted in the back- wards direction (Afzelius, Simon,et al.,2009), or by placing the AFC inside an optical cavity (Afzelius and Simon,2010). In such configurations, reabsorption is suppressed by destructive interference between all possible photon paths.

3 . 2 . 3 Multimode capacity

As long as the number of atoms comprising the AFC is much larger than the number of photons stored, that is, as long as the probability for any single atom to be excited is negligible, subsequent single-photon input pulses will trigger the same time evolution and will be stored with the same efficiency. This is most useful when all the pulses arrive before the first echo emission or the first control pulse. The maximum number of temporal modes that can be stored in this way is roughly equal to the number of peaks in the AFC, as can be seen as follows. Suppose that we prepared an AFC withnpeaks corresponding to a total bandwidth ofn∆. The shortest possible input pulses that can be stored efficiently will make full use of the available bandwidth and have a duration of aboutτ= (n∆)⁻¹. These can be sent in sequence for a maximum time oft_e=2π/∆, which is the time until the first echo is emitted. The total number of temporal modes is hence given byt_e/τ∼n.

A remarkable feature of the AFC protocol is that the multimode capacity does not depend on the available optical depth, which only af- fects the efficiency. This is a significant difference with respect to quantum memories based on CRIB or EIT, for which we shall briefly estimate the multimode capacity in the following (for a rigorous derivation, see Nunnet al.(2008)).

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3 . 3 important experimental results for the afc 21

In CRIB a narrow absorption line of widthγ and optical depthd is artificially broadened to a width Γ. The emission of the photon echo is achieved by reversing the artificial broadening, i.e. lettingδj→ −δj

after the first half of the storage time, which leads to a rephasing after the second half. The maximum storage time is given by T_max ∼ 1/γ, because the initial width leads to a dephasing that cannot be undone by the protocol. The shortest possible duration of the input pulses isτ∼ 1/Γ, such that the number of temporal modes to be stored is T_max/τ∼ Γ/γ. However, the efficiency of CRIB is determined by the optical depth after the broadening, which is _d˜ = dγ/Γ. This means that in order to keep the efficiency constant when increasing the multimode capacity, the optical depth needs to be increased proportionally.

In EIT, a control laser opens up a transparency window in an atomic transition. Inside the window the group velocity of the light is reduced by a factor n_g ∼αcΓ/(4Ω²)with Γthe linewidth of the transition, Ω the Rabi frequency of the control field andα=d/L(Lvovsky, Sanders, and Tittel,2009). Storage is achieved by reducing the control field intensity to zero before the pulse leaves the medium of length L, giving T_max∼Ln_g/c. For efficient storage, the pulses need to spectrally fit inside the transparency window, which has a width ofW ∼4Ω²/(Γ^p^d), so the number of modes that can be stored isT_maxW∼p

d.

To summarize, the number of temporal modes that can be stored in EIT is proportional to the square root of the optical depth. CRIB performs slightly better by having a linear dependence. In the AFC protocol the number of modes depends on spectral properties of the atomic frequency comb only, which for the right material permits temporal mode numbers of 100 or more (Afzelius, Simon,et al.,2009). The AFC quantum memory is hence the ideal building block for quantum repeaters, where a high mode capacity provides an important gain in the achievable communication rate.

3 . 3 important experimental results for the afc

The strong potential of the AFC protocol for quantum communication can be deduced from the many important experimental results that have been obtained in a time span of only five years.

The protocol was first demonstrated by Riedmatten et al. (2008).

They showed coherent storage and retrieval of four temporal modes at the single photon level. Storage times of up to 1µs were achieved with a maximum efficiency around0.3% for the shortest storage time of100ns. Significantly higher efficiencies were reported already during the next few years. By using a highly absorbing material an efficiency of35% could be reached (Amariet al.,2010). In a less absorbing material, the realization that the optimum efficiency for a given optical

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depth can be reached by using square-shaped combs led to an increase in efficiency from9to17% (Bonarota, Ruggiero,et al.,2010).

The complete AFC scheme including spin-wave storage has been demonstrated for the first time by Afzelius, Usmani,et al.(2010). Be- sides enabling on-demand readout, this also allowed to reach storage times of20µs³.

The multimode capacity of the protocol has been impressively demonstrated first by Usmani et al. (2010), who coherently stored and retrieved 64 well-defined temporal modes at the single-photon level, which would allow for the simultaneous storage of 32 time-bin qubits.

The current record of 1060 stored and retrieved modes was obtained one year later by Bonarota, Le Gouët, and Chanelière (2011), but not at the single-photon level.

The original AFC protocol only allows to reach near-unity efficiencies for very high optical depth and readout in the backward direction.

These constraints, which are difficult to meet in practice, can be lifted by embedding the quantum memory into an impedance-matched optical cavity (Afzelius and Simon,2010). This discovery has been applied recently leading to an efficiency of 58(5)% for an otherwise weakly absorbing sample (Sabooniet al.,2013).

Missing from this list are results concerning the quantum nature of the storage and retrieval processes, that is, the quantum memory’s ability to accept single photon inputs that carry true quantum correlations.

Such a charaterization has been performed as part of this thesis and the results are going to be presented in Chapter5.

3 . 4 the neodymium quantum memory

The quantum memory used for the experiments that are going to be presented in the later chapters of this thesis is based on an yttrium- orthosilicate (Y₂SiO₅) crystal of 1cm length. The actual carrier of the quantum information is an ensemble of Nd³⁺(neodymium) ions doped into the crystal at a concentration of30ppm in natural abundance.

The relevant transition, shown in Fig.3.3, connects the⁴I_9/2ground state to the⁴F_3/2excited state at a wavelength of883.2nm. The crystal is mounted between two permanent magnets which create a magnetic field of approximately300mT at an angle of30° to the crystal D₁-axis.

The presence of the magnetic field leads to a splitting of the ground and excited states of approximately11GHz and2GHz, respectively. These values for the direction and strength of the magnetic field give a time- constant on the order of 100ms for the redistribution of population between the two Zeeman ground states. This is sufficient for our purposes. Configurations with longer Zeeman lifetimes may exist, but are

3 Similar results have also been reported recently (Timoney et al.,2012; Gündo˘gan, Mazzera,et al.,2013)