Spin dynamics in rare-earth-ion-doped crystals for optical quantum memories

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Thesis

Reference

Spin dynamics in rare-earth-ion-doped crystals for optical quantum memories

ZAMBRINI CRUZEIRO, Emmanuel

Abstract

Spin dynamics are investigated in several rare-earth-ion-doped crystals, both for so-called Kramers ions and non-Kramers ions. I have studied two illustrative examples of such systems, Neodymium (Kramers) and Europium (non-Kramers). The main motivation behind these studies is the search for long relaxation and coherence times in solid state atomic ensembles.

To this end I employ spectroscopic techniques to probe different kinds of interactions and relaxation mechanisms taking place in the samples. I perform spectroscopy of Neodymium (Kramers) in two different crystal lattices, yttrium orthosilicate and yttrium orthovanadate. I propose new methods to improve the relaxation lifetimes, and make precise characterizations of the interactions taking place within the atomic ensemble. In particular we achieve the most precise measurement of a kind of interaction which has received considerable attention recently in other areas of physics, the Dzyaloshinskii-Moriya interaction. These spectroscopic studies are of course of fundamental interest, but they can also be applied to quantum communication and computation. In this context, [...]

ZAMBRINI CRUZEIRO, Emmanuel. Spin dynamics in rare-earth-ion-doped crystals for optical quantum memories. Thèse de doctorat : Univ. Genève, 2017, no. Sc. 5171

DOI : 10.13097/archive-ouverte/unige:115799 URN : urn:nbn:ch:unige-1157994

Available at:

http://archive-ouverte.unige.ch/unige:115799

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

Spin dynamics in rare-earth-ion-doped crystals for optical quantum memories

THÈSE

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

par

Emmanuel Zambrini Cruzeiro de Parede (Portugal)

THÈSE N. 5171

GENÈVE 2017

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To my parents, who inspire me every day. Para a Nana, um grande chi cadan cadan, estás sempre no meu coração. Para o Vôvô, as saudades são tantas, gostava que estivesses aqui. Pour Alexandre qui a choisit de nous laisser récemment, merci de m’avoir donné un aussi bon père. Pour Isabel et les tourtes au citron sans lesquelles je n’aurais pas pu grandir.

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Abstract

Spin dynamics are investigated in several rare-earth-ion-doped crystals, both for so-called Kramers ions and non-Kramers ions. I have studied two illustrative examples of such systems, Neodymium (Kramers) and Europium (non-Kramers). The main motivation behind these studies is the search for long relaxation and coherence times in solid state atomic ensembles. To this end I employ spectroscopic techniques to probe different kinds of interactions and relaxation mechanisms taking place in the samples.

I perform spectroscopy of Neodymium (Kramers) in two different crystal lattices, yttrium orthosilicate and yttrium orthovanadate. I propose new methods to improve the relaxation lifetimes, and make precise characterizations of the interactions taking place within the atomic ensemble. In particular we achieve the most precise measurement of a kind of interaction which has received considerable attention recently in other areas of physics, the Dzyaloshinskii-Moriya interaction. These spectroscopic studies are of course of fundamental interest, but they can also be applied to quantum communication and computation. In this context, we were able to increase the efficiency of a quantum memory based on a Kramers ion (Nd) to be on par with the best non-Kramers (Eu,Pr) memories. Using Neodymium ions, I have certified that multi-dimensional entanglement can be stored in a quantum memory. The second part of this thesis concerns the study of Europium (non-Kramers), starting with a characterization of the Zeeman and quadrupole interactions. I believe these studies will pave the way for long duration (100 ms or more) quantum storage in a rare-earth-ion-doped crystal.

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Résumé de la Thèse

La découverte de la physique quantique au début du siècle passé nous a apporté des concepts et des questions qui défient toute intuition humaine. L’un de ces concepts, le spin, découvert en 1922, nous permet de mieux comprendre les phénomènes magnétiques. Dans l’état solide, les spins interagissent entre eux et donnent lieu à des phénomènes magnétiques comme l’intéraction d’échange, qui sont aujourd’hui étudiés à travers le monde. L’intéraction d’échange est un phénomène quantique et n’a aucun analogue classique. Ceci dit, il est clair que le magnétisme est d’origine quantique, ce qui explique la frustration de Richard Feynman quand il doit expliquer à un journal- iste ce qu’il se passe lorsqu’on essaye d’approcher deux aimants et on ressent une force [1]. La seule façon de développer une intuition de ces phénomènes, c’est de les utiliser expérimentalement.

La dynamique des spins dans des crystaux dopés avec des ions terre-rare est l’objet d’étude de cette thèse. On y étudie les ions terre-rare dits de type Kramers et de type non-Kramers, ce qui permet d’obtenir une perspective générale concernant la dynamique des spins dans les crystaux dopés avec des ions terre-rare. La motivation principale de ces études est la recherche de temps de relaxation et de cohérence longs pour des ensembles atomiques dans l’état solide. Les ions terre-rare dopés dans des crystaux sont intéressants dans ce contexte car ils possèdent naturellement des longs temps caractéristiques. Dans le but d’étendre autant que possible les temps de relaxation et cohérence, j’utilise des techniques de spectroscopie pour explorer différents types d’intéraction et de mécanismes de relaxation qui ont lieu dans nos échantillons.

J’utilise ces techniques pour étudier le néodyme (Kramers) dans deux mailles crys- tallines différentes, l’orthosilicate d’yttrium et l’orthovanadate d’yttrium. Dans mes articles, je propose des nouvelles méthodes pour étendre les temps de vie de relaxation, et on caractérise de façon précise les intéractions qui ont lieu dans l’ensemble atomique. Notamment j’ai fait la mesure la plus précise jusqu’à la date d’un type d’intéraction qui reçoit beaucoup d’attention dans d’autres domaines de la physique, l’intéraction Dzyaloshinskii-Moriya ou échange antisymmétrique. Nos études spectro- scopiques sont non seulement d’un intérêt fondamental, mais les résultats sont aussi applicables à la communication et au calcul quantique. Dans ce contexte, j’ai augmenté l’éfficacité d’une mémoire quantique basée sur un ion de Kramers (Nd). Ceci nous à

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permis d’atteindre des efficacités comparables aux meilleures mémoires non-Kramers (Eu, Pr) qui ont des temps de vie en général plus longs dû au fait qu’ils n’ont pas de spin électronique effectifS. En utilisant des ions Nd, nous avons vérifié que l’intrication multidimensionnelle peut être stockée par une mémoire quantique. La deuxième partie de cette thèse concerne l’étude de l’europium (non-Kramers), en commençant par la caractérisation des intéractions Zeeman et quadrupolaire. Ensuite je décris des expéri- ences faites dans le but de progresser vers un stockage à plus long terme en utilisant des séquences d’impulsions dites de découplage dynamique. À mon avis, ces études pour- raient amener à une extension du temps de stockage de plusieurs ordre de grandeur (>100 ms par rapport à 1 ms aujourd’hui).

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Introduction

Since its inception, quantum mechanics has brought us a multitude of problems which defy our intuition and applications that have only just begun to change our lives. The story of quantum mechanics can be divided into two parts, called the first and second quantum revolutions. The first revolution was started in the beginning of the 20th century, with the discovery of black body radiation and the photoelectric effect, the concept of quanta and the wave-particle duality, which lead to the well-known Heisenberg uncertainty principle. The second quantum revolution, a term coined by Alain Aspect [2], one of the first physicists to measure a Bell inequality violation, describes the period from the 1960s to today. This period is characterized by a change in paradigm: quantum mechanics is now explored also through experiments and technology. Ever since its creation, quantum mechanics has defied our most basic intuitions.

The best way to become familiar with quantum mechanics, or anything else for that matter, is to get used to it. How do you get used to something? You use it. What better way than using technology based on quantum mechanics on an every day ba- sis? Note that already in the first quantum revolution, technology based on quantum mechanics was also developed, such as transistors (1947), but at the time the focus was more on fundamental issues of quantum mechanics. The year 1960 marked the invention of the laser and the start of the second revolution. John Bell said it very well:

“I am a quantum engineer, but on Sundays I have principles”, implying on Sundays he would think about fundamental issues. Today “quantum engineers” exist. Their goal is to build devices which generate, manipulate and characterize quantum states for both fundamental and technological reasons. These quantum states can then be used for information transmission and processing [3, 4]. To this end, they build single photon sources, single photon detectors and memories for such quantum states, just to name a few examples in the optical domain. They can generate, store and measure entanglement, a key property found in certain quantum states which allows them to be used as resources for communication or computing. They are able to measure correlations found in nature, and compare them to the best existing theoretical models. They can make weak measurements, measurements thatalmost do not perturb quantum states.

Imagining that one can measure properties of a single particle without perturbing it is the sort of thing that makes quantum mechanics such an intriguing and fascinating

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topic.

In this thesis, I focus on the development of solid state quantum memories for light. These are some big words, let us deconstruct them. The physical system I use for storage is rare-earth (RE) ion doped crystals. RE ions in crystals fulfill the basic requirements for a quantum memory: strong light-matter coupling (efficient input absorption and output emission), on-demand read-out and potentially long storage times. Since we want to store information encoded in light, I work with materials possessing optical transitions. In this thesis, I have characterized the spin properties of two species of RE ions doped into crystal lattices. The two species, Neodymium and Europium, are illustrative of the spin properties of RE ions in general, as they belong respectively to the Kramers and non-Kramers groups of ions. RE ions either belong to one group or the other, depending on the number of free electrons contained in the electronic shell used for optical transitions. If this number is odd, the ion belongs to the Kramers group, if it is even it is non-Kramers. Electronic and nuclear spin dynamics depend strongly on which group the ion belongs to, and the question of which system is the best for quantum storage is still open.

The first part of Chapter 1 is dedicated to introducing the subject of this thesis to someone who is not familiar with quantum theory. It is an attempt to familiarize the reader, no matter his/her background, with the motivation behind the work done in the field. The rest of Chapter 1 is a more technical introduction to quantum repeaters, the motivation behind the work done in this thesis. In Chapter 2, I describe in detail our strategy to build a quantum memory. Chapter 3 introduces RE-ion-doped crystals, our choice of physical media to store information, and their properties. Chapter 4 concerns the study of Neodymium-doped crystals, while Chapter 5 is a study of strongly-coupled pairs of Neodymium ions, an exotic system in the context of quantum storage. Chapter 6 discusses work done using Europium ions. At the end, I conclude by summarizing the results obtained during this PhD thesis and discuss avenues for future research.

In summary, my thesis consists of the following research efforts:

– I studied materials doped with Neodymium, a Kramers ion, and improved storage efficiencies in such materials. I also proposed new optical pumping techniques which are of interest for storage with any Kramers ion.

– I studied more exotic systems for quantum storage with Kramers ions, namely an isotope of Neodymium which possesses a nuclear spin and coupled pairs of Neodymium ions.

– I studied in detail the spin properties of the excited state of ¹⁵¹Eu³⁺:Y₂SiO₅ to pave the way for long duration storage experiments. Storage experiments are already under way as this thesis is being written.

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Chapter 1 The microscopic world: spins, photons, and other entities

The macroscopic life is simpler

Life is amazing. There are so many things to do, that you can be certain that in your lifetime, you will never get bored unless you want to. Just think of the number of jobs you could try in a lifetime, from driving an airplane to professional surfing or taking interest in the problems of others and being an activist. If you wanted to try them all, you would never be able to, because the number of possibilities seems, or is, infinite. In other words, the complexity of the world we live in is huge, and we can convince ourselves of this using our five senses. Now imagine that you had more senses, or that each of your sensory organ was much more evolved than it is right now.

For example, if your eyes were able to distinguish one particle of visible light (i.e. a photon) from another. Imagine how you would feel if you could see every single photon arriving to your eye. How would your brain process this? For sure with the version of brain we have right now, one would go mad. It is safe to say that our brain would need upgrading too.

Go back 117 years in time, to the year 1900, and ask the most notorious physicsists (Henri Poincaré, Niels Bohr, ...) how they would go about detecting a single photon of light, and they would not even understand what you mean. It is thanks to the work of A. Einstein on the photoelectric effect [5] that physicists discovered that light is quantized, and exists in discrete packets of energy called photons. Very quickly after this discovery, came the invention of the photoelectric tube by Elster and Geiter [6].

This allowed physicists to detect light of very low intensities, but still not at the single photon level. Nevertheless, physicists already considered thought experiments with single particles of light, such as the Einstein box. It took more than 20 years until the photomultiplier tube (PMT) was invented. From then on, single photons were

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detectable, and the thought experiments considered previously could enter the realm of experimental physics. Without technological progress, the advancement of physics is hindered. Well, now we have such single photon detectors! Although they are not implanted in our eyes, and our brain still needs an upgrade, we can use them to do experiments and probe the microscopic world! This is not the only example of course, because our understanding of pressure waves such as the ones which make up sound or thermal vibrations in materials is also getting better and better. To the point that we can also now do experiments with “single particles” of sound - calledphonons [7, 8, 9].

As a more far fetched example, imagine you could taste every single molecule you ingest during dinner. If your brain could process all this information, what would it change about your experience of the world?

So we see very clearly that the macroscopic world is simpler to describe than the microscopic world. Classical physics, the theoretical description that we had of physical processes before the beginning of the 20th century, is sufficient to describe most aspects of our day-to-day macroscopic life. In the microscopic case, we need more advanced theories: special relativity and quantum mechanics. Complexity is good, as it leads to new phenomena and applications. In this thesis, we work mostly with the quantum theory of light and matter and their interaction.

The microscopic world

In the modern times, most people have heard about atoms at some point in their lives. So most people know that they are the building blocks for the macroscopic world we are so familiar with. A simple example is the H₂O molecule (water), which is composed of two hydrogen atoms and one oxygen. This is a very simplified view of reality, as an atom is itself composed of smaller building blocks of different species:

physicists call them particles. Every atom that we know of (not considering anti- matter, which I will not speak about in this thesis) is made up of electrons, protons and neutrons, and even these particles are said to be composite as they are made of even smaller particles (elementary particles). Even if you have never followed a university course, you have probably also heard about the electron. Electric currents are composed of electrons. Typical household currents (for example 1 A of current) normally carry about 10¹⁸ electrons per second. An electron is a particle with a mass me− = 9.11.10⁻³¹ kg, and an electric charge e = −1.6.10⁻¹⁹ C. It may also possess angular momentum and spin, two important properties which we will talk about later.

Protons and neutrons also have a mass, but the proton has a positive charge−eand the neutron has no charge. There are several other kinds of particles, but I will only speak about one more: the photon. The photon is special as it is the only massless particle we know of, and that is also why it is the fastest particle. The photon is particular as special relativity theory is required to describe it, quantum mechanics is not enough.

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Now that we have introduced the particles we will be working with, we can start having some fun. As mentioned before, in the microscopic world classical physics breaks down and one has to use quantum mechanics. But quantum mechanics can be extremely non-intuitive. According to this theory, in the microscopic world, objects such as particles can have surprising properties. For example, they do not need to be in a well-defined state of the system (in more technical terms, an eigenstate). Instead, they can adopt several states at the same time. Let’s say we define a given state by saying that it corresponds to an electron being at position x = 0, and another state corresponding to the same electron at the positionx=L. Quantum mechanics allows the electron to be in the two states at the same time, and this is called asuperposition state. Physicists write

|ψi= 1

√2(|x= 0i+|x=Li) (1.1) to describe the corresponding superposition state, where|ψiis the state of the particle, or the wave function. |x = 0i is the state where the particle is at position x = 0, and

|x = Li the state for x = L. The 1/√

2 is a probability amplitude, meaning that there is a 50% chance for the particle to be found in one position or the other when measured.

Let us define these quantum states more formally. A quantum two-level system is called a qubit. When considering a qubit, one can represent the quantum state in a visually guiding way. The representation used for this is called theBloch sphere.

Since the system is two-dimensional, we can describe each state by a vector belonging toC². We denote the ground and excited states respectively as

|0i= 1

0

and |1i= 0

1

In this case, the system is two-dimensional so we can read out only two distinct values when performing a measurement. A state describing the position of a particle

|xi has a continuum of outcomes for the measurement, so such a state is infinite- dimensional. When the qubit state lays in the equator, we say we have a coherence - a superposition state. The coherence time is then the lifetime of this superposition state. In this thesis, a qubit is usually either a photon with two states or a spin with two possible projections (values) up and down when under a magnetic field. We will often make use of the Bloch sphere to help us understand the techniques we use.

Quantum coherence is fragile, vulnerable to any external perturbation. Therefore, to each quantum state we associate a characteristic coherence time. In general, the lifetime of a qubit is described by two characteristic times: T₁, the longitudinal (or population) lifetime and T₂ the coherence lifetime. In the Bloch sphere, T₁ implies a change of relative population in |0i and |1i and coupling to an external reservoir (dissipation), while T₂ also implies precession about the z axis, that is a change of the

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Figure 1.1: Bloch sphere representation, made using Mathematica, the MaTeX pack- age and Sjoerd C. de Vries’ “splineCircle” function. States on the equator of the Bloch sphere only differ by a relative phase. |Li,|Ri,|−i and |+i correspond the to relative phases −i,+i,−1 and +1 respectively.

relative phase of the ground and excited state components (dephasing). In general, 1

T₂ = 1 2T₁ + 1

T₂^∗ (1.2)

where T₂^∗ is an effective coherence time, describing the effect of dephasing processes on the coherence lifetime. The true coherence time T₂ also depends on T₁ and is thus bounded by T₂ 62T₁.

Let us look at another intriguing state quantum mechanics allows us to write:

|ψi= 1

√2(|0iA⊗ |1iB+|1iA⊗ |0iB) = 1

√2(|01iAB+|10iAB), (1.3) where the expression on the right serves to simplify the notation. This state is said to be entangled. One way to interpret this state is to imagine two particles A and B, which can have values 1 or 0 upon measurement. We do not know in which state each

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particle is, but we do know that if one of the particles is in state |0i, the other one will necessarily be in state|1i. ⊗represents the tensor product of two states belonging to different Hilbert spaces. The Hilbert space is the space where quantum states exist [10]. In the example I just gave, each particle lives in a different Hilbert space. What a tensor product of two states means is that we are considering two different channels (physicists call them modes). What the modes represent physically depends on the experimental setting. They can represent two different particles, two different paths on an optical table, two different times, the examples are uncountable.

John Stewart Bell had a funny example of a classical version of such a state [11]. His colleague Reinhold Bertlmann used to come to work every day with socks of different colors, never a matching pair. So if John Bell had the opportunity to catch a glimpse of one of Bertlmann’s socks, he would know immediately that the state of the other sock was different. This is a classical example because Bertlmann chose different socks in his room in the morning, meaning that each sock already had a determined state before John Bell looked at them. In particular, one could spy on Bertlmann’s private life and discover the true state of his socks at any time. If we do not spy on Mr. Bertlmann, there is an apparent randomness due to the lack of information we have.

In the quantum version, each particle is in a superposition state (i.e. in a non- determined state) until the measurement. For this particular state, this means that we do not know the state of each particle (one could say mode in more general terms), but we do know something about the global state of the two particles even before the observing them. We have this information because we know that there is acorrelation between the states of the particles of|ψi.

In mathematical terms, we say a pure quantum state is entangled if and only if it is not separable, meaning that we cannot decompose this state into the product of a state describing only the subsystem A and a state describing only the subsystem B, i.e. |ψi 6= |φiA⊗ |ξiB. There also exist states which are probabilistic mixtures of pure states, the so-called mixed states, which require a more general definition of entanglement: a (general) quantum state is entangled iff it cannot be written in the form ρ_AB = P

ip_iρ⁽ⁱ⁾_A ⊗ρ⁽ⁱ⁾_B where p_i are probabilities such that P

ip_i = 1, ρ⁽ⁱ⁾_A/B =

|ψ_iiA/Bhψ_i|A/B and |ψ_iiA/B is a pure state. Since in quantum mechanics correlations can exist no matter the distance, they are called non-local correlations. Note that the randomness in the quantum case is no longer due to a lack of knowledge, it is a fundamental property of quantum states, therefore of nature too. Non-local correlations allow for more funny stuff to happen. I will give another example which is related to the motivation behind this thesis. Non-local correlations allow teleportation of quantum states from point A in space to point B. It is quite remarkable that this happens in a way that does not violate relativity, as one can show observers located at A and B cannot use teleportation to communicate faster than the speed of light [12].

An important theorem in quantum mechanics is the no-cloning theorem, which

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states that one cannot perfectly and deterministically copy a quantum state [13, 14]. In fact, the linearity of quantum mechanics (which allows for the existence of superposition states) implies no-cloning [15]. This is yet another phenomenon which intuitively is not obvious, as we cannot compare it to anything that we are familiar with, in the macroscopic, classical world.

The definition of measurement itself also has to be revised when dealing with quantum mechanics. If I look at a coin after tossing it in the air and letting it fall, I will know on which side the coin fell, either I see heads or tails. No matter how many times I look at it or for how long, it will remain the same state. Well in quantum mechanics, if you measure a state you perturb it. Depending on the strength of a measurement, you can perturb it more or less.

In this introduction I have given a brief overview of the properties of quantum states which distinguish them from classical states that we are used to. In summary, quantum states are characterized by (not only) the following properties:

– superposition of states and entanglement;

– no-cloning;

– weak measurements.

Bringing the microscopic to the macroscopic

A fundamental question one could ask at this point, is why can’t we see the kind of phenomena described by quantum mechanics with our own eyes? In other words, what is the limit in size of quantum mechanical superposition states? Could you have a superposition of macroscopic objects? In quantum theory nothing seems to prevent it. This question was famously illustrated by Schrödinger’s cat gedankenexperiment.

A cat is inside a closed room, which is completely isolated from the outside world.

Next to it sits a bottle of poison, connected to a device which can open the bottle and therefore kill the cat. The device works with a quantum random number generator, such that each time a random number is generated, there is a 50% chance of the cat dying and a 50% chance of the cat living. For an observer outside of the room, the cat is entangled with the poison device, and it is both dead and alive at the same time. Is this realistic? Could we prepare a dead and alive cat in the lab? And if the answer is no, what prevents us from doing it? Is it due to interactions with the environment? Or is it because by the simple act of observation we destroy the superposition? Surely answering these questions is important for a better understanding of the difference between the microscopic and the macroscopic worlds, and ultimately, for the advancement of science.

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Even if these questions are still unanswered, macroscopic devices taking advantage of quantum physics already exist. Research in experimental quantum information is focusing on quantum metrology, quantum computers and quantum communication networks. All these are real life applications of quantum physics, and although they do not contain macroscopic superposition states yet, one can already take advantage of the complexity of the quantum world to build better technology.

What does quantum have to do with communication?

In this section I would like to give a very brief introduction to the topic of quantum communication, the goal being not to discuss technicalities, but simply to place this work in a larger context - so that the reader can fully appreciate the motivation behind the holy grail in long-distance quantum communication, the quantum repeater. If the reader is interested in knowing more, he/she can read any quantum information textbook [16] or quantum information related PhD thesis [17].

Cryptography does not mean much to most people, but it has shaped the history of the world [18]. Cryptography is present in our daily lives. Today most of us are used to the internet and massive information sharing. Every day millions of people share sensitive information about their personal lives on the internet. We can access our bank account from the web for example. As a consequence, there is an obvious need to secure information transfer in telecommunications. Security is guaranteed by cryptography. Let us think about this information transfer in the simplest way possible. Alice wants to send a message to her friend Bob, let us say a bit value for example, 1 or 0. Alice prepares the state and sends it to Bob. Somewhere between Alice and Bob, someone malicious called Eve is trying to intercept the signal. In classical telecommunications, Eve could simply copy the state, keep one copy for her and send the other one to Bob. In this way, Eve can eavesdrop on Alice and Bob’s conversation. Due to the no-cloning theorem, with quantum states this is not possible, making quantum communication intrinsically more secure. Also, the simple act of measurement will perturb the quantum state that Alice and Bob share, such that they can detect the presence of an eavesdropper in a subsequent classical communication.

In practice, quantum mechanics can lead to secure telecommunications. What is new is the change in paradigm: security is guaranteed by physics, not necessarily by the algorithm or protocol we use. The other main application of quantum mechanics, which is not the subject of this thesis but is nonetheless related, is solving problems which classical computers cannot solve, or solving problems more efficiently than using classical systems. The most notorious example is RSA encryption, which secures most financial transactions around the world today. With classical computers, one can only crack an RSA encrypted key in a time which grows exponentially with the key size.

With a quantum computer, the time required to crack the key grows only linearly with

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the key size. In layman terms, if someone had a quantum computer today, he would be able to hack to any encrypted communication taking place around the world in a reasonable amount of time.

Motivation and long-term dream

The dream, and I would add a very realistic one, is to build a quantum network.

You can imagine it as any other network, a collection of nodes and links. At the nodes we would have quantum memories (maybe one day quantum computers too), and other apparatuses to generate photons and perform measurements/detections on them. In my thesis, we focus on the nodes. The links between the nodes would be optical fibers, through which photons would propagate. Why choose photons to transmit information through the links? This is because in terms of loss with respect to the distance, photons propagating in optical fibers are one of the best physical system we know of.

There is also the option to propagate them through free space, but this is technically challenging and costly as it requires satellites and large telescope infrastructures on the ground. The best commercially available optical fibers today have a transmission loss of 0.16 dB/km and are typically used for transoceanic optical communications (for example Corning’s SMF-28 ultra-low-loss fiber). This implies that such an optical fiber of 100 km has a transmission of 2.5%. So already at these distances, which are very modest for telecommunications, the transmission loss of optical fibers is a serious problem. A way to circumvent this problem is to use quantum repeaters, analogous to their classical counterpart in telecommunications. In classical telecommunications, repeaters are devices that receive a signal and reemit an amplified version of it. The difficulty of building a quantum repeater is that in the quantum world, states cannot be copied (amplified) perfectly (no-cloning). Hence, one must use entanglement in order to adapt to this fundamental limitation.

The concept of a quantum memory

An essential ingredient for quantum networks, in particular for quantum repeaters, is the quantum memory. In a general way, a quantum memory is defined as a black box (black because we don’t know what is inside) which has an input and an output, see Fig. 1.2. The input is a quantum state |ψ_ini, while the output can be a different state |ψ_fi. These could be photons, electrons or other quantum systems. One of the main goals of a physicist developing a quantum memory is to minimize the difference between the two states.

Glossary:

fidelity F: a better fidelity means a better overlap of the input and output states. It is given by F =|hψ_out|ψ_ini|².

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Figure 1.2: Illustration of the general concept of quantum memory. The black box is the memory, nothing is specified about what is inside: i.e. the physical support, the protocol used to store, etc...

efficiencyη: the total probability to store and recover the quantum state.

storage time T_M: this one is self explanatory.

multimode capacityC_m: the capacity to store information in different modes (information channels) in order to increase information density in the memory.

wavelength λ: quantum memories typically work at well-defined wavelengths, so it is important to chose a wavelength which is interesting for the application.

In telecommunications, people use generally 1550 nm as in this region propagation losses in optical fibers are minimal. However, to date, it has proven difficult to make a good quantum memory at telecommunication wavelengths.

The quantum repeater scheme

In Fig. 1.3, we show an illustration of the original quantum repeater scheme, proposed by in 1998 by Briegel et al. [19]. The idea is that we have several pairs of nodes that share some quantum information locally (at short distances), in this case AB, CD, ... WX and YZ. Entanglement is generated with two particles in a lab between A and B for example, and each particle is then sent to each node (A and B). The distance between the lab and the nodes must be short enough such that entanglement is not destroyed during propagation. We then want to perform local

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operations (because we don’t know how to perform non-local operations) such that in the end nodes A and Z share the quantum information.

Let’s say we have two entangled states ψ_AB and ψ_CD. Entanglement swapping converts the entanglement between AB and CD to entanglement between A and D.

One can see this process as teleportation of entanglement [12, 20]. One can go on until the entanglement is between nodes A and Z. The teleportation is achieved through a special kind of measurement, performed at BC, called a Bell state measurement.

In practice, the generation of entanglement and entanglement swapping are both imperfect. The associated decrease in fidelity can be countered using entanglement purification [21]. Entanglement purification allows the generation of n maximally entangled states (singlets) from mnon-maximally entangled states, where m > n. Hence it suffices to generate enough pairs to form a singlet, and then one can go on distributing the entanglement. In the scheme of Briegel et al., also called BDCZ protocol, the time

QM QM

BSM

Figure 1.3: Illustration of an elementary link in the BDCZ quantum repeater. Each node has a quantum memory (QM). The photons are sent to an intermediate station where a bell state measurement (BSM) is performed.

and resources needed depend polynomially on the distance between A and Z, instead of exponentially as is the case for direct transmission. This is possible because each initial link (AB, CD, ...) can be prepared independently, i.e. we can prepare them at the same time. Also, once entanglement is created in a link it can be kept for the time required to finish the entanglement purification. In practice it is quite hard to keep the quantum properties of a quantum system for an indefinite amount of time, due to the phenomenon of decoherence - variations of the physical properties of the environment of the quantum system destroy coherences. In order to satisfy the condition we must turn to quantum memories, i.e. we must store the entanglement. Quantum memories would also allow for the synchronization of signals coming from different nodes, which is of course necessary to realize the entanglement swapping operation.

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The BDCZ protocol is a bit abstract in the sense that the practical implementation is not entirely clear. In 2001, a more practical scheme was devised, by L.M. Duan, M.

Lukin, I. Cirac and P. Zoller [22], which makes use of the storage medium as a single photon source too. They propose to implement their scheme using rather “simple”

components, such as atomic ensembles and single photon detectors. The idea can be summarized in three steps:

1. Generating entanglement between a collective atomic excitation and a photon in each node. The single photon is sent to a central station located between the nodes;

2. Performing a single photon measurement at the central station after mixing the paths on a beam splitter. If a single photon is detected then we do not know from where it came from, thus projecting the memories onto a single delocalized spin excitation;

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

The entanglement generation is based on spontaneous Raman scattering in an atomic ensemble. To understand how it works let us consider N identical atoms with three energy levels, as shown in Fig. 1.4. The excited state |ei is connected to the other two states via optical transitions, while|giand|siare two spin states so the transition

|gi − |si is in the micro-wave, or radio-frequency regime, depending on the physical system. This kind of three-level system is known in the literature as aΛ-system. The atomic ensemble must first be prepared such that all atoms are in state |gi. Then, a laser pulse slightly off-resonant with the |gi − |ei transition is sent to the atomic ensemble. With a small probabilityp 1, this induces a Raman transition of one of the atoms from|gito|siwith the simultaneous emission of a photon, called theStokes photon. An interesting consequence of the absorption process is that if there is no way to know which atom has absorbed the photon, then the N atoms become entangled.

It is important thatpis kept much smaller than 1, as we want only a single excitation in the atomic ensemble, but this means that it will take a long time to generate the entanglement. The advantage with this scheme is that by detecting the Stokes photon, we know for sure the collective excitation has been prepared, i.e. the Stokes photon heralds the creation of entanglement between two atomic ensembles. Another key point is that the excitation is stored in long-lived atomic ground states, allowing for the preservation of entanglement until the neighbouring links are ready for the next step. The atomic ensembles themselves here act as quantum memories, although the stored quantum state has been created directly inside the memory. For the read-out,

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we again apply a laser pulse, to transfer the coherence back to the optical transition.

When the excitation decays to the |gi state, an anti-Stokes photon is emitted. The detection of the anti-Stokes photon heralds the success of the entanglement swapping operation.

N should be large, in which case there will be a strong constructive interference to emit the photon in the direction ~k_AS =~k_w +~k_r −~k_S, where ~k_i (i = S,AS for Stokes and anti-Stokes, i = r, w for read and write) is the momentum, therefore facilitating the collection of the re-emitted states. In order to increase the rates with the DLCZ

Figure 1.4: Illustration of the DLCZ scheme taken from reference [23].

protocol we need some kind of multiplexing, which is technically challenging although some impressive advances have been made in this direction recently [24]. It is easy to see how the multiplexing improves the rate. Consider that the memory can only store one excitation (mode), then we have to wait for a classical signal telling us whether the Stokes photon was detected at the central station or not. The classical signal will take a time L/2c to propagate in vacuum, and this duration limits the rate. Multiplexing allows us to do N simultaneous attempts (N Stokes modes), thereby increasing the rate linearly with N.

Today, we are still a bit far from realizing a quantum repeater. Indeed the perfor- mance requirements for the quantum repeater still haven’t been reached. The requirements for a transmission rate of 1 bit per second through a distance of 800 km are efficiencies of at least 90% and storage times of seconds for 1000 modes [25]. Entan- glement swapping has been performed already [26], but at short distances. In order to go to long distances, one needs better quantum memories. Although the DLCZ quantum repeater scheme is a valid solution for the direct transmission problem, there are alternatives. One proposal by N. Sinclair et al. [27], discusses using delay lines instead of quantum memories to the same end.

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Chapter 2 Quantum memories in

inhomogeneously broadened ensembles

In the first chapter, we have introduced the concept of a quantum memory. There are several possibilities for the choice of physical medium, under study by research groups around the world, for example: nitrogen-vacancy centers in diamond, semi- conductor quantum dots, trapped atoms, room-temperature and cold atomic gases, and optical phonons in bulk diamond. In the work described in this thesis, we use rare-earth-ion-doped crystals, which will be described in the next chapter. Therefore, we work with inhomogeneously broadened transitions in the solid state. This chapter explains how we use such transitions to store a quantum state, using the Atomic Frequency Comb (AFC) protocol.

2.1 Homogeneous and Inhomogeneous broadening

In general, a transition between two energy levels in an atom is never a delta function, i.e. it has some spread, especially if this atom happens to be inside a solid-state material. The linewidth of a transition defines the distribution of absorption versus frequency for a particular transition. The linewidth of any transition is given by theVoigt profile, a convolution of Gaussian and Lorentzian distributions. There are two kinds of broadening: broadening which is common to all the atoms and broadening which depends on the local environment of each atom. The first kind is called homogeneous broadening, the second kind inhomogeneous.

The homogeneous linewidth Γ_h is related toT₂ by Γ_h = 1

πT₂ = 1

2πT₁ + 1

πT₂^∗ (2.1)

where we have used equation 1.2 and whereT₂^∗describes pure dephasing processes, such as spectral diffusion or elastic scattering. Using this formula, T₂ is given in seconds

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and Γ_h in Hz. Homogeneous broadenings in rare-earth (RE) can be very narrow, for example 50 Hz in Erbium [28].

Figure 2.1: Illustration of an inhomogeneously broadened transition, taken from reference [29].

The inhomogeneous linewidth Γ_inh is due to deformations of the lattice, originat- ing for example from strain and the presence of defects. More precisely, it is due to deformations in the lattice which make each ion experience a different local electric field (i.e. crystal field) environment, leading to broadening through the Stark effect.

The optical transition frequency depends on the local electric and magnetic field environment, so this leads to a distribution of frequencies for ions of the same species.

The more deformation the lattice has, the larger the inhomogeneous broadening. This will of course be a stronger effect in amorphous materials such as glass, but already in crystals the inhomogeneous broadening can be very large compared to the homogeneous broadening. In the case of Nd³⁺:Y₂SiO₅, Γ_inh= 6 GHz while Γ_h = 3.5 kHz [30]

at 3 K. In Eu³⁺:Y₂SiO₅, Γ_inh = 3 GHz and Γ_h = 796 Hz at 3 K [31]. The record for Γ_h in Eu³⁺:Y₂SiO₅ is 122 Hz at 1.6 K [32]. To improve the multimode capacity, it can be useful to increase the inhomogeneous broadening in rare-earth-ion-doped crystals (REIDC), which can be done by co-doping the sample with another RE ion besides the one you are interested in, in order to increase the lattice distortion [33].

2.2 Atomic ensemble quantum memories

2.2.1 General description of a stored state

When a single photon is absorbed by an ensemble of n atoms, the atoms become entangled, in a superposition state where each component corresponds to the photon being absorbed by one of the atoms. The resulting quantum state |ψi is called a W state, and it is a particular case (only one excitation) of the more general Dicke states

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obtained by addition of collective angular momenta in quantum mechanics.

|ψi= 1

√n Xn

j=1

c_je^iδ^j^t|g₁· · ·e_j· · ·g_ni (2.2) wherej labels the atom (j = 1, . . . , n), g and e represent respectively the ground and excited states of a two-level atom.

Atoms can have different optical detuningsδj, for optical transitions in solids, meaning that if we wait, the components of Eq. 2.2 will dephase and we will lose the coherence of the state. Hence, in any quantum memory protocol based on inhomogeneously broadened systems the goal is to let it dephase for a while and then make the ensemble rephase. In order to make the atoms rephase, we need some kind of photon echo technique. In particular, the protocol that we use is based on photon/spin echo techniques, therefore before going into the details of the protocol, we will explain the idea of the spin (or Hahn) echo.

2.2.2 The Hahn echo

Throughout this thesis, we will use variations of the most basic spin echo pulse sequence, the Hahn echo discovered in 1950 [34]. In order to picture what goes on during the application of the pulses up to the echo emission, it is useful to consider the Bloch sphere, which is shown in Fig. 1.1. π/2-pulses and π-pulses make spins rotate by 90^◦ and 180^◦ respectively on the Bloch sphere, later we will give a more formal definition. We start with all the spins in the ground state, and apply a firstπ/2-pulse.

This initializes the spins in the equator (the particular state, such as|Ri,|Li,|+i,|−i or any other state lying on the equator, depends on the phase of the pulse). Before the application of the second pulse, when we do nothing, the spins dephase due to the inhomogeneous broadening: this is called the free evolution. To undo this dephasing, one must apply a π-pulse. The π-pulse reverses the free evolution of the spins. After a free evolution time equal to the first one, the spins will rephase and emit an echo.

π-pulse rephasing initial state

Figure 2.2: Illustration of the Hahn echo rephasing on the Bloch sphere (looking at the equator only). Free evolution leads to dephasing, which is compensated by the application of a π-pulse.

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The Hahn echo was an important discovery because it revealed useful to measureT₂ coherence times in NMR experiments. The echo’s intensity depends on the coherence time T₂ in the following way

I_echo(τ) =e⁻^T^2τ². (2.3) By varying the delay between the pulsesτ and measuring the echo intensity each time, one can have a good estimation of T₂.

2.3 State-of-the-art of ensemble based quantum mem- ories

Several schemes are used for quantum memories. By scheme, we mean a protocol detailing how to store a quantum state in a physical medium. The medium will be specified later, but it is good to know that some media are more adapted to certain protocols. Quantum memory protocols can be divided in two categories: those designed specifically for inhomogeneously broadened media and those that work best with the homogeneous broadening of atoms.

In Fig. 2.3, I show a comparison from 2016 of the performances of existing quantum memories, and discuss the improvements our group has made since then. This figure was presented in reference [35]. Electromagnetically Induced Transparency (EIT) is the most commonly used quantum memory protocol, as can be observed on the graph. It relies on the creation of a narrow transparency window in the absorption profile, which changes the dispersion of light in the material. This creates the well-known slow light effect. A pulse of light can be slowed down to1 m.s⁻¹. EIT also allows forstopped light, but this should not be seen as the limiting case of slow light when the speed of light goes to zero, it is a coherent transfer of information from the photon to the spins and back, as in other quantum memory protocols. The main drawback of the EIT protocol is its low temporal multimode capacity [36], because it requires unrealistic values of optical depth for multimode storage. The optical depthdis defined as the product of the absorption coefficient of the material α and it’s length d = αL. The Controlled and Reversible Inhomogeneous Broadening (CRIB) protocol is better, but the required optical depth is still too high (e.g. 3000 for 100 modes and 90% efficiency[37]). Note that we are only discussing temporal multimode capacity, as other type of multiplexing schemes could be used. The Gradient Echo Memory (GEM) protocol is similar to CRIB, in the sense that the inhomogeneous absorption is broadened by applying a field (electric or magnetic). After absorption of the signal, reversing the field allows spin rephasing.

In this thesis we work with the Atomic Frequency Comb (AFC) protocol, so after this paragraph we will focus on this one. The AFC protocol relies on shaping the δ_j distribution into a comb, instead of reversing it (δ_j −→ −δ_j) as in CRIB and GEM.

The AFC protocol has the advantage of having a multimode capacity that does not

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Figure 2.3: Comparison of existing optical memories in 2016. “This work” refers to reference [35].

depend on the optical depth. In the AFC protocol, in order to have an on-demand quantum memory, one has to use an additional ground state usually denoted |si to freeze the spin free evolution. In order to transfer the excited state population to the

|si state and back, strongπ-pulses are used.

Not all the experiments shown in this graph were performed in the quantum regime, but the ones we have discussed are theoretically compatible with quantum storage.

The colored region indicates the region accessible by a simple optical fiber at1550 nm, neglecting the losses associated with coupling the light in input/output.

Finally, this figure does not take into account several figures of merit which are important for a quantum memory. It only takes efficiency and storage time into account.

We see from the figure that there are three contributions from the Geneva group. Two of them are actually from the same work (Geneva14), reference [38]. One of the results has a much higher efficiency because a cavity was used to increase the optical depth.

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The other work (Geneva15) is from reference [39]. Since the publication of this figure, we have realized an AFC-DLCZ experiment in Geneva [40]. The physical medium used for these experiments is Eu³⁺:Y₂SiO₅ and the author of this thesis has worked towards the extension of its storage time, as discussed later in chapter 5.

2.4 The atomic frequency comb protocol

Let us now describe the AFC protocol. The protocol was introduced in 2009 in our group [41]. We will start by describing the main idea, then explain the steps of the protocol and finally introduce a technique used to increase the storage time called dynamical decoupling.

2.4.1 Main idea

In an inhomogeneously broadened medium, atoms will have different optical transition frequencies. The optical detuningsδ_j, for each atomj, follow an inhomogeneous distribution which is typically several GHz wide (in FWHM) for RE ions in crystals, as the precession frequency is given by the optical detuning. This leads to rapid dephasing of the individual spin terms in Eq. 2.2. In the CRIB protocol, the detunings are controlled through the use of an electric field in order to allow rephasing. In the AFC protocol, the spins naturally rephase, one only has to wait. This is achieved using optical pumping, in particular spectral hole burning, to create a comb-like structure in the absorption profile (i.e. the distribution of δj’s). A comb with teeth spaced by ∆ changes the phase terms δ_jt to ∆t, leading to a photon echo at the time T_M = 2π/∆.

At this point in time, the time-dependent phases of each spin component of Eq. 2.2 all become multiples of 2π.

What we have just described is called an AFC echo memory. In order to have an on-demand quantum memory, we transfer the optical coherence to a spin transition where the coherence time is much longer than for the optical transition. To do this, we apply two optical π-pulses that we call the transfer pulses. The application of transfer pulses makes it on-demand as the user chooses when to apply the second pulse. In this case, the user is limited by the inverse of the inhomogeneous spin linewidth. If one applies a spin (Hahn) echo sequence, the storage time will be limited by the spin coherence lifetimeT₂. By using dynamical decoupling techniques, the storage time can become limited by the spin population lifetime T₁.

In order for the protocol to work, the photon width γ_p must be smaller than the AFC bandwidth Γ, but at the same time larger than the teeth separation ∆, i.e.

∆ γ_p Γ. Also, the peak separation should be larger than the peak width γ, i.e.˜

˜

γ ∆. Once these requirements are met, the uncertainty in time for the absorption event is given by 1/γ_p. This means that the light-atom system reacts with a Fourier-

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limited resolution γ_p, which effectively smears out the structure of the AFC to an essentially flat distribution, since ∆ γ_p. This means one can approximate the absorption of light by the comb as absorption by a flat spectrum with some effective absorption depth.

2.4.2 Implementation

The AFC protocol consists in four steps: the preparation of the AFC, the absorption of the input (write process), the transfer of the coherence in and out of the spin transition and the emission of an echo after TM = 2π/∆ (read-out).

preparation: consists in tailoring the inhomogeneously-broadened optical transition, or usually a portion of it, into a frequency comb.

write: the input is sent to the sample. The input can be either a single photon, a weak or a strong coherent state depending on the experiment.

transfer: Some time after the input absorption, an intense pulse (called the control pulse) is sent to the sample to transfer the optical coherence into a spin coherence between the |giand |si states shown in Fig. 2.4.

read-out: After a timeTM = 2π/∆ +T_spin, the spins rephase and emit an echo.

One can choose not to apply the transfer pulses, and in this case we have only a delay-line memory, not an on-demand memory. In this situation, the excitation is stored in |ei, not in the spin state |si.

2.4.3 Dynamical decoupling

Dynamical decoupling (DD) is a technique used to decouple a spin system from its environment, i.e. protect it from decoherence [42, 43, 44]. The best way to understand dynamical decoupling is to be familiar with the Bloch sphere, which we have already introduced. Four years after Hahn’s work, Carr and Purcell [45] showed that using a secondπ-pulse and repeating the sequence would compensate for the effects of diffusion.

In the case of the Hahn echo, there are only two pulses in the sequence (counting the firstπ/2-pulse used to initialize the state), and the delay between the pulses is changed for each measurement. The CP sequence uses a fixed time delay between the pulses, but repeats the sequence until there is no more signal. In 1958 Meiboom and Gill [46]

pointed out that if the phase of the second π-pulse is shifted with respect to the first, this cancelled out the effect of systematic errors in amplitude of the pulse, i.e. some

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e

g

Atomicdensity aux

Atomic detuningd Inputmode

Outputmode

C on

tro lfie

lds

s

D g (a)

Intensity

Time Input

mode

Output Control fields mode

/ 0

2p D-T T_s T0

(b)

Figure 2.4: a)Λ-system with an extra auxiliary state to allow for on-demand storage.

b) AFC sequence: the input is sent, after a while a control field is turned on, bringing the stored coherence to the |gi-|si transition. After a total time2π/δ+Ts, an echo is emitted.

constant error such that the pulse becomes a θ-pulse, with θ = π +. Already in these studies repetition of these pulse sequences were considered.

Although inspired by CP and CPMG experiments in nuclear magnetic resonance (NMR), DD was introduced much later, in the context of quantum information [42].

In DD, the goal is to “freeze” the evolution of quantum states and beat their natural coherence times. Sequences can be very complex, and cancel out the effect of decoherence processes. DD was originally thought to be applicable only to single spins [47] and not to ensembles storing a single excitation (see Fig. 2.5). Heshami et al. [48] showed that for sufficiently small errors in the areas of the pulses, the Hahn echo sequence could be successfully applied in spin ensembles. We made a detailed study of the effect of area and phase errors in the pulses of several known DD sequences, and quantified the amount of error resulting from the application of each sequence.

In order to gain an intuition of how errors in the pulses of DD sequences affect the AFC protocol, let us consider the Λ-system shown in Fig. 2.5. The input is absorbed in the|gi − |eitransition. After the application of the first control pulse, the input excitation is stored in the |si level. Each imperfect pulse of the DD sequence introduces extra population in the |si state, as the pulse does not make a perfect population inversion. When the second control pulse is sent to the sample, the extra population will be pumped to the excited state, resulting in optical noise upon read-out

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Figure 2.5: The idea of combining DD with spin ensemble quantum memory is illustrated here. The absorbed single photon is transferred to the shelving state|si for storage. DD consists in applying radio frequency or microwave pulses to do repeated population inversions between the|gi and|sistates. If the population inversion is not perfect, then each repetition introduces extra population in |si as can be seen in step b). In c), we see that the extra population introduced by DD will be read-out along with the single photon as noise. This gives an intuition of why errors in DD pulses are very detrimental in spin ensemble quantum memories.

of the quantum memory. In Paper V, we studied in detail the signal-to-noise ratio due to the application of different DD sequences.

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Chapter 3 Rare-earth-ion-doped crystals

The lanthanide group of chemical elements contains the fifteen metallic elements of the periodic table with atomic numbers ranging from 57 to 71. Rare-earth (RE) elements include the lanthanides, and two additional elements - Scandium and Yttrium.

Their most common oxidation state is RE³⁺. Thanks to their electronic structure, rare earth elements have interesting properties, for example a strong prevalence for optical transitions in the visible/near-infrared range. We can thus use these ions to produce coherent light via lasing, to amplify optical signals and also use them as medium to store the light.

Today, rare earth ions are a big part of our daily lives. Many commercial lasers use rare-earth-ion-doped crystals as gain media. Permanent magnets are usually made of alloys containing Neodymium. Also, rare-earth elements such as yttrium and Europium are used to emit bright light in display screens. Europium can for example be doped into a host lattice containing Yttrium. These phosphors then absorb energy from electrons arriving at the display, and use this energy to excite the optical transition of Eu³⁺ which then leads to fluorescence of red light. A modern smartphone for example contains a large amount of rare-earth ions inside.

In this thesis, we use crystals doped with rare-earth ions to store information encoded in light. The crystals that we use, such as yttrium orthosilicate, have very good optical properties and are transparent in the range of wavelengths we consider (visible to telecom,400−1600 nm). The RE ions themselves are highly absorptive to a specific wavelength. For example, in our work we use Neodymium which strongly absorbs 883 nm when doped into yttrium orthosilicate and Europium which absorbs at 580 nm when placed in the same crystal environment. The electronic transitions of RE ions in crystals are naturally protected from decoherence, which implies very good coherence times for RE³⁺ ion-doped crystals. The reason for this is that the 4f shell, with electronic configuration [Xe]4fⁿ, of trivalent RE³⁺ ions is not full, and the n electrons in this shell are shielded by electrons in the full, outer 5s and 5p shells. Typically it is said that the 5s and 5p orbitals provide a screening effect for the electrons in

Spin dynamics in rare-earth-ion-doped crystals for optical quantum memories

Thesis

Reference

Spin dynamics in rare-earth-ion-doped crystals for optical quantum memories

Spin dynamics in rare-earth-ion-doped crystals for optical quantum memories

THÈSE

Emmanuel Zambrini Cruzeiro de Parede (Portugal)

Abstract

Résumé de la Thèse

Contents

Introduction

Chapter 1

The microscopic world: spins, photons, and other entities

The macroscopic life is simpler

The microscopic world

Bringing the microscopic to the macroscopic

What does quantum have to do with communication?

Motivation and long-term dream

The concept of a quantum memory

The quantum repeater scheme

Chapter 2

Quantum memories in

inhomogeneously broadened ensembles

2.1 Homogeneous and Inhomogeneous broadening

2.2 Atomic ensemble quantum memories

2.2.1 General description of a stored state

2.2.2 The Hahn echo

2.3 State-of-the-art of ensemble based quantum mem- ories

2.4 The atomic frequency comb protocol

2.4.1 Main idea

2.4.2 Implementation

2.4.3 Dynamical decoupling

Chapter 3

Rare-earth-ion-doped crystals