Thesis
Reference
Solid-state optical quantum memory: from large entanglement to telecom wavelengths
STRASSMANN, Peter Clemens
Abstract
Quantum mechanics is leading to many modern technologies, e.g. based on laser physics.
Emerging fields are for example nonlinear optics, hybrid systems, or quantum communication.
Despite these applications, there are still limitations of our understanding of quantum mechanics. Even though it is broadly accepted that decoherence limits the size of entangled states, we currently do not know how to overcome these limitations. How large can commonly entangled states be? Can we observe macroscopic quantum phenomena with coarse-grained detectors or even with our bare eyes? This thesis contributes towards answering these fundamental questions and to technological advances. The experiments are based on a light-matter platform of quantum optical states and atomic ensembles. The ensembles consist of rare-earth ion-doped crystals. Additionally, the atomic frequency comb protocol turns the ensemble into a quantum memory. In the scope of quantum communication we work on a quantum repeater. First, we investigate fundamental quantum mechanical states and their validity at macroscopic scales. We show that the detection of a collectively [...]
STRASSMANN, Peter Clemens. Solid-state optical quantum memory: from large entanglement to telecom wavelengths . Thèse de doctorat : Univ. Genève, 2019, no. Sc.
5367
DOI : 10.13097/archive-ouverte/unige:123606 URN : urn:nbn:ch:unige-1236060
Available at:
http://archive-ouverte.unige.ch/unige:123606
Disclaimer: layout of this document may differ from the published version.
UNIVERSITÉ DE GENÈVE FACULTÉ DES SCIENCES
Groupe de Physique Appliquée - Optique Professeur Nicolas Gisin
Solid-State Optical Quantum Memory:
from Large Entanglement to Telecom Wavelengths
THÈSE
présentée à la Faculté des sciences de l’Université de Genève pour obtenir le grade de Docteur ès sciences, mention Physique
par
Peter C. STRASSMANN de
Mosnang (SG)
Thèse N
◦5367
GENÈVE 2019
Abstract
Quantum mechanics is leading to many modern technologies, e.g. based on laser physics.
Emerging fields are for example nonlinear optics, hybrid systems, or quantum communication.
Despite these applications, there are still limitations of our understanding of quantum mechanics.
Even though it is broadly accepted that decoherence limits the size of entangled states, we currently do not know how to overcome these limitations. How large can commonly entangled states be? Can we observe macroscopic quantum phenomena with coarse-grained detectors or even with our bare eyes?
This thesis contributes towards answering these fundamental questions and to technological advances. The experiments are based on a light-matter platform of quantum optical states and atomic ensembles. The ensembles consist of rare-earth ion-doped crystals. Additionally, the atomic frequency comb protocol turns the ensemble into a quantum memory. In the scope of quantum communication we work on a quantum repeater.
First, we investigate fundamental quantum mechanical states and their validity at macroscopic scales. We show that the detection of a collectively emitted excitation can be used to measure the size of the entangled ensemble. The entangled ensemble shared this excitation in a coherent superposition. This superposition of an atomic excitation in an ensemble is called W-state. We certify a new lower bound to the possible size of an entangled W-state between at least 16 million atoms. The proof of entanglement of this size relies on two measures: one determines the number of atoms involved in the storage process. The other one determines the state “quantumness” by examining the re-emitted light. Another experiment studies a macroscopically distinguishable superposition state. The superposition survives the storage in an atomic ensemble.
Second, we perform two more experiments that contribute to a quantum repeater. One experiment probes the quantum states simultaneously stored in a multi-mode quantum memory. In this work we also develop the concept of indirect entanglement witness. For the other experiment, a frequency down-conversion platform is set up. The conversion overcomes the wavelength compatibility issue between some quantum memories and the telecommunication C-band. The narrow spectral filtering of the conversion performs better than expected.
In summary, our work analyzes large entangled states and building blocks for a quantum repeater. The presented quantum memories are capable of temporal multi-mode storage. Its temporally precise re-emission is due to the re-phasing of the atoms of the W-state. We also show a good way to set up frequency down-conversion for the sake of quantum repeaters. This work emphasizes the atomic frequency comb protocol as a viable and reliable platform for quantum repeaters.
Résumé
La mécanique quantique est en train de se développer avec de nombreuses technologies modernes basées sur la physique des lasers ou l’information quantique. Par exemple, quelques domaines émergents sont l’optique non-linéaire, l’étude de l’interaction entre lumière et matière dans des systèmes hybrides, ou l’optique quantique avec la cryptographie quantique basée sur la distribution des clés quantiques. Malgré toutes ces percées et diverses applications, nous sommes confrontés à des limitations fondamentales de notre compréhension contemporaine de la mécanique quantique.
Même si la décohérence est acceptée comme phénomène limitant la taille des états intriqués, nous ne savons pas de nos jours comment dépasser ces limitations. Quel taille les états intriqués peuvent-ils avoir typiquement ? Pouvons-nous observer des phénomènes quantiques sur une échelle macroscopique avec des détecteurs classiques ou même nos yeux ?
Cette thèse donne des pistes pour répondre à ces questions fondamentales et propose des mises en œuvre d’applications technologiques telles que les répéteurs quantiques. Les expériences présentées ici explorent aussi bien le volet fondamental qu’appliqué, et utilisent des états quantiques de la lumière ainsi qu’une plateforme d’interaction lumière-matière à état solide. Celle-ci est réalisée à l’aide de cristaux dopés avec des ions de terres rares.
Premièrement, nous étudions des états fondamentaux de la mécanique quantique dans l’interac- tion lumière-matière et leur existence à l’échelle macroscopique. Nous démontrons la possibilité de mesurer la notion théorique de « profondeur d’intrication » dans un ensemble dans lequel une excitation délocalisée existe, grâce à la détection de l’émission collective de cet ensemble. En appliquant cette mesure nous trouvons une nouvelle borne inferieure à la taille de l’état intriqué, qui consiste en une excitation photonique délocalisée dans une superposition cohérente appelée état W, entre au moins 16 millions d’atomes. La preuve de la taille de cet état intriqué dépend de deux mesures expérimentales, qui déterminent d’une part le nombre d’atomes en jeu et de l’autre part à quel point l’état est quantique. Dans une autre expérience, un état de superposition distinguable macroscopiquement est généré entre un photon unique déplacé et un état cohérent.
Cette superposition reste largement insensible au stockage dans l’ensemble atomique est peut être déplacé en retour dans le domaine du photon unique.
Deuxièmement, nous réalisons deux expériences qui participent à l’élaboration d’applications telles que le répéteur quantique. L’une des deux développe le concept du témoin d’intrication indirecte pour certifier l’intrication entre deux paires de photons distinguable en temps, qui sont stocké simultanément dans la même mémoire. L’autre consiste en l’implémentation d’une plateforme de conversion par différence de fréquence pour rendre compatible les longueurs d’onde de mémoires quantiques avec la bande de télécom. Le bruit généré par la pompe de conversion est réduit de manière significative par l’utilisation d’un filtrage ultra fin, qui est implémenté derrière la conversion de fréquence.
En résumé, notre travail analyse des grands états quantiques et des éléments pour développer un répéteur quantique, qui est un objet essentiel pour la communication quantique sur des longues distances. La réémission avec un délai précis pour une mémoire avec un peigne de fréquences atomiques est le résultat du rephasage des atomes qui contribuent à l’état W et permet le multiplexage temporel. Au cours de la dernière expérience nous démontrons comment établir la conversion de fréquence pour des grandes différences de fréquence à très bas bruit dont par exemple le répéteur quantique peut profiter. Ce travail met l’accent sur les mémoires des terres-rares avec des peignes de fréquences atomiques pour mettre en lumière leur possible implémentation en tant que plateforme pour des répéteurs quantiques.
Zusammenfassung
Die Quantenmechanik ermöglicht eine Reihe moderner Technologien, basierend auf der La- serphysik und der Quanteninformation. Als aufstrebende Anwendungsfelder seien hier z.B. die nichtlineare Optik, die vertiefte Untersuchung der Wechselwirkung zwischen Licht und Ma- terie in hybriden Systemen oder die Quantenoptik mit Quantenkryptographie auf Basis des Quantenschlüsselaustausches genannt. Trotz genannter Errungenschaften und den vielfältigen Anwendungen ist unser heutiges Verständnis der Quantenmechanik an sich begrenzt. So ist zum Beispiel allgemein anerkannt, dass Dekohärenz die Größe verschränkter Zustände begrenzt, hingegen ist derzeit nicht bekannt, ob und wie diese Limitierung überwunden werden können.
Wie groß können kollektiv verschränkte Zustände sein? Können wir Quantenphänomene in makroskopischen Größenordnungen beispielsweise mit grobkörnigen Detektoren oder sogar mit bloßem Auge beobachten?
Diese Arbeit trägt einerseits zur Beantwortung dieser grundlegenden Fragen und andererseits zur Entwicklung von Anwendungen, wie einem Quantenrepeater im Rahmen der Quantenkom- munikation, bei. Das Zusammenspiel von Quantenspeichern und quantenoptischen Zuständen ermöglicht Experimente sowohl in der Grundlagenforschung als auch in der angewandten Richtung.
Zunächst untersuchen wir grundlegende quantenmechanische Zustände in der Licht-Materie- Wechselwirkung und ihre Existenz hin zur makroskopischen Skala. Wir zeigen, dass die Detektion einer kollektiv emittierten Einzelanregung aus einem atomaren Ensemble verwendet werden kann, um die Größe der Verschränkungstiefe im Ensemble zu messen. Dieser kohärente Überlagerungszu- stand des Ensembles, das die Anregung teilt, wird als W-Zustand bezeichnet. Wir belegen eine neue untere Schranke für die mögliche Größe verschränkter W-Zustände, die im vorliegenden Fall mindestens 16 Millionen Atome umfasst. Der Nachweis der Verschränkung dieser Größe beruht auf zwei Messungen: Die Erste bestimmt die Anzahl der am Speicherprozess beteiligten Atome und die Zweite quantifiziert die minimale Größe des darin enthaltenen Quantenzustands mit Hilfe der statistischen Analyse der emittierten Photonen. In einem weiteren Experiment wird ein makroskopisch unterscheidbarer Superpositionszustand zwischen einem verschobenen Photon und einem kohärenten Zustand. Diese Superposition überdauert auch die Speicherung in zuvor genanntem Ensemble und kann immer noch zurück in den mikroskopischen Bereich verschoben werden.
Des weiteren wurden zwei Experimente durchgeführt, welche wichtige Beiträge zu Anwendungen wie Quantenrepeatern leisten. In einem Experiment konnte erstmals die Verschränkung unabhän- giger Zustände, die sich gleichzeitig in einem multimodalen Quantenspeicher befinden, bestätigt werden. In dieser Arbeit entwickeln wir auch das Konzept der indirekten “Verschränkungszeugen”.
Für das andere Experiment wurde ein Aufbau zur Frequenzkonversion errichtet, um die Kompati- bilität der Wellenlängen von Quantenspeichern und dem C-Band der Glasfaserkommunikation zu gewährleisten. Der für die Differenzfrequenzkonversion implementierte schmalbandige Filter hat einen entscheidenden Einfluss auf das mit der Konversion einhergehenden Rauschen, da das Rauschen eine Frequenzkonversion in die entgegengesetzte Richtung erfährt.
Abschliessend lässt sich sagen, dass diese Arbeit große verschränkte Quantenzustände und Bausteine für einen Quantenrepeater analysiert, wobei letzterer eine Quantenkommunikation über weite Strecken ermöglicht. Die Quantenspeicher, basierend auf einem Atomfrequenzkamm, können gebündelt Information in vielen Moden und Freiheitsgraden speichern und ihre zeitlich präzise Reemission ist auf die Rephasierung der Atome, die zum verschränkten W-Zustand beitragen, zurückzuführen. Der Atomfrequenzkamm ist kompatibel mit dem zeitlichen Bündeln mehrerer verschränkter Photonenpaare und mit der Kommunikation über weite Distanzen. Durch das letzte Experiment zeigen wir, wie man eine rauscharme Differenzfrequenzkonversion implementiert.
Diese Arbeit verdeutlicht, dass der Quantenspeicher mit dem auf seltenen Erden basiertem Atomfrequenzkamm eine praktikable und zuverlässige Plattform für Quantenrepeater ermöglicht.
List of Publications
• Peter C. Strassmann, Anthony Martin, Nicolas Gisin, and Mikael Afzelius. Spectral noise in frequency conversion from the visible to the telecommunication C-band.Opt. Express,27 (10), 14298. May 2019.
• Florian Fröwis, Peter C. Strassmann, Alexey Tiranov, Corentin Gut, Jonathan Lavoie, Nicolas Brunner, Félix Bussières, Mikael Afzelius, and Nicolas Gisin. Experimental certifica- tion of millions of genuinely entangled atoms in a solid.Nat. Comm. 8(1), 907. October 2017.
• Alexey Tiranov,Peter C. Strassmann, Jonathan Lavoie, Nicolas Brunner, Marcus Huber, Varun B. Verma, Sae Woo Nam, Richard P. Mirin, Adriana E. Lita, Francesco Marsili, Mikael Afzelius, Félix Bussières, and Nicolas Gisin. Temporal multimode storage of entangled photon pairs.Physical Review Letters,117(24), 240506. December 2016.
• Alexey Tiranov, Jonathan Lavoie, Peter C. Strassmann, Nicolas Sangouard, Mikael Afzelius, Félix Bussières, and Nicolas Gisin. Demonstration of light-matter micro-macro quantum correlations.Physical Review Letters,116(19), 190502. May 2016.
Contents
Abstract iii
Résumé v
Zusammenfassung vii
List of Publications ix
1. Introduction 1
1.1. Quantum communication . . . 1
1.2. Frequency conversion . . . 2
1.3. Large quantum states . . . 2
1.4. Outline of the thesis . . . 3
2. The world of quantumness 5 2.1. Quantum states . . . 5
2.2. Entanglement measures . . . 6
2.3. Quantum communication . . . 8
2.3.1. Quantum repeater . . . 8
2.3.2. Quantum memories . . . 9
3. Nonlinear optics 13 3.1. Three-wave mixing . . . 13
3.2. Photon pair sources . . . 15
3.2.1. A source of heralded single photons . . . 15
3.2.2. A source of polarization entangled photon pairs . . . 16
4. Large entanglement in a solid 17 4.1. Large entanglement within the atomic frequency comb . . . 17
4.1.1. Ensemble size measure . . . 19
4.1.2. Experiment . . . 19
4.1.3. Entanglement depth . . . 22
4.1.4. Conclusion . . . 22
4.2. Displacement-induced micro-macro entanglement . . . 23
4.2.1. Experiment . . . 26
4.2.2. Results . . . 27
4.2.3. Conclusion . . . 29
4.3. Appendix: Derivation of the implicit entanglement depth relation . . . 29
5. Multimode quantum-state-preserving storage 31 5.1. Experiment . . . 32
5.2. Indirect entanglement witness . . . 34
5.3. Results . . . 35
5.4. Conclusion . . . 36
5.5. Appendix: Data analysis of four-folded coincidence events . . . 36
6. Spectral noise in quantum frequency down-conversion 39 6.1. Conceptual context . . . 39
6.2. Classical frequency conversion . . . 40
6.2.1. Experimental set-up . . . 41
6.2.2. Results . . . 42
6.2.3. Pump power dependence analysis . . . 44
6.3. Narrow-band filtering . . . 47
6.4. Conclusion . . . 48
6.5. Appendix: Comparison with literature . . . 48
7. Summary and perspectives for the future 51 A. Published articles 63 A.1. Demonstration of Light-Matter Micro-Macro Quantum Correlations . . . 64
A.1.1. Supplementary material . . . 70
A.2. Temporal Multimode Storage of Entangled Photon Pairs . . . 77
A.2.1. Supplementary material . . . 83
A.3. Experimental certification of millions of genuinely entangled atoms in a solid . . 88
A.3.1. Supplementary material . . . 94
A.4. Spectral noise in quantum frequency down-conversion from the visible to the telecommunication C-band . . . 102
1. Introduction
In the last decade technological advances have been strongly shaped by quantum mechanics and the new field of quantum information technology. Classical cryptosystems, for example, are vulnerable to Shor’s algorithm; one of the first quantum algorithms that theoretically outperforms any comparable classical algorithm today. This is where the quantum cryptography system became important as conceptually it is completely secure. Quantum cryptography relies on quantum communication to generate a secret one-time pad. It established on an academic level and proved invulnerable [1] to attacks [2] and can now be used commercially [3].
Quantum technology has received attractive applications from quantum cryptography [4] to quantum computation [5] to quantum sensing [6] to quantum simulation [7]. Enormous effort has been involved in all these advances: for example, in sharing and transferring entanglement or (equivalently) general quantum states over distance. The simultaneous storage of quantum states
and photonic quantum system interfaces are of enormous importance here.
1.1. Quantum communication
Quantum communication is the exchange of quantum information over distance. Photons can connect quantum systems with each other independently of their implementation.1 They have promising properties as they can travel long distances despite of exponential losses in a medium such as air or glass fiber. On an academic level the main application of quantum communication, quantum key distribution, has been shown to be feasible over distances of 421 km in fiber [11] and alternatively satellite-to-ground communication over 1200 km [12]. Free-space communication is limited by the absorption in air, which depends highly on water absorption and condensation, and restricts ground-to-ground application to distances of up to 144 km [13]. The current free space restriction to night time with low illumination can be avoided by shifting to telecom wavelength [14].
There are two ways to overcome these exponential losses with distance: first, in the case of quantum cryptography, trusted nodes can be used and have been implemented between Shanghai and Beijing [15], and a second more generally applicable possibility is entanglement swapping [16].
In addition, entanglement swapping between three photon pairs was performed by Goebel et al. in 2008 [17]. A first demonstration of the second level entanglement swapping involved four photon pair sources [18]. These experiments were performed without quantum state storage and on-demand rerouting so the count rates still decay exponentially with distance.
Quantum repeater and memories The storage of photons is a core ingredient of quantum repeaters which overcome the exponential losses over long distances [19]. Several requirements were formulated by Bussières et al. in 2013: storage efficiency, fidelity and duration, on demand read-out, multi-mode capacity, and the robustness/ease-of-use [20].
• For reasonable entanglement distribution rates, a storage efficiency of 90 % was claimed to be important, but it is not a strict requirement, similar to high detector efficiencies [19].
High efficiencies are achieved with ensembles, by multiple passages [21], cavity enhanced light-matter interaction [22], or waveguides on the ensemble [23]. In the lab efficiencies of up to 53 % have been achieved in atomic frequency comb [22], 76 % in electrically induced transparency [21] and 87 % with controlled reversible inhomogeneous broadening memories [24, 25].
1In the case of optical wavelength mismatch, frequency conversion efficiently overcomes this issue. For supercon- ducting or microwave qubits, the interface can in principle be implemented by quantum transduction to the optical domain [8, 9, 10].
• State fidelities above F = 2/3 are required to avoid QKD vulnerability by an adversary [26]. Experimentally, fidelities of 99.9 % have been shown with rare-earth based solid-state systems [27].
• The storage time should exceed the time required for entanglement distribution, which is estimated to be of the order of seconds for continental distances [19]. Long quantum storage times have been achieved with 0.1 s in cold vapor [28] and 1 ms in Eu3+:Y2SiO5 [29], which was extrapolated to be up to 6 h [30]. With classical light 1 min was achieved in electromagnetically induced transparency [31].
• The readout on demand can be guaranteed by a protocol based on the Duan-Lukin-Cirac- Zoller spin-storage [32] and combined with the atomic frequency comb scheme [33]. The latter storage protocol was implemented to include the photon pair generation in the AFC spin-storage [34, 29]. The spin-storage of single photons in an AFC memory was also combined with a quantum correlations to telecom photons [35].
• The multi-mode capacity was studied with coherent states [36, 37, 38, 39, 40] but also single photons [41, 42, 29]. Nevertheless none of them tested the simultaneous storage of quantum states, which is essential for quantum repeaters. As a part of this thesis, we study the capacity of retrieving entanglement from such a multi-mode memory.
• Another requirement is the photon wavelength compatibility with the telecom bands in order to maximize fiber transmission between the loss contribution of Rayleigh scattering and infrared absorption [43]. We have two examples to make the quantum memory state compatible with fiber transmission: first, the photon pairs can be generated asymmetrically, one photon for the quantum memory and the other at telecom, and, second, one of the generated photons can be frequency converted.
Most of these properties are optimized individually at the expense of other properties. It is of great importance to bring these building blocks together at the same time and test materials and techniques that would allow this effectively.
1.2. Frequency conversion
After the first coherent experiment of quantum frequency conversion [44], the field of quantum frequency conversion prosperously demonstrated applications from detector range enhancement [45, 44, 46, 47] to quantum information processing [48, 49, 50].
Quantum frequency conversion has different kinds of applications such as up-conversion detectors or quantum information processing. Even though great efforts have been made on telecom single photon detectors [51], efficient telecom detectors require cryogenic temperatures, compared to the range of visible light. The field if detectors mainly requires three properties as high conversion efficiencies, high count rates and low noise rates, while the field of quantum information processing additionally demands a coherent conversion. Dependent on the conversion efficiency the input beam undergoes an interaction between the two wavelengths [48], which is similar to a beam splitter.
Photon exchange with frequency conversion allows the compatibility between quantum systems, e.g. memories and allows fiber communication with lower losses than in direct transmission. With temporal multimode storage and frequency conversion for compatibility with other quantum systems at far distance since telecom wavelengths have few losses in optical fibers available internationally.
1.3. Large quantum states
We just saw that today quantum physics has various applications. However, the foundational understanding between quantum mechanics and the macroscopic world is still puzzling in nature and technical applications. Possible macroscopicity of quantum states has been a fascinating topic since the discovery of quantum mechanics. There are several facets of entangled states in
1.4. Outline of the thesis large systems such as Dicke states between a large number of particles [52] or Schrödinger cat micro-macro entangled states [53] or GHZ states [54]. The role of quantum mechanics in our daily life still remains an unsolved mystery.
1.4. Outline of the thesis
The thesis is based on two introductory chapters followed by the results of experiments on large entangled states and towards the quantum repeater. In chapter two I familiarize the reader with the topics of quantum mechanics and especially with the idea of a quantum repeater and quantum memories based on the atomic frequency comb. Chapter three is dedicated to nonlinear optics as a resource for quantum repeaters. The results are grouped in two topics: large entanglement in chapter four and quantum repeater technology, such as the temporal multimode capacity (chapter five) and the frequency conversion (chapter six). In conclusion, I summarize the results of my research and suggest further directions for investigation in chapter seven. In this thesis the author uses an existing quantum memory as is.
2. The world of quantumness
In this chapter we familiarize ourselves with some of the concepts and tools used in the experiments throughout this thesis. The major categories are quantum states, their analysis with appropriate measurements and quantum communication based on quantum repeaters.
2.1. Quantum states
In this thesis we use different quantum states, which are based on the quantum bit or simply qubit. To stress their analogy, a classical bit consist of a two-level system, which has no coherence between the levels, whereas the qubit also includes the coherence between them. In other words, a qubit has the possibility to cover not only two states |0⟩,|1⟩as the classical bit but complex, coherent superpositions of them like pure states as written in the Dirac notation [55],
|ψ⟩=α|0⟩+β|1⟩
= cos(︁θ
2
)︁|0⟩+eiϕsin(︁θ
2
)︁|1⟩, (2.1)
where the probability amplitudesα, β∈Care normalized as|α|2+|β|2= 1 or in the implicit versionθ∈[0, π] andϕ∈[0,2π). In the Bloch sphere representation these pure states correspond to the surface of the unit sphere, while the qubit states inside the sphere are called mixed states, which lost or share their coherence with other qubits. Every pure state|ψ⟩has an associated density matrixρ=|ψ⟩ ⟨ψ|. Mixed states can be written in terms of such pure states|ψi⟩(e.g.|0⟩,|1⟩), ρ=∑︁
ipi|ψi⟩ ⟨ψi|, with the condition on the probabilities to sum to unity,∑︁
ipi = 1. The three bases of the Bloch sphere relate to the so-called Pauli basis, which consists of the identity matrix 1and the Pauli matrices,
σx= (︃0 1
1 0 )︃
, σy=
(︃0 −i i 0
)︃
, σz= (︃1 0
0 −1 )︃
. (2.2)
The density matrix reads in this basis with real coefficientsmx, my, mz as follows, ρ= 1
2(1+mxσx+myσy+mzσz) =:1+m⃗ ·⃗σ
2 , (2.3)
where the coefficients vectorm⃗ represents the expectation values of each Pauli basis,mi= Tr(σiρ).
We call the central state (m⃗ = 0) of the Bloch sphere maximally mixed.
For example single photons possess several degrees of freedom, which can serve as a two-level qubit: the spatial (wave-vector) and temporal modes, frequency, and optical (spin and orbital) angular momentum. Also electronic states can be represented as qubits such as two of their energy levels e.g. at optical transition in atoms including the spin state in atoms which can be combined to hybrid quantum systems [56].
Entangled Bell states and generalization Looking at multi-qubit systems allows to study the property of entanglement. Let’s first consider two-qubit systems. In maximally entangled states, the qubits completely share their coherence. There are four such states, called Bell states, that form a basis, in which every maximally entangled two-qubit state can be written,
|Φ±⟩= 1
√2(|00⟩ ± |11⟩), |Ψ±⟩= 1
√2(|01⟩ ± |10⟩). (2.4) These states can be generalized tonqubits, where we have the so-called GHZ state, (|0⟩⊗n+
|1⟩⊗n)/√
n[54], or the Dicke states [52], which we discuss later.
2.2. Entanglement measures
The quantum state measurement is equally important as the existence of the quantum states themselves since the insights come only from studying these states, which is often harder than the state generation. In chapters 4 and 5 we discuss different methods how to analyze quantum states also indirectly such as via state transfer.
Measuring entanglement is at the core of quantum information technology. As such it is of major interest and developments in this direction is still on-going since the recognition of the
“spooky” feature later called entanglement [57]. Several criteria manifest entanglement in different degrees of freedom [58]. The measures differ in the statement, in the entanglement limits and in the number of measures required. Herein we discuss only methods with projective measurements, where after one single measurement one gets a definitive answer but the state after projection has irreversibly lost all information about the initial state.1
Bell and CHSH inequality What makes a two-particle state quantum? For illustration purposes we consider two players A and B as in figure 2.1. Each holds one of the particles, which they can manipulate and measure. Bell’s theorem forms an inequality saying that the observations of quantum mechanics cannot be predicted by any physical theory based on local hidden variables [59]. This inequality was initially developed to certify non-local or non-classical correlations, which are in contradiction with the locality.
A x∈ {0,1}
a∈ {−1,+1}
B y∈ {0,1}
b∈ {−1,+1} λ
Figure 2.1.: Bell scenario. A Bell inequality takes an input at each sidex, yand returns an output a, b, which depend on the corresponding input and the local hidden variableλ.
The four authors Clauser, Horne, Shimony, and Holt developed an explicit type of Bell inequality that distinguishes classical from stronger non-classical correlations [60]. The CHSH inequality defines a set of correlators based on projective single qubit measurements that can be directly applied on the quantum state. This criterion leaves the choice of basis free and it can be optimized for the given state. The CHSH parameter is given by,
S = ∑︂
x,y∈{0,1}
(−1)xyEx,y, where Ex,y= ∑︂
a,b∈{−1,+1}
ab p(a, b|x, y), (2.5) where the input parametersx, y∈ {0,1} represent the settings on each of the particles of the measurement with the outcomesa, b∈ {−1,+1}, respectively. Classical non-signaling strategies, i.e. without communication between the parties A and B, are limited by a value ofS= 2.
This paragraph describes an explicit example of the CHSH inequality for polarization entangled states of a quantum state. For consistency with the next chapters we substitute here the|0⟩,|1⟩by the polarization qubit basis with horizontal|H⟩and vertical|V⟩polarization. Theσx eigenstates are of diagonal|D⟩and anti-diagonal|A⟩polarization. In this basis our favorite Bell-state writes
|Φ+⟩=√1
2(|HH⟩+|V V⟩) and a possible measurement basis is given by the following set,
|A0⟩=|H⟩, |B0⟩= 1
√2(|H⟩+|D⟩), (2.6)
|A1⟩=|D⟩, |B1⟩= 1
√2(|H⟩+|A⟩), (2.7)
1In contrast several concatenated weak measurements in total can retrieve more information about the state but this is out of the scope of the present work.
2.2. Entanglement measures
(a) |1i
|0i A1
A0
B0
B1
(b) |1i
|0i
|+i
|−i
|+ii
|−ii
(c) |1i
|0i
|+i
|−i
|+ii
|−ii
Figure 2.2.: Illustration of qubit measurements. (a) CHSH measurement bases for qubitAare A0, A1 andB0, B1 on the other side for qubitB (4 outcomes each). (b) Complete quantum state tomography measurement bases on a single qubit with the minimally required set of 4 outcomes. (c) Over-complete quantum state tomography bases with the measured 6 different outcomes. The absolute direction is arbitrarily chosen.
where the first qubit is measured in the basis{|A0⟩,|A1⟩}and the second qubit is measured in the basis{|B0⟩,|B1⟩}, cf. figure 2.2 (a). The correlations are too strong to be compatible with any classical description without signaling (communication). The maximally entangled Bell states reach a value ofS= 2√
2, which is the maximal value achievable by any two-qubit quantum state.
For Werner states the valueS scales with the state visibilityV,S= 2√
2V [61].
The CHSH inequality can also be used as an entanglement measure in the framework of quantum mechanics [62]. We will come back on this in chapter 4.2. The violation of this adapted CHSH inequality has a lower bound ofS=√
2. Non-local correlations (V >1/√
2) are a stronger statement about the shared state than the application of CHSH as an entanglement measure (V >1/2).
Entanglement witnesses Entanglement is detected and witnessed by collective measurements on the quantum state. An entanglement witness is given by an observable W living in the joint-system Hilbert space which satisfies the following condition,
⟨W⟩ρ
sep = tr(Wρsep)≥0, (2.8)
for all separable statesρsepand for some entangled stateρent,
⟨W⟩ρ
ent <0. (2.9)
The set of separable states is convex and therefore its mixtures fulfill again the first condition of eq. (2.8). Experimental entanglement detection schemes are summarized by Gühne and Tóth [63].
Later in the chapter 4, we use different entanglement witnesses that are the CHSH inequality [62], the concurrence [64], the PPT criterion (positive under partial transposition) [65, 66], and the negativity [67]. The latter three criteria certify entanglement for visibilities above 1/3.
Quantum state tomography The single qubit quantum state tomography analyzes the quantum state in theσx, σy andσz qubit bases and reconstructs the quantum state from the measurements in these bases [68]. This concept can be generalized to several qubits where the tomography is applied on every qubit individually in collective measurements. There is a difference between a complete and an over-complete measurement set per qubit. The complete set consists of measuring four outcomes in three “orthogonal” projection bases, as shown in figure 2.2 (b), assuming an equally balanced number of qubits at the measurement input. An over-complete measurement includes measuring both outcomes in each of the three bases as in figure 2.2 (c) which allows to relax the assumption of balanced numbers of qubits. This measurement approach requires either more time to be measured or one more detector, which is the case we are interested in.
The analysis with quantum state tomography requires more measurements than the CHSH measurement, which can even certify non-locality for entangled photon pairs.
Cross-correlation function The cross-correlation functiongA,B(2) (τ = 0) determines to the strength of the correlation between detection events in two pair-wise qubit modes (e.g. the|H⟩modes of qubitsAandB) such as in the example of the CHSH inequality. The cross-correlation function is defined as,
gA,B(2) (τ= 0) = ⟨a†b†ab⟩
⟨a†a⟩ ⟨b†b⟩, (2.10)
where τ is the temporal delay between the measurements of AandB with their annihilation operatorsaandb, respectively. For example the cross-correlation of a thermal state is two and for a coherent state it is one. But quantum states can have higher correlations due to their strong non-classical correlations.
There is an empiric relation between the cross-correlation function and the certification of entanglement. The certification methods for entanglement, as seen right before, all require a certain minimal visibility. An empiric relation of the cross-correlation function and the Werner state visibilityV [69, 70], motivated by a previously found relation of the cross-correlation function and the quantum state coherence [71],
V =gs,i(2)−1 gs,i(2)+ 1
. (2.11)
This indicates that it should be feasible to prove entanglement for cross-correlations directly above the classical threshold,gs,i(2)>2 (forV = 1/3), with the entanglement witnesses based on quantum state tomography such as shown in chapter 4.2. With the CHSH based entanglement witness, entanglement should be certifiable forgs,i(2)>3 (forV = 1/2). The proof of non-locality requires a higher visibility, which results ings,i(2)>(√
2 + 1)/(√
2−1)≈6 (forV= 1/√ 2).
Monte Carlo method For the measures of CHSH inequality and quantum state tomography, the error propagation is not obvious and therefore the Monte Carlo method can be applied to sample the input variance and extrapolate the according outcome result variance. It is well-known that the photon number statistics follow the Poisson distribution. The resulting variance is in general no more Poissonian as the input distribution for the photon counts. The Monte Carlo method is used to create randomized samples of the input parameters around their mean number with the according standard deviation according to Poisson distribution. The outcome’s standard deviation or variance are read from the distribution of the samples’ outcome. Plotting the standard deviation as a function of the iterations gives a measure of how many samples are required for a stable solution.
2.3. Quantum communication
In quantum communication, it is important to have a high fidelity and repetition rate especially when it comes to sharing quantum entanglement over some distance which is limited by the loss in optical fibers. In contrast to classical communication the quantum state cannot be amplified nor copied [72]. To overcome these limitations quantum repeaters are a promising solution.
2.3.1. Quantum repeater
The scheme of quantum repeater works based on entanglement swapping [73] similar to quantum teleportation [74]. The entanglement swapping allows it to overcome the photon transmission losses as shown in figure 2.3. For example, in the case of two initial Bell-state qubit pairs, a simple mathematical regrouping allows to see the result on the remaining global state when one qubit of each pair undergoes a full joint Bell state measurement. The remaining qubit on each side needs to be stored in a quantum memory until the joint Bell state measurement is successful because the measurement is done at a distance between the two memories.
The total distance between two parties should be split by quantum repeater nodes into distances with reasonable losses. The mentioned entanglement swapping is applied between the memories at each node. At a next step the repeater then swaps entanglement between successfully entangled
2.3. Quantum communication
Figure 2.3.: The quantum repeater scheme. The Bell state measurement (BSM) together with quantum state storage allows for synchronization only between the successful mea- surements, which allow for entanglement swapping. This swapping theoretically allows to duplicate the blocks of BSM and storage for many intermediate steps and long distances. This figure is taken from [20].
stored qubits, which scales the benefit of this scheme exponentially on long distances to compensate the losses.
Performance criteria
The performance of a quantum repeater based on atomic ensembles and linear optics was summarized [19] and according criteria were postulated [20] similar to the Di Vincenzo criteria for quantum computers [75]. These performance criteria on the quantum memories concern the storage efficiency, duration and fidelity, on demand read-out, appropriate wavelength, multi-mode capacity, and the robustness/ease-of-use. The chapter 1.1 contains citations of current state-of-the-art performance results. The required performances depend on the repeater scheme, and it is out of the scope of this thesis to discuss these further.
2.3.2. Quantum memories
The storage of quantum information is an important ingredient for the long-distance quantum communication as shown for the quantum repeater scheme. The first paragraph starts with a comparison to the storage of classical information, followed by an overview of recent materials studied for quantum storage. In the following part we section a very brief introduction to the protocol used in this thesis and another section about the explicit implementation.
The storage of classical information has an enormous demand and as such the storage of light has been a investigated. Light was commercialized with optical discs such as re-writable compact discs (CD-RW), where laser light changes the reflectivity of sectors or simply reads the written state. However, this technology has reduced fidelity after a few writes [76]. These techniques are unfortunately not applicable to qubits as these do not store the superposition state. The fact that quantum states like photons cannot be copied nor amplified without errors requires new methods and materials, which on the contrary can be used to store classical light.
Alternative systems under study include hot and cold atomic vapor clouds of rubidium or color defects in crystals such as nitrogen-vacancy (NV) centers in diamond or rare-earth ions in crystals such as Y2SiO5. Among the rare-earth ions the studies currently focus on praseodymium, neodymium, europium, erbium, thulium, and ytterbium. Most of the systems are studied in atomic ensembles even though it is possible to work with single atoms or crystal defects such as NV centers [77]. Several storage protocols have been examined using a variety of materials, e.g. based on the controlled reversible inhomogeneous broadening [78, 79], the electrically induced transparency [80, 81], the DLCZ protocol [32] Some protocols can even take advantage of the
inhomogeneous broadening effect of the ensemble such as the atomic frequency comb [82]. For a review see e.g. the following review articles [83, 84, 20].
Atomic frequency comb protocol
The atomic frequency comb (AFC), a photon-echo type protocol, takes advantage of the ensemble’s inhomogeneous broadening while the atomic homogeneous broadening needs to be small allowing such a comb structure. Figure 2.4 shows the two concept versions of the protocol;
an optical delay line implementation and a full scheme including the spin-storage, which is investigated by Afzelius et al. [82]. Additionally a protocol adaptation allows to include the photon pair generation, the so-called AFC-DLCZ scheme [33].
The stored state is given by a Dicke state,
|D1⟩= 1
√n
N
∑︂
j=1
ei(k⃗·r⃗j−fjt)|01, . . . ,0j−1,1j,0j+1, . . . ,0N⟩, (2.12)
where⃗kis the wavevector and⃗rj are the atom positions. All atoms have their individual frequency fj=f0+l∆, wherel∈Zand ∆ is the frequency comb spacing. Without the frequency comb, the frequency distribution would be continuous and the de-phasing is rapid for a broad inhomogeneous system as in the rare-earth materials. Nevertheless thanks to the comb structure the atoms re-phase at the time delay oft= 1/∆ for the collective echo. The delay line scheme allows for a high storage fidelity together with a high efficiency and a multimode capacity.
The original DLCZ scheme [32] can be considered as a spin-storage quantum memory protocol that consists of a sequence of an off-resonant write pulse and a resonant read pulse. The write pulse provokes the spontaneous emission of a photon with the energy corresponding to the spin level. The photon is entangled with the spin state of the quantum memory. The same is valid for the AFC-DLCZ scheme except that the pulses are on resonance with the AFC so the re-emission time corresponds to the temporal photon delayTs−1/∆.
The absorption in the memory isηabs = 1−e−d˜, where the effective optical depthd˜ is reduced by the finesseF of the AFC,d˜ =d/F. The comb finesse is given in analogy to the finesse of a cavity by the ratio,F = ∆/Γ, between the tooth spacing ∆ and the FWHM of the teeth Γ.
Additionally, the re-phasing efficiency of the atomic excitations is given byηreph= sinc2(π/F) [87].
In this thesis we use a simple model for the AFC delay-line scheme which is valid in the case of low optical densityd˜≲1. This model shows an analogy to a fiber-loop by using twice the same interaction efficiency during absorption and re-emission (ηabs ≈ηre-emd˜) similar to the beam-splitter with a fiber-loop, which can be formulated as an interaction Hamiltonian. The re-absorption during the re-emission interaction ise−d˜. Hence we define the total AFC storage efficiency as,
ηtot=d˜2e−d˜ηreph. (2.13)
Neodymium doped quantum memory based on the atomic frequency comb
An AFC quantum memory was implemented in a Nd3+:Y2SiO5crystal and characterized by Clausen et al. [88], which we use in the experiments of the chapters 4 and 5. The storage protocol in delay-line scheme sets the storage time 50 ns for all quantum memory experiments presented in this thesis. The storage time corresponds to a fixed comb spacing of 20 MHz. This memory consists of two crystals separated by a λ2-waveplate to compensate for the absorption difference while the constant magnetic field still points in the same direction, which allows the faithful storage of polarization qubits. The crystals have a length of 5.8 mm and a Nd3+ concentration of 75 ppm. On the order of 10 billions atoms participate in the collective storage.
In a single passage through this memory the total storage efficiency is 4.6(2) %. The storage efficiency is determined by the absorption efficiency of 55(1) % and a re-phasing efficiency of 30 % (theoretical value) for 50 ns storage time. When used in double-pass configuration, the total in-out memory efficiency is 7(1) %, consisting of the absorption of 82(1) % and the same re-phasing efficiency.
2.3. Quantum communication
(a) AFC preparation
|ei
|gi
|auxi
|si
Absorption
Detuning
(b) Storage and retrieval
|ei
|gi
|si
|auxi
∆ input output con
trol
Absorption
Detuning
(c) Timing. AFC delay-line scheme preparation
input output
1/∆
time
(d) Timing. Full AFC scheme
input output
preparation
1/∆−τ control Ts control τ
time (e) Timing. AFC-DLCZ scheme
Stokes Anti-Stokes
preparation
1/∆−τ Ts τ
control control
time Figure 2.4.: The AFC protocol. (a) The preparation of the AFC consists of optical hole burning
with a spacing period ∆ by optical pumping the population via the excited state|e⟩
to an|aux⟩state. (b) The input photon excites the atoms contributing to the AFC which determines the timing of the photon output. The full AFC scheme involves two controlπ-pulses that map the excitation before the re-phasing to the|s⟩state for long storage and back. (c) The AFC delay-line scheme has a storage timetdl= 1/∆
fixed by the preparation sequence. (d) The full AFC scheme includes the two control beams that allow for on-demand read-out at a storage time oftsw = 1/∆ +Ts with a variable spin-wave storage timeTs. (e) The AFC-DLCZ scheme contains of an initial control beam from the ground state|g⟩with the AFC where some atoms decay to the|s⟩level emitting a photon entangled with the spin state which can be mapped on a photon by the read-out control pulse analogously to the full scheme. This image is adapted from [85, 86].
3. Nonlinear optics
This chapter provides a brief introduction to the nonlinear optical processes used in the rest of this thesis. Nonlinear optics cover processes outside the linear regime of absorption, dispersion, and refraction. The dielectric polarization has a nonlinear electric field dependence, in the form of a Taylor expansion,
PiNL=ϵ0(︁
χi,jEj+χ(2)i,j,kEjEk+χ(3)i,j,k,lEjEkEl+. . .)︁
. (3.1)
The linear susceptibilityχis responsible for the index of refraction,n=√︁
ℜ(χ) + 1≈1 +ℜ(χ)2 , and the absorption or gain of the medium,α≈ −kℑ(χ)2 . We are however interested in the first nonlinear termχ(2)i,j,k, which represents the three-wave mixing. Even though there are higher order interactions, other terms are negligible in our experiments.
Section 3.1 summarizes the fundamental characteristics and the versions of three-wave mixing, which are relevant for understanding and setting up such experiments. In section 3.2, I describe a photon pair source, which we used and improved for several experiments.
3.1. Three-wave mixing
The three-wave mixing is characterized by energy conservation and phase-matching, which is equivalent to conservation of momentum. These conservation laws are employed in two ways in three-wave mixing, in frequency conversion and in parametric down-conversion (PDC), as discussed in the following paragraphs. Another important aspect is the efficiency of the three-wave mixing process. The discussion of three-wave mixing here is restricted to the one dimensional case with co-propagating waves.
Frequency and phase matching Frequency matching is a synonym for energy conservation between photons while phase matching corresponds to conservation of momentum. In the case of three-wave mixing, frequency matching boils down to the basic equation of three frequencies, which are equivalent to the photon energies,
ℏω1+ℏω2=ℏω3. (3.2)
The phase matching condition depends on the index of refraction at each of the frequencies involved.
The index of refraction of the material usually has a frequency and temperature dependency.
The Sellmeier equation with the material specific coefficients describes these dependencies. For example, lithium niobate (LN) has been characterized in several articles such as [89]. The phase matching condition can be written as,
⃗k1+k⃗2=k⃗3, (3.3)
where⃓
⃓k⃓⃗
⃓= nωc , and in the case of co-propagating beams as,
n1ω1+n2ω2=n3ω3. (3.4)
We define a potential phase mismatch as ∆k=k3−k2−k1. The conversion efficiency depends on the effective wavelength mismatch ∆kand the crystal lengthL,
η(∆k)∝sinc2(∆kL
2 ). (3.5)
Ideally the mismatch vanishes indicating that the conversion is maximally efficient. Generally, the conditions eq. (3.2) and eq. (3.4) have no common solution because the index of refraction changes with the wavelength and is given by the material.
There are two ways to fulfill the frequency and phase-matching conditions at the same time;
either by a material with an orientation-dependent index of refraction or by artificial compensation of the phase-matching with periodical alternating susceptibility. Uni- or bi-axial materials have a polarization-dependent index of refraction that can be used to achieve phase matching. This approach is limited in the one dimensional case to the degeneracy of two wavelengths, such as second harmonic generation 2ω1=ω3. Another approach consists of quasi phase-matching the waves and compensating the imbalance in index of refraction over domains with an alternating second-order susceptibilityχ(2). The so-called poling period, Λ, is the length of two domains including the reversal of theχ(2) orientation between them, which allows a compensation of the wavelength mismatch, ∆k=k3−k2−k1−Λ/2π. The quasi-phase matching also allows us to adjust the phase matching point by the temperature. The poling period Λ can be chosen such that the quasi phase-matching is achieved closely above the room temperature. This is because the refractive index depends on the temperature. Frequently used media are, for example, uniaxial crystal lithium niobate (LN), potassium titanyl phosphate (KTP) with biaxial symmetry. These materials are used in this thesis with periodical poling.
Efficiency In addition to phase matching, the efficiency of the interaction between the input fields is given by the overlap integral of the field amplitudes. There are two approaches to maximizing the efficiency by increasing the field amplitudes: cavity enhancement in a bulk crystal and confinement in a waveguide by an adjusted index of refraction. A great advantage of a waveguide over cavity enhancement is its independence of cavity length stabilization. The functional principle of a waveguide is to confine the light in a cross-section of a higher index of refraction with respect to the surrounding host material. Waveguides are manufactured in two ways: buried waveguides by dopand injection and ridge waveguides. The latter constrains the light with three surfaces confining all beams in highly overlapping modes, which makes them more attractive. In buried waveguides the mode overlap can be rather low for far separated wavelengths even though each of them is strongly confined [90]. The cause for the low overlap is the asymmetry in the index of refraction because the buried waveguides lie at the surface of the host crystal. In this thesis we apply both types of waveguides to three-wave mixing.
Parametric processes in three-wave mixing This section discusses the processes of three-wave mixing that are relevant for this thesis. Figure 3.1 illustrates the two fundamental processes of sum and difference frequency generation (SFG and DFG) and parametric down-conversion (PDC). The SFG or DFG convert an input signal with the means of a high intensity pump wave into to an output frequency that is the sum or difference,ω3=ω1±ω2, of the two frequencies, respectively. The dashed arrows in figure 3.1 indicate the remaining input fields at the output, where they can be filtered out, e.g. with a dichroic mirror. In the PDC process there is only one input, the pump laser, which generates photon pairs (signal and idler) according to the frequency and phase matching conditions.
The experiments with the Nd3+:Y2SiO5 quantum memory involve a photon pair source based on the effect of PDC. In another experiment, DFG is implemented to convert the wavelength of the photons of an Eu3+:Y2SiO5 quantum memory from 580 nm to the telecom band.
(a)signal,ω1
pump,ω2
SFG or DFG
signal
output,ω3 pump
(b)
pump,ω3
SPDC
signal,ω1
pump
idler,ω2
Figure 3.1.: Optical parametric three-wave mixing processes. (a) Sum and difference frequency generation, abbreviated by SFG and DFG, respectively, have a different phase matching point according to the resulting wavelength. (b) Parametric down-conversion (PDC).
3.2. Photon pair sources
3.2. Photon pair sources
The parametric down-conversion (PDC) transforms the pump laser light into photon pairs. The resulting bandwidth of the signal and idler photon is often much broader in the PDC than for frequency conversion, where the input photon is usually more narrow-band than the spectral acceptance of the phase-matching. The signal and idler photon bandwidths are given by the spectral acceptance in the same condition of eq. (3.5). For convenience signal photons refer to those stored in the quantum memory.
The generated photonic state is given by a Taylor series of numbers of photon pairs,
|ϕ⟩=
∞
∑︂
n=1
pn|n, n⟩=|0,0⟩+p|1,1⟩+p2|2,2⟩+O(p3). (3.6) The probability to generate a photon pairpdepends on the nonlinear interaction strength and the input pump power. In reality we generate|ϕ⟩instead of |1,1⟩, which are similar for a very low generation probabilityp(once a pair is generated). Note that the Werner state visibility depends on pand on the losses after the generation. Higher order pair contributions act as white noise on the desired photon pair because the photons are pairwise entangled.
The PDC source, developed by Clausen et al. [91], generates polarization entangled photon pairs. Let us begin by discussing the simpler case, when this source is used to generate heralded single photons. The experiment in 4.1 is based on this simpler photon pair source as indicated in figure 3.2.
DM PPKTP
AOM Pump laser
532nm
Idler
Signal
Figure 3.2.: A photon pair source. The pump laser is pulsed by an acousto-optic modulator (AOM) and injected into a waveguide on a periodically poled potassium titanyl phosphate (PPKTP) crystal. Inside the waveguide pairs of signal and idler photons are generated by PDC. These photons are then separated by a dichroic mirror (DM) in two different optical paths.
3.2.1. A source of heralded single photons
In this paragraph we discuss the properties of the simpler photon pair source, indicated in figure 3.2. A periodically poled potassium titanyl phosphate (PPKTP) crystal acts as a nonlinear host material with a buried waveguide, which enhances the conversion from the photon pump into photon pairs. The pump is a continuous-wave laser at 532 nm, which is slowly modulated by an acousto-optic modulator (AOM). The photons are energy-time entangled since the pump pulse duration is much longer than the signal/idler pair coherence time. The signal photon matches the optical transition frequency of Nd3+ in Y2SiO5 at 883 nm. The detector of the idler photon at 1338 nm heralds the corresponding signal photon. The idler photon wavelength is situated in the telecom L-band, so they can be sent over long distances with low transmission losses (0.35 dB/km).
In the experiments the signal photons are sent to the Nd3+:Y2SiO5quantum memory introduced in chapter 2.3.2. In the source setup an AOM pulses the pump laser in order to reduce the noise to the signal photons during the memory readout.1 After the waveguide the photon pairs are split using a dichroic mirror.
1The undesired detected noise after the memory originates in the direct memory transmission. Due to the finite absorption efficiency of the memory, the transmission of a later photon pollutes the re-emitted signal if the delay between the two is equal to the storage time. This noise can be reduced by pulsing the pump laser with the AOM. Compared to a pulsed pump laser by Q-switching or mode-locking this method allows for higher photon rates due to a higher duty cycle.
Separated into the two modes the photons are filtered down to different bandwidths. The filtered signal photon fits in the bandwidth of the atomic frequency comb (AFC) and the filtered idler photon heralds the stored signal photon. Both signal and idler photons are filtered in two stages to bandwidths of 600 MHz and 240 MHz, respectively. Each filter stage consists of a volume Bragg grating and a cavity. The photon pair source described in the next section also uses this filtering.
A photon can only be heralded if the single-photon detector has a low jitter compared to the coherence time of the photon for a continuous-wave or slowly pumped source. In the case of the aforementioned filtering, this condition is satisfied.
3.2.2. A source of polarization entangled photon pairs
This source aims to generate a polarization encoded Bell state. The polarization entanglement is generated by equally splitting the pump laser via a polarizing beam splitter (PBS) and injecting it into two waveguides, which generate photon pairs in opposite polarization modes|H⟩and
|V⟩. The photon pairs of each waveguide are recombined via a PBS to generate a polarization entangled state of the form √1
2
(︁|HH⟩+e−iθ|V V⟩)︁
. The phase,θ, of the interferometer needs to be stabilized, as it defines the Bell state. The signal and idler photons undergo the same filtering as in the previous version.
PPLN DM PPKTP
PBS AOM
Pump laser 532nm
Idler
Signal PBS
Figure 3.3.: A PDC source of polarization entanglement. This setup enhances the version in figure 3.2. The pulsed pump laser is split on a polarization beam-splitter (PBS) to equalize the superposition of photon pairs generation between two nonlinear crystals.
For technical reasons the second nonlinear crystals are a periodically poled lithium niobate (PPLN) and potassium titanyl phosphate (PPKTP) waveguide. On a second interferometric PBS the photon pair superposition is merged to a Bell state, then separated by a dichroic mirror (DM) in signal and idler photons.
In chapter 5, I describe the work I undertook to increase the photon pair rates and the visibility related to the storage. During the experiments in the following sections, this photon pair source is analyzed with quantum state tomography and its non-local resource characterization via a CHSH violation.
4. Large entanglement in a solid
Scientists have been aware of the quantum physical phenomenon of entanglement for almost hundred years [57]. Today the frontier between the quantum and the macroscopic classical physics still puzzles us greatly. The frontier between relativity and classical physics was comparably straight forward to resolve and accept whereas the phenomenon of entanglement requires a change of paradigm. With the technical progress made during recent decades it has become possible to experimentally address the question of how the limits of quantum mechanics extend to the classical regime. Nevertheless whether it is possible to observe quantum effects on a macroscopic scale remains an open question, and decoherence is widely accepted as a limitation [92]. There is not a single way for entangled states to become large in size. Different facets of quantum states on a large scale are investigated here, such as an excitation shared between a large atomic ensemble (W state) or the micro-macro state of a cat, which is conditionally dead or alive, in Schrödinger’s
Gedankenexperiment [53].
For large quantum systems, the entangled state verification is much more difficult than its preparation as usually only certain collective measurements are possible [93]. The entangled states studied in this chapter are photonic states stored in an ensemble-based quantum memory as introduced in chapter 2.3.2.
This chapter contains two experiments, one that searches for the limiting size of entangled states while the other analyzes the possible size of quantifiable entanglement. In chapter 4.1, the experiment consists of the analysis of a stored single photon, which is equivalent to a highly multi-partite entangled state. In theory this state is as large as the ensemble. The second experiment, presented in chapter 4.2, examines the entanglement in a micro-macro light-matter state with a stored displaced entangled state. The state is generated by a single photon displaced with a coherent pulse on a beam-splitter, after which it is mapped into a quantum memory.
4.1. Large entanglement within the atomic frequency comb
This section is based on the results published in the following article:
F. Fröwis, P. C. Strassmann, A. Tiranov, C. Gut, J. Lavoie, N. Brunner, F. Bussières, M. Afzelius, and N. Gisin,“Experimental certification of millions of genuinely entangled atoms in a solid”, Nat. Commun.8, 907 (2017).
In this experiment we quantify the minimal size of multi-partite entanglement between the atoms involved in the storage of a photon by a lower bound. A single photon absorbed by an atomic ensemble creates a superposed excitation between the atoms, which is called W state. The most challenging part is the analysis of the state, especially without making assumptions about the initially created W state. The experiment as sketched in figure 4.1 has two novel conceptual parts:
• the analysis of the size of multi-partite entanglement, in the form of a W state, with single and two photon re-emission events and
• the ensemble size measurement via the ratio between the collectively re-emitted directional echo of an atomic frequency comb (AFC) memory and the isotropic spontaneous emission.
Our method to analyze the W state is based on the atomic excitation number statistics and the concept of entanglement depth. This concept is defined as the smallest number of genuinely entangled particles compatible with the measured data [94]. It applies state-independently to quantum systems, independent of their size. The access to the state and its size is limited to the optical re-emission. We cannot measure the state directly but only the re-emission related to the ensemble state and therefore we suppose linear interaction between the ensemble and the
