GaAs quantum dots: A Dephasing-Free Source of Entangled Photon Pairs / submitted by DI Daniel Huber, BSc

Submitted by

DI Daniel Huber, BSc Submitted at

Institute of

Semiconductor and Solid State Physics Supervisor and First Examiner Univ.Prof. Dr. Armando Rastelli Second Examiner Univ.Prof. Dr. Gregor Weihs Co-Supervisor Assoz.Univ.Prof. Dr. Rinaldo Trotta February 2019 JOHANNES KEPLER UNIVERSITY LINZ Altenbergerstraße 69

GaAs quantum dots:

A Dephasing-Free Source

of

Entangled

Photon

Pairs

Doctoral Thesis

to obtain the academic degree of

Doktor der technischen Wissenschaften

in the Doctoral Program

Eidesstattliche Erkl ¨arung

Ich erkl¨are an Eides statt, dass ich die vorliegende Dissertation selbstst¨andig und ohne fremde Hilfe verfasst, andere als die angegebenen Quellen und Hilfsmittel nicht benutzt bzw. die w¨ortlich oder sinngem¨aß entnommenen Stellen als solche kenntlich gemacht habe. Die vorliegende Dissertation ist mit dem elektronisch ¨ubermittelten Textdokument identisch.

Acknowledgements

A doctoral thesis depends on many people to be completed successfully. First of all, I want to thank my supervisor Armando Rastelli for his help over the last years and all the efforts he invested to help me during my thesis despite his extremely dense schedule. I also want to thank him for establishing the right connections when problems seemed to be not understandable and for organizing interesting seminar talks.

I would also like to thank Rinaldo Trotta for his trust in me and my skills and to start with me the doctoral thesis. Without his unstoppable passion for semiconductor quantum dots, insomnia, nightly discussions to keep the blood pressure high and the quantum repeater, we would not have been able to publish so many nice papers with an impact far beyond what I have ever expected.

A thesis like this requires a good team of scientists to run the experiments and keep the motivation alive. During my time at the institute I had the best lab homies I could imagine. Marcus, without your permanent help during the endless measurement nights, your unique sense of humor, your ability not to be grumpy and your disease-resistant body there would have been no chance to finish this thesis even in a thousand years. Christian, you are one of the smartest physicists I have ever met. Your passion to understand and verify all the details in detail is a blessing for all of us. Many thanks also to Johannes Aberl, your accurate way of working was helpful for us all and I really enjoyed the discussions with you. I also would like to thank Johannes Wildmann for giving me my first photoluminescence lab experiences.

A photoluminescence lab does not work without suitable samples. Therefore, I want to thank Saimon for spending countless hours next to the MBE to provide the best samples possible for my work. Further, I want to thank Javier for teaching me how to work in a cleanroom and to show me how to transform a sample to a (most of the time) working device. Since you left Linz sala limpia is not the same anymore. Many thanks also to Xueyong and Ursula for helping me during the device fabrication and for fruitful discussions. Further, I would like to thank Fritz for his amazing skills in the workshop. I also want to thank my office colleges Huiying, Lada, Martin, Magdalena and Hannes for establishing such a nice working environment and for ignoring my yelling voice. Further, I would like to thank Patrick for providing his office for cocoa discussions and forstanding all my complaints.

stu-to Thomas Fromherz and Julian Stangl for fruitful discussions and doing the exercises classes together with me.

I would like to thank Susi for making order out of the omnipresent chaos.

Ich m¨ochte mich ganz besonders bei meiner Familie bedanken. Liebe Alexandra, lieber Markus danke f¨ur eure Unterst¨utzung in den letzten Jahren und euer Interesse an meiner Arbeit. Liebe Mama, lieber Papa, es ist nicht selbstverst¨andlich, so gute Eltern wie euch zu haben. Ihr habt mich immer mit allem, was euch m¨oglich war, unterst¨utzt, und das nicht nur um mir ein Studium zu erm¨oglichen, sondern auch um meine anderen Interessen zu f¨ordern. Ich bin unendlich dankbar, dass es euch gibt! Ich m¨ochte mich auch bei meinen Schwiegereltern Oskar und Gerlinde bedanken, die mich in ihre Familie aufgenommen haben und mir immer Unterst¨utzung gegeben haben.

Finally, I would like to thank my loving girlfriend Irina, who shared with me all the ups and downs during the last years. Thank you for all the motivation, joy and love you give to me. I have never met anyone like you and I am looking forward to an amazing future together with you.

Contents

1 Introduction 13

1.1 Motivation and Introduction . . . 13

1.2 Thesis overview . . . 16

2 Droplet etched GaAs quantum dots fundamentals 17 2.1 Fabrication and optical properties . . . 17

2.2 Fine structure splitting . . . 20

2.3 Optical excitation of a quantum dot . . . 21

2.4 Resonant two-photon excitation . . . 23

2.5 Engineering of the emission properties of a quantum dot via strain tuning 24 2.6 Extraction efficiency . . . 26

2.7 Single photon emission . . . 27

3 GaAs quantum dots as a source of near maximal polarization entangled photon pairs 31 3.1 Fundamentals on entanglement and quantum state tomography . . . 31

3.1.1 Quantification of entanglement . . . 35

3.1.2 Error calculation . . . 38

3.2 Entanglement degrading effects in quantum dots . . . 38

3.2.1 Fine structure splitting . . . 38

3.2.2 Recapture processes . . . 40

3.2.3 Valance band mixing . . . 41

3.2.4 Exciton spin scattering processes . . . 41

3.2.5 Overhauser field . . . 45

3.3 Entanglement study on GaAs quantum dots . . . 47

3.3.1 Sample structure . . . 47

3.3.2 Experimental setup . . . 47

3.3.3 Resonant two photon excitation of a GaAs QD . . . 50

3.3.4 Single photon purity . . . 52

3.3.5 Decay time measurements . . . 54

3.3.6 Entanglement measurements . . . 54

3.3.7 Discussion . . . 57

3.4 Observation of nearly maximal entangled photons in a GaAs quantum dot 59 3.4.1 Device structure and fabrication . . . 59

3.4.2 Improved measurement setup . . . 60

3.4.3 Erasure of the fine structure splitting . . . 61

3.4.5 Background light correction . . . 66 3.4.6 Spin scattering model . . . 69 3.4.7 Discussion . . . 74

4 Two photon interference experiments with GaAs quantum dots 77

4.1 Fundamentals of two-photon interference . . . 77 4.2 Two-photon interference experiments with GaAs quantum dots . . . 83 4.2.1 Study of the two-photon interference visibility at short time scales 83 4.2.2 Study of the two-photon interference visibility at long time scales . 87 4.2.3 Two-photon interference from remote quantum dots. . . 89

5 Summary and Conclusion 95

6 Outlook 97

7 Appendix 103

7.1 Pauli matrices . . . 103 7.2 Fidelity measurement with a reduced set . . . 103

Abstract

The development of a scalable source of non-classical light is fundamental for unlocking the potential of quantum technology. The ideal source of quantum light should simultaneously deliver single and entangled photons deterministically, with high purity, high efficiency, high indistinguishability and high degree of entanglement, and it should be compatible with current photonic integration technologies. Semiconductor quantum dots are currently emerging as near-optimal sources of indistinguishable single photons. However, their performances as sources of entangled-photon pairs are still modest in comparison to state of the art sources of entangled photon states, which are based on probabilistic parametric-down-conversion processes. In general, it is believed that semiconductor quantum dots can not generate maximally entangled photon pairs, since the solid state system underlies various decoherence effects.

In this thesis we use experimental and theoretical methods to investigate a material system that has received limited attention so far: droplet etched GaAs quantum dots. By coherent population of the biexciton, which is the fundamental state for the generation of entangled photon pairs in a quantum dot, we observe a near perfect single photon purity and highly entangled - albeit non maximally entangled - photon pairs. Theoret-ical considerations revealed that the exciton fine structure splitting is dominating the entanglement degradation. By the use of an on-chip piezoelectric strain actuator, we are finally able to cancel the fine structure splitting and perform an examination of the residual decoherence mechanism in quantum dot entanglement. The experimental work in combination with theoretical considerations allows us to demonstrate for the fist time that quantum dots can be considered as a nearly dephasing-free source of polarization entangled photons pairs on-demand.

Beside the study of entanglement properties, we also investigated the indistinguishability of the emitted photons via two-photon interference experiments, where we find high, but not perfect, values of interference visibility. Based on the data we discuss the effects limiting the indistinguishability.

The results presented in this thesis are of great scientific interest for future research, making quantum dot entanglement resources usable in future quantum technologies.

Zusammenfassung

Die Entwicklung skalierbarer Quellen f¨ur die Erzeugung von nicht klassischen Photonen-zust¨anden ist grundlegend, um das technologische Potenzial der Quantentechnologie zu erschließen. Eine solche Quelle muss folgende Eigenschaften besitzen: Die Quelle muss auf Abruf ein einzelnes verschr¨anktes Photonenpaare emittieren mit einer verschwindenden Wahrscheinlichkeit f¨ur die simultane Erzeugung mehrerer Photonenpaare. Außerdem m¨ussen die emittierten Photonenzust¨ande ununterscheidbar in ihren Freiheitsgraden sein. Die Quelle muss zus¨atzlich mit hoher Effizienz operieren, integrierbar auf einem Chip und mit g¨angigen Technologien der Photonik kompatibel sein.

In den letzten Jahren wurde gezeigt, dass Halbleiterquantenpunkte ausgezeichnete Quellen f¨ur ununterscheidbare einzelne Photonen sind. Bei der Erzeugung von ver-schr¨ankten Photonen haben Quantenpunkte bisher nur mittelm¨aßige Ergebnisse im Vergleich zu den besten probabilistischen Quellen, die mit parametrischer Fluoreszenz arbeiten, erzielt. Im Allgemeinen wurde bisher davon ausgegangen, dass Quanten-punkte aufgrund diverser Dekoh¨arenzeffekte im Festk¨orper nicht als ideale Quelle f¨ur Verschr¨ankung genutzt werden k¨onnen.

In dieser Arbeit nutzen wir experimentelle und theoretische Verfahren um ein bisher kaum verwendetes Materialsystem n¨aher zu untersuchen: tropfen ge¨atzte GaAs Quanten-punkte. Durch koh¨arentes optischen Pumpen der Quantenpunkte zeigen wir, dass diese eine ausgezeichnete Quelle f¨ur einzelne stark verschr¨ankte, wenn auch nicht maximal verschr¨ankte, Photonenpaare sind. Durch mathematische Modellierung k¨onnen wir zeigen, dass hier der maßgebliche Dekoh¨arenzeffekt die Feinstrukturaufspaltung der Exzitonen-zust¨ande im Quantenpunkt ist. Die Fabrikation eines Chips, welcher es uns erm¨oglicht mechanische Spannungen auf die Quantenpunkte mittels eines piezoelektrischen Aktua-tors auszu¨uben k¨onnen, erlaubt es uns die Feinstrukturaufspaltung zu eliminieren. Wir k¨onne dadurch erstmalig nachweisen, dass Quantenpunkte als nahezu dephasierungsfreie Quelle verschr¨ankter Photonen genutzt werden k¨onnen.

Neben der Erforschung von Quantenpunkten als Quelle verschr¨ankter Photonen haben wir im Rahmen dieser Arbeit auch die Ununterscheidbarkeit der erzeugten Photonen mittels zwei-Photoneninterferenz Experimenten untersucht. Dabei hat sich gezeigt, dass die emittierten Photonen nur teilweise ununterscheidbar sind. Eine genaue Analyse der Daten erlaubt es uns ¨uber die verantwortlichen Effekte zu spekulieren und m¨ogliche Verbesserungen f¨ur zuk¨unftige Experiment zu diskutieren.

Die Resultate dieser Arbeit sind von großem Interesse f¨ur die Forschung Quantenpunkte als integrierbare Quellen von verschr¨ankten Photonen in zuk¨unftigen Quantentechnologien nutzbar zu machen.

1 Introduction

1.1 Motivation and Introduction

Entanglement is one of the most fascinating aspects of quantum mechanics. The idea, originally developed from a Gedankenexperiment (see Ref. [1] and Ref. [2]), took several decades to be experimentally realized, henceforward opened up a steadily growing field of research expanding to a manifold of sources generating “spookily” interacting quantum states. Further, innovative concepts of entanglement based computation and communication protocols were developed, leading to a dramatic increase of interest in the topic [3].

If we focus our discussion on the successful distribution of locally generated quantum states to any remote destination — a fundamental requirement of quantum communication — photon states are the optimal solution due to their low environmental interaction.

Nowadays, widely developed optical fiber networks could provide the communication basis of such photonic quantum states and finally pave the way towards envisioned concepts featuring secure communication via quantum cryptography — more precisely quantum key distribution (QKD) — [4] or even quantum computational networks [5]. However, this is a task far from being easy, as fiber channels are associated to decoherence and losses of transmitted states degrading the quantum information, thus crucially limiting the communication distance [6, 7]. A quantum state can not be amplified likewise to a classical communication channel and demands more advanced non-classical technology. Therefore, several concepts of so-called quantum repeaters have been developed over the past years [8, 6, 9, 10, 11, 12]. A fundamental building block of such a quantum repeater is a quantum relay as shown in Fig. 1.1 (a). Here two single photons emerging from two different entangled photon-pair sources are used to perform a Bell state measurement, eventually entangling the left-over photons. This procedure, which is referred to as entanglement swapping, allows distant and independent photons, which have never interacted before, to be entangled. After the swapping the entangled photons can be stored in a quantum memory. Ultimately, a chain-shaped arrangement of such quantum relays principally enables the distribution of entangled states over arbitrary long distances. In the following, we want to illustrate the benefits of a quantum repeater according to Ref. [13]: If Alice sends photons via a lossy fiber channel with the length L to Bob, the transmission probability is given by: P (L) = e

−L

L0_{, where L0 is the distance over}

which the intensity drops by a factor 1/e [13]. If Bob does not receive any photon at all, the transmission has to be repeated. On average e

L

L0 _{repetitions are required for}

a successful quantum state exchange. In a repeater scheme, as shown in Fig. 1.1 (b), the fiber channel is divided in N segments, where each segment with a length of L/N is

equipped with a quantum relay. Therefore, the average number of repetitions for the whole chain is given by N e

L

N L0_{, which has a minimum at N = L/L0. Consequently, a}

repeater becomes beneficial for L > L0. In a typical telecommunication fiber channel this

is fulfilled for a length of a few tens of km. It is obvious from the concept in Fig. 1.1 (b) that after a successful swapping procedure Alice and Bob share an entangled photon pair. Finally, Alice can perform a quantum state teleportation to transmit a local generated quantum state to Bob [14]. We want to especially emphasize the substantial advantage to work with pairs of locally generated entangled photons-pairs in the subject of quantum repeater schemes, instead of the conditional and probabilistic NOON-state generation which are relying on pure single photon sources [15]. The concept illustrated in Fig. 1 demands for sources of high quality entangled photons, to fulfill several requirements. In

a) Bell measurement S1 _S2 quantum memory b)

A relay relay relay B

1 2 N

L/N

L

Figure 1.1: Quantum repeater concept. (a) In a quantum relay the photon source S1 and S2 emits pairs of entangled photons, where one photon of each pair is guided to the Bell state measurement apparatus enabling entanglement swapping. As a consequence, the outer photons, which can be stored in a quantum memory, remain in an entangled state. (b) In order to construct an entire quantum communication channel over a distance L, N chained quantum relays have to be interfaced for a long-distance distribution of entangled states among individual parties A (Alice) and B (Bob).

particular, each source has to emit single and entangled photons deterministically, with high purity, high efficiency, high indistinguishability and high degree of entanglement [16]. In addition, it would be desirable to have sources to be integrable on chip-like infrastructures at competitive levels to state-of-the-art photonic technologies. In this regard, solid-state sources of entangled photons are of special interest [17]. In the past years, the concept of parametric down conversion [18, 17] has been used for the majority of photon based entanglement experiments. However, besides the obvious advantages of the source, like its outstanding degree of entanglement and simplicity, one has to deal with severe drawbacks. Firstly, a high degree of photon indistinguishability must be paid

1.1 Motivation and Introduction

by a loss of source brightness. Secondly, the photon generation is a probabilistic process and follows Poissonian statistics, hence, there is a non-zero probability of generating multiple photon pairs during a single excitation cycle. Envisioning a quantum repeater scheme according to Ref. [8] it would be vulnerable to errors, thus lowering its overall efficiency [19]. It is rather clear that an alternative source is necessary.

In 2000 O. Benson et al. presented the concept of a device using a single semiconductor quantum dot (QD) for the generation of polarization-entangled photon-pairs via the biexciton-exciton cascade [20]. QDs posses discrete quantum states similar to single atomic systems and therefore are also referred to as artificial atoms. In contrast to a single atom, QDs can be easily integrated within photonic chips and are compatible with actual semiconductor technologies. Recent research promoted QDs as emitters of single indistinguishable photons at high extraction efficiencies [21, 18]. Nonetheless, it was commonly believed for a long time that QDs are not capable to reach a perfect degree of entanglement because of degradation effects related to the solid-state environment. Indeed measurements showed entanglement fidelities barely exceeding 80% [22], which are mainly limited by: (i) the presence of an energy splitting between the two intermediate X states, the so-called fine structure splitting (FSS) [23], (ii) re-excitation (recapture) of the intermediate X level to the XX level before its decay to the ground state. (i) can be addressed via temporal post-selection [24, 25, 26, 27] that, however, lowers the effective brightness of the source. Alternatively, external optics could be used to “compensate” for the evolving character of the entangled state [28, 29, 30]. However, the need for complex and bulky optics in combination with post-selection techniques makes QDs less appealing for a scalable quantum technology. All these drawbacks can be avoided by using external perturbations to eliminate the FSS [31]. To avoid (ii) resonant two-photon excitation can be used [32, 33] that, in turn, ensures on-demand generation of entangled photons. Beside this, it is believed that the effective nuclear magnetic field (Overhauser field) negatively influences the degree of entanglement [34, 22, 35, 36], which is supposed to be dominant in QDs containing high nuclear spin atoms like In. In this work, we investigate a hardly studied material system: droplet etched GaAs QDs. In contrast to strained InGaAs these QDs are quasi strain free, of high symmetry and free of intermixing [22] leading to small average values of the FSS [37]. Further, the use of Ga instead of In drastically reduces the Overhauser field [38], which makes the GaAs material system to an excellent platform to study the entanglement performance of QDs. In combination with resonant excitation we show that droplet etched GaAs QDs can emit mostly indistinguishable highly entangled polarization-entangled photon-pairs on-demand, with a high single pair purity [35]. However, the entanglement is limited by a small, albeit non-zero FSS. To overcome this hurdle, we make use of a patterned piezoelectric actuator [39, 40, 41], which allows us to fully erase the FSS. Finally, we are able to show for the first time that QDs can be considered as a nearly dephasing-free source of entangled photons on-demand [35]. Based on our results we are convinced that QDs will play a major role in future quantum technology.

1.2 Thesis overview

This thesis reports on (i) the generation of entangled photon pairs from droplet etched GaAs QDs under resonant two-photon excitation and (ii) two photon interfercence ex-periments with droplet etched GaAs QDs. The work, which is focused on quantum optical experiments with QDs, is organized in four main chapters, where the fundamental theoretical considerations are given in each chapter separately.

In chapter 2, the fundamental properties of semiconductor QDs are discussed. Fur-ther, we give a focused summary about the fabrication of droplet etched GaAs QDs. In the first part of chapter 3 we discuss how to quantify polarization entanglement of a two-qubit quantum state. Further, experimental entanglement measurements and basic characterization measurements of the GaAs QDs are presented (decay time mea-surement, single photon purity, spectral properties). In the second part we present the device structure of our strain-tunable QDs and perform a comprehensive investigation of the entanglement and possible degrading effects.

In chapter 4, we present the results of the two-photon interference experiments with our GaAs QDs. Again some fundamental theoretical concepts are included.

In the final chapter 5, the reader can find a comprehensive summary and conclusion highlighting the most important findings of this work. Further, we provide an outlook.

2 Droplet etched GaAs quantum dots

fundamentals

2.1 Fabrication and optical properties

Optically active semiconductor QDs are nanostructures, which confine the motion of charge carriers (electrons (e−) and holes (h+)) in all three spatial dimensions. Such a confinement potential is typically realized by embedding a structure on the nanometer scale, made of a low direct bandgap semiconductor, into a matrix of a high direct bandgap semiconductor [34]. The confinement leads to atom like discrete energy levels, thus QDs are often referred to as artificial atoms. The QDs investigated within this work are fabricated by solid state molecular beam epitaxy (MBE) via the Al droplet etching technique [42]. In this method Al is evaporated on an AlGaAs substrate, where the Al forms droplets on the surface. Due to an As gradient at the interface between the AlGaAs layer and the overlying Al droplet, nanoholes are etched into the surface (see Fig. 2.1 (a)). The obtained holes are highly symmetric as shown in Fig. 2.1 (b), under optimized conditions, which subsequently leads to QDs with a high in plane symmetry. The QDs are then obtained by depositing GaAs that fills the nanoholes during an annealing process [43]. Finally, this layer is capped with AlGaAs acting as top barrier. Beside of a high structural symmetry [37] the obtained QDs are almost free of strain and composition gradients. For this reason they have been used as model systems for studying the impact of structural parameters on their optical emission properties [44, 45, 46]. For the investigated QDs the host material Al_0.4Ga_0.6As with a band gap of 1.92 eV (at room temperature) is grown on a GaAs wafer. The GaAs forming the QD core has a band gap of 1.42 eV (at room temperature). The difference in band gap and the small dimensions of the QDs ( ≈ 40 nm in the lateral and 5 nm in the growth direction) allows for 3-dimensional carrier confinement and leads to a type I band alignment as sketched in Fig. 2.1 (c).

In this work, we are mainly interested in the optical transitions involving carriers in the first energy level in the conduction band (CB) and valance band (VB). A single energy level can be occupied by two e− and two h+, respectively, due to Pauli’s exclusion principle, where the first h+ level is typically a heavy-hole (HH) like state due to the quantum confinement effects [34]. Since the e− and h+ are strongly localized in the QD, multi-particle complexes are formed by Coulomb interaction between the confined carriers. The fundamental complex is the neutral exciton (X), which consists of a single

e− - h+ pair. If the s-shell is fully occupied a neutral biexciton (XX) is formed, which is made up by two e− - h+ pairs bound by Coulomb interaction. Beside this, charged complexes can be formed - the so called trions - which are either complexes of two e− and one h+ (X−) or one e− and two h+ (X+).

AlGaAs

1.2 nm

-5.6 nm

Al droplet

Nanohole

Inverted GaAs quantum dot

a

b

c

Figure 2.1: Droplet etched GaAs quantum dots. (a) Fabrication of a GaAs quantum dot by solid state molecular beam epitaxy, using the aluminum droplet etching technique. (b) Atomic force microscopy measurement with color scale reflecting the local height of a representative nanohole. The highly symmetric shape is crucial to obtain a high degree of entanglement. (c) Schematics of the conduction band (CB) and valance band (VB) edges in an optically active quantum dot. The modulation of the energy band gap on a nanometer scale leads to confined states for electrons and holes.

2.1 Fabrication and optical properties

The fundamental process for the generation of polarization-entangled photon pairs in a QD is the biexciton-exciton cascade [20], which is illustrated in Fig. 2.2. Initially

R R L L Energy |-1> |+1> EXX EX |XX> |X> |G>

Figure 2.2: Biexciton-exciton cascade in a quantum dot. The biexciton state

(|XXi) decays under the emission of a circular right (left) polarized photon with an energy EXX and leaves the quantum dot in one of the degenerated

single exciton states (|Xi). The momentum state of the exciton (|±1i) is given by the polarization state of the emitted biexciton photon. Subsequently, the remaining exciton decays to the ground state (|Gi) by emitting a circular left (right) photon with an energy E_X.

the QD is prepared in the XX state, which radiatively decays via the recombination of an e−-h+ pair under the emission of a single photon. The QD is in a single X state now. Subsequently, the X decays via the emission of a second photon. Because of the flat morphology of the used QDs, the growth direction (z) defines the natural quantization axis for the total angular momentum. The two electrons in the XX state have a spin quantum number of Se,z = +1₂ and Se,z = −1₂, while the HH valance band states have

a spin of J_h,z = +3₂ and J_h,z = −3₂. As a consequence, the total angular momentum along the quantization axis of the XX complex sums up to M = 0. The polarization of the emitted photons is therefore determined by the total momentum of the recom-bining electron-hole pair, where only transitions with a change in the total momentum of M = ±1 are optically active and designated as bright states. States with a total angular momentum M = ±2 are called dark states and do not couple to light via dipole transitions. Hence, the cascade follows either the left or right decay channel — which are degenerate in energy — leading to a sequence of circular right (R) biexciton and circular left (L) exciton photons, or vice versa. The resulting two-photon state can be written as:

|ψ+_{i = 1/}√_2(|L

XXi |RXi + |RXXi |LXi), (2.1)

where LXX (RXX) and LX (RX) are XX and X photons in the circular left (right)

polarization basis, respectively. This state is a maximally entangled Bell state and will be discussed in all details in Sec. 3.1. We want to point out that in general the emission

energy from the XX, EXX, differs from the one of the single X, EX, by the XX relative

binding energy E_B= E_X − E_XX. Depending on the strength of the Coulomb interaction and correlation effects, the XX can be “antibinding”, with E_B < 0 or “binding”, with EB > 0. In the investigated droplet-etched GaAs QDs only the latter configuration has

been observed, which is qualitatively attributed to the relatively large size of the QDs and consequent impact of correlation effects.

2.2 Fine structure splitting

In general, the energy degeneracy of X state in a QD is lifted by an energetic splitting, the so called FSS, as illustrated in Fig. 2.2 (a). The excitonic FSS stems from the

|XX> |X> |G> H V V H FSS

a

b

Figure 2.3: Fine structure splitting in a quantum dot. (a) Energy level scheme for the biexciton-exciton cascade in presence of fine structure splitting. The degeneracy of the intermediate exciton states |Xi is lifted by the energy FSS. (b) Polarization resolved micro-photoluminescence measurement of the biexciton (XX) and exciton (X) transition of an InGaAs quantum dot. The horizontal (violet) and vertical (green) polarized components of the XX and X, respectively, are split by FSS.

exchange interaction between the e− and h+ forming the X state and can be described by the electron-hole exchange Hamiltonian:

Hexch= −

X

i=x,y,z

(aiSˆh,i· ˆSe,i+ biSˆh,i3 · ˆSe,i), (2.2)

where Sh,i (Se,i) is the hole (electron) spin operator and ai and bi are the spin-spin

coupling constants in the x, y and z spatial axis. By using the total X momentum eigenstates (|+1i,|−1i,|+2i,|−2i) as a basis, we can write Eq. 2.2 in the following matrix representation [23]: Hexch=      +δ₀ +δ₁ 0 0 +δ1 +δ0 0 0 0 0 −δ₀ +δ₂ 0 0 +δ₂ −δ₀      , (2.3)

2.3 Optical excitation of a quantum dot

where δ0 = 1.5(az+ 2.25bz), δ1= 0.75(bx− by) and δ2 = 0.75(bx+ by). From the matrix,

it is obvious that the bright (|±1i) and dark states (|±2i) do not couple with each other. Further, the bright and dark states are separated by an energy δ₀. If the rotational in-plane symmetry of the wavefunction has a symmetry of at least D2D, it follows that

bx = by. In this case, the bright states |±1i are eigenstates of the Hexch and their

eigenenergies E±1 = +1₂δ0 are degenerate. However, if the wavefunction symmetry is

lower, the bright states couple, which leads to the eigenstates √1

2(|+1i ± |−1i) with the

eigenenergies E1,2= +1₂δ0± 1₂δ1. The cascade under presence of a FSS is illustrated in

Fig. 2.3 (a). Due to the coupling of the bright states, a sequence of either two linear horizontal (H) or linear vertical (V) polarized photons is emitted. In contrast to the bright states, it can be seen from Eq. 2.3 that the dark states always split. We want to point out that in Eq. 2.2 only the short range exchange interaction is involved. The long range exchange interaction has two additional effects [23]: (i) the splitting between bright and dark states is increased to ∆0 = δ0+ γ0, where γ0is the contribution of the long range

terms. (ii) The bright state splitting increases to ∆1 = δ1+ γ1 with γ1= γx− γy, where

γi are the long range coupling terms. In case of a D2D symmetry, it holds that γx = γy

leading to ∆1 = 0. In contrast to the X, the XX does not show any FSS as the e− and h+

are in a singlet state. However, in polarization resolved micro-photoluminescence (µ-PL) measurements (see Fig. 2.3 (b)) one will observe a splitting in the X as well as in the XX spectral line, simply due the fact that the XX decays into a X state. The FSS of the X states can be a dominant source of decoherence when it comes to the generation of entangled photons with a QD. We will discuss this in all details in Sec. 3.2.

2.3 Optical excitation of a quantum dot

The population of an excited state in a QD can be achieved in various ways. If we restrict our discussion to pulsed optical excitation one has three major options: (i) non-resonant excitation, (ii) quasi-resonant excitation and (iii) resonant excitation, which we will discuss in more detail:

(i) In non-resonant excitation the laser is mainly exciting carriers in the barrier material of the QD to photo-generate e−-h+ pairs (see Fig. 2.4 (a)). Subsequently, these e− and h+ _{are captured by the QDs and relax to the lowest energy levels, the s-shell.}

Arguably, the excitation regime is not favorable for an entangled photon generation because of the appearance of pronounced recapture processes (re-excitation) [47, 48, 49] and spin scattering processes [34]. Besides the negative effects on the entanglement fidelity, additional drawbacks are, first, the possibility to achieve on-demand population of the biexciton state is inhibited [32, 33], and second, the indistinguishability of the emitted photons suffer from the carrier-relaxation-induced time-jitter and fluctuating electric fields [50, 21].

(ii) In contrast to non-resonant excitation, quasi-resonant excitation is a more favorable approach in generating entangled photon pairs. In this regime the e−-h+ pairs are generated via excitation in a higher QD shell, e.g. p-shell, and the carriers normally undergo a fast relaxation process to the s-shell (see Fig. 2.4 (b)). The excitation allows

Energy EL EL EL

a

b

c

Figure 2.4: Optical pumping of a quantum dot. (a) Above bandgap excitation, (b) p-shell excitation and (c) resonant excitation.

for a “near” coherent population upon resonantly driving the p-shell [34]. Further, no adverse effects on the entanglement related to the excitation could be identified [30]. However, a recent study by Kirˇsanskˇe et al. in 2017 showed that for quasi-resonant excitation condition the indistinguishability is essentially limited by the time-jitter introduced via p-shell to s-shell relaxation processes [51]. For the QDs investigated here, quasi-resonant excitation is completely unsuitable for observing entangled photon generation. On one hand the p-to-s relaxation times turn out to be much longer than the radiative recombination time. On the other hand, XX emission is poorly observable under non-resonant or quasi-resonant excitation because of the competition with other charge configurations.

(iii) In resonant excitation the e−-h+ pairs are directly created in the s-shell of the QD as shown in Fig. 2.4 (c). The resonant population of the XX state requires a resonant two-photon absorption process due to dipole selection rules [52, 53]. According to current research, however, the resonant and quasi-resonant excitation do not show appreciable difference on the degree of entanglement, but only on the indistinguishability of the emitted photons. In fact, current studies proved that QDs are only exhibiting near-perfect degree of photon indistinguishability under resonant conditions [21, 54]. Further, the resonant excitation allows for almost near unity population probability of the XX state [55, 33, 56] - a condition of high importance in terms of quantum efficiency of the entangled photon source. Hence, to simultaneously optimize the degree of entanglement, photon indistinguishability, single photon purity and on-demand generation a resonant excitation regime is obligatory [36].

2.4 Resonant two-photon excitation

2.4 Resonant two-photon excitation

In the following, we want to discuss how to drive the XX-X cascade resonantly. For the creation of a XX state a two-photon absorption process is required to satisfy dipole selection rules [57]. The two photons, which we label as 1 and 2, have to match the energy condition E₁+ E₂= E_X + E_XX [57]. However, to keep our experimental setup as simple as possible, we use two single frequency laser pulses, where we tune the laser energy in a way that EL= E1 = E2 = EXX₂+EX (see Fig. 2.5). Furthermore, we have to

Energy |XX> EL πx πx |0> |X>

Figure 2.5: Concept of resonant two-photon excitation. A resonant population of the biexciton state is achieved via the absorption of a photon pair with linear polarization π_x. The energy of each absorbed photons is E_L= EXX+EX

2 .

determine the best polarization conditions for the resonant two-photon excitation. As already discussed in this chapter, the XX complex is formed by two e−-h+ _{pairs with}

angular momenta of +1 and −1, respectively. Such state can be achieved by a consecutive absorption of either a σ+ and a σ− circular polarized photon or by two-photons with linear polarization π_x, which in the end is a superposition of σ+ _{and a σ}−_{. We will use}

the latter configuration, as it is easier to implement in the experiment. The interaction between the QD and the laser light field can be described via a Hamiltonian of the form [52]:

HT P E = Hqd+ Hint, (2.4)

where, in a rotating frame approximation,

Hqd=

EB

2 (|+1i h+1| + |−1i h−1|) (2.5)

is the unperturbed QD and

Hint=

1

2√2[f (t)(|Gi + |XXi)(h+1| + h−1|) + H.c.] (2.6)

is the interaction term1. The states |XXi, |±1i and |Gi describe the XX, X and ground state in a rotating frame approximation. For the laser pulse we use a sech function to

describe the temporal intensity distribution: I(t) = 2¯hΩ · arcosh( √ 2) πτ0 · sech 2t · arcosh( √ 2) τ0 ! , (2.7)

with a pulse duration τ₀ and pulse area Ω. We obtain via numerical calculation (see Ref. [52]) that the state of the system after interacting with the laser becomes:

|ψi = e−iΛ(∞)2  cos _Λ(∞) 2  |0i + sin _Λ(∞) 2  |XXi  . (2.8)

Here Λ is given by:

Λ(∞) = 1 4¯h Z ∞ −∞  EB− q E_B2 + 8I2_{(τ )}  dτ ≈ A Ω 2 EBτ0 , (2.9)

where A is a constant. It is obvious from Eq. 2.8 that the system is in a superposition of the ground and XX state now, where the weights depend on the parameters of the interacting laser pulse. We directly obtain the XX state population probability via:

N = | hXX|ψi |2 = sin2 _Λ(∞) 2  ≈ sin2 2A Ω 2 EBτ0 ! . (2.10)

In the measurements presented in chapter 3 and 4 the binding energy of the QD E_B as well as the laser pulse duration τ₀ are fixed. However, by varying the laser intensity

I, which is proportional to Ω2, we can control XX population. In a power dependent measurement N — and in consequence the recorded photoluminescence signal — will oscillate, while increasing I. The exact trend of these oscillations, which are called Rabi-oscillations, depend on the binding energy EB and non-trivial parameters of the

laser pulse described by the constant A. Rabi-oscillations are only present while the system is resonantly driven. If the laser energy deviates from the resonant condition the oscillations are damped and disappear. In the following experiments we will make use of this circumstance to ensure a resonant population of the XX-X cascade (see Sec. 3.3).

2.5 Engineering of the emission properties of a quantum dot

via strain tuning

The formation of self-assembled QDs relies on stochastic processes, leading to fluctuations in the optical properties of the QDs. These fluctuations lead to: (i) a size distribution (ii) shape and composition gradients and (iii) strain anisotropies in the QDs. Qualitatively, (i) leads to a distribution of the X emission energies for different QDs, (ii) and (iii) give rise to the FSS. Especially in standard Stranski–Krastanow (SK) grown InAs QDs (ii) and (iii) are dominant effects, while these effects are reduced in the highly symmetric

2.5 Engineering of the emission properties of a quantum dot via strain tuning

droplet-etched GaAs QDs investigated within this work [34, 37, 46]. However, we observe also here a distribution in X emission energy as well as E_B and small, albeit non-zero, values of the FSS [37]. Having in mind the quantum relay scheme, as discussed in Sec. 1.1, matching the emission energies between different QDs as well as restoring the X level degeneracy is crucial [22]. Arguably, the desired condition is not achievable with realistic growth processes. Nevertheless, it was shown that emission energy and FSS of a QD can be precisely engineered using external perturbations such as the optical Stark effect, electric-, magnetic fields as well as strain fields or a combination of them [31, 58, 59, 49]. In this work we focus on strain engineering, which allows us to cancel the FSS (see Sec. 3.4 ) and to engineer the X emission energy (see Sec. 4.2.3). To

e

e||

+

Figure 2.6: Piezoelectric actuator. On top of the piezoelectric material (PMN-PT) a membrane hosting the QDs is bonded via an epoxy-based photoresist (SU-8). By applying a voltage (VP) along the z-direction ([001] crystal direction) leads

to a deformation of the piezo (strain ⊥ and k). The strain k is transferred

to the membrane and allows to engineer the optical properties of the QDs. introduce post-growth variable strain-fields into a QD a successful procedure is to embed the QDs into a membrane, which is firmly connected with a piezoelectric material [58]. The concept of such a device is sketched in Fig. 2.6. On top of the piezoelectric [Pb(Mg_1/3Nb_2/3)O3]0.72-[PbTiO3]0.28 (PMN-PT) (001) substrate a membrane hosting

the GaAs QDs is bonded via a hard baked epoxy-based photoresist (SU-8). Applying a voltage along the z direction leads to an out-of-plane ⊥ and an in-plane k strain

field. The strain _k is transferred to the membrane and consequently also to the QDs. The magnitude as well as the direction (tensile _k > 0 and compressive k < 0) of the

resulting strain-field can be controlled by the applied electric field F_p = Vp

d, where d is

the thickness of the piezoelectric material. For describing the strain state, we use the Voigt notation:

 = (xx, yy, zz, 2yz, 2xz, 2xy)T, (2.11)

where xx,yy,zz are the normal and yz,xz,xy are the shear-components. The effect of strain

on the semiconductor can be described by the theory presented by Pikus and Bir [60] in 1974. However, a comprehensive analysis of strain effects on a semiconductor material

is beyond the scope of this work and we refer the interested reader to Ref. [61], where a detailed analysis is provided. The effect of strain on the optical properties of a QD is manifold and depends on the components of . The tuning of the emission energy is mainly related to the strain induced change of the bandgap energy of the semiconductor material, which is proportional to the hydrostatic strain _hyd= xx+ yy+ zz [58]. In

contrast to that, the tuning of the FSS requires an anisotropic in-plane strain field (for example xx 6= yy) [62]. However, in an ideal PMN-PT (001) substrate the

in-plane piezoelectric constants are isotropic, leading to an isotropic biaxial strain field. Nevertheless, also here a strain anisotropy can be observed as the available substrates are typically polycrystalline. Further, the bonding leads to an anisotropic strain transfer [58]. We want to point out that in general a monolithic piezoelectric actuator is not able to erase the FSS [62] since the direction of the strain anisotropy is fixed and only the magnitude can be varied. To fully cancel the FSS for any arbitrary QD, at least two independent perturbations are required [31]. In this work, we use a micro-machined piezo structure to achieve the condition FSS=0 (see Sec. 3.4).

2.6 Extraction efficiency

As already discussed, a QD is embedded in a host matrix for achieving a proper 3D confinement. The host material AlGaAs has a high refractive index of about n ≈ 3.5 at the emission energy of our QDs. Hence, for planar as-grown samples the extraction efficiency is limited by total internal reflection. In fact, only a small fraction (1/(4n2)) of generated photons exit the sample. Taking into account a finite numerical aperture (NA) of the collection optics, commonly only less than 2% of the emitted light can be collected from the top surface [63]. However, for a real application as a deterministic photon source a near-unity extraction efficiency is required. Starting out with the definition of the extraction efficiency according to Ref. [64]:

ηee= ηceβ =

ηceΓtm

Γ_tm+ Γ_om (2.12)

where ηce describes the fraction of the emitted light collectable via the objective lens

(with a given NA) and the β-factor accounts for the spontaneous emission rates in a specific target mode Γ_tm or into all other modes Γ_om, respectively, we find three main approaches for its improvement:

(i) The spontaneous emission rate Γtm can be enhanced by embedding the QD into a

photonic structure with Purcell enhancement. The emission rate into a targetmode is then given by:

Γ_tm= Γ₀FP, (2.13)

where Γ0 is the original spontaneous emission rate of the QD and FP the Purcell factor,

which is given by:

FP =

3Q(λ_c/nef f)3

4π2_V

ef f

2.7 Single photon emission

Here Q is the quality factor (Q factor), λc is the central wavelength, nef f the effective

refractive index and V_{ef f} is the effective mode volume of the structure. Significant Purcell enhancement can be achieved by using structures with high Q factor and/or low mode volumes such as cavities or waveguides [64, 65].

(ii) The rate Γ_om can be lowered by inhibition of the emission into modes that cannot be collected via a shaping of the electromagnetic field around the QD. For this purpose structures as photonic crystal cavities, nanowires and photonic crystal waveguides are suitable [64, 22]. (iii) The value of η_ce can be increased by levering the TIR limitation. This can be achieved by shaping the sample surface at the emitter position, for example via microlenses [22].

However, the fabrication of a structure with high extraction efficiency is beyond the scope of this work and we refer the interested reader for an comprehensive overview to Ref. [22] and Ref. [64]. In this work we use a simple broad band planar cavity structure as discussed in Sec. 3.3 and 3.4. The QD layer is therefore place at the center of a λ cavity mode. Additionally, we make use of a solid immersion lens (SIL) with a Weierstrass geometry, which theoretically enhances the NA of the objective by a factor of n2, where

n is the refractive index of the SIL. Although we do not observe any Purcell enhancement

by our structure design, we estimate the photon extraction efficiency to be about 10%. This finally gives us a XX-X pair extraction efficiency of ≈ 1%, which is sufficient for our experiments.

2.7 Single photon emission

According to the decay scheme presented in Fig. 2.2 it is obvious that the XX-X cascade emits a pair of single photons during one emission cycle. More precisely, the transition from the XX to X and the X to ground state leads to the emission of a single photon, respectively. Since this property is fundamental for the measurements discussed in chapter 3 and 4, we will give a short introduction to the quantification of this non-classical state of light. The single photon emission can be verified by using an experimental concept presented by Hanbury Brown R. and Twiss R. Q. [66] in 1956, in the following called HBT setup. Such a HBT configuration is shown in Fig. 2.7 (a), which allows us to measure a function proportional to the second-order correlation function defined by:

g(2)(τ ) = < I(t)I(t + τ ) >

< I(t) >< I(t + τ ) >, (2.15)

where I(t) is the light intensity and τ a time delay. At first, we want to analyze the outcome of the HBT measurement assuming a classical light field. The 50/50 beam splitter ensures that the average intensity < I > is equal for both detectors. The time dependent intensity is then given by:

-2 0 2 0 1 2 g (2 )(t ) t/t_c coherent chaotic D1 D2

∫

τ

a

b

Figure 2.7: Hanbury Brown–Twiss setup. (a) Schematics of a Hanbury Brown–Twiss setup. The investigated light is split by a 50/50 beam splitter and subsequently detected by the detectors D1 and D2. The correlation electronics filters the AC part of the detector current and multiplies and subsequently integrates the photo current, where in the signal from D1 the delay τ is introduced. (b Theoretical trend of the second order auto-correlation function (g(2)(τ )) versus the _ττ

c, where τ is the delay and τc the coherence time, for a coherent

and a chaotic light source.

where ∆I(t) are the intensity fluctuations acting on a time-scale determined by the coherence length τ_c of the light and < ∆I >= 0. It can be shown that for any I(t) [13]:

g(2)(0) ≥ 1 (2.17)

and

g(2)(0) ≥ g(2)(τ ). (2.18)

The second order correlation function for coherent light (for example a laser beam) and chaotic light are shown in Fig. 2.7 (b). For the latter one we assume a Doppler-broadened Gaussian line shape, which simulates an atomic discharge lamp. While the coherent laser beam yields a constant value g(2)(τ ) = 1, the chaotic light increases within the coherence time τ_c to a maximum value of g(2)(0) = 2, which is called a photon bunching. Having in mind a single photon emitter, one would expect from the experimental setup in Fig. 2.7 (a) to observer g(2)(0) = 0, as always only one detector can detect a photon at a delay of τ = 0. However, according to Eq. 2.17 and Eq. 2.18 we can not find a classical expression describing this property. Therefore, we have to rewrite Eq. 2.15 by using the concepts of quantum mechanics so that [34]:

g(2)(τ ) = < ˆa

†_(t)ˆa†_{(t + τ )ˆa}†_{(t)ˆa(t + τ ) >}

< ˆa†_{(t)ˆa(t) >} , (2.19)

where ˆa† (ˆa) is the photon creation (annihilation) operator in the Heisenberg picture.

2.7 Single photon emission

before [34]. However, for a photon number state containing n photons we find that:

g(2)(0) = 1 − 1

n. (2.20)

As a consequence, this leads to a photon antibunching where g(2)(0) < 1, which is in contradiction to the classically obtained result.

For the g(2) measurements, we use a slightly different approach as the one presented by Hanbury Brown and Twiss. We perform a so called start and stop measurement. The used detectors are avalanche photodiodes. If a single photon is detected on one detector (e.g. D1) the correlation electronics starts a counter, which measures the time till a second photon is detected on D2, while all other events on D1 are ignored. The recorded time is quantized an then stored in a histogram where all measured start-stop intervals are integrated over a given measurement time. In Fig. 2.8 (a) a representative

-80 -60 -40 -20 0 20 40 60 80 0 100 200 300 400 C o in c id e n c e s (c o u n ts ) t ns 0 2000 4000 6000 t ns

a

b

Figure 2.8: Autocorrelation measurement under pulsed excitation. (a) The au-tocorrelation measurement of a single photon emitter (QD exciton transition). Due to pulsed excitation we observe several peaks, with a distance of 12.5 ns corresponding to the repetition rate of the laser. At τ = 0 ns the peak is much weaker than all the other indicating a high photon antibunching. (b) Autocorrelation measurement on a long time scale. Due to a blinking of the quantum dot we observe a decay in the peak height versus τ .

HBT measurement in the following also refereed as autocorrelation measurement -of a X to ground state transition in a QD is shown. Due to a pulsed excitation the resulting histogram shows several peaks instead of a continuum. The shape and width of these peaks is determined by (i) the decay rate of the measured transition and (ii) the pulse response function of the HBT setup. (ii) is dominated by the detector time jitter. The separation between the peaks is given by the inverse of the repetition rate of the excitation laser, which is in our case 12.5 ns. The missing peak at τ = 0 indicates strong photon antibunching. According to Eq. 2.20 we expect for n = 1 that g(2)(0) = 0. However, due to effects like recapture processes (see Sec. 3.2) or scattered laser light (see Sec. 3.3.4) we typically obtain values g(2)(0) > 0. For quantification of the single photon

purity we calculate the g(2)(0) via:

g(2)(0) = A0

As

, (2.21)

where A0 is the area of the central peak and Asis the average area of a side peaks, which

is estimated by averaging over a few peaks around τ = 0. We want to point out that this formula only holds, because of the low photon extraction efficiency (<< 1) of the used QD sample. For a system with an extraction efficiency equals to 1, we would observe only two peaks at τ = ±12.5 ns in a start and stop measurement. For an extraction efficiency < 1 the height of the peaks drops with increasing value of |τ |, where the decay is related to the extraction efficiency of the source. On a timescale in the µs range we observe an exponential drop in the peak height (see Fig. 2.8 (b)). However, this effect is not related to the source efficiency, but shows the effect of QD blinking [67, 68, 69]. Here the QD switches between a bright and dark state with a characteristic on and off time. Due to this, the probability of having a coincidence at τ 6= 0 is increased for low τ [70], since a coincidence can only occur in an “on” period. The origin of the blinking is not fully understand yet, however, in Sec. 3.2 we speculate about a possible cause.

3 GaAs quantum dots as a source of near

maximal polarization entangled photon

pairs

3.1 Fundamentals on entanglement and quantum state

tomography

For understanding the basic concept of entanglement, we start out by defining the properties of an entangled state. We consider two independent quantum systems A and

B defined by the finite-dimensional Hilbert spaces HA and HB. The product space is

then given by HAB = HA⊗ HB, where ⊗ is the Kronecker product. As a consequence,

it holds that |ψ_Ai ⊗ |ψ_Bi ∈ H_AB for the states |ψ_Ai ∈ H_A and |ψ_Bi ∈ H_B. Let the systems A and B be photons in a basis |Hi , |V i, where H (V ) denotes a linear horizontal (vertical) polarization. We can write the product state in the most general case as:

|ξi = |ψAi ⊗ |ψBi = (a |HiA+ b |V iA) ⊗ (c |HiB+ d |V iB), (3.1)

where a, b, c and d are the weights. The state in Eq. 3.1 is a so called separable state, as the systems A and B are independent from each other and only connected via ⊗. In the following we will use a short notation for a product state so that |Hi_A⊗ |V i_B= |HV i._b Beside the separable state in Eq. 3.1 we can also find non-separable states. If we restrict our discussion to a two qubit system the following four states are non-separable:

|φ+_{i = √}1 2(|HHi + |V V i) (3.2) |φ−i = √1 2(|HHi − |V V i) (3.3) |ψ+i = √1 2(|HV i + |V Hi) (3.4) |ψ−i = √1 2(|HV i − |V Hi). (3.5)

These are the so called Bell states and represent maximal entangled states. The Bell states form a basis for bipartite quantum states of two-dimensional Hilbert spaces. It is obvious from Eq. 3.2 - 3.5 that the polarization states of photon A and B in one of the four states are not independent from each other and a measurement of A determines the

state of B and vice versa. A pair of photons in one of the above states is also known as an Einstein-Podolsky-Rosen pair [1]. In general, the statement is true, that all non-separable states are entangled states. In contrast to the maximal entangled Bell an example for an partly entangled state is:

|κi = cos(θ) |HV i − sin(θ) |V Hi), (3.6)

where | cos(θ)| 6= {√1

2, 1, 0}.

Starting from the two-photon state in Eq. 2.1, which is a non-separable state made up by the X and XX single photon states, we use |Hi = √1

2(|Li + |Ri) and |V i =

i

√

2(|Li − |Ri)

to rewrite the entangled state in the linear basis and obtain: |χi = √1

2(|HHi + |V V i), (3.7)

which is equivalent to a φ+ Bell state. To study the entanglement of photons stemming from a QD — which is one of the main subjects of this work — we have to consider two points: Firstly, we can not determine the two-photon state of our quantum system by just measuring one photon pair. Therefore, we have to look at an ensemble of photon pairs emitted by the QD. Secondly, due to entanglement degrading effects and measurement errors (see Sec. 3.2) the photon ensemble will not contain only the state |χi and is indeed a mixture of entangled and separable states. Hence, it is obvious that for a full description of the system a density operator formalism is obligatory. The density operator for a statistical ensemble of states is defined according to:

ˆ

ρ =X

i

pi|φii hφi| , (3.8)

where p_i is the portion of the ensemble being in the state |φ_ii, whereP

ipi = 1. In case

of an ideal QD entangled photon emitter the density operator reduces to a pure state. Using the linear basis (|HHi,|HV i,|V Hi,|V V i), the matrix representation of the density operator is given by:

ˆ ρ0 = |χi hχ| = 1 2      1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1      . (3.9)

As a consequence, the density matrix ˆρ0 fulfills the relation T r( ˆρ0) = T r( ˆρ02) = 1, which

is always true for a pure state [71].

In the following, we will briefly discuss how to measure a density matrix and use it to verify entanglement. For this procedure —which is called quantum state tomography— we use a framework based on the theoretical work done in 2001 by James et al. [72]. For simplicity we start out with a single photon qubit state. Therefore, we find a definition

3.1 Fundamentals on entanglement and quantum state tomography

of the density matrix by using a four dimensional Stokes vectors S = (S0, S1, S2, S3)T

according to ˆ ρ = 1 2 3 X i=0 Siσi, (3.10)

where σi denotes the Pauli matrices (see Appendix 7.1) and Si are the elements of S

called Stokes parameters. The Stokes vector, which describes the polarization state of a photon ensemble, can be obtained by projective measurements according to:

S0 = P|Hi+ P|V i≡ 1 (3.11)

S1 = P|Di− P|Ai (3.12)

S2 = P|Ri− P|Li (3.13)

S3 = P|Hi− P|V i, (3.14)

where P|Ii is the probability to find a photon in the polarization state I and |Di = 1

√

2(|Hi + |V i) the linear diagonal and |Ai = 1 √

2(|Hi − |V i) the linear anti-diagonal

polarization state, respectively. For example, if the photons are purely |Hi, we obtain a density matrix in the |Hi, |V i base of the form:

ˆ ρ = " 1 0 0 0 # . (3.15)

In contrast to Eq. 3.9, the density matrix has a dimension of two instead of four, which is simply due to the fact that we use a single qubit state.

The definition of the density matrix in Eq. 3.10 can be expanded to a n qubit state. However, we are only interested in n = 2, where the matrix is then given by:

ˆ ρ = 1 4 3 X i1,i2=0 Si1,i2σi1 ⊗ σi2. (3.16)

Determining the Stokes parameters for a 2-qubit system requires 36 projective measure-ments in total. However, in the work of James et al. the authors present a mathematical formalism to use only 16 projective measurements for determining the density matrix. These 16 measurements have to build up a tomographical complete set so that the matrix

B given by:

Bν,µ= hψν| |ˆΓµ| |ψνi (3.17)

is non-singular. Here ψν denotes the projective state ν and ˆΓµ are 16 linearly

inde-pendent 4×4 matrices given in Ref. [72]. An example for such a set is given in Tab. 3.1. The same set is used for the measurements presented in the following sections.

ν ζXX ζX 1 |Hi |Hi 2 |Hi |V i 3 |V i |V i 4 |V i |Hi 5 |Ri |Hi 6 |Ri |V i 7 |Di |V i 8 |Di |Hi 9 |Di |Ri 10 |Di |Di 11 |Ri |Di 12 |Hi |Di 13 |V i |Di 14 |V i |Li 15 |Hi |Li 16 |Ri |Li

Table 3.1: Tomographically complete measurement set. The table shows a rep-resentative tomographically complete measurement set required for the re-construction of the density matrix according to the formalism in Ref. [72].

ζXX and ζX denote the projective state for the biexciton and exciton photons,

where ψν = ζXX,ν⊗ ζX,ν.

The density matrix ˆρ can be reconstructed via a mathematical framework, by

measur-ing the coincidence counts nν = N hψν| ˆρ |ψνi, where N is a constant. However, this

mathematical framework is beyond the scope of this work and we refer the interested reader to Ref. [72]. We will focus now on the experimental considerations to obtain

nν. The experimental setup required for the quantum state tomography is shown in Fig.

3.1. The QD emits a single photon pair, which is then separated in X and XX via a monochromator. A set of a half-wave plate (HWP), a quarter-wave plate (QWP) and a polarizer allows us to project the photons in any required state ν. The detectors D1 and D2 in the XX and X optical path, respectively, are connected to the correlation electronics. The construction presented here is similar to the HBT setup, as discussed in Sec. 2.7. However, instead of measuring an auto-correlation of a single transition, we now are interested in the cross-correlation between XX and X. The obtained normalized co-incidence rates at a time delay τ = 0 in the projective base ψ_ν finally give us a value for n_ν. We want to point out that due to experimental imperfections the obtained density matrix can violate physically important properties like positivity, normalization and Hermiticity. To compensate this, a maximum likelihood estimation can be performed. In this work, we use an approach presented in Ref. [72]. The calculation follows a three step process: (i) A physical density matrix ˆρp(t1, t2, ..., t16) is generated fulfilling the

3.1 Fundamentals on entanglement and quantum state tomography monochromator QD X XX X XX D1 D2 HWP QWP P HWP QWP P electronics

Figure 3.1: Quantum tomography setup. The quantum dot (QD) emits a biexci-ton (XX) - excibiexci-ton (X) phobiexci-ton pair, which subsequently is separated via a monochromator. The XX (X) is guided to a single photon detector, in front of which a polarization optics consisting of a half-wave plate (HWP), a quarter-wave plate (QWP) and a polarizer (P) is placed. By changing the angles of the fast axis of the HWP and QWP the photons are projected in a certain base. The correlation between XX and X is then recorded via the correlation electronics.

requirements of positivity, normalization and Hermiticity. The ˆρp matrix is a function

of the 16 real parameters ti. (ii) A “likelihood” function L(t1, t2, ..., t16; n1, n2, ..., n16)

quantifies how well ˆρp fits to the measurement data. Here nj denotes the measured

coin-cidences in the projective measurement j. (iii) By numerical calculation an optimal set of parameters t(opt)₁ , t(opt)₂ , ..., t(opt)₁₆ is estimated, so that L has a maximum. The parameters

t(opt)_i define the resulting density matrix. For mathematical details on this method we refer the interested reader to Ref. [72]. Additionally, the “likelihood” estimation can be improved by using a set of 36 projective measurements, whereby 16 of them are redundant according to the measurement process discussed above.

3.1.1 Quantification of entanglement

In the following, we will discuss several quantities one can obtain from the two-photon density matrix to quantify the degree of the entanglement and benchmark different sources of non-classical light. Over the years different figures of merit have been established. However, we will focus here on three important parameters commonly used in the QD entanglement community:

Fidelity: The fidelity measures the state overlap between a measured density matrix

ˆ

ρm and a model density matrix ˆρ0. The fidelity is then defined by:

f =  Tr q p ˆ ρ0· ˆρm· p ˆ ρ0 2 . (3.18)

To quantify the level of entanglement it is obvious that ˆρ0 has to originate from a pure

entangled state. Hence, Eq. 3.18 simplifies to (see Ref.[73]):

f = Tr[ ˆρm· ˆρ0]. (3.19)

For our QDs we use the matrix given in Eq. 3.9 as a reference system to calculated the fidelity, which we from now on write as f+. The entanglement fidelity yields values between 0 and 1, where f+= 1 corresponds to a perfect state overlap and in consequence maximal entangled photons. If the fidelity drops to a level of f+ ≤ 0.5 the correlation between the photons is of classical nature and the state is not entangled. We want to point out that in the literature sometimes the square root of Eq. 3.19 is used as a measure of the fidelity. However, Eq. 3.19 can be interpreted as the probability of measuring the target state, e.g. φ+ _{in an ensemble given by ρ}

m. The square root of Eq. 3.19 has

no such interpretation. A disadvantage of the entanglement fidelity is that the presence of entanglement can be hidden if the measured state in ρ_m does not correspond to the expected one in ρ₀, for example if ρ_m originates from a |ψ+_{i state, while ρ}

0 is a |ψ−i.

Further, a rotation of the polarization state by the measurement setup can lead to similar problems (see Sec. 3.4).

A full reconstruction of the density matrix is not always required to determine the fidelity f+. It is sufficient to calculate it according to:

f+= (PHH+ PV V + PDD+ PAA− PRR− PLL)/2, (3.20)

where P_ii is the joint-measurement probability in the polarization base ii [74]. If the source is unpolarized and the measurement setup does not introduce any additional phase shifts, we can assume that P_HH = P_{V V}, P_DD = P_AA and P_RR = P_LL. The joint-measurement probability is given by [74]:

Pij = Tr[ρm|iji hij|] = hij| ρm|iji . (3.21)

Hence, the fidelity in Eq. 3.20 can be written via the density matrix formalism according to:

f+= hHH| ρm|HHi + hDD| ρm|DDi − hRR| ρm|RRi . (3.22)

The interferometric setup presented in Fig. 3.1 can be used to perform this simplified fidelity measurement by measuring the correlations visibilities C in the linear, diagonal and circular base so that [75]:

f+= 1 + Clinear+ Cdiagonal− Ccircular

4 . (3.23)

The correlation visibilities are then defined by:

Cµ= g2_XX,X− g2 XX,X g2 XX,X+ g_XX,X2 , (3.24)