Towards quantum teleportation with remote quantum dots / submitted by Ing. Christian Schimpf, B.Sc.

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Submitted at

Institute of Semiconductor and Solid State Physics

Supervisor

Prof. Dr. Armando

Rastelli

Co-Supervisor

Ass. Prof. Dr. Helga Böhm November 2017 JOHANNES KEPLER UNIVERSITY LINZ Altenbergerstraße 69 4040 Linz, Österreich

Towards quantum

teleportation with

remote quantum dots

Master Thesis

to obtain the academic degree of

Diplom-Ingenieur

in the Master’s Program

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Eidesstattliche Erklärung

Ich erkläre an Eides statt, dass ich die vorliegende Masterarbeit selbstständig und ohne fremde Hilfe verfasst, andere als die angegebenen Quellen und Hilfsmittel nicht benutzt bzw. die wörtlich oder sinngemäß entnommenen Stellen als solche kenntlich gemacht habe. Die vor-liegende Masterarbeit ist mit dem elektronisch übermittelten Textdokument identisch.

Linz, November 2017

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Acknowledgement

First of all, I want to thank my supervisor Rinaldo Trotta for providing me the opportunity of performing this extraordinary master thesis. His visionary attitude and sympathetic manner should never be taken for granted in the world of science.

In equal measure I want to thank Javier Martín-Sánchez for teaching me the trades of process-ing in the cleanroom durprocess-ing my bachelor thesis. His enthusiastic and incredibly humble and courteous behaviour enriches our community beyond words.

Thanks to our valued head Armando Rastelli for providing me all the support and a never ending pool of visions, enriching our minds.

Further, I want to thank Munise Cobet for all our fruitful discussions. Her caring kind and her great scientific expertise inspires me constantly.

Of course I want to thank my appreciated colleagues Marcus Reindl, Daniel Huber, Huiying

Huang and Xueyong Yuan for accompanying me throughout my thesis. My work would have

never been possible without their friendly manner and competent advices.

Last but not least I want to thank Susanne Schwind for taking care about all administrative issues and Alma Halilovic, Stephan Bräuer, Albin Schwarz and Ursula Kainz for helping me with any technical issues throughout my work.

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"Der Verstand vermag nichts anzuschauen, und die Sinne nichts zu denken. Nur daraus, dass sie sich vereinigen, kann Erkenntnis enspringen."

Immanuel Kant Kritik der reinen Vernunft, Transzendentale Elementarlehre

"The mind does not see, and the senses do not think. Only when uniting both, knowledge can arise."

Immanuel Kant Critique of Pure Reason, Transcendental Elements

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Abstract

In dieser Diplomarbeit erklären wir das Konzept der Teleportation eines Polarisations-Zustands von Photonen mittels zweier örtlich getrennter Quanten-Punkte. Wir führen den Leser durch die Theorie der Teleportation, welche das Konzept von Verschränkung und der Bell-State-Messung beinhaltet. In diesem Zusammenhang beschreiben wir den Weg, die Teleportation statistisch zu überprüfen und inkludieren numerische Simulationen für eine bessere Einsicht in die vielen Herausforderungen, die sich dadurch ergeben.

Anschließend untersuchen wir die GaAs Quanten-Punkte, welche determistische Quellen für verschränke Photonen-Paare darstellen. Wir zeigen auf, warum diese unseren Anforderungen für die Teleportations-Experimente genügen. Wir beschäftigen uns mit deren Herstellung und deren energetischer Struktur, und weiters mit deren Emissions-Eigenschaften und den inher-enten Fehlerquellen, die die Qualität der Teleportation beeinflussen.

Wir beschreiben die Herstellung einer Baugruppe, die durch Verformung der Quanten-Punkte und deren Umgebung die Emissions-Energien zweier entfernter Quanten-Punkte anzugleichen vermag.

Weiters beschreiben wir die Maßnahmen, unseren experimentellen Aufbau für die ersten Ver-suche der Teleportation mit zwei Quanten-Punkten vorzubereiten.

Wir beschließen die Arbeit mit der Präsentation erster Resultate, die durch eine selbsgeschriebe Software ausgewertet wurden.

In this thesis we explain the approach of teleporting the polarisation states of photons emitted by two remote quantum dots. The reader is guided through the theory of teleportation, including the concepts of entanglement and the Bell state measurement of two photons. In this context we explain the way of statistically proving successful quantum teleportation. We include numerical simulations, which allows us to gather a better insight to the numerous challenges arising. Subsequently we investigate the GaAs quantum dots as deterministic sources of entangled single photons and why they are suitable for our purpose. We address their fabrication and energetic structure, as well as their excitation, the emission properties and imperfections, affecting the quality of teleportation.

We describe the fabrication of a device capable of exerting elastic stress to the host matrix of the quantum dots. This allows us to precisely match the emission energies of two remote quantum dots. Further, we point out the measures taken to prepare the experimental optical setup, enabling the first experiments of teleportation done so far in our institute. We conclude the thesis by showing preliminary results, evaluated by a self written software.

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1 Introduction

y -y A f+ f + -Bell state measurement y Classical channel Unitary transformation c Source of entangled photons BC Source of input photon Output

2

3 U

1

A B C

ALICE

BOB

Figure 1.1: The concept of teleporting the polarisation state of a photon from Alice to Bob, divided into three

basic building blocks. (1) The generation of an entangled pair of photons B and C by a suitable source. (2) The Bell state measurement of the input photon A to be teleported and photon B of the entangled pair. (3) The unitary transformation of photon C according to the Bell state measurement and the subsequent prove of teleportation by an experimental setup.

Since the end of the previous century, optical communication gained more and more impor-tance. Fiber optics paved the way for broad band data transfer with low losses, overcoming the electronic systems in many regards. Apart from the high efficiency, the exchange of photons also opens the gates for new fields of technology, arising from quantum mechanics: In classical information transfer schemes, information is conveyed by digital states in the form of bits (true or false). Photons, on the other hand, are also characterised by a polarisation, which can be described by any superposition state of horizontal and vertical polarisation. As a consequence, the information they carry is no longer only available in binary states, like for classical bits, but in so called qubits ("quantum bits").

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Consequently, another intrinsically safe method has to be found to connect two or more quan-tum channels with each other. Unfortunatley, direct replication of one quanquan-tum state to another is forbidden by the No Cloning Theorem[6] [7]_{. This is where quantum teleportation comes to} play.

The technique of quantum teleportation takes advantage of basically two peculiar phenomena of quantum mechanics:

1. Quantum systems can be entangled in certain degrees of freedom. In our case, we talk about entangled polarisation states of photon pairs. Due to their special creation mecha-nism in the semiconductor quantum dots we use, the polarisations of both photons possess certain mutual correlations: As long as none of them is measured, their polarisations are completely undefined, i.e. they remain in a superposition state. As soon as the polarisa-tion of one photon is defined (e.g. by a polariser), the polarisapolarisa-tion of the other photon is determined instantaneously, regardless of their current distance. This astounding

non-local feature is known as the Einstein-Rosen-Podolsky paradox[8] and was proven by a

manifold of experiments by showing the violation of Bell’s inequality[9]

2. Whenever two photons match in all possible parameters and degrees of freedom, i.e. in energy and polarisation, they are considered as indistinguishable. Whenever these photons meet at the very same time and place at a beam splitter, they interfere with each other. As a consequence, both photons will always leave the same output of the beam splitter, while both outputs are equally probable. This phenomenon is known as the Hong-Ou-Mandel

(HOM) effect[10]. Also polarisation-entangled superposition states of two photons, called

the Bell states, interfere at a beam splitter and yield particular possible output patterns. Quantum teleportation combines both aforementioned effects in order to transfer the polari-sation state of Alice’s photon A to photon C received by Bob (see figure 1.1). We use two quantum dots, where one produces the photon A to be teleported, and the other provides the entangled photon pair B and C. Photons A and B have to match in energy to enable the Bell

state measurement. The Bell state measurement yields information about how to manipulate

photon C arriving at Bob’s site in order to reproduce exactly photon A’s polarisation state. This information is sent by a classical channel, i.e. by simple bits.

Teleportation is not restricted to the transfer of single qubits. The concept also allows the teleportation of entangled states, which opens the gate for long range quantum networks by interlinking numerous teleportation subsystems[11]_{. Furthermore, in contrast to large scaled} quantum systems, teleportation emerges also in small scale application like integrated photonic circuits[12]_.

In this thesis we present the concept of quantum teleportation in the framework of photons from two semiconductor quantum dots.

Chapter 2 treats the theory of quantum teleportation, widely avoiding assumptions about any underlying physical system. We explain the general idea behind teleportation, followed by the description of the two major parts of the scheme: The entangled photon pair and the Bell state measurement as a special application of two-photon interference at a beam splitter. The final part presents the way of proving the successful implementation of quantum telepotation by deriving an important statistical quantity: The teleportation fidelity. Numerical simulations support each of the sections in order to suggest the crucial parameters for obtaining a highly functional teleportation system.

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teleportation experiments. This part introduces the consecutive chapters as the building blocks for overcoming these obstacles.

Chapter 4 introduces GaAs quantum dots as bright and deterministic sources of indistinguishable and entangled photon pairs. All these attributes, which will be addressed in detail through-out the chapter, make the quantum dot as a useful tool for our purpose. We will explain the resonant excitation scheme by a pulsed picosecond laser and the emission properties of the ex-tracted light. We will briefly discuss the production procedure of the quantum dots and how it inherently produces certain internal fluctuations and asymmetries. These consequently vary the emission energy, introduce decoherence effects and open a fine structure splitting between the dot’s excited levels.

Chapter 5 demonstrates the manufacturing of a piezo-electric device, capable of tuning the strain fields within the quantum dots and their host material. Biaxial in-plane strain perpendicular to the emission direction alters the energy levels. The device therefore allows the matching of the emission energies of two dots from remote samples, enabling the Bell state measurement between photon A and photon B.

In chapter 6 we explain the experimental setup used during our work. The section contains the various measures taken in order to provide a well operating arrangement for the first teleporta-tion attempt in the optics lab. We will demonstrate the realisateleporta-tion of the two-photon resonant excitation scheme by femto-second laser pulses. For separating the spectral components of the extracted light signal, we employ a transmission grating with high diffraction efficiency. Much effort was invested in the conservation of the polarisation state of the entangled photon pair: The polarisation-altering effects of the setup were quantified and subsequently corrected by a liquid crystal retarder (LCR). We show the full concept of measuring the polarisation state and of determining the necessary configuration of the LCR based on the results. The density matrix of the entangled photon pairs after passing the setup was recorded, proving the conservation of the original state from the quantum dot. Furthermore, a transformation in two other Bell states was performed to support the evidence.

We will illustrate results from interfering indistinguishable photons from two remote quantum dots at a beam splitter, recently published by our group[13]_{. With this work, the foundation of} the Bell state measurement is laid.

The final part of the chapter introduces the actual teleportation experiment and how to extract the teleportation fidelity from the measurement data.

In chapter 7 we present the software, written by the author, for processing the data from the teleportation experiment. The data basically consisting of a large list of detector events from three avalanche photo diodes (APDs), saved in a file. The software extracts the 3rd order

corre-lationfunction from the provided data, enabling the calculation of the teleportation fidelty. Due

to the recursive programming style, the software is prepared for future applications, demanding the calculation of even higher order correlations.

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and |φi are arbitrary quantum states (see appendix A.1 for a mathematical proof). However, there is a way of performing the transfer by using specific peculiarities of quantum mechanics, which we will describe in this chapter.

Figure 2.1: The Poincaré sphere as a graphical representation of a qubit in the state |ψi. The rectilinear basis

{|hi , |vi}and the diagonal basis {|di , |ai} span the equatorial plane, the component in the circular basis {|ri , |li} defines the angle in the polar plane. A pure state of the qubit |ψi represents a point on the surface of the sphere.

Throughout this thesis, we will treat the teleportation of the polarisation state of a single photon in particular. This state can be written as an element of a two-dimensional Hilbert space in an orthonormal basis spanned by the horizontal (|hi) and vertical (|vi) polarisation:

|ψi= a |hi + b |vi ∈ H2 (1)

with a, b ∈ C. Such a superposition state is called qubit in information technology. In contrast to a classical bit, which can only carry a binary information (on or off), the qubit’s information is defined by the photon’s arbitrary and continuous polarisation state. The qubit can therefore be represented by a point on the Poincaré sphere, as shown in figure 2.1. The elements of the diagonal and the circular basis are given by particular superposition in the rectilinear basis {|hi , |vi}: |di= √1 2(|hi + |vi) |ai= √1 2(|hi − |vi) |li= √1 2(|hi + i |vi) |ri= √1 2(|hi − i |vi) (2)

For quantum information processing, the preservation of this superposition state during a trans-fer is crucial. Determining the state directly by a single measurement (e.g. by a polariser) and subsequently sending a reconstructed photon to Bob is impossible: Born’s rule tells us, that the probing of the photon’s polarisation state by a projection in hA| leaves the photon in the state |Ai with probability of | hA|ψi |2_{, or absorbs it. The superposition state is therefore destroyed.}

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Quantum teleportation (the scheme is shown in figure 1.1) overcomes this problem by using a pair of entangled photons as a "mediator". The concept of entanglement will be treated in the following section (2.2). A suitable source owned by Alice sends out photon A in an arbitrary polarisation state |ψiA, which is intended to be "copied" to photon C from the entangled pair

and received by Bob for further use.

With photon A and the remaining photon B of the entangled pair, a so called Bell state

mea-surement is performed. The details about this step and its requirements will be treated in

section 2.3. Depending on the outcome of the Bell state measurement, Bob gains information about how to treat photon C in order to transform it into the polarisation state of photon A. Mathematically speaking, the measured mutual Bell state of photon A and B defines a unitary transformation ˆU to be performed on photon C in the state |φi_C, so that ˆU |φi_C = |ψi_A. The

information about the adequate transformation is conveyed by 2 bits (equals four possibilities) via a classical channel, excluding superluminal communication.

In real quantum systems, various effects originating from the used components lower the quality of teleportation, in the sense of how accurately the transferred state |φi_C resembles the input state |ψiA. The statistical quantity teleportation fidelity FT describes the quality of quantum

teleportation. It will be derived in section 2.4, showing the dependencies on the aforesaid effects and conclude this chapter by combining the treated concepts. We will use few assumptions on the actual physical system, keeping the calculations and simulations general.

2.2 Quantum entanglement

2.2.1 Product states

The state of two independent quantum systems A and B can be written as a product of their individual states, yielding an element of the product space of their original Hilbert spaces:

|ψi_A∈ Hn, |ξi_B∈ Hm→ |ψi_A⊗ |ξi_B ∈ Hn⊗ Hm (3) In the specific case of two independent photons in polarisation states written in the basis {|hi , |vi}, so that |ψi_A= a |hi_A+ b |vi_A and |ξi_B = c |hi_B+ d |vi_B, the product state is:

|vi_B =: |hvi_AB. Now the product state reads:

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(thus forming an entangled 2-qubit system), four distinct states are possible. These states are called the Bell states:

|φ+i= √1 2(|hhi + |vvi) |φ−i= √1 2(|hhi − |vvi) |ψ+_i₌ _√1 2(|hvi + |vhi) |ψ−i= √1 2(|hvi − |vhi) (6)

As quantum mechanical states they are a superposition of their basis states ({|hhi , |vvi} or {|hvi , |vhi}. They posess a non-local correlation, known as the Einstein-Rosen-Podolsky

para-dox[8] and violate Bell’s inequality[9]. This property predestine them as the fundament of

teleportation, which will be clarified at the end of this chapter (section 2.4).

2.2.3 The 2-qubit density matrix

For teleportation, any maximum entangled state can be used, as long as it is well known as a fixed property of the source. However, in real systems, the environment alters the entangled state, affecting the degree of entanglement. In particular we will now introduce a spurious effect which generates a time dependent relative phase shift Φ(t) between the two eigenstates of the system. The origin of this effect will be treated in detail in chapter 4. A time dependent state |χi(t) with the initial state |χi (0) = |φ+_i_{, for example, now reads:}

|χi(t) = √1 2

|hhi+ eiΦ(t)|vvi (7)

We assume the relative phase shift to evolve with a constant angular frequency Ω, so that Φ(t) = Ωt.

In our experiments, we deal with a huge ensemble of photons. If the special interest lies in the statistical composition of quantum states of a system, the density matrix plays an important role. It allows, amongst others, the deduction of a variety of quantities defining a systems entanglement properties.

In general, the density matrix is defined as

ρ=X i pi|χii hχi| , X i pi= 1 (8)

where pi is the probability to find the system in the quantum state |ψii.

If we now consider the 2-quibit state 7, with the time t as a continuous variable, the density matrix reads: ρ= bmax Z bmin P(t) |χi hχ| (t) dt (9)

where P (t) is the probability distribution function of finding the system in the state |χi (t) at a certain time t. The parameters bmin and bmax give the time window, in which the density

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matrix is evaluated. P (t) is normalized, so that

bmax Z

bmin

P(t) dt = 1 (10)

Due to the dyade product in the integral, ρ forms a complex 4x4 matrix in the basis {|hhi , |hvi , |vhi , |vvi}.

(a) |φ+i (b) |φ−i (c) |ψ+i (d) |ψ−i

Figure 2.2: 2-qubit density matrices of the pure Bell states

In the absence of a phase evolution, so that Φ(t) = 0, a perfectly entangled system remains in its pure initial Bell state for all times. The corresponding density matrices of the four Bell have no imaginary part and are represented graphically in figure 2.2.

The experimental construction of a two photon system’s density matrix will be explicitly ex-plained in subsection 6.5.5.

2.2.4 Entanglement fidelity and concurrence

The fidelity F defines the probability of finding a system, described by a density matrix ρ, in a certain reference state. The reference state itself may as well be a mixed state described by a density matrix. In our case we are especially interested in the resemblance to a certain Bell state. The fidelity regarding a Bell state |Bi , which is a pure state, is given by

F|Bi= T r (ρ |Bi hB|) = hB|ρ|Bi (11)

For completely uncorrelated photons, the fidelity regarding any Bell state yields 1/2 in the limit of measuring an infinite number on photons. This gives the fidelity threshold (or "classical limit") to overcome when proving the presence of entanglement. A fidelity value of one implies a perfect matching of the system to the Bell state. A value below 1/2 means an anti-correlation

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and Σ is the spin flip matrix for the chosen basis {|hhi , |hvi , |vhi , |vvi}: Σ =      0 0 0 −1 0 0 1 0 0 1 0 0 −1 0 0 0      (14) (a) (b)

Figure 2.3: Temporal fidelity f(t) (black lines) of the time dependent entangled state |χi (t) at a certain time t

for angular frequencies of (a) Ω = 3 GHz and (b) Ω = 15 GHz. The red line shows a fixed exponentially decaying probability density. The gray area visualizes possible integration boundaries when calculating ρ.

For a better understanding of the fidelity F and concurrence C arising from an ensemble of photons, we will investigate their dependencies on the integration boundaries bmin and bmax

and the angular frequency Ω. For this purpose, we introduce the temporal fidelity f(t) as a measure of how well |χi (t) resembles the specific Bell state |φ+_i_{at a certain time t:}

f(t) := T r|χi hχ|(t) |φ+i hφ+| (15)

The choice of P (t) as an exponential decay already approaches a real source of entangled photons, based on spontaneous emission (see chapter 4):

P(t) = 1 τ0

e−τ0t (16)

with a chosen fixed decay constant of τ0 = 150 ps.

Figure 2.3 shows the oscillating f(t) for two different Ω. In figure 2.3 (a) f(t) has values close to unity within a large region in which P (t) is significantly larger than zero. This means, photons in the state |χi (t) arriving at a detector will resemble |φ+_i_{with a high probability, thus giving} high statistical F and C. The gray area indicates an example of an integration time window when calculating ρ. For low angular frequencies, like in (a), also high integration times yield high F and C due to the fast drop of P (t) with respect to the temporal fidelity oscillations. In contrast, in the case shown in figure 2.3 (b), |χi (t) deviates from |φ+_i _{within the region} where P (t) is significantly higher than zero, and thus the expected values for F and C for large integration times are lower. However, for lower integration time windows, still high values for F and C are possible. This circumstance gives rise to so called post selection techniques

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when processing experimental data. With detector resolutions well within the temporal fidelity oscillations of the system, one can select only the photons with states close to the desired Bell state.

(a) (b)

Figure 2.4: (a) Fidelity F and (b) concurrence C for different angular velocities Ω as a function of the upper

integration limit bmax at fixed bmin= 0.

Simulations of F and C as functions of bmax (at fixed bmin= 0) were performed for varying Ω,

as shown in figure 2.4. Note that in principle any time window may be used, not necessarily starting from zero. This allows selecting time windows with high f(t), yielding large values for

F and C even for relatively high Ω. A drawback of this approach is the drop in the overall

performance, as a large part of the photons is simply discarded. Without any post selection, the result is practically given by the limiting case of bmax→ ∞.

2.3 Bell state measurement

Two photons in independent polarisation states form a superposition of all four Bell states with equal probability. By interference of these photons at a 50:50 beam splitter and particular measurements at the outputs, one can determine a distinct Bell state, and thus gain information about the system. How this information is used for teleportation will be clarified in section 2.4. This section will focus solely on the procedure of measuring the Bell state.

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where ωc _{is the center emission frequency and ∆ω the line width, defined as the full width}

at half maximum (FWHM) of the spectral distribution. We define the wave packet ζ(t) of a photon as the temporal distribution of the electric field, given by the Fourier transform of the spectral distribution function Φ(ω), so that

ζ(t) = √1 2π ∞ Z −∞ Φ(ω)e−iωt_dω ₍₁₈₎

Two photons are considered as indistinguishable, if

ωc₁= ω₂c

∆ω1= ∆ω2 (19)

as their wave packets are equal under these conditions (ζ1 = ζ2) (neglecting random dephasing mechanisms) and thus perfect interference is feasible if the photons are spatially well overlapped at the beam splitter.

2.3.2 Interference of two perfectly indistinguishable photons at a beam splitter

1

2

3

4 50:50

BS

Figure 2.5: Two indistinguishable photons arriving at the same time at the input ports 1 and 2 of a 50:50 beam

splitter. Due to quantum interference the two photons leave the beam splitter at the same output port 3 or 4.

For an convenient mathematical handling of single photons at a beam splitter, we employ the picture of number states[15] _{(a.k.a. Fock states) |n}

σi with n ∈ N0 photons in the polarisation

state σ. Using the photon creation and annihilation operators ˆa†

i,σ and ˆai,σfor the beam splitter

port i ∈ {1, 2, 3, 4} and the polarisation σ, the following equations hold: â† i,σ|nσii = √ nσ + 1 |(n + 1)σii âi,σ|nσii = √ n |(n − 1)σii âi,σ|0i = 0 [â†

i,σ,ˆaj,σ0] = δi,jδσ,σ0

[ˆa†

i,σ,ˆa

†

j,σ0] = 0

[ˆai,σ,ˆaj,σ0] = 0

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the creation operators transform like: â† 1,σ → 1 √ 2(â † 3,σ+ iâ † 4,σ) â† 2,σ → 1 √ 2(iâ † 3,σ+ â † 4,σ) (21) Assuming two indistinguishable photons in the same polarisation (we reduce â†

i,σ to ˆa

†

i and ˆai,σ

to ˆai) at the inputs 1 and 2 yields the following result for the outputs 3 and 4:

|1i₁|1i₂ = ˆa†₁ˆa₂†|0i₁|0i₂ (21)= √1

2(|2i3|0i4+ |0i3|2i4) (22) The latter equation reveals an important phenomenon, known as the Hong-Ou-Mandel (HOM)

effect[10]: If perfectly indistinguishable photons arrive at the same time with perfect spatial

overlap at the beam splitter inputs 1 and 2, both leave the very same output (3 or 4, with equal probability), as depicted in figure 2.5.

In the next step, we consider input photons in an entangled superposition state. As the term

Bell state measurement suggests, we are especially interested in the transformation of the four

Bell states (6). We still assume indistinguishability for photons arriving in the same polarisation. Applying the same rules as in 22, but now for superposition states in the base σ ∈ {|hi , |vi}, the four Bell states transform at the beam splitter like:

|φ+i₁₂= √1 2 â† 1,hâ † 2,h+ â † 1,vâ † 2,v |0i₁|0i₂ → 1 2(|2hi₃+ |2vi₃+ |2hi₄+ |2vi₄) |φ−i₁₂= √1 2 â† 1,hâ † 2,h−â † 1,vâ † 2,v |0i₁|0i₂ → 1 2(|2hi3− |2vi3+ |2hi4+ |2vi4) |ψ+i₁₂= √1 2 â† 1,hâ † 2,v+ â † 1,vâ † 2,h |0i₁|0i₂ → √1 2(|1h1vi3+ |1h1vi4) |ψ−i₁₂= √1 2 â† 1,hâ † 2,v−â † 1,vâ † 2,h |0i₁|0i₂ → √1 2(|1hi3|1vi4+ |1vi3|1hi4) (23)

Equation 23 reveals that only |ψ−_i _{yields two photons at two different outputs. Further, the} state |ψ+_i _{is the only one delivering a pair of photons at the same output, but with different} polarisations. We will later find these facts to be the basis of the Bell state detections we perform.

2.3.3 Interference of real photons at a beam splitter

In real applications, perfect matching of the photonic wave packets ζ1 and ζ2, as defined in subsection 2.3.1, is hard to achieve. Various environmental effects alter the spectral shape, the

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The field operators Ê₁+(t) and Ê₂+(t) at the inputs of the beam splitter transform at the outputs like: ˆ E₃+(t) = √1 2 ˆ E₁+(t) + Ê₂+(t) ˆ E₄+(t) = √1 2 ˆ E₁+(t) − Ê₂+(t) (25) We consider two single photons with same polarisation arriving at the same time at the beam splitter, so that the input state reads |ψiin = â

† 1ˆa

†

2|0i. The joint probability to measure a photon at output 3 and at output 4 with a time delay τ is then given by:

P34(τ) = h0| â1â2Eˆ₃−(0) Ê₄−(τ) Ê₄+(τ) Ê₃+(0)â†₁â†₂|0i (26) Using 25 and restricting the system to single photon states, so that h1| â†_â† _{= â â |1i = 0, the} probability reads:

P34(τ) = 1

4|ζ1(τ)ζ2(0) − ζ2(τ)ζ1(0)|2 (27)

This equation holds for coherence times τc  τ, so that the mutual phase relations between

the wave packets remain. In the opposite case, for τc  τ, for an ensemble of many photons,

the incoherent joint probability P34(τ) is composed of the probabilities of measuring individual photons: P34(τ) = 1 4(P1(0)P2(τ) + P2(0)P1(τ)) (28) where Pi(t) = h1|iEˆ − i (t) ˆE+i (t) |1ii (29)

The probability P34(τ) represents the case of two fully distinguishable photons, and thus also describes photons in orthogonal polarisations at the beam splitter.

(a) (b)

Figure 2.6: Evolution of P34(τ) and the visibility V for (a) different spectral detunings ∆EC at the same

linewidth (∆l = 0) and (b) for different linewidth differences ∆l at the same center emission energy (∆EC_{= 0).}

For a better visualisation, the dashed line shows P34(τ) only for ∆l = 0, although it also evolves with changing

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The results of numerical calculations of P34(τ) are shown in figure 2.6. The center emission energy and the linewidth for photon 1 were fixed at EC

1 = ~ωC1 = 1.5 µeV and l1 = ~∆ω1 = 50 µeV. In figure 2.6 (a) the line widths of both photons were kept equal (∆l = l2− l1 = 0) and the detuning ∆EC _{= E}C

2 − E1C was altered. Figure 2.6 (b) shows the evolution for fixed center energies (∆EC _{= 0) and changing the line width differences ∆l. The dashed line depicts the}

probability P34(τ) for completely distinguishable photons. P34(τ) also evolves for ∆l 6= 0, but in order not to overload the figure, only the case for ∆l = 0 is plotted.

The area under the curves represent the probability of measuring a photon at both outputs within a certain time interval. Note that for τ = 0 P34is always zero, regardless of the detuning ∆E and the linewidth differences ∆l. For high detector resolution, the shown interference curves can be resolved in coincidence measurements. For lower detector resolutions, only single peaks around τ = 0 will be visible, with shapes mainly dependent on the temporal distribution of the photons and the hardware’s response.

An important measure for evaluating the interference quality at a beam splitter is the visibility, sometimes also denoted as mean mode overlap, defined as:

V := A − A

A (30)

where A is the area under P34(τ) and A the area under P34(τ). The values for V for the simulated cases are also denoted in figure 2.6.

The visibility V can be interpreted as the probability of not getting a double click at the detectors compared to the number of double clicks from an equal stream of distinguishable photons at the beam splitter inputs. The experimentally measured value for V includes all effects accompanying the detuning of the photons, like random dephasing or bad single-photon purity of the signal. As a consequence, even without knowing the exact origin of a damped V , its value can be used for all further calculations.

Section 6.6 shows the results of experimentally performed two photon interference from remote quantum dots.

2.3.4 Detection of the bell state

As explained in the introductory part of this chapter, the Bell state measurement has to provide a certain information to enable reconstructing the original state |ψi_Aof photon A on the output photon C (defined in figure 1.1). It has to deduce, in which distinct Bell state configuration (defined in 6) the Photons A and B are.

Although in principle the detection of all different Bell states in one setup is feasible, for practical reasons we focus on isolated detections of |ψ−_i_{or |ψ}+_i_{, so that in the end only one fixed unitary} transformation has to be performed on photon C. On the other hand, this proceeding discards

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(a)

1

2

3

4 50:50

BS

D2

D1

(b)

1

2

3

4 50:50

BS

PBS

D2

D1

6

5

(c) (d)

Figure 2.7: Scheme of detecting the bell states (a) |ψ−iand (b) |ψ+_{i. The Bell state |ψ}−

iis the only one to yield photons at output 3 and 4 at the same time, thus one beam splitter (BS) is sufficient. The polarising beam splitter (PBS) for detecting |ψ+_i_{filters the photon pairs from output 3 by their polarisation. (c) and (d) show}

the probability of generating a double click by the different Bell states as a function of V for a |ψ−_i_{and a |ψ}+_i

detection setup, respectively.

Detecting |ψ−_i

Equation 21 reveals, that |ψ−_i _{is the only Bell state causing double clicks (bunching) at the} detectors D1 and D2 at the outputs 3 and 4, like arranged in figure 2.7 (a). Hence, for perfect interference at the beam splitter (V = 1), a detector coincidence ensures the detection of an incoming photon pair in the |ψ−_i _{configuration. As V decreases, the probability of a false} detection, i.e. the probability that the detection origins from another Bell state, increases. In the limit of V = 0, the probability for all four Bell states are distributed equally, so we can write them as:

p|ψ−_i= 1 4 + 3 4V p|φ+_i= p|φ−_i = p|ψ+_i= 1 4 − 1 4V (31) where p|Bi is the probability that the double click at the detectors is caused by the Bell state |Bi.

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Detecting |ψ+_i

For detecting |ψ+i, as shown in figure 2.7 (b), a combined probability has to be considered. Obviously, a double click at detector D1 and D2 can only occur if two photons left the output 3 of the first beam splitter in the first place. The probabilities for this event to origin from a specific Bell state are:

˜p|φ+_i= ˜p|φ−_i= ˜p|ψ+_i = 1 3− 1 4 V +1 4 ˜p|ψ−_i = 1 4− 1 4V (32) The Bell states |φ+_i _{and |φ}−_i _{only contain photon pairs with equal polarisation, thus a double} click after passing the polarising beam splitter can never origin from these. The probabilities for a double click from the remaining possible Bell states |ψ+_i _{and |ψ}−_i _{then read:}

p|ψ+_i= ˜p|ψ+_i ˜p|ψ+_i+ ˜p|ψ−_i p|ψ−_i= ˜p|ψ−_i ˜p|ψ+_i+ ˜p|ψ−_i (33) 2.4 Teleportation fidelity

In order to prove successful teleportation, the "similarity" of the state |φi_C of the output photon C (see teleportation scheme 1.1) to state |ψiA of the input photon A has to be quantified. For

this purpose, we derive a statistical quantity for an ensemble of photons, the teleportation

fidelity, defined as:

FT = T r (ρCρA) (34)

where ρC is the density matrix of the statistically mixed output state and ρAthe density matrix

of the input state, typically prepared as a pure state, so that ρA= |ψiAhψ|A.

For constructing ρC, we calculate the state |ψiABC of the whole quantum system formed by

photons A, B, and C. Therefore we derive the product state of the input state |ψiA of photon

A and the entangled state |χiBC of photon B and C. The arbitrary state |ψiA can be written

in the rectilinear base {|vi , |hi}, so that:

|ψi_A= a |hi_A+ b |vi_A (35)

The entangled state |ψiBC ideally represents a stationary Bell state ∈ {|φ+i , |φ−i , |ψ+i , |ψ−i}.

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The combined state then reads:

|ψi_ABC(t) = |ψi_A⊗ |χi_BC(t) = 1/2 [ |φ+i_AB⊗(a |hi_C + eΦ(t)b |vi_C)+

|φ−i_AB⊗(a |hi_C − eΦ(t)_{b |vi}

C)+

|ψ+i_AB⊗(b |hi_C+ eΦ(t)a |vi_C)+

|ψ−i_AB⊗(b |hi_C− eΦ(t)a |vi_C) ]

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Equation 38 contains an important message, representing the mathematical fundament of quan-tum teleportation: As soon as the mutual Bell state |BiAB of the photons A and B is well known,

the state |φiC of photon C is also defined instantaneously, together with the unitary

transfor-mation ˆU needed to reconstruct the input state |ψi_A, so that ˆU |φi_C = |ψi_A. The equation

reveals the important requirements for a successful teleportation, thus for a high teleportation fidelity FT:

1. Controlling the relative phase shift Φ(t) of the entangled photon pair (see section 2.2 ) 2. Measuring the Bell state of photons A and B with high reliability (see section 2.3) We name the possible outcomes of |φi|Bi

C (t) according to their linked Bell states |BiAB:

|ψi|φ_C+i(t) = σ0(t) |ψi0 _A |ψi|φ_C−i(t) = σ_z0(t) |ψi_A |ψi|ψ_C+i(t) = σ_x0(t) |ψi_A |ψi|ψ_C−i(t) = σ_y0(t) |ψi_A (39) with σ00(t) = 1 0 0 eΦ(t) ! , σ0_x(t) = 0 1 eΦ(t) 0 ! , σ_y0(t) = 0 1 −eΦ(t) ₀ ! , σ_z0(t) = 1 0 0 −eΦ(t) ! (40) In statistics, the contributions of the respective states are weighted by their detection probabil-ities as derived in subsection 2.3.4, denoted as p|Biin the following. So we can write the density matrix as: ρC = bmax Z bmin P(t)X B p|Bi|φi |Bi C hφ| |Bi C (t) dt (41)

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with the probability distribution function P (t) for the entangled photon pair, and bmin, bmaxas

defined in subsection 2.2.3. As |φiC lives in the Hilbert space H2, the result is a 2 × 2 matrix.

For an efficient experimental extraction of the teleportation capabilities of a system by a reason-able number of measurments, we consider the following set of input states we want to teleport[17]_: |ψii_A∈ {|hi , |vi , |di , |ai , |li , |ri} (42)

A later discussion will reveal the benefits of this particular set of three orthogonal pairs of states, as it exhibits some useful symmetry properties.

For investigating the effect of different parameters on the quality of the teleportation, we perform several simulations of FT. We assign a decay time of τ0= 250 ps to the probability distribution

P(t). The respective missing unitary transformation σito be performed on the output state |φiC

in order to reconstruct the input state |ψiA was taken into account. Hence, the teleportation

fidelity F|Bi

T regarding the detection of the |ψ−i and the |ψ+i Bell states, respectively, read:

F_T|ψ−i= T rρCσy|ψi_Ahψ|_Aσ†y F_T|ψ+i= T rρCσx|ψi_Ahψ|_Aσx† (43) with σx = 0 1_{1 0} ! , σy = ₋0_{1 0}1 ! (44)

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(a) |ψi_A= |hi (b) |ψi_A= Rπ 8|hi

(c) |ψi_A= |di (d) |ψi_A= |ri

Figure 2.8: Evolution of the teleportation fidelity FT as a function of the angular frequency Ω in the relative

phase evolution of the entangled photon pair. The evolution is shown for different input states |ψiA, where Rπ

8 |himeans a rotation of π/8 in the linear base w.r.t. |hi. The colors distinguish the varying beam splitter visibilities V of the Bell state measurement. The full lines show the results for a |ψ−i, the dashed lines for the |ψ+i Bell state detection. The green dash-dot line represents the classical limit of the average fidelity FC for the

applied set of input states.

Due to the high symmetry of our problem, arising from the chosen set of input states 42, it is sufficient to simulate F|ψ−i

T and F

|ψ+i

T for a reduced set: {|hi , Rπ8, |hi , |di , |ri}. The input

state Rπ

8 means a rotation of π/8 in the linear basis w.r.t. |hi and is included for pointing out

the trend when rotating |ψiA from |hi towards |di. Figure 2.8 (a)-(d) show the fidelities FT as

a function of the angular frequency Ω in the relative phase shift of the entangled photon pair. The different curves represent different visibilities V for Bell state measurements of both |ψ−_i (full lines) and |ψ+_i _{detection (dashed lines).}

Figure 2.8 (a) states that FT is completely independent from Ω, justified by the relative phase

shift between |hi and |vi of the eigenstates of the entangled photon pair 36. Furthermore, FT

for a |ψ+_i _{Bell state measurement does not depend on V at all, as long as the assumption of} single photons at the beam splitters still hold.

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The more the linear angle is turned towards |di, the more the |ψ−_i _{Bell state measurement} overcomes the |ψ+_i _{measurements in terms of fidelity.}

(a)Ω = 2 GHz (b) Ω = 10 GHz

Figure 2.9: Evolution of the teleportation fidelity FT as a function of the rotation angle in the linear base w.r.t.

|hi. It is shown for two different angular frequencies Ω in the relative phase shift of the entangled photon pair. The colors distinguish the varying beam splitter visibilities V of the Bell state measurement. The full lines show the results for a |ψ−i, the dashed lines for the |ψ+i Bell state detection. The green dash-dot line represents the classical limit of the average fidelity FC for the applied set of input states.

Figure 2.9 reinforces this statement by showing the dependency of FT on the linear angle of the

input state for two different Ω. The green line in both figures 2.8 and 2.9 refers to the maximum classical average fidelity of FC = 2/3, dedicated to the applied set of input states. For a valid

experimental evidence of the very quantum nature of the teleportation setup, this threshold has to be overcome by the individual fidelities measured for each of those input states. A detailed derivation of FC can be found in the appendix A.2.

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3 Challenges

Matching of excitonic emission energy Resonant excitation of biexciton level

Correct evaluation of measured data

High teleportation fidelity results High visibility for Bell state measurement

High detector count rate High entanglement Strain tuning device Experiments and optical setup Data processing software GaAs quantum dot Device developement and processing

Quality sample for indistinguishable single photons High extraction efficiency by DBR structure and SIL Polarisation correction by liquid crystal retarder Efficient algorithm for correlating measured data

Elements

Measures taken

Goals

Chapter 5 Chapter 4 Chapter 6 Chapter 7

Low fine structure splitting by low prestress Application of low loss transmission grating

Figure 3.1: The graphical summary lists the elements treated in the rest of this thesis. All the measures taken

aim towards the goal of proving the teleportation capabilities of our setup with two remote quantum dots.

The main goal of our work is the experimental prove of quantum teleportation, which is accom-plished by maximising a statistical value, the teleportation fidelity FT (see section 2.4). Figure

3.1, from top to bottom, summarizes the four major elements we treat in this thesis and which

measures we take order to achieve all the partial goals, heading towards a high teleportation

fidelity FT.

The previous chapter already suggested the fundamental goals to reach for achieving a high teleportation fidelity, arising from theoretical considerations:

• A high entanglement is crucial to preserve the relative phase between the basis states of the output photon C. Entanglement spoiling effects, like fine structure splitting (FSS) and decoherence decrease FT

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• High visibility V at the beam splitter ensures a high probability of a reliable Bell state measurement. Lower V increases the probability of false detections, thus lowering FT.

We will see that the above features are mostly met by the GaAs quantum dots (QDs) we use, which will be introduced in chapter 4. We will recognize the FSS and the decoherence effects as intrinsic properties of the QDs. The applied resonant excitation scheme will be explained, as well as the arising benefits compared to above-band excitation.

A high visibility requires matching emission energies of the interfering photons from the remote QDs. The piezoelectric strain tuning device, presented in chapter 5, is capable of fulfilling this task.

The experimental framework creates the following additional requirements:

• The excitation laser creates pulses with a pulse duration of about 100 fs at a fixed rep-etition rate of R = 80 MHz. The overall efficiency η gives the probability, that a pulse yields a photon, which reaches the designated detector and produces a click. It therefore includes all effects, from the extraction efficiency of the sample to the detector efficiency. The detector count rate is then given by r = Rη. The calculation of FT is done by

evalu-ating the third order correlation function g(3) _{of three photon channels. The probability} for a coincidence count in the g(3) _{after an excitation pulse is proportional to η}3_. Conse-quently, high detector count rates are crucial for a successful statistical proof of quantum teleportation.

• The evaluation of FT relies on well defined polarisation states of all photons

through-out the setup. Polarisation-altering effects change the mutual correlations, making an experimental prove of successful teleportation hard to achieve.

• The calculation of the third order correlation function g(3) _{requires efficient algorithms,} keeping the calculation time short while avoiding errors.

One part of chapter 4 will focus on the enhancement of the extraction efficiency by growing a distributed Bragg reflector (DBR) and applying a solid immersion lens (SIL), both yielding higher count rates.

Optimising the optical setup by appropriate optical elements, like a transmission grating with high diffraction efficiency, additionally contributes to this goal. These measures and the correc-tion of the polarisacorrec-tion by a liquid crystal retarder (LCR) are covered in chapter 6.

A self-written software for processing the measured data will be presented in chapter 7. It meets our requirements and is prepared for future applications, demanding the calculation of even higher order correlations.

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4 GaAs quantum dots as sources of polarisation entangled

pho-ton pairs

4.1 General properties and manufacturing

(a) (b)

Figure 4.1: (a) Sketch of GaAs quantum dot formed within the Al0.4Ga0.6As bulk material. (b) Atomic force

microscope (AFM) picture of a nano hole in the Al0.4Ga0.6As bulk before the filling with GaAs.

Quantum dots (QDs) in general are nanostructures of a semiconductor material embedded in a matrix with larger energy bandgap. The name quantum dot originates from the quantum confinement effects arising from their small geometries, forming quantum-well-like electronic structures due to the band gap mismatch between QD and host material. The resulting atom-like discrete energy levels justify the frequently used term artificial atoms. We will investigate the confinement effects in GaAs QDs in Al0.4Ga0.6As bulk material in detail throughout this chapter. Eventually we will understand, why the GaAs QDs represent a suitable deterministic source entangled single photon pairs and how to enhance the extraction efficiency for a high photon yield. Also the downsides of the QDs with respect to other sources of entangled photons will be pointed out, arising from the solid state environment of the dots, and how can deal with them.

Prominent alternative sources of entangled photon pairs are single atoms[9] _{and spontaneous}

parametric down conversion (SPDC)[18] crystals. The advantage of single atoms are their

in-herently sharp electronic transitions, as they are energetically isolated and thus free from charge fluctuations. A major drawback, however, is their complicated storage in bulky arrangements, while QDs are scalable to the nano-range by semiconductor processing techniques.

Parametric down conversion systems are the brightest frequently used sources of entangled pho-tons. However, their are not deterministic, but probabilistic. That means, the number of photon pairs resulting from an excitation pulse is statistically distributed, while QDs yield at most one photon pair per excitation, due to well defined optical decay channels.

The GaAs quantum dots are produced by the aluminum droplet etching technique[19]_{. By} molecular beam epitaxy (MBE) a host layer of Al0.4Ga0.6As is grown. In this layer, nano-holes of about 50 nm radius and 6 nm depth are etched by Al droplets. The holes are filled by diffusing GaAs, forming the QD, and the structure is finally capped in Al0.4Ga0.6As. The sketch of the procedure and an atomic force microscopy (AFM) picture of a nano-hole before the filling with GaAs are shown in figure 4.1. The optimized procedure results in structures with morphology close to a D2d symmetry, i.e. in a radial symmetry in the perpendicular plane of the growth direction, as long as no uniaxial stress is exerted in this plane.

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4.2 Energetic structure and radiative transitions in GaAs quantum dots (a) Energy x,y,z CB VB e -h+ (b) XX X G -1 +1 l r r l 0

Figure 4.2: (a) Simplified band alignment scheme of a QD in a host material. Due to the formation of quantum

wells in all spatial directions, discrete energy levels form. Each energy level can be occupied by two electrons (e-_{) or two holes (h}+_{), respectively. (b) Biexcitonic decay cascade of the QD. The biexciton (XX) can decay via}

two paths into two different energetically degenerated exciton (X) states, distinguished by their total angular momentum (-1,+1). The photons emitted during the transitions possess a certain circular polarisation, depending on the decay path.

The quantum dot material GaAs and the host material Al0.4Ga0.6As possess different band gaps of 1.92 eV and 1.42 eV (at room temperature), respectively. Incorporating nano-dots of GaAs in Al0.4Ga0.6As leads to a quantum-well-like band alignment in all spatial directions, as sketched in figure 4.2 (a). In first approximation, we assume perfect spherical symmetry of the dot. The small dimensions of the quantum well lead to a quantum confinement effect, yielding the discrete energy levels in the conduction band (CB) and in the heavy-hole valence band (VB) 1_{. Each of the energy levels can be occupied by two electrons (e}-_{) and holes (h}+_{), respectively,} due to Pauli’s exclusion principle. We will focus exclusively on the first energy levels closest to the band gap, often denoted as the s-shell levels, as we use the radiative transitions between them as our photon source.

The e-_{and h}+_{are spatially localized in the dot by the confinement potential. As a consequence,} multi-particle complexes, also known as quasi-particles, form due to Coulomb attraction between e- _{and h}+_{. The relevant quasi-particles for us are the neutral exciton (X), formed by one e} -and one h+_{, and the neutral biexciton (XX), formed by two e}- _{and two h}+_{. Charged complexes} like the negatively charged trion (X-_{), formed by two e}- _{and one h}+_{, are irrelevant for our}

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The specific quantum-well-like band alignment, arising from the chosen material combination and the size of the used QDs and the band gap mismatch of the GaAs and the Al0.4Ga0.6As host material, yield a center emission energy of the X photons of about EX = 1.58 meV,

cor-responding to a center wavelength of λX = 785 nm. The XX emission energy is similar, but

significantly lower, as explained in section 4.4.

We briefly recapitulate the radiative transition rate Γ between quantum mechanical states in order to derive the lifetime of the X and XX states and the corresponding spectral profile of the emitted light. The rate represents the transition probability per unit time of an initial state |ii to decay to a final state |fi. The time dependent, first order perturbation theory yields

Fermi’s golden rule. Using it in the dipole approximation for perturbations by weak light-matter

interaction yields:

Γ ∝ | hf|e · ˆp|ii |2_δ_(E

i− Ef− EP) (45)

where e is the unit polarisation vector of the emitted light, ˆp the electron momentum operator and Ef, Ei and EP the energies of the final state, the initial state and the emitted photon,

respectively. Having the transition rate, we can define the rate equation for the population of an initial state by

dNi(t)

dt = −ΓNi(t) (46)

We define the lifetime of the initial state as τ = 1/Γ. The solution of 46 then reads:

Ni(t) = Ni(0)e− t

τ (47)

Concluding, the X and XX states underlie exponential decay with respective decay constants

τν (ν ∈ {X, XX}), and thus also the intensity Iν(t) of the emitted light follows the same

probabilistic distribution. A Fourier transform of the corresponding electric fields with a center frequency of ων = Eν/~ yields the spectral distribution of the intensity:

Iν(ω) = 1 2πτν 1 (ω − ων)2+ 1 2τν 2 (48)

which represents a Lorentzian distribution with a full width at half maximum (FWHM) of ∆ων = 1/2τν, often denoted as the natural linewidth, as it directly arises from the wave nature

of light and thus represents the minimum linewidth.

Apart from the homogeneous spectral broadening by the finite lifetime of the states and by dephasing, also inhomogeneous broadening effects occur, randomly shifting the emission energy in time scales below common spectrometer integration times. As a consequence, the usual QD emission lines rather appear as a Gaussian distributed superposition of many Lorentzian lines. The inhomogeneous broadening origins from random fluctuations of charges, strain fields or temperature in the environment of the QD, as well as from phonon interaction.

2_{The |−1i and |+1i X are called bright excitons, due to their strong transition dipole. Also |−2i and |+2i X}

exist, but their radiative decay is dipole-forbidden. These are therefore called the dark excitons and not further relevant in this thesis.

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4.3 Polarisation entanglement and coherence in GaAs QDs

Reconsidering the biexcitonic decay cascade in figure 4.2 (b), we recognize that both possible decay paths are degenerate in energy. Measuring the energy of the X and XX transition photons yields no which-path information about the decay sequence. What persists is the correlation between the X and XX photons regarding their polarisation: Knowing the polarisation of one immediately determines the polarisation of the other. Before measuring the polarisation, the X/XX photon pair remains in a superposition state of both possible outcomes, so that the joint state reads:

|χi= √1

2(|riXX⊗ |liX + |liXX⊗ |riX) =:

1 √

2(|rli + |lri) (49)

which is an entangled state as introduced in section 2.2. Rewriting the state in the rectilinear basis {|hi , |vi} gives:

|χi= √1

2(|hhi + |vvi) (50)

which is exactly the |φ+_i_{Bell state as defined in equation 6. Concluding, under ideal conditions,} the QD provides consequent pairs of entangled photons in the |φ+_i_{state, hence it is inherently}

deterministic 3_.

Excitonic fine structure splitting (FSS), random and polarisation dependent decoherence in QDs and limited single photon purity impact differently the entanglement properties and the two-photon-interference. In the following, we will investigate the aforementioned effects and draw useful mathematical conclusions.

4.3.1 Fine structure splitting

(a) XX X -1 v h 0 S 1/Ö2 +1 + -1 1/Ö2 +1 -(b)

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Two prominent effects occurring in QDs can lift the degeneracy and open a fine structure splitting (FSS) between the two possible bright X states. One is related to the hyperfine interaction between the spins of the electrons and a collective spin orientation of the ion cores, causing internal magnetic Overhauser fields[21] _{up to 2 T.}

The dominant effect in GaAs, however, is related to the exchange interaction between electrons and holes[22]_{. The electron-hole exchange Hamiltonian reads:}

H_ex= − X

i=x,y,z

aiJh,iSe,i+ biJh3iSe,i (51) where ai and bi are the spin-coupling constants, Jhi the holes’ and Sei the electrons’ total angular momentum. The index i denotes the spatial direction. We assign the z-direction as the crystal growth direction and define the bright X states by the z-projections Jh,z and Se,z of the

angular momenta, so that |+1i = |+3 2, − 1 2iand |−1i = |− 3 2,+ 1

2i. In the basis {|+1i , |−1i} the Hamiltonian 51 then reads:

Hex= h+1|H_ex|+ 1i h+1|H_ex| −1i h−1|Hex|+ 1i h−1|Hex| −1i ! = ∆0_∆1 ∆1_∆0 ! (52) where ∆0= 3 4(az+ 9 4bz) and ∆1= − 3 8(bx− by)

[23]_{. Under optimised conditions, the fabrication} of the GaAs QDs result in a D2d-symmetry with the symmetry axis along the z-direction. In this case, bx = by holds and ∆1 vanishes, leaving |−1i and |+1i as the degenerate eigenstates

of Hex with the exchange energy ∆0.

For lower symmetry, mainly caused by irregularities in the shape of the nanoholes, interface disorder, and possibly by internal anisotropic strain in the x/y-plane, the symmetry decreases and thus bx 6= by. The radial symmetry is lifted, hence the angular momentum M along the

z-direction is no longer preserved and |−1i and |+1i are no longer eigenstates of Hex. Instead,

by diagonalising the Hex, the new basis can be found as √1₂(|+1i + |−1i) and √1₂(|+1i − |−1i)

with their respective exchange energies 1

2(∆0+ ∆1) and 1

2(∆0−∆1).

Figure 4.3 (a) shows the corresponding XX decay path with the now energetically split X eigen-states, separated by a FSS of S = ∆1. For the XX, the angular momenta of the electrons and holes add to zero, respectively, so no splitting due to exchange interaction occurs.

The optical transitions from the XX to the new X eigenstates and to the ground state correspond to the basis states {|hi , |vi}, as the the photons in the linear base carry no angular momentum. Figure 4.3 (b) shows a polarisation resolved PL measurement of a QD with S = 21.3 µeV, per-formed with a linear polariser. The clear shift in energy indicates the linear polarisation of the FSS components and the orientation of the {|hi , |vi} coordinate system.

We reconsider the 2-qubit state of the emitted XX and X transition photons under ideal condi-tions (50). The time dependency of the stationary XX state is given by a global phase evolution, not effecting the relative phase between the basis states |hhi and |vvi. The FSS between the X states, on the other hand, leads to different phase evolutions of the X states during their lifetimes, due to their energy difference S. The accumulated relative phase is reflected in the photons pair’s state:

|χi(t) = √1 2 |hhi+ e−~iSt|vvi (53) Now, an ensemble of many emitted photon pairs is no longer a pure state, but a statistical mixture. The temporal distribution of the time dependent states is defined by the the excitonic lifetime, as derived in subsection 2.2.3. The angular frequency of the relative phase shift in 7 can now be identified as Ω = S/~.

Towards quantum teleportation with remote quantum dots / submitted by Ing. Christian Schimpf, B.Sc.

Towards quantum

teleportation with

remote quantum dots

Eidesstattliche Erklärung

Acknowledgement

Abstract

Contents

1

Introduction

2

3

U

1

ALICE

BOB

1

2

3

4

50:50

BS

1

2

3

4

50:50

BS

D2

D1

1

2

3

4

50:50

BS

PBS

D2

D1

6

5

3

Challenges

Elements

Measures taken

Goals

4

GaAs quantum dots as sources of polarisation entangled

pho-ton pairs