Photonic cluster states with quantum dots

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Photonic cluster states with a

quantum dot

THESIS

submitted in partial fulfillment of the requirements for the degree of

BACHELOR OF SCIENCE

in

PHYSICS

Author : Edward Hissink

Student ID : 1850245

Supervisor : Dr. W. Löffler

2ndcorrector : Dr. P.H. Denteneer

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Photonic cluster states with a

quantum dot

Edward Hissink

Huygens-Kamerlingh Onnes Laboratory, Leiden University P.O. Box 9500, 2300 RA Leiden, The Netherlands

July 8, 2019

Abstract

Cluster states are often investigated in the field of quantum information. We consider two photon interference and two photon entanglement in a setup consisting of linear optical elements in which single photons are produced by a quantum dot in a microcavity. We measure correlations between two detectors for different polarizations which we compare with

theoretical predictions. Understanding of the predictions and the experimental results should lead to an extension to three or more photon

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4

Acknowledgements

For producing this work, I would like to thank Gerard Westra for intensive cooperation. We have been working together in both making predictions and doing experiments. Large parts of chapters 4, 5 and 6 could never have been produced in this time without his help.

Also, Henk Snijders and Petr Steindl deserve credit for helping with the production of single photons by use the quantum dot and creating a stable setup (chapter 4, especially sections 4.1 and 4.4). Without their help, the experiments would never have been performed as succesfully as now. Last but not least, I thank Wolfgang Löffler for taking responsibility as a su-pervisor and for the many discussions about the ever upcoming questions and problems during the project.

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6.1 Old setup 60 6.1.1 VV configuration 60 6.1.2 VH configuration 61 6.1.3 HV configuration 64 6.1.4 HH configuration 65 6.1.5 DD and AA configuration 67 6.1.6 AD and DA configuration 69 6.2 New setup 71 6.2.1 HV configuration 71 6.2.2 VV configuration 73 6.2.3 HH configuration 74 6.2.4 VH configuration 75 6.3 Further discussion 76 7 Conclusion 79 8 Appendix 81 8.1 Quantum gates 81

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CONTENTS 7

8.1.1 Quantum gates acting on one qubit 81

8.1.2 Quantum gates acting on two qubits 82

8.2 The PPT criterion 83

8.3 Example of a cluster state: a qubit chain 84

8.4 Two photon output state for the CW-laser in the new setup 86 8.5 Phase-dependent and vacuum-dependent quantum

interfer-ence in the setup 87

8.5.1 Quantum interference at a waveplate under an angle

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Chapter

1

Introduction

More and more progress is made by the field of quantum information. Not only from theoretical point of view knowledge is growing, but also experimental research is thriving. One important concept used in quantum information is quantum entanglement, which studies a specific states of sys-tems consisting of several particles. Many composed states are separable, which means that the particles can be considered independent from each other: the evolution of one particle state does not influence the evolution of another particle state in any way. Entangled states are exactly the opposite: we can not separate particles from each other anymore and we have to consider all the subsystems together, since the outcome of measuring one state will definitely influence the outcome of another. By the way, the distance between the particles does not even matter!

Quantum entanglement is often visualized in the realm of atomic systems. For example, it was shown that gravitational acceleration can cause entan-glement between particles [1]. This is an important step for the use of (fast developing) quantum technology in space. Furthermore, by use of quan-tum entanglement a super-sensitive detector for measuring gravitational waves was invented [2]. But quantum entanglement is also been scaled up to larger scale: quantum entanglement of the motion of objects, consistings of many billions of atoms, was created [3].

We can also consider systems of multiple quantum bits (qubits), basic units of quantum information. The qubit is the quantum variant of the classical bit in the context of a two-level system. These qubits can be identified with many physical systems, e.g. polarization (with horizontal and vertical polarization as basis states) and spin 1₂-particles (which can either have

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10 Introduction

spin up or spin down). A state of multiple qubits that is highly entangled is called a cluster state.

Until recently, cluster states are often produced via a proces that is called spontaneous parametric down-conversion (SPDC). This is a nonlinear op-tical process where one photon (a pump photon) with higher energy pro-duces a pair of photons (a signal photon and an idler photon) with lower energy [4]. The entanglement of photon pairs produced by PDC is related to conservation of energy and momentum. Qualitatively speaking, these conservation laws relate the energies and momenta of the two created pho-tons to each other, which imply that the two phopho-tons are entangled.

In SPDC, no input signal or field is present to stimulate the process. The photons are generated spontaneously in a medium, often a crystal, where interactions between the three photons and the medium are required. We call the process parametric because it is dependent of the electric field of the photons (not just on their intensities), where that the phases of the input and output electric field are related. The term down-conversion refers to the fact that the signal and idler photon will always have a lower frequency than the pump photon, which follows from conservation of energy.

However, producing many entangled photons via PDC turned out to be difficult, since it required many different sources, entangling operators and other materials [5].

Also other methods are possible to produce cluster states. Several people tried to produce cluster states consisting of single photons. For example, it is possible to produce a stream of single photons by use of a quantum dot [6]. People have also shown that quantum entanglement of these pho-tons can be created by implementation of only linear optical elements and photodetectors [5, 7]. However, quantum entanglement in cluster states can be affected or even disappear if qubits interact with the environment or lose contact with the system [6].

The development of a single photon source such as the quantum dot was a very important step in quantum electronics research, since it shows the pure quantum nature of the process in which single photons are gener-ated. One of the things that we can observe from a single photon stream is antibunching: the photons are separated in such a way that it is very unlikely to detect to photons at the same location at the same time (from a qualitative point of view). We can observe antibunching via a correlation measurement, by use of a setup which we will consider later in more detail. In this work, we will demonstrate an experiment in which a quantum dot in combination with a delay loop is used to create polarization-entangled photon pairs. The entanglement of photons can be created via quantum interference, such as in the Hong-Ou-Mandel effect which we will explain

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11

further in this thesis. We will consider the following modes: position, po-larization and time. A challenge in producing entangled photons in this way lies in the fact that the photons must be fully indistinguishable and can interfere at exact the same location and at exact the same time.

We will give an answer to the following question:

How can cluster states consisting of single photons be produced and measured? This thesis has the following structure.

• First, we will mathematically and physically explain the meaning of quantum entanglement and cluster states (chapter 2).

• After explaining a few concepts related to a beamsplitter (that are important to understand our experiment, chapter 3), we will describe the setup and materials of the experiment in which cluster states are produced by use of single photons (chapter 4).

• We will discuss thoroughly how quantum entanglement can be mea-sured and present predictions for the measurements (chapter 5). • In the end, we discuss the results and we conclude if the quantum

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Chapter

2

Definition of entanglement and

cluster states

In this section, we will explain the mathematical and physical meaning of quantum entanglement. From this we can construct an appropriate defini-tion of cluster states. Before we give definidefini-tions and further descripdefini-tions, we introduce some mathematical notation relevant for this subject.

2.1 Notation of cluster states

2.1.1 Tensor product

Before we can explain the meaning of a cluster state, knowledge and un-derstanding of the tensor product is needed. With a tensor product, it is possible to describe the state of a system that is composed of several subsystems with a Hilbert space (or tensor space), if the subsystems are also described with a Hilbert space.

Consider two Hilbert spacesH1andH2. The tensor product ofH1andH2

is the Hilbert spaceH =H₁⊗ H₂which satisfies the relations

(|αi+|βi)⊗ |γi=|αi ⊗ |γi+|βi ⊗ |γi, (2.1)

|αi ⊗(|γi+|δi) =|αi ⊗ |γi+|αi+|δi ⊗ |αi, (2.2) λ(|αi ⊗ |γi) = (λ|αi)⊗ |γi= |αi ⊗(λ|γi), (2.3)

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14 Definition of entanglement and cluster states

with|αi,|βi ∈ H1,|γi,|δi ∈ H2and λ ∈C.

It is also possible to construct the tensor product of n Hilbert spaces H₁,H₂,· · ·,Hn, where n ∈ N_>3. We can write this tensor product as

H=H1⊗ H2⊗ · · · ⊗ Hn. Just as the tensor product for two Hilbert spaces,

this tensor product is linear in each component.

It is also possible to define operators on a tensor space, where each oper-ator acts on one or several subsystems. Consider for example the Hilbert space H = H1⊗ H2 and the operators T and U which act on |αi ∈ H1

and|βi ∈ H2, respectively. Then the tensor product T⊗U acts on states

defined inHas follows:

(T⊗U)(|αi ⊗ |βi) = T(|αi)⊗U(|βi). (2.4)

Finally, we can define an inner product on a tensor spaceH=H₁⊗ H₂. If hα|βiH₁ is an inner product onH1with|αi ∈ H1and|βi ∈ H1andhγ|δiH2

is an inner product onH2 with |γi ∈ H2 and|δi ∈ H2, then the tensor

product between the elements|αi ⊗ |γi ∈ Hand|βi ⊗ |δi ∈ His defined

as

hαγ|βδiH1⊗H2 =hα|βiH1 · hγ|δiH2. (2.5)

tensor product between|αiand|βi.

2.1.2 Two-state systems and qubits

In this thesis, we will consider many situations with a two-state system, i.e. a two-dimensional Hilbert space. In general, the Hamiltonian of such a system can be written as H = a0I2+ a1σx+ a2σy+ a3σzwith a0, a1, a2, a3 ∈ C,

I2the 2×2-identity matrix and

σx =0 1 1 0 , σy = 0 −i i 0 , σz = 1 0 0 −1 (2.6) the Pauli matrices. For these matrices, the following identities hold: σx2= σ_y2 = σ_z2 =−iσxσyσz = I2.

We can identify the two (eigen)states of a two-state system with two basis states that we will write as |0i and |1i. These states form a so-called computational basis of the system. A pure state of that system can then be written as

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2.2 Pure states 15

The normalization condition then reads |κ|2+|λ|2= 1. Also, the states |0i

and|1iare orthogonal soh0|1i=h1|0i = 0.

We can also express κ and λ in terms of spherical coordinates, with κ = cos(θ

2) and λ = eiφsin(θ2) where θ, φ∈ R [8]. All the possible states lie on a

so-called (three dimensional) Bloch sphere.

We usually write the tensor product of several Hilbert spaces (with the same bases) with qubit states without the⊗-symbol. For example, if subsystem A is in state|0iand subsystem B is in state|1i, we can write|0i_A|1iBfor the

state of the whole system. The notations|0i|1iand|01iare also possible.

Example: polarization of photons

An important case in which qubits can be used (or ’encoded’) is in describ-ing the (electric field) polarization of photons. There exist three different so-called mutually unbiased bases to describe this polarization:

1. The{|Hi,|Vi}-basis, with|Hifor polarization in a horizontal direc-tion and|Vifor polarization in the vertical direction.

2. The{|Di,|Ai}-basis, with|Di = √1

2(|Hi+|Vi) and|Ai = 1 √

2(|Hi −

|Vi) for polarization in (anti)diagonal directions. The {|Di,| Ai}-basis is obtained from the{|Hi,|Vi}-basis after a rotation about 45◦. 3. The{|Ri,|Li}-basis, with|Ri= √1

2(|Hi+ i|Vi) right-handed circular

polarized light and|Li = √1

2(|Hi −i|Vi) left-handed circular

polar-ized light.

In qubit notation, we often choose|0i=|Hiand|1i=|Vifor the{|Hi, |Vi}-basis. For the{|Di,|Ai}-basis, we often write|+i=|Diand|−i=|Ai.

2.2 Pure states

A pure state is a state that is a linear combination of basis states. The states|0iand|1i in section 2.1.2 are examples of pure states. Pure states are defined in both Hilbert spaces and tensor spaces. In Hilbert spaces, pure states are linear combinations of basis states. For the tensor space H = H1⊗ H2 of the Hilbert spaces H1 and H2, a pure state |ψi can be

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16 Definition of entanglement and cluster states written as |ψi= d1,d2

∑

i,j aibj|αii ⊗ |βji. (2.8)

Here|αiiand|βjiare the basis vectors of the Hilbert spaces H1andH2,

re-spectively, and aiand bjare the corresponding coefficients. The dimensions

ofH1andH2are equal to d1and d2.

For each state, we can define a so-called density matrix ρ. For a pure states |ψi, the density matrix is given by

ρ= |ψihψ|. (2.9)

For this matrix, the following identities hold: ρ2 = ρ (the matrix is a pro-jection matrix), ρ† = ρ (the matrix is hermitian), Tr(ρ) = 1 (the matrix is normalized because the state is normalized) andhψ|ρ|ψi >0 for all states ψ

with the same basis (the matrix is positive semidefinite so it has no negative eigenvalues).

Now we can define entanglement for pure states. Consider again the tensor spaceH =H1⊗ H2of the Hilbert spacesH1andH2and the state|ψi ∈ H.

Such a tensor space of two Hilbert spaces is called a bipartite system. The state|ψiis called separable if there exist states|αi ∈ H1and|βi ∈ H2such

that|ψi=|αi ⊗ |βi(this is also called a product state). Otherwise, the state

|ψiis entangled. In the same way, we can define entanglement for states

in the tensor spaceH₁,H₂,· · ·,Hn with n ∈N>3. Examples of entangled

states are |αi = √1 2(|00i+|11i), |βi = 1 √ 2(|000i+|111i), |γi = 1 √ 3(|001i+|010i+|100i). The state|αiis known as a maximally entangled Bell state, the state|βias

the Greenberger-Horze-Zeilinger state (GHZ state) and the state|γias the

W state. Note that|βiand|γiare elements of a tensor space of three Hilbert

spaces, a so-called tripartite system. For now, we will focus on bipartite systems withH= H₁⊗ H₂and after that we will consider entanglement for tripartite systems separately.

Physically, (dis)entanglement of multiple systems implies that the states in the different systems are (not) correlated to each other. If|ψiis separable,

the outcome of a measurement inH₁will not depend on the outcome of a measurement in H2. The outcomes and states can just be considered

separately: we can just write what happens inH₁ and after that we can write what happens inH2, or the other way around. But if|ψiis separable,

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2.3 Mixed states 17

the other way around. If we measure|αifor example, we already know

that we will measure the state|1iinH2if we measure the state|1iinH1.

Now a tensor product is certainly needed to describe the state.

An important property of quantum entanglement, which can be easily proven, is that if a quantum state is entangled, it is entangled in each basis that can be used to describe the state (’basis independence’). It does not matter in which basis the state is measured, it will never be separable.

2.3 Mixed states

In most of the cases, we have to deal with mixed states. Roughly speaking, a mixed state is a state that is constructed from multiple pure states. These states are described by a density matrix, as follows:

ρ=

∑

i

pi|ψiihψi|. (2.10)

Here 0 < pi < 1 (pi ∈ R) is the probability for the system to be in the

corresponding state|ψiiwith ∑ i

pi = 1.

Density matrices of mixed states do not correspond to projectors, so ρ2 6= ρ. Furthermore, one could show that for density matrices of mixed states we have

Tr(ρ2) =

∑

i

p2_i <1. (2.11)

By contrast, for density matrices of pure states we have ρ2= ρ = 1, Tr(ρ) = 1 and Tr(ρ2) = 1. This is a very simple manner to distinguish density matrices of mixed states from density matrices of pure states.

The definition of entanglement for mixed states is very analogous to the definition of pure states. Consider a bipartite systemH=H₁⊗ H2of two

Hilbert spacesH1andH2with state ρ ∈ H. The state ρ is called separable

if there exist pi ∈ R with 0 < pi < 1 and product states ρ1i ⊗ρ2i with ρ1i ∈ H1and ρ2i ∈ H2such that

ρ=

∑

i

piρ1i ⊗ρ2i. (2.12)

If this is not the case, the state|ρiis called entangled.

There exists a useful criterion to check if a state is entangled or not, which we call the PPT criterion. We will discuss this criterion in section 8.2 of the appendix.

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2.4 Tripartite systems

Now we turn to composite systemsH=H1⊗ H2⊗ H3that consist of three

different subsystems described by Hilbert spacesH₁,H₂andH₃. The struc-ture of entangled states of a tripartite system is much more complicated than in the case of a bipartite system. We will see that there exist several inequivalent classes of entangled states.

2.4.1 Pure states

For pure states with three subsystems, there exist three different kinds of pure states: fully separable states, biseparable states and genuine entangled states. [9] A state|ψi ∈ His fully separable if we can write

|ψi=|αi ⊗ |βi ⊗ |γi, (2.13)

with|αi ∈ H1,|βi ∈ H2and|γi ∈ H3. Furthermore, a state is biseparable

if we can write it in one of the following ways:

|ψi =|αi1⊗ |δi23, |ψi=|βi2⊗ |δi13, |ψi=|γi3⊗ |δi12 (2.14)

For clearity, subscripts are used with|αi1 ∈ H1,|δi23 ∈ H2⊗ H3,|βi2 ∈ H2,

|δi13 ∈ H1⊗ H3,|γi3∈ H3and|δi12 ∈ H1⊗ H2. If|ψi ∈ His neither fully

separable nor biseparable, it is called genuine entangled. Two examples of genuine entangled states are the GHZ state and the W state, mentioned in section 2.2. If we consider this from physical perspective, we see that only for the creation of genuine entangled states interaction between all the subsystems is needed.

Sometimes it is possible to transform a state|ψiinto a state|φivia so-called

stochastic local operatorations and classical communication (SLOCC) [10]. ’Stochastic’ implies that it is not required that the operation can be done

with certainty. Such operations can be used if we have

|φi= A⊗B⊗C|ψi, (2.15)

with A, B and C invertible operators acting on Hilbert spacesH1,H2and

H₃, respectively. It turns out that there exist two different equivalence classes which can not be transformed into each other by SLOCC. These are the class of the GHZ states and the class of the W states.

It is useful to note that the W state is more durable than the GHZ state. This means that if one system is removed (so if one particle is lost), the W state remains entangled, while the GHZ state becomes separable.

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2.4 Tripartite systems 19

2.4.2 Mixed states

The definition of mixed states in a tripartite system is analogous to the definition of mixed states in a bipartite system. A mixed state of a tripartite system with density matrix ρ is fully separable if we can write

ρ=

∑

i

pi|ψiihψi| (2.16)

for the density matrix, where each pure state|ψii is fully separable and

∑

i

pi= 1. Likewise, we say that a mixed state is biseparable if we can write

the density matrix ρ in the same form as in the equation above, also with ∑

i

pi= 1 but now with each state|ψiibiseparable.

If a mixed state of a tripartite system is not fully separable or biseparable, it is fully entangled. Also for mixed states, there exist two different equiva-lence classes of fully entangled states. [10] If we can write the matrix in the same form as in eq. (2.16), also with∑

i

pi = 1 but now with each state|ψii

equivalent to a pure W state, the mixed state is of the W type. Otherwise the mixed state is part of the GHZ class.

2.4.3 Cluster states

We can now turn to the definition of cluster states. Consider a systemHof multiple qubits. As we saw in secion 2.1.2, the state of each qubit can be written as a superposition of basis states|0iand|1i. Also, each state ψ ∈ H can be writtten as a superposition of tensor product of these basis states. For example, each state|φiin a system of two qubits can be written as

|φi= a|00i+ b|01i+ c|01i+ d|11i, (2.17)

with a, b, c, d∈ _{C. Roughly speaking, a cluster state is a highly entangled} state of a system of multiple qubits.

The more formal definition of a cluster state is as follows. Consider a connected subset C of a d-dimensional lattice (a cluster) with d∈ _N_>₁. A pure state of qubits located on C is called a cluster state. Mathematically, cluster states|φ{ι}iCobey the following eigenvalue equations:

K(a)|φ{ι}iC = (−1) ια|

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with K(a)the so-called correlation operators: K(a) = σx(a)

O

b∈N(a)

σz(b), (2.19)

with σxand σz Pauli matrices (see section 2.1.2), N(a) the neighbourhood

of a and{ια ∈ {0, 1}|α ∈C}parameters characterizing the instance of the

state. In figure 2.1, we show a clusterC with two qubits in it. In section

Figure 2.1:Entangled clusterC. The black dots represent two-state particles. Two qubits are labeled with c0and c00and connected with each other by a pathP. These qubits (or any other pair qubits of the cluster) can be projected into a Bell state if we do measurements on other qubits of the cluster BriegelHJ2001Peia.

8.3 of the appendix, we give an example of a cluster state by introducing a one-dimensional chain of N qubits for arbitrary N ∈N>1.

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Chapter

3

Theory of a beamsplitter

For understanding the experiment, knowledge about physical effects and measurements at beamsplitters is crucial. In this chapter, we discuss the Hong-Ou-Mandel effect at a beamsplitter and correlation measurements on a beamsplitter.

3.1 Hong-Ou-Mandel effect

In this section, we will discuss the Hong-Ou-Mandel effect which is related to the quantum interference of photons. The Hong-Ou-Mandel effect occurs if two (identical) indistinguishable photons enter a 1:1 beamsplitter, i.e. an optical device where one half of the photons of the light is transmitted and the other half is reflected (so both the transmission probability and the reflection probability is 50%).

3.1.1 The Hong-Ou-Mandel effect from mathematical

per-spective

We start with a mathematical desciption, considering so-called Fock states (or number states) |Ni where N is the number of photons. These states carry annihilation operators ˆA and creation operators ˆA†, which act on a Fock state|Nias follows:

ˆ

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22 Theory of a beamsplitter ˆ

A†|Ni =√N + 1|N + 1i, (3.2)

with n ∈ _{N. We see that an annihilation operator destroys a photon and} that a creation operator creates a new photon. Note that the annihilation operator ˆA working on a vacuum state|0igives zero.

Now consider two identical indistinguishable photons arriving at a 1:1 beamsplitter and input ports A and B. For the initial state of the two photons (so the state of the photons before they arrive at the beamsplitter) we can write

|1iA|1iB = ˆA†Bˆ†|0iA|0iB, (3.3)

with ˆA†and ˆB† the creation operators acting on a photon at input port A and B, respectively [11]. If we choose an appropriate coordinate system, we can say that these operators transform in the following way when the photons pass through the 1:1 beamsplitter:

ˆ A† → Cˆ †_{+ ˆ}_D† √ 2 , ˆB † _→ Cˆ†_√−Dˆ† 2 .

Here we use that the phase difference between the transmitted and reflected beam is equal to π. The resulting Fock states will be, where C and D refer to the different output ports,

(3.4) ˆ A†Bˆ†|0iA|0iB → ˆ C†+ ˆD† √ 2 ˆ C†−Dˆ† √ 2 |0iC|0iD = 1 2( ˆC †2₋_D_ˆ†2_)|0i C|0iD = √1

2(|2iC|0iD− |0iC|2iD).

In words, the photons will leave the same output port, either C or D. This is a nice example of quantum interference of photons.

We can also consider interference of states that are a superposition of a vacuum state and a one photon state, with relative phase α. If the two input states have relative phase φ, we can write them as

|ψai = ( √ p0+ √ p1eiαAˆ†)|0iA, |ψbi= ( √ p0+ √ p1ei(α+φ)Bˆ†|0iB, (3.5)

with p0, p1 ∈R and 0 6p0, p1 61. The expression of the resulting output

state|ψoutiafter interference can be written as

(3.6) |ψouti= p0|0iC|0iD+

p1

√ 2e

i(2α+φ)_(|2i

C|0iD − |0iC|2iD)

+p2p0p1ei(α+

φ

2)(cos(φ

2)|1iC|0iD −i sin(

φ

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3.1 Hong-Ou-Mandel effect 23

where C and D refer to the different output ports [12]. Note that our earlier result for a ’pure’ two photon input is obtained for p0 = 0, p1 = 1 and α = φ = 0.

3.1.2 The Hong-Ou-Mandel effect from physical

perspec-tive

We can also give a physical explanation for the Hong-Ou-Mandel effect. Consider figure 3.1, where the two input ports of the beamsplitter a1and

b1and the output ports as a2and b2[13].

Effect.png

Figure 3.1: Sketch of the possible paths that two photons with different input ports could take after a 1:1 beamsplitter.

We see that we have four possibilities:

(a) The photon at input port a1is transmitted and the photon at input

port b1is reflected.

(b) Both photons are reflected. (c) Both photons are transmitted.

(d) The photon at input port a1is reflected and the photon at input port

b1is transmitted.

Because we have no further constraints on these possibilities, we can use Feynman rules, which say that we can add all four possibilities to obtain the

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24 Theory of a beamsplitter

probability amplitudes. Here we again have to note that reflection gives an additional relative phase shift of π, which corresponds to a factor of−1 in the superposition of the probability amplitudes. Since the photons are iden-tical and indistinguishable, the relative minus sign between possibilities (b) and (c) imply that the corresponding terms cancel in the superposition (destructive interference). Thus both photons will always leave the same output port, b1or b2.

3.2 Second order correlation function

In order to measure entangled states of photons, one should know how to derive some information about the photon state from the detection. If the so-called second order correlation function is determined, we can obtain that information. In this section, we will explain how the second order correlation function can be interpreted for various light sources.

Entangled states can be produced by the use of single photons. These photons have to be indistinguishable and for that, single photon purity is required. This purity can be measured via the so-called Hanbury Brown Twiss (HBT) setup in figure 3.2.

Figure 3.2:Hanbury Brown Twiss setup. A stream of photons from a light source (left) enters a 1:1 beamsplitter. Instead of measuring (and comparing) the time at which each photon is detected, it is also possible to use a coincidence counter. This counter can be used in the following way: if a photon is detected at detector 1, the counter is started and it stops if a photon is detected at detector 2. (From: Picoquant.)

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3.2 Second order correlation function 25

g(2)(∆τ) defined by g(2)(∆τ) = ha

†

1(t)a†2(t + ∆τ)a2(t + ∆τ)a1(τ)i

ha†

1(t)a1(t)iha2†(t + ∆τ)a2(t)i

= hn1(t)n2(t + ∆τ)i

hn1(t)ihn2(t + ∆τ)i, (3.7)

where ai and a†_i (i = 1, 2) are annihilation and creation operators,

respec-tively, and ni = a†_iaiis the number operator [14]. Physically, we can think

of this function as follows: it gives the (normalized) probability of detecting N photons at detector 2 at time t + ∆τ if N photons are detected at detector 1 at time t (where N ∈ _N_>₁). If we do not select states, the second order correlation function is even, i.e. g(2)(∆τ) = g(2)(−∆τ) for all ∆τ ∈R.

Now assume that we use an unknown light source, so we do not know where and when the photons of the source arrive. We only measure the time delay ∆τ for detecting a photon at a detector after detecting a photon at the other detector. Then the following results are possible for the second order correlation function (see also figure 3.3 for a clear picture):

1. g(2)(0)>1. In this case the probability to detect two photons at both detectors at the same time is (relatively) high. Then the light consists of bunched photons. This is the case for thermal light.

2. g(2)(0)<1. In this case the probability to detect two photons at both detectors at different times is (relatively) high. Then the light consist of antibunched photons. This is also what we measure for the second order correlation function if we use single photons. In the ideal case, when using single photons and in absence of background noise, we have g(2)(0) = 0. This is also the case that we will consider further in this work.

3. g(2)(0) = g(2)(∆τ) = 1 for all ∆τ ∈ R (also negative values for ∆τ are possible here). In this case, the probability of detecting a photon at a detector (say at time t) is completely independent of detecting a photon at the other detector (say at time t + ∆τ). Then the light is coherent.

Note that for large ∆τ, the probability of detecting a photon at time t at one detector will depend less and less on the probability of detecting a photon at the other detector at time t + τ so

hn1(t)n2(t + ∆τ)i ≈ hn1(t)ihn2(t + ∆τ)i ⇒ g(2)(∆τ) =

hn₁(t)n2(t + ∆τ)i

hn1(t)ihn2(t + ∆τ)i

≈1, (3.8) for ∆τ 1. From this, we can define a coherence time τc: the time

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26 Theory of a beamsplitter

difference ∆τ for which the second order correlation function approaches unity.

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Chapter

4

The experiment: preparation

In this section, we will describe an ’old’ and a ’new’ experiment in which cluster states of photons are created and investigated. Later on (in the next chapter) it will become clear why we choose to do another new experiment after doing the old one. Before we explain what the setups of the old and new experiment look like, we will describe shortly how the single photons used in our experiment are produced and consider some aspects of the quantum dot that could play a role in our measurements. After introducing both setups, we will explain how the loop of the setup can be aligned and stabilized so that (stable) interference in the setup can occur.

4.1 Production of single photons by a quantum

dot

Of course, to create entanglement between photons, a light source is needed. This light source must produce a stream of single photons, in order to produce photon-entangled pairs after possible interference. In this work, we will use a quantum dot (QD) in combination with a laser as light source. Quantum dots are very small semiconductor crystals which vary from one to a few tens of nanometers in size [15]. In our experiment, we will use InAs quantum dots of approximately 10 nm in width and around 2 nm in height. Sometimes, quantum dots are called ’artificial atoms’ [16], for their discretized energy levels similar to atoms leading only to discrete optical transitions. In this thesis, we consider quantum dots as two-state

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28 The experiment: preparation

systems, each with a ground state and an excited state. If laser light enters the cavity and interacts with the quantum dot. The quantum dot will now be in the excited state. As a result, an electron-hole pair is created and upon recombination a single photon can be emitted. In order to transport energy through the cavity, the frequency of the laser must be equal to the frequency that corresponds to the frequency of the energy transition: the resonance frequency.

We will not discuss the production of single photons by the quantum dot into detail, since we are focused on producing and measuring cluster states with our setup. Important aspects of the QD (that could be relevant for our measurements) are listed below.

Figure 4.1:Reflectivity scan of the micropillar that we use for the quantum dot [17]. The cavity is observable in the middle of the radioactive sign.

• The lifetime T of the quantum dot. This lifetime tells us how much time it takes for an excited electron to return to the ground state, when a photon is emitted. If we measure the second order correlation function g(2)of the produced single photon stream with an HBT-setup, the width of the single photon dip decreases for higher lifetimes. [18] The lifetime of the quantum dot that we will use is T = 130±10 ps. However, the single photon dip is strongly narrowed by the cavity. • The linewidth ∆λ of the quantum dot, or frequency range ∆ f [19].

This linewidth or frequency range can be very small, so it is very important to use light with a constant wavelength (or frequency), to obtain a high count rate of single photons (so that background radiation or extra (dark) counts from a detector can be neglected) [20].

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4.2 Old setup 29

The laser light that we use has a wavelength λ = 934.3 nm.

• Filtering of the excitation laser light from the single photon light. After the light is reflected from the cavity, the single photons are filtered from the coherent laser light by use of polarizers to ensure that our single photon source is as pure as possible. One should be very careful in the alignment here [21, 22]. For our quantum dot, we use a 30-1.5 cavity with an A19 sample [17]. In figure 4.1, the shape of the cavity is shown.

More aspects of the quantum dot can be found in [18].

4.2 Old setup

In figure 4.2, the setup for the ’old’ experiment is given.

In the figure, several locations are labeled with several letters. At first, only horizontally polarized incoming single photons will be transmitted into the setup (by means of an H-polarizer, labeled as POL) and after that, we place a λ/2-waveplate (labeled as WP1) under an angle of 22.5◦so that all the photons will have diagonal polarization|Di= √1

2(|Hi+|Vi) (A). Then,

these photons travel to a polarizing beamsplitter (labeled as PBS) where H-polarized (horizontally polarized) light is transmitted in the direction of (B) and the other half, V-polarized (vertically polarized) light, is reflected in the direction of (D). The H-polarized light travels through an extra loop with a time delay of 3.5 ns. It goes through an extra λ/2-waveplate (labeled as WP2), again placed under an angle of 22.5◦so that H-polarized photons will be in the diagonal state at (C). Now the V-polarized component is reflected in the direction of (B) and H-polarized light is transmitted in the direction of (D). The V-polarized photons will be in the antidiagonal state |Ai = √1

2(|Hi − |Vi) after passing the waveplate, at (C). After passing (D)

and (E), the photons enter a detection zone containing a 1:1 beamsplitter and two detectors for the corresponding paths, that both measure the number of photons as a function of time. This detection zone, consisting of one beamsplitter and two detectors, follows the HBT-setup.

Detection

In our experiment, we will set the following combinations of polarizers before the detectors:

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Figure 4.2:Setup of the ’old’ experiment.

1. Two vertical polarizers, one polarizer before each detector (VV con-figuration).

2. One vertical polarizer before one detector and one horizontal polar-izer before the other detector (VH or HV configuration).

3. Two horizontal polarizers, one polarizer before each detector (HH configuration).

We say that the corresponding four measurements are done in the{|Hi,|V i}-basis. This will give us not only information about the time differences between the detected photons, but also about the polarization of the pho-tons. Our goal is to detect the entangled photons, so photons in one of the states (related to the interference) that clearly show evidence that we produced a cluster state. Furthermore, it is important to know that the

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4.3 New setup 31

detector has a dead time of 60 ns, so only two photons that are detected by different detectors are important. This means that if two photons enter the same detector in a time interval of less than 60 ns, only one photon will be detected.

In the setup, an extra waveplate WP3 is placed before the non-polarizing beamsplitter. In the first four measurements, we place this waveplate under an angle of 0◦, so that the polarization will not change. In the next four measurements, we place WP3 under an angle of 22.5◦. In that case, an H-polarized photon will be diagonally polarized (D-polarized) and a V-polarized photon will be antidiagonally V-polarized (A-V-polarized). We could say that we measure the polarization of photons in the{|Di,|Ai}-basis in this case. The reason why we want to measure the polarization in different bases, is to verify entanglement: as we saw in section 2.2, the entanglement of a photon state is independent of the basis choice.

For each combination, we will measure the second order correlation func-tion g(2)(τ). Then we will compare the dips or peaks of this function with our expectations and determine if we can deduce entanglement from these results. The expectations will be described in more detail in the next chap-ter.

4.3 New setup

For the setup of the ’new’ experiment, see figure 4.3.

In the figure, several locations are again labeled with several letters. This setup has many similarities with the old one. The photons at (B) enter a 1:1 non-polarizing beamsplitter (BS), where the half of the photons will be reflected in the direction of (C) and detectd by the detector D1. The other half will be transmitted in the direction of (D). Since this beamsplitter is placed in the loop, we will call this the loop beamsplitter, or in short LBS. The V-polarized photons in the loop, reflected by the PBS after (E), that are transmitted by the beamsplitter will be in the antidiagonal state|Ai =

1 √

2(|Hi − |Vi) after passing the waveplate, at (C). Just as in the old setup,

we measure correlations between detector D1 and detector D2.

Only the half of the photons at (F) will be detected by the other detector D2; the other half will be reflected by a beamsplitter and is not used for detection. The reason for not removing this beamsplitter is to minimize the difference in photon count rates (i.e. the number of photons per second

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Figure 4.3:Setup of the ’new’ experiment.

that is counted by a detector) between both detectors. This is important, because the coincidence count rate is affected by the detector with the lower photon count rate. We will work this out in more detail in the next chapter. Instead of one extra waveplate, we now need two extra waveplates (WP3a and WP3b placed before the polarizers before D1 and D2, respectively) for measurements in the{|Di,|Ai}-basis. (Of course, both waveplates are placed under the same angle in each measurement.)

4.4 Stabilization of the setup

In order to prevent high fluctuations in the photon count rates, the follow-ing is done to stabilize the setup:

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4.4 Stabilization of the setup 33

used to emit light with a constant phase and a constant frequency.[23] Locking of the laser is important for keeping a constant excitation condition for of our QD light source and stabilizing the count rate of photons coming into the setup, since the frequency range in the emission spectrum of a quantum dot is very small as we saw in section 4.1.

• Locking of the loop. This means that a feedback control system is used to ensure that the phase change of light going through the loop is constant. In this feedback system, another (red light emitting) laser is used to control the phase changes [24]. In our setup, we can use a filter to prevent that the light of this laser reaches the detectors, since this laser emits light of another wavelength. Locking of the loop is especially important to prevent fluctuations due to first-order inter-ference. These interference effects are very sensitive for fluctuations in temperature or sound in the environment.

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Chapter

5

The experiment: expectation

5.1 Expectations in terms of operators and Fock

states: general description

In this chapter, we will describe analytically what happens with the polar-ization, location and time of photons used in the experiment. To make this description more clear, we first calculate two photon output states in the old setup. The different polarizations are not described with kets, but in terms of creation operators acting on a vacuum state|00i. These creation operators (say A† and B†) satisfy the relation

A†B†|N Mi =√N + 1√M + 1|N + 1, M + 1i, (5.1) with A†acting on the Hilbert space containing elements|Niand B†acting on the Hilbert space containing elements|Mi. The creation operators act on the so-called Fock states|N Miof the photons, where N is the number of photons with horizontal polarization and M is the number of photons with vertical polarization. Note that only the operators change when a photon enters another location in the setup: all the operators act on the same Fock state|00i. The reason for using this description is that we can easily take account for the quantum interference that can happen in the experiment, as we will see later.

We will identify horizontal polarization with the operator ˆH†and vertical polarization with the operator ˆV†. It follows that diagonal polarization can be identified with the operator √1

2( ˆH

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36 The experiment: expectation

with the operator √1 2( ˆH

† ₋_V_ˆ†_{). (All these operators are acting on the}

vacuum state.)

After a waveplate, the polarization is rotated by 45◦. Then the operators associated with horizontal and vertical polarization change as follows:

ˆ H† WP−−→ √1 2( ˆH †_{+ ˆ}_V†_{), ˆ}_V† WP_−−→ _√1 2( ˆH †₋_V_ˆ†_). _(5.2)

From now on, we will also write the time and location in the setup of each operator in the subscript. We refer to the locations with the same symbols as in figures 4.2 and 4.3. For example, we write ˆH†_A,tfor the operator that is associated with horizontally polarized light acting at time t, directly coming from the light source and entering the polarizing beamsplitter (PBS). The PBS reflects vertically polarized light and transmits horizontally polar-ized light. We can write the following for the associated operators in the old setup:

ˆ

H†_A,t −−→PBS Hˆ_B,t† , ˆV_A,t† −−→PBS Vˆ_D,t† , ˆH_C,t† −−→PBS Hˆ_D,t† , ˆV_C,t† −−→PBS Vˆ_B,t† . (5.3) We can also write how the creation operators change in the new setup at the LBS: ˆ H_B,t† −−→LBS √1 2( ˆH † C,t+ ˆH†D,t), ˆVB,t† LBS −−→ √1 2( ˆV † C,t+ ˆVD,t† ). (5.4)

5.2 Expectation for the old experiment

5.2.1 Two photon entanglement

Now we have enough information to obtain an expectation with a simple analytic calculation. First, we will do a calculation for a two photon input for which we obtain the simplest entangled state. This input can be de-scribed as follows:

- one horizontally polarized photon entering the setup at t = 0 ns,

- another horizontally polarized photon entering the setup at t = 3.5 ns, so that the time difference between these photons is exactly equal to the loop delay time.

Here we do not write the vacuum state|00i for simplicity, except at the end (because only the operators change, not the state itself). Furthermore, WP3a and WP3b are set at 0◦. Also, we write the times not as numbers

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of nanoseconds, but as numbers of timesteps, where each timestep corre-sponds to 3.5 ns.

A way to realize this input is to use a pulsed laser, which emits two suc-cessive pulses of two photons with a time interval of 3.5 ns. Such a pair of pulses can be emitted many times. In order to prevent entanglement between photons of different pairs of pulses, the time between two pairs of pulses should not be equal to a multiple of the delay loop time and not too small. For example, we can choose a time interval of 12.5 ns between two pulses, corresponding to experimental options in our lab.

At t = 0 ns, the first photon is rotated about 45 degrees after the first waveplate: ˆ H_in,0† −−→WP1 √1 2( ˆH † A,0+ ˆVA,0† ). (5.5)

We then obtain a superposition of horizontally polarized light, which is transmitted, and vertically polarized light, which is reflected:

1 √ 2( ˆH † A,0+ ˆVA,0† ) PBS −−→ √1 2( ˆH † B,0+ ˆVD,0† ). (5.6)

In the next step (or evolution) horizontally polarized photons travel through the loop, while the other photons enters the setup and travel through the first waveplate. So for the next two evolutions, we can write the following, where we use a tensor product of the operators acting on the first and second photon: 1 √ 2( 1 √ 2( ˆH † C,1+ ˆVC,1† ) + ˆVD,0† ) 1 √ 2( ˆH † A,1+ ˆVA,1† ) PBS −−→ (5.7) 1 2[ 1 √ 2( ˆH † D,1HˆB,1† + ˆHD,1† VˆD,1† + ˆVB,1† HˆB,1† + ˆVB,1† VˆD,1† ) + ˆVD,0† HˆB,1† + ˆVD,0† VˆD,1† ]. (5.8) (In the last evolution, both the first and second photon travel through the polarizing beamsplitter and the expression is rewritten without factor-izations.) After this, horizontally polarized photons at location B again travel through the second waveplate (which again takes 3.5 ns), so after the second roundtrip of the first photon we can write:

(5.9) 1 2[ 1 2Hˆ † D,1( ˆHC,2† + ˆVC,2† ) + 1 √ 2 ˆ H_D,1† Vˆ_D,1† + 1 2√2( ˆH † C,2−VˆC,2† )( ˆHC,2† + ˆVC,2† ) + 1 2( ˆH † C,2−VˆC,2† ) ˆVD,1† + √1 2 ˆ V_D,0† ( ˆH_C,2† + ˆV_C,2) + ˆV_D,0† Vˆ_D,1† ].

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Now focus on what happens with the state at the waveplate after 7 ns for the following terms (so in fact slight before location C, but here we just use C because of clear labels):

( ˆH_C,2† −Vˆ_C,2† )( ˆH_C,2† + ˆV_C,2† ) = ˆH_C,2† Hˆ_C,2† −Hˆ_C,2† Vˆ_C,2† + ˆV_C,2† Hˆ_C,2† −Vˆ_C,2† Vˆ_C,2† . (5.10) For the tensor product of operators acting on two indistinguishable photons, we have

ˆ

H_C,2† Vˆ_C,2† |00i=|11i= ˆV_C,2† Hˆ_C,2† |00i,

leading to destructive (quantum) interference at the second waveplate after 7 ns:

( ˆH_C,2† −Vˆ_C,2† )( ˆH_C,2† + ˆV_C,2† ) = ˆH_C,2† Hˆ_C,2† −Vˆ_C,2† Vˆ_C,2† . (5.11) In the next evolution, remaining photons again travel through the PBS. If we combine this with the quantum interference at the second waveplate, we get the following expression:

(5.12) 1 2[ 1 2Hˆ † D,1HˆD,2† + 1 2Hˆ † D,1VˆB,2† + 1 √ 2 ˆ H_D,1† Vˆ_D,1† + 1 2√2 ˆ H†_D,2Hˆ†_D,2− 1 2√2 ˆ V_B,2† Vˆ_B,2† + 1 2Hˆ † D,2VˆD,1† −1 2Vˆ † B,2VˆD,1† + 1 √ 2 ˆ V_D,0† Hˆ_D,2† + √1 2 ˆ V_D,0† Vˆ_B,2† + ˆV_D,0† Vˆ_D,1† ].

(Note that the resulting Fock state|ψi is normalized (hψ|ψi = 1) and

con-tains an extra factor √2 in the terms with ˆH_C,2† Hˆ_C,2† (or ˆH†_D,2Hˆ†_D,2) and V_C,2† V_C,2† (or V_B,2† V_B,2† ) which follows from the identities of the creation oper-ators which can be found in sections 3.1.1 and 5.1.) We see that, because of the quantum interference, our final state is entangled. [REVISION]

5.2.2 General expectation for two photons (continuous wave

laser)

Instead of using a two-photon input where the time interval between the photons is equal to the loop delay time (such as in section 5.2.1), we can also consider the situation in which the second incoming photon has an arbitrary time delay ∆T with respect to the first incoming photon (see drawing).

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If we use a continuous wave laser, the time intervals between each pair of photons will be random. In that case, ∆T can in principle be any real number except zero (note that cases with ∆T <0 can also be considered: then the ’next photon’ enters the setup before the ’photon in’). To prevent further loss of generality, we assume that the delay time of the loop (the ’loop length’) is L, where L is fixed. If we now use the same method and the same number of evolutions as in section 5.2.1, the resulting superposition of operators will look as follows:

(5.13) 1 2[ 1 2Hˆ † D,L( ˆH†D,∆T+L+ ˆVB,∆T+L† ) + 1 √ 2 ˆ H_D,L† Vˆ_D,∆T† + 1 2√2( ˆH † D,2L−VˆB,2L† )( ˆH†D,∆T+L+ ˆVB,∆T+L† ) + 1 2( ˆH † D,2L−VˆB,2L† ) ˆVD,∆T† + √1 2 ˆ V_D,0† ( ˆH†_D,∆T+L+ ˆV_B,∆T+L† ) + ˆV_D,0† Vˆ_D,∆T† ].

Note that for ∆T = L, quantum interference takes place and we get eq. (5.12) back, with L = 3.5 ns. We already saw that the output state is entangled in this case. Quantum interference can also take place for ∆T = 2L: if we plug in ∆T = 2L in eq. (5.13), one of the terms can be developed as follows:

−1 4Vˆ † B,2LHˆB,2L† → − 1 8( ˆH † C,3L−VˆC,3L† )( ˆHC,3L† + ˆVC,3L† ) = 1 8( ˆV † C,3LVˆC,3L† −HˆC,3L† HˆC,3L† ). (5.14) In the same way, we can show that quantum interference will also take place for ∆T = 3L, 4L,· · ·. By symmetry, we can conclude that quantum interference takes place for ∆T = x·L with x ∈_Z\ {0}, which always will eventually result in an entangled state. However, ∆T = L (or ∆T = −L) is still the most interesting case since the probabilities to produce these states with interference is the highest. (We can easily see that the probability to produce states with interference is ₃₂1 ·(1₂)n if the time interval between the photons is ∆T = n·L with n ∈N>1.)

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5.2.3 Detection of entanglement

Now the question arises if we can really detect the entangled parts of the resulting state. For this, focus on the resulting correlation function g(2)(τ) (recall its physical meaning from section 3.2), that we can measure. If we place identical polarizers before the detectors, we should observe, due to the use of single photons, a great dip for time differences τ = 0. This implies that we can use postselection in these cases [5]. We only consider the pho-tons that are detected at different times, because only for time differences unequal to zero we can see dips or peaks that are not caused by antibunch-ing. For example, if we again consider eq. (5.12), we see that √1

2VVL is a

possible output state with time combination 0−7 (here we use no subscript for photons arriving at a detector and the subscript L for photons that are still in the loop, so we just detect one photon here). We see that the result-ing state, and also the resultresult-ing states with time combination 0−3.5 and 3.5−7, are clearly separable, so we can not detect entanglement in this way! Let us take a step back to eq. (5.8), focusing on the first four terms. Using postselection, we can ignore the second and third term. (Note that the third term will also guarantee that the photons will arrive at a detector at the same time, due to quantum interference in eq. (5.11).) Now we see that the two remaining terms will give an entangled state! In fact, this is the Bell state, which we can associate with a so-called Bell state operator

ˆ φBD ≡ 1 √ 2(H † D,3.5H†B,3.5+ VB,3.5† VD,3.5† ). (5.15)

Now you may ask what happens with this entangled state and why we don’t detect it. Well, the problem lies in the fact that each of the two possible resulting states consists of a detected photon entangled with a photon in the loop. As soon as the photon in the loop travels through the PBS to the detector, we will detect a separable state since the time difference changes (see also eq. (5.12)). This gives rise to the question if it is possible to detect a photon that is inside the loop, to confirm entanglement, which is exactly the reason for creating another experiment with a ’new’ setup.

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5.3 Expectation for the new experiment

5.3.1 Count rates

Before we turn to the calculations for entangled states in the new setup, we first have to comment on the count rates in the new setup. Because the detectors are placed in different locations in the setup, the photon count rates (the total number of photons detected per second) at both detectors are not equal for each polarization configuration. (In comparison, in the old setup, the detectors are just connected to identical output ports from the beamsplitter outside the loop). Below, we will explain with quantitative details why a 1:1 LBS is chosen. For convenience, we will consider perfect detectors in a perfectly aligned setup.

First, note that the total count rates (the count rates measured if no polariz-ers are placed before the detectors) at both detectors are equal. This can be simply understood by considering the D-polarized incoming photons: 50% of the light is transmitted and 50% is reflected by the PBS. Then 50% of the transmitted light will be reflected by the LBS and detected at (C). 50% of the reflected light will be transmitted through the beamsplitter outside the loop (BS) and detected by the other detector. Thus both detectors detect 25% of the light, while 25% of the light remains in the loop and 25% is lost. In the same way, it follows that after one roundtrip, each detector detects 25% of the light in the loop, 25% of this light is lost and the other 25% again goes through the loop. This process goes on till all the light has left the loop, so we can conclude that the total count rates at both detectors are equal.

Now, consider the HV configuration. Using the same reasoning as above, it follows that 25% of the incoming light, which is H-polarized, is directly detected at (C) before making a roundtrip through the loop. Also, another 25% of the light is detected as reflected V-polarized photons outside the loop by the other detector. Now note that all the other light, that goes through the loop, will not be detected since the remaining light inside the loop will be V-polarized and the remaining light outside the loop will be H-polarized. So also for the HV configuration, the count rates are equal. For the VH configuration, we can consider the situation the other way around. Now the directly incoming H-polarized light and the directly reflected V-polarized light will not be detected anymore, but the remaining V-polarized photons in the loop and the H-polarized photons outside the loop will. As we saw above, those amounts are equal to each other, so the count rates are also the same for the VH configuration.

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However, the VV configuration gives another result. We already saw that 25% of the incoming light is detected outside the loop as V-polarized pho-tons. However, only 25% of the light in the loop is detected inside the loop as V-polarized photons (the light must be reflected by both the PBS and the LBS, where both reflection probabilities are 50%). Also, only 25% of the incoming light will actually go through the loop (50% will be reflected by the PBS and the other 25% are H-polarized photons reflected by the LBS, which won’t be detected). It follows that the ratio between the count rates of the detectors in and outside the loop is 1 : 4, respectively. In the same way, we find that these ratio is 4 : 1 for the HH configuration.

In table 6.1, we give an overview of the ratios of the count rates of the detectors (the first number refers to the detector connected with the LBS and the last number to the other one).

Configuration Ratio HV 1 : 1 VV 1 : 4 HH 4 : 1 VH 1 : 1 Total 1 : 1

Table 6.1: ratios between the count rates of both detectors in the new setup for

different polarization configurations.

Because most count rates are equal to each other (especially the total count rate), we chose a 1:1 LBS. In principle, a lower count rate does not affect the position and the height of the dips. However, the dips will be less visible due to less coincidence counts. (Also, a lower photon count rate will negatively affect the signal-to-noise ratio.)

5.3.2 Two photon entanglement

Also for the new setup, we will do a calculation example for two photon entanglement. Here we will calculate the resulting Fock state if we use the same (pulsed laser) input as in the old experiment. WP3a and WP3b are again set at 0◦.

Below, we write the operators in the system acting on the vacuum state after each evolution. (Note that the operators acting on the second pho-ton change at the moment that the second phopho-ton enters the polarizing beamsplitter.) Above the arrows, we write which optical components are

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involved in the evolution.

(Note that ’in’ refers to an incoming photon just entering the setup slightly before location A.)

ˆ

H_in,0† WP1−−→ √1 2( ˆH

†

A,0+ ˆVA,0† )⊗Hˆin,1† (5.16)

PBS −−→ √1 2( ˆH † B,0+ ˆVF,0† )⊗Hˆin,1† (5.17) BS −→ √1 2( 1 √ 2( ˆH † C,0+ ˆH†D,0) + ˆVF,0† )⊗Hˆin,1† (5.18) WP2,WP1 −−−−−−→ 1 2( 1 √ 2( ˆH † C,0+ 1 √ 2( ˆH †

E,1+ ˆVE,1† )) + ˆVF,0† )⊗( ˆH†A,1+ ˆVA,1† ) (5.19)

(5.20) PBS −−→ 1 2[ 1 √ 2( ˆH † C,0HˆB,1† + ˆHC,0† VˆF,1† + √1 2( ˆH † F,1Hˆ†B,1+ ˆHF,1† VˆF,1† + ˆVB,1† HˆB,1† + ˆVB,1† VˆF,1† )) + ˆV_F,0† Hˆ_B,1† + ˆV_F,0† Vˆ_F,1† ]. (5.21) BS −→ 1 2[ 1 2Hˆ † C,0HˆD,1† + 1 2Hˆ † C,0HˆC,1† + 1 √ 2 ˆ H_C,0† Vˆ_F,1† + 1 2√2( ˆH † F,1HˆC,1† + ˆH†F,1HˆD,1† ) + 1 2Hˆ † F,1VˆF,1† + 1 4( ˆV † D,1Hˆ†D,1+ ˆVD,1† HˆC,1† + ˆVC,1† HˆD,1† + ˆVC,1† Hˆ†C,1) + 1 2√2( ˆV † C,1VˆF,1† + ˆVD,1† VˆF,1† )+ 1 √ 2( ˆV † F,0HˆC,1† + ˆVF,0† HˆD,1† )+ ˆVF,0† VˆF,1† ].

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Applying two more evolutions, we get the following result:

(5.22) WP2,PBS −−−−−→ 1 2[ 1 2√2( ˆH † C,0HˆF,2† + ˆHC,0† VˆB,2† ) + 1 2Hˆ † C,0HˆC,1† + √1 2 ˆ H_C,0† Vˆ_F,1† + 1 2√2 ˆ H†_F,1Hˆ_C,1† + 1 4( ˆH † F,1Hˆ†F,2+ ˆHF,1† VˆB,2† ) + 1 2Hˆ † F,1VˆF,1† + 1 8( ˆH † F,2Hˆ†F,2−VˆB,2† VˆB,2† ) + 1 4√2( ˆH † F,2HˆC,1† −VˆB,2† HˆC,1† + ˆVC,1† Hˆ†F,2+ ˆVC,1† VˆB,2† ) + 1 4Vˆ † C,1HˆC,1† + 1 2√2 ˆ V_C,1† Vˆ_F,1† + 1 4( ˆH † F,2VˆF,1† −VˆB,2† VˆF,1† ) + √1 2 ˆ V_F,0† Hˆ_C,1† + 1 2( ˆV † F,0Hˆ†F,2+ ˆVF,0† VˆB,2† ) + ˆVF,0† VˆF,1† ].

Again, we see that quantum interference took place at the waveplate in the loop (WP2).

5.3.3 Overview of detected states in the case of two

pho-tons (

{|

H

i

,

|

V

i}

-basis)

In table 6.2, we give an overview of the detected states for the situation described in section 5.3.2, including time differences in terms of timesteps of 3.5 ns. We can apply one more evolution on the resulting state from eq. (5.22), to determine more states where a photon is detected by the detector D1. However, this will produce an even larger superposition with states we are not interested in. Of course, only the states for which the detectors (D1 and D2) both detect one photon are relevant for the experiment. These states are given in table 6.2 in terms of kets. In each combination, the left number or state is measured by D1 and the right number or state is measured by D2. So the state 1₈|HViwith time combination 0−1 means that D1 detects an H-polarized photon after zero timesteps and D2 detects a V-polarized photon after one timestep (with probability amplitude ₂√1

2).

For determining this overview, it is sufficient just to consider the state that follows from eq. (5.22) and apply a factor √1

2 to photons at location (B) (this

corresponds to the part of the photons at (B) that is detected by detector D1 after being reflected by the LBS). We will identify location (C) with the detector D1 and location (F) with the detector D2, after reducing the

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probability amplitude of the second photon with √1

2(it must be transmitted

through the extra beamsplitter outside the loop).

(Keep in mind that with ’detected states’, we mean states that can be de-tected, the states are of course not detected in all the three cases with different combinations of polarizers before the detector.)

Time combination Detected state 1−1 1₈(|HHi+|VVi) 2−1 ₁₆1(|V Hi − |VVi) 0−2 1₈|HHi 1−2 ₁₆1(|HHi+|V Hi) 1−0 1₄|HVi 2−0 1₈|VVi 0−1 1₄|HVi

Table 6.2: Produced polarization states in the{|Hi,|Vi}-basis including prob-ability amplitudes with corresponding detection times in the case of two photon entanglement.

We see that the state 1

4√2(|HHi+|VVi) with time combination 1−1 is

entangled! However, there is no difference between the times at which the two photons are detected. This can be solved by e.g. varying the distance of D1 to the beamsplitter, which gives an extra (or shorter) time delay ∆t for which we can account when considering time differences in a g(2)-measurement. (Note that in this example, it takes no extra time for photons coming from the light source to travel to D1, compared to photons that travel to D2. In fact, this time can be adjusted by adjusting the setup.)

5.3.4 Overview of detected states in the case of two

pho-tons (

{|

D

i

,

|

A

i}

-basis)

Now consider the case in which the two waveplates WP3a and WP3b are placed under an angle of 22.5◦. Recall that in this situation, each H-polarized photon at (C) or (F) will be D-H-polarized and each V-H-polarized photon at (C) or (F) will be A-polarized. So in this case, we apply two more evolutions on the state that follows from eq. (5.22). We can use the same identifications as described in the previous case. In table 6.3, the detected states for this situation are listed.

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Time combination Detected state 1−1 1₈(|HHi+|VVi) 2−1 ₁₆1(|HVi − |VVi)

0−2 ₁₆1(|HHi+|HVi+|V Hi+|VVi) 1−2 ₁₆1(|HHi+|HVi)

1−0 1₈(|HHi − |HVi+|V Hi − |VVi) 2−0 ₁₆1(|HHi − |HVi − |V Hi+|VVi) 0−1 1₈(|HHi − |HVi+|V Hi − |VVi)

Table 6.3: Produced polarization states in the{|Di,|Ai}-basis including prob-ability amplitudes with corresponding detection times in the case of two photon entanglement.

Note that these results can also be obtained by applying the operations |Hi → √1

2(|Hi+|Vi) and|Vi → 1 √

2(|Hi − |Vi) on each polarization state

given in the overview for the other case separately. If we compare this overview with the other one, we see a remarkable result: the probability of the (entangled) Bell state (corresponding to the Bell state operator) is the same in both cases, while the probabilities of many other involved states decrease if we change the angle of the waveplates WP3a and WP3b from 0 to 22.5◦. We can say that the relative probability of this state increases, so it becomes even more easier to measure entanglement in the second case!

5.3.5 Overview of detected states in the case of three

entan-gled photons (

{|

H

i

,

|

V

i}

-basis)

We could also do a calculation for the case that each laser pulse contains three photons, each separated by a time interval of 3.5 ns, without the use of extra waveplates WP3a and WP3b outside the loop. We use four detectors to measure three photon correlations, see figure 5.1. The three photon entangled state can be calculated in the same way as the two photon entangled state. However, now the calculation becomes very lengthy, so instead of doing it analytically, we will do this numerically, by use of the program Mathematica.

In tables 6.4 and 6.5, we will list the states that are detected after four timesteps. Here, we don’t give the probability amplitude but just the corresponding probability for each separate state (so not for the whole superposition). (Signs do not matter here.) The two detectors after loca-tion (C) are labeled by α and β and the two detectors after localoca-tion (F) are labeled by γ and δ. We use these letters as subscripts in the different

Photonic cluster states with quantum dots

Photonic cluster states with a

quantum dot

Photonic cluster states with a

quantum dot

Edward Hissink

Abstract

Acknowledgements

Contents

Chapter

1

Introduction

Chapter

2

Definition of entanglement and

cluster states

2.1

Notation of cluster states

2.1.1

Tensor product

2.1.2

Two-state systems and qubits

2.2

Pure states

∑

2.3

Mixed states

∑

∑

∑

2.4

Tripartite systems

2.4.1

Pure states

2.4.2

Mixed states

∑

2.4.3

Cluster states

Chapter

3

Theory of a beamsplitter

3.1

Hong-Ou-Mandel effect

3.1.1

The Hong-Ou-Mandel effect from mathematical

per-spective

3.1.2

The Hong-Ou-Mandel effect from physical

perspec-tive

3.2

Second order correlation function

Chapter

4

The experiment: preparation

4.1

Production of single photons by a quantum

dot

4.2

Old setup

4.3

New setup

4.4

Stabilization of the setup

Chapter

5

The experiment: expectation

5.1

Expectations in terms of operators and Fock

states: general description

5.2

Expectation for the old experiment

5.2.1

Two photon entanglement

5.2.2

General expectation for two photons (continuous wave

laser)

5.2.3

Detection of entanglement

5.3