On Quadripartite Entanglement in High-Dimensional Hilbert Space

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On Quadripartite Entanglement in

High-Dimensional Hilbert Space

THESIS

submitted in partial fulfillment of the requirements for the degree of

BACHELOR OF SCIENCE in

PHYSICS

Author : Andreas Lepidis

Student ID : S1692003

Supervisor : Wolfgang L ¨offler

2ndcorrector : Michiel de Dood

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On Quadripartite Entanglement in

High-Dimensional Hilbert Space

Andreas Lepidis

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

July 8, 2019

Abstract

In this thesis we investigate multi-photon entanglement in high dimensions, where the high-dimensional Hilbert space is given by the transverse-mode space of photon fields. We first introduce the reader to

the mathematical framework and the quantization of light fields. Thereafter we discuss the possible partitions of the transverse-mode space, and the generic structure of photons created via type-I collinear

degenerate spontaneous parametric down conversion. Finally, we present to the reader strong evidence that such entangled photons can be

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Chapter

1

Introduction

The Einstein-Podolski-Rosen gedankenexperiment [1] introduced a rather peculiar consequence of quantum theory, namely that the measurement of one particle can determine the measurement outcomes of another particle with unit probability. This phenomenon has been given the name quan-tum entanglement and it was later shown that this nonlocal effect is not caused by the transfer of information between the two particles [2]. Even though Einstein-Podolski-Rosen used the polarization degree of freedom in their gedankenexperiment, the described effect is not limited to polar-ization and can also occur in momentum space, or any other degree of free-dom for that matter. This means that momentum and momentum derived degrees of freedom are also subject to this phenomenon. One of these de-grees of freedom is the orbital angular momentum of light. Although any electromagnetic wave has orbital angular momentum, it is only paraxial waves which have orbital angular momentum which is well-defined so that a quantum description of this property is possible [3]. The quantum number associated with the orbital angular momentum ranges from−∞

to ∞ and, therefore, the dimension of its associated Hilbert space is infi-nite. Though, in practice it is not feasible to encapsulate the entire Hilbert space, and thereby one is confined to a finite region. Despite this, the— effective—orbital angular momentum Hilbert space is vastly greater than its spin counterpart, making it a more powerful tool for quantum infor-mation applications.

Up until now there has been intense research on orbital angular mo-mentum entanglement of light, but most of the current research focuses on the properties of a photon pair that is entangled in the orbital angular momentum degree of freedom, and in some cases this is extended to

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mul-Figure 1.1:The experimental setup. The lenses L1, L2 and L3 have focal lengths of 25 cm, 4 cm and 75 cm, respectively. The GaP and BPF plates filter out the pump laser. Image courtesy: Charlie Bender.

tiple pairs of entangled photons. Currently, the largest of such photon pair clusters realized consists of 6 photon pairs [4]. More that two photons en-tangled in their orbital angular momentum degree of freedom have only recently been observed. In fact, the first of such observations has been con-ducted in 2016 by our group [5]. This makes it such that further research is needed to deepen the understanding of this phenomenon.

Brief Overview of Experimental Context

In order to understand the concepts and reasoning in the remaining parts of this thesis, we need to discuss the experimental context on which this thesis is based. Although my collegue and I have both worked on creat-ing the setup, this thesis mainly focuses on the theoretical aspects of the experiment, and thus we will give a brief—yet sufficient—overview of the experiment. Readers interested in a more thorough discussion of the prac-tical aspects of the experiment are encouraged to read the thesis written by my colleague, Charlie Bender [6]. For a schematic overview of the ex-periment, the reader is referred to Fig. 1.1.

That being said, let us begin by specifying two useful abbreviations: • by pump laser we mean a 826.2 nm,∼_{2 ps pulsed laser.}

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3

• by crystal we mean a periodically poled potassium titanyl phosphate (PPKTP) crystal, cut out for degenerate collinear type-I SPDC. The main mechanism driving our experiment is spontaneous paramet-ric down conversion. This is a non-linear optical effect, which we will discuss in extensive detail in chapter 6. For now it suffices to state that this effect arises when a pump laser is focused in the crystal, which then causes the spontaneous creation of photon pairs which are entangled in their orbital angular momentum degree of freedom. This effect can also happen twice—or any number of times for that matter—in one instant of time, thereby creating a photon quadruplet. Said quadruplet then consists of two entangled photon pairs.

However, it is also possible to create quadruplets for which its con-stituent photons are all entangled. This happens when a created photon pair causes the stimulated emission of yet another photon pair.1 It is the latter quadruplets that we aim to observe.

The intensity of the quadruplets is very small in comparison to that of the pump laser, requiring us to filter out the latter. At some point in the setup, the photons will have to be focused on spatial light modulators, which only work for horizontally polarized light. But, the pump laser is horizontally polarized, and thus, since we study type-I SPDC, the created photons are vertically polarized, requiring us to use a λ/2 plate to rotate the polarization of the photons by an angle of 45◦. We then use a horizon-tal polarizer to rotate the photons the remaining 45◦ needed for them to become horizontally polarized, albeit at the cost of the intensity.

In order to detect the quadruplets, we need a way to isolate the indi-vidual photons. This is accomplished by the use of three beam splitters, to probabilistically distribute the photons to four detectors.

The single photons are then sent onto spatial light modulators, which are configured to convert a certain spatial mode into the fundamental Gaussian mode, which can be detected by sending them into single-transverse-mode optical fibers connected to single-photon detectors. For an extensive discussion on transverse modes and the measurement process, the reader is referred to chapter 5 and 7, respectively.

Finally, there are three lenses placed throughout the setup, the first is needed to focus the laser pump into the crystal, while the second and third are used to image the photons onto the spatial light modulators.

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Chapter

2

Preliminaries

Throughout this thesis we will discuss a plethora of subjects, most of which require a basic understanding of certain mathematical concepts. To equip the reader with the needed background, we will use this chapter to lay the foundations needed to understand the remaining parts of this thesis. Still, there are some concepts that the reader is assumed to be familiar with, such as bachelor’s level linear algebra and calculus. Other than that, we will define and derive most of the tools that we use during the course of the thesis, if these have not already been defined in this chapter. In the rare occasion that a concept is used without a thorough explanation or deriva-tion, the reader will be referred to literature which can be used for further reading.

We will start this chapter by introducing a concept which is a funda-mental building block of all else that is to come, namely the set. As it is an elementary construct, its definition is not as clear or restrictive as that of most derived constructs. We will therefore cite the definition given by one of the founders of set theory.

“A set is a gathering together into a whole of definite, distinct objects of our perception or of our thought—which are called elements of the set.”

— Georg Cantor There are some basic operations defined with respect to sets:

1. The unification of two sets S∪ A is again itself a set, which contains the combined elements of the initial sets.

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2. The intersection of two sets S∩A is a set which contains the elements which all elements which lie in both initial sets.

3. The relative complement of a set with respect to another set S\ A, is the set containing all elements of the first set minus the elements of the second set.

4. The symmetric difference of two sets S4A is the set of all elements which lie exclusively in one of the two initial sets.

5. The Cartesian product of two sets is a set containing all pairwise com-binations of elements from the original sets. Which is to say that the Cartesian product of S and A, denoted as S×A, is the set of all pairs

(s, a)for s and a elements of S and A, respectively.

We denote the set without any elements, or rather the empty set, as∅. The cardinality of a set is the number of elements it contains, so the cardinality of the empty set is #∅ =0, whereas the cardinality of the natural numbers and of the integers is #N = #Z = ℵ0. In fact, all sets that are countably

infinite have cardinalityℵ0. One the other hand, the set of real numbers if

not countable, and it has cardinality #R=2ℵ0_.

Note that throughout this thesis we will use the following notation: 1. We say that two sets are equal, S = A, if both contain the same

ele-ments.

2. We say that S is a subset of A, if A contains all the elements of S. We denote this as S⊆ A. In the case that we are certain that A contains more elements which are not part of S, we say that S is a perfect subset of A, denoted as S⊂ A. Conversely, we say that in these cases A is a superset, or perfect superset of S, respectively.

3. If s is an element of S, we write s ∈ S, whereas we write s /∈ S if s is not an element of S.

Finally, the power set of a set S is the set of all subsets of S. For example, suppose we have a set S= {1, 2, 3}, its power set then is

P (S) = {∅,{1},{2},{3},{1, 2},{1, 3},{2, 3}, S}.

2.1 Groups

Now that we know what a set is, we have all the tools needed to define groups.

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2.2 Hilbert Space 7

Definition 2.1.1. Let G be a set, and let◦be an operation. G is a group under◦

if

1. for all g, h∈ G⇐⇒ g◦h∈ G.

2. there exists an identity element e ∈ G such that e◦g = g◦e = g for all g ∈ G.

3. for all g ∈ G there exists an inverse g−1 ∈ G such that g◦g−1 = g−1◦

g =e.

4. for all g, h, i ∈ G we have that(g◦h) ◦i= g◦ (h◦i)

There is one concept related to groups which is of great importance to physics, namely the Lie bracket.

Definition 2.1.2. Let G be a group, and let ◦ : G×G → G be a unitary oper-ation. For h, g ∈ G we write h◦g =hg. The Lie bracket is a binary operation defined by

[·,·] : G×G→G,

(h, g) 7→hg−gh.

In physics it is customary to say that[a, b]is the commutator of a and b. If

[a, b] =0 we say that a commutes with b, and vice versa. Another construct that is of great importance in quantum physics, namely the space in which all quantum states live, is the Hilbert space.

2.2 Hilbert Space

LetH be a (complex) vector space equipped with a scalar product

h·|·i: H ×H →C

possessing the following properties: 1. ha|bi = hb|ai,

2. ha|λb1+µb2i = λha|b1i +µha|b2i,

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It is customary in physics to consider the constituents of this scalar prod-uct to be objects. This notation was first introduced by Paul Dirac, and it is called the bra-ket notation. The bra stands for the left side of the product, namelyh·|, which is another way to represent a complex conjugated vec-tor, and the right side of the product|·iis called the ket, which represents a regular vector.

Definition 2.2.1. Ametric d :H ×H →R is an image which has the

follow-ing properties:

1. d(a, b) ≥0, where d(a, b) = 0⇐⇒ a=b, 2. d(a, b) =d(b, a),

3. d(a, b) ≤d(a, c) +d(c, b).

Corollary 2.2.2. Let|| · || : H ×H → R be the norm induced by the scalar

product

||a−b|| =

q

ha−b|a−bi. Then, the norm is a metric.

Proof. In order to prove that the norm || · || is a metric, we have to show that it has the properties as described in definition 2.2.1

1. ||a−b|| ≥ 0, furthermore, ||a−b|| = 0 ⇐⇒ a = b, both of which follow from the definition of the scalar product.

2. ||a−b|| = || − (b−a)|| = | −1|||b−a|| = ||b−a||,

3. Since ||a+b||2 _{= ||}_a_||2_{+ ||}_b_||2_{+ h}_a_|_b_{i + h}_b_|_a_{i ≤ ||}_a_||2_{+ ||}_b_||2₊

2||a||||b|| = (||a|| + ||b||)2, it follows that

||a−b|| = ||a−c+c−b|| ≤ ||a−c|| + ||c−b||.

This proves our claim.

Definition 2.2.3. Ametric space is a space H equipped with a metric d. We denote this pair as(H , d).

Definition 2.2.4. Let (H , d) be a metric space. A sequence (an)∞_n=1 ∈ H

is called a Cauchy sequence if, for each e > 0, there is N > 0 such that d(an, am) < e for all n, m≥ N.

Definition 2.2.5. A metric space (H , d) is called complete if every Cauchy sequence inH has a limit in H .

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2.3 Hilbert-Schmidt Space 9

Definition 2.2.6. A vector spaceH is called a Hilbert space if the metric space

(H ,|| · ||)is complete.

Intuitively one can say that a metric space is complete if there are no points missing from it or its boundary. One might then wonder why it is necessary for a Hilbert space to be complete. But, it is precisely this requirement which ensures that states which lie in this space, and the op-erators which act on them, have a linear time evolution. If it were not for the completeness of the Hilbert space, there would be operators for which their action would not necessarily lie in the Hilbert space, and therefore would not be linear. Thus, it is the completeness of the Hilbert space which enables one of the fundamental assumptions of quantum mechanics, and thereby provides the mathematical framework for one of the most effec-tive models of reality thus far.

2.3 Hilbert-Schmidt Space

Definition 2.3.1. Let V, W be vector spaces over the same field K, with norms

|| · ||V,|| · ||W respectively, and let A : V →W be a linear operator. We say that

A is bounded if ∃c>0 such that||Ax||W ≤c||x||V ∀x ∈V.

Definition 2.3.2. Let H be a Hilbert space, and let A ∈ A , where A is the

vector space of bounded operators onH . The Hilbert-Schmidt norm|| · ||_HSis a map defined by

|| · ||_HS: A →R,

A 7→√Tr A†_A.

We say that a bounded operator A on a Hilbert space H is a Hilbert-Schmidt operator if it has a finite Hilbert-Hilbert-Schmidt norm.

Definition 2.3.3. Let A be the set of Hilbert-Schmidt operators on a Hilbert spaceH , equipped with a scalar product defined by

h·,·i: A ×A →R, (A, B) 7→Tr A†B.

Then, the scalar product is an inner product, and thus,A is a vector space. Proof. In order to prove that A is a vector space, we need to prove that

h·,·iis an inner product. Recall that a scalar product is an inner product if it is (1) positive-definite, (2) conjugate symmetric, and (3) linear in its second argument. Let A, B ∈A , with elements aij, bij respectively.

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1. positive-definite: hA, Ai =Tr A†A=

∑

i,j a∗_jiaji =

∑

i,j |aji|2≥0, hA, Ai =0⇔ aij =0 ∀i, j⇔ A=0. 2. conjugate symmetry: hA, Bi =Tr A†B=Tr(A†B)T =Tr BTA∗ =Tr(B†A)∗ =Tr B†_A_{= h}_{B, A}_i_,

where Tr(B†A)∗ =Tr B†_{A follows from the fact that}₍_a₊_b₎∗ ₌ _a∗₊ b∗ ∀a, b∈ C. 3. linearity: hA, λB+µB0i = Tr A†(λB+µB0) =Tr λA†B+Tr µA†B0 =λTr A†B+µTr A†B0 =λhA, Bi +µhA, B0i

∴ h·,·iis indeed an inner product, which proves our original claim. Corollary 2.3.4. LetA be the the vector space of Hilbert-Schmidt operators on a Hilbert spaceH . If we equip A with the Hilbert-Schmidt norm induced metric, then the metric space(A ,|| · ||_HS)is a Hilbert space.

Proof. In order to prove that (A ,|| · ||_HS) is indeed a Hilbert space, we need to prove that(A ,|| · ||HS)is complete. Let us start by noting that, for

an orthonormal basis{|ii}ofH , we have that

||A||2_HS =Tr A†A=

∑

i

1. v1⊗w+v2⊗w= (v1+v2) ⊗w

2. v⊗w1+v⊗w2=v⊗ (w1+w2)

3. λ(v⊗w) =λv⊗w=v⊗λw

The pairs(v, w) ∈V⊗W are denoted as v⊗w.

As a corollary, the tensor product of two vectors v∈V and w ∈W has elements(v⊗w)ij =viwj.

Remark. Note that at first glance thetensor product may seem trivially similar— or equivalent—to the Cartesian product. However, this is not the case. Firstly, the Cartesian product is defined for any arbitrary space, whereas the tensor prod-uct is defined only for vector spaces. Secondly, the Cartesian prodprod-uct of two spaces V, W is defined as V×W = {(v, w) : v ∈ V and w ∈ W}, which inherits the basis elements from its constituent spaces, namely

B(V×W) = {(v, 0),(0, w) ∈V×W : v∈ B(V)and w∈B(W)}, while the basis of the tensor product of two vector spaces is defined as

B(V⊗W) = {(v, w) ∈V⊗W : v∈B(V)and w ∈B(W)}.

As a corollary, we have that dim(V×W) =dim(V) +dim(W)and dim(V⊗

W) =dim(V) ·dim(W).

Definition 2.4.2. Let V, W be vector spaces over the field K, and let{vi},{wj}

be bases of V, W respectively. The direct sum of V, W is a vector space V×W over the same field K spanned by the pairs {(v_i, 0),(0, wj)}, and is denoted as

V⊕W. The direct sum is a bilinear map

⊕: V×W →V⊕W,

(v, w) 7→v⊕w, which has the following properties

1. (v1, w1) + (v2, w2) = (v1+v2, w1+w2)

2. c(v, w) = (cv, cw)

If V∩W =∅, we denote the pairs(v, w) ∈ V⊕W as v⊕w.

Now that we have a basic understanding of the tensor product and direct sum, we can proceed to give a definition of the Fock space.

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2.4 Fock Space 13

Definition 2.4.3. Let H be a single-particle Hilbert space. The Fock space Fν(H )is the direct sum of tensor products of copies of the Hilbert spaceH

Fν(H ) = ∞ M n=0 SνH ⊗n_,

where Sν is the operator which symmetrizes (ν = +) or antisymmetrizes

(ν = −) a tensor for bosonic and fermionic particles, respectively. Since photons are bosons, we have that ν = +. For our purposes it is suffices to have the effect of Sν be accounted for implicitly, whence we neglect the

term leading us to write

F+(H ) = ∞ M n=0 H ⊗n . In general, a state inF+(H )is written

|Ψi =

∑

n

|Ψni = a0|0i ⊕a1|ψ1i ⊕

∑

i,j

aij|ψ2i, ψ2ji ⊕. . .

where the a are complex coefficients,|0iis called the vacuum state,|ψ1i ∈

H is a single particle state, and|ψ2i, ψ2ji = (|ψ2ii ⊗ |ψ2ji + |ψ2ji ⊗ |ψ2ii)/2

is a bipartite state and so on. For two states|Ψi,|Φi ∈ F+(H )the inner product onF+(H )is defined as hΨ|Φi ≡

∑

n hΨn|Φni = a0∗b0+a∗1b1hψ1|φ1i +

∑

ijkl a∗_ijbklhψ2i|φ2kihψ2j|φ2li +. . . (2.3) When the underlying Hilbert space and particle is clear we simply denote the Fock space asF . It turns out that the Fock space is itself also a Hilbert space, for which the proof is beyond the scope of this thesis.

We now have the needed background to start our journey. When at any point the reader is unsure of his or her knowledge on a certain topic, the reader can return to this chapter to freshen up on the subject at hand.

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Chapter

3

Quantum Optics

A natural way to continue our journey is by noting that in quantum theory— and thereby in nature—it is the case that a particle cannot have any arbi-trary real number as its energy. Rather, energy exists in quanta, which is to say that the energy is described by a real number line that is partitioned into infinitely many discrete values. Since the possible values that energy can take on are discrete, they are countable, and we can label each energy by a natural number n. We denote the nth energy value as En, which is

an eigenvalue of the Hamiltonian for the eigenvector |ni. This statement has a plethora of meanings, the simplest of which relating to the harmonic oscillator, which has energy eigenvalues given by

En =¯hω n+1 2 .

As photons represent quasi-monochromatic excitations of the electromag-netic field, they are in fact quantum harmonic oscillators.1

3.1 Photon-Number States

In this context, we say that |niis a Fock state, which is a representation of the number of photons in a certain spatial mode, possibly with a complex polarization structure. For instance, a single photon is written |1i. It is thereby the quantum of the electromagnetic field. The reader familiar with the teachings of quantum mechanics may recall that it is possible to ’raise’ 1_{In principle, frequency is another degree of freedom, so not all photons need to} be quasi-monochromatic. Though, in our discussion we will only consider quasi-monochromatic photons.

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or ’lower’ the eigenstate onto the next or previous energy eigenstate, re-spectively. The operators which accomplish this task are the creation and annihilation operators, which we will define in this section, and we will de-rive some of their properties. These operators are of great importance to describe the process of spontaneous parametric down conversion, which will be explained in chapter 6. But for our current purposes it is sufficient to consider a more general context than that of chapter 6. The creation and annihilation operators therefore create and annihilate a quantum of en-ergy, respectively. As we stated at the start of this section, a photon is the excitation of the electromagnetic field, which carries the energy of a har-monic oscillator of frequency ω [8]. In this framework, one can see that the creation operator creates a photon, and thus raises the photon number by 1, whereas the annihilation operator annihilates a photon, and thereby lowers the photon number by 1. This can be summarized by the following relations

ˆa†|ni = √n+1|n+1i, (3.1)

ˆa|ni = √n|n−1i. (3.2)

When one is dealing with multiple photons which are not in the same spa-tial mode, it is customary to label the creation and annihilation operators to specify the spatial mode they operate on. The operators are then writ-ten ˆaks, and ˆa†ks, where k is the wave vector and s represents either one of

two independent polarization directions. The commutation relations for the creation and annihilation operators are given

[ˆaks, ˆak0s0] =0, (3.3)

[ˆa†_ks, ˆa†_k0_s0] =0, (3.4)

[ˆaks, ˆa†k0s0] = δkk0_δ_ss0. (3.5)

Since this relation vanishes unless both k = k0 and s = s0, one may combine both labels into one number which uniquely denotes the spatial mode. Considering this, we have that kisi → i, and the commutator of

Eq[3.5] becomes

[ˆai, ˆa†j] =δij. (3.6)

Finally, we must keep in mind that while the nthenergy eigenstate of a har-monic oscillator represents a single particle, the Fock state|ni represents n distinct photons in a certain spatial mode. Although this may seem like a trivial distinction at the moment, it plays a crucial part in the interpreta-tion of photon states. This can be understood by considering the follow-ing. Suppose that we have a single photon|1i ∈ H , and two other

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3.1 Photon-Number States 17

represent m and n photons, they are elements ofH ⊗m andH ⊗n, respec-tively. Therefore, in order to correctly describe photon number states, we need to use the Fock space.

We now have the tools to describe photon number states, which we will use in chapter 6 to describe entangled four-photon states. But, we still need a way to describe our laser in quantum optical terms. Since, in gen-eral, conventional light is not in a Fock state, we introduce the notion of a coherent state.

3.1.1 Coherent States

Coherent states are needed to give a quantum mechanical description of macroscopic light. But, what exactly do we mean by macroscopic light? One way to look at it, is by noting that macroscopic light consists of a macroscopic number of photon number states. In fact, one should not be able to notice the annihilation or creation of a single photon. As a corollary, one should also not be able to notice the annihilation or creation of two photons, or any number of photons for that matter.2 Therefore, we are interested in finding eigenstates of the creation and annihilation operators. From our previous comments, it follows that we require the eigenstates to be of the form3 |αi = ∞ M n=0 cn|ni,

Suppose now that the eigenstates|αi of the creation and annihilation

op-erators satisfy the following eigenvalue equations ˆa|αi = α|αi,

hα|ˆa† =α∗hα|.

From the first eigenvalue equation we derive ˆa|αi = ∞ M n=1 cn √ n|n−1i = ∞ M n=0 αcn|ni,

2_{While there might be a point at which coherent light would be reduced into a finite} sum of Fock states via the creation and annihilation of its constituent photons, this has never been observed, and thus, we assume that coherent light consists of infinitely many Fock states.

3_{One might notice the resemblance to the definition of e}x_{, but instead of invariance} under ∂x, we have invariance under ˆa and ˆa†.

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which after equating the components of the sums leads to cn

√

n =αcn−1.

Therefore, we have that

|αi = c0 ∞ M n=0 αn √ n!|ni.

Furthermore, as with all state vectors, we require |αi to be normalized,

which leads us to find

1= hα|αi = |c0|2 ∞ M n=0 α2n n! = |c0| 2_eα2_, and therefore c0 =e− α2

2 . Finally, the coherent state is written

|αi =e− α2 2 ∞ M n=0 αn √ n!|ni. (3.7)

Now that we have derived the coherent state, we would like to know which space it resides in. Since it contains elements from infinitely many different Hilbert spaces, it certainly does not reside in either one of them exclusively. Instead, it is an element of the Fock spaceF .

We would now like to remind the reader of something that we had stated at the start of this section. Namely, that photons are in a certain spa-tial mode. The spaspa-tial mode consists of the polarization and wave vector of the photon. The polarization of the photon is related to the orientation of the electric field of the photon with respect to its direction of propagation, and it is the quantum optical analogue to spin. We now consider parax-ial light fields propagating along a particular central wave vector, so that we can consider only a two-dimensional polarization space perpendicular to the direction of propagation. The state space related to polarization is isomorphic toC2, and can be categorized as follows:

1. Linear polarization: for which we have horizontal |Hi, and vertical

|Vipolarization.

2. Diagonal polarization: for which we have

|Di = √1

2(|Hi + |Vi),

|Ei = √1

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3. Circular polarization: which consists of right- and left-handed circular polarization.

|Ri = √1

2(|Hi +i|Vi),

|Li = √1

2(|Hi −i|Vi).

One may keep in mind the form of these three categories, as we will make an analogy to in later on in the thesis. Besides this, we will not discuss polarization to great extent. The other component of the spatial mode, the wave vector, is related to the momentum of the photon. As the orbital angular momentum of photons is of great importance to us, we dedicate the next section to discuss this property.

3.2 Angular Momentum of Light

Readers that are familiar with the teachings of classical electrodynamics might recall that electromagnetic fields carry energy density

u= 1 2 e0E2+ 1 µ0 B2 , and momentum density

π =e0E×B.

As a corollary, it is also the case that electromagnetic fields carry angular momentum density

jjj=r×π =e0[r× (E×B)]. (3.8)

Since there exist local conservation laws for both the energy and mentum, one would suspect that such a law also exists for angular mo-mentum. In fact, if one were to define a angular momentum flux density

Mli =εijkxjTkl,

where Tkl are the components of the electromagnetic stress tensor, and εijk

is the Levi-Civita pseudo tensor, one would find the continuity equation for angular momentum

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As one can see from Eq. 3.8, all light beams that have non-zero momentum perpendicular to the axis of rotation have angular momentum. Though, in our case we are interested beams that have non-zero angular momentum in its direction of propagation. This can arise in one of two ways. Either the beam has a spin angular momentum, which is related to the rotation of the field for circularly polarized light, or the wavefront of the beam is helical, in which case we relate to it the orbital angular momentum. Since we are only interested in orbital angular momentum, we will omit entirely the discussion on spin angular momentum.

3.2.1 Orbital Angular Momentum

Orbital angular momentum arises from the azimuthal dependence of the distribution of the field in a plane perpendicular to the direction of prop-agation. Therefore, any electromagnetic wave has a well-defined orbital angular momentum, classically speaking. Though, it is not the case that every electromagnetic wave has a well-defined orbital angular momen-tum, speaking in terms of quantum theory.

But what is it that we mean when we say that a electromagnetic wave has a well-defined orbital angular momentum in a quantum mechanical sense? In order for a property to be well-defined for quantum mechanical use, it should have a quantization so that one can distinguish a countable (or countably infinite) set of values which make up the parameter space of said property. If this is not possible for a given property, then it is not possible to define a Hilbert space related to this property, and thus, it is not possible to give a rigorous quantum description of it. Although all electromagnetic waves have a well-defined orbital angular momentum, for certain types of waves there may arise problems when in the frame of single photons, which is to say locally.

When we constrain ourselves to the case of paraxial waves—waves which have a negligible dependence on their direction of propagation—we see that one can speak definitively about the orbital angular momentum of the wave in the direction parallel to the propagation axis.

Having this in the back of our minds, it is important for us to be clear about which types of waves we consider and which we do not. For our purposes, we require the wave to be transverse, which is to say that the wave has negligible dependence on its direction of propagation, except the

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harmonic oscillation at frequency ω. In chapter 5 we will further discuss this topic, but for current purposes this is not needed.

Orbital angular momentum in paraxial waves is well-defined, and even has a natural quantization due the cyclical nature of the phase dependence of the field density of the wavefront. Considering that φ ∈ [−π, π),

conti-nuity at the boundaries requires that the electromagnetic wave ψ is cyclic. This can be summarized as

ψ(φ) = ψ(φ+2kπ), ∀φ∈ [−π, π).

Therefore, the wave has a φ-dependence of the form

ψ(φ) ∝ ei`φ. (3.9)

Since the exponential function forms a complete basis for the space of square integrable functions, it is possible to express any wave as a su-perposition with components of the form given in Eq. 3.9. Therefore, any waveΨ can be written as

Ψ(φ) = √1

2π `∈

∑

Z

A`ei`φ,

where A`is the complex amplitude of the component with orbital angular momentum`, which satisfy the normalization condition

∑

`∈Z

|A`|2=1.

Written in this form, it is easy to see that the orbital angular momentum is dual to the phase, in the sense that they are related via the Fourier trans-form A` = 1 √ 2π Z π −π Ψ(φ)e−i`φdφ,

which means that their relation to each other is similar to that of mo-mentum and position, although they are not completely analogous. The main difference arises when considering the values which the position and phase can take on. For instance, the position x can take on any real number as its value, whereas φ∈ [−π, π). The distinction then arises from the fact

that[−π, π)is not homeomorphic toR. Rather,[−π, π)is homeomorphic

to the unit circle, which gives rise to the cyclic nature of properties that de-pend on φ, and hence the discrete nature of the orbital angular momentum

`. This can be seen from the fact that the map exp(iφ)is a homeomorphism between[−π, π)and S1.

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With this knowledge, we can state that the orbital angular momentum Hilbert spaceL is spanned by the orthonormal basis{|`i}`∈Z, for which the elements satisfy the orthogonality relation

h`|`0i∝

Z _π

−π

e−i`φ_ei`0φ_dφ₌

δ``0.

It forms a complete basis of the Hilbert space L , which is analogous to the exponential functions forming a complete basis of the space of square integrable functions. It is therefore clear that the orbital angular momen-tum Hilbert space is in fact the space of square integrable functions over the range[−π, π).

Finally, the orbital angular momentum modes are eigenmodes of oper-ator ˆLz = −i¯h∂φthrough the eigenvalue equation

ˆLz|ì = −i¯h∂φ|ì = `¯h|ì,

where we have taken the z-axis as the direction of propagation. This con-vention will be used in the remainder of this thesis.

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Chapter

4

Quantum Information

In general, a quantum state is completely, and uniquely described by its wave function|ψi, which can be written as the sum of its constituents

|ψi =

∑

i

λi|ψii.

Still, there are quantum states which cannot be described by the above rep-resentation, but instead require a matrix representation. We say that these are mixed states, and we call states which can be described as above pure states. In order to correctly describe mixed states, we have to introduce the concept of a density operator.

4.1 Density Operators

LetA be the Hilbert-Schmidt space over the Hilbert space H . Now, sup-pose that we have a state which is represented by the wave function we defined above. We can then define the density operator as [8]:

Definition 4.1.1. Let |ψi = ∑iλi|ψii ∈ H be a wave function. The density

operator ˆρ∈ A is defined by

ˆρ= |ψihψ| =

∑

i

∑

j

λiλ∗j|ψiihψj|

Besides potentially being a representation of more states than the vector representation, the density operator has some other interesting properties. For instance, we have that Tr ˆρ =∑_i|λi|2 =1. As a corollary, it is

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Let A∈ A be an operator that satisfies the eigenvalue equation

where pi 6=0 for at least two i.

Just as for pure states, we have Tr ˆρ = 1, and in turn one calculates estimation values in the same manner as for pure states. Still, there are differences between pure and mixed states apart from the fact that the first can be represented by a state vector, while the latter cannot. For one, the coefficients pi represent the probability of the system being in state |Ψii.

Therefore, they represent classical probabilities. This is not to be confused with the coefficients of a pure state that is a superposition of wave vectors. Although the pure state|Ψ_ii is still a superposition of wave vectors, the state itself is known with 100 % certainty. Furthermore, maybe one of the most important differences between the two forms of states—a difference that is used to determine if a state is pure or mixed—is that for pure states one has ˆρ2 = ˆρ =⇒Tr ˆρ2 =Tr ˆρ=1, while for mixed states Tr ˆρ2<1.

In general, a density operator ˆρ has the following properties

ˆρ† = ˆρ, Tr ˆρ2≤1, ˆρ ≥0. (4.3) The first property states that all density operators are Hermitian, which is a direct consequence of their definition. The second property follows from

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the normalization condition, and the third means that the density matrix strictly has eigenvalues greater than or equal to zero. Density operators are an important tool in the description of quantum states and quantum entanglement, of which the latter will be discussed in the following sec-tion.

4.2 Quantum Entanglement

As was touched briefly in the introduction, we say that a quantum state of a collection of particles is quantum entangled if its constituent states cannot be described independent of each other. In this section we will extradite a more rigorous definition of this phenomenon, and introduce some nomen-clature used in the remaining parts of this thesis.

Definition 4.2.1. LetH ≡H _A⊗H _Bbe a bipartite Hilbert space. A quantum state |ψiAB∈ H is said to be entangled if @|ψiA ∈ H A and @|ψiB ∈ H B

such that

|ψiAB = |ψiA⊗ |ψiB

In contrary, when a state|ψiAB ∈H can be written in the above

man-ner, we say that it is separable. By definition it follows that quantum en-tanglement is basis independent. Since this is a strict requirement, one is required to perform measurements in multiple bases before one can defini-tively say that a state is entangled.

In other words, a quantum state is said to be entangled if the wave function of the state cannot be determined by merely looking at the con-stituents Hilbert spaces, and thus, the state contains information that can-not be attained by combining states from the constituent Hilbert spaces in a classical manner.

This is most easily explained by an example.

Let H A, H B be Hilbert spaces induced by the polarization. It is the

case that H A = span{|HiA,|ViA}. The same holds for H B, albeit we

now use B as a subscript instead of A. We denote|Hi_A⊗ |Vi_B as|H, Vi. Now, consider the following entangled state

|ψi = |H, Hi + |V, Vi.

Suppose that a researcher performs a measurement on system A, and finds it to be in|HiA. By the mere act of measurement, the researcher has now

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collapsed the wave function, which now is written

|ψi → |AhH|ψi||H, Hi.

This means that if the researcher were to measure the state of B at this instant, he would find|HiBwith unit probability.

Let us now consider the non-entangled state

|ψi = (|Hi + |Vi)A⊗ (|Hi + |Vi)B,

and suppose that the researcher again finds A to be in the state|Hi_A, then the wave function becomes

|ψi → |AhH|ψi||HiA⊗ (|Hi + |Vi)B.

We see that the measurement of A has not affected the possible outcomes of our measurement of B.

But this is not the whole story. The attentive reader might have no-ticed that the definition of quantum entanglement given in definition 4.2.1 is not applicable to mixed states, since they cannot be written as regular state vectors. Instead, we require a more general definition for mixed state entanglement.

Definition 4.2.2. Let A ≡ A A⊗AB be a bipartite Hilbert-Schmidt space,

and let ˆρ(_Ai) ∈ A_A, ˆρ_B(i) ∈ A_B ∀i ∈ I. A mixed state ˆρAB ∈ A is said to be

entangled if it cannot be written as the sum of product states ˆρAB6=

∑

i∈I piˆρ (i) A ⊗ ˆρ (i) B .

On the contrary, if it is possible to write a state as a sum of product states, we say that it is separable.

Now that we know what entanglement is, how do we go about deter-mining if a certain state is entangled? Since a state vector representation is not always possible, we certainly need more general ways of determining if a state is entangled, other than just looking at its state vector. Moreover, although it is theoretically possible to represent a pure state by a state vec-tor, it is the physical reality that throws a spanner in the works.

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4.2.1 Entanglement Witnesses

Suppose that we have a system composed of two subsystems, which we will label A and B. For simplicity we will take as their Hilbert spaces H A = H B = C2. Now, suppose that our system is in a pure state

|ψiAB ∈ H ≡H A⊗H B defined by

|ψiAB=

1

√

2(|0iA⊗ |0iB+ |1iA⊗ |1iB), where span{|0i,|1i} =C2_{. We would like to determine if}_|

ψiABrepresents

an entangled state—which as a matter of fact we are confident that it is—but how would we go about showing this? It turns out that in our case this is a remarkably simple task. But, before we can proceed we need to define the partial trace.

Definition 4.2.3. Let V, W be finite-dimensional vector spaces over some field K, and let F(A) denote the space of linear operators on a space A. The partial trace over V is a mapping defined by

TrV : F(V⊗W) → F(W),

TrVB⊗C =C Tr B ∀B∈ F(V) ∀C ∈ F(W).

Now, recall that a pure state |ψiAB ∈ H is said to be entangled if it

cannot be written as the tensor product of elements ofH AandH B. As a

consequence, when one performs a measurement on either one of the sub-systems, the state collapses into a form which does not contain all of the in-formation about the other subsystem that was originally in|ψiAB. Another

way to visualize this concept, is by saying that the collapsed wave vector is equal to the wave vector that does contain all information about the sub-system minus a wave vector containing the missing information. Therefore, by noting that these hypothetical wave vectors are pure, it follows that the collapsed state must be mixed. But, how can we determine if the collapsed state is indeed a mixed state? This is where the partial trace we had defined earlier comes into play. This can be best explained by an example.

Let |φiAB = ∑i∑jλij|iiA⊗ |jiB. Now, suppose that, in order to

de-termine the observable ˆOA, we were to perform a measurement on

sys-tem A. From Eq. 4.1 it can be seen that the expectation value in terms of ˆρAB = |φiABhφ|is written

hOˆAi =Tr ˆρABOˆA=

∑

i

∑

j

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Now, since ˆOA only acts on system A, we can write the estimation value

ashOˆAi =Tr ˆρAOˆA. Which in turn leads us to find

ˆρA=

∑

j

hj|ˆρAB|jiB ≡TrB ˆρAB. (4.4)

Thus, it turns out that we can find the density operator of a subsystem by taking the partial trace of the density operator of the composite system with respect to the other subsystem. Now, in order to find out if |ψiAB

we defined earlier on is in fact entangled, recall that Tr ˆρ2 < 1 if the state is mixed. Considering that the collapsed form of an entangled pure state is always a mixed state, all that is left is to find the trace of the squared density operator.

First, note that the density operator of|ψiABis written as

ˆρAB = 1

2(|0iAh0| ⊗ |0iBh0| + |1iAh1| ⊗ |1iBh1|

+ |1i_Ah0| ⊗ |1i_Bh0| + |0i_Ah1| ⊗ |0i_Bh1|), and thus, the density operator of system A is

ˆρA =TrB ˆρAB =

1

2(|0iAh0| + |1iAh1|). A simple calculation then shows that

Tr ˆρ2_A = 1

2 <1,

which leads us to conclude that, since ˆρA represents a mixed state, |ψiAB

is entangled.

But what about composite states that are mixed? How can we deter-mine if these states are entangled? Well, since mixed states have no state vector representation, it is not possible to use the above method to deduce if the state is entangled. Therefore, we need to find other methods to test entanglement in mixed states. It turns out that this is not a trivial task. Mixed State Entanglement Witnesses

Consider a bipartite Hilbert-Schmidt space A ≡ AA⊗AB on a discrete

finite-dimensional Hilbert spaceH ≡H _A⊗H _Bof dimension d=dAdB.

Now, letE denote the subspace of A containing all entangled states, and letS =A \E be the subspace of all separable states.

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There exists a criterion [9], namely the entanglement witness criterion, which states that a state ˆρ∈ A is entangled if and only if for some

Hermi-tian operator A the following holds

hˆρ, Ai =Tr ˆρA <0, (4.5)

hˆσ, Ai =Tr ˆσA ≥0 ∀ˆσ ∈ S . (4.6) In that case, we say that A is an entanglement witness. The first condition (Eq. 4.5) is called the entanglement condition, while the second (Eq. 4.6) is called the separability condition. Moreover, if there exists a state ˜σ ∈ S for

whichh˜σ, Ai =0, then A is called an optimal entanglement witness. In order for us to proceed, we need to define some concepts.

Definition 4.2.4. Let V be a vector space over some body K, and let S ⊂V. We say that S is convex if tv+ (1−t)w∈ V ∀v, w∈ S and∀t∈ (0, 1).

Corollary 4.2.5. The subspaceS of A is convex. Proof. Recall that all ˆσ∈ S can be written

ˆσ=

∑

i piˆρ (i) A ⊗ ˆρ (i) B , where ˆρ(_Ai) ∈ A Aand ˆρ (i)

B ∈ AB. Now, let ˆσ1, ˆσ2 ∈ S , t ∈ (0, 1), and I, J

index sets. We calculate t ˆσ1+ (1−t)ˆσ2=t

∑

i∈I piˆρ (i) A ⊗ ˆρ (i) B + (1−t)

∑

j∈J pjˆρ (j) A ⊗ ˆρ (j) B =

∑

i∈I∆J p0_iˆρ(_Ai)⊗ ˆρ(_Bi), where p0_i = ( (1+t)pi, if i∈ J\I (1−t)pi, if i∈ I\ J

∴ t ˆσ1+ (1−t)ˆσ2 ∈ S , which proves our claim.

In order to fully appreciate the consequences of this result, consider the following theorem.

Theorem 4.1(Hahn-Banach separation theorem). Let A and B be two dis-joint nonempty convex sets. If A is open, then there exists a nonzero vector v and real number c such that

ha, vi > c andhb, vi ≤c

for all a ∈ A and b∈ B. If both A and B are open, this reduces to

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Figure 4.1:This image is a two-dimensional representation of the topology ofA . One sees that the red line separatesS from the largest convex subspace of E .

As a consequence of the Hahn-Banach separation theorem, and since S is convex, we can draw a ’line’ that separatesS from the largest convex subspace of E . This principle is illustrated in Fig[4.1], and can be made rigorous by considering the operator A ∈A defined by

A≡ ˆσ0− ˜ρ− hˆσ0, ˆσ0− ˜ρi1,

where ˜ρ ∈ E and ˆσ0 ∈ S . It can be shown that A defines a

hyper-plane [9]—the red ’line’ we discussed earlier—including the state ˆσ0that

is tangent toS , and has the following properties

h˜ρ, Ai = Tr ˜ρA<0, (4.7)

hˆσ, Ai =Tr ˆσA ≥0 ∀ˆσ ∈ S , (4.8) where the equality in (4.8) only holds for ˆσ0. Clearly A is Hermitian, and

thus it is an entanglement witness for the state ˜ρ. Moreover, since the hy-perplane is tangent toS , the operator A is an optimal entanglement witness. Entanglement witnesses constructed in the above manner are called ge-ometric entanglement witnesses. It is precisely the Hahn-Banach separation theorem that allows for the existence of such witnesses, and for that matter all witnesses [9].

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Chapter

5

Transverse Modes

We now discuss the transverse modes that span the orbital angular mo-mentum Hilbert space. Therefore, we are interested in finding solutions to the paraxial wave equation. Before we can go into further detail, we will have to specify what we mean by paraxial, and why we are interested in this regime. First, let us consider the general form of the wave equation in vacuum

∇2ψ− 1

c2∂ 2

tψ=0 (5.1)

It may not seem obvious at first glance, but with little manipulation it is possible to write this in the form of the Laplace equation, ∇2

ψ = 0. To

see this, consider that we can define a four-vector differential operator into which the time derivative is absorbed. Namely, the d’Alembert operator which is defined by = 1_c∂t, ∂x, ∂y, ∂z

. Using this, the wave equation can now be written

2ψ=0, (5.2)

which is of the form of the Laplace equation with the d’Alembert operator taking the place of the Laplace operator.1

Since in our experiment we consider quasi-monochromatic light, we are interested in solutions of the form ψ(r, t) = A(r)e−iωt, where A(r) is known as the complex amplitude of the wave, and ω is the angular frequency of the wave. From now on we will omit the word angular and will simply 1_{One might wonder if it is not the Laplace equation that is a special case of the} Helmholtz equation, but rather that the Helmholtz equation is a special case of the four-vector Laplace equation. Specifically, when ∂t→const. one recovers the Helmholtz equa-tion.

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call ω the frequency. Moreover, since laser light is of the form of a Gaus-sian wavefront, we have that all waves that are of our interest have a small dependence on their direction of propagation. It is customary to take the z-axis as the direction of propagation. Therefore, we are only interested in solutions which are of the approximate form of a plane wave in the z⊥ plane. Putting it more formally, we are interested in solutions of the form

ψ(r, t) = u(r)ei(kz−ωt), where k ≡ |k| = 2π_λ is the wave number. We have

now arrived at the paraxial regime. Within this regime we have that

∂z →ik, ∂t → −iω, 1

c∂t → −∂z.

Substituting our assumptions into the wave equation leads us to find

2ψ= ∇2⊥uei(kz−ωt)+∂2z(u)ei(kz−ωt) +2ik∂z(u)ei(kz−ωt)+u(∂2z− 1 c2∂ 2 t)ei(kz−ωt) = (∇2_⊥u+∂2_zu+2ik∂zu)ei(kz−ωt) =0. Therefore, ∇2_⊥u+∂2zu+2ik∂zu =0,

where∇⊥ = (∂x, ∂y)is the transverse Laplace operator. Now, since we have

assumed that our wave is almost invariant in its direction of propagation, we have that|∂2_zu| |2ik∂zu|. We therefore neglect the ∂2zu term, which

leads us to the paraxial wave equation

∇2_⊥u+2ik∂zu=0, (5.3)

or written in terms of the complex amplitude A(r) =u(r)e−ikz

∇2_⊥A+2ik∂zA+2k2A =0. (5.4)

By rearranging the terms one can see that this equation is of the form of the inhomogeneous Helmholtz equation∇2

ψ+a2ψ=h, namely

∇2_⊥A+2k2A= −2ik∂zA,

where the transverse Laplace operator plays the role of the ordinary Laplace operator.

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Since the paraxial wave equation is of the form of the inhomogeneous Helmholtz equation, we will take the homogeneous Helmholtz equation as our starting point

∇2ψ+a2ψ=0.

It suffices to find the separable solutions, from which all other solutions can be constructed. To see this, consider that we can rewrite the problem to be of the form Dψ =0, where D ≡ ∇2₊_a2_{is a differential operator. Since}

differential operators are in fact linear operators, we can use linear algebra to solve this problem. Considering this, we see that Dψ = 0 can simply be solved by finding the kernel of D (ker D). Now, let us turn our attention to the inhomogeneous case, which is to say Dψ = λψ. Suppose that ψ0 is a

particular solution to this equation, it is then easy to see that ψ0+ψis also

a solution∀ψ∈ker D. Moreover, since D is a linear operator, there exist d

linearly independent solutions to Dψ =λψ, where d =dim(im D). We can

then combine solutions from both families to construct any solution to the given differential equation.

In order to understand why this is relevant to our claim that we only need the separable solutions, note that by separation of variables one splits a differential equation of multiple variables into multiple differential equa-tions of one variable, one per variable to be precise. It is important to note that this is merely a restatement of the original differential equation, and thus they are mathematically equivalent. Therefore, since we already know how to find all solutions to the individual separated differential equations, we can combine them to find all solutions to the original differential equa-tion.

It is the case that paraxial wave equation admits separable solutions, al-beit for certain coordinate bases. If a coordinate basis allows for separable solutions, we call it a Helmholtz basis. For a discussion on Helmholtz bases, the reader is referred to appendix A. In the next section we will solve the paraxial wave equation for three of such bases.

5.1 Particular Solutions to the

Paraxial Wave Equation

In this section we will discuss particular sets of solutions to the paraxial wave equation. For reasons that will become clear at the end of this sec-tion, we will confine our discussion to three families of modes, namely:

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1. Hermite-Gauss, 2. Laguerre-Gauss, 3. Ince-Gauss.

As we will see later, each family of solutions arises when rewriting the paraxial wave equation into the form of some well known differential equation, for which the solutions are members of a family of orthogo-nal polynomials. To be particular, Hermite-Gauss modes involve Her-mite polynomials, Laguerre-Gauss modes involve generalized Laguerre polynomials, and Ince-Gauss modes involve Ince polynomials.2 As their names suggest, these mode families are all related to a Gaussian solution of the paraxial wave equation.

Therefore, we will start by solving the paraxial wave equation to find a solution of the form of a Gaussian wavefront

u(r) = A0eik

(x2+y2+z2) 2q(z) _eip(z)_,

where q(z)and p(z)are functions to be determined. On substitution into the paraxial wave equation we find

A0 k 2 q2(x 2₊_y2₎ dq dz −1 −2k dp dz − i q =0.

This equation will only be satisfied if dq dz =1, dp dz = i q.

Which results in the following forms of q(z)and p(z)

q(z) = q0+z, p(z) = i ln

q0+z

q0 ,

where we have assumed that p(0) = 0. In general, q(z) is a complex number, and therefore it is convenient to write it in the form

1 q(z) = 1 R(z) + iλ πw2(z),

2_{The derivations of the Hermite-Gauss and Laguerre-Gauss modes are based heavily} on the work found in [10], while the derivation of the Ince-Gauss modes is based on [11].

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where R and w are real valued functions, representing the curvature and radius of the wavefront, respectively. The width of the beam w0 in the

plane z=0 is called the beam waist. With this choice, we may write

eip(z) = −q0+z

q0

= 1

1+z/R0+iλz/πw2₀

. We now find that

1 R(z) =

<(q0) +z

|q0|2+2z<(q0) +z2.

By letting R0 = ∞, which corresponds to a flat wavefront, we find that

q0= −iz0, where z0≡πw2₀/λ. We may then write

R(z) = z+z 2 0 z, (5.5) w(z) = w0 q 1+z2_/z2 0. (5.6)

The parameter z0 is known as the Rayleigh range and is the distance at

which the beam width is √2 times the beam waist. With its definition, and the specification of R0 =∞, we further find that

eip(z) = 1 1+iz/z0 = 1 1+z2_/z2 0 e−iΦ(z), where Φ(z) =arctan z z0 , is the Gouy shift. Our solution may then be written

u(r) = A0e−iΦ(z)   1 q 1+z2_/z2 0 eikzeik(x2+y2)/2R(z)  e−(x 2₊_y2₎_/w2 0_. _(5.7)

This solution is known as the fundamental Gaussian mode. As it is the base-line for all Gaussian derived beam modes, we are now able to proceed.

5.1.1 Hermite-Gauss Modes

We now look for other shape-invariant beams of the form Ψ(r) = f "√ 2x w(z) # g "√ 2y w(z) # u(r)eiΦ(z),

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where f and g are the propagation-dependent transverse profiles to be de-termined. The functionΦ(z) is a propagation dependent phase shift, as-sumed to be more general than the Gouy shift. Through substitution into the paraxial wave equations we find three independent partial differential equations ∂2_ξf −2ξ f +2m f =0, ∂2νg−2νg+2ng=0, ∂zΦ= m +n kw2₀ 2 1+z2_/z2 0 ,

where ξ ≡ √2x/w, ν ≡ √2y/w, and m and n are separation constants. The equation forΦ can be directly integrated, to find

Φ(z) = (m+n)arctan z z0

.

The equations for f and g are now both in the form of the Hermite equa-tion. In order for the functions to converge for large values of ξ and ν, m and n are constrained to integer values [10].

We find that there exist an infinite number of solutions to the paraxial wave equation of the form

Ψmn(r) = Hm "√ 2x w(z) # Hn "√ 2y w(z) # u(r)eiΦ(z),

where Hiare Hermite polynomials. In the plane z=0, they take on the form

Ψmn(x, y, 0) = Hm "√ 2x w0 # e−x2/w20H_n "√ 2y w0 # e−y2/w20.

The intensities |Ψ_mn(x, y, 0)|2 of several of these modes are illustrated in Fig. 5.1. Requiring the modes to be normalized leads us to find

Ψmn(x, y, 0) = CmnHm "√ 2x w0 # e−x2/w20H_n "√ 2y w0 # e−y2/w20, (5.8) where Cmn = r 2 π2 −m+n₂ 1 q n!m!w2₀ .

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Figure 5.1: The intensity distribution of some Hermite-Gauss modes. The tuples denote(m, n).

These modes now satisfy the orthogonality relation

Z Z

R2Ψmn(x, y, 0)Ψm0n0(x, y, 0)dS =δmm0δnn0.

Just as an arbitrary function may be expanded in terms of a Hermite series of functions Hm(x)e−x

2_/2

, an arbitrary paraxial wave in the plane z = 0 may be expanded in terms of functions of the form of Eq. 5.8.

Finally, as one would expect, since Hermite-Gauss modes are repre-sented using Cartesian coordinates, their intensity distributions possess the following symmetries:

1. mirror symmetry along the x and y axes,

2. in the case that n = m they are in addition also symmetric under rotations of multiples of 45◦around the z-axis.

5.1.2 Laguerre-Gauss Modes

In the previous section we derived the set of Hermite-Gauss modes by looking for Gaussian-like solutions to the paraxial wave equation in Carte-sian coordinates. We may also look for GausCarte-sian-like solutions in cylindri-cal coordinates, which are in fact the Laguerre-Gauss modes.

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In order to use cylindrical coordinates(ρ, φ, z), we make the apply the

following coordinate transformations

ρ = q x2₊_y2_, φ=arctany x, z =z.

We assume a solution of the form Λ(r) = F "√ 2ρ w(z) # G(φ)u(r)eiα(z).

Upon substitution into the paraxial wave, we find 1 F∂ 2 ζF+ 1 F ikw2 R −2 ζ∂ζF+ 1 ζ F∂ζF −ikw 2 R ζ F∂ζF+ 1 ζ2G∂ 2 φG−kw 2 ∂zα=0,

where ζ ≡ √2ρ/w. We now let the last term be equal to C, whence α(z)

may be written α(z) = C 2 arctan z z0 .

The second to last term of the paraxial equation we set equal to−`2, as G is of the form

G(φ) = A`ei`φ+B`e−i`φ, with` ∈Z. We may now write

d2F dζ2 + 1 ζ −2ζ dF dζ + C− ` 2 ζ2 F =0.

Now, if we make the coordinate transformation of y = ζ2, our equation

becomes of the form yd 2_F dy2 + (1−y) dF dy + 1 4 C− ` 2 y F =0.

The singularity at y=0 is regular, which suggests that there exists a Frobe-nius solution of the form

F(y) =

∞

∑

k=0

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Now, the indicial equation gives s= ±`/2. We choose the positive root as the negative root diverges at the origin. We define

H(y) =√yF(y).

Our differential equation then becomes yH00+ (` +1−y)H0+ C 4 − ` 2 H =0,

which is of the form of the associated Laguerre equation. Our result may now be written

Λ` p(r) = s 2p! πw2₀(` +p)! √ 2ρ w(z) !` L`p 2ρ2 w2₍_z₎ ei`φ_u₍_r₎_eiα(z)_, _(5.9)

where L`pare generalized Laguerre polynomials, and

α(z) = (2` +p)arctan z

z0.

The functions have been normalized such that

Z Z R2Λ ` p(ρ, φ, 0)Λ` 0 p0(ρ, φ, 0)dS =δ``0_δ_pp0.

The intensity of these beams in the plane z =0 can be seen in Fig. 5.2. Since Laguerre-Gauss modes are represented using cylindrical coordi-nates, their intensity distributions are rotationally symmetric, which makes them eigenmodes of the orbital angular momentum operator ˆLz = −i¯h∂φ

through the eigenvalue equation

ˆLzΛ`p = ¯h`Λ`p.

Since LG modes are more constrained than HG modes, they are harder to generate, and thus laser light mostly exists in HG modes [12].

5.1.3 Ince-Gauss Modes

We may also look for Gaussian-like solutions in elliptic coordinates, which leads us to the family of Ince-Gauss modes [13].

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Figure 5.2:The intensity distribution of some Laguerre-Gauss modes. The tuples denote(`, p).

In order to use elliptic coordinates (ξ, ν, z), we apply the coordinate

transformation

x = f(z)cosh ξ cos ν, y = f(z)sinh ξ sin ν, z =z,

where ξ ∈ [0,∞) and ν ∈ [0, 2π) are the radial and the angular elliptic variables, respectively. We now assume a more general solution of the form

Ξ(r) = E(ξ)N(ν)eiZ(z)u(r),

where E, N, and Z are real functions. This results in three differential equations d2E dξ2 −esinh 2ξ dE dξ − (a−me cosh 2ξ)E =0, d2N dν2 +esinh 2ν dN dν + (a−me cosh 2ν)N =0, −z 2₊_z2 R zR dZ dz =m,

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Figure 5.3:The intensity distribution of some even Ince-Gauss modes. The tuples denote(m, n), and ε is the ellipticity parameter.

where m and a are separation constants, and e = 2 f₀2/w2₀ is the ellipticity parameter. Note that the first two of the differential equations are in the form of the Ince equation, albeit the latter one has iξ instead of ν.

Its solutions are written Ξ e mn(r, e) = Cw0 w(z)C n m(iξ, e)Cmn(ν, e)e−r 2_/w2₍_z₎ ei[kz+2R(z)kr2 −Z(z)]_, _(5.10) Ξ o mn(r, e) = Sw0 w(z)S n m(iξ, e)Snm(ν, e)e−r 2_/w2₍_z₎ ei[kz+2R(z)kr2 −Z(z)]_, _(5.11)

where C and S are normalization constants, Z(z) = (m+1)arctan z/z0,

and Cnm and Smn are the nth order even and odd Ince polynomials,

respec-tively. The superindices e and o respectively refer to even and odd modes. The modes satisfy the orthogonality relation

Z Z R2 Ξ σ mnσΞ 0 m0n0dS =δσσ0δmm0δnn0.

Since the Ince-Gauss modes are represented using elliptic coordinates, the symmetries of their intensity distributions are not as straightforward as those of the other two mode families we have derived. For now it is sufficient to say that the Ince-Gauss modes possess the same symmetries

On Quadripartite Entanglement in High-Dimensional Hilbert Space

On Quadripartite Entanglement in

High-Dimensional Hilbert Space

On Quadripartite Entanglement in

High-Dimensional Hilbert Space

Andreas Lepidis

Abstract

Contents

Chapter

1

Introduction

Brief Overview of Experimental Context

Chapter

2

Preliminaries

2.1

Groups

2.2

Hilbert Space

2.3

Hilbert-Schmidt Space

∑

∑

∑

∑

∑

∑

2.4

Fock Space

∑

∑

∑

∑

Chapter

3

Quantum Optics

3.1

Photon-Number States

3.1.1

Coherent States

3.2

Angular Momentum of Light

3.2.1

Orbital Angular Momentum

∑

∑

Chapter

4

Quantum Information

∑

4.1

Density Operators

∑

∑

∑

∑

∑

∑

∑

∑

∑

4.2

Quantum Entanglement

∑

4.2.1

Entanglement Witnesses

∑

∑

∑

∑

∑

∑

∑

Chapter

5

Transverse Modes

5.1

Particular Solutions to the

Paraxial Wave Equation

5.1.1