Single photon interaction and entanglement

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Thesis

Reference

Single photon interaction and entanglement

BARBOSA DOS SANTOS GUERREIRO, Thiago

Abstract

Quantum mechanics not only revolutionized physics, but also provided an important tool allowing the development of unprecedented technology in the past century. Today, the quantum revolution continues to shape the development of new technologies through the fields of quantum information and communication. This thesis deals with the generation, manipulation and detection of quantum information using single photons. Creating single photons with precision, interacting individual photons and generating quantum entanglement are some of the questions we will deal with in detail.

BARBOSA DOS SANTOS GUERREIRO, Thiago. Single photon interaction and entanglement. Thèse de doctorat : Univ. Genève, 2016, no. Sc. 4921

URN : urn:nbn:ch:unige-844950

DOI : 10.13097/archive-ouverte/unige:84495

Available at:

http://archive-ouverte.unige.ch/unige:84495

Disclaimer: layout of this document may differ from the published version.

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Universit´ e de Gen` eve Groupe de Physique Appliqu´ ee

Facult´ e des Sciences Professeur Nicolas Gisin

Single photon interaction and entanglement

Th` ese

pr´ esent´ ee ´ a la Facult´ e des Sciences de l’Universit´ e de Gen` eve pour obtenir le grade de Docteur ` es Sciences, mention physique

par

Thiago Barbosa dos Santos Guerreiro du Br´ esil

Th` ese N

^◦

4921

GEN` EVE

2016

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I dedicate this thesis to Jos´e Antonio Guerreiro, among the first to discuss science with me.

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Abstract

Quantum mechanics not only revolutionized physics, but also provided an important tool allowing the development of unprecedented technology in the past century. Today, the quantum revolution continues to shape the development of new technologies through the fields of quantum information and communication.

This thesis deals with the generation, manipulation and detection of quantum information using single photons. Creating single photons with precision, interacting individual photons and generating quantum entanglement are some of the questions we will deal with in detail.

In the first part of the thesis, we describe the phenomenon of spontaneous parametric down-conversion and how it can provide a source of heralded single photons. Furthermore, we study how to engineer the emission of parametric down-conversion to perfectly match standard single-mode fibers.

In the second part, we discuss how to detect single photons and how to measure with precision the efficiency of a single photon detectors. This touches upon the modern field of quantum metrology, and serves as a nice example of how quantum physics can help improving measurement precision.

We then focus on the problem of making single photons interact. In particular, we demonstrate the first photon-photon interaction based on sum frequency generation.

In the last part, we are concerned with how to create and distribute quantum entanglement with delocalized single photons for use in various quantum information experiments. As we will show, this opens the way to an entire new framework for doing quantum information, where a single excitation of the electromagnetic field is used to generate and distribute entanglement to distant parties. We demonstrate the effectiveness of these methods via experiments to certify entanglement and verify the steering of quantum states.

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R´ esum´ e

La mécanique quantique a non seulement révolutionné la physique, mais elle a également fourni les outils nécessaires pour permettre le développement des nouvelles technologies qui ont marqué le siècle dernier. Aujourd’hui, la révolution quantique continue de fa¸conner le développement de nouvelles technologies dans le domaine de l’information et de la communication quantique.

Cette thèse porte sur la création, la manipulation et la détection de l’information quantique en utilisant des photons uniques. La génération de photons uniques avec des propriétés bien spécifique, l’observation d’interaction entre photons individuels et la préparation d’états intriquées sont quelques- unes des questions que nous traiterons en détail tout au long de ce manuscrit.

Dans la première partie de la thèse, je vais décrire le phénomène de la conversion paramétrique spontané et comment implémenter des sources de photons uniques annoncés avec. De plus, je présenterai comment faire de l’ingénierie des modes d’émission de ce processus non-linéaire afin d’adapter parfaitement leurs profils spatiaux à ce d’une fibre monomode standard.

Dans la deuxième partie, je discuterai des méthodes employées pour détecter des photons uniques et de comment caractériser avec précision leurs efficacités de détection. Ceci est en relation avec le nouveau champ de recherche sur la métrologie, et est un bel exemple de la fa¸con dont la physique quantique peut aider permettre d’améliorer la précision de la mesure.

Par la suite, je me concentrerai sur le probléme de faire interagir entre eux des photons uniques. En particulier, je présenterai la première observation expèrimentale d’un processus de somme de fréquence entre deux photons uniques issus de sources indépendantes.

Dans la dernière partie, je m’intèresserai à la problématique de la création et de la distribution d’intrication basée sur des photons uniques délocalisés afin de pouvoir utiliser dans diverses expériences d’information quantique.

Comme je vais le montrer, cela ouvre la voie à tout un nouveau domaine pour faire l’information quantique, où un seule photon est utilisée pour générer et distribuer l’intrication entre des partenaires distants.

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Introduction

Why quantum information

At the beginning of the last century, quantum mechanics revolutionized our understanding of the Universe. As the years went by, physicists started to realize that quantum theory is not only a good framework for describing fundamental phenomena, but also a whole new tool for developing new technology. Feynman and Deutsch were the first to propose that the laws of quantum physics could be used to do things that would otherwise be very hard, if not impossible, within the framework of classical physics [1,2]. This led to the fields of quantum information and communication [3,4]. Quantum information promises great revolutions in the way we process and transmit data. Today, we know that a computer that exploits the fundamental laws of quantum mechanics can outperform standard classical computers [5, 6], and information encoded in quantum matter can be transmitted in provably secure ways [7, 8].

Behind the many advantages of processing and transmitting quantum information lies the phenomena of interference and entanglement. These effects are responsible for the advantages quantum information offers with respect to its classical counterpart, classical information. The smallest unit of quantum information is the qubit, while for classical information, the smallest unit is the bit, encoded in the usual 0’s and 1’s. A pair of bits can only be in either one of four possible combinations (“00”, “01”, “10”, “11”).

We can do logical operations on these bits, and to achieve any possible logical operation on pairs of bits we only need one universal gate, the NAND [4].

Pairs of qubits, on the other hand, can assume superposed states such as

|Ψi= |01i − |√ 10i

2 ,

so-called entangled states. The existence of entangled states requires a more general set of logical operations. For example, we could start with a pair of qubits of the form|ai|biand by some operation end with the entangled state

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|Ψi. As a consequence of that, we need not only one, but four operations to achieve any possible logical operation on pairs of qubits [3]. It is intuitive to think that because of this higher complexity (that ultimately originates from the existence of entangled states), quantum information will allow us to do more things than classical information. That intuition turns out to be correct.

Interference and entanglement also allow quantum information to be transmitted in a secure way. Classically, unknown bits can be copied. Un- known qubits, on the other hand, cannot [9, 10], and the random but never- theless correlated measurement results on a state of the form |Ψi allow the distillation of a secret key at a distance.

Generating, manipulating and distributing quantum information and entanglement in effective ways is one of the great challenges quantum physicists face today. To address these challenges, different physical systems have been proposed and are currently being studied in laboratories around the world.

Cold atomic ensembles [11], trapped ions [12,13], superconducting electron- ics [14], optomechanical systems [15] and photons [16, 17] are some of these physical systems with promising future. In this thesis, we shall focus on photons.

Why photons

Photons are massless particles Fadeev-Popov, and because of that they must travel at the speed of light [18]. This makes them perfect carriers of information, and therefore ideally suited for distributing quantum states over long distances.

In the near future, when quantum data centers are everywhere and quantum information must be broadcast between different locations, photons will provide the ideal platform for a quantum network.

Additionally, from a practical point of view optics is a very advanced experimental science [19]. Quantum opticians can make use of several existing technologies such as lasers, fiber optics and photodiodes, making the manipulation of photons in the laboratory relatively simple. This simplicity allows experiments in quantum optics to be rapidly implemented (mainly in comparison to other quantum fields such as cold atomic ensembles or trapped ions), and as a consequence, a lot of demonstrations of quantum information are typically done first using photons.

All these considerations are not to say that creating and distributing entangled photons is an easy task. As we will see, there are a series of outstanding problems that must be solved to accomplish the goal of a fully- fledged photonic quantum information network. In particular, how to create,

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CONTENTS 3 interact and detect single photons are not straightforward tasks, and are issues that we will address within this work.

This work

In this thesis we will be concerned with how to create, interact and distribute single photons. In presenting these issues we will follow this exact logical order with a small (useful) deviation about how to detect single photons and how to characterize single photon detectors.

When dealing with the creation of photons, we will work with the phenomenon of spontaneous parametric down-conversion [20, 21], whereby a strong laser pumps a dielectric crystal or waveguide and generates photon pairs. In particular, we will build upon existing work [22, 23, 24] to demonstrate how toengineer the emission of photon pairs to perfectly match standard single-mode fibers. As will be clear later on, this knowledge is of great use if one wants to distribute entanglement efficiently over long distances.

These matters will be addressed in chapter 1.

We will then briefly discuss, in chapter 2, how to detect single photons, and perhaps more importantly, how to measure with precision the efficiency of a single photon detector. This will touch upon the modern field of quantum metrology, and serves as a nice example of how quantum physics can also help to improve measurement precision and sensitivity.

We will then discuss, in chapter 3, the problem of making single photons interact. This has been addressed within the atomic physics literature [25,26, 27,28,29,30]. Here we demonstrate for the first time an interaction between independent single photons based on sum frequency generation [31]. This was a challenging experiment, and as we will show, it opens the possibility of distributing entangled pairs in a heralded way [32].

What is the simplest way to create and distribute entanglement is also a questions we will devote considerable attention. We will discuss how a heralded single photon source can be turned into the simplest heralded entanglement source (a topic much studied in [33, 34, 35, 36]), and how the generated entanglement can be measured using interference in phase space followed by highly efficient single photon detectors. This will open the door to an entire new framework for doing quantum information science, where a single excitation of the electromagnetic field is used to generate and send entanglement to distant parties. We will demonstrate the effectiveness of these methods via experiments to certify entanglement and verify the steering of quantum states.

To close the work, we will give a discussion of possible future research

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directions and open problems. The publications that resulted from this work can be found in the appendix.

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Chapter 1 Engineering photons

Engineering states of photons is a challenging problem in experimental physics.

To address this problem, several solutions have been proposed ranging from simple attenuated laser beams, to artificial atoms in the form of quantum dots [37], to cold atomic ensembles [38]; see [21] for a complete review. De- pending on the needs of the experiment, one method or the other may provide a good enough approximation of a single photon [39], but none of the methods known to date is able to create an exact single excitation of the electromagnetic field upon demand.

The most robust way to create a state close to a single photon, and indeed the most popular among quantum opticians, is given by theparametric amplifier [40, 41, 42, 43]. After briefly introducing the physics behind this process and explaining why it gives good approximations to single photons in the lab, we discuss what needs to be improved in order to make parametric sources suitable to distributing photonic quantum information.

The publications that resulted from this work are reprinted in the appen- dices P.1,P.2, P.6and P.7.

1.1 Parametric down-conversion

The simplest way to obtain a quantum state that approximates a single photon, or even more generally, the simplest way to obtain a non-classical state of light, is through the phenomenon known as parametric down-conversion (PDC).

To understand optical PDC from a microscopic viewpoint, we need to understand what a dielectric medium is. It consists of an insulator that can be polarized by an external field. In other words, when a dielectric is placed in an electric field, charges only shift from their equilibrium position,

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inducing a polarization vector of the form [31]

P_a(t, ~r) =χ⁽¹⁾_abE_b(t, ~r) +χ⁽²⁾_abcE_b(t, ~r)E_c(t, ~r) +..., (1.1) where repeated indices are summed over. The quantitiesχ⁽¹⁾_ab, χ⁽²⁾_abc, ...are ten- sors known as optical susceptibilities. While the first optical susceptibility χ⁽¹⁾is associated to the material’s refractive index, the second optical suscep- tibilityχ⁽²⁾ is responsible for a nonlinear response of the medium’s electrons in the presence of an applied electric field.

The physical picture behind PDC is that when an oscillating electric field E(t, r) = X

i

E(ω_i)e^i(kⁱ^z−ωⁱ^t)+c.c. (1.2) passes through the dielectric medium, it induces a motion (via the polarization) of the charges in the material proportional to

χ⁽²⁾_abcX

i,s

(E_b(ω_i)e^i(kⁱ^z−ωⁱ^t)+c.c.)(E_c(ω_s)e^i(k^s^z−ω^s^t)+c.c.) (1.3) and this motion has frequency components at the sum of the frequencies ω =ωi +ωs (SFG), at the double of the frequencies ω = 2ωi (SHG) and at the difference of the frequenciesω =ω_i−ω_s(DFG) of the initial applied field.

Consequently, light is emitted from the motion of the electrons at all these frequency modes. Since all the frequency modes are emitted at the same time, they interfere, and only those modes which interfere constructively propagate and exit the dielectric medium. This happens when momenta of the emitted modes sums up to the momentum of the initial electric field,

∆~k=~k−~k_i−~k_s= 0, (1.4) where|~k|=n(ω)ω/c. This condition is known as the phase-matching condition. Typically, phase-matching will be achieved for either the sum, or the difference of the frequencies, but not for both.

One can exploit the birefringence of the dielectric material to achieve the phase-matching condition. This leads to two different types of phasematch- ing. Type I corresponds to when the two generated electric fields have the same polarization, but orthogonal to the polarization of the initial electric field. Alternatively, Type II phase-matching corresponds to the situation where the generated fields have orthogonal polarization. It is also possible to achieve phase-matching by modulating the sign of the second-order susceptibility tensor χ⁽²⁾ along the material, with a period Λ. The phase-matching condition is thus modified to

∆~k =~k−~ki−~ks+ 2πz/Λ = 0,ˆ (1.5)

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1.1. PARAMETRIC DOWN-CONVERSION 7 making birefringence unnecessary. This leads to the so-called Type 0 phase- matching, where the generated fields have the same polarization as the input one. Throughout this work we will be mostly concerned with Type 0 phase- matching.

Typically, the dielectric material will consist of a square piece of crystal with a length L on the order of a cm. If we inject coherent monochromatic beams in the crystal, these will generate coherent light at the output, with a square waveform. In the wavevector domain this waveform corresponds to the usual product of sinc functions

S(~k, ~k_i, ~k_s) =S₀

"

sin(∆~kL)

∆~kL

#

, (1.6)

so we see that the length of the crystal determines thebandwidth of the generated light. IfL→ ∞, S(~k, ~k_i, ~k_s)→δ(∆~kL), which is just a manifestation of momentum conservation. If the input beams are not monochromatic, but have non-zero bandwidths, we must multiply their wave-forms in the wavevector domain toS(~k, ~ki, ~k_s). We will sometimes refer toS as thetwo-photon wavefunction.

In a second quantized theory of electromagnetism, the electric field is written in terms of creation and annihilation operators [44], and the presence of a nonlinear polarization termχ⁽²⁾ in the equations of propagation of light induce an effective nonlinear term in the hamiltonian of the form

H_I =χ⁽²⁾ Z

dω dω_s dω_i d~k d~k_s d~k_i S(~k, ~k_i, ~k_s) (1.7) a(ω, ~k) a^†(ω_s, ~k_s) a^†(ω_i, ~k_i) +h.c. ,

so we see that effectively, PDC acts in such a way as to destroy one photon of the initial beam with frequency ω and momentum ~k, and creates two photons with frequencies ω_i and ω_s and momenta~ki and ~ks weighted by the probability amplitude S(~k, ~k_i, ~k_s) which is governed by the phase-matching condition (1.5).

If the initial beam is a single-mode coherent state we can treat a(ω, ~k) as a complex number since the coherent state is an eigenstate of the annihilation operator. Dropping the integrals over frequency also greatly simplifies the hamiltonian (1.7), allowing us to schematically write it as

H_I =χ⁽²⁾(a^†_ia^†_s+h.c.) . (1.8) Spontaneous PDC (SPDC) happens when only the coherent state at frequency ω is initially present, and the state of the field evolves according

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to

|ξi = e^iH^I^t/~|0i

= sech(ξ) X∞ n=0

tanhⁿ(ξ)|niⁱ|ni^s

= p

1−p X∞ n=0

p^n/2|niⁱ|ni^s, (1.9)

where ξ=χ⁽²⁾t/~, and p = tanh²(ξ). From (1.9) see that photons in SPDC are always created in pairs, and the probability of generating n photons in each mode is given by P(n) = (1−p)pⁿ. Moreover, the mean number of photons in the state (1.9) is hni = p/(1−p), and we can re-write the probability of having n photons in each mode as a function of this mean number of photons,

P(n) = hniⁿ

(1 +hni)ⁿ⁺¹ . (1.10)

The probability of finding n photons in each mode follow the Bose-Einstein distribution (1.10), meaning that each individual mode is populated by a thermal state. It is interesting to note that, historically, the statistics of parametric light was first studied by Zeldovich and Klyshko in [20].

Due to the pair-wise emission of SPDC, we can use states generated via (1.8) to herald single photons. One of the modes, say i, is sent to a single photon detector, and we only look at modesprovided there was a detection event in modei. With this technique, we obtain a state of the electromagnetic field that approximates very well a single photon. To see how good this approximation is, we plot in Figure 1.1 the photon number statistics of a coherent state (given by a Poissonian distribution), a thermal state (given by the Bose-Einstein distribution) and the statistics resulting from heralding a photon via SPDC. The mean number of photons in the plot is set to 10⁻³, and the efficiency of the heralding detector is 80%, typical values found in a quantum optics lab. We see that thermal states with a low mean number of photons are much closer to ideal single photons than coherent states with the same mean number of photons. It is then possible to remove most of the vacuum component of a thermal state by using the heralding technique with an SPDC source, yielding something even closer to an ideal single photon.

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1.2. WHAT NEEDS TO BE IMPROVED? 9

0.0 0.5 1.0 1.5 2.0

Number of photons 10

^-6

10

^-5

10

^-4

10

^-3

10

^-2

10

^-1

10

⁰

Probability thermal

coherent Heralded

Figure 1.1: Comparison of the photon number statistics of a coherent state, a thermal state and a heralded single photon resulting from SPDC and non- photon-number-resolving detection. The mean number of photons in both the thermal and coherent states is set to hni = 10⁻³, and the detection efficiency of the heralding detector is 80%.

1.2 What needs to be improved?

We would like to use the heralded single photons from SPDC to encode, distribute and manipulate quantum information over long distances. In particular, we would like to distribute entanglement over long distances using these heralded states.

We can achieve this goal by using optical fibers, but to do that we need to engineer the two-photon wavefunctionS(~k, ~k_i, ~k_s) resulting from SPDC to match existing communication technologies. In particular we need to learn how to prepare heralded single photons appropriate for coupling into single mode fibers at telecommunication wavelengths. We will call this problem the coupling problem.

To define the problem precisely, we need to explicitly state what do we mean by coupling into an optical fiber. Classically, the coupling efficiency is defined as the ratio between the intensity of a light beam after the fiber and before the fiber. For single photons, we could make a similar definition, substituting intensities by probabilities, but since we are interested in using heralded states from SPDC, we define the heralding efficiency as the ratio between the probability ofdetecting a photonsin the mode of an optical fiber and the probability of detecting a correlated photoniin the mode of another

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optical fiber. The heralding efficiency is written in terms of experimentally measurable numbers as the ratio between detections in mode s (which we call coincidences) and detections in modei (called singles),

ηheralding =Ncoincidences/Nsingles . (1.11) We can correct for the detection efficiencies and losses over the optical setup used in a heralding source, and obtain a number that characterizes only the efficiency of coupling correlated photons in the fiber. This is defined as the coupling efficiency of an SPDC source,

η_coupling =Ncoincidences/(N_singlestcoincidences) , (1.12) wheretcoincidencescorresponds to all the excess loss in optical components and detection efficiency in mode s of an SPDC source. Since high transmission optical components are readily available and high efficiency detectors have been recently developed [45,46,47,48], heralded single photon sources based on SPDC are mostly limited by the coupling efficiency, and knowing how to increase η_coupling as much as possible is an important challenge for optical quantum information.

High coupling of SPDC photons has been extensively addressed in the literature [22, 23], and even demonstrated in different setups [24, 49, 50].

Prior to this thesis, however, no practical systematic way of obtaining a high coupling (and consequently a high heralding efficiency) was known. In the remainder of this chapter, we will mostly focus on presenting the complete solution to the coupling problem, and all the analysis of parametric down conversion it took to obtain this solution.

1.3 Solving the coupling problem

To achieve the goal of generating heralded single photons in the mode of an optical fiber, we will be mainly concerned with the two-photon term in (1.9). We wish to engineer the two-photon wavefunction in such a way as to maximize the heralding efficiency as defined above. Before going into the details of the two-photon wavefunction S(~k, ~k_i, ~k_s), however, we make the following simple observation.

Suppose we pump a monochromatic plane wave coherent state into our dielectric crystal. As explained in the previous section, SPDC conserves energy and momentum, and therefore the sum of k-vectors of the generated photon pair must add up to the k-vector of the pump. Quantum mechanically, this

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1.4. TWO-PHOTON WAVEFUNCTION 11 means that the generated two-photon state will have the form

|ψi= Z

dk_s |k−k_si|k_si . (1.13) If we project one of the photons in a spatial profile hφ|, the spatial profile of the other photon will be given by the Fourier transform ofhk−k_s|φi:

hφ|hx|ψi = Z

dk_s hφ|k−k_sihx|k_si

= Z

dk_s e^ik^s^x hφ|ki . (1.14) If|φiis the Gaussian profile of a single-mode fiber, so will be the profile of the other photon. This implies that if one of the photons gets inside an optical fiber and is detected, it will herald the presence of another photon in the mode of an optical fiber, yielding a high coupling efficiency of the heralded photon. This simple observation suggests that high heralding efficiency can be obtained by pumping the dielectric crystal with a plane wave.

As a strategy to solve the coupling problem, we will try to devise experimental situations in which this simple argument is valid. To do this we need to eliminate any existing correlation between the k-vector and any other de- gree of freedom of the emitted photons. We now turn to the study of the full two-photon wavefunction generated from SPDC.

1.4 Two-photon wavefunction

z

k

s

k

i

k

p

Figure 1.2: Geometry of SPDC.

When studying the two photon wavefuntion of SPDC, we will work in a regime where the cross-section of the crystal is much bigger than the waist of the involved beams, so we can assume rotational symme- try with respect to the z-axis and treat the problem in only two dimen- sions, as represented in Figure 1.2.

In this approximation the quantity S given in (1.6) becomes a func-

tion of the frequencies, modulus of wave-vectors and emission angles, S = S(ω, k, ω_s, k_s, ω_i, k_i, θ_s, θ_i). The phase-mismatch can be written as

∆k =k−k_scosθ_s−k_icosθ_i+ 2π/Λ , (1.15)

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When calculating the phase mismatch, one must remember that the wave- vectors depend on the crystal’s refractive index, which in turn depends on the energy of the photons and on the temperature of the crystal according to the so-called Sellmeier equations of the material. Assuming the pump has Gaussian waveform with bandwidth ∆ω, the two-photon wavefunction becomes

S=e^−(ω−ω^s^−ωⁱ⁾²^/∆ω²

sin(∆kL)

∆kL

(1.16) We wish to plot the various reduced density matrices that can be obtained from S, characterizing spectral-spectral, spatial-spatial and spectral-spatial correlations between the generated photon pair. These are

Z

dθs dθi |S(ω, k, ωs, ks, ωi, ki, θs, θi)|² (spectral−spectral) (1.17) Z

dω_s dω_i |S(ω, k, ω_s, k_s, ω_i, k_i, θ_s, θ_i)|² (spatial−spatial) (1.18) Z

dω_i dθ_i |S(ω, k, ω_s, k_s, ω_i, k_i, θ_s, θ_i)|² (spatial−spectral) (1.19) Z

dω_s dθ_s |S(ω, k, ω_s, k_s, ω_i, k_i, θ_s, θ_i)|² (spatial−spectral) (1.20) To proceed further, we need to be specific about the material to be used for SPDC, and the phase-matching conditions we wish to obtain. For that reason we will focus on one particular material, but as will be clear along the discussion, the methods to be presented are valid over a wide range of possible SPDC setups.

1.5 Experiments

We will focus on Periodically Poled Lithium Niobate crystals (PPLN), with Type 0 phase-matching given by

532 nm→1550 nm + 810 nm .

The PPLN crystal is 1 cm long and temperature stabilized at 443.15 K to achieve the desired phase-matching condition. As a pump laser, we use a pulsed (Time-bandwidth passive mode-locked) laser generating 8 ps pulses at a repetition rate of 430 MHz. The pump is first coupled to a single mode fiber (3.5µm core diameter) in order to ensure a Gaussian profile, and a pair of aspheric lenses are used to image a beam waist of 220µm at the center

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1.5. EXPERIMENTS 13

nam e.’a.txt’

0.807 0.808 0.809 0.81 λ_s(µm) 1.548

1.551 1.554 1.557 1.56

λi(µm)

nam e.’b.txt’

−1.5 −1 −0.5 0 0.5 1 1.5 Θ_s(°)

−1.5

−1

−0.5 0 0.5 1 1.5

Θi(°)

nam e.’c.txt’

0.807 0.808 0.809 0.81 λ_s(µm)

−1.5

−1

−0.5 0 0.5 1 1.5

Θs(°)

nam e.’d.txt’

1.548 1.551 1.554 1.557 1.56 λ_i(µm)

−1.5

−1

−0.5 0 0.5 1 1.5

Θi(°)

a b

c d

Figure 1.3: Plots of the (a) spectral-spectral, (b) spatial-spatial and (c-d) spectral-spatial density matrices from the total two-photon wavefunction S.

Notice the existence of correlations between spectral and spatial degrees of freedom.

of the crystal (we refer to Figure 1.6 for a schematics). We choose such a large beam waist to have the pump as close as possible to a plane wave, as suggested by the simple argument above. With a beam waist of 220µm at 532 nm, the Rayleigh length of the pump beam is much larger than the crystal length, guaranteeing that the pump is collimated along the crystal, and therefore well approximates a plane wave. The maximum size of the beam waist is limited by the cross-section of the bulk crystal, which is of 0.5×0.5 mm².

With these details about the system, we can plot the density matrices (1.17)-(1.20), as can be seen in Figure 1.3. Notice, in Figure 1.3a) the diag- onal orientation of the two-photon state in the energy-energy space, indicat- ing high correlation between the photons’ energies. Figure 1.3b) shows the spatial-spatial two-photon density matrix, where it is also possible to see the presence of high correlation between the photons’k-vectors. This is a consequence of pumping the PPLN crystal with a loosely focused pump (close to a plane wave).

Finally (and perhaps most importantly), Figure 1.3c) and d) show the

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DM PPLN

iris CCD

Lens CCD

DG

Figure 1.4: Setup for studying spectral-spatial correlation of parametric down-conversion. DM: dichroic mirror, DG: diffraction grating.

spectral-spatial density matrices for the s and i photons, respectively. In these images it is also possible to see the existence of correlations between each photons’ energy and k-vector.

We can confirm the calculations plotted in Figure 1.3c)-d) by imaging the spectral-spatial correlations with a setup as shown in Figure 1.4. A combination of dichroic mirror and high-pass filters (not shown) are used to extinguish the pump coherent state and select the generated photons at 810 nm. These photons are then collimated with a 150 mm aspheric lens and directed to one of two possible paths: an iris diaphragm followed by a CCD camera (Atik) or a ruled diffraction grating (DG) with 1200 lines per mm, followed by an iris diaphragm and a CCD camera. We can select either one of the two paths by turning a flip-mount where the grating is placed.

We then image the photons in both paths for each different iris radius. By taking thedifference between each subsequent image, it is possible to directly resolve the correlations between k-vector (angle of emission) and energy (or wavelength). The results of these measurements are shown in Figure1.5. The left column shows the profile of the photons directly out of the crystal, in the Fourier plane. Each emission ring corresponds to a different wavelength, which is measured with the use of the diffraction grating on the other path.

Each image has been normalized to unity, and the scale is in pixels.

As predicted by the plots shown in Figure1.3c), the lower energy photons (higher wavelengths) are emitted at the center ring at 0^◦. To the best of our knowledge, this is the first time in the literature where such kind of spatial- spectral correlations in two-photon states from SPDC is measured directly.

Once we understand the spectral-spatial correlations of both generated photons given in Figure 1.3c) and Figure 1.3d), we can select a range of

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1.5. EXPERIMENTS 15

100 200 300 400 500 600

Lower energy (Pixels) (Pixels)

(Pixels)

Figure 1.5: Imaged spectral-spatial correlations of the 810 nm photons. Left column: The emission angles for different wavelengths are obtained by subtracting two images taken after an iris. Right column: The corresponding wavelengths are measured using a diffraction grating, and subtracting subsequent wavelength-resolved images after the same iris. As predicted by the calculations of the two-photon wavefunction and associated density matrices, the lower energy photons are emitted in the central ring. The scale of all the graphs are pixels.

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PPLN DM L1

L2 L3 L4

L5

TDC P

Figure 1.6: Experimental setup for coupling photons into optical fibers.

emission angles over which the two-photon state can be approximated as a product state, and thus, wavelength and angles become uncorrelated. With this the simple argument for obtaining high coupling efficiency into optical fibers described in the previous section applies, and we can achieve a high coupling (and consequently heralding) efficiency of the photons. As can be readily seen from the plots in Figure 1.3c), the spectrum of the photons becomes uncorrelated from their respective emission angles if we restrict our- selves to a collection mode between −0.1^◦ and 0.1^◦ for the 810 nm photons and between−0.2^◦ and 0.2^◦ for the 1550 nm photons. From this angle range, it is possible to calculate the size of the optimal collection waist for each photon using

w' 2λ

π∆θ (1.21)

We obtain ws = 145µm for the 810 nm photons and wi = 140µm for the 1550 nm photons. With these numbers one can choose the lenses to image the required waists in the core of the optical fibers. We have tested this method with a variety of nonlinear crystals, pumped both with CW and pulsed lasers.

To image the calculated collection waists on the optical fibers, we use two lenses on each path. On the 1550 nm path, an f = 150 mm lens (L2) is used to collimate the beam, and an f = 7.5 mm lens (L3) is used to focus the beam on the optical fiber. These lenses were chosen to match the mode field diameter of a single mode fiber at 1550 nm (5.1µm) and were verified to produce the desired waist at the crystal position during the alignment using a CCD camera. The 810 nm path has a similar setup, with anf = 150 mm lens (L4) to collimate and an f = 3.1 mm lens (L5) to focus. Again, lenses were chosen according to single mode fiber diameter (2.8µm) and the produced

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1.5. EXPERIMENTS 17

14 16 18 20 22 24 26 28 30

Delay (ns) 10²

10³ 10⁴ 10⁵ 10⁶

Counts

Figure 1.7: A coincidence measurement between detection of 1550 nm and 810 nm photons. Log scale on the vertical axis. The clear peak indicates that photons are always created in pairs, at the same time, as described by the state (1.9).

waist was verified with a CCD. After the crystal, the remaining pump light is filtered by a high-pass filter and a prism (P) on the 810 nm path.

When coupled into fibers, the photons are then directed to detectors, connected to a time-to-digital converter (QuTools TDC). For the 810 nm photon we use a free running Si-APD (Laser components), with efficiency of (0.480±0.025) at 810 nm. On the 1550 nm side we use an InGaAs detector (ID200) with efficiency of (0.240±0.020). With this we can measure the correlation between detection in modes s and i. A typical coincidence measurement can be seen in Figure 1.7. Logarithmic scale is employed so it is possible to see the background noise, which is mostly composed of coincidences between photons coming from different laser pulses, as the delay between the background peaks is spaced in time intervals of approximately 2.3 ns (∼1/430 MHz).

To estimate the coupling efficiency as defined previously, we measured the values for overall transmissions in the setup of Figure 1.6. These are

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Crystal Length Pump Wavelengths (nm) Type η_coupling

KNbO₃ 1 cm CW 532→810 + 1550 I (0.80±0.04)

KNbO₃ 1 cm Pulsed, 8 ps 532→810 + 1550 I (0.80±0.04) PPLN 1 cm Pulsed, 8 ps 532→810 + 1550 0 (0.77±0.04) PPKTP 3 cm Pulsed, 2 ps 780→1560 + 1560 II (0.88±0.04) Table 1.1: Different setups where the proposed method for achieving high- coupling efficiency was tested.

t1550 = (0.870±0.002) and t810 = (0.780 ± 0.002). These values are limited mainly by the transmission of the high pass filter used to remove the pump, which was measured to be (0.880 ± 0.002). The value t₈₁₀ is smaller than t1550 due to the prism in that path. As already explained, these values can be improved by using higher coated optical elements.

We can then proceed to the measurement of the coupling efficiencyηcoupling

from the rates of singles and coincidences. With a pump power of 5 mW the rate of singles at 810 nm was 39.0 kHz, while the coincidence rate was 7.0 kHz.

Integrating for 30 s a coupling efficiency ofη¹⁵⁵⁰_coupling= (0.860±0.070) was measured, where the error is dominated by the systematic error in the measurement of the detector efficiency. If we reverse the detection scheme, heralding the 810 nm photons with those at 1550 nm, the rate of singles was 3.2 kHz, while the coincidence rate was 0.9 kHz. Integrating also for 30 s a coupling efficiency ofη_coupling⁸¹⁰ = (0.750±0.050) was measured. This yields a geometric averaged coupling efficiency ofhη_couplingi '0.8.

The same coupling efficiencies are obtained for both CW and pulsed pump lasers, as well as with a variety of dielectric materials. The results of various experiments are summarized in Table 1.1.

Relaxing the constraints of symmetric coupling it is possible to obtain higher values for one specific wavelength. Further optimization of the 810 nm coupling, at a pump power of 0.6 mW and an integration time of 100 s yielded a coincidence rate of 5.45 kHz, single rates of 15.10 kHz, corresponding to a coupling efficiency ofη⁸¹⁰_coupling= (0.930±0.050). The corresponding coupling in the 1550 nm arm was η_coupling¹⁵⁵⁰ = (0.640±0.050).

1.6 Summary

We have described the phenomenon of parametric down conversion, and how it can be used to generate heralded single photons. After analyzing how close to a single photon the generated states can be, we discussed a key

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1.6. SUMMARY 19 requirement to implement heralded single photon sources to the distribution of quantum information: coupling into single mode optical fibers. We then went on to discuss the solution to the “coupling problem”, that is, how to create photons with the spatial mode adapted to propagate from free space to a fiber efficiently. A series of experiments implementing a solution to the coupling problem were then presented. Most importantly, we have devised a practical method for obtaining high coupling, and consequently high-heralding efficiency with SPDC sources of any type.

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Chapter 2 Quantum radiometry

Detecting single photons is a fundamental task for future quantum information networks. Throughout this work, we deal with different types of detectors, so a brief review of single photon detection techniques is in order. We also require a good knowledge of the detection efficiencies and their associated error bars. This leads us to the main topic of this chapter, the metrology problem of quantifying detection efficiency in an absolute way.

After briefly introducing the different types of single photon detectors relevant in this work, we will show how quantum radiometry can serve to establish an absolute standard of optical radiance, and therefore be used for the absolute characterization of a detector efficiency.

The publications that resulted from this work are reprinted in the appen- dices P.3and P.5.

2.1 Brief review of single photon detectors

The first type of detector used in this thesis, known as single photon avalanche diode (SPAD) uses the photoelectric effect as detection mechanism [51,52].

The SPAD consists of a p-n junction biased above its breakdown voltage, a regime known as Geiger mode [53,54]. Once a photon is absorbed, it creates an electron-hole pair which triggers a macroscopic avalanche current that can be detected by a readout circuit.

Electron-hole pairs in a SPAD can also be created and amplified in the absence of a photon. This may be caused by thermal fluctuations or by charges that get trapped inside the detector from prior detection events, being released only after some relaxation time (often called afterpulsing).

The creation of an avalanche in the absence of an incident photon is known as a dark count. To reduce dark counts due to thermal fluctuations, one may

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Material T (K) λ (µm) η_peak DCR (Hz) Jitter (ps) Ref.

Thick Si (SPAD) 200 0.4 - 1.0 70% 3 300 [55]

InGaAs/InP (SPAD) 200 0.9 - 1.7 30% 10 100 [56]

NbN (SNSPD) 2 cavity 80% <1 150 [57]

WSi (SNSPD) <1 cavity 90% <1 150 [58]

Table 2.1: The different detectors relevant throughout this thesis.

cool the detector. Cooling, however, is ultimately limited by afterpulsing, as the time it takes to release trapped charges increases at low temperatures.

The quantum efficiency of a SPAD strongly depends on the wavelength of the incident light, and different materials and structures have peak efficiencies at different wavelengths. To limit the afterpulse probability, a deadtime is often applied to SPAD detectors. This leads to a maximum count rate the detector can achieve. The SPAD is non-photon-number resolving (NPNR), meaning that it cannot discriminate how many photons were incident in a detection event.

The second type of detectors used in this thesis, namely superconducting nanowire single photon detectors (SNSPD), work upon physical principles different from those of a SPAD [59, 60]. The SNSPD detector consists of a wire made of superconducting material, for example Niobium Nitride (NbN) or Tungsten Silicide (WSi), with a width typically on the order of 100 nm [61,62]. The wire is cooled to the material’s superconducting state - normally on the order of 2 K - and a bias current is passed through the wire. Typical values of a bias current for an SNSPD are around tens ofµA. Once a photon is absorbed by the detector, it creates hot-spots in the form of vortex-anti- vortex pairs [63]. It costs more energy for the flowing current to traverse the vortex than it does for the current to deviate around it. If a sufficient number of vortices are created, the current surpasses the so-called critical value, and the material transitions out of the superconducting phase. The resistance created by the absorption of a photon can be measured with a simple readout electronic circuit. Since they rely on the binary process of breaking or not breaking the superconducting state, superconducting nanowire detectors are also NPNR.

The intrinsic absorption probability of the superconducting material form- ing the nanowire itself is not very high, for example typically around 12%

for WSi [62]. For this reason it is important to have a cavity around the detector. Once the photon enters the cavity, it can bounce a number of times increasing the detection efficiency. For infrared photons, a system detection efficiency of 93% has been measured [58]. The need for a cavity makes the

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2.3. THE CONCEPT BEHIND QUANTUM RADIOMETRY 23 detector wavelength-dependent, even tough the nanowire itself may have a broader absorption spectrum. Table2.1 summarizes the detector types used throughout this thesis and their main characteristics.

2.2 A metrology problem

With the development of single photon detection, it is essential to devise trustworthy ways of quantifying single photon detector efficiency. This pri- mary task turns out to be a challenge for the following reasons. To well characterize a detector, we need a stable light source. On the other hand, to make sure we have a stable light source, we need a stable trusted detector to verify the stability of the source.

This chicken-and-egg type of problem is recurrent in metrology and solving it requires the definition of an absolute standard. For the definition of the standard of optical radiance, scientists have normally relied on bulky experimental setups called cryogenic radiometers [64]. The construction of these radiometers is highly non-trivial, and the resulting equipment often cannot be made available where a measurement of optical radiance is needed [65].

It is therefore interesting to seek different approaches to the measurement of optical radiance.

2.3 The concept behind quantum radiometry

Quantum radiometry is a set of techniques to measure spectral radiance that explore fundamental laws of quantum mechanics. Here we give a brief review of the concept behind these techniques, first proposed in [66, 67], and later on explored in the context of quantum information by [68].

The main objective is to determine the radiance of a given light source in an absolute way, that is, independently of the calibration of the detectors used in the measurement. To do so, let us consider an atomic medium such as an erbium-doped fiber (EDF) [69]. The EDF has a gain G defined by

G= ∂µout

∂µ_in , (2.1)

where µ_in and µ_out are the number of input and output photons per mode.

The smallest possible gain is G = 1, corresponding to a medium for which the output light equals the input light. with this definition, the number of output photons per mode µ_out is

µ_out =Gµ_in+G−1 , (2.2)

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whereGµ_in corresponds to light generated by stimulated emission andG−1 to light coming from spontaneous emission, or vacuum fluctuations. These values are proportional to the optical power, the proportionality constant being the photon energyhν times the number of modes per secondN, namely the inverse coherence timeN = 1/τ_c [70].

When measurements of spontaneous and stimulated emissions with an uncalibrated detector are made, the reading is proportional to the actual value, with an unknown proportionality constant k. The measured values, denoted with an asterisc, P_sp^∗ and P_st^∗ are

P_sp^∗ = (G−1)hν

τ_ck (2.3)

P_st^∗ = (Gµ_in+G−1)hν

τ_ck (2.4)

and we can now use these equations to obtain the number of input photons per mode independently of any detector calibration factor k,

µin= (1−1/G)(P_st^∗/P_sp^∗ −1) (2.5) The value of G can also be measured absolutely using (2.1). When sending light into a medium like EDF, by measuring its spontaneous and stimulated emission power spectra, we can infer in an absolute way the input number of photons per spectral mode. We may say that spontaneous emission provides an absolute standard of optical radiance.

With this method we can assert the stability and optical radiance of a light source, and then use this light source to calibrate a detector absolutely. In the remainder of this chapter, we present two experiments, one for constructing a stable light source and another for developing the quantum radiometer described in this section.

2.4 Stable light source: experiment

As means of achieving a stable light source we consider amplified spontaneous emission. The gain medium can also be an EDF. The atomic structure of erbium can be approximated as a three-level system, shown in Figure 2.1 [69]. An electron in the ground state|1iabsorbs a pump photon and goes to level |3i. It then rapidly decays into the meta-stable level |2ithat decays to

|1ispontaneously emitting a photon at 1530 nm. Given that the pump power P_P is much higher than the output power of the fiber, a condition known as

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2.4. STABLE LIGHT SOURCE: EXPERIMENT 25

pump

fast

slow SE

1

2 3

Figure 2.1: Three-level model for Er³⁺ spontaneous emission: a pump field brings atoms from the ground state |1i to an excited state|3i from which it rapidly decays to a meta-stable state|2i. Atoms decay from level|2ito level

|1i by spontaneous emission (SE).

“no pump depletion”, the proportion N^∗/N of excited atom in level |2i with respect to level|1i is determined from the power P_P as [71]

N^∗/N =PP/(PP +C) (2.6)

where C = hν/τ σf, h the Planck constant, ν the pump laser frequency, σ the pump absorption cross section,τ the decay time from level|2ito|1iand f the pump intensity profile at the fiber. This results in the spontaneous emission powerP_SE ∝N^∗/N tending towards a constant as the pump power is increased. This saturation effect yields an intrinsic stability of the output light independently of fluctuations in the pump power. The shorter the fiber, the easier it is to saturate the EDF, and therefore the more stable the output amplified spontaneous emission will be. Of course, the shorter the fiber, the less power we will also be able to obtain.

A measurement of the saturation of the spontaneous emission light as a function of the pump power is shown in Figure 2.2 a). A typical spectrum of such amplified sponetaneous emission light is shown in Figure 2.2 b).

The setup of the source is shown in Figure 2.3. A 3 mm erbium doped fiber sits inside a connector head, and is pumped by a 980 nm laser injected using a WDM. The spontaneously emitted output light is collected in the backwards direction, and an Ytterbium doped fiber is employed to absorb any residual pump light.

To characterize the stability of the source we use a the Allan deviation [72]. The normalized Allan deviation of our light source is shown in Figure 2.4. In one hour of measurement the normalized Allan deviation was 8.4 × 10⁻⁶ ± 3.0 × 10⁻⁶ whereas in days of measurement we found 1.5×10⁻⁵±8.4×10⁻⁶.

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a) b)

0 200 400 600 800 1000

current (mA) 20

40 60 80

power (nW)

1450 1500 1550 1600 1650

wavelength (nm) 0.5

1.0

normalized power

Figure 2.2: a) Saturation of the atomic medium: as the pump laser powerP_P is increased the spontaneous emission powerP_SEscale asP_SE ∝P_P/(P_P+C).

b) Typical spectrum of amplified spontaneous emission light.

a)

Undoped ﬁber Erbium doped ﬁber

Splice Pump light in

Spontaneous emission out

Angle polish Pump light

b)

EDF

980 nm pump WDM

SE out YDF

980 nm 1530 nm Powermeter Spontaneous emission source

Figure 2.3: a) Schematic view of the spontaneous emission source: a 3 mm Erbium doped fiber sits inside a connector head, and its pumped using a 980 nm laser injected through a WDM. The light which is spontaneously emitted in the backwards direction that couples into the fiber is directed to the output by the WDM. An Ytterbium-doped fiber is used to absorb any residual 980 nm pump. The pump laser, the source and powermeter are temperature controlled. b) Complete setup schematics.

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2.5. RADIOMETER: EXPERIMENT 27

10^-3 10^-2 10^-1 10⁰ 10¹ 10² 5.0x10^-6

1.0x10^-5 1.5x10^-5 2.0x10^-5 2.5x10^-5

Normalized Allan deviation

Time (hour)

Figure 2.4: Normalized Allan deviation of the amplified spontaneous emission light source.

2.5 Radiometer: experiment

Once we have an intrinsically stable light source, we can pursue the construction of the absolute radiometer described in the beginning of this chapter.

The experimental implementation of the quantum radiometer is fairly simple. A schematics of the experimental setup can be seen in Figure 2.5. A broadband unpolarized light source (in our case the stable source developed in the previous section) is filtered to define a central wavelength and bandwidth. The light produced by this source is then amplified into an inverted single-mode erbium-doped fiber amplifier (EDF) and the power spectrum is subsequently measured by a spectrometer and compared with the power spectrum of spontaneous emission in the EDF.

The erbium-doped fiber amplifier we use consists of a short section of single-mode erbium-doped fiber in which the input light and a 980 nm pump are injected via a wavelength division multiplexer (WDM). At the output most of the pump light is eliminated using another WDM and an isolator.

For our application it is important that the medium remains fully inverted during measurement, especially at the beginning of the amplifier where losses directly influence the measurement accuracy. We measured a gain ofG= 7.9, which was found to be flat over the entire spectrum of the input light. On one hand, a gain with this value is low enough to guarantee that the fiber remains fully inverted, even at relatively high input powers. On the other hand, the precision required on the gain measurement is relaxed for higher gains.

To acquire the spectra we used an Anritsu MS9710C spectrometer which was calibrated against a Bristol Instruments 621 wave-meter. The calibration curve is shown in Figure 2.6. The relative wavelength error due to an uncalibrated spectrometer is of the order of 10⁻⁴ and will become relevant

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Figure 2.5: Setup for absolute radiance measurement based on spontaneous emission. A stable ASE source is filtered and injected into a fully inverted erbium-doped fiber. At the output of this fiber the spectrum is measured with a spectrometer. The power of the source is controlled by a variable attenuator which also contains a shutter. The diode symbols represent optical isolators which both stop back-reflections and remove any remaining 980 nm light.

only once the used power meter is improved. Once calibrated the error is of the order of 2 ppm. The linearity of the spectrometer’s vertical scale was measured with respect to the calibrated power meter and found to be linear to better than 0.01 dBm, over a 30 dBm range.

We used standard fibers of two different core diameters at different points of the experiment. On the parts of the setup in which only 1542 nm light cir- culates, SMF28 fiber was used, whereas in elements which have both telecom and 980 nm light we employed fiber components which guarantee that both wavelengths propagate in only in a single fundamental mode. This results in insertion loss at the interfaces between the two types of fibers. However, internal losses of the telecom light in the fibers was negligible over the short lengths used in the experiment. In order to achieve the best accuracy one should splice all the fibers, as the connections, especially APC, have loss which might change with time due to mechanical and thermal stress effects, and additionally, water condensation effects. In our case most fibers were connectorized in the interest of flexibility, and to better assess the limitations and stability of the individual elements of the experiment. The spectral radiance µ_in measured by the radiometer corresponds to the amount of light present exactly at the input of the EDF. For this reason it is important to characterize insertion losses at the connectors C1 and C2, shown in Figure 2.7 a).

When a radiometric measurement is made, the source is in position A shown in Figure 2.7 a). The reference light with which a power meter is calibrated is taken from C1. It is therefore important to estimate the loss at this connection. This is done with the source in position A and measuring the amount of light at C1 and then at C2. We closed the connection C1 multiple times to evaluate its repeatability and found a standard deviation between

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2.5. RADIOMETER: EXPERIMENT 29

Figure 2.6: Spectrometer calibration curve vs a wave-meter. The calibration error is of the order of 10⁻⁴. Once calibrated (linear fit) the standard deviation of the residuals is 2 pm.

consecutive connections of 0.4%, as shown in Figure 2.7 b). It is possible to improve this by only accepting the highest values for transmission. Using this method the error is reduced to 0.16%. To calibrate the loss in C2 we inject the reference light back through the EDF by putting our source in position B. The loss at C2 is measured after the characterization of connector C1 and introduces no random error as this connection is only made once.

To measure output loss, light is fed through the EDF with the pump laser turned off. The amount of light at the output of the EDF (C3) is measured to be 14.12 nW. Connecting C3 we measure 9.889 nW at the output, so that the output transmission is T_out = 0.7004. We evaluate the input loss at connector C1 by measuring the amount of light present in the input fiber before and after making the connection, obtaining 133.3 nW and 102.3 nW respectively, corresponding to a transmission of T_C1 = 0.7674. This loss is mainly due to the WDM insertion loss. The input loss at C2 is measured by turning off the pump laser, injecting light from our source backward through the EDF and measuring the amount of light before and after making the C2 connection. We obtain 20.12 nW and 17.63 nW resulting in T_C2 = 0.8762.

The total input transmission will be Tin= TC1TC2 = 0.6725. This relatively high loss is due to the different fiber mode field diameters employed in the

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a)

b)

Figure 2.7: a) In order to calibrate the input loss of the EDF at connector C2 one injects light backwards through connector C3 and measures the amount of light P₂ out of the EDF at C2. After this, one makes the connection at C2 and measures the amount of light P1 coming out of C1. The input loss is thenP₁/P₂. This characterization is made only once since C2 will remain connected during the entire experiment. b) Connection statistics. Making a connection without measuring its loss each time will result in a random error.

We measured this random error to be 0.4%. However it can be reduced by only accepting connections with the highest values of transmission. Rejecting values below a certain threshold we improve the repeatability to 0.16%.

setup.

Having discussed all the setup, from radiometer to stable source, we can now describe the final characterization of optical radiance using our spontaneous emission radiometer integrated with the stable source.

We measure the input power and spectrum at connector C1 with the power meter and spectrometer respectively. Although the power meter (Thor- labs PM100A with S154C head) is stable to four digits of precision, its reading is influenced by multiple reflections and interference between the fiber con-

Single photon interaction and entanglement

Thesis

Reference

Single photon interaction and entanglement

Universit´ e de Gen` eve Groupe de Physique Appliqu´ ee

Facult´ e des Sciences Professeur Nicolas Gisin

Single photon interaction and entanglement

Th` ese

pr´ esent´ ee ´ a la Facult´ e des Sciences de l’Universit´ e de Gen` eve pour obtenir le grade de Docteur ` es Sciences, mention physique

par

Thiago Barbosa dos Santos Guerreiro du Br´ esil

Th` ese N

4921

GEN` EVE

2016

Abstract

R´ esum´ e

Contents

Introduction

Why quantum information

Why photons

This work

Chapter 1

Engineering photons

1.1 Parametric down-conversion

0.0 0.5 1.0 1.5 2.0

Number of photons 10

10

10

10

10

10

10

Probability thermal

coherent Heralded

1.2 What needs to be improved?

1.3 Solving the coupling problem

1.4 Two-photon wavefunction

z

k

k

k

1.5 Experiments

1.6 Summary

Chapter 2

Quantum radiometry

2.1 Brief review of single photon detectors

2.2 A metrology problem

2.3 The concept behind quantum radiometry

2.4 Stable light source: experiment

2.5 Radiometer: experiment