Single photon path entangled states for quantum communication

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Thesis

Reference

Single photon path entangled states for quantum communication

MONTEIRO, Fernando

Abstract

The main work done along my PhD is related to entangled states composed of a single photon. This state generation requires fewer resources when compared to other entangled states generation. Moreover, it's easily scaled to multipartite scenarios and is also robust to setup inefficiencies of quantum repeaters. Our first work with this state was related to its generation and detection, encouraging us to pursue more evidences that this state is useful for quantum communication. The following project was related to demonstrate how this state can be used in a semi-device independent scenario. Our last project regards the use of a heralded photon amplifier that is intended to compensate the loss that is present during state distribution. All these advances makes us confident that this kind of state can find its application in quantum communication, and also encourage us to pursue new practical implementations such as device independent quantum key distribution.

MONTEIRO, Fernando. Single photon path entangled states for quantum communication. Thèse de doctorat : Univ. Genève, 2016, no. Sc. 4998

URN : urn:nbn:ch:unige-895990

DOI : 10.13097/archive-ouverte/unige:89599

Available at:

http://archive-ouverte.unige.ch/unige:89599

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

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UNIVERSITÉ DE GENÈVE FACULTÉ DES SCIENCES Groupe de Physique Appliquée Professeur Hugo ZBINDEN

Single Photon Path Entangled States for Quantum Communication

THÈSE

Présentée à la Faculté des Sciences de l’Université de Genève Pour obtenir le grade de Docteur ès Sciences, mention physique

par

Fernando Henrique do Rêgo M ^ONTEIRO de Rio de Janeiro (Brésil)

Thèse N

^◦

4998

GENÈVE

2016

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I dedicate this thesis to my wife Paula Alvarez, who is always by my side.

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Abstract

During the PhD I worked on multiple topics that covered many aspects of quantum communication. This work resulted in several scientific publications, while one of them is still under preparation. Some of these topics are covered in this thesis and organized as follow:

The initial projects were related to the development and characterization of a intrinsically stable light source in power. This source is meant to be used on the precision characterization of optical components’ transmission and single photon detector efficiency. Such carefully characterization is important to quantum communication, as some experiments demand a high overall transmission. If used with a precise powermeter, this source allows detection efficiency characterization within 1% of error bar, while typical methods of characterization usually result in error bars around 10%.

Also developed during this thesis is a source of narrowband photon pairs.

This source is based on a below threshold Optical Parametric Oscillator, in which the bandwidth of the generated photons is limited by a cavity finesse.

Nice features of this source are that it generates single mode indistinguishable photons and that it allows interface with the telecom DWDM technologies.

The long coherence length of the generated photons makes them less sensitive to fiber length fluctuations when compared to single photon sources operating in the pulsed regime.

The main work done along my PhD is related to entangled states composed of a single photon. This state generation requires fewer resources when compared to other entangled states generation. Moreover, it is easily scaled to multipartite scenarios and is also robust to setup inefficiencies of quantum repeaters. Our first work with this state was related to its generation and detection. This project showed the feasibility of the detection scheme that we developed and encouraged us to pursue more evidences that this state is useful for quantum communication. The following project was related to the demonstration of how this state can be used in a semi-device independent scenario. The most recent project regards the use of a heralded photon amplifier that is intended to compensate the loss that is present during state

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distribution. All these advances makes us confident that this kind of state can find its application in quantum communication, and also encourage us to pursue new practical implementations such as device independent quantum key distribution.

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R´ esum´ e

Au cours de ma thèse j’ai travaillé sur plusieurs sujets qui ont couvert de nombreux aspects des communications quantiques. Ce travail a donné lieu

`

a plusieurs publications scientifiques, dont l’une est encore en pr´eparation.

Certains de ces sujets sont abordés dans cette thèse et organisés comme suit:

Les premiers projets ont porté sur le développement et la caractérisation d’une source de lumière intrinsèquement stable en puissance. Cette source est destiné à être utilisée afin de caractériser précisément la transmission de composants optiques ainsi que l’efficacité de détecteurs de photons uniques.

Une telle precision est importante pour les communications quantiques, car certaines expériences exigent une transmission globale élevée. Lorsqu’elle est utilisée avec un puissance-mètre de précision, cette source permet de déterminer des efficacités de détection avec une précision de 1%, tandis que les méthodes standard de caractérisation permettent généralement d’obtenir une précision de l’ordre de 10%.

En outre, au cours de cette thèse, une source de paires de photons à bande étroite a été développée. Cette source est basée sur un oscillateur paramétrique optique pompé en-dessous du seuil d’oscillation, dans lequel la bande passante des photons générés est limitée par la finesse d’une cavité.

Les atouts de cette source sont les suivants. Elle génère des photons uniques monomodes indiscernables et elle permet d’être intégrée dans un système de multiplexage utilisé dans les télécommunications. Par ailleurs la grande longueur de cohérence des photons générés les rend moins sensibles aux fluctuation de la longueur des fibres par rapport à des sources de photons uniques opérant en régime pulsé.

La partie la plus importante de ma thèse concerne les états intriqués composés de photons uniques. La génération de tels états requiert moins de ressources que celle d’autres états intriqués couramment utilisés et peut être aisément étendue à des arrangements multi-parties. Ces états sont par ailleurs robustes aux inefficacités des répéteurs quantiques. Le premier travail effectué sur ces états a porté sur leur génération et leur détection. Ce projet a démontré la faisabilité du modèle de détection développé et nous a

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encouragés à rechercher d’avantage de preuves que ces états sont utiles pour les communications quantiques. Le projet successif concernait la faisabilité et la démonstration de l’utilisation de tels états dans un contextesemi-device independent. Enfin, le projet le plus récent a porté sur l’utilisation d’un am- plificateur de photon annoncé destiné à compenser les pertes présentes lors de la distribution de l’état. Ces avancées nous confortent dans l’idée que ce type d’états peuvent trouver des applications dans les communications quantiques et nous encouragent à développer de nouvelles implémentations pratiques telles que le protocole device independent de distribution de clé quantique.

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Chapter 1 Introduction

1.1 Quantum Communication

The past century and the beginning of this millennium were marked by tech- nological advances that culminated in the age of information. The amount of data transfer is increasing at an exponential rate and by the end of 2016 it will exceed the zettabyte “10²¹” threshold for Internet traffic per year [1].

Security of exchanged data has become fundamentally important, not only because private information is online, but also because automated technologies can become compromised. The majority of cryptographic schemes, which aim is to provide safer ways to communicate between two or more parties, are not 100% secure, as they rely on the current technical difficulty of performing certain operations, such as factorization of large numbers [2].

This security issue can be circumvent by advances brought by quantum mechanics in what is known as quantum key distribution (QKD) [3, 4, 2]. The advantage of this new quantum framework is that its security proof relies on established physical grounds and very few assumptions.

Quantum computation is another advance brought by quantum mechanics [5,6]. These computers use a quantum version of a bit, the qubit, which is a quantum superposition of classical bits. Quantum computers capabilities are doubling every year [7], and one day, they will be capable of performing the difficult tasks that can potentially break the majority cryptographic schemes used nowadays [8]. This increase in computational power serves as another motivation for the development of QKD.

More stringent types of QKD use what is called Device-Independent protocols (DIQKD), in which information can be transferred in a safe fashion even if the devices used by the involved parties are not trusted [9,10,11]. An- other quantum communication protocol allows for qubit distribution between

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12 CHAPTER 1. INTRODUCTION different locations in what is known as quantum teleportation [12, 13, 14].

A common feature of both protocols is that both require the distribution of a quantum state known as entangled state. Much of the recent evolution of quantum communication is related to generation, distribution and detection of this state. As much progress have already been made, there is still much to be done, as technologies such as DIQKD are still not available.

1.2 Entanglement: Generation, distribution and detection

An entangled state is one of the most important resources used in quantum communication. One of the simplest forms of this state can be written as:

Ψ⁻

= |0iA|1iB√− |1iA|0iB

2 (1.1)

This state is shared between two parties, here denoted as A and B. The 1 and 0 are symbols that represent properties such as particle’s polarization or number of particles.

A fundamental property that characterizes this state is that it cannot be written in the separable form|xiA⊗ |yiB, and that it has strong correlations that cannot be reproduced by classical physics [15, 11]. The presence of this state can be certified by comparing the joint probabilities measured by parties A and B with the probabilities that are expected in a classical system. The comparison that certifies the presence of this state is made by the violation of a so called Bell inequality.

A photon pair source is one of the most reliable tools used for entanglement generation, and as a consequence, a very important tool for quantum communication. The reason to use entangled states made of photons is because light is broadly used in classical communication. This means that once entanglement is produced, its distribution can be made using existing communication optical fibers [16] and/or free space technologies such as satellites [17,18]. Due to its importance, a very big part of current research is dedicated to find the best scheme able to generate single photons with the properties that are required for quantum communication protocols. These properties are fast rates, photons indistinguishably and deterministic generation.

Entanglement distribution is limited by a characteristic common for all optical elements: loss. This lossy nature induces the decoherence of entangled states and limits the distance in which quantum communication protocols can be executed. To deal with loss in a real world scenario, a so called

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1.3. THIS THESIS 13 quantum repeater architecture can be used to distribute entanglement over increasing distance. Other approaches include the development of new forms of entangled states, such as the ones composed of one photon, as they are less sensitive to quantum repeaters imperfections when compared with other typically used entangled states [19].

Single photon detectors are important tools used to characterize entanglement. This device, as the name suggests, is able to detect a single quantum of light. One limitation that was recently overcome was the low photon detection probability, specially at telecom wavelengths [20]. This recent advance allows the execution of experiments that were unthinkable before [21,22,23].

1.3 This Thesis

This Thesis addresses single photon source, entanglement creation, distribution and detection.

Chapter 2 is dedicated to a metrology device that finds its application in the characterization of optical elements’ transmission and single photon detection efficiencies. It consists of an intrinsically stable light source at telecom wavelengths, which emits unpolarized and broadband light with high degree of stability [24].

Chapter 3 is dedicated to a narrowband source of single photons based on spontaneous parametric down conversion (SPDC) in an OPO configuration [25]. This source has a high brightness and is able to herald pure narrowband photons. This purity is suitable for all protocols that require interference effects between different sources, such as quantum teleportation and entanglement swapping. This source can also be used to generate photons in selected channels of the telecom ITU grid.

Chapter 4 is dedicated to single-photon path entangled states, which are states composed of one photon entangled in two or more spacial modes. This state is relatively simple to generate and has some advantages compared to other entangled states that are typically used. A user-friendly detection scheme was also demonstrated in a series of experiments that showed the feasibility of this state in quantum communication protocols [26, 23].

The outlook goes beyond these results, looks at what challenges remain and what can be done to improve these schemes. The appendix has sup- plemental information regarding the work of chapter 4, and the annex is dedicated to have all the publications that were done during this Thesis. An exception is the work regarding path entanglement amplification, for which the paper is currently under preparation.

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14 CHAPTER 1. INTRODUCTION

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Chapter 2 Stable Light Source

2.1 Introduction

Being able to properly characterize loss is of fundamental importance for quantum communication, as most quantum information experiments and protocols are affected by loss in one way or another. As an example, loss puts limits on quantum key distribution rates [2], and some experiments, especially the ones involving several parties, can not be performed, as loss results in fewer events, hence longer measurement time. Some other experiments, such as the violation of Bell inequalities without the detection loophole, which are fundamental for Device Independent Quantum Key distribution, can not even be executed for transmission below a certain threshold [11].

All optical elements are lossy, including Single Photon detectors, as they are unable to detect photons 100% of the time [27]. While most optical elements can be characterized using standard light sources and powermeters, Single Photon detectors need a special treatment.

The standard way to characterize the efficiency of these detectors is by comparing the number of measured photons per second to the amount of power displayed by a calibrated powermeter. The problem is that powermeters and Single Photon detectors operate at input powers that are orders of magnitude different, requiring the input power to be attenuated before being sent to the detectors. This attenuation also needs to be characterized prior to the whole measurement, and because of the required level of attenuation and powermeter power range operation, three attenuators placed in series are usually required for this task. The characterization start to become compli- cated as standard commercial powermeters have a measurement uncertainty of the order of 5%, and the uncertainty propagation along all the previously described steps lead to an imprecision of the order of 10% at the measured

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16 CHAPTER 2. STABLE LIGHT SOURCE detection efficiency.

Another important device to be used along the calibrated powermeter is the light source used during all steps of the optical elements and single photon detection characterization. This source must be intrinsically stable on the long and short time scales, be unpolarized, so that it eliminates any polarization dependence present along the measurements and broadband to eliminate interference effects that can limit the experimental precision. Fiber sources are ideal due to their mechanical stability, single mode operation and easy construction. However, they usually operate using stimulated emission, suffering from interference and polarization effects [28, 29, 30, 31] that must be avoided in a precision measurement.

Presented here, is a power stable light source with all the important properties listed above [24]. This source is ideal to be used with the absolute radiometer developed in [32], as they can be used together in the characterization of optical elements and single photon detectors withing a precision of 1%. This source is also of easy implementation as it uses optical elements and devices that are well known in optics.

The present scheme is based on amplified spontaneous emission of an in- verted atomic medium, more specifically, an erbium doped fiber whose emission is in the telecommunication range. The use of spontaneous emission as a fixed measurement standard is not something new, as this process, stimulated by the vacuum, is omnipresent at each space-time point [33,34,35,36].

Some experiments that uses this concept were carried out using spontaneous parametric down conversion in bulk crystals, but they lack accuracy due to the free space nature of the used setups [37,38].

2.2 Stable Source

2.2.1 Principle of Operation

The rate of spontaneous emission of an excited atom depends on its structure and environment [39, 40, 36]. In an erbium doped fiber, the position of Er³⁺ ions is fixed and the fiber has a constant number N of erbium atoms.

As depicted in Fig 2.1, the erbium atomic structure can be modeled as a three level system. The electron ground state |1i goes to level |3i after the absorption of a pump photon. This level rapidly decays to the metastable level |2i that decays to level |1i while emitting a photon at ∼1530 nm. In case there is no pump power P_p depletion, that is, if the pump power loss is negligible compared to the pump output power, the proportion of excited atoms at level |2i with respect to level|1i is:

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2.2. STABLE SOURCE 17

Figure 2.1: Level model for the Er³⁺ ions. A pump photon brings an atom from the ground state|1ito state|3i, which decays to the metastable level|2i, before decaying to level|1iby spontaneously emitting a photon at∼1530 nm.

N^∗

N = P_p

P_p+C (2.1)

Where N^∗ is the number of excited ions, the constant C is equal to hν/τ σf, where hν is the energy of the pump photon, τ is the lifetime of the metastable level|2i,σ is the absorption cross section for the pump, and f is the pump intensity profile of the fiber, such that 2πRR_f

0 f rdr = 1 whereR_f is the fiber radius. The result of this model is that the spontaneous emission power saturates at a constant value as the pump power increases.

To test this model, a laser at 980 nm is used to pump a short∼3 mm and a long ∼10 cm erbium doped fiber. Fig 2.2 shows the spontaneous emission power as a function of the pump power for these two fibers. The emitted power PSE goes as:

P_SE =αN^∗

N (2.2)

Where α is proportional to the length of the fiber. It takes into account the density of Er³⁺ ions per fiber length and coupling of the spontaneous emitted light to the fiber.

The stability of the emitted spontaneous power P_SE is higher than the stability of the pump laser P_P. The quantification of this behavior is made by defining a stability enhancement parameter S as the relative change of the pump power divided by the relative change of the amplified spontaneous power:

S=

∆PP

P_P

∆PES

P_SE

= ∂P_P

∂PSE × P_SE PP

(2.3)

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18 CHAPTER 2. STABLE LIGHT SOURCE

Figure 2.2: Amplified spontaneous emission as a function of the pump power.

The emitted power of the short erbium doped fiber saturates faster than the emitted power of the long erbium doped fiber. This behavior is due to the fact that the short fiber has fewer of Er³⁺ ions than the long one, demanding less pump power in order to become saturated.

Equations 2.1 and 2.2, show that the stability enhanced parameter S depends linearly on the pump power:

S(P_P) = 1 +τ σf

hν ×P_P ≈ τ σf

hν ×P_P (2.4)

This parameter is calculated for the short and long fiber at PP = 0.35W using σ = 2.58×10⁻²⁵m² [41], τ = 10ms [42, 41], assuming f to be homo- geneous over the fiber’s core and zero outside, and using fiber specification from Thorlabs Er30-4/125.

The theoretically found S for the short fiber is S(0.35W) = 134±18, while Fig 2.2 experimentally gives S(0.35W) = 99±10 for the short fiber.

The same measurement on the long fiber gives a ofS(0.35W) = 44±5. The discrepancy for the long fiber can be explained by the fact that the model only takes spontaneous emission in account, while the output power of this fiber has non-negligible amounts of stimulated power. This happens because the decay from level |2i to level|1ican be stimulated by the produced amplified spontaneous emission in a process that grows exponentially with fiber length.

The short fiber has a scaling that can still be considered linear, resulting in a small amount of stimulated emission and a reduced discrepancy in the measurement of S.

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Figure 2.3: Schematic of the stable source based on spontaneous emission.

The pump laser at 980 nm is sent to a WDM and then to a ∼3 mm erbium doped fiber. The emitted spontaneous light back-propagates to the WDM and then goes to an ytterbium doped fiber that acts as a filter for any pump laser light that has propagated up to this point. The lower part of this figure shows the ∼3 mm erbium doped fiber placed inside a fiber connector and with a angle polishing used to avoid reflection at its end.

2.2.2 Stable source components

To guarantee stability, the source must be of simple construction and only use spliced fibers, as any fiber connector can lead to instabilities. The schematic of the source is shown in Fig 2.3, where the ∼3 mm erbium doped fiber is pumped by a 980 nm diode laser. As shown in this figure, most of the pump laser light leaves our setup in an angle polished fiber into a metallic beam block, resulting in pump back-reflection below −60 dB. Part of the spontaneous emission will couple into the fiber and back-propagate up to a wavelength division multiplexer¹ (WDM) that sends the light into an ytterbium doped fiber²that is used to prevent any residual pump laser from going to the output of our source. This fiber transmission to the pump laser was characterized to be 2×10⁻⁴, for incident powers below 70µW.

Fig 2.3 shows that the ∼3 mm erbium doped fiber is located inside an angle polished fiber connector. This is obtained by first splicing the erbium doped fiber to standard single mode telecom fiber, inserting it in the connector and cleaving the excess fiber outside the connector in the standard way.

This scheme is mechanically stable, not suffering from polarization effects.

The small erbium doped fiber has a broadband emission, minimizing possible interference effects that can affect the source stability. Another advantage of the smaller size is the low output power∼100 nW, requiring low attenuation

1Model AFW WDM-2-9815-L-1-L-0

2Model Thorlabs YB1200-4/125

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Figure 2.4: Spectrum measured from our∼3 mm erbium doped fiber. 99% of the power is located from 1400 nm to 1700 nm, and the absence of a peak at 980 nm shows that the pump laser is being filtered by the ytterbium doped fiber. The small peak located between ∼1050 nm and ∼1100 nm could not be identified due to its signal to noise ratio.

and making it suitable to be used in detection efficiency characterization.

A low coherence interferometer was used to characterize the coherence length of the emitted light to be 20µm, which, by considering the size of the optical elements used during the source construction, is enough to avoid interference effects. This number can be used to calculate the coherence time τ_C and the number of modes per secondµ_out = 1/τ_C = 1.4×10¹³ [43]. Given that the output power is ∼100 nW at ∼1530 nm, it is possible estimate the number of photons per mode to be 0.04 1, showing that the source is indeed working in the spontaneous emission regime.

The measured coherence time is in agreement with the spectrum shown on Fig 2.4. There is no evidence of the pump laser at the source’s output, showing that it is being filtered by the ytterbium doped fiber. There is also a small peak located between ∼1050 nm and ∼1100 nm that could not be identified due to its signal to noise ratio. Also tested was how the spectrum behaves for a pump power ranging from 100 mW to 400 mW, but the measurements were not only indistinguishable, but had an integral overlap of 1.002 among themselves.

The temperature stability was tested on each optical element used in the source. The temperature behavior of the spontaneous emission power of the erbium doped fiber is ∆PES/PES =−71 ppm/^◦C. Similar behavior was observed by changing the temperature of the WDM and the ytterbium doped fiber, resulting in an output power fluctuation of ∆P_ES/P_ES =4.2 ppm/^◦C and ∆PES/PES =−2.2 ppm/^◦C respectively. All elements were then temperature controlled in order to further improve the source stability during

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Figure 2.5: Stability tests were performed by increasing the temperature of the source and powermeter. Also tested was the behavior of the electronics’

power supply and pump power. The long decay time after decreasing the pump power indicates that it affects the overall temperature of the source.

operation.

The laser used to pump the source is a JDS Uniphase³. This laser used a current controller⁴to maintain the operation at a constant current of 900 mA.

At the same time, the laser temperature is maintained at 20.000±0.001^◦C.

The normalized standard deviation of the pump power is measured to be 10⁻⁵ over 1 h of measurement at ∼500 mW. Taking into account the stability enhancement factor S, the predicted source’s output power is going to be limited to a standard deviation of 10⁻⁷ over 1 h of measurement.

Finally, a home-made powermeter was also developed using stable elec- tronic components such as the LTC1052 operational amplifier, with a spec- ified long term stability of 100 nV/√

month and an input noise current of 0.6 fA/√

Hz. The designed powermeter setup is such that the input light is sent in a small angle to the diode⁵, in order to avoid multiple reflections between the diode and the fiber.

2.2.3 Results

After mounting, the stable source was completely characterized by measuring the power fluctuations caused by different settings of the source’s and powermeter’s temperature, electronics power supply and pump laser power. As

3Model S30-7402-660

4Model SRS LDC502

5Model Hamamatsu G8605-12

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22 CHAPTER 2. STABLE LIGHT SOURCE shown in Fig 2.5, this characterization was made by increasing the source’s temperature by 1^◦C, followed by another increase of 1^◦C at the powermeter.

Then the powermeter’s power supply was decreased from 4.9 V to 4.5 V and finally the pump power was decreased from 0.32 W to 0.29 W. The data was fitted with an exponential functionP₀+A×e^−(t−t⁰^)/τ and the time constant τ were found to be 72.5±3.7s, 119.2±1.8s and 62.3±4.4s for the source, powermeter and pump power respectively. The time constant value regarding the decrease of the pump power is expected to be much smaller, since the time the laser takes to change the power is much faster than 60 s. This is an indication that the pump power causes temperature dependence at the optical elements of the stable source. Another feature to be noticed is that the overall output power drift measured by this procedure is bigger than the drift observed at each individual component. Also, each component’s instability can not be added in quadrature, as they are all correlated by temperature changes.

Figure 2.6: Normalized Allan deviation for our stable source measured along four days. It is possible to observe that measured power has a standard deviation of (8.4±3.0)×10⁻⁶ within 1 h of measurement. This result shows that the measured power is sufficiently stable during the time it takes for a single photon detector to be characterized.

Fig 2.6 shows the normalized Allan deviation [44] of the output power of the stable source and powermeter. A standard deviation was measured to be (8.4±3.0)×10⁻⁶ during 1 h of measurement and (1.5±0.8)×10⁻⁵ during 2 days of measurement. By assuming that the temperature fluctuation of source and powermeter is the same, it is possible to calculate the temperature fluctuation along the measurement to be (1.7±0.6)×10⁻² ^◦C in 1 h and (3.1± 0.2)×10⁻² ^◦C in approximately 2 days of measurement, which are

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2.2. STABLE SOURCE 23 both consistent with expected temperature fluctuations. The increase in power fluctuation beyond 10 h is consistent with the temperature daily cycle.

2.2.4 Conclusion

The constructed stable source generates unpolarized and broadband light at telecom wavelengths. A measured output power with standard deviation of (8.4±3.0)×10⁻⁶ was achieved in 1 h of operation. It is possible to further improve the stability to 10⁻⁷ by applying a better temperature control and a better environment insulation. Stable sources at other wavelengths could be achieved by the use of other fiber amplifiers and by a generalization of the method above described.

This stable source is ideal to be used in detection efficiency characterization when used together with the radiometer developed in Ref. [32]. This task was put forward by [45] in a construction of a fully automated version of these devices. This constructed device not only can be used in a field operation, but it was also able to characterize single photon detection efficiency with a precision better than 1%. As already mentioned, the use of standard light sources with standard powermeters are only able to provide a precision on the order of 10%.

Finally and still related to the present discussion about stable light sources, radiometers and photon detectors, is a possible redefinition of a SI base unit known as the Candela. This redefinition would use standards, such as spontaneous emission, that are commonly used in the quantum world [46, 47].

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Chapter 3 Narrowband Photon Pair Source

3.1 Introduction

Photon sources are used in the generation of most entangled states used in quantum communication. Even though there are several schemes that are able to generate single photons [48,49,50], the most reliable one is based on a process called Spontaneous Parametric Down Conversion (SPDC) [51, 52, 53, 16, 27]. Among the advantages found in this scheme when compared to quantum dots [54, 27,50] is the ease of operation, which consists of a pump laser passing through a nonlinear material that is close to room temperature, the production of photon pairs at high rates, the tunable wavelengths and bandwidths and the high coupling of the produced photons into optical fibers [55, 56,57, 58].

The objective of this chapter is to provide a brief introduction to SPDC sources and show a flexible SPDC source mounted in a cavity configuration that allows the generation of narrowband photon pairs [25]. Among the advantages of this source, it is possible to list the high brightness, the ease with which the SPDC nonlinear crystal and cavity mirrors can be changed without misaligning the optical setup, the pure photon generation and the fact that the wavelength distance between produced photons is close to the ITU grid of telecommunication wavelengths. Due to its purity, this source can be used for quantum communication protocols that require two or more independent photon sources and can also be used with commercially available ultra dense wavelength division multiplexing in order to simultaneously distribute the generated photons among different places.

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26 CHAPTER 3. NARROWBAND PHOTON PAIR SOURCE

3.1.1 SPDC

Whenever a dielectric material is placed in the presence of an electric field, its electric charges suffer small displacements and the material becomes po- larized. Summing over repeated indices, this polarization is expressed by

P_i =₀χ¹_ijE_j +₀χ²_ijkE_jE_k+₀χ³_ijkwE_jE_kE_w+... (3.1) whereχⁿis then_thorder optical susceptibility tensor,P_i is the component of the polarization vector in the ith direction and Ej is the component of the electric field vector in the j_th direction. The figure of merit for the SPDC process is the second order optical susceptibility tensor χ². A non- zero value for the second order optical susceptibility can only be found in a material lacking inversion symmetry, as opposed to gases, liquids, amorphous materials and even for some crystals [59].

The second order nonlinear response for an oscillating field that enters the material is going to be given by

₀χ²_ijkX

n,m

(E(k_n, ω_n)e^i(kⁿ^x−wⁿ^t)+c.c.)(E(k_m, ω_m)e^i(k^m^x−w^m^t)+c.c.) (3.2) where the resulting field will have components that are given by the prod- uct of the Fourier components shown above, with frequencies that can be written as 2ω₁, 2ω₂, ω₁ +ω₂ or ω₁ −ω₂, on processes called, Second Har- monic Generation (SHG), Sum frequency generation (SFG) and Difference frequency generation (DFG) respectively. Spontaneous parametric down conversion is the opposite process of SHG and SFG, in which one photon with frequencyω is converted in two photons such that ω=ω_s+ω_i.

During SPDC, the initial photon with momentum k = n(ω)ω/c is converted into two photons, such that k = k_s +k_i and n(ω) is the refractive index of the medium. This condition imposed by momentum conservation, called phase-matching, can be engineered to generate the desired down converted properties, resulting in what is called Type-I SPDC, in which the photons leave the crystal with the same polarization, Type-II in which the photons leave with perpendicular polarization and Type-0, in which all three photons have the same polarization. Among other properties that can be tuned, the generated photons can leave the crystal in a colinear fashion or in a non-colinear fashion.

The spectral envelope of the produced photons in an interaction is given by the volume size in which the interaction takes place [36]. In a nonlinear interaction, the volume is reduced to the nonlinear crystal lengthL, and the

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3.1. INTRODUCTION 27 generated envelope in the momentum space is given by sinc(∆k), which has a bandwidth proportional to 1/L.

A proper description of SPDC, and light in general, is provided by the second quantization of the electromagnetic theory [40,36]. In this theory, the effective interaction Hamiltonian governing SPDC is written as [60, 59,61]:

H =g(a^†_sa^†_ia_p+a_sa_ia^†_p) (3.3) wherea^†is the photon creation operator in a specific mode,gis a coupling parameter ands, i, prepresent the signal, the idler and the pump laser modes respectively. The above relation can be written taking into consideration the multiple frequencies involved in the process.

H =g Z

dω_sdω_idω_pΦ(ω_s, ω_i, ω_p)δ(ω_s+ω_i−ω_p)[a^†_s(ω_s)a^†_i(ω_i)a_p(ω_p) +h.c.]

(3.4) Where Φ is a function that takes into account the phase-matching condition and δ(ω_s+ω_i−ω_p) is consequence of energy conservation.

What actually makes the SPDC process useful as a photon pair source, and as a heralded single photon source is the resulting photon statistics. The resulting state in the photon number basis, obtained by inputting a strong coherent state |αi in the nonlinear material is

|Ψi=e^iHt|αip|0is|0ii ≈

≈ |αip{|0is|0ii+αgl|1is|1ii+|α|²|g|²l²|2is|2ii+...} (3.5) where l is the nonlinear crystal length and αthe amplitude of the coherent state. The probability of generating a pair of photons is given byP_pair ≈

|α|²|g|²l², while the probability of generating a double pair isP_double =P_pair² . As a side comment, note that P_pair ≈ 10⁻³ for the most experiments presented in this thesis.

The fact that with a probability P_pair the crystal generates a pair of photons is extreme useful, as one of the photons can be used to announce the presence of the other, transforming the SPDC process into a heralded single photon source.

3.1.2 Photon Purity

Equally important as being able to herald single photons, is to be able to herald pure photons [62]. It happens that not all heralded photons from

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28 CHAPTER 3. NARROWBAND PHOTON PAIR SOURCE the same source are in the same frequency state, in other words, the heralded photons are in a mixed state. If the generated photons are in a pure state, then it is possible to use two or more independent sources in quantum communication protocols that rely on photon interference, as an example, entanglement swapping.

The approaches used to herald pure photons in the CW regime are different from the approaches used when working in the pulsed regime [63, 64, 65, 66, 67, 68]. One of the possible CW approaches uses spectral filtering before the detection of the photon. Reference [69] showed that the heralded photon state remains pure whenever the filter bandwidth ∆ω produces a photon with a coherence time 1/∆ω longer than the detection temporal resolution ∆T. This filtering approach was implemented in several situations [70,71], but has the disadvantage of being a lossy procedure, hence lowering the experimental rates.

Another approach avoids this spectral filtering by directly generating photons in a spectral bandwidth such that 1/∆ω > ∆T. This can be achieved by placing the SPDC crystal inside an optical cavity, in a configuration that is called Optical Parametric Oscillator (OPO) and is operated below the lasing threshold. In this way, the two photons will only be produced in the allowed cavity modes, and as a consequence, they will have a coherence time that will depend on the cavity finesse.

3.2 Optical Parametric Oscillator - Cavity SPDC

Optical cavities confine the electromagnetic field between two mirrors. The boundary condition at each mirror surface implies that only specific frequency modes are allowed inside the cavity [72]. The idea of using nonlinear crystals surrounded by a cavity, in a device called Optical Parametric Oscil- lator (OPO) is not new [73], and has gained applications such as narrowband photon and squeezed states [74] generation.

3.2.1 Purity, Bandwidth and Wavelength

While the coherence time of the photons produced using standard SPDC pumped with pulsed laser are typically .1 ns, OPO based sources obtain bandwidths on the order of 10 MHz to 100 MHz depending on the cavity, which corresponds to coherence time of 10 ns to 100 ns [75]. These narrowband sources can be used to generate pure photons when combined with standard detection jitter of 60 ps to 400 ps at telecom wavelengths. Another advantage of the narrowband operation is that the generated photons can be

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3.2. OPTICAL PARAMETRIC OSCILLATOR - CAVITY SPDC 29 matched to atomic transitions, which is suitable to be used with quantum memories [76,75], at the same time that is robust to fiber length fluctuation and chromatic dispersion [16,77].

A short review is necessary to better understand how the cavity con- straints affect the generated bandwidth and wavelength. The boundary con- ditionsEk(x, t) = 0 at the mirrors’ interfaces induce destructive interference for all modes [72] with wavelength different than

λ_N = 2L

N (3.6)

where, L is the optical cavity length and N is an integer. The fact that L is much longer than the optical telecommunication wavelengths not only implies that N 1 but, also determines a mode to mode distance called Free Spectral Range (FSR):

∆λ_N =λ_N −λ_N+1 = λ²_N

2L (3.7)

Any internal cavity loss induces a mode broadening [72], such that:

F = ∆λN

δλ_N ≈ π

1−√ρ (3.8)

where F is the cavity finesse, δλ is the allowed mode’s bandwidth and 1−ρ is the fraction of the power lost per cavity round trip.

The bandwidth of the generated photons in an OPO-based source is not only going to be limited toδλ, but the photon pair is only going to be generated if it satisfies energy and momentum conservation and if the signal and idler photons simultaneously match an allowed cavity mode. This last condition is very restrictive, not only because the allowed modes are scarce, but because chromatic dispersion induces different optical paths Li,sfor different wavelengths, causing the FSR around the idler photon to be different from the FSR around the signal photon. This FSR mismatch causes many SPDC allowed modes satisfying energy and momentum conservation to become forbidden ones. This effect is known as clustering [78,79,80] and is depicted in Fig 3.1. The OPO generated photons are usually grouped together forming photon clusters, in which the distance from one cluster to the next, when operating far from degeneracy, is approximately given by [78]:

± ∆ω_i∆ω_s

∆ω_i−∆ω_s (3.9)

where ∆ωi,s is the FSR in the frequency space. The clustering effect not only constrains the output frequency modes, but also renders OPO sources

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Figure 3.1: Allowed cavity spectral modes for the idler ωi and signal ωs

photons. Due to chromatic dispersion the FSR at the idler frequencies is different from the FSR at the signal frequencies, inducing regions where an allowed idler mode does not correspond to an allowed signal mode, even though they satisfy energy conservationω_i+ω_s=ω_p. Frequencies associated with forbidden down conversion process are depicted by the dashed rectangles while clusters of allowed modes are depicted by the solid squares.

to be highly temperature dependent. As shown in reference [78] and verified by references [79,80], even small temperature changes, that do not affect the phase-matching condition, are able to change the position of allowed modes by values greater than the bandwidth δλ. That is because the signal and idler optical pathsL_i,s depend on the cavity and crystal’s temperature. This change in the mode position together with the clustering effect is able to convert an allowed down conversion mode into a forbidden one even for a slight temperature fluctuation of ∼0.01^◦C [25].

3.2.2 Brightness

One main feature of OPO based source when compared to the filtered SPDC is the generated brightness, defined as the number of generated photon pairs per spectral bandwidth per mW of pump power per second _∆ωmW^{N s}⁻¹ . The typical procedure to obtain narrowband operation without a cavity is to spectrally filter the signal photon [81,70,71]. While the source brightness is not affected by a square filter with peak transmission 1, a Gaussian filter with a bandwidth that is matched to the photon will have an effective transmission given by

R(e⁻^(ω−µ)2^2σ² )²dω R e⁻^(ω−µ)2^2σ² dω

= 1

√2 (3.10)

Nevertheless, all filters are lossy, which constrains the overall brightness

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3.3. EXPERIMENT 31 produced by a filtered SPDC source. On the other hand, OPO based sources not only produce the photons in the desired modes, but also enhance the overall brightness by some orders of magnitude, as the density of states of the allowed modes inside the cavity becomes higher than the free space density of states [39, 82,83, 84].

3.3 Experiment

3.3.1 Experimental Description

The desired OPO source must be of easy operation, in a design that allows a quick exchange between different nonlinear crystals and mirrors without affecting the alignment between the pump laser, crystal and cavity modes.

Also desired is a setup that is able to herald pure photon states whose wavelengths are tunable by small temperature increments. Finally, the OPO source should have a FSR close to the ITU grid of commercially available ultra dense wavelength division multiplexing (U-DWDM). In this way, the generated pure photons could be distributed along the several available outputs of the U-DWDM, allowing one source to simultaneously distribute photons between multiple locations.

3.3.2 OPO Design and Alignment

Fig 3.2 shows the used cavity mount design. A crystal holder is able to support a crystal with standard dimensions of 1 mm×1 mm×10 mm, while also following a track that not only allows easy insertion inside the cavity, but also allows an easy selection of the desired crystal’s poling period. The cavity design was carefully made such that the center of the crystal is always aligned with the center of the cavity mirrors within a distance less than 100µm.

The nonlinear crystal is a PPLN MgO-doped¹ with poling periods of 19.20 , 19.50 , 19.80 , 20.10 and 20.40µm. Its transmission was characterized to be 0.997 at 1560 nm. Following eq. 3.8, high transmission is as important as high mirror reflectivity as narrowband operation can only be achieved by a low loss cavity.

The cavity consisted of a concave dielectric mirror with a 12 mm focal length and a dielectric plane mirror. While both mirrors were transparent to the pump laser, the measured reflectivity at 1560 nm are 0.995 and 0.985 for

1Covesion MSH1550-1.0

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Figure 3.2: Flexible mount design of our OPO based source. The crystal holder follows a track that allows easy crystal insertion and always main- tains the alignment between the cavity mirrors and nonlinear crystal within a distance of 100µm. This mount also allows easy mirror replacement with minor misalignment, and to further improve stability, the mirrors are held by three contact points. The whole mount is grounded to avoid charge ac- cumulation, which can also influence the clustering effect [78].

the concave and plane mirrors respectively. The spacing between each reflec- tive surface along the central cavity axis is 3.9 mm, resulting in a resonant spatial Gaussian mode with 65µm waist at the center of the crystal.

The temperature of the cavity mount and crystal is controlled by a re- sistor² fixed to the lower plate. A finely tuned temperature control not only avoids output mode fluctuation caused by the clustering effect, but also allows passive control of the allowed frequency modes. Nevertheless, the crystal’s phase matching is not significantly affected by temperature fluctuation as it has a temperature bandwidth of∼80 K.

The schematics used for the cavity alignment can be seen in Fig 3.3.

A CW pump laser³is frequency locked on a Rubidium transition at 780.24 nm, as a stable laser frequency is as important to the cluster’s allowed modes as the cavity temperature stabilization. This laser is sent through an isolator that is used to avoid back reflection and a single mode optical fiber that acts as a mode cleaner. After the fiber, the laser is transmitted through a dichroic mirror (D) and is aligned to the cavity’s central axis. Following this procedure, the cavity’s fundamental mode is aligned to the fiber that is used to collect the generated photons. This goal is achieved by injecting a tunable telecom laser⁴ in this fiber, which outputs a Gaussian beam that goes to

2EPCOS

3Toptica DL 100

4NetTest Tunics Plus

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3.3. EXPERIMENT 33

Figure 3.3: This setup is divided in two parts: 1)A tunable telecom laser is used to measure the FSR and to align the cavity to the fiber used to collect the generated photons. 2) A CW laser at 780 nm is used to pump the OPO based source. The generated photons are coupled to the previously aligned optical fiber and go to a channel selector followed by single photon detectors that are used to characterize the source statistics.

a 15 cm focal length lens followed by reflection on the dichroic mirror (D).

This lens serves to match the beam diameter to the cavity mode, while the dichroic mirror serves to split the generated photons from the pump laser. A powermeter (PM) is placed at the cavity’s output, so that, as Fig. 3.4shows, it can measure the transmitted wavelengths while the laser is being scanned around∼1 560 nm.

Fig 3.4 not only shows that the FSR of the system composed by cavity and nonlinear crystal is 0.241±0.008 nm, but also shows that the optical fiber used to collect the photons is well aligned to the cavity’s fundamental Gaussian mode. This is a consequence of the fact that any small misalignment induces the population of non-gaussian modes that are slightly detuned to higher frequencies [72], corresponding to additional peaks in between the ones shown in Fig 3.4 ⁵.

3.3.3 Photon pair spectrum

The generated photon pair spectrum was measured using a spectrometer composed of a grating and an InGaAs CCD sensor that is sensitive at the single photon level. Fig3.5(a) shows the spectrum of the generated photons when the cavity is at 43.77^◦C. The 0.5 nm spectrometer resolution does not allow a detailed inspection of the spectral structure of the cavity itself, but only the cluster’s envelope. The observed amplitude asymmetry between the corresponding envelopes is due to a higher detection efficiency towards lower wavelengths. Fig 3.5 (b) shows the envelope’s behavior when the optical

5See reference [72] for a detailed analysis of the modes in cavities and waveguides

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Figure 3.4: Powermeter response to the tunable telecom laser as the wavelength is scanned. The obtained FSR for the composite cavity and crystal is measured to be 0.241±0.008 nm, while the absence of other peaks also shows that the optical fiber used to collect the generated photons is well aligned to the OPO based source.

path for the signal and idler photons changes by the increase of cavity’s and crystal’s temperature.

The ability to tune the wavelength position of the cluster, and the fact that there are several cavity modes inside each envelope, allows for a cavity mode selection that is done by carefully aligning the modes to a DWDM channel or filter. Once aligned, single photon detectors and the time to digital converter (TDC)⁶ are used to characterize the correlated photon pairs. The filters’ bandwidth is not only smaller than the FSR, but also has no effect on the photon’s bandwidth, as they act as top hat filters that selects the desired modes. This top hat behavior allows for a photon pair selection with a reduced loss.

3.3.4 Coincidence Measurements and Enhancement fac- tor

A standard DWDM selects one cavity mode which has the singles count rates maximized on a free running detector (D1)⁷. This detector is placed after a tunable filter with 24.6 GHz of bandwidth, whose purpose is to mimic a U- DWDM. Coincidence measurements are then performed between (D1), and a gated detector (D2)⁸ that is positioned after the other output of the DWDM.

6Qtools

7IDQ - id220; 15% efficiency, 600 Hz of noise, 20µs deadtime

8IDQ - id210; 20% efficiency and 20 ns gate

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3.3. EXPERIMENT 35

Figure 3.5: Measured spectrum of the generated photon pairs. Fig. a) shows the spectrum when the cavity and crystal are at 43.77^◦C. This spectrum is dominated by two cluster peaks around 2λ_p. Each peak has a bandwidth of 2.5 nm and the height asymmetry between the peaks is due to different detection efficiencies. The smaller peaks to the sides are due to other clusters of modes which can arise due to the extremely large phase-matching bandwidth of such a short crystal. Fig. b) shows the envelope position of the clusters with respect to the temperature of the crystal and cavity. The data consisting of 13 horizontal stripes of different temperature settings has been smoothed.

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36 CHAPTER 3. NARROWBAND PHOTON PAIR SOURCE Figure3.6 (a) shows the coincidence peak on a logarithmic scale and with a linear fit in order to highlight the exponential decay typical of this kind of source [84, 85, 86, 76, 87, 88]. The decay is fitted with e^±δωT, where T is the time delay, and δω/2π =116 MHz is bandwidth for both photons, corresponding to a finesse of 255, or Q-factor of 1.66×10⁶ [72]. This bandwidth also corresponds to a coherence time of 8.6 ns, that is not only bigger than the detection jitter of∼0.3 ns, but is also in good agreement with the expected value of ∼8.2 ns. This mismatch on the coherence time can be explained with an error of less than 1% for the measured cavity’s internal loss.

Correcting for detection efficiency and the external losses, the measured brightness is 134 pairs(s×mW×MHz)⁻¹, corresponding to an enhancement factor of≈500 when compared to the crystal without cavity. This enhancement allows the generation of 5×10⁻³ pair per mode, with 40 mW of pump power.

By analyzing the coincidences and singles rates, correcting for loss in fiber components and detector efficiency, a source-fiber coupling efficiency is found to be ∼25%. Using the mirror reflectivities and crystal transmission described above and the equations from [79], it is possible to calculate that the escape probability of a photon through the flat mirror is∼45%, implying that the free space to fiber coupling efficiency is ∼60%. This flat mirror escape probability can be improved by a higher reflectivity of the concave mirror and higher crystal’s transmission accompanied by a slight reduction of the flat mirror reflectivity.

3.3.5 Purity

As stressed above, the condition to obtain pure photons when using a CW pump laser, is that the photon’s coherence time is longer than the detection jitter [69]. In the present case, the coherence time of the produced photons is 8.6 ns and the detectors used to measure the purity have a jitter of 0.3 ns and a good signal to noise ratio as the singles rate is∼5 KHz, and the noise rate is∼30 Hz.

The purity can be measured by a g²(0) measurement. More specifically, the relationship g²(0) = 1 + 1/N, where N is the effective number of modes [88]. In case the source has thermal statistics, N = 1 and g²(0) = 2. To perform such measurement, a 50/50 beam splitter was used in a Hanbury Brown and Twiss configuration that was placed after the mode selection.

As shown in Fig. 3.3, the coincidences were recorded using two custom low-noise free running SPDs [89] D₂ and D₃ while Fig. 3.6 (b) shows the g² measurement as a function of the arrival time delay. Theg²(0) was measured to be 1.84±0.11, which corresponds to an effective number of modes N =

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3.4. CONCLUSION 37

Figure 3.6: Fig. (a) the coincidence measurement for two correlated modes of the OPO based source. The log scale serves to show the exponential behavior of this kind of source. Fig. (b) shows the performedg² measurement, where photons from the same mode are split on a beamsplitter and sent to two detectors that makes the measurement without any previous heralding. The measured effective number of modes is calculated to be N = 1.19±0.15.

1.19±0.15 and a high level of purity for the OPO based source. The purity can be limited by two factors: The first is related to cavity misalignment, which would excite some unwanted transverse modes [72]. However Fig. 3.4 does not show any unwanted spatial modes above the noise level. The second limiting factor is measurement error, as any detector jitter or temporal drift smooths the g² peak.

3.4 Conclusion

Here was presented a simple and compact photon pair source that is both well suited to produce narrowband photons and interface with telecom DWDM technologies. This OPO based source has a high level of purity, enabling it to be used in experiments that require entanglement swapping from photons created at different sources [70]. Another advantage of the narrowband operation is that source synchronization becomes more relaxed as fiber length fluctuations barely affect the arrival time of photons with long coherence lengths [70,79]. Apart from the pump laser, this device does not require ac- tive temperature control, which together with the characteristics mentioned above, make it interesting for complex quantum communication networks.

A promising approach is to use small nonlinear crystals as the one used here, but in a waveguide configuration with coated ends as mirrors. Such device would ensure compatibility with U-DWDM technologies and would be

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38 CHAPTER 3. NARROWBAND PHOTON PAIR SOURCE robust against misalignment, temperature and pressure fluctuation, as these quantities are able to change the signal and idler optical paths. The waveguide’s mode confinement would also allow for a higher brightness. Some examples include the use of intrinsically stable and highly tunable whis- pering gallery [88, 90] and ring [91, 92] resonators, while other approaches have demonstrated a robust cluster control, by engineering a type II phase- matched OPO based source with only one mode per cluster [75]. Approaches like these can allow new devices that are compatible with DWDM technologies and even devices that can be coupled to quantum memories [76, 75].

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Chapter 4 Single Photon Path Entanglement

Distributing entangled states between two or more distant parties is of fundamental importance to quantum communication, as this allows protocols such as quantum teleportation [12,13,14], in which a qubit is sent from one place to another, and device independent quantum key distribution (DIQKD), in which the security of the protocol can be certified by the violation of Bell inequalities [9, 93, 11].

This chapter presents the generation and detection of a so called single photon path entangled state, which consists of one photon superposed between two or more paths. This state generation not only requires fewer resources when compared to other entangled states, but also can be easily scaled to multipartite scenarios. Among other features, this state can be heralded by a photon pair source and its single photon character makes it more robust to loss and detector inefficiency than the polarization entangled state [19].

4.0.1 Path Entangled states

Since its conception, quantum correlations always involved quantum states with at least two particles. However, as argued by Tan et al., [94] a state with only one photon delocalized between two or more spatial modes must also demonstrates quantum correlations. The first proposal to detect the quantum behavior of this state, now known as single photon path entangled state, or just path entanglement, was made by [94] and further improved by [95, 96, 97, 98], while its detection was experimentally verified by photon counting [99, 26, 23] and continuous variables techniques [100, 101].

Path entangled states are suitable to be used on quantum networks and 39

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40 CHAPTER 4. SINGLE PHOTON PATH ENTANGLEMENT

Figure 4.1: Fig (a) shows the procedure to announce a bipartite path entangled state by sending a heralded photon on a beam splitter. Fig (b) shows the procedure to scale from bipartite entangled state to multipartite entangled state just by adding beam splitters in the photon’s optical path. The advantage of this procedure is that the rate in which entanglement is generated only depends on the heralding rate of the initial photon.

present several advantages when compared to other entangled states composed of two photons (now denoted 2-photon entangled state). Among these advantages, there is the ease of production, exemplified in Fig 4.1 (a), in which a heralded single photon path entangled state is generated after a heralded photon is sent to a beam splitter (BS). The rate in which the entanglement is heralded only depends on the rate of the heralding photon, which for SPDC process can be at the MHz range [56]. This approach generates heralding rates that are order of magnitudes higher than recent works that heralded 2-photon states [102, 103].

As shown in Fig 4.1 (b), path entangled states are easy to scale from bipartite to multipartite scenarios by just concatenating more BSs along each photon path. This procedure does not affect the rate of generated entangled states, which is only given by the rate of the heralded photons.

Another advantage of this state compared to any 2-photon entangled states is the robustness against loss. This robustness is due to its single photon nature, in which the probability of finding the state after it travels through a medium with transmission η is proportional to η while the same probability is η² for any state composed of two photons. This property is clear at eq. 4.1, which shows how the state |Ψi = ^|10i^ab^√^+|01i^ab

2 , written in the {0,1} photon number subspace, evolves when it travels through a channel with transmissionη_a and η_b in paths a and b, respectively. Not only the h00|ρ|00i element, corresponding to the vacuum component, is linearly affected by the non ideal transmission to become 1−^η₂^a−^η₂^b, but the coherence terms h10|ρ|01i and h01|ρ|10i are also linearly affected, and become

√ηaηb

2 . The transmissionsηa,b can also include detection efficiency, and in this case, eq. 4.1 also shows how ρ is affected by the detectors.

Single photon path entangled states for quantum communication

Thesis

Reference

Single photon path entangled states for quantum communication

Single Photon Path Entangled States for Quantum Communication

THÈSE

Fernando Henrique do Rêgo M ONTEIRO de Rio de Janeiro (Brésil)

Thèse N

4998

2016

Abstract

R´ esum´ e

Contents

Chapter 1 Introduction

1.1 Quantum Communication

1.2 Entanglement: Generation, distribution and detection

1.3 This Thesis

Chapter 2

Stable Light Source

2.1 Introduction

2.2 Stable Source

2.2.1 Principle of Operation

2.2.2 Stable source components

2.2.3 Results

2.2.4 Conclusion

Chapter 3

Narrowband Photon Pair Source

3.1 Introduction

3.1.1 SPDC

3.1.2 Photon Purity

3.2 Optical Parametric Oscillator - Cavity SPDC

3.2.1 Purity, Bandwidth and Wavelength

3.2.2 Brightness

3.3 Experiment

3.3.1 Experimental Description

3.3.2 OPO Design and Alignment

3.3.3 Photon pair spectrum

3.3.4 Coincidence Measurements and Enhancement fac- tor

3.3.5 Purity

3.4 Conclusion

Chapter 4

Single Photon Path Entanglement

4.0.1 Path Entangled states

Fernando Henrique do Rêgo M ^ONTEIRO de Rio de Janeiro (Brésil)