PALACKY UNIVERSITY

FACULTY OF SCIENCE

DEPARTMENT OF OPTICS

DIPLOMA THESIS

Optical frequency conversion and non-classical lightgeneration

Thesis supervisor:

Mgr. Miroslav Jezek, Ph.D.

May 17, 2012

Author:

Bc. Ivo Straka

Optics and Optoelectronics

Abstract

The thesis presents a non-classical light source based on a frequency conversion. The

source produces correlated photon pairs, which are generated via spontaneous paramet-

ric down-conversion in a BBO crystal. Such pairs may also be employed as a source of

single photons, which are heralded by a presence of another photon. The thesis gives a

brief theoretical overview of the down-conversion process and describes three measure-

ments: Hong-Ou-Mandel dip, anticorrelation parameter and quantum non-Gaussianity

criterion. An experimental part follows, where the optical setup is presented and the

aforementioned measurements discussed. The results show good quality of generated

photon pairs, exhibiting up to 99 % of Hong-Ou-Mandel visibility. Non-classical nature

of the single-photon source is then demonstrated by anticorrelation measurement, which

indicated strong anti-bunching, and quantum non-Gaussianity criterion, which witnesses

non-Gaussian character with a certainty of hundreds of standard deviations.

Keywords

Photon pairs, down-conversion, single-photon source.

UNIVERZITA PALACKEHO

PRIRODOVEDECKA FAKULTA

KATEDRA OPTIKY

DIPLOMOVA PRACE

Frekvencnı konverze a zdroje neklasickeho svetla

Vedoucı prace:

Mgr. Miroslav Jezek, Ph.D.

17. kvetna, 2012

Autor:

Bc. Ivo Straka

Optika a optoelektronika

Abstrakt

Prace pojednava o zdroji neklasickeho svetla zalozenem na frekvencnı konverzi. Zdroj

vytvarı korelovane pary fotonu generovane spontannı sestupnou parametrickou konverzı

v krystalu BBO. Tyto pary mohou byt pouzity jako zdroj jednotlivych fotonu, jejichz

prıtomnost je podmınena detekcı dalsıho fotonu. Prace predklada strucny teoreticky

prehled o procesu sestupne konverze a popisuje tri merenı: Honguv-Ouuv-Mandeluv dip,

antikorelacnı parametr a kriterium kvantove negausovskosti. Nasleduje experimentalnı

cast s popisem optickeho usporadanı experimentu a jsou diskutovana zmınena merenı.

Vysledky ukazujı na dobrou kvalitu generovanych paru, ktere vykazujı az 99 % visi-

bilitu dipu. Neklasicky charakter jednofotonoveho zdroje je pak demonstrovan antiko-

relacnım parametrem, ktery ukazuje na silne antishlukovanı fotonu, a kriteriem kvantove

negausovskosti, ktere je poruseno na stovky standardnıch odchylek.

Klıcova slova

Fotonove pary, sestupna konverze, zdroj jednotlivych fotonu.

Declaration

I hereby state that I have written this thesis myself under the supervision of Mgr.

Miroslav Jezek, Ph.D. All used resources are listed under References.

Acknowledgements

Above all, I thank my supervisor Mgr. Miroslav Jezek, Ph.D. for his endless patience,

support and fruitful discussions. I also thank the rest of the lab team, Mgr. Martina

Mikova, Bc. Helena Fikerova, Mgr. Michal Micuda, Ph.D. and prof. RNDr. Miloslav

Dusek, Dr. for valuable cooperation and stimulating environment. My unending grati-

tude belongs to my family, which has been supporting me during my studies.

In Olomouc, May 17, 2012

Contents

1 Introduction 1

2 Theory 3

2.1 Spontaneous parametric down-conversion . . . . . . . . . . . . . . . . . . 3

2.2 Phase-matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3 Walk-off . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.4 Focus of the pump . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.5 Hong-Ou-Mandel effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.6 Single-photon source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.6.1 Anticorrelation parameter . . . . . . . . . . . . . . . . . . . . . . 13

2.6.2 Non-Gaussianity criterion . . . . . . . . . . . . . . . . . . . . . . 16

3 The experiment 18

3.1 The experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.1.1 Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.1.2 Down-conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.1.3 Compensation and filtering . . . . . . . . . . . . . . . . . . . . . 18

3.1.4 Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.1.5 Modelling the layout . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2 Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2.1 Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2.2 Detection electronics . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.3 Discussion of losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.4 SPDC alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.5 Hong-Ou-Mandel dip measurement . . . . . . . . . . . . . . . . . . . . . 24

3.6 Single-photon source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.7 Anticorrelation parameter . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.8 Non-Gaussianity criterion . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.9 Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.10 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4 Conclusion 35

1 Introduction

The aim of this thesis is to design, experimentally implement and characterize a source

of photon pairs using optical frequency conversion. Such source should ideally produce a

pair of time-correlated photons, which are indistinguishable from each other. That means

we can select some degree of freedom such as polarization, in which information will be

stored. Apart from such degrees of freedom, the photons must be able to coherently

interfere. This is vital for photonic implementations of quantum logic gates, which are

the main goal of constructing such sources. Quantum gates usually need at least two

photons carrying information and perform an operation using some sort of interaction

between them. This interaction includes interference on beam splitters, the visibility of

which limits the operation of the gate. It is therefore essential to produce photons which

are capable of perfect interference. Furthermore, quantum gates require a specific photon

statistics. Quantum gates can work properly only with certain expected number of input

photons. For two-photon gates it means we need to produce a Fock state |1〉⊗ |1〉, which

exhibits highly non-classical properties and represents a significant challenge in the field

of experimental quantum optics. While the vacuum itself does not diminish the quality

of the prepared photons, it affects overall photon rate and poses a practical limitation

for the time frame of the experiment. To reconcile, we have to strive not only for ideal

photon indistinguishability and highly non-classical statistics, but also keep the photon

rate as high as possible.

This thesis presents an approach which allows us to design and construct such source

using spontaneous parametric down-conversion in a non-linear BBO crystal. In the first

part, an introduction is given regarding the physical basis of spontaneous parametric

down-conversion. Factors relevant to the experiment are presented, which include phase-

matching by crystal alignment and spatial walk-off requiring compensation. In the next

part, a Hong-Ou-Mandel effect is described, as well as its significance to the photons’

undistinguishability. Furthermore, a single-photon regime is described, where the de-

tection of one converted photon heralds the presence of another. Such photons exhibit

anti-bunching statistics, which is characterized by anticorrelation parameter—a measure

akin to second-order correlation function—presented in the following part. A condition

for single-photon quantum states, the non-Gaussianity criterion, is subsequently pre-

sented as the second measure of the single-photon source. In the experimental part,

the optical setup is described with its components and detection electronics. Next, all

measurements—Hong-Ou-Mandel dip, anticorrelation and non-Gaussianity—are evalu-

ated with respect to their limits. In the end, the results favorably point out the high

quality of the source and non-classical character of the converted light, which fulfills the

experimental goal of the thesis.

1

2

2 Theory

2.1 Spontaneous parametric down-conversion

The spontaneous parametric down-conversion (SPDC) is a second-order nonlinear pro-

cess, a three-wave interaction, where two modes are populated by a conversion of a

pumping mode. It occurs in a nonlinear material with a sufficiently large second-order

susceptibility χ(2). We assume only one populated optical mode, which we call the pump,

with a strong amplitude V, wave vector k0 and angular frequency ω0. Interaction with the

medium causes other modes, previously in vacuum states, to be populated, hence sponta-

neous generation. The particular mode structure is given by the interaction Hamiltonian

[1]

HI(t) =1

L3

∑k′,s′

∑k′′,s′′

Vl χ(2)lij (ω0, ω

′, ω′′)(ε∗k′′,s′′)j(ε∗k′,s′)i (2.1)

×∫Vei(k0−k′−k′′)rei(ω

′+ω′′−ω0)t a†k′,s′ a†k′′,s′′ d3r + h.c.

Here we consider an interaction volume V with conversion to frequency/spatial modes

denoted by vectors k′,k′′, frequencies ω′, ω′′ and polarizations s′, s′′. The annihilation

operator of the pump mode has been replaced by classical amplitude due to its strong

nature, thus making the interaction non-resonant or parametric. The resulting state is

then obtained in

|ψ(t)〉 = Exp

[− i~

∫ t

0

HI(t′)dt′

]|ψ(0)〉 . (2.2)

The integration over time and interaction volume gives

|ψ(t)〉 ≈ |0〉 |0〉+∑k′,s′

∑k′′,s′′

φ(k′,k′′, s′, s′′)3∏

m=1

sinc

[1

2(k0 − k′ − k′′)mlm

](2.3)

× t ei(ω′+ω′′−ω0)t/2 sinc

[1

2(ω′ + ω′′ − ω0)t

]|k′, s′〉 |k′′, s′′〉+ . . .

We can see that both the expression ω′ + ω′′ − ω0 and k0 − k′ − k′′ must be very close

to zero, or else these modes are quickly attenuated and do not significantly contribute.

These effective conditions are referred to as phase-matching and can be viewed as energy

and momentum conservation for interacting photons. As a result, generated frequencies

are lower than the pumping frequency, which gives the name down-conversion. We can see

that the length of the crystal (and interaction time) place a boundary on phase-matching

sensitivity. For relatively long crystals, we need spectrally narrow pumping in order to

use the whole body of the crystal. After long interaction times t, only a very narrow

3

spectrum contributes to the desired modes due to the sinc term in (2.3).

Since the nonlinear medium employed by our experiment (a β-barium borate) is

also anisotropic, the wave vector condition is determined not only by the spectra, but

direction-dependent refractive index as well. This yields a complex spatial mode structure

depending on the type of generation, the medium and its alignment. In our experiment,

we have used a type-II down-conversion in a uniaxial BBO crystal, which means the

pumping wave must enter the crystal in extraordinary polarization, producing s′, s′′ as

ordinary and extraordinary polarization modes, respectively. The desired result to be

achieved is generating spectrally degenerate photons

ω′ = ω′′ =1

2ω0, (2.4)

called signal and idler, in a single coaxial spatial mode (see Fig. 2.1). These modes would

be split on a polarizing beam-splitter and coupled into single-mode fibers (see Fig. 2.2).

Figure 2.1: The collinear type-II spontaneous parametric down-conversion.

For theoretical models in this thesis however, the spatial and spectral mode structure

is not paramount. By fiber coupling and spectral filtering we are able to reduce these

effects significantly. In further discussion, the photon statistics will be the main focus

and a reasonably simple model is required.

Let us assume that we have filtered the perfectly phase-matched collinear state and

that the pump is single-mode. The exponentials in (2.1) vanish, making the Hamiltonian

time-independent. If we expand (2.2) in a series, we can express all remaining factors by

a single gain coefficient√g, getting

|ψ〉 =1

1− g∑n

1

n!

√gna′na′′n |00〉 =

1

1− g∑n

√gn |nn〉 , (2.5)

where we have let out the phase factors, which do not play any role. This is a very

4

Figure 2.2: The schematic setup of the source of correlated photon pairs.

simplified statistical model, yet extremely useful for some calculations.

2.2 Phase-matching

To obtain the necessary crystal alignment (θ in Fig. 2.1) that would enable this ideal

collinear generation, we need to meet the phase-matching condition

k0(θ) = k′ + k′′(θ). (2.6)

Using (2.4), we get the condition for refractive indices

2ne(ω0, θ) = n0(ω0/2) + ne(ω0/2, θ), (2.7)

which are dependent on polarization (e/o), optical frequency and the tilt of the crystal.

The manufacturer provides the Sellmeier equations for λ[µm]

n0(λ) =

√2.7405− 0.0155λ2 +

0.0184

λ2 − 0.0179, (2.8)

ne(λ) =

√2.373− 0.0044λ2 +

0.0128

λ2 − 0.0156, (2.9)

where ne denotes the extremal case for ne(90). For uniaxial crystals, the angular depen-

dence of ne takes form of an ellipsis [2]

1

n2e(θ)

=cos2(θ)

n20

+sin2(θ)

n2e

. (2.10)

5

By substituting our pump wavelength 407 nm to (2.7), we get θ = 41.2, while the crystal

is cut for 42.7, requiring a tilting alignment.

2.3 Walk-off

In anisotropic media, the propagation of energy (Poynting vector) does not have the

same direction as the wavefront normal. This causes the pump beam to divert from the

optical axis inside the crystal by an angle β (see Fig. 2.3). Since the generated modes

Figure 2.3: The walk-off affects generated modes.

have orthogonal polarizations, the directin of propagation varies as well. For a Gaussian

beam, the (extraordinary) signal beam retains its profile, while the idler gets transversally

convoluted with a square function, since the generation happens at different points along

the way (Fig. 2.3). The resulting width is proportional to β and crystal length. This is

called the walk-off effect and reduces the coupling efficiency into single-mode fibers for

idler photons because of the deformed beam profile.

We can quantify this effect by calculating the walk-off angle. The Poynting vector is

always normal to the refraction-index ellipse [2]. Therefore, we can approximate

β ≈ − 1

n(θ)

∂n(φ)

∂φ

∣∣∣∣φ=θ

≈ 4. (2.11)

2.4 Focus of the pump

It has been shown for some nonlinear processes to be dependent on the focus of the

pump [3]. For spontaneous down-conversion, there is a theoretically optimal Gaussian

waist size, which gives maximum generated intensity to all output spatial modes [4].

However, if we focus the beam too strongly, the walk-off effect will considerably impair

6

single-mode-fiber coupling efficiency. Therefore, we need to combine these two effects to

calculate the optimal focus.

First, it is necessary to make certain approximations. We will consider a closely phase-

matched case, ∆k k0,k′,k′′. An achromatic material response will be assumed. Then,

from the interaction Hamiltonian, we can calculate the amplitude [4]

ψ(ω′, ω′′) ∝ 1√A+B+

∫ 1

−1

√ξ exp(iΦl/2)

1− iξldl, (2.12)

A+ = 1 +k′ξ′

k0ξ0+

k′′ξ′′

k0ξ0, (2.13)

B+ =

(1− ∆k

k0

)(1 +

k′ + ∆k

k0 −∆k

ξ0ξ′

+k′′ + ∆k

k0 −∆k

ξ0ξ′′

), (2.14)

ξ =B+

A+

ξ′ξ′′

ξ0, (2.15)

where the parameters ξ0, ξ′, ξ′′ are defined as a ratio of the respective confocal parameter

and crystal length L,

ξA =L

kAw2A

. (2.16)

The idler amplitude distorted by the walk-off can be approximated as

ι(x, y) =

∫ ∞∞

1√Aπw2

e−x2+y′2

w2 U

(y − y′

A

)dy′, (2.17)

where U(y) denotes a unit rectangular function on an interval (−12, 12) and A is the

vertical walk-off, A ∝ β, where β is the walk-off angle (2.11). The coupling mode overlap

is Gaussian,

o =

∫ ∞∞

ι(x, y)1√πw2

e−x2+y′2

w2 dxdy. (2.18)

The intensity coupled to the fibers is then proportional to the square of both the generated

amplitude and mode overlap,

I ∝ |ψ|2 o2. (2.19)

The numerically obtained result is shown in Fig. 2.4 with the optimal waist size of

18 µm for non-linear material thickness of 2 mm and walk-off angle of 4. Because of

narrow slope of the total coupled output signal for sub-optimum waist size and quite large

uncertainty of design parameters, we prefer to set the waist slightly above the optimum

value where the overall stability is better. Another reason is to balance the signal and

idler rates. The idler coupling efficiency in the optimum focus is 43 % relative to the

signal, while for 28 µm waist it is 63 %.

7

10 20 30 40

0.000

0.002

0.004

0.006

0.008

0.010

w@ΜmD

coup

led

inte

nsity

@a.u.

D

Figure 2.4: The dependence of the intensity coupled to single-mode fibers with respectto the waist size of the pump beam.

2.5 Hong-Ou-Mandel effect

This quantum effect was demonstrated by Hong, Ou and Mandel in [5]. It is the basic

measure of photon indistinguishability [6]. Since our source should ideally provide pairs

of identical photons, one at each arm, this measure allows us to establish a certain merit

of quality. In the following section, the Hong-Ou-Mandel effect will be explained and

discussed.

Let us consider two photons incident on a beam splitter, one in each mode. The input

modes are populated by creation operators a†, b†, while the output modes are denoted by

c†, d†. The beam splitter transformation takes the form

a† =√T c† +

√1− T d†, (2.20)

b† =√

1− T c† −√T d†,

where T is the transmittance of the beam splitter. The single-pair portion of SPDC state

then becomes

|11〉 BS−→√T (1− T )(|20〉 − |02〉) + (1− 2T ) |11〉

1− 2T (1− T ). (2.21)

For a balanced beam splitter (T=1/2), there is zero probability of detecting the presence

of a photon in both modes simultaneously. However, our detectors measure many modes

simultaneously, such as spectral, polarization and time modes. The beam splitter, meant

as an optical component, carries out the transformation (2.20) for all these modes, with

only two of them interfering in one such transformation. If the incident photons are in a

general superposition of different modes, the result will be more complicated than (2.21).

8

Let us assume the input states

|ψa〉 =∑k

√pa,k a

†k |0〉 , (2.22)

|ψb〉 =∑l

√pb,l b

†l |0〉 . (2.23)

Here we assume a complete set of orthogonal modes for each arm, where a†n and a†n are

transformed by (2.20) to c†n and d†n. Applying the trasformation, we get

|ψ〉 =∑k,l

√pa,kpb,l a

†kb†k |0〉 =

=∑k,l

pa,kpb,l

(√T (1− T )c†kc

†l − T c

†kd†l + (1− T )d†kc

†l −√T (1− T )d†kd

†l

). (2.24)

Figure 2.5: The Hong-Ou-Mandel effect.

Only the middle two terms represent a photon in each arm (see Fig. 2.5). Since[c†k, c

†l

]= δkl and

[d†k, d

†l

]= δkl, these terms are orthogonal for k 6= l and we can sum

their probabilities to get a probability of a simultaneous detection, called a coincidence.

However, for k = l, these terms coherently add together. Expressed in one equation, the

probability of a coincidence is

Pc =∑k,l

pa,kpb,l [1− 2(1 + δkl)T (1− T )] . (2.25)

Alternatively, we can measure coincidences for both photons in completely distinguishable

modes, each of which is subject to its own transformation 2.20. Thus, we can determine

the probability of a coincidence from the sheer statistical analysis: either both photons

9

must be transmitted or reflected, which gives a probability

P0 = T 2 + (1− T )2. (2.26)

This state can be reached by introducing a time difference between the incident photons.

By increasing the difference, the overlap of the photons’ wavefunctions after the beam

splitter decreases, until the time largely exceeds their coherence time, making the photons

arrive at the beam splitter independently. Having the probabilities (2.25) and (2.26) for

these extremal cases, we define Hong-Ou-Mandel visibility

V =P0 − PcP0

=2

T1−T + 1−T

T

Tr [ρaρb] , (2.27)

which is valid for incident mixed states as well. Here, both density matrices are considered

to have a common Hilbert space and to be separable. We can see that the term containing

beam splitter transmissivity accounts for experimental imperfections, while the trace

represents our measure of indistinguishability—ideally close to 1. The visibility detected

will always be lower than the trace itself, giving us a safe estimate for photon-pair quality.

2.6 Single-photon source

By the means of conditional detection, we are able to generate an approximation of

attenuated single-photon state. We choose only those time windows in the signal arm,

where the photons coincide with idler detections. The idler serves as trigger, which would

ideally herald a state with one or more photons present in the signal arm within a certain

time (coincidence) window. The probability of n photons detected in the coincidence

window decreases exponentially with n, while the losses γ in the signal arm contribute to

the vacuum portion of the state. In small gain approximation, we can consider our scheme

as a heralded single-photon source, which corresponds to a triggered state γ |0〉〈0|+ (1−γ) |1〉〈1|.

The generated state in the co-axial coupled modes before the losses reads

|ψ〉 =√

1− g∞∑n=0

√gn |nn〉 ≈ 1√

1 + g + g2(|00〉+

√g |11〉+ g |22〉) , (2.28)

where g denotes down-conversion gain, which is linearly dependent on pumping power

and assumed to be g 1. Let η and τ be overall transmittances of signal and trigger

(idler) modes. We consider these parameters to contain all losses in the experiment,

including coupling and detector efficiency. Let us denote the signal mode S, trigger T ,

signal loss mode LS and trigger loss mode LT . The initial state from (2.28) is

10

Figure 2.6: Single-photon source.

ρ0 = |ψ〉〈ψ|S,T ⊗ |00〉〈00|LS ,LT. (2.29)

Losses are introduced as an imaginary beam splitter between a photon channel and the

corresponding loss mode. In bra-ket notation, the operator applying losses to mode M

with transmittance γ is

LM(γ) =∞∑n=0

n∑k=0

√γn−k(1− γ)kn!

(n− k)!k!|n− k〉〈n|M ⊗ |k〉〈0|LM

. (2.30)

We obtain the resulting state by tracing over both loss modes. We get

ρ = TrLS ,LT

[LS(η)LT (τ)ρ0L

†T (τ)L†S(η)

]. (2.31)

Finally we perform a detection in the trigger using POVM operator

Π = 1− |0〉〈0| , (2.32)

which yields a heralded state

ρS =TrT [ΠTρ]

Tr [ΠTρ]. (2.33)

11

Substituting particular parameters, we get

ρS = p0 |0〉〈0|+ p1 |1〉〈1|+ p2 |2〉〈2| , (2.34)

p0 =[1 + g(2− τ)(1− η)] (1− η)

1 + g(2− τ),

p1 =[1 + 2g(2− τ)(1− η)] η

1 + g(2− τ),

p2 =g(2− τ)η2

1 + g(2− τ).

with a trigger probability

pT = Tr [ΠTρ] =g(1 + g(2− τ))τ

1 + g + g2, (2.35)

where τ > 0 and g > 0 are assumed. In the limit of g → 0, τ → 0,

p0 = 1− η, (2.36)

p1 = η,

p2 = 0.

This is the ideal single-photon state, which is the desired result. Unfortunately, in this

limit the probability of trigger pT = 0. Next, the influence of the parameters on ap-

proaching this state with a reasonable success rate will be discussed.

Losses in the signal η are responsible for increasing the portion of vacuum at the

expense of higher Fock states. Since the ideal output state only consists of vacuum and

single-photon state (in any proportion), it is theoretically beneficial that higher Fock-state

contributions vanish. With increasing losses, they decrease with higher power of η than

single-photon state, which improves the output state. Experimentally, we need a certain

minimum photon rate to successfully employ the source in an experiment. Here we are

usually restrained by the time available, but detector noise can be an issue as well. With

a relatively high dark count rate, the single detections contain a large portion of false

positives, which exhibit entirely different statistical properties, and of course represent no

real photons. Therefore, we cannot decrease η arbitrarily. Depending on measuring time

requirements, there is always a trade-off between quality and measurement duration. For

a fixed measurement time, the trade-off can be formulated as a statistical confidence of

the state quality.

Losses in the trigger affect overall rate as well as the signal state. With higher losses,

12

the rate drops, while the p2 probability rises. For small g, p2 can be doubled by lowering

τ from 1 to nearly 0. Unfortunately, p2 is critical for the quality of the single-photon

source, as will be shown in further criterions. It is the dominant contribution in the

deviation from the perfect single-photon state (2.36). In contrast with η, it is therefore

convenient to keep the τ transmittance as high as possible.

The overall rate (2.35) is proportional to gτ (neglecting smaller quadratic contribu-

tion), while the term appearing in the probabilities is g(2− τ). As mentioned above, we

wish to lower p2, while maintaining satisfactory rate. In contrast with τ , g has the same

sign in both terms, which gives rise to another trade-off: by lowering gain, we improve

the state, but decrease the trigger rate.

In the following sections, methods will be presented, which provide characterization

of the single-photon source.

2.6.1 Anticorrelation parameter

Single-photon sources exhibit photon anti-bunching, 〈(∆n)2〉 < 〈n〉, which cannot be

explained by classical optics. Let us demonstrate this by a cirrekatuib measurement [7]

shown in Fig. 2.7.

Figure 2.7: Anticorrelation measuring.

The input signal is split on a 50:50 beam splitter and detected by photodetectors 1

and 2, which measure count rates in N time windows, each of length τ . If we denote

single rates on the detectors N1, N2 and coincidences as NC , then the probabilities of

such events are directly proportional to the rates:

p1 =N1

N, p2 =

N2

N, pC =

NC

N. (2.37)

Let us express these probabilities in terms of semi-classical photodetection. In the n-th

13

window opened at the time tn, the probability of photon detection is proportional to the

integrated intensity

in =1

τ

tn+τ∫tn

I(t)dt. (2.38)

The probability of a coincidence is a product of both singles’ probabilities, therefore being

proportional to i2n. By averaging over all time windows, we obtain the probabilities

p1 = γ1τ 〈in〉 , p2 = γ2τ 〈in〉 , pC = γ1γ2τ2⟨i2n⟩. (2.39)

Proportion constants γ1,2 include detection efficiencies and beam splitter imbalances. The

variance of in is non-negative,

⟨(∆in)2

⟩=⟨i2n⟩− 〈in〉2 ≥ 0, (2.40)

implying

pCp1p2

=〈i2n〉〈in〉2

≥ 1. (2.41)

We define the anticorrelation parameter α by means of count rates [7]

α =pCp1p2

=NcN

N1N2

≥ 1. (2.42)

If α < 1, the coincidence rates are below the classical limit. This intuitively corresponds

to low multi-photon contributions in the analyzed state, which are the only ones that can

split on the BS and cause a coincidence event.

Since we wish to work only with experimentally measurable count rates, we need to

eliminate N from (2.42). This can be achieved by coinciding distant time windows n and

m, so that |tm − tn| = ∆t is much greater than coherence time. Then we can assume no

correlation between the intensities

pC(∆t) = γ1γ2τ2 〈inim〉︸ ︷︷ ︸〈in〉〈im〉

= p1p2, (2.43)

NC

N=N1N2

N2, (2.44)

α(∆t) = 1. (2.45)

From the corresponding rates, we can calculate N and redefine

14

α(t) =Nc

N1N2

, α(t 0) = 1 (2.46)

with N1, N2 and NC normalized with respect to their values for t 0.

If we assume pn+1 pn in accordance with (2.28), we can calculate the detection

probabilities based on photon-number statistics assuming beam splitter transmittance T .

By a straightforward analysis of transmission/reflection cases we obtain

N1 = Tp1 + p2[1− (1− T )2

], (2.47)

N2 = (1− T )p1 + p2(1− T 2

), (2.48)

NC = 2p2T (1− T ), (2.49)

which leads to

α ≈ 2p2(p1 + p2(2− T ))(p1 + p2 + p2T )

≈ 2p2p21. (2.50)

In this approximation, it can be calculated from (2.34) that

α ≈ 2g(2− τ), (2.51)

with g being the gain of SPDC and τ being losses in the trigger arm. This result shows

that if we have certain inevitable losses in each arm, it is always better to choose the less

lossy one as trigger. Not only it provides better count rate, it improves the non-classical

statistics in the view of anticorrelation parameter.

This approach to measuring non-classical statistics is a simplification of quantum

second-order correlation function measurement [8]. It is defined as

g(2)(τ) =

⟨a†(t+ τ)a†(t)a(t)a(t+ τ)

⟩〈a†(t)a(t)〉 〈a†(t+ τ)a(t+ τ)〉

, (2.52)

where the averaging is carried out over time. For τ τc, the time modes become inde-

pendent, making [a(t), a(t+ τ)] = [a(t), a†(t+ τ)] = 0. Like α, we get the normalization

g(2)(∞) = 1. (2.53)

For τ = 0, we get

g(2)(0) =

⟨a†

2a2⟩

〈a†a〉2. (2.54)

Assuming a density matrix ρ with a photon-number distribution pn = 〈n| ρ |n〉 and keep-

15

ing the approximation pn+1 pn, we arrive at

g(2)(0) =

∑∞n=2 n(n− 1)pn

(∑∞

n=1 npn)2 ≈ 2p2

p21. (2.55)

This shows that the anticorrelation parameter is a simplified measurement, which corre-

sponds to g(2) for our weak-pumping approximation (2.50).

2.6.2 Non-Gaussianity criterion

If we analyze the output signal mode of the heralded single-photon source, we are able to

determine whether its quantum state is beyond a classical mixture of Gaussian states. A

criterion introducing a sufficient condition for quantum non-Gaussianity was proposed by

R. Filip and L. Mista in [9], with an experimental test published by M. Jezek, I. Straka,

M. Micuda, M. Dusek, J. Fiurasek and R. Filip in [10].

Gaussian state is defined as having Gaussian Wigner function; for a single-mode pure

state it can be defined in terms of displacement and squeeze operators:

|λ〉 = S(r, ψ)D(β) |0〉 . (2.56)

If there exists a (non-negative) probability distribution P (λ) of the multiindex λ, such

that we can express the state by a density matrix

ρc =

∫P (λ) |λ〉 〈λ| dλ, (2.57)

we say that the state is a classical mixture of Gaussian states. If no such distribution

exists, it is quantum non-Gaussian.

In the space of photon-number probabilities p0 and p1, there is a convex set G, which

contains such photon number probabilities, which are achievable by convex Gaussian

mixtures (see Fig 2.8). Its boundary is parametrised by

p0 =e−d

2[1−tanh(r)]

cosh(r), p1 =

d2e−d2[1−tanh(r)]

cosh3(r), (2.58)

where d2 = (e−4r − 1)/4 and r ≥ 0 is the squeezing parameter. Therefore, for a given p0,

there is a maximum p1,max for which ρ ∈ G. If p1 > p1,max, the state cannot be expressed

in the form (2.57) [9].

In order to have an experimentally feasible measure of quantum non-Gaussianity, we

define a witness

W (a) = ap0 + p1, a ≤ 1. (2.59)

16

Figure 2.8: All states outside G are non-Gaussian [10].

The purpose of this witness is to quantify the “distance” of a given point from G, in

pursuit of which we parametrise all tangents to the boundary of G using (2.59). In the

p0 − p1 space of Fig. 2.8, it represents a line containing the point [p0, p1] with a slope

−a. We can see that W represents vertical position of this line on the axis p0 = 0.

Let us substitute the border probabilities (2.58), fix the slope a and maximize over the

parameter r. Since the border is represented by upper maximum of p1 for each p0, such

maximization yields witnesses WG of all tangents to G in a form of extremal equation

(1 + e2r)a = e2r(3− e2r). (2.60)

Now we define the measure as a maximum difference

∆W (p0, p1) = maxa∈(−∞,1〉

[W (a, p0, p1)−WG(a)] . (2.61)

This maximization represents scanning all tangents to G while leading a parallel line

through tbe measured point. ∆W then represents vertical distance between these two

lines. Since G is convex, it is easy to see that the maximum occurs for the tangent line

touching the border at the p0 measured. Hence the interpretation for ∆W is a difference

between the measured p1 and its maximum possible value for Gaussian mixtures with

the same p0. This conveniently corresponds with the interpretation of the criterion itself

mentioned above, and quantifies the single-photon contribution exceeding the Gaussian

boundary.

17

3 The experiment

3.1 The experimental setup

3.1.1 Lasers

The down-conversion photon source is pumped by 407 nm Coherent Cube laser diode

at 50 mW of continuous power. The beam waist (1/e2 diameter) at the laser output is

1.4 mm with divergence of 0.4 mrad, which gives M2 = 1.08.

A probing beam is used for adjusting the components designed for down-converted

photons at 814 nm. It is provided by a fiber-coupled OZ Optics laser diode at 1 mW

continuous power and 814 nm central wavelength. It enters the experiment through the

collimator C3.

A dichroic mirror DM directs both pumping and probe laser beams to the BBO

crystal.

3.1.2 Down-conversion

The pumping beam is focused by a lens L1 (f = 15 cm, 40.0 cm after the laser) to a BBO

crystal, which is 2 mm thick and cut for type-2 generation. The beam enters the crystal

as an extraordinary wave, generating two orthogonal polarization modes bound with the

crystal symmetry—ordinary (H) and extraordinary (V) linear polarization. This requires

the crystal to be rotated so its optical axis is in a plane that includes polarization of

the pump and the axis of the pump beam. In order to achieve collinear generation, the

BBO optical axis has to form an angle ϑ = 41.2 with the pump axis. The beam waist

in the crystal (1/e2 width) is 55 µm. After the generation, the down-converted beam is

collimated by a second lens L2 (f = 17.5 cm, located 18.4 cm after the crystal).

3.1.3 Compensation and filtering

Due to group velocity dispersion and spatial walk-off inside the generating crystal, another

BBO is needed that would compensate these differences between the polarization modes.

First, the modes are switched by a half-wave plate HWP1. Then they pass through a

second BBO crystal, which has the same alignment as the generating crystal. The first

goal is that each polarization mode would cover an equal distance inside crystals as an

ordinary and extraordinary wave. The second goal requires that both generated modes are

transversally centered (see Fig. 2.3), since the convoluted ordinary mode has the highest

power density in its center. This yields the required thickness of the compensating BBO

crystal to be exactly half the thickness of the generating crystal, that is 1 mm.

18

Figure 3.1: The scheme of the experimental setup.

With dispersion and spatial modes corrected, the next issue is spectrum. It is neces-

sary to cut off the 50 mW of pumping power, which is achieved by 3 cut-off filters reducing

the pump power by 10−14, reducing the intensity below dark counts of the detectors. The

first Semrock filter F1 is placed between the generating crystal and collimating lens to

prevent parasitic reflections on the lens coating. The other two (Semrock and Eksma)

are placed at position F2.

Additionally, introducing an Andover interference filter, only a narrow spectral band

is allowed to pass, increasing the coherence of the pairs. The filter has a high transmission

band centered in 814 nm, FWHM 2.0 nm. Outside this band, transmission is negligible.

All employed interference filters utilize thin layers coating.

3.1.4 Coupling

By having the pump cut off, the generated light can be coupled to optical fibers. The

two polarization modes are split on CVI Melles Griot PBS1, with horizontal polarization

transmitted and vertical polarization reflected. In order to separate the pairs well, the

polarization separation must be as good as possible. The PBS has a very high extinction

ratio for the transmitted mode, effectively making at least 99.8 % of transmitted light

horizontally polarized. Unfortunately, the reflected mode contains parasitic polarization

in magnitude of 2 %. This is solved by flipping the polarization by a half-wave plate

HWP3 and guiding the light through PBS2, which is the same type as PBS1. After

being separated, both polarization modes need to be coupled to optical fibers in order to

19

maintain perfect spatial mode correlation. Each photon is focused by an aspherical lens

(Thorlabs C220-TME-B, f = 11 mm) to a face of an optical fiber. The required Gaussian

mode, which has the maximum overlap with the fiber mode, has a waist (1/e2 width) of

5 µm located on the face of the fiber.

The photon time correlation needs to be adjusted for purposes of measurements. For

this purpose, the length of one arm is adjusted by motorized corner reflector made of two

mirrors mounted on a Newport MFA-CC motorized linear stage with spatial resolution

of 0.0175 µm and bi-directonal repeatability of 1.5 µm.

The fiber outputs are connected to the target experiment that requires correlated

photons, or a measurement setup.

3.1.5 Modelling the layout

The primary factor that needs to be taken into consideration is the desired beam waist

in the generating crystal. Since the laser output beam waist is fixed and there is not

much space available, the desired focus is determined only by the focusing lens L1. The

most suitable focal length from our lens repertoire is 150 mm, which focuses the pump

to a waist radius of 28 µm. This is sufficiently close to the optimum focusing, while

remaining on the safe side of the slope in Fig. 2.4. The next components to choose are

the collimating and coupling lenses. Between them there is a sufficient amount of space

needed, but since the collimated beam has low divergence, it is of little significance. The

respective focal lengths are bound together by the coupling condition, which dictates

that the beam needs to be focused to 5 µm waist diameter in order to be coupled into

the fiber. Based on the selection of lenses available, the coupling lens’ focal length was

chosen as 11 mm. The corresponding collimating lens should have a focal length of 180

mm, we have used 175 mm.

When we have solved the beam model with fixed positions of the lenses, we can

consider the appropriate placement of other components. Since the collimating lens is

AR-coated for 600–900 nm, the strong UV pumping would reflect back and scatter, which

is undesirable. Therefore, a cut-off filter needs to be placed between the generating crystal

and the collimating lens. At the filter’s position, the beam already has waist in the order

of 10−1 milimeters, so the filter can withstand the power without damage. The horizontal

tilt of the filter was adjusted in a way that reflects the beam slightly off-axis, so it can be

blocked by an iris diaphragm. The transferred power is reduced by 5 orders of magnitude,

collimated and further attenuated by two other filters. The overall transmission lowers

the pumping well below dark count level of the avalanche photodiodes.

Between the crystal and the collimating lens a compensation block is needed. It is

situated after the first cut-off filter and before the L2 lens. Also there is a half-waveplate

20

HWP2 for adjustment purposes. The optical path beyond the PBS is determined by

the idler arm, where there are more components: half-wave plate, PBS, delay line and

walk-in mirrors. In the signal arm, there are only walk-in mirrors, but both arms have

approximately equal length for the delay line positioned in the middle of its range.

3.2 Measurement

The most simple measurement is to connect an avalanche photodiode (PerkinElmer

SPCM-AQ4C) to each arm. Apart from a count rate from each avalanche photodi-

ode (singles), we measure coincidences as well. They are either triggered by randomly

correlated photons or a photon pair that has been generated in the BBO crystal. With

various losses introduced to each generated mode, the coincidence rate is lower than

singles, diminishing the amount of usable photons.

Generally, as we narrow the bandwidth of the interference filter, all rates drop. It will

be revealed in the following section, that this has positive effect on the photons’ quality.

It is therefore important to show dependency of said rates on the filter bandwidth (see

Table 1).

3.2.1 Detection

All data acquisition relies on detecting single photons by avalanche photodiodes (APD).

These diodes are kept under high reverse voltage to trigger an avalanche current by the

means of inner photoelectric effect. Unfortunately, the current needs to be quenched,

making the diode unable to detect photons for a short period of time—dead time. This

mechanism sends out a signal when one or more photons have been detected, unable to

distinguish the number of photons triggering the current. The process of detection only

takes place with a certain probability: quantum efficiency. It may also occur even though

no photon is detected, making APDs exhibit a noisy offset called dark counts. This makes

the measurement of low photon rates difficult and imprecise. When an APD registers a

photon, it sends out a TTL electronic pulse with amplitude and FWHM depending on

the APD device. The pulse is then electronically processed using detection modules.

The most straightforward approach is to use the TTL pulses directly. To measure

coincidences, a time-to-amplitude converter (TAC) can be used. It operates by measuring

a time interval between two pulses. Once a voltage threshold is passed, the converter

waits a pre-set period of time for a second event. If a second pulse arrives, the converter

generates a voltage level proportional to the time delay. A single channel analyzer (SCA)

is then used to evaluate the voltage and register a coincidence event, if the delay falls into

a pre-set coincidence time window. However, TAC/SCA is easily saturated, having dead

21

time in the order of microseconds [11]. We experienced significant saturation effects with

count rates in the order of 105/s, which led us to use a discriminator with a coincidence

logic module described in the next section.

3.2.2 Detection electronics

All our electronic modules operate in NIM-logic. The NIM-logic uses current levels with

negative true of -16 mA (-0.8 V at 50 Ω). A description of the connection follows.

First, the APD outputs are inverted and connected to delay modules (Phillips Sci-

entific 792) capable of aligning the pulses in time. A delay line can be set from 0 to

63.5 ns with a resolution of 0.5 ns. After the time adjustment, the pulses are shortened

by a disciminator (Phillips Scientific 708), which emits a double-amplitude NIM pulse

(-32 mA to 50 Ω load) of adjustable width (set to 2.4 ± 0.1 ns FWHM) upon registering

a voltage below the threshold of -0.9 V. Depending on particular detection needs of the

experiment, we may need to replicate this pulse in a fan-out module (Phillips Scientific

748). The pulses are then either detected by a counter or brought into a coincidence logic

module. The counters simply register an event if they receive a standard NIM pulse,

having the threshold at -250 mV.

The logical module accepts NIM pulses of -500 mV to 50 Ω. It can be set to perform

logical operations on its 4 inputs. For measuring coincidences we use the AND operation,

which is true if there is sufficient overlap of the input pulses. Using this, we can set the

coincidence window by adjusting the input pulse width on a discriminator. Once the

logical operation’s output is true, it emits a double-amplitude NIM pulse. In all following

measurements, the coincidence window is 5 ns, double the width of the discriminator’s

output.

3.3 Discussion of losses

Photon losses are present throughout the experiment. None of the optical components

used have perfect parameters and they introduce losses either directly or indirectly. Di-

electric mirrors have been used in order to increase reflectivity and where possible, the

transmission components were coated anti-reflexively.

Losses concerning pumping are due to mirror losses from the first steering module,

dichroic separator and from transmission losses of the focusing lens and the first surface

of the BBO crystal. Pumping losses affect the power density inside the generating crystal

and thus have effect on generation efficiency. These losses can be effectively included in

the SPDC gain.

Losses concerning the generated photons have two main causes – filtering and tech-

22

nical. Some filtering losses are introduced by narrowing the spectrum by an interference

filter. Since single-mode fiber coupling covers wide spectrum, every spectral filtering nec-

essarily affects the photon rate. There are also coupling losses that are systematic due

to spatial walk-off in the generating crystal. The ordinary mode is a transverse convo-

lution of extraordinary spatial mode and spatial walk-off function, which permits only

a fixed portion of the mode to be coupled to a single-mode fiber. Technical losses are

caused by imperfect transmissivity of the cut-off filters and non-ideal coupling efficiency.

There are also minor factors including coating losses on the compensating crystal, half-

waveplates, beams plitters, lenses and mirrors. All losses mentioned can be expressed as

two transmissions τ and η (see Sec. 2.6) for each mode. This becomes convenient in later

calculations and photon statistics predictions.

3.4 SPDC alignment

To properly adjust the generating crystal output coupling, we rely on single count rates

instead of the coincidences. When misaligning the tilt of the generating crystal and

shifting towards non-collinear generation, the paraxial non-degenerate photon pairs can

still be coupled due to large spatial mode overlap with the coupling modes. Moreover,

there is no way to guarantee that both fiber couplers will couple the same generated

spatial mode, which is essential in determining the alignment of the generating crystal. In

the non-collinear case, we can still couple coinciding modes and obtain a coincidence rate

comparable with the collinear case. It is therefore uncertain whether a slight misalignment

of the crystal would decrease coincidences at all (when both couplers are adjusted to

maximum coinc. rate).

Our approach consists of three steps, the significance of which will be discussed below.

1. Set the half-wave plate HWP2 to 22.5.

2. Adjust both fiber couplers to maximum single rate consecutively.

3. Set the half-wave plate HWP2 back to 0.

The half-waveplate setting will rotate the H/V polarization modes by 45. For each mode,

the PBS then effectively acts as a 50:50 beam splitter. Then we are able to couple both

modes to a single fiber coupler. Consequently, when measuring single count rate, the

global maximum is reached in the collinear case, when both modes indicated in Fig. 2.3

join in a single mode. This allows us to determine the proper alignment of the crystal

(apart from Hong-Ou-Mandel interference) and alignment of the couplers as well.

23

3.5 Hong-Ou-Mandel dip measurement

The coinciding photons must be indistinguishable in order for the experiments utilizing

the photon source to work properly. This quality can be conclusively measured via the

Hong-Ou-Mandel (HOM) effect discussed in Sec. 2.5. The measuring setup is shown in

Fig. 3.2.

Figure 3.2: The Hong-Ou-Mandel dip measuring setup.

Sifam fiber coupler has been used, with splitting ratio 47.5 : 52.5. As we approach

evenly balanced arm lengths by the motorized delay line, the coincidences registered by

APDs should drop due to HOM effect. The coincidence rate at the dip is not zero due

to the coupling ratio, multiple-pair contributions and spectral distinction of the down-

converted photons, as discussed below. With the maximum and minimum rate Cmax, Cmin,

we define HOM visibility

V =Cmax − Cmin

Cmax

. (3.1)

In Table 1 results are shown that have been measured. S1 and S2 denote singles rates

and C are coincidences detected after the beam splitter. The measured Hong-Ou-Mandel

dips are shown in Figure 3.3 with errorbars showing 5 standard deviations.

The shape of the dip is determined by the temporal probability amplitude of both

photons, which is a Fourier transform of their spectrum. The spectrum is largely de-

24

-0.2 -0.1 0.0 0.1 0.20

5000

10 000

15 000

20 000

25 000

30 000

delay @psD

coin

cide

nce

rate

@s-1 D

no filter

-0.4 -0.2 0.0 0.2 0.40

5000

10 000

15 000

20 000

delay @psD

coin

cide

nce

rate

@s-1 D

10 nm filter

-1.0 -0.5 0.0 0.5 1.00

500

1000

1500

2000

2500

3000

delay @psD

coin

cide

nce

rate

@s-1 D

2 nm filter

Figure 3.3: Measured Hong-Ou-Mandel dips.

25

Filter FWHM [nm] S1 [103 s−1] S2 [103 s−1] C [103 s−1] V

– 480 480 45 93.8 %10 229 227 19 96.6 %2 58 58 2.5 98.7 %

Table 1: Hong-Ou-Mandel dip measurement.

termined by the interference filter, which has approximately rectangular-shaped spectral

transmittance for 10 nm filter, while 2 nm filter has Gaussian shape. For square-shaped

spectra, the Fourier transform reveals sinc-shaped oscillations, while Gaussian shapes

remain Gaussian in the dip (Fig. 3.3). In the regime without any filter, the spectrum is

determined by projection to the fiber mode. Furthermore, the broader the spectrum, the

more narrow the dip becomes. Its visibility is theoretically determined by SPDC gain

and channel losses, which affect the photocount distribution. Multiple-pair contributions

cause the visibility to drop under 1, but we can estimate their effects. Based on mea-

surements in Section 3.7, we can determine gain and the two-pair contribution to pose

a limit of about 99.9 %. The beam splitter employed in this measurement was measured

to have transmissivity of about 52.3%, which also affects the visibility by the limit of

99.5 %. Taking these imperfections into account, theoretical predictions are still > 99%

visibility for all filters (99.3% for 2 nm filter, 99.1% for 10 nm, 99.0% without filter).

Since the temporal overlap is guaranteed by single-mode fibers and time-correlation can

be adjusted, the main limitation must rise from the spectral distinguishability of the

photons. If we increase the spectral width of the filter, more non-degenerate photon pairs

are transmitted. These pairs have non-zero mode overlap with the modes coupled into

the fibers and affect our visibility. The wider the filter, the more spectrally distinct the

photons are, which yields lower visibility, as can be clearly seen from Table 1. It is also

apparent in the differences between the estimated visibilities and the values in Table 1.

While the 2 nm filter case has only 0.6% left to estimate, the most spectrally broad case

(without filter) lacks 5.2%.

Theoretically, we could eliminate some of the technical imperfections, such as imbal-

anced beam splitter, channel losses or imperfect coupling. But the physical nature of the

generation process will still fundamentally limit us (in a small-gain approximation) by

V ≈ 4 + 5g

4 + 7g. (3.2)

In our conditions, this limitation alone is negligible, since g is in order of 10−3. Taking

g as the major limiting factor, we can calculate the theoretical visibility and evaluate

a series with respect to g. We can assume that the quantum state after the losses and

26

before the beam splitter is (2.31). A convenient way for a software calculation is to

express the transformation (2.20) in operator form, because it allows us to use symbolic

linear algebra. Basically, we need to transform every ket |mn〉, so we take the unity on

signal and idler

1 =∑m,n

|m〉〈m| ⊗ |n〉〈n| (3.3)

and transform all ket-sides of the projectors. This gives us a two-mode beam splitter

transformation

BS(T ) =∞∑

m,n=0

m∑k=0

n∑l=0

(−1)n−l

√m!n!

(m− k)! (n− l)! k! l!(k + l)! (m+ n− k − l)! × (3.4)√

T k+n−l(1− T )l+m−k |k + l〉〈m| ⊗ |m+ n− k − l〉〈n| .

The resulting density matrix is

ρ′ = BS(T ) ρBS†(T ). (3.5)

Now we can calculate the coincidence detection probabilities, which are proportional to

coincidence rates appearing in (3.1), and can be used instead. The minimum coincidence

probability (in the dip) rises from the transformed matrix ρ′,

pmin = Tr [(Π⊗ Π) ρ′] , (3.6)

with projector Π defined in (2.32). The maximum probability appears when we misalign

both modes, interference disappears and we can consider the cases shown in Fig. 2.5

incoherently. Generally, if there are m and n incident photons, a coincidence occurs only

if they are not all transferred only to one mode, giving probability

pC(m,n) = 1− Tm(1− T )n − T n(1− T )m, (3.7)

which is valid, if either m or n is non-zero. Now we can use ρ to calculate

pmax =∑m,n

〈mn| ρ |mn〉 [1− Tm(1− T )n − T n(1− T )m] . (3.8)

The first two terms of resulting visibility (3.1) with a perfectly balanced beam splitter

are

V ≈ 1−(η

τ− η +

τ

η− τ +

ητ

2

)g. (3.9)

It is clear, that even for a very weak pumping, where the two-pair contribution almost

27

vanishes, the visibility is limited by individual losses and, intriguingly, their imbalance.

However, it should be noted that in this approximation, we cannot let τ or η approach

zero beyond g, because the first term would diverge and higher-order contributions would

have to be taken into account.

3.6 Single-photon source

To measure higher Fock state contributions, we connect a fiber beam splitter to the

signal arm with a detector on each output (Hanbury Brown and Twiss scheme). In

order to adjust the detections properly in time, each signal detector is followed by an

electronic delay line. Then, each electronic pulse is turned to NIM pulse in a discriminator

and fanned out. Then, two-fold coincidences (N1, N2) with the trigger are detected, as

well as three-fold coincidences (Nc). These rates correspond to singles and coincidences

measured on a non-heralded single-photon source. The quality of the heralded state can

be quantified by two criterions: anticorrelation parameter and non-Gaussianity criterion.

Figure 3.4: The scheme of the measurement with detection electronics.

3.7 Anticorrelation parameter

We measured the anticorrelation parameter, described in Sec. 2.6.1, for two interference

filters with FWHM of 2 nm and 10 nm (Fig. 3.5). The time delay was scanned using

28

only delay 1 in Fig. 3.4.

æ æ ææ

æ

æ

æ

æ

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æ

æ

æ

ææ æ æ æ

à

à

à

à

à

à

à

àààà

à

à

à

à

à

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à

à à

-15 -10 -5 0 5 10 15

1

1

10

1

100

1

1000

delay @nsD

Α

à 2 nm

æ 10 nm

Figure 3.5: Results of the α measurement.

The results show a very sharp decrease of the anticorrelation parameter in the origin

(zero delay): α(0) < 1% by approximately 4 standard deviations for 10 nm interference

filter and α(0) < 0.5% by 1 standard deviation for 2 nm filter. The STDs are based on

Poisson variance of detected count rates.

The undesirable three-fold coincidences at delay 0, which are responsible for α(0) > 0,

are mainly due to multiple pair contributions from the parametric down conversion (2.5).

These contributions are lowered by losses, which means increasing the quality (non-

classical statistics) of the single photon source at the expense of its count rate. These

losses are considered in the model (2.34) and discussed in section 3.3.

The finite length of the coincidence time window (5 ns) contributes with random

coincidences that rise from distinguishable pairs: the HOM dip time width is in order

of picoseconds, therefore two generated pairs with distance > 1 ns can still cause a

coincidence event. Finally, there are dark coincidences, which are caused by dark counts

contributing with false positives to the coincidence rate.

The effect of random and dark coincidences can be seen when one of the delays is com-

pletely misaligned with the trigger. For 2 nm filter measurement, three-fold coincidences

drop from ∼ 40/100 s at delay 0 to ∼ 24/100 s at delay > 4 ns. We can therefore see that

nearly half of the three-fold coincidences at delay 0 are due to multi-pair generation. For

29

an ideal single-photon source, three-fold coincidences at delay 0 should be 0/s, whereas

for our source, their rate rises as we approach zero. However, the decrease in α(0) is

secured by increasing two-fold coincidences with the signal arm, the delay of which is

being changed.

The shape of the function has been numerically fitted with a convolution of a Gaussian

curve and a rectangular function. The exact character of the shape is unknown. It is

determined by many factors: electronic pulse shape, photon spectrum, jitter rising from

detection and electronics, all combined together in a way that cannot be easily anticipated

or theoretically modelled. The shape of the fit has been chosen to be plausible with the

data, with the value of α(0) being the significant result.

3.8 Non-Gaussianity criterion

Experimentally, the criterion (Sec. 2.6.2) can be measured using the setup depicted in

Fig. 2.7. This provides us with a statistical model, but further discussion is required to

account for losses [10].

In general, a lossy channel maps Guassian states onto Gaussian states. Therefore, if

the input state is a classical mixture, so must be the output. If we measure a non-Gaussian

state after the losses, we know that the input state is non-Gaussian as well. This way,

we can assume there are no detection losses, cover the imbalance of detector efficiencies

in the beam splitter and calculate the probabilities p0 and p1 without any compensation.

If we obtain a non-Gaussian state, its witness can be considered a conservative estimate.

The input state has photon-number probabilities pn = 〈n| ρ |n〉. We measure single-

detections N1,2 and coincidences N12 that are heralded by the trigger channel. Experimen-

tally, “single-detections” are two-fold coincidences with the idler arm and “coincidences”

are three-fold coincidences. The probabilities of these events can be simply expressed as

ratios of respective rates and the rate of the trigger N0. With perfect detectors, only

vacuum triggers no response (note that we exclude the coincidences N12 from N1,2):

p0 = 1− N1 +N2 +N12

N0

. (3.10)

Assuming effective transmittance T of the beam splitter, the photon-number probabilities

contribute to the singles:

N1

N0

=∞∑n=1

pnTn,

N2

N0

=∞∑n=1

pn(1− T )n. (3.11)

To obtain a positive non-Gaussianity result, we need to get a lower estimate of p1 for

30

P [mW] w [nm] p0 p1 ∆W [×10−5] STDs

50 2 0.9124 0.0875 412± 1 40050 10 0.8589 0.1410 1666± 3 50050 − 0.7095 0.2901 14252± 17 800

Table 2: The non-Gaussianity criterion.

measured p0 (3.10). If we take our simplified model (2.28), we obtain

p1,est =N1 +N2

N0

− T 2 + (1− T )2

2T (1− T )

N12

N0

. (3.12)

Using (3.10) and (3.11), we get

p1,est = p1−∞∑n=3

pnT 2 − T n + (1− T )2 − (1− T )n

2T (1− T ), (3.13)

which guarantees that p1,est < p1, giving a lower bound on the real value. Since p1,est

decreases with increasing ratio T : (1− T ) for T ≥ 12, having an upper bound on T fixes

a certain minimum value of p1,est,

Test =N1

N1 +N2

≤ T. (3.14)

If we can show that the state is non-Gaussian for this value, it holds for the real state as

well.

The results are shown in Table 2 with dependence on pumping power P and full

spectral width w at half maximum of the interference filter. The last column shows

the number of standard deviations by which the criterion is violated. These standard

deviations are calculated from uncertainties of detected count rates.

It is interesting that although the measured witnesses are small, the measurement is

still very precise. Wigner functions of these states would show no negativity, containing

a major portion of vacuum and being very similar to Gaussian states. That is what

accounts for low ∆W . Still, the violation of the criterion is almost certain, which is

presumably due to reasonably long measurements which lead to smaller uncertainties in

the count rates. Clearly, this method gives robust results for down-conversion heralded

single-photon sources and requires a very simple measurement setup.

An intriguing property of the non-Gaussianity witness is its difference from the an-

ticorrelation parameter. Experimentally, we always face a question, which arm of the

SPDC source to use as a trigger. For small gain, which is the case with continuous

31

pumping, the anticorrelation parameter does not depend on signal losses, whereas the

non-Gaussian witness does not depend on trigger losses. If the two photon channels have

different losses, optimization by α(0) dictates that the less lossy one is better for trigger-

ing in terms of heralded state quality and count rate as well. However, the non-Gaussian

witness decreases with signal losses significantly (in the non-Gaussianity area, to be pre-

cise), thus preferring the signal channel to be less lossy. It is therefore important to

determine, which measure represents a valid merit of quality for the output state, which

depends on the experiment that employs the heralded single-photon source.

3.9 Development

Due to Cube laser malfunction, the pumping is presently provided with a 100 mW Cube

equivalent. Its spectrum is specified to be centered at 405 nm, which creates a 4 nm

difference in the down-converted signal (810 nm), whereas the original setup was designed

for 814 nm. First, the BBO crystals needed to be readjusted, because their tilt angle

depends on the wavelength. Then another interference filter was chosen to properly filter

the degenerate spectrum. Minor changes were made due to different beam propagation

and laser displacement, involving adjustment of steering mirrors, focusing and coupling

lenses.

Since we were allowed to chose from a wider variety of purchased interference filters,

we were able to make a better match (Semrock MITC 810.0 nm /2.7 nm FWHM). Also,

100 mW of pumping power offers better coincidence rate, a factor crucial for the pro-

grammable phase gate [12], which operates with overall transmissivities in the order of

percents. The development of measured characteristics are shown in Table 3, compared

to Table 1 it shows a significant improvement. Fig. 3.6 shows the Hong-Ou-Mandel dip

for the second case, with 99 % visibility (errorbars show 5 standard deviations).

Date pump [mW] S1 [103 s−1] S2 [103 s−1] C [103 s−1] η2ph V

7. 10. 2011 100 220 152 43 23.5 % 98.5 %28. 2. 2012 89 218 140 42 24.0 % 99.0 %

Table 3: The development of rate and visibility.

η2ph is a two-photon coupling efficiency defined as C/√S1S2. It is important to note

that the counts in Table 3 were measured directly after the couplers, while in Table 3

there was a 50:50 fiber beam-splitter present (see Fig. 3.2).

Fig. 3.7 shows the measured anticorrelation parameter (α(0) = 0.76%) with a 5

ns coincidence window. The same measurement has given a non-Gaussianity witness

∆W = (1155.8± 0.7)× 105 with 1600 STDs for a 1000 s measurement time.

32

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.50

5000

10 000

15 000

20 000

delay @psD

coin

cide

nce

rate

@s-1 D

Figure 3.6: The new Hong-Ou-Mandel dip

æ ææ

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ææ

ææ

æ

-30 -20 -10 0 10 20 30

1

1

10

1

100

1

1000

delay @nsD

Α

Figure 3.7: The new anticorrelation parameter.

33

3.10 Outlook

In the future, my efforts regarding photon sources will include both photon-pair and

multiple-photon generation. In the field of correlated photon-pair generation, there are

still some factors left to optimize. We intend to employ periodically-poled crystals, which

allow us to exploit a high effective nonlinearity due to quasi-phase-matching. The relative

focus size of the pump can be further optimized with respect to spatial walk-off, which

offers an optimum regime for coupling to single-mode fibers. Experimentally, the position

of the collimating lens can also be subject to optimization and has not yet been carried

out due to time limitations.

Single-photon source optimization will be performed using our efficient measures,

simultaneously offering broad possibilities for testing new criterions and studying their

dependence on noise or losses. The parameters to examine include coincidence window

and noise statistics.

Producing multi-photon states presents an entirely new enterprise utilizing femtosec-

ond pulse pumping. By second-harmonic generation and one down-conversion, it is pos-

sible to generate photon pairs and an attenuated coherent state, which serves as the

third photon channel. Alternatively, two down-conversions can be employed to produce

two correlated photon pairs. While the experiment is in its early stages, in the future

it shows promise of designing complex experiments utilizing more than two photons,

possibly quantum gates operating on many qubits.

34

4 Conclusion

The thesis presented an implementation of a correlated photon source. The source em-

ployed a type-II spontaneous parametric down-conversion in collinear mode and was

pumped by a continuous-wave multimode laser diode. Methods have been described,

which allow measuring the quality of the generated photons.

The Hong-Ou-Mandel visibility was presented as a key measure of photon undistin-

guishability, a crucial property for quantum information processing. The visibility was

shown to be adjustable by interference filter bandwidth, depending on the specific coinci-

dence rate and interference quality required. It was possible to reach visibility of 99 % with

a count-rate sufficient even for highly lossy experiments [12], which makes the source a

valuable asset for further experimental undertakings. As a heralded single-photon source,

it performed very well in the measurements of both anticorrelation parameter and non-

Gaussianity criterion. Using a simple Hanbury-Brown-Twiss setup, we were able to assess

the experimental parameters, including SPDC gain, as well as measure two key criterions.

A non-classical anti-bunching with anticorrelation below 1% was measured, which shows

a high portion of single-photon state relative to higher contributions. Provided we can

rely on detection post-selection, the source offers a high-quality heralded single-photon

state. The non-Gaussianity witness virtually eliminated the possibility that the heralded

single-photon state would rise from a classical Gaussian-state mixture. Despite showing

a large portion of vacuum, the two-photon contributions were almost negligible, allowing

the witness to be violated by more than 1000 standard deviations. Since both these

measures exceed desired results, the single-photon source is an excellent platform upon

which to test various new criterions, their breaking and liabilities.

The source was used as a laboratory resource for experiments involving Toffoli gate,

the non-Gaussianity criterion [10], programmable quantum gate [12], quantum reading

[13], and noiseless communication [14].

35

Figure 4.1: The photograph of the experiment.

36

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37