Download Quantum Optics and Photon Correlations in Astronomical

Bachelor project
Astronomical Institute Anton Pannekoek
Quantum Optics and Photon
Correlations in Astronomical
Observations
Report of ‘Bachelorproject Natuur- en Sterrenkunde’, credit load 12 EC,
completed between 10-01-2011 and 13-10-2011.
Author:
Mathijs Elbers
(SN: 9817352)
Supervisor:
Dr. Alessandro Patruno
Second grader:
Dr. Anna Watts
December 15, 2011
Abstract
This report investigates the feasibility of measuring the second order coherence function g (2) (τ ) of light from astronomical objects with today’s technology and to explore the technique of measuring it. First it introduces the
theory behind second order coherence, intensity interferometry (as developed
by Hanbury Brown and Twiss), light classifications based on photon statistics or g (2) (0), signal loss in the photodetection processes and coherence time.
Then, using data taken from the Crab pulsar with the Iqueye (courtesy of
the University of Padua), it attempts to measure g (2) (τ ). The accuracy of
the measurement is increased until shot noise occurs after which the results
and feasibility of g (2) -measurements with today’s technology are discussed.
Chapter 1
Populair Wetenschappelijke
Samenvatting
In dit project onderzoeken wij de haalbaarheid van het gebruik van de techniek van ‘intensity interferometry’ met hedendaagse technologie. Deze techniek is in de jaren ’50 ontwikkeld door de sterrenkundige Hanbury Brown en
de wiskundige Twiss om de ‘angular size’ (grofweg de grootte) te meten van
twee astronomische objecten die door de net-onwikkelde radio-astronomie
ontdekt waren.
Standaard ‘interferometry’ is het optellen van (twee of meer) golven wat
vaak leidt tot complexe patronen. Wat er bij standaard interferometry
typisch gebeurt is dat twee verschillende lichtbundels met zeer nauwkeurig
opgestelde spiegels worden samengevoegd tot een gecombineerde lichtbundel,
die dan wordt opgevangen in een detector. Je kan ook zeggen dat de amplitudes van de twee verschillende lichtbundels worden opgeteld en daarna naar
een detector geleid. ‘Standaard interferometry zou je daarom ook ‘amplitude
interferometry kunnen noemen.
Bij intensity interferometry combineren we de meetsignalen van twee
intensiteits-meters die de twee lichtbundels opvangen. En deze meetsignalen
kan je met simpele COAX kabels naar een elektronische vermenigvuldiger
leiden voor interferometrie. Veel eenvoudiger dan het precies uitlijnen van
spiegels, wat over langere afstanden heel ingewikkeld wordt en voor het meten
van zeer kleine of zeer vergelegen sterren heb je juist langere afstanden nodig.
Ten tijde van Hanbury Brown en Twiss was de technologie nog niet zo ver
dat dit mogelijk was en de intensity interferometry bracht uitkomst. Uiteindelijk zouden Hanbury Brown en Twiss een intensity interferometer op grote
schaal bouwen in Australië die van zo’n dertig sterren de tot dan toe onbekende angular size zou meten. Deze metingen waren destijds zo revolutionair
dat het tientallen jaren duurde voordat ze konden worden gereproduceerd.
1
Figure 1.1: Drie lichtbundels met dezelfde intensiteit (20 fotonen in de afgebeeldde tijd) maar met verschillende tussenpozen tussen de individuele fotons. Source: wikipedia.
Na dit aanvankelijke succes werd het echter stil rondom de intensity interferometry. Ontwikkelingen in de technologie maakten dat standaard interferometry om verschillende redenen superieur werd aan intensity interferometry
en de sterrenkunde verloor de techniek uit het oog.
Maar intensity interferometry heeft in potentie veel meer toepassingen
dan alleen het meten van de ‘angular size’ van sterren. Wanneer je de intensiteiten van twee lichtbundels vergelijkt door de signalen te vermenigvuldigen
dan vergelijk je in wezen het arriveren van fotonen in de twee lichtbundels
(want intensiteit is een maat voor de hoeveelheid fotons die per seconde door
een doorsnede van de bundel heen vliegen). Fotonen die, in het geval van de
sterrenkunde, met dezelfde tussenpozen aankomen op aarde als die waarmee
ze zijn vrijgekomen in de ster die geobserveerd wordt en daarmee wellicht
nog meer informatie over de ster met zich dragen. Maar om de techniek
van intensity interferometry te gebruiken om de tussenpozen van individuele fotonen in de lichtbundels te vergelijken moet je extreem geavanceerde
meetapparatuur hebben. Zo geavanceerd dat het in de jaren ’70/’80 totaal
ondenkbaar was.
Nu, dertig tot veertig jaar later, lijkt het erop dat de technologie de intensity interferometry (bijna) ingehaald heeft. In dit verslag onderzoeken wij
de mogelijkheid om met hedendaagse technologie aan intensity interferometry te doen op het niveau van de individuele fotons. We doen dit door de
techniek toe te passen op data die genomen is van de Crab pulsar door de
Iqueye (de snelste fotometer ter wereld) tijdens een ijkrun. Deze data is ons
vrijelijk ter beschikking gesteld door de Universiteit van Padua, de eigenaar
van de Iqueye.
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Contents
1 Populair Wetenschappelijke Samenvatting
2 Introduction
2.1 Historical background .
2.2 Photonic consequences
2.3 Motivation . . . . . . .
2.4 Overview Report . . .
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3 Theory
3.1 Astronomical observations revisited . . . . . . . . . .
3.1.1 Intensity of the light: Bolometer . . . . . . . .
3.1.2 Spectrum of the light: Spectrometer . . . . .
3.1.3 Coherence of the light: (Phase) Interferometer
3.2 Hanbury Brown Twiss experiment . . . . . . . . . . .
3.2.1 Classical explanation of the HBT experiment .
3.2.2 Quantum explanation of the HBT experiment
3.3 From spatial- to temporal coherence of light . . . . .
3.4 The ‘semi-quantized’ atomic model . . . . . . . . . .
3.5 Photon Counting & Statistical fluctuations . . . . . .
3.6 Coherent light: Poissonian photon statistics . . . . .
3.7 Classification of light by photon statistics . . . . . . .
3.8 Degradation of photon statistics by losses . . . . . . .
3.9 g 2 (0): a different light classification . . . . . . . . . .
3.10 τc : the need for good time resolution . . . . . . . . .
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4 Project, Results & Discussion
4.1 Description Project . . . . . .
4.2 The Experiment . . . . . . . .
4.2.1 Telescope . . . . . . .
4.2.2 Detector . . . . . . . .
4.2.3 Astronomical Object .
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4.3
4.4
4.2.4 The raw data . . . . . . . .
The Results . . . . . . . . . . . . .
4.3.1 The modified data . . . . .
4.3.2 Calculating g (2) (0) . . . . .
4.3.3 Calculating g (2) (τ ) for τ ≥ 0
Discussion . . . . . . . . . . . . . .
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5 Special Mentions
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A Mathematica source code
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4
Chapter 2
Introduction
2.1
Historical background
It is commonly accepted among physicists that the mathematical birth of
the photon (and through that the ‘quant’) was when Planck used discrete
energy states to derive his famous formula describing black body radiation. It
is equally commonly accepted that the true, conceptual birth of the photon,
and the notion of quantization, was when Einstein published his paper[7]
about the photo-electric effect five years later during his Annus Mirabilis.
Subsequent development of quantum mechanics and analysis of the physics
involved however showed that the photo-electric effect could also be explained
by taking a quantized nucleus-with-electron-cloud atom but treating light as
a classical (Maxwellian) electromagnetic wave. And the same holds true
for ‘photon counting‘ devices like photomultiplier tubes or Geiger counters.
This meant that it had still not been conclusively proven that light consists
of particles which behave in a wave-like manner rather than simply being a
wave.
The search for specific properties of light which could only be explained
by the existence of photons turned out to be a difficult one, and it wasn’t until
the 1970s that such properties were actually observed in physical experiments
[13] [8]. These discoveries sparked the birth of a new, and still young, field
of physics aptly called ‘Quantum Optics’, in which the consequences and
potential of light particles are investigated.
Classically we know light through its properties of intensity, spectrum,
and coherence. All three of which are extensively exploited in astronomy.
With quantum optics discovering new properties of light this is obviously of
major interest to astronomy since it means potentially that more information
might be gathered from the same light which is caught in the telescopes. That
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is, if the new properties can be observed and detected.
2.2
Photonic consequences
With light consisting of quantum particles which behave in a wave-like manner we need to review what that means for astronomy and photodetection.
If light is considered a wave then the process of photodetection (the release
of a photo-electron in the photosensitive part of the detector) is a random
(Poissonian) process which has a fixed probability of happening based on the
energy density of the light at the detection point. This also means that the
distribution of the photo-electrons over the detection time could never tell us
something about the source of the light since that distribution is generated
in the detector.
If light consists of discrete photon particles however, then the release of
a photo-electron can only happen if a photon from the light beam passes
through the photosensitive part of the detector. Even though there are still
probabilities involved in this process, the chance of releasing a photoelectron
will be maximal at the position of the photons in the beam, and zero at
the empty spaces in the beam between two photons. And because these
photons have been released at the light source, their distribution might give
us information about the source.
In classical astronomy we count the number of photons which impinge
on the telescope during the detection interval. At the detection level this
constitutes a (very large) collection of single-photon detections (or photoelectron releases). But collections of single photon detections won’t give us
information about the photon distribution in the light beam since, as we have
discussed above, photodetection can also be explained with classical light.
What we need to do instead is to look at two-photon events rather than
single photon events. It is in the time between successive photons that the
classical and photonic model can really differ: for classical light at constant
intensity the chance at a photo-electron is constant. Such a process will
generate a Poisson distribution in the time between successive photons. But
with photonic light this distribution can be pretty much anything depending
on what the photon distribution in the light beam is.
In theoretical terms when we shift our attention from single photon events
to two-photon events we go from the first-order coherence function (g (1) )
to the second-order coherence function (g (2) ). The second order coherence
function is an insightful and adaptable formalism with many applications in
quantum optics and astronomy. We cover the theory extensively in Chapter
3 and discuss possible applications in Chapter 4.
6
The very first time that g (2) was used in an astronomical context was
when the astronomer Hanbury Brown and the mathematician Twiss used
it in the 50s to measure the radial size of two newly discovered gas giants
(see Section 3.2 for details). They went on to use the same technique to
successfully measure the size of 32 more unknown star radii between 1961
and 1972 [12]. But after these initial successes g (2) has fallen into disfavor and
hasn’t been used in astronomy since, despite the development of quantum
optics.
The reason that astronomers haven’t looked at g (2) since the 70s is that it
is a property of the second order as opposed to g (1) being of the first. And as
with most things in mathematics and physics second order effects are more
difficult to detect or measure than first order effects. Hanbury Brown and
Twiss (HBT) did their g (2) measurements in an era where technology wasn’t
able to allow measurements of those stars using ‘classical’ g (1) , and the roadblocks preventing the g (1) measurements were not present in the g (2) case,
despite its higher systemic difficulty. But once technology caught up with
the demands of g (1) there was no longer a reason to use g (2) for measurements
where g (1) would suffice. And those measurements which fundamentally required g (2) as they transcend the scope of g (1) required at-the-time impossible
telescope sizes, observation times, and detector specifications. And this has
been the case for the past 40 years. After four decades technology is catching up with g (2) , and in this project we will investigate the feasibility of
measuring g (2) (τ ) (a version of g (2) ) with today’s technology.
2.3
Motivation
We do this project because the time is right (as explained above), and because
the topic is an interesting and newly developing field within astronomy with
a big potential.
If technological developments and available telescopes succeed in making
g (2) consistently measurable, that will constitute a revolution in astronomy
comparable with extending the detectable frequencies in astronomy to infrared light and X-rays, or the development of photometers with microsecond resolution. We think that maybe astronomy has neglected g (2) for longer
than was strictly necessary from a technological point of view. It is always
easier to do what has been done before than to venture into unknown territories, also in astronomy, and it is our hope that we’ll be able to quantifiably
demonstrate that measuring g (2) will lead to new science even if measured
with today’s technology.
In 1971 when HBT made their proposal for an improved intensity inter7
ferometer they listed as possible research areas
• Measuring radii of single stars (extension of the work at Narrabi)
• Observation of the radial pulsations of Cepheid variables
• The shape of rotating stars
Even though the proposal was submitted 40 years ago, those areas are still
worthy of research to this date. But a lot has changed in 40 years, and an
SII built with today’s technology could in extremis give optical imaging at
micro-arcsecond resolution, and sub-milliarcsecond resolutions are certainly
reachable. Such a modern SII would at once catapult stellar imaging to the
forefront of astronomy. To this day next to the Sun only Betelgeuse and
Altair have been imaged, and with a maximum of ten resolution elements.
A modern SII would easily increase that number to 100: it would extend
solar physics to the realm of the stars. And next to resolving galactic sources
feasibility studies indicate several other potential research areas [1]:
• Stellar surface phenomena and dynamo action
• Conditions for planet formation around young stars
• Cepheid properties and the distance scale
• Mass loss and fast rotation of massive hot stars and supernova progenitors
• Nuclear optical emission from AGN
• Structure of gamma-ray bursts
Of course the mileage will vary once a modern SII would be built. But
just as some of these areas will turn out to be undetectable with an SII so
will a modern SII undoubtably make discoveries that no one has thought of
beforehand.
The aim of the project is to check the feasibility of measurements of
(2)
g (τ ) with current instrumentation and to explore the technique. We will
develop a numerical algorithm to measure g (2) (τ ) and apply it to the Crab
pulsar data recently observed at the NTT with the Iqueye.
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2.4
Overview Report
In this project we attempt to measure g (2) (τ ) by using data taken with the
Iqueye, a newly developed 0.1ns time resolution photometer during its calibration run. We do the measurement by using existing formalisms to convert
astronomical raw data into the second order coherence function. The results
of this project were generated with the software package ‘Mathematica 8’
(the raw, undocumented, code of the Mathematica notebooks which generated the results of this project can be found in Appendix A)
In Chapter 3 we cover the theory behind the project. We first give a small
summary of the applications of the classical first-order coherence function
(Section 3.1). We then move on to the original HBT experiment which we
treat both at a classical and a quantum level after which we introduce g (2) (τ )
- the property we are trying to measure - (Section 3.3) and define properties
of light which distinguish discrete particle light from classical light (Section
3.4 through 3.9).
In Chapter 4 we first provide a description for our project (Section 4.1),
we then describe the experimental setup which generated the data (Section
4.2), we then display the results of our g (2) (τ ) calculations (Section 4.3), and
lastly we discuss the results and try to place them in an astronomical frame
of reference (Section 4.4).
9
Chapter 3
Theory
In this chapter we will introduce the theory behind the scientific theories and
techniques involving this project. We will first revisit (classical) astronomical
observation and place it in a more rigid theoretical framework. Then we will
provide an explanation of the theory behind the original Hanbury Brown
Twiss (HBT) experiment. We will go over both the classical and the quantum
explanation of the results in the experiment, discuss the results (also in the
light of the 1950s timeframe) and particular where they are relevant for the
project.
In this report, and this chapter in particular, are mostly based on the
book ‘Quantum Optics’ by Mark Fox [10], an article by Gorden Baym [2],
and work of Dainis Dravins [5][6].
3.1
Astronomical observations revisited
In this project we will be looking at astronomical observations at the quantum level, and at the individual photon arrival times at the detector. With
this in mind it follows that it is important to understand what happens at
a theoretical, quantum level when a telescope detects a photon. In modern day astronomy one measures the intensity, spectrum, and coherence of
light. We classify these measurements as one photon experiments (because
the measurement of these physical quantities requires the detection of (multiple) single photons) described by special cases of the ‘first order correlation
function’ G(1) [6]:
G(1) (r1 , t1 ; r2 , t2 ) = hE ∗ (r1 , t1 ) · E(r2 , t2 )i
(3.1)
In this formula the angular brackets stand for the average value over a
long period of time. And between the brackets we take the inner product of
10
the electric field E at a certain space and time with its conjugate at a certain
(not necessarily same) time and space. G(1) can also be normalized into the
‘first order coherence function’ g (1) :
hE ∗ (r1 , t1 ) · E(r2 , t2 )i
g (1) (r1 , t1 ; r2 , t2 ) = p
h|E(r1 , t1 )|2 ih|E(r2 , t2 )|2 i
(3.2)
We see in g (1) that the numerator G(1) , which is the time-average over the
product of E1 and E2 , is divided by a denominator which is the squareroot
of the product of the time averages of E1 and E2 . Let’s have a quick look at
what values one might expect g (1) to have. If E1 and E2 are constant then
g (1) = 1. If E1 and E2 are both fluctuating around their respective averages
independent of each other we would expect those fluctuations to cancel each
other out over the long time averages and g (1) = 1. If on the other hand
E1 and E2 are correlated then we would expect the numerator to exceed the
denominator due to their fluctuations being multiplied with each other and
g (1) > 1. With g (1) ≥ 1 we have a nice way to quantify coherence. Later on
in the report we will discuss how, taking into account the photonic nature of
light, it is possible to get values for g (1) smaller than one.
We proceed to look at the special cases of G(1) which cover the standard
astronomical observations. We choose to go with G(1) rather than g (1) here
because all interesting information is in the ‘numerator’. We don’t need
normalization to gain an understanding of the mechanisms involved, but
further on in the report we will use g over G almost exclusively.
3.1.1
Intensity of the light: Bolometer
Measuring the intensity of the light makes up for the vast majority of astromical observations: mount a camera (CCD) at the focal point of a telescope
and point it at a bright object and detect the photons coming off it. Mathematically this is done by measuring the square of the amplitude of the local
electric field. The special case of the first order correlation function which
describes this is as follows:
G(1) (0, 0; 0, 0) = hE ∗ (0, 0) · E(0, 0)i
(3.3)
A device which measures this quantity is termed a ‘bolometer’. The onephoton nature of this observation is very intuitive: the CCD counts the
number of (single) photons it detects in its pixels, and that number of photons
divided by the observation time will give you the intensity of the source.
11
3.1.2
Spectrum of the light: Spectrometer
To measure the spectrum of the light one takes half the signal impinging
on a telescope and takes the inner product of its conjugate with the other
half of the signal delayed by a time t. If you plot this function against the
delay time t and then take the Fourier transform of the plot you will get the
spectrum of the observed light. The special case of the first order correlation
function which describes this is as follows:
G(1) (0, 0; 0, t) = hE ∗ (0, 0) · E(0, t)i
(3.4)
A device which measures this quantity is termed a ‘spectrometer’.
3.1.3
Coherence of the light: (Phase) Interferometer
To measure the interference of the light one takes the inner product of the
conjugate of the electric field at a point of interest with the electric field
displaced with a distance r. If you plot this function against the displacement
distance r you will see the function drop off at a certain value of r which will
translate into a value for the coherence of the light. The special case of the
first order correlation function which describes this is as follows:
G(1) (0, 0; r, 0) = hE ∗ (0, 0) · E(r, 0)i
(3.5)
A device which measures this quantity is termed a ‘(phase) interfermometer’.
3.2
Hanbury Brown Twiss experiment
Now that we placed standard astronomical observations within a framework
we will proceed to discuss the Hanbury Brown and Twiss (HBT) experiment
and in particular where it stands apart from those standard observations.
Hanbury Brown and Twiss were a radio astronomer and a mathematician
respectively who in the early fifties were interested in measuring the radial
size of stars. In particular the bright ‘radio stars’, such as Cassiopeia A and
Cygnus A, which had been discovered by the then newly developed field of
radio astronomy. The accepted method of measuring the radial size of stars at
the time (and still to this day) was through a Michelson Stellar Interferometer
(MSI). The MSI had been developed 70 years earlier in the 1890s and is an
amplitude interferometer : it combines the amplitudes of the light from the
same source at two different positions and translates the resulting interference
pattern in a radial size. The MSI is a phase interferometer as mentioned in
Section 3.1.3 and described by the special case G(1) (0, 0; r, 0) of the first order
12
correlation function. Even though the MSI is to this day the accepted method
of measuring the radial size of stars, and has produced over the years the most
accurate measurements of star radii, it does suffer from drawbacks: the light
from the two different positions has to be carefully combined with lenses and
that makes the system increasingly more sensitive to outside disturbances
as the distance between those is increased. As that distance needs to be
increased to be able to resolve smaller sources the MSI can be very demanding
technically. Hanbury Brown realized that the technical difficulties of the MSI
might not be possible to overcome (in the 1950s) if the stars of interest were
either too small or too far away and proceeded to work on an alternative:
intensity interferometry. Later on he called in the help of the mathematician
Twiss to assist him with the mathematics (and particular statistics) of their
interferometry.
The fundamental step in the development of intensity interferometry was
the realization that if radiation received at two locations is coherent, then the
intensity fluctuations of the received signals will also be correlated. This, in
short, reduces the problem of interferometry to having two (photo-)detectors
at two different positions pointed at the same source and then combine the
signals of both detectors through multiplication (as opposed to combining
the actual radiation from the source arriving at both positions). With the
number of photons counted being proportional to the intensity we get this
way a signal which is sensitive to the intensity fluctuations and relative phase
of the respective streams. Hence the name ‘intensity interferometry’. In the
case of G(1) or g (1) we were looking at the electric field. We now look at
intensity in a similar way through the ‘second order coherence function’ g (2) :
g (2) (r1 , t1 ; r2 , t2 ) =
hI(r1 , t1 )I(r2 , t2 )i
hI(r1 , t1 )ihI(r2 , t2 )i
(3.6)
We see that the g (2) treats intensity the same way as g (1) does the electric
field minus the dot product since intensity is already a scalar by itself. With
|E|2 = I we see that g (2) roughly the ‘square’ of g (1) which ties into it being
called the second order coherence function. All the special cases which have
been laid out for g (1) are also applicable to g (2) . Knowing that the number
of photons counted is proportional to the intensity we can rewrite g (2) as a
function of the photons counted, which will be instrumental into converting
experimental data into a numerical value for the coherence:
hn1 n2 i
hI1 I2 i
=
(3.7)
hI1 ihI2 i
hn1 ihn2 i
with n1 and n2 being the number of photons counted at detector 1 and 2
respectively.
g (2) =
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Figure 3.1: Schematic of the HBT experiment. Source: [2].
In the following subsections we will go further into detail on the theory
behind the HBT experiment and discuss how it can be understood both from
a classical and from a quantum point of view. The latter being an astonishing
result in the fifties triggering the birth of ‘quantum optics’, an entirely new
field of physics.
3.2.1
Classical explanation of the HBT experiment
A classical treatment of the HBT experiment is more than just a historical
exercise. At long wavelengths (infrared and beyond) light behaves very much
as a classical wave, and we’re only able to detect it as waves with antennas
rather than photons with photon-counters. Since a complete treatment of
the classical theory behind HBT carries too far we will only give a very
general outline of the derivation, along the lines of [2]. A complete theoretical
treatment can be found - among others - in [3].
The simplest way to look at a HBT experiment is by taking two random
(distant) sources of radiation a and b and two independent detectors 1 and 2
which can catch the light of either sources. The distance between the sources
is R, the distance between the detectors is d, and the distance between the
sources and the detectors is L as depicted in Figure 3.1.
Let source a produce a spherical electromagnetic wave of amplitude
αe|~r−~ra |+iφa / (|~r − ~ra |)
and source b a similar wave of amplitude
βe|~r−~rb |+iφb / (|~r − ~rb |)
with φa and φb random phases. We get for the amplitude at detector 1:
A1 =
1
(αeikr1a +iφa + βeikr1b +iφb )
L
14
(3.8)
with r1a the distance between source a and detector 1, etc. From this we get
for the intensity in 1:
I1 =
1
(|α|2 + |β|2 + α∗ βei(k(r1b −r1a )+φb −φa ) + αβ ∗ e−i(k(r1b −r1a )+φb −φa ) ) (3.9)
L2
And similarly for I2 . The random phases in the complex exponential
terms will average out to zero when taken over a large enough time and we
get:
1
(h|α|2 i + h|β|2 i)
(3.10)
L2
Making hI1 ihI2 i independent of R and, particularly, d. If we try and calculate
hI1 I2 i we get non-vanishing cross terms:
hI1 i = hI2 i =
hI1 I2 i = hI1 ihI2 i +
=
2
|α|2 |β|2 cos(k(r1a − r2a − r1b + r2b ))
L4
(3.11)
1
[(|α|4 + |β|4 ) + 2|α|2 |β|2 (1 + cos(k(r1a − r2a − r1b + r2b )))]
L4
(3.12)
We now have enough to calculate the coherence using the special case of the
second order correlation function in the case of the HBT experiment:
h|α|2 ih|β|2 i
hI1 I2 i
= 1+2
cos(k(r1a − r2a − r1b + r2b ))
hI1 ihI2 i
(h|α|2 i + h|β|2 i)2
(3.13)
with d being the aforementioned detector separation. Let’s look at this result
more closely: we see that g (2) (d) is unity in absence of the crossterm, and
that that crossterm is a cosine with an amplitude of
g (2) (0, 0; d, 0) =
2
h|α|2 ih|β|2 i
(h|α|2 i + h|β|2 i)2
(3.14)
we instantly notice that if either α or β is zero that the amplitude itself will
be equal to zero and that the coherence of a monochromatic spherical plane
wave (described by such a case) is equal to one. In the special case where
α = β we get:
2
h|α|2 ih|β|2 i
h|α|2 ih|α|2 i
h|α|2 i2
=
2
=
2
=1
(h|α|2 i + h|β|2 i)2
(h|α|2 i + h|α|2 i)2
2h|α|2 i2
15
(3.15)
and for the other cases of α and β we write out the denominator and get
(h|α|2 i + h|β|2 i)2 = 2h|α|2 ih|β|2 i + h|α|2 i2 + h|β|2 i2 ≥ 2h|α|2 ih|β|2 i
(3.16)
and see that the amplitude will be smaller than one for those cases. Regardless of the argument to the cosine we see that g (2) (d) is 1 for a single
source, and will fluctuate between 0 and 2 for the other case of α and β. It
is important to remember that this is still a classical calculations where light
is being treated as a perfect mathematical wave.
We now proceed to look at the phase of the cosine where there are variables which correspond to the geometry of our HBT model (Fig. 3.1). We
can rewrite the phase of the cosine as:
k(r1a − r2a − r1b + r2b ) = k((r1a − r2a ) − (r1b − r2b ))
(3.17)
and immediately see that if the phase is equal to an integer multiple of 2π,
then cos(φ) = 1 and the value of g (2) (d) will be maximized. This corresponds
to a situation in which the sources a and b are not resolved and they appear
a point source to the distant observer (i.e. R=0 in the Fig. 3.1 model).
As an unrealistic, but insightful, theoretical exercise we will now look
at the complete opposite situation: source a and b at opposite ends and at
distance L of the two detectors. In the case of our model this translates into
r1a = r2b = L and r2a = r1b = L + d. We plug these values into 3.17 and get:
−4πd
(3.18)
λ
with λ the wavelength of the light. With the cosine symmetrical around zero
we can drop the minus sign. We see that, maybe surprisingly, for a situation
of d = 0 coherence is maximized, while the sources could be literally at
opposite ends of the galaxy with no way to communicate with one another.
The cosine will drop from 1 to 0 over the course of 12 π which translates in
our example to a d of
k((L − (L + d)) − ((L + d) − L)) = −2kd =
λ
(3.19)
8
We see that increasing the distance between the detectors from 0 to λ/8 will
cause a drop in the coherence (roughly a drop from 2 to 1) and it is that drop
which tells us that we have resolved a source and which is the mechanism
behind intensity interferometry. Of course with the wavelength of optical
light being around 100 nm it would be rather difficult to find two detectors
which are not separated by more than λ/8. But then it can hardly come
d=
16
as a surprise that it is easy to distinguish two sources which are at opposite
sides of the observer. And of course two sources which are at opposite ends
of the observer will never be unresolved, with the light coming from opposite
directions. Again: this was only a hypothetical example.
The above example is on the other hand quite similar to looking into
deep space. If we assume R large enough so that a and b are not causally
connected over the short timescale of the observation, but L large enough
that the sources are spacially unresolved (i.e. their images overlap for the
observer) then we can still observe the cross term in Eq. 3.16. A typical
example would be looking at two very distant galaxies. Apparently light
sources don’t need to be able to communicate with one another for their
intensity to be correlated.
Of course in the typical scenario in which we observe two points on the
surface of a star a and b will be close together so that we can assume that
R L. Using this assumption we can derive that the correlated signal in
3.13 varies as a function of the detector separation d on a characteristic lengh
scale
λ
(3.20)
θ
with λ the wavelength of the light and θ = R/L the angular size of the
sources as seen from the detector. In such a case one would start measuring
g (2) (d) with the detectors directly next to one another and then slowly move
one of the detectors further and further away from the other detector until
one sees a drop in the measured correlation and the distance between the
detectors at that point will constitute d. Knowing the wavelength of the
detected light will then give you the angular size of the source via Eqn. 3.20.
d=
3.2.2
Quantum explanation of the HBT experiment
HBT successfully tested the workings of their newly developed technique
by measuring the radial size of the newly found radio sources Cassiopeia
A and Cygnus A with a prototype intensity interferometer. During said
testing however they noticed that when the sources were strongly scintillating
due to atmospheric disturbances the measurements of correlation were not
significantly affected. Looking into this effect Twiss found that this was
actually to be expected and that intensity interferometers could be made to
work unaffected through a turbulent medium such as the atmosphere.
This insight led them to investigate the possibility of developing an intensity interferometer for light waves. Even though at first glance it seems that
what works for one energy range of photons should also work for another it
17
wasn’t that trivial in this particular case. The energy of individual photons
in the radio frequency band is so small that for any magnitude the number
of photons is so large that the energy arrives smoothly (i.e. wave-like) as
opposed to in bursts as might be expected from a stream of discrete energy
carriers. In the visible light range however that is no longer the case and it
becomes necessary to take into account the particle nature of photons, or, in
other words, develop a quantum theory for the intensity interferometer.
As in the previous subsection we will not do a full derivation but give
the reader a rough outline of how the theory works. For this outline we
will use the same model of a HBT experiment as depicted by Fig. 3.1.
Now that we consider light to consist of photons in this derivation we can
interpret g (2) (d) in an insightful way: a value proportional to the probability
of detecting two photons, one in each detector, at the same time (or: within
a very small window). It is also said that g (2) (d) stands for a two-photon
event, which ties neatly into it being the second order coherence function. A
consequence of which is that one cannot say anything about the intensity of
detected light being coherent or not by detecting single photons. Classical
astronomy consists of observations described by special cases of the first order
coherence function and exploit only the single photon events which occur in
their telescopes. We see here a first hint of how by explicitly taking into
account the photonic nature of light astronomy can gather more information
from the ‘same old’ light which falls on its telescopes.
In contrast with the rest of this report the h and i below will denote
the bra- and the ket notation as used in quantum theory as opposed to the
average over a long time. As before we have two sources a and b, separated
by a distance R, and, at a distance L, our two detectors 1 and 2 (as depicted
by Fig. 3.1). We will write the amplitude of the wave function for the case
of detecting a particle from source a in detector i as ha|ii, and conversely for
b. We shall assume that:
ha|1i = ha|2i ≡ a
hb|1i = hb|2i ≡ b
Let’s assume that detector 1 and 2 are at a distance d so that sources a
and b are unresolved. This means that from the observer’s point of view
the light coming from a and b seems to come from the same point. Having
interpreted g (2) (d) as a value proportional to the occurrence of a two-photon
event we calculate the probability of such an event in the described setup.
There are two ways in which we can have a two-photon event: a to 1 and b
to 2 or a to 2 and b to 1. With a and b unresolved the photons impinging on
18
the detectors are indistinguishable and as they are bosonic particles the two
possibilities will interfere constructively and we have to add their amplitudes
before taking the square:
Pi = |ha|1ihb|2i + ha|2ihb|1i|2 = |2ab|2 = 4|a|2 |b|2
(3.21)
with Pi the probability of a two-photon event in the case of indiscernable
bosonic particles. Now let’s assume that the observer increases the distance
d until sources a and b are (spatially) resolved. This means that we can
distinguish photons coming from a from photons coming from b by their
direction. Even if we don’t actually take the effort to look into the focal
plain of the focusing lens of our telescope to see whether the photon hits the
CCD in the a-pixel or in the b-pixel, the fact that they can be distinguished
destroys the interference and the probability at a two-photon event becomes
a straighforward sum of the square of the amplitudes of the two ways one
can have a two-photon event:
1
(3.22)
Pd = |ha|1ihb|2i|2 + |ha|2ihb|1i|2 = 2|a|2 |b|2 = Pi
2
with Pd the probability of a two-photon event in the case of distinguishable
particles. We find that in the case of indiscernable bosonic particles that it
is twice as probable to have a two-photon event than in the case of distinguishable particles. With Pd and Pi proportional to the coherence we find
that, similar to the classical case, that g (2) (d) will lose half its value when
the sources go from being unresolved to being resolved. But now that we
have given a quantum treatment at the particle level for the HBT effect we
can say that indiscernable bosonic particles (such as photons from an unresolved star) have a tendency to arrive together as opposed to on their own.
This phenomenon is called photon-bunching and, as we have seen, it is an
intrinsic (quantum) property of light rather than a property of the source.
At the time of HBT applying quantum physics on photons in such a way was
very revolutionary and many physicists of the time were unwilling to accept
their quantum explanation of intensity interferometry. In fact: many papers
were published trying to prove HBT wrong, but in the end HBT were vindicated and they went on to build a large-scale stellar intensity interferometer
in Narrabi, Australia where they successfully measured the radial size of 32
stars for the first time.
In this report we will now depart somewhat from HBT and their experiment and focus on a different special case of the second order coherence
function: g (2) (τ ).
19
3.3
From spatial- to temporal coherence of
light
So far in our project we have been working with the special case of the second order coherence function g (2) (0, 0; d, 0) = g (2) (d) where we compare the
datastreams of two detectors at the same time but separated in space by a
distance d. This special case of coherence is also called spatial coherence and
by exploring the shape of the function for different values of d (particularly
starting at 0 and then increasing the distance) it will give you information
about the geometry of the source(s) under investigation. It was - as mentioned - successfully used between the 50s and 70s by HBT to measure the
radial size of distant stars which Michelson interferometry couldn’t touch at
the time [3], and these days is mostly used to probe high energy nuclear and
particle collisions for information about the space-time geometry of those
collisions [2].
In quantum optics however, a field of physics which was for an important
part sparked by the insights gained from the work of HBT on intensity interferometry, another special case of second order coherence is most commonly
used: g (2) (0, 0; 0, τ ) = g (2) (τ ). To calculate g (2) (τ ) we compare two datastreams taken at the same location from a source with one of the streams
shifted by a time (delay) τ . In this project we will work exclusively with
temporal coherence.
In essence spatial- and temporal coherence work in the same way. In
both cases we take two datastreams from a source and compare those by
introducing an offset to one of the two datastreams. We then plot g (2) as a
function of the offset and the resulting graph (typically containing a ‘drop’
at a certain value of the offset) gives us information about the source we
are investigating. Except that with spatial coherence we ‘probe’ the source
by varying the distance between two detectors and this gives us information
about the geometry of the source, and with temporal coherence we ‘probe’ by
varying the time delay and this gives us information about the arrival time
of the photons at the detectors and consequently about the emission times
of those photons at the source. Furthermore: we can convert these times
to distances by multiplying them with the light speed and potentially gain
information about small scale structures at distances that would be completely impossible to resolve with any type of interferometry (interferometry,
whether with amplitudes or with intensities, is in the end limited by the
maximum distance, or baseline, between the detectors making the earth’s diameter effectively the upper limit). But a time difference of one nanosecond
between two simultaneously released photons will constitute 30 centimeters
20
regardless of our distance to the source! We shall look further into temporal
coherence.
As g (2) (τ ) is a special case of the second order coherence function, it
stands for a two-photon event. We can interpret g (2) (τ ) as a function being
proportional to the chance that a second photon arrives within a time τ
after detecting a first one. This forces us to look at incoming light in a
new way. Apparently next to the intensity and the frequency of the light
we also need to look at the distribution of the photons in the incoming light
stream. With the above interpretation we can easily think of two light beams
of equal intensity and frequency with very different g (2) (τ ) graphs. Imagine
for example a light stream where the photons are very evenly spread, to the
point that the time between two successive photons is always larger than ∆t,
then it follows that g (2) (τ ) = 0 for that stream for 0 ≤ τ ≤ ∆t. On the
other hand, imagine a light stream with identical intensity as the previous
one but with all photons clumped together into ‘bursts’ of ∆t followed by
a stretch of nothing until the next burst. It is obvious that in this case
g (2) (τ ) will be very high for values of τ between 0 and ∆t. So here we have
two hypothetical photon streams with identical intensity and frequency, but
very different g (2) (τ ). And indeed such differences between photon streams
exist and, as we shall see in the next sections, they even lead to an insightful
classification of light based on the distribution of its photons. More important
for astronomy though is the fact that as we observe the photon distribution
of light from a distant source we simultaneously measure the distribution of
photon emission at that source. If we perceive that a certain source emits its
photons at regular time intervals or, conversely, in small bursts then theory
has to account for that observation. And the more experimental constraints
there are, the more accurate the theory can become.
3.4
The ‘semi-quantized’ atomic model
Less than fifty years after the Maxwell equations had seemingly proclaimed
Huygen’s wave principle the victor over Newton’s ‘corpuscles’ Einstein seemed
to place the victory firmly back in the hand of the Englishman. But subsequent development of quantum mechanics and analysis of the physics involved showed that the photo-electric effect could also be explained by taking
a quantized nucleus-with-electron-cloud atom and treating light as a classical (Maxwellian) electromagnetic wave [10], also called the semi-quantized
atomic model. And the same holds true for ‘photon counting‘ devices like
photomultiplier tubes or Geiger counters. Even minute adjustements to
atomic energy levels due to the presence of electric- (Stark effect) or magnetic
21
fields (Zeeman effect or spin-orbit coupling) are calculated by introducing a
wave-perturbation into the appropriate equations. This meant that it had
still not been conclusively proven that light consists of particles which behave
in a wave-like manner rather than simply being a wave.
It turned out to be very hard to find properties of light which could not be
explained by the ‘semi-quantized atomic model’, and it wasn’t until the 1970s
that properties of light were observed in physical experiments which could
only be explained by light consisting of particles [13][8]. These discoveries
sparked the birth of a new field of physics aptly called ‘Quantum Optics’, in
which the consequences and potential of light particles are investigated and
exploited. In the following sections we will look at how ‘wave light’ can be
distinguished from ‘photonic light’.
But first we shall give the reader a qualitative argument why it was so
hard to distinguish ‘wave light plus quantized atom’ from ‘photonic light plus
quantized atom’. If we have quantized atoms bombarded with monochromatic light with constant amplitude and the right frequency so that one
or more electrons can be freed from the atoms thereby ionizing them, then
we know that (detectable) photoelectrons will be freed from the atoms over
time. And the ‘time’ here is essential. The classical light falls upon the atom
in a continuous stream, but the atoms can only make discrete absorptions
and emissions. This means that sometimes the wave will pass harmlessly
over the atoms and sometimes it will interact with an atom and ionize it.
One could say that the transition from classical (light) to quantum (atom) is
moderated by giving ‘classic’ and ‘quantum’ a certain chance per unit time
to interact with each other. And if we now were to measure these individual
photoelectrons as a current and plot that current against time we would see
a zero current with an occasional ‘spike’ indicating the arrival of an electron.
And that obviously looks like light consisting of particles! From Einstein’s
publication until the 70s, physicists were not certain if their photo-multipliertubes were showing them the intrinsic photon statistics of the light beam or
the statistical nature of photodection of a classical light wave. With both
models of light leading to the same type of discrete signal what is left to us
is to compare the distributions of those detections over time, to look at their
statistics.
22
3.5
Photon Counting & Statistical fluctuations
Before we can say anything about the distribution of photons we need to be
able to detect and count them. We assume monochromatic light of constant
itensity I with angular frequency ω. In the photonic model of light we define
the photon flux Φ as the average number of photons passing through a crosssection of the beam in unit time. In the wave model of light we would define
the flux (without ‘photon’) Φ as the energy density passing through a crosssection of the beam in unit time.
We calculate Φ by dividing the energy flux by the energy of the individual
photons:
P
IA
≡
photons s−1
(3.23)
~ω
~ω
with A the area of the beam and P the power.
A photon-counting detector is characterized by its quantum efficiency η
which is defined as the ratio of the number of photocounts to the number of
incident photons. We calculate the average number of counts registered by
the detector N in a counting time T :
Φ=
N (T ) = ηΦT =
ηP T
~ω
(3.24)
from which follows count rate R:
ηP
N
= ηΦ =
counts s−1
(3.25)
T
~ω
We shall now work under the assumption that light consists of photons. In
the case of photons Eqn. 3.24 and Eqn. 3.25 are average properties of the
beam. If we would have a photon flux of one million photons per second, and
we would take a beam segment of 3 · 108 m (one lightsecond) then we know
that there are one million photons within the segment on average; maybe
we just had two photons enter the beam segment at the front, and had none
leave at the back, thereby increasing the total number of photons by two, but
over time such fluctuations will even out and regardless those fluctuations
are small when compared to a million photons. If however we decrease the
size of our beam segment then the number of photons we will expect to find
will decrease accordingly, while our uncertainty about the number of photons
(arrivals and departures at the front and back of the segment) remains the
same. Once our beam segments becomes smaller than 3 · 102 m (which is a
million times smaller than our original segment and has a one photon average)
R=
23
we will expect a fractional number of photons to be in the segments. Since
photons can evidently occur only in integer multiples we will expect some
of the segments to have no photons and others one or more. If we decrease
the beam segment size further then we will get to a point when the beam
segments will have either one or zero photons. And whether one of those
very small beam segments contains a photon or whether its direct neighbor
contains one makes no difference to the average photon flux if the majority
of segments is empty. Whether a sequence of beam segment photon counts
looks like:
0, 1, 0, 0, 0, 0, 0, 0, 0, 1
(3.26)
0, 0, 1, 0, 0, 1, 0, 0, 0, 0
(3.27)
or like:
on the macro-level of total photon fluxes it amounts to exactly the same.
Consequently we have no way to predict which of the small segments might
be filled with photons and which might be empty.
By decreasing our beam segment from one lightsecond to 300 m and
less we have gone from a situation where we had a fair idea of where the
photons were located to being increasingly uncertain about where they are.
Even if the average photon flux might be a very well-defined value, once we
decrease the size of the beam segments to the point that we start ‘seeing’
invidual photons (or: once we start working with small timescales) we will
start seeing statistical fluctuations in photon distribution caused not by any
physics in the source but by having discrete light particles. We will even see
this phenomenon for ourselves in section 4.3.3 of this report.
3.6
Coherent light: Poissonian photon statistics
In the previous section we have introduced the statistical fluctuations in the
photon count of small beam segments (or short timescales). In this section
we will derive the mathematical properties of the distribution of photons
in the most stable type of light that we can imagine in classical physics: a
perfectly coherent light beam with constant angular frequency ω, phase φ,
and amplitude E0 :
E(x, t) = E0 sin (kx − ωt + φ)
24
(3.28)
with E(x, t) the electric field of the light wave and k = ω/c in free space.
With E0 and φ being constant in time the intensity I of the beam, which is
the long time average of the square of the electric field, will be constant as
well. Eqn. 3.23 tells us that a constant intensity I and angular frequency ω
means a photon flux Φ which is independent of time. Let us now look at a
light beam of constant power P . The average number of photons within a
beam segment of length L is given by:
n̄ = ΦL/c
(3.29)
We take L large enough so that n̄ is well defined (similar to the one million photons in the one lightsecond beam segment in our example). We now
divide the beam segment into N subsegments of length L/N , with N large
enough that p = n̄/N , the probability of finding a photon within a particular
subsegment, is small enough that we can neglect the probability of finding
two or more photons in any one single subsegment. We will now proceed to
calculate the probability P(n) of finding n photons within a beam of length L
containing N subsegments. Having assumed that there are no subsegments
with two or more photons the calculation breaks down to calculating the
chance of finding n subsegments containing one photon and (N − n) subsegments containing no photons. This probability is given by Newton’s binomial
distribution:
P(n) =
N!
pn (1 − p)N −n
n!(N − n)!
(3.30)
substituting p = n̄/N this becomes
n̄ N!
n̄ N −n
P(n) =
1−
n!(N − n)! N
N
(3.31)
we rewrite Eqn. 3.31
1
P(n) =
n!
N!
(N − n)!N n
n̄ N −n
n̄n 1 −
N
(3.32)
Using Stirling’s approximation
ln N ! ≈ N ln N − N
(3.33)
we proceed to look at the natural logarithm of the first part between parentheses of Eqn. 3.31. Because we are working with large N and Nn 1 we
can safely neglect the extra terms in Stirling’s approximation:
25
ln
N!
(N − n)!N n
= N ln N −N −(N −n) ln (N − n)+N −n−n ln N (3.34)
we rewrite the righ hand side of Eqn. 3.34:
N
+ (n ln (N − n) − n ln N ) − n
N ln
N −n
(3.35)
for big N the part between parentheses will go to zero. We are left with
investigating the first logarithmic term
N
n
1
N ln
=
−N
ln
1
−
= N ln
(3.36)
N −n
1 − Nn
N
we take one of the ‘standard’ Taylor series:
ln (1 − x) = −
∞
X
xn
n=1
n
for − 1 ≤ x < 1
,
(3.37)
plug it into Eqn. 3.36 and get
N
n n 2 n 3
+
+
+ ...
N
N
N
=n+
n2
n3
+ 2 + ...
N
N
(3.38)
for large N all the terms except the first one will go to zero and the expression
evaluates to n. We enter this result into Eqn. 3.34 and get
N!
ln
=n−n=0
(3.39)
(N − n)!N n
if the natural logarithm of an expression is 0 then it follows that the expression itself is equal to 1. We find that, for large N and Nn 1:
N!
=1
(N − n)!N n
(3.40)
we now proceed to look at the second part between parentheses of Eqn.
3.31. Again we are assuming that N is large and Nn 1. Taking the Taylor
expansion of the expression of interest we find:
1−
n̄ 2
n̄ N −n
n̄
1
= 1 − (N − n) + (N − n)(N − n − 1)
− . . . (3.41)
N
N 2!
N
26
at every power of n̄ we neglect those terms which have a positive power of
N in the denominator since thsoe will go to zero for large N . The resulting
expression becomes:
n̄2
− . . . = exp (−n̄)
(3.42)
2!
Substituting the above two results in Eqn. 3.31 we find for P(n) for large N
and Nn̄ 1:
1 − n̄ +
P(n) = n̄n n!e−n̄ , n = 0, 1, 2, . . .
(3.43)
this particular distribution is well-known and it is called the Poisson distribution. The Poisson distribution is a
“discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time and/or space if these events
occur with a known average rate and independently of the time since the last
event.”
In the above calculation we have derived the distribution of photons in a
beam from a coherent source impinging on our detector and what we have
found is that the photons behave as if their detection is a random event which
occurs with a constant ‘chance per time’. But in section 3.4 we found that
in the semi-quantized atomic model we have an identical mechanism: the
constant chance per time that the quantized atom has to interact with the
classical light wave. And indeed for an identical perfectly coherent source
with constant intensity and phase both models lead to exactly the same
Poissonian distribution of ‘spikes’ in the photodectors. Here we have the
mathematical formulas behind the difficulty of definitively proving the existence of photons as discrete particles. In the next section we will see how we
can use this result to classify different types of light and what a central place
the Poisson distribution has in this.
3.7
Classification of light by photon statistics
In the above section we found that in the case of a perfectly coherent light
source of constant intensity and phase both photonic light and classical light
lead to a Poisson distribution of detected events in the photodetectors. But
the Poisson distribution (given in Eqn. 3.43) has some intrinsic properties
which, apparently, will also have to apply to the distributions found in the
detectors. We look in this case at the property variance, defined as:
27
∞
X
Var(n) ≡ (∆n) =
(n − n̄)2 P(n)
2
(3.44)
n=0
it gives a measure for deviations from the average, with ∆n called the standard deviation. It is a well-known mathematical result that the variance of
the Poisson distribution is
(∆n)2 = n̄
(3.45)
and consequently the standard deviation:
∆n =
√
n̄
(3.46)
and this gives us a benchmark to classify light with. We distinguish three
different types of light:
√
• sub-Poissonian light: ∆n < n̄
√
• Poissonian light: ∆n = n̄
√
• super-Poissonian light: ∆n > n̄
If we take the same coherent lightsource as before but start modulating its
intensity over time then we expect larger fluctuations in the photon spread
than with constant intensity. Consequently all classical light sources with
time-dependent intensity will produce super-Poissonian light. Examples of
super-Poissonian light are: partially coherent (chaotic) light, incoherent light,
or thermal light.
Sub-Poissonian light however does not exist in a world with classical light.
We can’t improve upon an already perfect wave and this is indeed our first
example of non-classical light. Even if the theory currently put forward in
this report has said nothing about how we could generate such light, it is
not hard to conceive of light with sub-Poissonian statistics. All it takes is
a beam of light with the photon-spread having a narrower distribution than
a Poissonian one. A very obvious example would be a photon stream with
∆n = 0, meaning the photons ‘marching in lockstep’ at identical distances
of c∆t. Photon streams with this characteristic are called photon number
states and it is obvious that they are the ultimate form of sub-Poissonian
light. But any light with a photon distribution with less spread than a
Poisson distribution is sub-Poissonian and indeed these types of light exist
and can be created and detected in the lab even though both creation and
detection are very technically demanding. In the next section we shall go
over some of the difficulties that arise when trying to detect sub-Poissonian
light.
28
3.8
Degradation of photon statistics by losses
In any detection process there are imperfections which cause loss of signal (in
this case photons). We have seen that we can detect sub-Poissonian light by
measuring the statistical distribution of the photons in such light. Therefore
in the case of the detection of sub-Poissonian light our ‘signal’ is ‘statistics’.
The problem here is that it is a well-known fact that the distribution obtained
by random sampling of a given set of data is more random than the original
distribution. And indeed these imperfections in the detection process will
tend to degrade the quality of the signal to the (random) Poissonian case.
Examples of such inefficiencies are:
• inefficient collection optics, whereby only a fraction of the light emitted
from the source is collected
• losses in the optical components due to absorption, scattering, or reflections from the surfaces
• inefficiency in the detection process due to using detectors with imperfect quantum effiency(η)
These processes all lead to a version of random sampling. The first randomly selects photons from the source, the second randomly deletes photons
from the source, and the third randomly selects photons to be detected.
Where the first two degrade the photon statistics themselves, the third degrades the correlation between the photon statistics and the photoelectron
statistics. This is a good example of the sort of difficulties that are intrinsic
in the photodection process.
In Fig. 3.2 the process of statistical degradation is graphically shown.
The process of degradation of a perfectly regular photon number state due
to a ‘lossy medium’ with transmission T = 50% is modeled there by a 50:50
beam splitter. It is clear to see how much more ‘random’ the resultant
stream is after the insertion of even one mechanism with 50% efficiency. It
is obvious that optical losses need to be minimized at all costs and detectors
with very high efficiencies will be necessary for us to properly detect subPoissonian light. The development of these highly efficient techniques was
one of the major hurdles that physicists needed to take to definitively prove
the existence of photons.
Particularly insightful because it provides a quantitative backing of this
so-far qualitative argument is a result from the ‘Quantum theory of photodetection’ where the aim is to relate the photocount statistics in an observed
experiment to those of the incoming photons. The relationship between the
29
Figure 3.2: (a) The effect of a lossy medium with transmission T on a beam
of light can be modelled as a beam splitter with splitting ratio T : (1 − T )
as shown in (b). The beam splitting process is probabilistic at the level of
the individual photons, and so the incoming photon stream splits randomly
towards the two outputs with a probability set by the transmission : reflection
ratio (50:50 in this case) as shown in part (c). Source: [10].
30
variance in the photocount number (∆N )2 and the variance in the number
of photons impinging on the detector (∆n)2 is given by:
(∆N )2 = η 2 (∆n)2 + η(1 − η)n̄
(3.47)
With η = N̄ /n̄ the detector efficiency as before. The derivation of this result
is far beyond the scope of this project. For further information on the topic
we refer to [15].
We see that for η = 1 the relation becomes ∆N = ∆n: a perfect correlation of the statistics. We see that if (∆n)2 = n̄ (incoming Poissonian light),
then (∆N )2 = ηn̄ ≡ N̄ for all values of η: No matter what the detector,
if Poissonian light impinges on it, then Poissonian distributed photocounts
will come out. And lastly we see that if η 1 that then (∆N )2 = ηn̄ ≡ N̄ :
irrespective of the photon statistics, if the detector is highly inefficient then
the photocounts will be Poisson distributed.
3.9
g 2(0): a different light classification
In the previous section we ordered light in three categories based on the
benchmark of Poissonian light. In this section we shall discuss a similar
classification but this time based on g (2) (0). Again we distinguish three types
of light and in analogy with the earlier classification we shall see that two
of those are compatible with classical optics, and one not. To understand
this classification we need to introduce the concept of coherence time τc and
coherence length Lc = cτc . The coherence time gives the time duration
over which the phase of a wave train remains stable. The appearance of
interference fringes is entirely dependent on the phases of the waves of interest
being constant in first-order coherence 3.1.3, and this tells us we can only say
something about the coherence of light between two points in time or space
if |t2 − t1 | / τc or |z2 − z1 | / Lc .
Even though strict derivations of the formulas involved are beyond the
scope of this project the concept of coherence time is not hard to picture.
In general light consists of a wave package of a collection of frequencies (and
conversely wavelengths). If the phase of this light is the same for all frequencies at a certain moment, meaning the different waves of different frequences
are in phase, then after a certain amount of time the waves will no longer be
in phase due to their different frequencies. And in the line of this argument
it is easy to see that the smaller the spectral width of light is, the longer its
coherence time will be. This relation between the two concepts turns out to
be so that the following holds:
31
1
(3.48)
∆ω
with ∆ω being the full width at half maximum (FWHM) of the spectral
lineshape [10]. This formula also shows why a perfect monochromatic light
source is the perfect classical case of (first order) coherence: with ∆ω = 0
the coherence time goes to infinity. More information on this topic can be
found in [15].
We return to the classification of light by means of g (2) (τ ). The driving
principle behind the HBT effect is that the intensity fluctuations of a beam
of light are related to its coherence: if light is coherent, then its intensity
fluctuations will be coherent. From a classical point of view let us assume
that we have a light source with constant average intensity such that hI(t)i =
hI(t + τ )i. We rewrite I(t) as a function of the average intensity with a
variation on that average:
τc ≈
I(t) = hIi + ∆I(t)
(3.49)
With intensity fluctuations being related to coherence we can say that coherence fluctuations which happen at different times such that |t2 − t1 | ≡ τ τc
will be completely unrelated, so that ∆I(t)∆I(t + τ ) will average out to zero:
h∆I(t)∆I(t + τ )iτ τc = 0
(3.50)
By definition h∆I(t)i = 0 and we get:
(3.51)
hI(t)I(t + τ )iτ τc = h(hIi + ∆I(t))(hIi + ∆I(t + τ ))i
= hIi2 + hIih∆I(t)i + hIih∆I(t + τ )i + h∆I(t)∆I(t + τ )i
(3.52)
= hIi2
(3.53)
Now we can calculate g (2) (τ ) (see Eqn. 3.6) for τ τc :
g (2) (τ τc ) =
hI(t)i2
hI(t)I(t + τ )i
=
=1
hI(t)i2
hI(t)i2
(3.54)
On the other hand if τ τc then the intensity fluctuations will be correlated,
and we get:
g (2) (0) =
hI(t)2 i
hI(t)i2
It can be shown that for any I(t) we get that
32
(3.55)
g (2) (0) ≥ 1
(3.56)
g (2) (0) ≥ g (2) (τ )
(3.57)
and that
for a mathematical derivation of these results we refer to [10].
Incidentally we find for a perfectly coherent monochromatic source with
time-independent intensity I0 :
g (2) (τ ) =
I02
=1
I02
(3.58)
for all values of τ .
Combining these results we get an idea of the shape of a classical g (2) (τ )
function: the function is at its highest value (≥ 1) at the origin and will go
to 1 in a timespan of roughly τc . We now go back to our classification of light
based on g (2) (0). We distinguish three types of light:
• bunched light: g (2) (0) > 1
• coherent light: g (2) (0) = 1
• antibunched light: g (2) (0) < 1
We have seen in the above section that g (2) (0) = 1 corresponds to perfectly
coherent light, which, as we have seen, has Poissonian photon statistics. So
we see that in both classifications perfectly coherent light is the benchmark
on which the classification is based. In the g (2) (0) ordening of light however,
we are not looking at the variance of the photon statistics but at whether
or not the photons come in clumps, or bunches, or whether they are evenly
spread (antibunched). Even though the concepts are of course not unrelated,
they are not identical (see: [10] and the original paper [16]).
Bunched light consisting of clumps of photons arriving at the same time
has, almost by definition, a variable intensity I(t). Earlier in this report we
have interpreted g (2) (τ ) as being proportional to the chance of detecting a
second photon within a time τ after detecting the first one. With this in
mind we can make a qualitative argument about what we expect g (2) (τ ) to
behave like for bunched light: with photons clumped chances of detecting
a second photon will be high for τ < τc (with τc being the average time
between intensity fluctuations). For τ > τc we expect g (2) (τ ) to decrease in
value. Summarizing for bunched light:
33
g (2) (0) > 1
g (2) (0) > g (2) (∞)
Classical light has to obey Eqn. 3.56 and Eqn. 3.57. Bunched light obeys
these equations and is therefore consistent with the wave model of light.
Using once more the two-photon-event interpretation we look at how we
expect g (2) (τ ) to behave for antibunched light: if the photons are evenly distributed over the light stream with gaps of c∆t between subsequent photons
then we expect g (2) (τ ) to be zero for τ < ∆t. In reality it won’t be exactly
zero (among other things due to nonzero vacuum energy) but the point is
that we expect it to be low, and particularly less than unity. Then as we
increase τ to values of ∆t and greater it becomes more and more likely to
detect a second photon and we expect g (2) (τ ) to increase. Summarizing for
antibunched light:
g (2) (0) < g (2) (τ )
g (2) (0) < 1
These to properties are in obvious violation of Eqn. 3.56 and Eqn. 3.57
and antibunched light can only be explained by the photonic light model.
In Fig. 3.3 one can see typical g (2) (τ ) graphs for Gaussian- and Lorentzian
chaotic light (g (2) (0) > 1), and for antibunched light (g (2) (0) < 1).
3.10
τc: the need for good time resolution
In the previous section we have classified light using the g (2) (0) benchmark
and we have introduced coherence time τc , and seen that light becomes incoherent on timescales longer than τc . This phenomenon puts rather severe
technical demands on detection technology when trying to measure g (2) (τ )
of optical light. The wavelengths of optical are in the order of 10−6 s. This
corresponds to angular frequences in the order of 1015 rad/s. This gives that
∆ω ≈ 1015 rad/s
(3.59)
and using Eqn. 3.48 we find that for optical light
τc ≈ 10−15 s
34
(3.60)
Figure 3.3: Three typical g (2) (τ ) graphs. Blue (—) line: Gaussian chaotic
light, black (· · ·) line: Lorentzian chaotic light, red (-·-) line: antibunched
light. Source: [9].
35
Figure 3.4: Antibunched(green), Random(red), and Bunched(blue) light
streams shown with τc . Source: wikipedia.
In our report we shall work with data taken with the highest time resolution
photometer to date and with its (very impressive) 100 picosecond accuracy
it is still five orders off the coherence time scales in optical light.
The need for small timescales is graphically explained in Fig. 3.4. In
this figure there are three light streams of equal intensity (20 photons per
the time shown) of the three different types of light. Because the intensity
of the streams is identical (20 photons per the time shown) the only way we
can distinguish them is by looking at the photon distributions. If we have a
time resolution of the order of τc then we can use a bin size (in analogy of
the earlier-mentioned subsegments) with a width of τc and the bin counts of
the different photons streams will look something like:
antibunched: 2, 2, 2, 2, 2, 2, 1, 2, 2, 1, 2
random: 2, 2, 2, 1, 3, 2, 1, 2, 1, 2, 2
bunched: 3, 1, 2, 1, 3, 3, 0, 2, 1, 2, 2
If however our time accuracy (and correspondingly our bin size) is five
orders of magnitude larger than τc then our bins will be filled with 100000
photons each but the ‘signal’ which would distinguish between the different
types of light would still lie in the single digits. And the only way to get that
signal with that much noise is by applying the law of great numbers. And
this makes the measurement of g (2) (τ ) not only very demanding technically,
but also statistically. And the latter part is yet another thing which ties into
g (2) being the second order coherence function. In the case of any type of
‘series’ it is almost always harder to get a result to the second order instead
of just to the first order. And it is no different here.
36
Lastly, the difficulties described above in measuring g (2) have been the
main reason that no astronomer since Hanbury Brown and Twiss has since
looked into using the second order coherence function once modern technology was advanced enough to use first order Michelson Stellar Interferometry
even on very distant stars. But 40 years since HB & T did their work technology has progressed to the point that measuring the second order coherence
of light from distant stars is coming within our reach. And in this project
we will try to see if it is already feasible with today’s technology.
37
Chapter 4
Project, Results & Discussion
This chapter contains the actual work done for this project. We will state the
goal of the work, we will describe the experimental setup which produced the
data and why this data was taken, we will explain the operations performed
on the data to acquire our results, and lastly discuss these results and their
implications.
4.1
Description Project
The goal of this project is to investigate the feasibility of using high efficiency
detectors with top-of-the-line time accuracy in astronomy, and through them
measure the second order correlation function g (2) (τ ).
The reason that we decided to investigate this possibility is that second
order coherence, once we can systematically measure it, stands for an entirely
untapped region of ‘parameter space’ in observational astronomy.
In the past whenever astronomy managed to increase the parameter space
over which it could investigate the universe it always turned out that interesting physics was to be found: the invention of the radar led to the discovery
of giant radio stars. The ability to detect infrared light allowed for the observation of background radiation. Having satellites in space around the
earth with x-ray-sensitive telescopes showed that the galaxy is full of objects
which are active in that wavelength. And increasing the time accuracy of
the photodetectors allowed for the discovery of many astronomical processes
on small timescales (such as pulsars and masars).
The accurate measurement of g (2) (τ ) requires, as we have laid out in this
report, a near perfect quantum efficiency of the detectors, and yet further
improvement of the time accuracies. These techniques if they are developed,
especially the further improvement of the time accuracy, will in all likelihood
38
lead to the discovery of new physics by themselves. But a case can be made
that the systematic measurement of g (2) (τ ) will lead to the most spectular
spinoff of these technological advancements: as it opens up the relevant parameter space it will provide extra constraint on the theoretical models, and
the (generally small-scale) processes in physics which depend on the second
order of coherence rather than the first (such as multi-step ionizations) might
be observed on stars thousands of parsecs away.
Fields in astronomy which so far are the exclusive domain of theoreticians
might become verifiable with astronomical observations. Actual experiments
will have to confirm the practical observability of these processes, but in
principle photon statistics of the radiation from any kind of source could
convey something about the processes through which the radiation was liberated. Whether light in a spectral line consists of short photon showers
with one spontaneously emitted photon leading a trail of ‘stimulated’ others,
whether a Doppler shift of light occurred due to motions of emitting atoms
or by those of scattering media, whether exotic ‘suggested to exist’ photon
bubbles exist in stars, or whether pulsars generate their pulsations through
stimulated synchrotron and curavture radiation, these are all the type of
questions which the accurate measurement of photon statistics (or g (2) (τ ))
can potentially address [5].
The relatively recent development of photodetectors with very high resolution time and relatively high quantum efficiencies (in the 30%-region) has
been an enormous step towards the consistent measurement of g (2) (τ ). In
most articles however the authors await (or actively participate in the development of) the arrival of the next generation of detectors or telescopes.
Even though there is no doubt that higher quantum efficiencies and bigger
telescopes will lead to a tremendous increase in the signal to noise ratio when
measuring g (2) (τ ), this report investigates the feasibility of doing it with today’s technology.
And if we want to observe something our signal to noise ratio will tell
us whether or not that is feasible, so let’s see what kind of signal-to-noise
we can expect. The theoretical expression for the signal-to-noise ratio of an
intensity interferometer is given by [3]:
p
(4.1)
(S/N )RM S ∼ α T0 10−0.4B
with S the signal, N the noise, α a constant dependent on the detector setup
with a value of around 1, with B the apparent magnitude of the star and
T0 the observation time in seconds. Let’s imagine that we’re looking at the
brightest star in the sky: Sirius, which has an apparent magnitude (V) of
−1.46, let’s say we have an α of 0.5, and let’s say that we have a target
39
signal-to-noise ratio of 1000. It follows that our observation time would have
to be:
2
1 S
0.4B
= 271681 seconds = 75.5 hours
(4.2)
T0 =
10
α N
and if we were willing to accept a signal-to-noise ratio of 100 then we’d be
done after 45 minutes of observation. Measuring the g (2) (τ ) of Sirius seems
fairly feasible. The calculation is quite different for the Crab pulsar however
with its apparent magnitude of 16.5. With an α of 1 and a target signal-tonoise of only 10 the necessary observation time would have to be 50 million
years. Quite impossible. And we knew that before we began the project, but
the goal of this project was (and is) to say something about the feasibility of
measuring g (2) (τ ) with today’s technology, not the measurement of g (2) (τ ) of
the Crab pulsar. And if we’ll find that the techniques are perfectly feasible
if only the telescope and the detector would be pointed at a brighter source
then we’d be happy with that result.
4.2
The Experiment
For us to try and say something about the feasibility of measuring g (2) (τ )
we needed data gathered with a relatively modern telescope-and-detector
combination. An opportunity for this presented itself in the form of data
taken with the Iqueye, a newly developed photometer, while being mounted
on the New Technology Telescope in Chile for testing and calibration. We
will provide some technical information on these instruments which gathered
the data, then discuss the data as it was delivered and how it was modified
to suit the needs of this project.
4.2.1
Telescope
The ‘New Technology Telescope’ (NTT) is a 3.58-meter Ritchey-Chretien
telescope in Chile that pioneered the use of active optics. The telescope and
its enclosure had a revolutionary design for optimal image quality when it
was built. The telescope was commissioned in 1989 and completely upgraded
in 1997.
40
Figure 4.1: The New Technology Telescope (NTT). Source: wikipedia.
Organization:
Location:
Coördinates:
Altitude:
Wavelength:
Diameter:
Mounting:
European Southern Observatory
La Silla Observatory, Chile
70◦ 44’01.5”W - 29◦ 15’32.1”S
2375m
Optical
3.58m
Alt-azimuth
The NTT is under normal cirumstances equipped with 2 instruments:
• Sofl (‘Son of ISAAC’, a VLT instrument), a near infrared camera and
low-resolution spectrograph, with polarimetric and high-time resolution
modes
• EFOSC2 (ESO Faint Object Spectrograph and Camera, v.2), a visible light camera and low-resolution spectrograph, with multi-object
spectroscopy, polarimetry and coronography modes
but for the purpose of its testing and calibration it allowed the newly developed Iqueye to be mounted on it (and take astronomical data) for several
days.
4.2.2
Detector
Iqueye is the second prototype (after its predecessor Aqueye) of a “quantum”
photometer, QuantEYE, being developed for future Extremely Large Telescopes of 30-50m aperture to explore micro- and nanosecond timescales in
astronomy. Iqueye is to date the photometer with the highest time resolution
in existence. We could say it is the ‘World Champion’ of photometry. The
Iqueye splits the telescope aperture into four sections each with their own
41
single photon avalanche diode. The arrival time of each photon is measured
with 100 ps accuracy, and it can sustain a count rate of 8 MHz for an entire
night without interruption [11].
Properties of the Iqueye:
Organization:
Pixels:
Accuracy (time):
Max. Count rate:
Dead time:
Quantum efficiency:
University of Padua
1
100 ps
8 MHz
30 ns
v30%
While mounted on the NTT for the purpose of testing and calibration the
Iqueye observed several astronomical objects, among which the ‘Crab pulsar’
was the brightest one observed. It was the data taken during this test- and
calibration run that we received from the University of Padua and with the
Crab pulsar being the brightest of the objects observed it was the natural
candidate for our g (2) (τ ) analysis.
4.2.3
Astronomical Object
The Crab pulsar is the central star in the Crab Nebula, a remnant of the
supernova SN 1054, which occurred in 1054 (and was observed by many on
earth at the time). The Crab is an ‘optical pulsar’ at a distance of 2000
parsecs, roughly 20 km in diameter, and it pulses 30 times per second. The
Crab’s rotational period is decreasing by 38 nanoseconds per day.
Information on the Crab pulsar:
Constellation:
Taurus
Right ascension:
05h 34m 31.97s
Declination:
+22◦ 00’52.1”
Apparent Magnitude (V):
16.5
Distance:
2000 pc
Rotations:
29.6/s
Age:
957 years
The Crab pulsar is not a particularly nearby, or fast pulsar, but the
supernova-remnant-cloud surrounding it is the only x-ray standard candle
currently known in astronomy which makes it an ideal target for intensity
calibration, and its pulsations introduce a well-known ‘ticking clock’ into the
data stream which allows for an easy first-level verification of the proper
working of the photometer (even though the Iqueye is easily four orders of
magnitude more sensitive than what is necessary to detect Crab ’pulsations’).
42
4.2.4
The raw data
With the Iqueye being a single pixel (meaning we can’t say anything about
the direction) photometer the only information it can provide about a photon
is the timestamp of its arrival (be that timestamp with the aforementioned
world-class accurary). This suits our project perfectly since we are investigating the feasibility of measuring g (2) (τ ), which - as we have seen in the
theory section - is a measurement of photon bunching (or anti-bunching) in
the stream: photon-arrival timestamps is all we need. As such the file with
the raw datastream we received consisted of (nothing but) a list of Modified
Julian Dates (MJDs) with every MJD standing for the time-of-arrival of a
photon. To give an idea of the raw data we give the first five entries in the
data-file used mostly for this report:
55178.16722324284934600436
55178.16722324297473500909
55178.16722324307445599056
55178.1672232432495769916
55178.16722324376537800461
The ‘day zero’ of the MJD is 17 November 1858. It follows that day
55178 corresponds with 13 December 2009, which is the date the data was
taken. The total number of datapoints in the file is 247, 389. The first and
last entry in the dataset are
first: 55178.16722324284934600436
last: 55178.1673653066085450114
the difference between these two MJDs is 1.42064 · 10−4 days which corresponds with 12.2743 seconds. Newly added: The reason that we use this 12
second subset instead of the several hours’ worth of Crab-data we received is
that the computational cost of our calculations depend strongly on the total
number of photons used in them. Using a smaller subset we can check the
feasibility of these calculations much more easily, and then later we can in
principle extend them to the entire data-set.
In Fig. 4.2 we see the MJD of the photons plotted against their place
in the datastream. We see that the photons arrive in such a regular way
43
TimeHsL
60
58
56
54
52
50
50 000
100 000
150 000
200 000
Photon Nr.
250 000
Figure 4.2: The MJD of the photon arrival plot against the photon number
in the stream.
that the fluctuations above and below the average time between photons are
entirely invisible in this plot of 250, 000 photons. Since the data is taken
off a pulsar there will be two types of fluctuations: intensity fluctuations
corresponding to the pulse profile of the Crab pulsar, and fluctuations due
to photon grouping. With the former (already invisible in the plot) being
much larger than the latter. Yet it is exactly those small photon-grouping
fluctuations we are interested in in this project. Maybe Fig. 4.2 functions as
a good qualitative (or figurative) argument of how a good time-accuracy is
prerequisite for the measurement of g (2) (τ ).
4.3
The Results
In this section we describe our efforts at calculating and plotting g (2) (τ ). We
will first look at the special case of g (2) (0) and then move on to the general
case of g (2) (τ ).
4.3.1
The modified data
Having familiarized ourselves with the raw data as it came off the Iqueye it
is now time to adapt the data to the needs of this project. First we convert
the modified Julian dates into modified Julian ‘seconds’ by multiplying with
24 · 60 · 60, second we cut off all identical digits left of the comma which are
deadweight, and third we cut off all digits beyond the ‘10 picosecond’ decimal
to prevent us finding physical results in digits beyond Iqueye’s accuracy. The
first five entries of the resultant data now look like:
44
TimeHsL
0.0008
0.0006
0.0004
0.0002
50 000
100 000
150 000
200 000
Photon Nr.
250 000
Figure 4.3: The time until the arrival of the next photon plot against the
photon number in the stream. The horizontal line stands for the mean time
between photons.
48.08818218349
48.08819301710
48.08820163300
48.08821676345
48.08826132866
Now that the data has been modified to be in seconds and at the appropriate precision we can proceed with the actual work of the project. We proceed
to calculate the time between photon arrivals for every photon (except the
last one). The resultant data are shown in Fig. 4.3 and some elementary
statistics of that data in Table 4.1. The statistics tell us that the average time
between photons is 50 microseconds and that the time between individual
photons can fluctuate between 0.26 nanosecond (close to the time-accuracy
limit of the detector) and 0.8 millisecond, almost six orders of magnitude
difference.
Fig. 4.3 shows by means of the horizontal (red) line indicating the average
value that the time between two photons is most likely to lie between zero and
50 microseconds (the average), and that fluctuations upwards become rarer
and rarer the further they are removed from that average. We investigate
the distribution of the time between photons further in Fig. 4.4, which
45
2.6 · 10−10 s
7.9275344 · 10−4 s
4.961582 · 10−5 s
minimum:
maximum:
mean:
Table 4.1: Minimum, maximum, and mean time difference between photon
arrivals in the stream.
Nr. of Photons
50 000
40 000
30 000
20 000
10 000
0.0002
0.0004
0.0006
0.0008
TimeHsL
0.0010
Figure 4.4: A histogram showing the number of photons against the time
between successive photons with a bin size of 10−5 s. Pay Note: the axes
don’t cross in the origin.
is a histogram showing the number of photons against the time between
successive photons.
We see that roughly half ‘between photon’ times lie inside the first three
bins (meaning between 0 and 3 · 10−5 s). In general this is what we want
when trying to measure g (2) (τ ). We have seen in the theory section that
once we increase our binning resolution to the point that we’re starting to
see individual photons (bins with one photon) that the particle-nature of
photons is going to introduce statistical fluctuations. For us to be able to
increase binning resolution and get a better signal we need lots of very short
‘between times’. Whether enough photons in our data file arrive close enough
together that we will be able to measure g (2) (τ ) for interesting (small) values
of τ remains to be seen, but Fig. 4.3 and Fig. 4.4 can’t make us discard the
possibility at least.
4.3.2
Calculating g (2) (0)
We calculate g (2) (0) using Eqn. 3.7, which we repeat here for readability:
g (2) =
hn1 n2 i
hn1 ihn2 i
46
(4.3)
except that we don’t have two streams of photons but only one. But since
we are trying to measure g (2) (τ ), which, as we have seen, is a measure for
the chance of detecting two photons simultaneously at the same place we
can approximate this by comparing the data stream of a single detector with
itself. Eqn. 3.7 becomes:
g (2) (τ ) =
hn(0)n(τ )i
hn(0)ihn(τ )i
(4.4)
with n(τ ) being the number of photons counted at the (Iqueye) detector at
time τ , or, in our specific case, the number of photons in the appropriate
bin. For τ = 0 Eqn. 4.4 simplifies into
g (2) (0) =
hn2 i
hni2
(4.5)
the ratio between the average of the square and the square of the average
(of the bin values). We proceed to divide our twelve seconds of data in
10, 000 bins with a width of roughly 1.2 milliseconds and count the number
of photons in each bin. In Fig. 4.5 we see the result of this operation: the
number of photons per bin plotted against time (or the place of the bin in
our 12 second data stream).
We see that the number of photons per bin varies roughly between 10
and 100 photons and that there is a thick ‘band’ between 10 and 30 photons.
The latter is not too surprising. With 10, 000 bins the average number of
photons per bin is 25 and that is - apparently - a high enough number that
the fluctuations between the individual bins are not too big, as we can see
from the clearly defined band.
Imagine now that we’d do the same calculation with only 100 bins. In this
case we’d expect there to be almost no fluctuation in n, and consequently
that g (2) (0) would be 1. Conversely imagine that we do the calculation with
one million bins. With an average of 0.25 photons per bin this would almost certainly lead to bins with either 0 or 1 photons in them (a maximally
fluctuating n) in which case we’d expect g (2) (0) to evaluate to (since n2 = n):
g (2) (0)106 bins =
1
1
hn2 i
=
' 1 =4
2
hni
hni
4
(4.6)
we see that by increasing the number of bins we increase the value of g (2) (0).
Which is what we expect from the average of the square over the square of
the average (fluctuations will be amplified). This effect is made visible in
Fig. 4.6 where g (2) (0) is plotted as a function of the number of bins used in
the calculation. We see indeed that the value of g (2) (0) goes to 1 for a small
47
Photons per bin
100
80
60
40
20
0
2
4
6
8
10
12
timeHsL
Figure 4.5: The number of photons per bin plotted against the left-side value
of the bin-interval.
number of bins, that it rises rapidly between 0 and 10, 000 bins, smoothes out
between 10, 000 and 20, 000 bins, and becomes what seems to be a straight
line after that.
Superimposed in Fig. 4.6 is a first-order polynomial fit (the red line)
through the data points from 20, 000 bins and onwards. The first- and secondorder polynomial fits (with 99% confidence levels) through these data points
are:
F it1st order (x) = 1.229(2) + 4.48(4) · 10−6 x
F it2nd order (x) = 1.222(5) + 4.9(3) · 10−6 x − 5(3) · 10−12 x2
(4.7)
(4.8)
with x the number of bins. With the second order coefficient being a million
times smaller than the first order coefficient this seems to be a straight line.
At one million bins the linear fit would give a g (2) (0) of 5.7 which is close, and
a little bit higher, to the value of 4 that we expect theoretically. It would be
interesting to see how the g (2) (0) function actually behaves near one million
bins, whether the linear increase will slow down due to diminished returns or
whether we overlooked something in theory, but this falls outside the scope
of the project.
Let’s see if we can understand the behavior of g (2) (0) as a function of the
number of bins. We have established that the function for g (2) (0) amplifies
fluctuations. Judging by the rapid increase of g (2) (0) between 0 and 10, 000
we expect that a lot of fluctuations become ‘visible’ in that interval. Obviously increasing the number of bins by four orders of magnitude will increase
48
g H2L H0L
1.5
1.4
1.3
1.2
1.1
1.0
0
10 000
20 000
30 000
40 000
50 000
Nr. of bins
60 000
Figure 4.6: The value of g (2) (0) plot against the number of bins used during
its calculation.
fluctuations and therefore the value of g (2) (0). But we also know that the
observed light comes from the Crab Pulsar which, as we have seen above,
pulses 30 times per second. Within a data stream of 12 seconds we will have
360 pulses, and to consistently catch all aspects (fluctuations) of the pulse
profile we’d certainly need a several bins per pulse. As such it would seem
likely that the rapid increase of g (2) (0) over the first 20, 000 bins is caused by
the combined effect of the statistical fluctuations and the pulsar fluctuations
becoming visible. In the high-end of the graph we see a very regular linear increase which we would assume is caused by the statistical fluctuations
being introduced by using more bins for the calculation.
To investigate the behavior of this function further we simulated a Poissoniandistributed photon stream with the same rate as the one from the detector
and calculated its g (2) (0) using an increasing bin number. Fig. 4.7 shows
the resultant data superimposed on the Iqueye data stream. The first- and
second-order polynomial fit through the simulated data points are:
F it1st order (x) = 1.0006(4) + 4.08(2) · 10−6 x
F it2nd order (x) = 0.9999(5) + 4.15(5) · 10−6 x − 1.2(7) · 10−12 x2
with x the number of bins.
We see that the g (2) (0) value of the simulated data increases linearly with
the total number of bins used in the calculation over all used bin numbers.
For reasons that are not clear to us the slope of the simulated data are slightly
lower than that of the experimental data, but we seem to be confirmed in
49
g H2L H0L
1.5
1.4
1.3
1.2
1.1
1.0
0
10 000
20 000
30 000
40 000
50 000
Nr. of bins
60 000
Figure 4.7: The value of g (2) (0) plot against the number of bins used during
its calculation. The experimental data is plot in blue, and the simulated data
is plot in red.
our hypothesis that the linear increase of g (2) (0) for increasing bin number is
a statistical effect. And the absence of a rapid increase of the g (2) (0) value
of the simulated data in the low bin numbers region also seems to confirm
that it is caused by fluctuations inherent to the signal.
This is a nice example of how careful one must be when analyzing data
gathered in photodetection, and secondly how analysis of second order coherence effects can tell us something about the source. For example: the above
exercise suggests that non-linear sections of the g (2) (0) versus the number of
bins plot are indicative of inherent fluctuations in the signal, and that the
place of the non-linearity gives us information about the frequency-range of
the effect. Second order coherence plots clearly contain information if read
properly.
4.3.3
Calculating g (2) (τ ) for τ ≥ 0
Calculating g (2) (τ ) for τ values greater than or equal to zero is conceptually
easy. We use Eqn. 4.4 from the previous section:
g (2) (τ ) =
hn(0)n(τ )i
hn(0)ihn(τ )i
and we calculate g (2) (τ ) for enough values of τ in the region of interest that
we can make an accurate plot of the function.
Let’s look more closely at this function and its components: n(0) is the
50
n(0)n(0):
n(0)n(∆τ ):
1
1
1
1
2
2
2
2
3
...
...
...
...
...
...
N −1
N −1
N −2
N −1
N
N
N −1
N
N
number of photons in each respective bin of the unmodified photon stream.
n(τ ) is the number of photons in each respective bin after adding a time τ to
the arrival time (MJD) of all photons in the stream and rebinning the data.
The only drawback is that it is computationally heavy: for every data-point
we need to rebin the data to generate the ‘n(τ ) bin values’ and calculate
hn(0)n(τ )i and hn(τ )i. And that means multiplying every n(0) bin-value
with its n(τ ) counterpart and taking the total average of these values, and
taking the average of the n(τ ) bin values.
To simplify these calculations, and make them computationally a bit less
heavy, we have chosen in our report to only make ‘τ -steps’ equal to the bin
size. This way we can avoid having to rebin the data for every data point
since introducing a delay of n times the bin size corresponds to dropping
the first n bins of the originally binned stream. We will try to visualize this
below:
with ∆τ the binsize, N the total number of bins, and the numbers inside
the fields denote the bin number (of the originally binned data), and the bin
contents which are multiplied with one another are paired in the columns.
As we can see in the second figure this method causes two bin-contents to
be exluded from the calculation (the excluded bins are highlighted in grey).
But since the timescales of the processes we’re investigating are much smaller
than any realistic observation time (even in our twelve second case) we can
stand to lose a couple of bins’ worth of data.
Now that we have introduced the method with which to calculate g (2) (τ )
for τ ≥ 0 we can proceed to the results of these calculations: Fig. 4.8 shows
g (2) (τ ) for τ -values up to 0.12 seconds, using 100, 000 bins, and consequently
making ∆τ -steps of 12/105 = 0.12 milliseconds.
This plot is obviously quite different from the typical g (2) (τ ) examples as we
have discussed them in the theory section (see: Fig. 3.3). We shall discuss
why this is so. The first thing to notice about Fig. 4.8 is the ‘lonely’ data
point (0, g (2) (0)) (indicated by an arrow), whose value is in accordance with
what we expect the value of g (2) (0) to be for a calculation using 100, 000 bins
(see Fig. 4.6 and Eqn. 4.7), followed by a large drop in value, after which
the function becomes continuous and starts to fluctuate. Both observations
51
will need to be discussed.
As we have seen in the theory section of this report g (2) (τ ) will always go
to 1 if τ τc . And with the coherence time of optical light being typically
10−15 seconds our ‘τ ’ in the second point on the graph is roughly a factor
1012 bigger (remember: ∆τ = 0.12 · 10−3 s). A drop is what we would
expect between the first and the second data point, in fact we’d expect
the g (2) (τ ) value to drop straight to 1 and stay there for the remainder of
the graph. Except that instead it starts fluctuating around g (2) (τ ) = 1
with a cycle length identical to the Crab pulsations (33.8 milliseconds). The
reason for this is obviously that we are looking at the light of a pulsar.
The explanation follows readily from Eqn. 4.4: with both terms in the
denominator effectively constant the fluctuations in the g (2) (τ ) graph have
to come from the numerator hn(0)n(τ )i. When two terms are multiplied (like
n(0) and n(τ ) in this case) simultaneous fluctuations will be amplified. And
with the pulsar signal being multiplied with itself shifted over a time τ this
means that every time τ is equal to an integer multiple of the pulsation time
all the pulse profiles in the photon stream will be ‘aligned’ and this will cause
the coherence value to spike through the multiplication of n(0) and n(τ ). The
exact shape of the pulse profile of the Crab will in this case determine the
exact shape of our ‘shifted by time τ and multiplied with itself’ profile. We
can see in Fig. 4.8 that our 12 second data file starts with the peak of a
pulse of the Crab, leading to a g (2) (τ ) value of around 1.2. This could mean
that if we were to investigate g (2) (τ ) at τ values relevant to the coherence
time of optical light the coherence function might drop from its g (2) (0)-value
to 1.2 instead of 1. Something to be very mindful of if we were to do the
appropriate calculation.
As we have seen in the theory section a bin size of the order of the
coherence time of the light would be ideal theoretically. Of course this is
not feasible in any way since this will for any realistic source lead to the
bins being filled with either one or zero photons which will cause statistical
fluctuations (also called: shot noise) in the signal. In choosing the right bin
size one must find a middle ground between signal-to-noise ratio on the one
hand, and shot noise on the other. To get an idea of how small we can make
the bin size we performed the same g (2) (τ ) calculation for 0 ≤ τ ≤ 0.12 s
for different total bin numbers, starting at small numbers and increasing the
number until we see shot noise. The results of these calculations can be found
in Fig. 4.9 through Fig. 4.16. Having explained in detail how to read these
type of graphs in the 100, 000-bin example above we have made the following
graphs more easily readable by drawing a line through the data points and
letting the (0, g (2) (0) point get outside the plot range when necessary.
We list the different plots in order of total number of bins and note what
52
g H2L HΤL
1.6
1.4
1.2
1.0
0.00
0.02
0.04
0.06
0.08
0.10
0.12
Τ HsL
Figure 4.8: g (2) (τ ) for our 12 seconds of data using 105 bins. Pay note: the
axes don’t cross in the origin and the x-axis is NOT arrival time, but ‘τ ’,
which is the delay with which the photon stream is compared with itself. The
data point (0, g (2) (0)) is indicated by an arrow.
53
is to be noticed about them:
• Fig. 4.9 & 4.10: at 10 and 100 bins respectively we see that the bin
number is too low to detect any of the fluctuations in the data with a
flat g (2) (τ ) function as a result.
• Fig. 4.11: at 1, 000 bins we start to see some fluctuations in g (2) (τ ),
but it would still be hard, if not impossible, to judge by this graph the
pulse frequency of the pulsar.
• Fig. 4.12: at 5, 000 bins (about 15 bins per pulse) we see the pulses
clearly, and accurate enough to have an idea of what the pulse profile
(in a g (2) (τ )-graph) looks like.
• Fig. 4.13: at 10, 000 bins we see the pulses in great detail and the
g (2) (τ ) graph is very smooth and well-defined. The Crab pulses becoming visible between Fig. 4.11, Fig. 4.12, and Fig. 4.13 confirm
our hypothesis that the nonlinear increase in Fig. 4.6 is caused by the
pulsations in the data stream.
• Fig. 4.14: not surprisingly, at 100, 000 bins the pulses are visible in
great detail, but at the low points of the graph (which will generally
correspond to a relatively low number of photons in the bin) we see a
tiny hint of what might be shot noise.
• Fig. 4.15: at one million bins the pulses are perfectly defined, but we
see clear shot noise in the lower half of the graph.
• Fig. 4.16: at ten million bins we see that the shot noise has spread
across the entire graph (even though much more present in the lower
half) to the point that the quality of the g (2) (τ ) graph has deteriorated
in quality because of it. This bin size (1.2 microseconds) is too much
to handle for our data set.
54
g H2L HΤL
1.3
1.2
1.1
1.0
0.9
0.00
ΤHsL
0.02
0.04
0.06
0.08
0.10
0.12
Figure 4.9: g (2) (τ ) for 0 ≤ τ ≤ 0.12 s using 10 bins for the calculation.
g H2L HΤL
1.3
1.2
1.1
1.0
0.9
0.00
ΤHsL
0.02
0.04
0.06
0.08
0.10
0.12
Figure 4.10: g (2) (τ ) for 0 ≤ τ ≤ 0.12 s using 100 bins for the calculation.
55
g H2L HΤL
1.3
1.2
1.1
1.0
0.9
0.00
ΤHsL
0.02
0.04
0.06
0.08
0.10
0.12
Figure 4.11: g (2) (τ ) for 0 ≤ τ ≤ 0.12 s using 1, 000 bins for the calculation.
g H2L HΤL
1.3
1.2
1.1
1.0
0.9
0.00
ΤHsL
0.02
0.04
0.06
0.08
0.10
0.12
Figure 4.12: g (2) (τ ) for 0 ≤ τ ≤ 0.12 s using 5, 000 bins for the calculation.
56
g H2L HΤL
1.3
1.2
1.1
1.0
0.9
0.00
ΤHsL
0.02
0.04
0.06
0.08
0.10
0.12
Figure 4.13: g (2) (τ ) for 0 ≤ τ ≤ 0.12 s using 10, 000 bins for the calculation.
g H2L HΤL
1.3
1.2
1.1
1.0
0.9
0.00
ΤHsL
0.02
0.04
0.06
0.08
0.10
0.12
Figure 4.14: g (2) (τ ) for 0 ≤ τ ≤ 0.12 s using 100, 000 bins for the calculation.
57
g H2L HΤL
1.3
1.2
1.1
1.0
0.9
0.00
0.02
0.04
0.06
0.08
0.10
0.12
Τ HsL
Figure 4.15: g (2) (τ ) for 0 ≤ τ ≤ 0.12 s using 1, 000, 000 bins for the calculation.
g H2L HΤL
1.3
1.2
1.1
1.0
0.9
0.00
0.02
0.04
0.06
0.08
0.10
0.12
Τ HsL
Figure 4.16: g (2) (τ ) for 0 ≤ τ ≤ 0.12 s using 10, 000, 000 bins for the calculation.
58
Total bin nr.
10, 000:
100, 000:
1, 000, 000:
10, 000, 000:
0
0%
12.2%
79.1%
97.6%
1
0%
23.2%
17.5%
2.3%
Number of photons per bin
2
3
4
5
0%
0%
0%
0%
23.9%
17.5%
10.4%
5.6%
2.8%
0.4% < 0.1% < 0.1%
< 0.1% < 0.1% 0.1% 0.1%
6
0.05%
2.9%
0.1%
0%
Table 4.2: Analysis of the percentage of bins that have a certain number
of photons per bin for four different total-bin-numbers. Pay note: for those
total-bin-numbers where there are bins with more than 6 photons the percentages will not add up to 100%.
4.4
Discussion
Having seen shot noise in Fig. 4.15 and particularly in Fig. 4.16 we expect to
have reached the point there that we are looking at individual photons, that
most bins will be empty, and that those that are not empty will have only
one photon in them. We have verified this by counting the number of bins
that contain a specific amount of photons for different total bin numbers.
The results of this analysis can be found in Table 4.2.
In the theory section we have seen that when we decrease the bin size to the
point that we see individual photons (and have many empty bins) that we
encounter shot noise. We can see in Table 4.2 that when we bin our data
subset with 1, 000, 000 and 10, 000, 000 bins that we have 79% and 98% empty
bins respectively. If we recall that measuring g (2) (τ ) comes down to analyzing
the photon spread at timescales of the order of the coherence time of optical
light (10−15 s) by comparing fluctuations between adjacent bins then in both
those cases it seems a lost cause. With 80% or more of the neighbouring bins
empty there will be nothing to analyze.
If we consider the case with 1, 000, 000 bins as the upper limit of how far
we could push the bin resolution (and that is already quite a stretch) then
that comes down to a time resolution of 12/107 = 1.2 · 10−6 seconds. Which
is a timescale nine orders of magnitude larger than the phenomena that we
are trying to observe. In short: that’s not going to happen. When Hanbury
Brown & Twiss measured their first distant star with their newly developed
technique they had such a good signal to noise ratio (with an α of 0.53) that
they could detect deviations in g (2) (d) of one part out of a million. At a
thousand times worse than that it is in no way feasible that we can measure
g (2) (τ ) at interesting values of τ with data from the Crab pulsar, no matter
the observation time.
59
But all is not lost. This project shows that the Iqueye works and the
methods for measuring photon spread through g (2) (τ ) are clearly robust. Analyzing only 12 seconds of ‘Crab data’ we see the pulsations clearly and there
is a clearly defined ‘pulse profile’ in the g (2) (τ ) graph. Furthermore Eqn. 4.1
tells us that it is entirely feasible to measure g (2) (τ ) with great accuracy (100
S/N ratio or better) for very bright stars (like Sirius). And these brightstar-measurements of g (2) (τ ) will allow us to investigate photon bunching,
antibunching, and emission mechanisms as described in this report. We are
of the opinion that even if that was the only thing to ever come off the measurement of g (2) (τ ) that it would be very worthwhile building advanced (and
expensive) detectors like the Iqueye for, as it would constitute experimental
data from as yet completely untapped region of parameter space.
But the measurement of the second order of coherence doesn’t need to
(and shouldn’t) end with a handful of very bright stars. Eqn. 4.1 also allows for improving the signal-to-noise ratio by increasing α (in short: get a
detector with better quantum efficiency and a bigger telescope). And here
the fact that we are looking at a second order effect will actually help us:
by increasing either quantum efficiency or telescope size we will increase the
intensity of the signal, which is the chance at the arrival of a photon. But
since g (2) (τ ) is proportional to the chance at a two-photon event it goes with
the square of the intensity. Therefore going from a 4m to a 40m telescope will
increase the signal-to-noise ratio by 10, 000. Add to that a hypothetical ultrafast detector with perfect quantum efficiency and the fifty million years of
required observation time to measure the g (2) (τ ) of the Crab pulsar doesn’t
seem quite so hopeless. But, contrary to the Sirius case, it will definitely
require tomorrow’s technology.
Tomorrow’s technology however is fully in development. In this very report we have used data from the Iqueye, which is a prototype for the to-bebuilt QuantEYE, which will be an even better ultrafast high resolution high
quantum efficiency detector [4]. And the 39.3 meter European Extremely
Large Telescope (E-ELT) is slated for completion in 2022. Powerful instruments such as these will extend the possibility of studying g (2) (τ ) to more
sources, but they also have a broad range of applications to astronomy: with
a prospected nanosecond (10−9 ) accuracy they will be able to investigate astronomical processes at those timescales and potentially resolve structures
in astronomical objects as small as 30cm. Potential areas of research in astronomy with such instruments would include, next to g (2) (τ ) measurements,
topics such as white dwarf accretion, quasi-periodic oscillations, terrestrial
atmosphere, millisecond pulsars, neutron star oscillations, and MHD instabilities in accretion [5]. Because of that there is a good chance that g (2) (τ )related research will be conducted within the near future, and an increasing
60
amount of articles and papers on the the topic reflects this. The same is
sadly not true for Stellar Intensity Interferometry (SII): the measurement of
g (2) (d).
After Hanbury Brown and Twiss built their first Intensity Interferometer
in Narrabi and observed what was to be observed within its detection limit
(most importantly the radial size of 32 stars whose size wasn’t verifiable by
any other instrument for many years [12]) they submitted a proposal to the
Australian government for a bigger and more expensive SII. The proposal was
well-received at the time and was greenlighted by several reviewing boards
[3]. Technological advances at the time however put Michelson stellar interferometry ahead of SII and the eventual telescope, the Sydney University
Stellar Interferometer (SUSI), was to be a modernized Michelson stellar interferometer. Essentially there has been no serious talk of building an SII since.
And considering that SIIs are up to two orders of magnitude cheaper to build
than regular telescopes (you only need to count the photons which impinge
on your telescope, there is no need for a focusing lens and the accompanying
precision requirements), and all the potential applications to astronomy (as
mentioned in Section 2.3) it does beg the question why.
Until the time when a modern SII is built it might be possible to adapt
existing instruments such that they will be able to engage in SII. A relatively recent example of that are the investigations into using atmospheric
Cherenkov telescope arrays as SIIs [14]. But in the end nothing will be as
good as a dedicated, designed just for that purpose, modern stellar intensity
interferometer. Fifty years after Hanbury Brown and Twiss did their pioneering work the time seems ripe for astronomers to do what hasn’t been
done since 1972: measure g (2) (d).
61
Chapter 5
Special Mentions
• This project would not have been possible without the University of
Padova for first building their amazing detector, the Iqueye, and then
graciously allowing us to use the data taken during its calibration for
this project. First thanks go to them.
• I would like to thank my supervisor Alessandro Patruno for coming
up with such an original project, for making my bachelor project fun
and intellectually engaging, and for dealing with my peculiarities in his
typical stoic fashion. You have a place in my heart.
• I would like to thank Anne for setting an example and being an inspiration.
• Lastly a word of thanks to Karri for his almost infinite patience while
I worked on this report.
62
Appendix A
Mathematica source code
63
Ÿ Data Import & Modification
SetDirectory@"D:\\data"D
Directory@D
$MaxExtraPrecision = 0
sf = ScientificForm;
rawimportMJD = SetAccuracy@ð, 21D & ž ReadList@"testdata.txt", NumberD
rawimportMJS = rawimportMJD * 24 * 60 * 60
Length ž rawimportMJS
Max ž rawimportMJS
Min ž rawimportMJS
Because the integer part is the same we can remove it without worry:
temp1 = rawimportMJS  100
temp2 = FractionalPart ž temp1
Clear ž temp1
temp3 = temp2 100
Clear ž temp2
fractImportMJD = SetPrecision@ð, 13D & ž temp3
Clear ž temp3
ListPlot@ð, AxesLabel ® 8Style@"Photon Nr.", 14D, Style@"TimeHsL", 14D<D & ž
fractImportMJD
SetDirectory@"D:Bachelor ProjectLatex"D;
Export@"MJDvsPhotonNr.eps", %%D
SetDirectory@"D:data"D;
data1 = Differences ž fractImportMJD
ListPlot@ð, PlotRange ® AllD & ž data1
Again we see the bad data point and again we drop it:
data2 = Drop@ð, - 1D & ž data1
Verification of removal of bad data point:
p1 = ListPlot@ð, PlotRange ® All,
AxesLabel ® 8Style@"Photon Nr.", 14D, Style@"TimeHsL", 14D<D & ž data2
Length ž data2
p2 = PlotA49 ´ 10-6 , 8x, 0, Length ž data2<, PlotStyle ® RGBColor@1, 0, 0DE
Show@p1, p2D
SetDirectory@"D:Bachelor ProjectLatex"D;
Export@"photon_arrival_time_differences.eps", %%D
SetDirectory@"D:data"D;
Min ž data2
Max ž data2  sf
Mean ž data2  sf
data3 = BinCountsAð, 90, 10-3 , 10-5 =E & ž data2
We plot the bin-counts against the left value of the bin intervals:
2
Bachelor_Project.nb
TableA9Hi - 1L 10-5 , data3@@iDD=, 8i, 0, 99<E;
Length ž %
ListPlot@ð, PlotRange ® All, AxesOrigin ® 8- 0.00005, - 2000<,
AxesLabel ® 8Style@"TimeHsL", 14D, Style@"Nr. of Photons", 14D<D & ž %%
SetDirectory@"D:Bachelor ProjectLatex"D;
Export@"number_of_photons_vs_time_between_photons.eps", %%D
SetDirectory@"D:data"D;
Ÿ Calculating g I2M H0L
Ÿ gH2L H0L for the experimental data
correctedFractImportMJD = Drop@ð, - 1D & ž fractImportMJD
cfiMJD = correctedFractImportMJD;
minCfiMJD = Min ž cfiMJD
maxCfiMJD = Max ž cfiMJD
lengthCfiMJD = Length ž cfiMJD
bindata2 = BinCountsAcfiMJD, 9Min ž cfiMJD, Max ž cfiMJD,
maxCfiMJD - minCfiMJD
10 000
We check if our hand-waving sort of bin interval definition really caught all the data:
=E;
Plus žž bindata2
l1 = Table@8i HmaxCfiMJD - minCfiMJDL  10 000, bindata2@@iDD<, 8i, 0, 9999<D;
ListPlot@ð, PlotRange ® 80, Max ž bindata2<,
AxesLabel ® 8Style@"timeHsL", 14D, Style@"Photons per bin", 14D<D & ž l1
SetDirectory@"D:Bachelor ProjectLatex"D;
Export@"photons_per_bin10k.eps", %%D
SetDirectory@"D:data"D;
First we calculate G(2)(0) for the single above-depicted case:
Mean ž Ibindata22 M
HMean ž bindata2L2
1.25266
 N
g2zero@list__, binSize_D :=
Mean ž ð2
HMean ž ðL2
& ž BinCountsAlist, 9Min ž list, Max ž list,
Max ž list - Min ž list
t1 = Table@8n, g2zero@cfiMJD, nD<, 8n, 1000, 60 000, 1000<D;
binSize
=E
p1 = ListPlot@ð, PlotRange ® 880, 60 500<, 81, 1.5<<D & ž t1
t1  N
It seems the graph becomes linear after a number of bins of 20,000 or more. Let's have a look at those data
points:
t2 = Drop@ð, 19D & ž t1  N
Let's try some fitting to see what we're talking about:
lmf1 = LinearModelFit@ð, 81, x<, x, ConfidenceLevel ® 0.99D & ž t2
lmf1@"ANOVATable"D
lmf1@"ParameterConfidenceIntervals"D
lmf2 = LinearModelFitAð, 91, x, x2 =, x, ConfidenceLevel ® 0.99E & ž t2
lmf2@"ParameterConfidenceIntervals"D
With a quadratic component a million times smaller than the linear component it strongly hints at the graph
becoming a straight line. Let's have a look at that line superimposed over the data points (maybe interesting to
see where it hits the y-axis?)
Bachelor_Project.nb
3
With a quadratic component a million times smaller than the linear component it strongly hints at the graph
becoming a straight line. Let's have a look at that line superimposed over the data points (maybe interesting to
see where it hits the y-axis?)
ShowAp1, Plot@lmf1@xD, 8x, 0, 60 500<, PlotStyle ® RedD,
AxesLabel -> 9Style@"Nr. of bins", 14D, StyleA"gH2L H0L", 14E=E
SetDirectory@"D:Bachelor ProjectLatex"D;
Export@"g2zero_vs_Nbins.eps", %%D
SetDirectory@"D:data"D;
Ÿ Comparison gH2L H0L experimental data and simulated Poissonian data
We calculate the rate of impinging photons:
lengthCfiMJD
rate =
maxCfiMJD - minCfiMJD
We simulate a Poissonian distributed variable with the above rate
simTime = 0;
-1
simData = TableAsimTime = simTime +
rate
Clear ž simTime
Log ž Random@D, 8i, lengthCfiMJD<E
tSim1 = Table@8n, g2zero@simData, nD<, 8n, 1000, 60 000, 1000<D;
pSim1 = ListPlot@ð, PlotRange ® 880, 60 500<, 81, 1.5<<, PlotStyle ® RedD & ž tSim1
ShowAp1, pSim1, AxesLabel ® 9Style@"Nr. of bins", 14D, StyleA"gH2L H0L", 14E=E
SetDirectory@"D:Bachelor ProjectLatex"D;
Export@"t1_vs_tSim1.eps", %%D
SetDirectory@"D:data"D;
line = LinearModelFit@ð, 81, x<, x, ConfidenceLevel ® 0.99D & ž tSim1
line@"ParameterConfidenceIntervals"D
parabola = LinearModelFitAð, 91, x, x2 =, x, ConfidenceLevel ® 0.99E & ž tSim1
parabola@"ParameterConfidenceIntervals"D
Table@8t1@@i, 1DD, t1@@i, 2DD - tSim1@@i, 2DD<, 8i, Length ž t1<D  N
ListPlot ž %
Ÿ g I2M HΤL
Ÿ Calculation gH2L HΤL for E1 - E5
To try and make the function a bit less intense CPU wise we'll (at least for now) remove the ability to give the
bin size as an argument to the function. We predefine a couple of data sets with specific bin numbers:
dataE1bins = BinCountsAcfiMJD, 9Min ž cfiMJD, Max ž cfiMJD,
Table@Count@ð, iD, 8i, 0, 20<D & ž dataE1bins
12
tg2tauE1bins = TableA9
101
Table@Count@ð, iD, 8i, 0, 20<D & ž dataE2bins
12
102
10
i, g2tau@ð, iD=, 8i, 0, 9<E & ž dataE1bins  N
dataE2bins = BinCountsAcfiMJD, 9Min ž cfiMJD, Max ž cfiMJD,
tg2tauE2bins = TableA9
Max ž cfiMJD - Min ž cfiMJD
Max ž cfiMJD - Min ž cfiMJD
100
i, g2tau@ð, iD=, 8i, 0, 99<E & ž dataE2bins  N;
=E;
=E;
4
Bachelor_Project.nb
dataE3bins = BinCountsAcfiMJD, 9Min ž cfiMJD, Max ž cfiMJD,
Max ž cfiMJD - Min ž cfiMJD
1000
Table@Count@ð, iD, 8i, 0, 20<D & ž dataE3bins
12
tg2tauE3bins = TableA9
103
i, g2tau@ð, iD=, 8i, 0, 999<E & ž dataE3bins  N;
data5E3bins = BinCountsAcfiMJD, 9Min ž cfiMJD, Max ž cfiMJD,
Max ž cfiMJD - Min ž cfiMJD
5000
Table@Count@ð, iD, 8i, 0, 20<D & ž data5E3bins
12
tg2tau5E3bins = TableA9
5 ´ 103
i, g2tau@ð, iD=, 8i, 0, 4999<E & ž data5E3bins  N;
dataE4bins = BinCountsAcfiMJD, 9Min ž cfiMJD, Max ž cfiMJD,
Max ž cfiMJD - Min ž cfiMJD
10 000
Table@Count@ð, iD, 8i, 0, 20<D & ž dataE4bins
12
tg2tauE4bins = TableA9
104
i, g2tau@ð, iD=, 8i, 0, 9999<E & ž dataE4bins  N;
dataE5bins = BinCountsAcfiMJD, 9Min ž cfiMJD, Max ž cfiMJD,
Max ž cfiMJD - Min ž cfiMJD
Table@Count@ð, iD, 8i, 0, 20<D & ž dataE5bins
12
tg2tauE5bins = TableA9
105
We define a preliminary g2tau:
100 000
i, g2tau@ð, iD=, 8i, 0, 1999<E & ž dataE5bins  N;
g2tau@list__, tauInBins_D := HHLength ž list - tauInBinsL *
Plus žž HHDrop@ð, - tauInBinsD & ž listL * HDrop@ð, tauInBinsD & ž listLLL 
HPlus žž HDrop@ð, - tauInBinsD & ž listL * Plus žž HDrop@ð, tauInBinsD & ž listLL
ListPlotAð, PlotRange ® 880, 0.12<, 80.9, 1.3<<, PlotJoined ® True,
AxesLabel ® 9Style@"ΤHsL", 14D, StyleA"gH2L HΤL", 14E=E & ž tg2tauE1bins
SetDirectory@"D:Bachelor ProjectLatex"D;
Export@"g2tauE1bins.eps", %%D
SetDirectory@"D:data"D;
ListPlotAð, PlotRange ® 880, 0.12<, 80.9, 1.3<<, PlotJoined ® True,
AxesLabel ® 9Style@"ΤHsL", 14D, StyleA"gH2L HΤL", 14E=E & ž tg2tauE2bins
SetDirectory@"D:Bachelor ProjectLatex"D;
Export@"g2tauE2bins.eps", %%D
SetDirectory@"D:data"D;
ListPlotAð, PlotRange ® 880, 0.12<, 80.9, 1.3<<, PlotJoined ® True,
AxesLabel ® 9Style@"ΤHsL", 14D, StyleA"gH2L HΤL", 14E=E & ž tg2tauE3bins
SetDirectory@"D:Bachelor ProjectLatex"D;
Export@"g2tauE3bins.eps", %%D
SetDirectory@"D:data"D;
ListPlotAð, PlotRange ® 880, 0.12<, 80.9, 1.3<<, PlotJoined ® True,
AxesLabel ® 9Style@"ΤHsL", 14D, StyleA"gH2L HΤL", 14E=E & ž tg2tau5E3bins
SetDirectory@"D:Bachelor ProjectLatex"D;
Export@"g2tau5E3bins.eps", %%D
SetDirectory@"D:data"D;
=E;
=E;
=E;
=E;
Bachelor_Project.nb
5
ListPlotAð, PlotRange ® 880, 0.12<, 80.9, 1.3<<, PlotJoined ® True,
AxesLabel ® 9Style@"ΤHsL", 14D, StyleA"gH2L HΤL", 14E=E & ž tg2tauE4bins
SetDirectory@"D:Bachelor ProjectLatex"D;
Export@"g2tauE4bins.eps", %%D
SetDirectory@"D:data"D;
ListPlotAð, PlotRange ® 880, 0.12<, 80.9, 1.3<<, PlotJoined ® True,
AxesLabel ® 9Style@"ΤHsL", 14D, StyleA"gH2L HΤL", 14E=E & ž tg2tauE5bins
SetDirectory@"D:Bachelor ProjectLatex"D;
Export@"g2tauE5bins.eps", %%D
SetDirectory@"D:data"D;
ListPlotAð, PlotRange ® 880, 0.12<, 80.9, 1.3<<, PlotJoined ® True,
AxesLabel ® 9Style@"ΤHsL", 14D, StyleA"gH2L HΤL", 14E=E & ž tg2tauE6bins
SetDirectory@"D:Bachelor ProjectLatex"D;
Export@"g2tauE6bins.eps", %%D
SetDirectory@"D:data"D;
ListPlotAð, PlotRange ® 88- 0.01, 0.12<, All<, PlotJoined ® False, AxesOrigin ®
8- 0.01, 0.9<, AxesLabel ® 9"Τ Hin secondsL", "gH2L HΤL"=E & ž tg2tauE5bins
Ÿ Calculation gH2L HΤL for E6
dataE6bins = BinCountsAcfiMJD, 9Min ž cfiMJD, Max ž cfiMJD,
Table@Count@ð, iD, 8i, 0, 20<D & ž dataE6bins
Max ž cfiMJD - Min ž cfiMJD
1 000 000
=E;
e6Stream = OpenWrite@"g2tauE6bins.txt"D
DoAWrite@e6Stream, ðD & ž 912 ´ 10-6 i, g2tau@dataE6bins, iD=;
If@Mod@i, 100D Š 0, Print ž iD, 8i, 0, 10 000<E
Close ž e6Stream
tg2tauE6bins = ReadList@"g2tauE6bins.txt"D  N
ListPlotAð, PlotRange ® 880, 0.12<, 80.9, 1.3<<, PlotJoined ® True,
AxesLabel ® 9Style@"Τ HsL", 14D, StyleA"gH2L HΤL", 14E=E & ž tg2tauE6bins
SetDirectory@"D:Bachelor ProjectLatex"D;
Export@"g2tauE6bins.eps", %%D
SetDirectory@"D:Data"D;
Ÿ Calculation gH2L HΤL for E7
dataE7bins = BinCountsAcfiMJD, 9Min ž cfiMJD, Max ž cfiMJD,
Table@Count@ð, iD, 8i, 0, 20<D & ž dataE7bins
Max ž cfiMJD - Min ž cfiMJD
10 000 000
=E;
Since we know that we'll be working with the same data set of E7 bins from now on we're going to adapt the
g2tau function with that (fore)knowledge in mind to make it more efficient:
We predefine the sums of the (shortened) dataE7bin array to cut down on calculation time:
Clear ž sumEndDroppedDataE7bins
Clear ž sumBeginDroppedDataE7bins
sumEndDroppedDataE7bins@0D = Plus žž dataE7bins
Do@sumEndDroppedDataE7bins@iD =
sumEndDroppedDataE7bins@i - 1D - dataE7bins@@10 000 001 - iDD, 8i, 1, 1 000 000<D
sumBeginDroppedDataE7bins@0D = Plus žž dataE7bins
6
Bachelor_Project.nb
Do@sumBeginDroppedDataE7bins@iD =
sumBeginDroppedDataE7bins@i - 1D - dataE7bins@@iDD, 8i, 1, 1 000 000<D
g2tauE7bins@list__, tauInBins_D := HH10 000 000 - tauInBinsL *
Plus žž HHDrop@ð, - tauInBinsD & ž listL * HDrop@ð, tauInBinsD & ž listLLL 
HsumEndDroppedDataE7bins@tauInBinsD * sumBeginDroppedDataE7bins@tauInBinsDL
e7Stream = OpenWrite@"g2tauE7bins.txt"D
DoAWrite@e7Stream, ðD & ž 912 ´ 10-7 i, g2tauE7bins@dataE7bins, iD=;
If@Mod@i, 10 000D Š 0, Print ž iD, 8i, 0, 1 000 000, 100<E
Close ž e7Stream
rl2 = ReadList@"g2tauE7bins.txt"D  N
It turns out that we have made a mistake causing the time-stamps in the generated file to be a factor of 10 too
high. We compensate:
tg2tauE7bins = Table@810 * rl2@@i, 1DD, rl2@@i, 2DD<, 8i, Length ž rl2<D
ListPlotAð, PlotRange ® 880, 0.12<, 80.9, 1.3<<, PlotJoined ® True,
AxesLabel ® 9Style@"Τ HsL", 14D, StyleA"gH2L HΤL", 14E=E & ž tg2tauE7bins
SetDirectory@"D:Bachelor ProjectLatex"D;
Export@"g2tauE7bins.eps", %%D
SetDirectory@"D:Data"D;
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