From point processes to quantum optics and back R emi Bardenet1 Alexandre Feller1 J er emie Bouttier2 Pascal Degiovanni3 Adrien Hardy4 Adam Ran con5 Benjamin Roussel6 Gr egory

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From point processes to quantum optics and back
R´emi Bardenet∗†1, Alexandre Feller∗‡1, J´er´emie Bouttier2, Pascal
Degiovanni3, Adrien Hardy4, Adam Ran¸con5, Benjamin Roussel6, Gr´egory
Schehr7, and Christoph I. Westbrook8
1Univ. Lille, CNRS, Centrale Lille, UMR 9189 – CRIStAL – Centre de Recherche en Informatique,
Signal et Automatique, F-59000 Lille, France
2Universit´e Paris-Saclay, CNRS, CEA, Institut de Physique Th´eorique, 91191, Gif-sur-Yvette, France
3Univ Lyon, Ens de Lyon, Universit´e Claude Bernard Lyon 1, CNRS, Laboratoire de Physique (UMR
5672), F-69342 Lyon, France
4Qube Research and Technologies, 75008 Paris, France
5Univ. Lille, CNRS, UMR 8523 – PhLAM – Laboratoire de Physique des Lasers, Atomes et Mol´ecules,
F-59000 Lille, France
6Department of Applied Physics, Aalto University, 00076 Aalto, Finland
7Sorbonne Universit´e, Laboratoire de Physique Th´eorique et Hautes Energies, CNRS UMR 7589, 4 Place
Jussieu, Tour 13, 5`eme ´etage, 75252 Paris 05, France
8Univ. Paris-Saclay, Institut d’Optique Graduate School, CNRS, UMR 8501 – Laboratoire Charles
Fabry, F-91127 Palaiseau, France
Abstract
[Disclaimer: this is Part I of a cross-disciplinary survey that is work in progress;
All comments are welcome.] Some fifty years ago, in her seminal PhD thesis, Odile
Macchi introduced permanental and determinantal point processes. Her initial moti-
vation was to provide models for the set of detection times in fundamental bosonic
or fermionic optical experiments, respectively. After two rather quiet decades, these
point processes have quickly become standard examples of point processes with non-
trivial, yet tractable, correlation structures. In particular, determinantal point pro-
cesses have been since the 1990s a technical workhorse in random matrix theory and
combinatorics, and a standard model for repulsive point patterns in machine learn-
ing and spatial statistics since the 2010s. Meanwhile, our ability to experimentally
probe the correlations between detection events in bosonic and fermionic optics has
progressed tremendously. In Part I of this survey, we provide a modern introduction
to the concepts in Macchi’s thesis and their physical motivation, under the combined
eye of mathematicians, physicists, and signal processers. Our objective is to provide a
shared basis of knowledge for later cross-disciplinary work on point processes in quan-
tum optics, and reconnect with the physical roots of permanental and determinantal
point processes.
Contents
1 Introduction 2
2 Point processes and the semiclassical picture of HBT 5
2.1 The correlation functions of a point process . . . . . . . . . . . . . . . . . . 6
2.2 Poisson and Cox point processes . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Permanental point processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4 Determinantal point processes . . . . . . . . . . . . . . . . . . . . . . . . . . 11
∗Joint first authorship and corresponding authors.
†remi.bardenet@univ-lille.fr
‡alexandre.feller@univ-lille.fr
1
arXiv:2210.05522v1 [math-ph] 11 Oct 2022
3 Elements of quantum field theory 12
3.1 The mathematical framework of quantum theory . . . . . . . . . . . . . . . 12
3.2 Two fundamental one-particle systems . . . . . . . . . . . . . . . . . . . . . 15
3.2.1 Thequbit................................. 16
3.2.2 The harmonic oscillator . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.3 Modelling a finite number of particles . . . . . . . . . . . . . . . . . . . . . 18
3.3.1 Subsystems, bosons and fermions . . . . . . . . . . . . . . . . . . . . 18
3.3.2 Example N-particlestates........................ 19
3.3.3 Occupation number representation . . . . . . . . . . . . . . . . . . . 20
3.4 Modeling an indefinite number of particles . . . . . . . . . . . . . . . . . . . 20
3.4.1 Fock spaces for bosons and fermions . . . . . . . . . . . . . . . . . . 21
3.4.2 Creation and annihilation operators . . . . . . . . . . . . . . . . . . 21
3.4.3 Modes................................... 23
3.4.4 Fieldoperators.............................. 23
3.4.5 Observables................................ 24
3.5 Wick’stheorem.................................. 25
3.5.1 Gaussian density matrices . . . . . . . . . . . . . . . . . . . . . . . . 25
3.5.2 Where permanents and determinants appear . . . . . . . . . . . . . 26
3.6 Bosonic coherent states model classical fields . . . . . . . . . . . . . . . . . 29
3.6.1 Definition and properties . . . . . . . . . . . . . . . . . . . . . . . . 29
3.6.2 The Husimi distribution and time-frequency analysis . . . . . . . . . 31
4 Photodetection and bosonic coherences 32
4.1 Modeling photodetection events . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.1.1 Modeling the radiation . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.1.2 Modeling the detector: the first-order coherence function . . . . . . 33
4.1.3 Role of the detector structure function . . . . . . . . . . . . . . . . . 37
4.2 Correlation between photodetection events . . . . . . . . . . . . . . . . . . . 38
4.2.1 Second-order coherence . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.2.2 Higher-order coherences . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.3 Turning coherence functions into correlation functions . . . . . . . . . . . . 40
4.4 Single-photon sources can lead to anti-bunching . . . . . . . . . . . . . . . . 43
5 Electrodetection and fermionic coherences 45
5.1 Modeling electrodetection events . . . . . . . . . . . . . . . . . . . . . . . . 46
5.1.1 Modeling the fermionic field . . . . . . . . . . . . . . . . . . . . . . . 46
5.1.2 First-order electronic coherence . . . . . . . . . . . . . . . . . . . . . 47
5.2 Correlation between electrodetection events . . . . . . . . . . . . . . . . . . 49
5.3 Electronanti-bunching.............................. 52
5.4 Recovering a classical current is not as easy as for bosons . . . . . . . . . . 53
6 Wrapping up and open questions 54
6.1 From a point process to free particles . . . . . . . . . . . . . . . . . . . . . . 54
6.2 Going further: from simple to open questions . . . . . . . . . . . . . . . . . 56
6.2.1 A quantum state is more than a point process . . . . . . . . . . . . . 56
6.2.2 A DPP from bosons in a non-Gaussian state . . . . . . . . . . . . . 58
6.2.3 Interacting field theories and point processes . . . . . . . . . . . . . 58
6.2.4 Constructive arguments for point processes . . . . . . . . . . . . . . 59
6.3 AteaserforPartII................................ 59
1 Introduction
The photoelectric effect is the release of individual electrons from a metal, when light falls
onto that metal. The empirical observation that no electrons are released if the light
frequency is beneath a certain threshold, along with an analogy with elastic collisions,
prompted Einstein to posit the existence of light quanta, also known as photons. A century
2
later, the full explanation of the photoelectric effect is considered a success of the quantum
theory of light.
In parallel, experimental devices have been designed to amplify the current resulting
from a small number of released electrons, eventually giving detectors of light so sensitive
that they are able to detect single photons. These detectors have led to investigations on
the quantum coherence properties of light. Coherence here means the ability of sources of
light to generate interference patterns, such as in Young’s celebrated double slit experiment
[Mandel and Wolf,1965, Sections 1-3]. One puzzling aspect of coherence was demonstrated
by Hanbury Brown and Twiss [1958], and is called photon bunching or the HBT effect,
after the initials of its discoverers. When placing a detector of single photons in the electric
field created by a thermal light source, such as incandescent matter, Hanbury Brown and
Twiss [1958] showed that the detector tends to produce clicks that are grouped in time.
Everything happens as if the photons were tightly bunched together when arriving at the
detector, hence the name of photon bunching. A few years later, with the invention of the
laser, it was realized that this bunching effect disappeared for (so-called coherent) laser
light.
At the turn of the 60s, there was intense research in finding the right mathematical
framework to describe such coherence phenomena. A celebrated contribution is that of
Glauber [1963], who introduced concepts like coherent states and coherence functions, which
quickly became textbook material for quantum optics. Having met Glauber at Les Houches
in 1964, French signal processing pioneer Bernard Picinbono launched a team in statistical
optics in Orsay, determined to explore the transition between bunching and non-bunching
measurements using the formalism of stochastic processes. Picinbono assembled a small
and diverse group, including PhD student Odile Macchi, who had just obtained a degree
in mathematics. The starting point of Macchi’s 1972 PhD thesis was to find the right
stochastic object to describe the detection times in HBT and explain photon bunching. Her
thesis turned out to be foundational in many respects. This justified a recent translation
of Macchi’s thesis (originally in French) by Hans Zessin [Macchi,2017], along with a large
appendix written by Hans Zessin and Suren Poghosyan.
Odile Macchi introduced what we now call correlation functions to describe point pro-
cesses, i.e., random configurations of points in a generic space. She showed how common
models for physical sources and detection led to point processes with closed-form correlation
functions. In particular, for a given model of the source used in the HBT experiment, she
showed that the resulting point process of detection times is a permanental point process,
for which bunching can be fully characterized and related to properties of the electric field.
Teaming up with fellow PhD student in physics Christine B´enard, they used Glauber’s
formalism to identify the point process describing the parallel situation of detection times
of electrons. Unlike for photons, the latter point process naturally exhibits anti-bunching,
with detection times being very regularly spaced. [B´enard and Macchi,1973] is one the
foundational stones1for what we call today determinantal point processes (DPPs).
Outside physics, determinantal point processes (DPPs) have since then become a cor-
nerstone of the theory of random matrices [Anderson, Guionnet, and Zeitouni,2010,Jo-
hansson,2006], with applications in combinatorics [Borodin et al.,2000] and number theory
[Rudnick and Sarnak,1996]. DPPs are also a popular model for repulsive point pattern data
in spatial statistics [Lavancier, Møller, and Rubak,2014] and machine learning [Kulesza
and Taskar,2012]. Many DPP users in these fields2have little idea of the physical origin
of DPPs. Given the late blooming of DPPs outside their original field, and some fifty
years after Macchi’s first generic formalization of DPPs, we felt it a natural endeavour to
reconnect DPPs with their physical roots, as models for non-interacting fermions and tools
to probe generalizations of the HBT effect. This has been the purpose of two workshops
so far.
1Together with a 1974 conference communication, that appeared later as [Macchi,1977], and earlier
work by Jean Ginibre; see the preface to the recent re-print [Macchi,2017] of Odile Macchi’s thesis for a
broader history.
2including some of us prior to this work!
3
We started with a two-day event3in 2019 in Lille, France, opened by Odile Macchi
and featuring both theoretical and experimental physicists exposing their view of fermionic
coherence to an audience of DPP users across mathematics, computer science, and signal
processing. The workshop having been met with cross-disciplinary enthusiasm, we made
two decisions. The first was to organize an ambitious, two-week follow-up to the workshop,4
which took place in 2022 in Lyon, France. The second decision was to unite forces and
write a survey on the links between point processes and optics, to pin down a common
ground for discussions. The current document is the first part of this survey. It is both
a joint introduction to point processes and quantum optics, and organized notes from a
modern reading of Odile Macchi’s thesis. The novelty of our document relies in its cross-
disciplinary target audience of mathematicians, physicists, and signal processers, with a
solid undergraduate background in probability and functional analysis. In particular, while
tackling topics in modern physics, we assume little physics knowledge from the reader
beyond undergraduate exposure to wave optics. One of our leitmotivs is to sketch the
thought process behind some fundamental arguments in quantum optics. Indeed, in our
experience, arguments thought as basic by physicists can be hard to grasp by people trained
in mathematics, mostly because the implicitly assumed lore differs across communities. In a
reverse movement towards physicists, and following the spirit of Macchi [1975], we motivate
most mathematical concepts by their need as models in optics, including point processes.
We have striven to maintain a balance between mathematical rigour, clarity, and brevity,
giving references whenever we had to take shortcuts. We expect that every reader will find
some of the material basic and some more exotic, depending on their background, and
we hope that all readers will eventually learn something useful. Our objective is to make
the potential barrier for crossing from one discipline to another as low as possible, so that
ideas can flow more easily. A second part of the document is in preparation, presenting
selected advanced topics from the Lille workshop, including experimental measurements
of the HBT effect, non-interacting trapped fermions in statistical physics [Dean et al.,
2016,2019], fermions in combinatorics, and electronic quantum signals. The two parts are
ultimately to be bound in a single manuscript.
The rest of the document is organized as follows. In Section 2, we introduce key examples
of point processes. Poisson, Cox, and permanental point processes are motivated there by
the so-called semi-classical derivation of the HBT photon bunching effect, treating only
the detector as a quantum object, not the electric field. Determinantal point processes
are also introduced, but their physical motivation requires to go beyond the semi-classical
picture, which justifies the next three sections. Section 3is a crash course in quantum
field theory, from the basics of quantum mechanics to Wick’s theorem on the average of
products of ladder operators. Wick’s theorem yields two very different results depending
on the commutation rules of the operators it applies to, which in turn derive from modeling
either bosonic particles (like photons), or fermionic particles (like electrons). Ultimately,
this dichotomy is at the origin of the appearance of permanental and determinantal point
processes. The section concludes with a discussion of the coherent states of Glauber [1963]
and their relation to time-frequency analysis in signal processing. Section 4covers the
modern view on photodetection, culminating in the full quantum justification of the HBT
effect using permanental point processes as a model for the coincidence measurements of
photons, as well as considerations on the role of the source in the bunching properties of the
measurements. Following the lines of Section 4, Section 5covers the detection of electrons,
finally resulting in the appearance of determinantal point processes. The section concludes
on the comparative difficulties of recovering non-quantum arguments from the quantum
treatment in the case of fermions. Finally, Section 6wraps up this first part, commenting
on the generic derivation of a permanental or determinantal point process from a model of
free fermions, and discussing open questions motivated by this construction.
A note on the style. Because of its cross-disciplinary objectives, the style of our docu-
ment is hybrid. We mostly follow a style inspired by texts in mathematics, with definitions,
3https://dpp-fermions.sciencesconf.org/
4https://indico.in2p3.fr/event/25182/
4
theorems, assumptions, and examples, sometimes merged with the main text, sometimes
fleshed out to draw the reader’s attention. Assumptions, in particular, often stand out. By
assumption we mean a statement for the reader to accept in order to progress in a discussion
or a computation. It can be, e.g., a modeling choice or a mathematical approximation.
Examples are often borrowed from physics; to make the text self-contained, some of
these examples are relatively long. To help the reader to visually isolate examples from
the rest of the text, we conclude each example with a symbol. Finally, footnotes abound,
and are usually meant as side remarks to one of the targeted scientific communities, e.g.
to discuss notational or minor conceptual differences between different domains.
2 Point processes and the semiclassical picture of HBT
In a physical experiment where a detector clicks when hit by a particle, the observation
consists of a set of real numbers, the times at which the detector clicks. The natural
probabilistic model for such a situation, where both the number and the location of the
observed points are uncertain, is a point process, i.e, a random configuration of points. In
Sections 2.1 to 2.4, we introduce some of the modern vocabulary of point process theory,
along with three key families of point processes: Poisson, permanental and determinantal
point processes.
While optical models that feature determinantal point processes will have to wait for
the fully quantum treatment of electron detection in Section 5, we already motivate Poisson
and permanental point processes in this section by the so-called semi-classical (as opposed
to fully quantum) derivation of the photon bunching effect. Our running examples are
based on the simple setup given here as Example 0.
Example 0 (A simple photodetection setup).As introduced in Section 1, photodetection
is based on the photoelectric effect: when light falls onto a metal, electrons are released,
generating a current that we can measure. Light is an electromagnetic field, so typically has
an electric and magnetic component, but the physics of the photoelectric effect is essentially
dependent on the electric component of the field.
The electric field at a point r∈R3in space and time tis modeled by a square-integrable
function E:R3×R→R3of space and time. The three components of E(r, t) correspond
to what is called in physics the polarization of light. Throughout this paper, for simplicity,
we assume that the field is linearly polarized. In this section,5linear polarization amounts
to assuming the existence of a unit vector u∈R3such that, for all (r, t)∈R3×R,
E(r, t) = E(r, t)u, with E(r, t)∈R. This simplification allows us to focus here on the
scalar function E:R3×R→R. Moreover, we consider a photodetector placed at a fixed
position r∈R3. The effect of the field on the detector is assumed to only depend on the
value E(r,·) of the field at the detector position, and we thus further focus on the function
t7→ E(r, t) in our examples, which we also denote6by Ein this section.
We are interested in the times at which our detector clicks, i.e., detects a single photon.
Because the measured times vary from one run of the experiment to the next, we want to
model them as a random set of (distinct) real numbers.
Finally, note that we only describe in this section an idealized version of the HBT
experiment, following [Macchi,1975, Section 4.2]. For physics arguments, we refer the
reader with little prior exposition to physics to [Mandel and Wolf,1995, Chapter 9], which
is a recent textbook treatment of the survey [Mandel and Wolf,1965] to which Macchi
[1975] originally referred.
5In later sections, the field will be modeled by a collection of operators, and linear polarization will thus
correspond to a different mathematical assumption.
6Overloading variable names is common in physics, and we shall follow this convention when possible
without confusion.
5
摘要：

FrompointprocessestoquantumopticsandbackRemiBardenet*„1,AlexandreFeller…1,JeremieBouttier2,PascalDegiovanni3,AdrienHardy4,AdamRancon5,BenjaminRoussel6,GregorySchehr7,andChristophI.Westbrook81Univ.Lille,CNRS,CentraleLille,UMR9189{CRIStAL{CentredeRechercheenInformatique,SignaletAutomatique,F-590...

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