Fundamental Limits to Single-Photon Detection Fundamental Limits to Single-
Photon Detection Department of Physics Department of Physics Doctor of
Philosophy September 2020 Steven J. van Enk Benjamin McMorran Brian Smith
Miriam Deutsch David Wineland Jeff Cina Kate Mondloch

###### Abstract

Quantum mechanics cements the intimate relationship between the nature of
light and its detection. Historically, quantum theories of photodetection have
generally fallen into two categories: the first tries to determine what
quantum field observable is measured when photoelectrons are detected, laying
the theoretical groundwork for photodetection being possible. The second type
are phenomenological theories, which take great care to model the details of
specific photodetectors. In this dissertation, we fill in the gap between
these two models in the modern literature on photodetection by constructing a
fully quantum-mechanical and sufficiently realistic model that includes all
stages of the photodetection process. We accomplish this within the framework
of quantum information theory using the language of positive operator valued
measures (POVMs).

A POVM provides the most general description of a quantum measurement. In the
context of single-photon detection, the photodetector POVM provides a complete
characterization of the single-photon detector (SPD) from which all figures of
merit can be calculated. Each element of the POVM is comprised of a weighted
sum over single-photon state projectors. To construct SPD POVMs, we identify
the states projected onto by a measurement and the associated weights. We
first accomplish this for a simple time-dependent system, laying out how to
efficiently project onto arbitrary single-photon states. Then, we describe the
three stages of a realistic SPD: transmission, amplification, and measurement.

In the transmission stage of photodetection a photon enters a network of
discrete energy levels, its energy propagating through that network and
finally exiting into another output continuum of modes, a process entirely
described by a single complex function. In the amplification stage of
photodetection, a single excitation is amplified into a macroscopic signal via
a nonlinear amplification mechanism. In the final stage, an inefficient
“classical” measurement is made on the macroscopic signal. By combining these
three stages we form a chain of inference from the final “click” to the input
photon and construct a POVM projecting onto the input Hilbert space. Lastly,
we discuss limits and tradeoffs that arise at each stage and implications for
photodetection applications.

This dissertation contains previously published and unpublished material.

University of Oregon, Eugene, Oregon College of Wooster, Wooster, Ohio Doctor
of Physics, 2020, University of Oregon Bachelor in Physics, Philosophy, 2015,
College of Wooster, Ohio Theoretical Quantum Optics and Quantum Information
Theory Graduate Teaching Assistant, University of Oregon, 2015-2019 Graduate
Research Assistant, University of Oregon, 2016-2020 Science Literacy Program
Fellow, University of Oregon, Fall 2017 & 2019 Adjunct Faculty Instructor,
Lane Community College, 2019-2020 Northstar Project Teacher, University of
Oregon 2016-2019 Queer Caucus Chair, University of Oregon GTFF, 2018-2020
LGBT+ in STEM Executive Board, University of Oregon, 2017-2020 John R. Moore
Scholarship, University of Oregon, 2020 Most Informative Talk, University of
Oregon Women in Graduate Science Science Slam, 2018 First Place, Oregon
Statewide 3 Minute Thesis Championship, 2017 First Place, University of Oregon
3 Minute Thesis Competition, 2017 Physics Department Teaching Award,
University of Oregon, 2017 Propp, Tz. B & van Enk, S. J. (2019). On nonlinear
amplification: improved quantum limits for photon counting. _Optics Express_
27, 16, 23454-23463. Propp, Tz. B & van Enk, S. J. (2019). Quantum networks
for single photon detection. _Physical Review A_ , 100, 033836. Propp, Tz. B &
van Enk, S. J. (2020). How to project onto an arbitrary single-photon
wavepacket. Submitted to _Physical Review A_. I would like to thank my advisor
Dr. Steven van Enk for his mentorship in completing this dissertation, for
supporting me through the hardest years of graduate school, and for inspiring
me to strive to be the best and happiest theoretical physicist I can possibly
be. I also want to thank my former advisor and committee chair Dr. Ben
McMorran for believing in me and helping instill in me the value of outreach,
along with PhD committee member Dr. Dave Wineland for invaluable revisions. I
would like to thank Andrea Goering, Brandy Todd, and Benjamín Alemán for
supporting me, providing mentorship, and enabling my physics outreach
endeavors. Similarly, I want to thank Dr. Elly Vandegrift who opened my eyes
to the science of science education, and whose mentorship has been
instrumental to my development as both teacher and learner. I would like to
acknowledge the leadership of the Graduate Teaching Fellows Federation, as
well as University of Oregon Lesbian, Gay, Bisexual, and Trans+ Education
Support Services (LGBTESS) coordinator Haley Wilson for their institutional
support. Special thanks are due to Saumya Biswas who–in his own words–did not
get in the way, along with our collaborators in the DARPA DETECT program for
thought-provoking discussions. I would like to acknowledge the Kalapuya tribe,
on whose stolen land this research was conducted. Lastly, I want to thank my
queer chosen family and friends for being there for me through the toughest
times, particularly Vé Gulbransen, Hales Wilson, and Liana Clark as well as
Deepika Sundarraman, Hayley Shapiro, Christianna Hannegan, Alice Greenberg,
Abby Pauls, Caden Valencia, and many others. This research was supported
financially in part by DARPA contract No. W911NF-17-1-0267 and the Emanuel OMQ
Scholarship. blank filler text. I dedicate this dissertation to my beloved cat
King Ubu who is also known as Small Bean Black Bean Moon Bean Sun Bean
Bean-O’Noire Tiabanie Beanie Baby Sneaky Bean and The Bean. This dissertation
is the product of unionized labor as part of the Graduate Teaching Fellows
Federation, AFT Local 3544.

##  Chapter 1 Introduction

###  1.1 History

Measurements are our window to the universe. Since the ancient Greeks, we have
accepted that we only have access to our perceptions, not direct access to the
_phusis_ or physical reality. The statement that we can only know what we can
measure has withstood the test of time, and is now enshrined and made explicit
in quantum theory’s separation of measurement and the quantum state. Yet even
though no measurement can reveal the complete underlying reality (if such an
underlying reality even exists), we can improve our measurements to better
reveal physical reality in analogy to Plato’s famous Allegory of the Cave [1],
depicted in Fig. 1.1; in quantum theory, it is as if we are trying to learn
about light (the quantum state) by studying the shadows different objects cast
(projective measurements). This is necessary, as there is a real sense in
which our measurements constitute the most foundational layer of the universe
to which we have direct access. In this way, if we wish to learn more about
the structure and constituents of physical reality, our study should include
consideration of our measurement schemes themselves.

Figure 1.1: Plato’s allegory of the cave [1], as illustrated on the cover of
_Diagnosis: interpreting the shadows_ [2]. By improving our measurements, we
can improve our understanding of the universe even if the ultimate level of
direct access to reality (for Plato, looking directly at the sun) is
unattainable.

The study of light and its measurement date back to at least $500$ to $300$
BCE [6]. These early works by Ptolemy and Euclid were preserved and expanded
upon in the Islamic world with the works of Ibn Sahl on curved mirrors and
lenses in $984$ CE [7], Ibn Al-Haytham on the correct identification of vision
as a three-part process involving a light source, a reflected object, and the
eye between $1028$ and $1038$ CE [8], and Kamāl al-Dīn al-Fārisī on the
mathematically rigorous description of the rainbow phenomena in $1309$ CE [9].
These concepts were introduced to Europe during the Renaissance, where the
well-developed science of spectacle construction (curved lenses) was used by
Hans Lippershay in $1608$ to create the first telescope, an invention which
was subsequently refined by Johannes Kepler and modified to magnify nearby
objects (that is, the invention of the microscope) in the subsequent three
years [10]. These developments in our ability to measure and manipulate light
paved the way for a series of early experiments in the mid-1600’s, most
famously by Isaac Newton and Christiaan Huygens, trying to elucidate the
nature of light which was believed to be a particle by the former and a wave
by the latter. These experiments were built upon in the following century and
a half by Thomas Young (wave theory: interference), Etienne Louis Malus (wave
theory: polarization), Augustin-Jean Fresnel (wave theory: polarization and
interference), Simeon-Denis Poisson (particle theory: shadows and attempting
to refute Fresnel) and Francois Arago (wave theory: diffraction and refuting
Poisson). During this same time period, parallel research by Hans Christian
Ørstedand and later Michael Faraday in unifying electricity and magnetism.
This all culminated in the unifying work by James Clerk Maxwell, who proved in
1864 that light is an electromagnetic wave with a speed exactly determined by
the electromagnetic permittivity and permeability of free space [11].

Despite the rigorous theoretical foundation of the wave theory of light and
the tremendous predictive success of classical physics111In 1900, Lord Kelvin
is quoted as saying, “There is nothing new to be discovered in physics now.
All that remains is more and more precise measurement” [11]., the wave theory
of light alone failed to explain several phenomena including the photoelectric
effect222The photoelectric can in fact be explained semiclassically, but this
was discovered later [12]. and the Rayleigh–Jeans ultraviolet catastrophe for
blackbody radiation. The resolution of these mysteries required the
elucidation of wave-particle duality, wherein both the particle and wave
properties of light are understood as differing aspects of a single quantum
theory [11]. The definitive evidence for wave-particle duality was the
discovery of the Compton effect in 1923, wherein an X-ray changes momentum
after colliding with a target [13]. While the wave nature of light was already
well established as essential to quantum theory, the Compton effect made it
clear that light must not only be quantized but also exhibit the properties of
classical particles as well at sufficiently small wavelengths.

The development of the modern understanding of light and its detection is
simultaneous with the advent of quantum theory in the early 1900’s [14]. In
the low-energy limit (that is, below $1.02MeV$ where a photon can
spontaneously produce an electron and positron), it is convenient to represent
light in the Fock basis so that it is comprised of photons, the quanta of the
electromagnetic field333At higher energies, interaction terms lead to non-
commutativity between number operators and the Hamiltonian so that Fock states
cease to be energy eigenstates [15].. Not all quantum states of light have a
definite integer number of photons, either because of unknown correlations
with an environment (for instance, a thermal distribution of photons) or
because of quantum necessity (for instance, coherent states of light, such as
those produced by a laser). In these cases, the state projected onto is not a
pure Fock state but a mixture (or a coherent superposition) of Fock
states444Similarly, a measurement that retrodicts a non-Fock input state may
rely on a Fock state as an intermediate stage, such as counting photons in
homodyne and heterodyne measurements [16, 17].. However as the quantum degree
of freedom of the electromagnetic field, any pure measurement that projects
onto Fock states will only retrodict555To retrodict is to utilize present
information or ideas to infer or explain a past event or state of affairs
[18]. an integer number of photons even though the internal state of the
photodetector may be partially unknown (mixed); that is, there may be _many_
internal photodetector states corresponding to a single detector outcome.

The lowest non-zero integer is one, so single-photon detection is the
intensity-frontier limit to detection of photon Fock states. For this reason,
we expect the fundamental quantum limits to general photodetector figures of
merit to be manifest in quantum descriptions of single-photon detectors
(SPDs). In this dissertation, we will construct what is (to our knowledge) the
first completely quantum description of the entire photodetection process in a
SPD in order to identify the fundamental limits and tradeoffs inherent to
single-photon detection.

In addition to the pure intellectual merit, answering the question “What are
the fundamental limits to single-photon detection?” is important for industry
applications of SPD-based technology. Recently, there has been progress in
achieving single photon detection with unprecedented performances in standard
photodetector figures of merit [19]: timing jitters or temporal uncertainties
in the tens of picoseconds in superconducting nanowire SPDs [20, 21], dark
count rates on the order of a single dark count per day [22, 23], and
robustness to thermal fluctuations nearing room temperature performance [24,
25]. This is in no small part due to the work done by the DARPA DETECT
collaboration, one of the aims of which is to jumpstart the next generation of
photodetection technology. Exploration of fundamental limits and tradeoffs
inherent to single-photon detection is essential to guiding this work, so
experimentalists can have ideas of what they will come up against and
decisions can be made about which areas of tradespace are important to
explore. Furthermore, they enable accountability; if a research group claims
their photodetector can simultaneously perform well in metric X and metric Y,
and we know metric X and Y are not simultaneously attainable from our generic
model, we have good reason to question their claims. While the DARPA DETECT
collaboration has resulted in a plethora of device-specific phenomenological
theories of photodetection [26, 21, 24, 27, 28], the universal limits to SPDs
are not explicit in such models (even though they are present). To construct a
model where the limits themselves are revealed (our goal), we must answer the
following two questions: What are the essential components of an SPD? What are
the fundamental limits of each component’s performance? We will answer these
two questions and construct a model where the fundamental limits to SPD
performance are manifest. However, it first behooves us to review
photodetection theory in the modern era.

The advent of a quantum theory of photodetection is simultaneous with the
advent of quantum theory, and has its origins in Einstein’s Nobel prize-
winning study of the photoelectric effect [14]. Here for the first time the
notion that light can only be absorbed (and thus detected) in discrete packets
of energy or quanta was rigorously incorporated into the foundation of a
theory, which was accomplished by building upon the more ad hoc treatment by
Planck [29]. With this, Einstein was able make statistical arguments for rates
of atomic transitions in the two-level approximation in the presence and
absence of an external electromagnetic field. In this way, Einstein was able
to describe the known processes of spontaneous emission and absorption, as
well discover stimulated emission [30]. This was done without a relativistic
quantum treatment of the electromagnetic field itself, the methods for which
did not exist at the time. Instead, Einstein made use of knowledge of the
Planck blackbody spectrum in order to demand agreement with quantum theory. In
a photodetector approximated as a two level system, the Einstein rate
equations are directly connected to the efficiency with which a photon is
absorbed and with how long it stays in the detector before leaking back out
(which can be connected to a gain factor in an amplification scheme such as
electron shelving [31]).

Although Einstein was able to correctly predict the rates of emission and
absorption for a two-level system, it was not until work by Dirac in 1927
(and, throughout the next two decades, Wignerl Oppenheimer, Fermi, Bloch,
Weisskopf, Tomonoga, Schwinger, and Feynman) formulating a theory of quantum
electrodynamics (QED) that a complete quantum treatment of the electromagnetic
field interacting with matter (here, only the electron) was developed [32, 33,
34, 35, 36, 37, 38, 39, 40]. One of the key developments was a quantum
treatment of the vacuum, clarifying spontaneous emission as stimulated
emission due to vacuum fluctuations by Wigner and Weisskopf [41]. Another key
development was a quantum method for treating the continuum of electromagnetic
field modes using Dirac-normalized creation and annihilation operators. These
well-known physicists, along with others, contributed to the extension of QED
in the development (and predictive success) of relativistic quantum field
theory, laying the foundation for modern particle physics which has dominated
the popular news spotlight ever since. Meanwhile and behind the scenes,
another revolution was brewing as the the initial question of an
electromagnetic field interacting with matter was further explored with the
new methods of QED.

In 1959, Hanbury Brown and Twiss published a controversial paper showing that,
for a few-photon signal collected from the star Sirius, an interference effect
was observable in intensity from which the angular size of Sirius was
calculable [42]. Purcell was quick to point out that this interference can be
understood as a manifestation of boson statistics applied to counting photons
[43]. However, the underlying mechanism behind the Hanbury Brown-Twiss effect
was partial coherence, a property of classical waves that had yet to be
understood at the quantum level.

In his seminal work in 1963 [44], Glauber connected the quantized formulation
of the electromagnetic field to experimentally measurable correlations using
the language of QED, introducing the field intensity correlation as a measure
of quantum coherence and calculating transition probabilities correlated with
the detection of individual photons. We will illustrate the method now,
focusing on a single polarization $\mu$ of the electromagnetic field and
following [45]. The first step is to separate the quantized (that is,
operator-valued) electric field into positive and negative frequency
components

$\displaystyle\hat{E}_{\mu}(\vec{r},t)=\hat{E}_{\mu}^{+}(\vec{r},t)+\hat{E}_{\mu}^{-}(\vec{r},t)$
(1.1)

where $\mu$ denotes the polarization we are describing and we note
$\hat{E}_{\mu}^{+}(\vec{r},t)^{\dagger}=\hat{E}_{\mu}^{-}(\vec{r},t)$. We can
represent the operator $\hat{E}_{\mu}^{+}(\vec{r},t)$ as a sum over discrete
plane wave modes with frequencies $\omega_{j}$

$\displaystyle\hat{E}_{\mu}^{+}(\vec{r},t)=-\sum_{j}\sqrt{\frac{\hbar\omega_{j}}{2\epsilon_{0}}}e^{\vec{k}_{j}\cdot\vec{r}-i\omega_{j}t}\hat{\epsilon}_{\mu}\hat{a}_{j}$
(1.2)

with $\hat{\epsilon}_{\mu}$ the component of the polarization vector,
$\vec{k}_{j}=\vec{p}/\hbar$ the wave number proportional to the momentum
$\vec{p}$, $\epsilon_{0}$ the permittivity of free space, $\hbar$ Planck’s
reduced constant, $i=\sqrt{-1}$, and $\hat{a}_{j}$ the annihilation operator
corresponding to the mode $j$. Since $[\hat{a}_{j},\hat{a}^{\dagger}_{j}]=1$,
the positive and negative electric field components will not commute. We can
associate a photodetection event with a transition in the state of the field
$\left|i\right\rangle\rightarrow\left|f\right\rangle$. For idealized single
photon detection, the difference between the initial and final state is that
one photon in mode $j$ was removed through an interaction at location
$\vec{r}$ and time $t$. For each particular initial field configuration, the
transition probability is described by unitary evolution

$\displaystyle\left\langle f\right|\hat{U}(i\rightarrow
f)\left|i\right\rangle$ $\displaystyle\propto$ $\displaystyle\left\langle
f\right|\hat{E}_{\mu}^{+}(\vec{r},t)\left|i\right\rangle$ (1.3)
$\displaystyle\propto$ $\displaystyle\left\langle
i\right|\hat{E}_{\mu}^{-}(\vec{r},t)\hat{E}_{\mu}^{+}(\vec{r},t)\left|i\right\rangle$

where proportionality with the full sum in (1.2) achieved due to only a single
term contributing to the to the matrix element and the sum over polarizations
$\mu$ is implicit. Since the initial state $\left|i\right\rangle$ is
arbitrary, we can carry out a sum over initial states and perform an average
so that the full transition probability is proportional to

$\displaystyle
P(t)\propto\langle\hat{E}_{\mu}^{-}(\vec{r},t)\hat{E}_{\mu}^{+}(\vec{r},t)\rangle$
(1.4)

where the angled brackets indicate an average or expectation value. This
expression for the photodetection probability motivated Glauber to define a
field correlation function. (1.4) is a special case of the first-order
coherence

$\displaystyle
G^{(1)}(\vec{r}_{1},t_{1};\vec{r}_{2},t_{2})=\langle\hat{E}_{\mu}^{-}(\vec{r}_{1},t_{1})\hat{E}_{\mu}^{+}(\vec{r}_{2},t_{2})\rangle$
(1.5)

By rewriting the electric field in terms of positive and negative frequency
components promoted to operators, Glauber was able to calculate measures or
degrees of coherence of the quantum field in terms of a series of coherence
functions, each measuring correlations at some order of the electric field
operators. Both (1.4) and (1.5) are normally ordered, so that, for the vacuum
state, $P(t)=G^{(1)}(\vec{r}_{1},t_{1};\vec{r}_{2},t_{2})=0$. First-order
coherence described by (1.5) is analogous to classical coherence in standard
optics, but Glauber showed that quantum systems can also exhibit higher order
correlations and coherences, making rigorous the theoretical foundation for
the maser and laser. Critically, Glauber also introduced the coherent state
representation for quantum states. These coherent states are minimum
uncertainty states in the quantum field quadratures and are denoted by a
single complex number identifying both the amplitude and phase of the
electromagnetic field (which are precisely what defines a particular laser
field). As the “most classical” states, the coherent states form a bridge
between quantum and classical descriptions of light.

To this day, the Glauber model of photodetection, given in terms of field
correlations and point-like interactions between a quantized electromagnetic
signal and a photoionization system (or other quantum electrical system) as in
(1.4), provides the foundation for modern photodetection theory. Although it
is mathematically rigorous, the Glauber model is highly idealized as a
practical theory of measurement; critically, Glauber merely posits that one is
able to count photoelectrons without describing any subsequent mechanisms.
This work was soon expanded to incorporate less-idealized descriptions of the
electromagnetic field-photodetection interaction [46, 47], with later
additions to the theory also incorporating the back action of the detector on
the detected quantum field [48, 49, 50, 51, 52]. Throughout, a challenge to
photodetection theory has been describing the interaction between a discrete
system (from a photon’s perspective, a system of discrete bosonic states) and
the full continuum of bosonic states of the electromagnetic field666See, for
instance, the Jaynes-Cummings model, describing only a single mode of the
electromagnetic field’s interaction with a two-level system [53].. In the
intermediate years, numerous quantum optical techniques have been developed
for describing the continuum of states, some of which we will discuss in
detail in the next chapter.

We will end this section with a brief survey of modern photodetecting
platforms amenable to single photon detection. (For in-depth review, see Refs.
[19, 54].)

Photoionization, the production of a current due to incident radiation, is the
method of photodetection longest studied quantum mechanically and has as its
fundamental theory the photoelectric effect [55]. First constructed by Hertz
in 1887 [56], the photomultiplier tube (PMT) is a cathode and vacuum-tube
device that uses the photoelectric effect coupled with secondary emission to
amplify signals as small as an single incident photon into a macroscopic
signal. Despite its quantum mechanical foundation and use in counting photons,
it is possible to generate a “classical” theory of a PMT where the external
quantum field is treated as a perturbation to which the PMT responds,
achieving a new steady-state. In this case, it is straightforward to calculate
the statistics of photodetection events [46]; the probability of detecting $n$
photons between a time $t$ and $T$ is calculated

$\displaystyle P(n,t,T)=\int_{0}^{\infty}\frac{1}{n!}W^{n}e^{-W}{\cal P}(W)dW$
(1.6)

with

$\displaystyle W=\eta\int_{t}^{t+T}I(t^{\prime})dt^{\prime}$ (1.7)

where $\eta$ is the photodetection efficiency, $I(t)$ is the intensity of the
electromagnetic field, and ${\cal P}(W)$ is the quasi-probability distribution
of the integrated intensity. While highly idealized, this expression can be
included into more realistic semiclassical models [57], and alone rapidly
gives useful predictions (even for a somewhat generic photodetector). For
instance, for a thermal distribution of light one finds that (1.6) yields a
Bose-Einstein distribution for photodetection events [55]. However, this
semiclassical formula does not describe the truly quantum nature of multi-
photon detection as made explicit in Hanburry Brown and Twiss’ work [42].

Another widespread SPD platform is the avalanche photodiode (APD), invented by
Jun-ichi Nishizawa in 1952 [58]. Like the PMT, APDs function based on the
photoelectric effect. Whereas in an light-emitting diode (LED) electrons and
holes annihilate converting current into light, in an APD an incident photon
with energy greater than the material band gap generates an electron-hole
pair, resulting in a current [59]. Generally, APDs can only operate in single-
photon detection mode at the expense of significant reset or dead time as the
circuit must be quenched unless an array is used [59, 60], so for an
individual APD pixel within an array Mandel’s formula (1.6) only applies for
$n\leq 1$ (higher order counts are highly suppressed). However, for single
photon detection we are indeed interested in the $n\leq 1$ regime and Mandel’s
formula will have validity.

Other single photon detecting platforms are less straightforward, but all
share a generic feature; the presence of a photon changes the state of the
system, either by shifting the equilibrium state or temporarily moving the
system out of equilibrium. Examples of the former include quantum dot field
transistor detectors [61] and superconducting transition-edge sensors (TES)
[62], both of which involve a measurable change in system conductance due to
absorption of a photon. (It is especially drastic for TES, the system is taken
out of the superconducting state!) Since the equilibrium state has changed,
resetting these mechanisms is resource intensive and SPDs built on these
principles have a longer dead time ($\sim 10\mu s$) compared to newer
technologies [19]. An interesting example of temporarily moving a system out
of equilibrium are superconducting nanowire single photon detectors (SNSPDs)
[63, 22, 23, 64], which in recent years have seen a rising popularity due to
their low dark count rates ($\sim 10Hz$) and dead times ($\sim 1ns$). However,
while theoretical modeling based on time-dependent Ginzburg-Landau theory has
produced qualitative agreement with experiment, quantitate models for SNSPDs
remain elusive [65]. In part, this is due to the complexity of system
dynamics; in an SNSPD a photon is collectively absorbed by many electrons and
phonons, resulting in a hot spot in which vortex anti-vortex pairs form with
local variations in the superconductivity. This results in a measurable
voltage when the vortex anti-vortex pair crosses the wire.

It is worth noting that one method of detecting single photons has been around
for thirty million years, namely, the African clawed frog _Xenopus laevis_
[66]. This species can detect single photons of visible light with moderate
($\leavevmode\nobreak\ 30\%$) efficiency and, as living organisms, they do
this at room temperature. This has motivated other participants in the DARPA
DETECT program to develop bio-inspired SPD platforms [25]. Although these will
suffer from larger dark count rates due to the increased thermal noise at
higher temperatures, the hope is that they still may have high enough signal-
to-noise ratio to be useful in room temperature experiments.

###  1.2 Very Brief Introduction to Quantum Information and POVMs

During the late 20th century and on into the 21st century, a revolution has
been taking place in quantum science; technological advancements are allowing
for increasing control over environmentally isolated single quantum systems
[67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78]. Simultaneously, physicists
were increasingly realizing the limits of classical computation both in terms
of scalability and applicability to quantum systems [79]; Moore’s law
describing the size of transistors _must_ break down when we approach the size
of single atoms. And if we are interested in simulating quantum systems,
shouldn’t we utilize a quantum system itself to do so [79]? Early pioneers of
quantum information science were quick to discover problems where a quantum
algorithm provided an exponential speedup [80, 81] and the so-called second
quantum revolution was born [75]. In many quantum algorithms, an essential
ingredient is measurement of an internal two-level system or qubit whose state
encodes information (generically, the answer to some input query with some
known probability). Thus the need for measurement theory, the rigorous
theoretical foundation for describing quantum measurement.

Measurement is a ubiquitous concept in quantum theory: examples include
measurements of an electron’s position or momentum (limits to the simultaneous
performance of which are enforced by Heisenberg’s uncertainty principle [82]),
measurement of the population of a multi-level system (such as an atom, simple
harmonic oscillator, optical cavity, or quantum dot), or detection of light
through the use of a photo detector (as we will explore in this thesis). The
most general quantum description of a measurement is in terms of a positive
operator-valued measure (POVM), a set of positive operators $\hat{\Pi}_{k}$
that sum to the identity, where each $k$ corresponds to a different
measurement outcome. Given an arbitrary quantum state $\hat{\rho}$ the
probability to obtain outcome $k$ is given by the Born rule

$P(k)={\rm Tr}(\hat{\rho}\hat{\Pi}_{k})$ (1.8)

where ${\rm Tr}(\hat{O})$ is the trace of the operator $\hat{O}$.

Generically, each POVM element $\hat{\Pi}_{k}$ can be written as a weighted
sum over projectors onto orthogonal quantum states

$\displaystyle\hat{\Pi}_{k}=\sum_{i}w_{i}^{(k)}\left|\phi_{i}^{(k)}\right\rangle\left\langle\phi_{i}^{(k)}\right|$
(1.9)

reducing to an ideal projective von Neumann measurement only when the sum
reduces to a single term with its weight $w^{(k)}$ equal to unity [83]. The
weight $w_{i}^{(k)}$ equals the conditional probability to obtain measurement
outcome $k$ given input state $\left|\phi_{i}^{(k)}\right\rangle$. The
_posterior_ conditional probability that, given an outcome $k$, we project
onto input $i$ is given by Bayes’ theorem [84]

$\displaystyle P(i|k)=\frac{w_{i}^{(k)}P(i)}{P(k)}$ (1.10)

with $P(k)=\sum_{i}w_{i}^{(k)}$. Here, $P(k)$ is _a priori_ probability for
observing measurement outcome $k$, and and $P(i)$ is the _a priori_
probability for the $i$th input state $\left|\phi_{i}^{(k)}\right\rangle$ to
be present [85]. Through Bayes’ theorem, an experimentalist is able to
_retrodict_ the likely input state or states; they can update the probability
distribution over the possible (past) inputs _if and only if they know what
measurement they actually perform_.

In the next chapter, we discuss applications of POVMs to single-photon
detection. First, we will review general properties of POVMs in the remainder
of this section, before giving an overview of the structure, format, and goals
of this dissertation in the final section of this chapter.

The POVM has several interesting properties it is worthwhile to review and
note: firstly, a POVM forms a partition of unity. If one sums over all
measurement outcomes (including the null outcome), from the Born rule one
finds $\sum_{k}{\rm Tr}(\hat{\rho}\hat{\Pi}_{k})=1$ for any state
$\hat{\rho}$; in other words, regardless of the input quantum state we can be
sure that _some_ outcome will occur since the probabilities of all outcomes
necessarily sum to unity!

Although the POVM is a complete description of a measurement, we note that
alone it is insufficient to determine the post-measurement quantum state. For
this we must additionally define Kraus operators $\hat{A}_{k}$ satisfying
$\hat{A}_{k}^{\dagger}\hat{A}_{k}=\hat{\Pi}_{k}$ [85]. Note that the POVM does
not uniquely specify $\hat{A}_{k}$, as for any unitary $\hat{U}$ the
substitution $\hat{A}_{k}\rightarrow\hat{U}\hat{A}_{k}$ leaves the POVM
unchanged. Having defined Kraus operators, the normalized post-measurement
state $\hat{\rho}^{\prime}$ is calculated

$\displaystyle\hat{\rho}^{\prime}=\frac{\hat{A}_{k}\hat{\rho}\hat{A}_{k}^{\dagger}}{{\rm
Tr}(\hat{A}_{k}\hat{\rho}\hat{A}_{k}^{\dagger})}.$ (1.11)

We note that repeated measurement will not guarantee the same outcome (unless
the measurements are von Neumann measurements), as the measurement outcomes
are not orthogonal.

Non-orthogonality is a general feature of POVM elements (except in the case of
the idealized von Neumann measurement, which always project onto orthogonal
[and pure] quantum states). Of particular interest in quantum information
theory are symmetric, informationally complete (SIC)-POVMs; the POVM is
considered symmetric if each of the elements of these POVMs share the same
mutual overlap ${\rm
Tr}(\hat{\Pi_{i}}\hat{\Pi_{j}})=\frac{d\delta_{ij}+1}{d+1}$ with $d$ the
dimension of the Hilbert space and $\delta_{ij}$ the Kronecker delta ($0$ when
the indices are not the same, unity when the indices are the same).
Additionally, the SIC-POVM is informationally complete if there are $d^{2}$
elements in the POVM, where $d^{2}-1$ are linearly independent. While they
will not be discussed further in this dissertation, they are of particular
importance to two applications of our work which we will discuss, quantum
state tomography [86] and quantum cryptography [87], where they illuminate the
most efficient measurement scheme for quantum state characterization.
Furthermore, there are interesting connections to be made between SIC-POVMs
and mutually unbiased bases of quantum states, with the latter playing an
essential role in quantum communication and cryptographic protocols [88].

###  1.3 Overview and Glossary

Having laid the theoretical foundation for our work in this first chapter, we
now review the structure of this dissertation. We also provide a glossary of
terms ubiquitous in this thesis at the end of this section (see Tbl.
LABEL:glossary).

The motivation and goal of this work is to uncover the fundamental limits
inherent to single photon detection across physical platforms, both current
and future. To accomplish this, we approach photodetection from the
perspective of information theory, recasting the process of detecting a photon
into the interpretation of a photodetection outcome in the absence of priors.
Interpretation of a measurement outcome requires sufficient information about
the measurement apparatus to identify the POVM implemented. As such, it is
necessary to develop a model of photodetection that is realistic in describing
all parts of the photodetection process, but sufficiently idealized and
general so that the limits to each part of the process are fundamental limits
across all platforms.

In the next chapter of this dissertation, we begin the construction of such a
model starting with a POVM corresponding to the time-dependent two-level
system. Here a detection event occurs when one “checks” whether the two-level
system is in the excited state at a particular time $T$. The two-level system
is an essential component to a description of an SPD, as it ensures the
measurement projects onto the Hilbert space of single-photon states. As we
show, the time-dependent nature of this system is sufficient to project onto
arbitrary single-photon wavepackets (with arbitrarily high efficiency),
including Gaussian wavepackets so that Heisenberg-limited simultaneous
measurements of time and frequency are implemented. At the end of the second
chapter, we introduce our three-stage model of the photodetection process:
transmission, amplification, and measurement (Fig. 2.4), and clarify the role
of the time-dependent two-level system in this model as the trigger for the
amplification mechanism.

In the third chapter of this dissertation, we analyze transmission functions
$T(\omega)$ describing the quantum network structure of the transmission stage
of photodetection. In the transmission stage, the photon has to interact with
one or more charged particles, its excitation energy will be converted into
other forms of energy which will later lead to a macroscopic signal. In this
chapter, we discuss the tradeoffs inherent to different network structures and
their implications for photodetecting systems.

In the fourth chapter of this dissertation, we analyze the amplification
process itself. In particular, we identify commutator-preserving
transformations that implement different nonlinear amplification schemes
outperforming linear amplification in terms of signal-to-noise ratios (SNRs)
as illustrated in Fig. 4.2. Additionally, we demonstrate that amplification
into a single bosonic mode not only outperforms all previously analyzed
amplification schemes, but provides the fundamental limit to amplification of
single and many-photon Fock states.

In the fifth chapter of this dissertation, we introduce the final measurement
stage and construct realistic SPD POVMs that include all three stages of
photodetection. In particular, we demonstrate that an SPD POVM retrodicts a
relatively pure external state of the electromagnetic field, even though the
internal state of the detector is highly mixed in general. In the remainder of
this chapter, we discuss applications to quantum cryptographic protocols,
quantum sensing, and quantum information science.

In the sixth and final chapter of this dissertation, we discuss the
fundamental limits and tradeoffs inherent to photodetection encapsulated by
the three-stage model of photodetection. We also discuss implications of this
work for the future development of novel SPD platforms.

##  Chapter 2 POVMs for Single-Photon Detection

In this chapter, we review techniques in quantum information theory amenable
to single-photon detection, in particular, the definition of photodetector
figures of merit. Then, we construct a simplified single-photon detector (SPD)
positive operator valued measures (POVM) capable of projecting onto arbitrary
single-photon wavepackets. Lastly, we discuss the three stages of
photodetection universal to all detector schemes, each of which we will
analyze in detail in its own chapter.

###  2.1 Quantum Information for Photodetection

Photodetection is at its core an information theoretic process; a measurement
outcome—a click—reveals information about the outside world quantifiable in
bits [90]. In the case of a single-photon detector (SPD), a click is
correlated (imperfectly) with the presence of a photon in a particular single-
photon state. (Here, the quantum state refers to the spectral, spin angular
momentum, and orbital angular momentum quantum degrees of freedom at every
point in spacetime). In this way, a detection reveals information about the
presence of photons in that particular state, along with whatever else in the
world is correlated with that photon state. For an example from fluorescence
microscopy, detection of ultraviolet light when a sample is illuminated with
red light is positively correlated with the presence of fluorine [91]. For
single photons, measurements that detect light are naturally described through
a SPD POVM.

Knowledge of the POVM is essential for both gaining information from a
measurement device and characterizing detector performance, hence the
experimental need for detector tomography [92, 93, 94, 95, 96, 97]. In
detector tomography, known quantum states are input to a device to determine
the quantum states projected onto by different outcomes (along with their
associated weights) and calculate the full measurement POVM. Commercial
photodetectors are characterized by industry-standard figures of merit [19],
which can be calculated from a POVM (for an in-depth review, see Ref. [98]).
Here we will concern ourselves mostly with two standard figures of merit,
detection efficiency and time-frequency uncertainty:

_Efficiency.—_ The maximum efficiency with which an SPD outcome $k$ (for
instance, a single click) can be triggered by input single-photon states is
exactly the maximum relative weight in (1.9): $\eta_{\max}={\rm
Max}_{i}[w_{i}^{(k)}]$. This maximum efficiency is achieved only when the
input quantum state is the single-photon state the measurement maximally
projects onto. This follows directly from the Born rule; $P(k|i)={\rm
Tr}[\hat{\Pi}_{k}\hat{\rho}]\rightarrow w_{i}^{(k)}$ if and only if
$\hat{\rho}=\left|\phi_{i}^{(k)}\right\rangle\left\langle\phi_{i}^{(k)}\right|$.

_Time-Frequency Uncertainty.—_ The spectral uncertainty (that is, the
uncertainty in measurements of photon frequency) and the input-independent
timing jitter (uncertainty in measurements of photon time-of-arrival) are
determined entirely by the spectral and temporal distributions of the single-
photon states projected onto by the measurement outcome $k$ [99], which form a
retrodictive probability distribution. For any continuous variable $X$ (here
either time $t$ or frequency $\omega$), we find it less convenient to use the
variance as measure of uncertainty and instead define the uncertainty
entropically [99, 100, 101, 98, 102]

$\displaystyle\Delta X^{(k)}=2^{H^{(k)}_{X}}\delta X.$ (2.1)

Here $H^{(k)}_{X}$ is the Shannon entropy defined as

$\displaystyle H_{X}^{(k)}=-\sum_{j}p(j|k){\rm log}_{2}p(j|k)$ (2.2)

with the sum over discretized $X$-bins of size $\delta X$. $p(j|k)$ is the _a
posteriori_ probability for the detected photon to be in bin $j$ given outcome
$k$, and is calculated as

$\displaystyle p(j|k)=\int_{(j-1)\delta X}^{j\delta
X}dX\sum_{i}P(i|k)|\phi_{i}(X)|^{2}$ (2.3)

where we have defined a normalized distribution over $X$ given by the norm
squared of the amplitude of the quantum state $|\phi_{i}(X)|^{2}$ (where we
have defined $\phi_{i}(X)\equiv\left\langle X|\phi_{i}\right\rangle$ for the
single-photon state $\left|\phi_{i}\right\rangle$). The conditional
probability $P(i|k)$ is precisely the one from Bayes theorem (1.10); $P(i|k)$
reduces to $w_{i}^{(k)}/\Omega^{(k)}$ in the case of a uniform prior111For
inclusion of priors in updating information about the quantum state, see Ref.
[103]., where $\Omega^{(k)}=\sum_{i}w_{i}^{(k)}$ is the channel bandwidth
defined in Ref. [98] (which characterizes the number of states the measurement
efficiently projects onto). Critically, $\Delta X^{(k)}$ is independent of the
bin size $\delta X$ in the small-bin limit, even though the entropy
$H_{X}^{(k)}$ is strongly dependent on the bin size. One can verify that this
definition of uncertainty yields an uncertainty relation [100]

$\Delta\omega\Delta t\geq e\pi.$ (2.4)

The construction of measurements projecting onto arbitrary single-photon
states is critical in quantum optical and quantum communication experiments.
Mismatch between the single-photon state generated and the state projected
onto by the measurement induces an irreversible degradation in efficiency.
(This is a direct consequence of the Born rule (1.8).) Furthermore, the
capacity to efficiently project onto single-photon states with orthogonal
wavepackets enables a wide range of quantum information and quantum optical
applications, as we will revisit in a later chapter. From a foundational
perspective, a procedure to build measurements projecting onto minimum-
uncertainty Gaussian single-photon wavepackets paves the way for future tests
of fundamental quantum theory.

###  2.2 Simple SPD POVMs

We will now discuss how to construct a simple POVM that efficiently projects
onto an arbitrary single-photon wavepacket222A wavepacket is a localized
envelope of wave actions, which can be decomposed into an infinite set of
complex sinusoidal waves with differing wave numbers. Here we define the
wavepacket $\psi(t)$ as the representation of the single-photon state
$\left|\psi\right\rangle$ in the temporal basis $\psi_{(}t)\equiv\left\langle
t|\psi\right\rangle$.. To aid us, we will now make four simplifying
assumptions. First, we will consider only the time-frequency degree of freedom
of the electromagnetic field, as the other degrees of freedom (e.g.
polarization) can be efficiently sorted prior to detection in a pre-filtering
process [104, 105, 106]. Second, we consider only a single photon incident to
the photodetector. Multiple photons can always be efficiently multiplexed to
achieve a photon number resolution using SPD pixels [107]. Third, we will not
model a continuous measurement (as briefly discussed in the appendix of
[108]), but instead a discretized measurement where at a particular time $T$
we ascertain whether or not a photon has interacted with the SPD, ending the
measurement. Lastly, we will consider only a binary-outcome photodetector,
“click” or “no click.” This simplifies the POVM so that it only contains the
two elements $\hat{\Pi}_{T}$ and $\hat{\Pi}_{0}$, both projecting onto the
Hilbert space of single-photon states and the vacuum state. Generalizations to
non-binary-outcome SPDs are straightforward: one can concatenate binary-
outcome POVMs to generate non-binary-outcome experiments.

We now begin construction of the POVM $\\{\hat{\Pi}_{T},\,\hat{\Pi}_{0}\\}$ in
earnest. We begin in the rotating frame by considering a two-level system with
time-dependent detuning (that is, the transition frequency in the rotating
frame) $\Delta(t)$333In the rotating frame, we define
$\Delta(t)=\omega_{e}(t)-\omega_{g}(t)$, with $\omega_{e}(t)$ and
$\omega_{g}(t)$ the time-dependent resonance frequencies of the excited and
ground states. and time-dependent coupling to a Markovian external
electromagnetic continua of states $\kappa(t)$ 444Non-markovianity of the
external continua can be included via couplings to fictitious discrete states
or pseudomodes, see Refs. [109, 110, 111, 89, 112].. Experimentally, a time-
dependent decay rate $\kappa(t)$ is induced by a rapid variation of density of
states [113, 114]555Since the rate of incoherent coupling to the continuum is
$\sqrt{\frac{\kappa(t)}{2\pi}}$, a time-dependent decay rate also induces a
time-dependent coupling., and a time-dependent resonance $\Delta(t)$ can be
varied with a time-dependent external electric field (Stark effect, [115]) or
through a two-channel Raman transition [116].

The general state of the two-level system and the continuum can be written in
the Schrödinger picture
$\left|\psi(t)\right\rangle=C_{0}(t)\left|0\right\rangle+C_{1}(t)\left|1\right\rangle$,
where the first term corresponds to the ground state of the two-level system
and the second term corresponds to the excited state of the two-level system.
In the quantum trajectory picture, there are two types of evolution of
$\left|\psi(t)\right\rangle$: Schrödinger-like smooth evolution with a non-
Hermitian effective Hamiltonian and quantum jumps (at random times) [117,
118]. A quantum jump will always correspond to the excitation leaking out of
the system and so, in the absence of a dark counts, we only need consider the
Schrödinger-like evolution of the system. In the quantum jump picture, the
quantum state of the two-level system remains will remain pure. We proceed to
calculate this in the rotating frame under the rotating wave approximation,
following the methods for time-dependent systems introduced in Refs. [119,
120] and letting $\hbar=1$666For similar treatments of universal quantum
memory and, more recently, a quantum scatterer, see see Ref. [121] and Ref.
[122], respectively..

Since we are working the Schrödinger picture, we begin the calculation by
invoking an additional system: a time-dependent optical cavity with time-
dependent detuning in the rotating frame $\delta(t)$ and time-dependent decay
rate $\Gamma(t)$ containing exactly one photon in the infinite past. This will
populate the continuum driving the two-level system with an (arbitrary) photon
wavepacket. The population of this optical cavity is described by a single
bosonic mode amplitude $C_{b}(t)$ which is in turn equal to $C_{0}(t)$; due to
our assumption of a single photon, the ground state acts as a bosonic mode.

In the Schrödinger picture, the evolution of the bosonic mode population and
two-level system population are given

$\displaystyle\dot{C}_{b}(t)$ $\displaystyle=$
$\displaystyle-i\delta(t)C_{b}(t)-\frac{\Gamma(t)}{2}C_{b}(t)$
$\displaystyle\dot{C}_{1}(t)$ $\displaystyle=$
$\displaystyle-i\Delta(t)C_{1}(t)-\frac{\kappa(t)}{2}C_{1}(t)-\sqrt{\kappa(t)\Gamma(t)}C_{b}(t).$
(2.5)

The decay rate $\kappa(t)$ appears in the second line of (2.2) twice due to
its dual role: acting as the system’s time-dependent decay rate and
determining the time-dependent coupling $\sqrt{\frac{\kappa(t)}{2\pi}}$ to the
external electromagnetic continuum of state. Let us consider a time in the
infinite past when there was no coupling to the external continua
($\kappa(-\infty)=0$), the excited state is unpopulated ($C_{1}(-\infty)=0$),
and bosonic mode is fully populated ($C_{b}(-\infty)=1$), so that we can solve
the first line of (2.2) for $C_{b}(t)$ directly

$\displaystyle
C_{b}(t)=\exp\left[-\int_{-\infty}^{t}dt^{\prime}(i\delta(t^{\prime})+\frac{\Gamma(t^{\prime})}{2})\right].$
(2.6)

Defining a driving term $f(t)=\sqrt{\kappa(t)}C_{b}(t)$ and substituting (2.6)
into the second line of (2.2), we find the time-dependent excited state
amplitude $C_{1}(t)$ will have the form of a Langevin equation

$\displaystyle\dot{C}_{1}(t)=-\frac{\kappa(t)}{2}C_{1}(t)-i\Delta(t)C_{1}(t)+\sqrt{\kappa(t)}f(t)$
(2.7)

with decay (the first term), resonance frequency (the second term), and
driving from the continuum (the last term).

We can now interpret $f(t)$ as the normalized input photon wavepacket
corresponding to the state

$\left|f\right\rangle=\int dtf(t)a^{\dagger}_{{\rm in}}(t)|{\rm vac}\rangle$
(2.8)

with $a^{\dagger}_{{\rm in}}(t)$ the standard creation operator for the
bosonic continuum of input states (as we will use extensively in the input-
output theory treatments to come in this thesis) [4]. We can solve this
equation exactly with the result

$C_{1}(t)=\int_{T_{0}}^{t}dt^{\prime\prime}f(t^{\prime\prime})\sqrt{\kappa(t^{\prime\prime})}\exp\left[-\int_{t^{\prime\prime}}^{t}dt^{\prime}D(t^{\prime})\right],$
(2.9)

where we have defined $T_{0}$ to be a (finite) time in the distant past where
our photodetector was still off so that there is no coupling to the external
continua ($\kappa(T_{0})=0$ [this could be a smooth transition as explored in
Fig. 2.2]) and thus the excited state is unpopulated ($C_{1}(T_{0})=0$), and
we have also defined

$D(t)=i\Delta(t)+\frac{\kappa(t)}{2}.$ (2.10)

Our measurement consists in checking if the system is in the excited state at
time $t=T$. The probability to obtain a positive result (corresponding to
detecting the incident photon wavepacket) is $|C_{1}(T)|^{2}$. We can write

$C_{1}(T)=\int_{T_{0}}^{T}dt\Psi^{*}(t)f(t)$ (2.11)

with

$\Psi^{*}(t)=\sqrt{\kappa(t)}\exp\left[-\int_{t}^{T}dt^{\prime}D(t^{\prime})\right].$
(2.12)

Figure 2.1: The retrodictive probability amplitudes ${A(t)=|\Psi(t)|}$ defined
in (2.12) are plotted for polynomial decays of the form
${\kappa(t)=\kappa_{0}\left(\frac{T-t}{\sigma}\right)^{n}}$ with polynomial
order $n$ varying from $0$ (violet, top) to $40$ (red, bottom) denoted by
color. The polynomial decays correspond to systems where the decay rate drops
to zero precisely at the time of detection. Here time is measured with respect
to (w.r.t.) time of detection ${T=0}$ and we set ${T_{0}=-\infty}$ so that
${{\cal W}=1}$ (2.13).

Only constant ${\kappa(t)=\kappa_{0}}$ yields non-zero probability amplitude
at ${t=0}$ (where it is maximum). This illustrates the two roles $\kappa(t)$
plays: determining the probability $\kappa(t)dt$ with which an excitation can
enter the system at a time $t$, and the rate $\kappa(t)$ at which the excited
state will decay.This latter effect drives down the probability amplitude for
the distant past. For ${\kappa(T)\neq 0}$, the most likely time that a photon
entered the system is _now_ (${t=0}$) whereas if ${\kappa(T)=0}$, there is
some time of maximum likelihood determined by competition between the two
effects (absorption and decay). For no order $n$ are the retrodictive
probability distributions continuously differentiable; these simple polynomial
couplings do not yield measurements projecting onto smooth wave packets unlike
the couplings plotted in Fig. 2.3. Note that $\kappa(t)$ is not normalized and
diverges as ${t\rightarrow-\infty}$, yet it results in well-behaved,
normalized retrodictive probability amplitudes.

Whereas $f(t)$ is a normalized wavepacket, $\Psi(t)$ is subnormalized for
finite $T_{0}$, since

${\cal
W}=\int_{T_{0}}^{T}dt|\Psi(t)|^{2}=1-\exp\left[-\int_{T_{0}}^{T}\\!dt\,\kappa(t)\right].$
(2.13)

We can interpret $\Psi(t)$ as a retrodictive probability amplitude (for simple
examples, see Fig. 2.1 and Fig. 2.2), identifying at which times a photon
likely entered the system given a detector “click” at $t=T$.

Figure 2.2: Retrodictive probability amplitudes $|\Psi(t)|$ defined in (2.12)
are plotted for polynomial decays of the form
${\kappa(t)=\kappa_{0}\left(\frac{t-T_{0}}{\sigma}\right)^{n}\left(\frac{T-t}{\sigma}\right)^{n}}$
for ${T_{0}\leq t\leq T}$, with the polynomial order $n$ varying from $0$
(violet, lowest peak) to $15$ (red, highest peak) denoted by the color. Time
is measured w.r.t. detection time ${T=0}$.

Except for ${n=0}$ (also in Fig. 2.1 and included here for reference), these
decays become non-zero when the photodetector is turned on at
${T_{0}=-2.5\sigma}$ so that ${\cal W}$ from (2.13) is less than unity. As in
Fig. 2.1, for no order $n$ are the retrodictive probability distributions
continuously differentiable (indeed, this is the source of the singularity in
Ref. [3], and do not project onto smooth wave packets as in Fig. 2.3. Also as
in Fig. 2.1, we observe a time of maximum likelihood determined by competition
between the two effects (absorption and decay).

We can define a normalized single-photon state

$\displaystyle\left|\Psi_{T}\right\rangle={\cal
W}^{-1/2}\int_{T_{0}}^{T}\\!dt\,\Psi(t)\hat{a}^{\dagger}_{\rm in}(t)|{\rm
vac}\rangle$

with the creation operator $\hat{a}^{\dagger}_{\rm in}(t)$ acting on the input
continuum of states (it’s Fourier transform $\hat{a}^{\dagger}_{\rm
in}(\omega)$ creates a monochromatic state with frequency $\omega$). The
arbitrary input single-photon state (which may have been created long before
our detector was turned on at $T_{0}$ or long after the measurement ended at
time $T$) is

$\displaystyle\left|f\right\rangle$
$\displaystyle=\int_{-\infty}^{\infty}dtf(t)\hat{a}^{\dagger}_{\rm in}(t)|{\rm
vac}\rangle.$ (2.15)

The commutator relation for the input field operator is $[\hat{a}_{\rm
in}(t),\hat{a}^{\dagger}_{\rm in}(t^{\prime})]=\delta(t-t^{\prime})$.

The probability for an arbitrary input photon wavepacket $f(t)$ to result in
the system being found in the excited state at a time $T$ is
$|C_{1}(T)|^{2}={\cal W}\left\langle
f|\Psi_{T}\right\rangle\left\langle\Psi_{T}|f\right\rangle$. The measurement
does not project onto times _after_ we have checked if the system is in the
excited state, nor onto times before the detector was turned on.

We rewrite this probability in terms of a POVM element containing a single
element

$\displaystyle\hat{\Pi}_{T}={\cal
W}\left|\Psi_{T}\right\rangle\left\langle\Psi_{T}\right|$ (2.16)

To the extent our detector has been open long enough, such that ${\cal
W}\rightarrow 1$, our detector could act as a perfectly efficient detector for
a specific single-photon wavepacket with temporal mode function
$\Psi(t)$777For measurements projecting onto a Gaussian wavepacket as in
(2.24) and Fig. 2.3, the weight (2.13) has the simple form ${\cal
W}=\frac{1}{2}{\rm Erf}[\frac{t_{0}-T_{0}}{\sqrt{2}\sigma}]+\frac{1}{2}{\rm
Erf}[\frac{T-t_{0}}{\sqrt{2}\sigma}]$, going to unity for $T,T_{0}\gg\sigma$..
This wavepacket is the time reverse of the wavepacket that would be emitted by
our system if it started in the state $\left|1\right\rangle$888Instead of
solving (2.7), one can find the Green’s function $G(t)$ of the time-reversed
problem: at $t_{0}$ the two-level system is started in the excited state and
at time $T$ we check whether the excitation has leaked out. Taking
$t\rightarrow T-t$, one arrives at (2.2) with $\Psi(t)=G(T-t)$. This clarifies
the role of the Green’s function; propagating back in time starting from $t=T$
(when the photon is detected) back to the infinite past, which indeed is what
the POVM does as well..

For this simple system, the POVM element is both pure (containing just one
term999As an auxiliary and less-conventional figure of merit, we define purity
of the POVM element ${\rm Pur}[\hat{Pi}_{k}]=\frac{{\rm
Tr}[\hat{\Pi}_{k}^{2}]}{\left({\rm Tr}[\hat{\Pi}_{k}]\right)^{2}}\leq 1$ where
the upper limit is reached only when the POVM element projects onto a single
state (von Neumann measurement).) and (almost) maximally efficient; the weight
${\cal W}$ may approach unity as close as we wish by lengthening the time the
detector is on for (that is, taking $T_{0}$ to the distant past and making
sure the projected wavepacket [approximately] ends before checking at time
$T$).

Here we observe an obvious tradeoff between efficiency and photon counting
rate: one cannot project onto a long single-photon wavepacket in a short time
interval without cutting off the tails, lowering the overall detection
efficiency101010The limitation to photon counting rate imposed by efficient
detection of long temporal wavepackets is avoided via signal multiplexing, see
Ref. [107]..

The two-level system described in Eq. (2.7) is a special case but an important
one; the two-level system is often a very good approximation of more
complicated systems near-resonance. Furthermore, the two-level system is the
foundation for generating more complicated network structures, as we will
discuss in more detail in the next chapter111111For an arbitrary multi-level
time-independent structure, we will end up with a system of equations
governing discrete state evolution
$\displaystyle\dot{\vec{C}}(t)=\mathbf{M}\vec{C}(t)+\vec{S}(t)$ (2.17) with
$\mathbf{M}$ a time-independent matrix and $\vec{S}(t)$ a time-dependent
(inhomogeneous) source term describing the input photon. The solution is then
always of the form
$\displaystyle\vec{C}(t)=e^{\mathbf{M}t}\vec{C}(t_{0})+\int_{t_{0}}^{t}dt^{\prime}e^{\mathbf{M}(t-t^{\prime})}\vec{S}(t^{\prime}).$
(2.18) Writing $e^{\mathbf{M}(t-t^{\prime})}$ as a Green’s matrix, we can
identify elements that correspond to transitions to the final monitored
discrete state (detector outcomes) through standard numerical techniques
[123].. In this chapter, we will focus on the simple time-dependent system
(2.7) as it is sufficiently general to perform a measurement described by any
time-independent system, and more121212In particular, time time-independent
systems cannot achieve Fourier-limited measurements of time and frequency.
This is because networks of discrete states experience a natural spectral
broadening that is Lorentzian. While Gaussian broadening can additionally
occur (for instance, due to Doppler shifts in atomic distributions [124]) this
only increases the product uncertainty further from the minimum of
${\Delta\omega\Delta t=e\pi}$ [100], attained _only_ by pure measurements
projecting onto Gaussian wavepackets.. Indeed, (2.7) is general enough to
project onto a _completely arbitrary_ single-photon wavepacket, a result we
will now prove.

_Proof.—_ Consider a photon with complex wavepacket
$\Psi^{*}(t)=A(t)e^{i\phi(t)}$, positive amplitude $A(t)$, and phase
$\phi(t)$. Inserting this into (2.12), we arrive at two separate expressions

$\displaystyle A(t)$ $\displaystyle=$
$\displaystyle\sqrt{\kappa(t)}e^{-\int_{t}^{T}dt^{\prime}\frac{\kappa(t^{\prime})}{2}}$
$\displaystyle\phi(t)$ $\displaystyle=$
$\displaystyle-\int_{t}^{T}dt^{\prime}\Delta(t^{\prime}).$ (2.19)

The second line is always solvable by $\Delta(t)=\dot{\phi}(t)$ up to a
constant global phase shift provided $\phi(t)$ is everywhere differentiable
(smooth). We now focus on the first line. Taking the natural logarithm we
arrive at an expression

$\displaystyle 2{\rm Ln}[A(t)]-{\rm
Ln}[\kappa(t)]=-\int_{t}^{T}dt^{\prime}\kappa(t^{\prime}).$ (2.20)

Taking the time derivative of both sides, we arrive at a Bernoulli
differential equation [125]

$\displaystyle\kappa^{-2}(t)\frac{d\kappa(t)}{dt}-\frac{2}{A(t)\kappa(t)}\frac{dA(t)}{dt}=-1.$
(2.21)

Provided $\frac{1}{A(t)}\frac{dA(t)}{dt}$ is continuous, this is solved by

$\displaystyle\kappa(t)=\frac{A^{2}(t)}{1-\int_{t}^{T}A^{2}(t^{\prime})dt^{\prime}}$
(2.22)

in agreement with Ref. [119] and Ref. [122]. Here, $\kappa(t)$ is given by the
square of the electromagnetic field, divided by a correction factor accounting
for the finite response time imposed by $\kappa(t)$ itself131313We observe
from (2.22) that now ${A(t)=0\iff\kappa(t)=0}$ for smooth wavepackets, whereas
for a general retrodictive distribution (2.12) we find ${\kappa(t)=0\implies
A(t)=0}$; to generate a smooth wavepacket, $\kappa(t)$ must go to $0$ in the
distant past.. From (2.22), we observe that the only condition imposed on
$A(t)$ is that $A^{2}(t)$ has an antiderivative (indefinite integral). We
simply require $A^{2}(t)$ be continuous, which in turn requires $A(t)$ to be
continuous. Thus, any wavepacket with smooth phase profile $\phi(t)$ and
smooth amplitude $A(t)$ is projected onto by _some_ physically realizable
single-photon detection scheme. ∎

_Special Case: Fourier-Limited Measurement.—_ A Fourier-limited simultaneous
measurement of time and frequency is achieved with a Gaussian time-frequency
distribution. We want a temporal wavepacket $\Psi^{*}(t)$ that is the complex
square root a Gaussian distribution

$\displaystyle\Psi^{*}(t)=\frac{1}{(2\pi\sigma^{2})^{\frac{1}{4}}}e^{-\frac{(t-t_{0})^{2}}{4\sigma^{2}}}e^{i\frac{\omega_{0}}{2}t}$
(2.23)

where $\sigma$ is the temporal half-width, and $t_{0}$ and $\omega_{0}$ are
the central time and frequency of the Gaussian distribution. We find that this
wavepacket is projected onto by a time-dependent system with constant
resonance $\Delta(t)=\omega_{0}$ and a time-dependent coupling

$\displaystyle\kappa(t)=\frac{e^{-\frac{(t-t_{0})^{2}}{2\sigma^{2}}}}{\sqrt{2\pi\sigma^{2}}(1+\frac{1}{2}{\rm
Erf}[\frac{t_{0}-T}{\sqrt{2}\sigma}]-\frac{1}{2}{\rm
Erf}[\frac{t_{0}-t}{\sqrt{2}\sigma}])}$ (2.24)

as in Fig. 2.3. Note that the coupling $\kappa(t)$ is $T$-dependent even
though the projected state (2.23) is $T$-independent, in agreement with the
general case (2.22).

Figure 2.3: Time-dependent couplings $\kappa(t)$ defined in (2.24) generating
a minimum uncertainty (Gaussian) wavepacket are plotted for times of detection
${T=2\sigma}$ (${{\cal W}=0.98}$) and ${T=7\sigma}$ (${{\cal W}=1-10^{-10}}$)
and detector on-time ${T_{0}=-\infty}$. The retrodictive probability amplitude
is now a well-defined wavepacket, with amplitude $A(t)$ depicted in the
rotating frame (without fast oscillations at central frequency $\omega_{0}$).
Time is measured w.r.t. the wavepacket’s central time $t_{0}$. For times near
${t\approx T}$ the coupling $\sqrt{\kappa(t)}$ is approximately Gaussian as an
excitation absorbed at this time will not have sufficient time to decay back
out. For earlier times, $\sqrt{\kappa(t)}$ is strictly larger than $A(t)$ and
is skewed towards earlier times to incorporate the time it takes the system to
respond (${\sim 1/\kappa(t)}$); one needs to prepare the two-level system for
the photon, accomplished via a “sharkfin” coupling that precedes the single-
photon wavepacket [121]. Near ${t=-T}$, we observe the coupling $\kappa(t)$
rapidly drops to zero, which is a direct consequence of the temporal symmetry
of the wavepacket about $t_{0}$. [From (2.12) one can verify that, if $A(t)$
is time-symmetric around ${t=t_{0}}$, the coupling $\kappa(t)$ must satisfy
${\kappa(t_{0}-t)=e^{\int_{t_{0}-t}^{t_{0}+t}dt^{\prime}\kappa(t^{\prime})}\kappa(t_{0}+t)}$.]

###  2.3 The Three-Stage Model of Photodetection

The model of a SPD as an isolated two-level system is highly idealized. In a
more realistic system, photodetection is an extended process wherein a photon
is transmitted into the detector, interacting with the system and triggering a
macroscopic change of the photodetector state (amplification) which can then
be measured classically. Many theories of single-photon detection have been
developed over the past century, [44, 47, 48, 49, 126, 50, 51, 127, 52, 26,
24, 25] and indeed there are numerous implementations of SPD technology [128,
129, 22, 66, 23].

Figure 2.4: The three stages of photodetection in a physical implementation:
transmission, amplification, and measurement. An input photon with frequency
$\omega$ is transmitted irreversibly via network of coherently coupled
discrete states into contact with the photodetection mechanism, here modeled
as a two level system. Once the amplification mechanism is triggered by a
photon, a macroscopic optical signal with frequency
${\omega^{\prime}\approx\omega_{L}}$ (the laser frequency) is irreversibly
created via amplification, in our model implemented with a three-level
electron shelving scheme in a single mode cavity; absorption of the single
photon at frequency $\omega$ results in many flourescence excitations at
frequency $\omega^{\prime}$. This is then converted to a classical electrical
readout through a photomultiplier tube (which could have low efficiency) in a
third irreversible step. Both initial transmission into the system and
absorption by the two-level system filter the light, changing which quantum
state is projected onto by the final measurement.

Across all systems, we identify these three stages of transmission,
amplification, and measurement as universal in Fig. 2.4. In the next three
chapters, we will study the transmission and amplification stages in detail
before deriving a POVM that incorporates all three stages quantum mechanically
and includes fluctuations of system parameters. The time-dependent two-level
system from the previous section enabling arbitrary wavepacket projection will
be incorporated into the three-stage model as the trigger for the
amplification mechanism. We will assume in this analysis that the system is
left on for a sufficient time such that the subnormalization of $\Psi(t)$ is
minimal and ${\cal W}\approx 1$ (2.13).

##  Chapter 3 Transmission

We now study in detail the topic of the “transmission” stage of photo
detection, wherein a photonic signal is transmitted into the photodetector,
potentially changing form (transduction) and being absorbed by the device. To
accomplish this, the photon has to interact with one or more charged
particles, its excitation energy will be converted into other forms of energy
which will (in the next stage) lead to a macroscopic signal (amplification),
and then a “click” (measurement). A simple example of this is a fiber optical
cable: here, a discrete fiber mode acts as intermediary between two continua.
More complicated forms of energy transport are also described by quantum
networks of coupled coupled discrete quantum states and structured continua
(e.g. band gaps) provide generic models for that first part of the detection
process, as we will discuss in this chapter. The input to the network is a
single continuum (the continuum of single-photon states), the output is again
a single continuum describing the next (irreversible) step. The process of a
single photon entering the network, its energy propagating through that
network, and finally exiting into another output continuum of modes can be
described by a single dimensionless complex transmission amplitude,
$T(\omega)$. Along with $T(\omega)$, we calculate a complex reflection
amplitude $R(\omega)$ which characterizes light reflected off of the
photodetector. In this chapter, we discuss how to obtain from $T(\omega)$ the
photo detection efficiency, how to find sets of parameters that maximize this
efficiency, as well as expressions for other input-independent quantities such
as the frequency-dependent group delay and spectral bandwidth of the
transmission portion of a single-photon detector. We then study a variety of
networks, discuss how to engineer different transmission functions $T(\omega)$
amenable to photo detection, and discuss implications for single-photon
detection technology.

The underlying basics of photodetection theory was developed in the early
1960s [130, 55, 47], with the quantum nature of light being taken into
account, and with later additions to the theory also incorporating the
backaction of the detector on the detected quantum field [48, 49, 50, 51, 52].
More recent additions to the theory have analyzed more deeply the
amplification process by itself [5] and its relation to the absorption and
transduction part of the process [26, 127]. In particular, it turns out that
for an ideal detector one should decouple the two processes, by having an
irreversible step in between the two, such that the amplification part does
not interfere negatively with the absorption (and possibly, transduction)
stage, here called transmission [26, 52]. This decoupling will be assumed in
this analysis, too.

In order to develop a useful fully quantum-mechanical theory we cannot be
completely general; or, rather, if we are completely general, then the only
statements on fundamental limits we can make are likely going to be merely
examples of Heisenberg’s uncertainty relations. So we will make three
restrictive but—we think—reasonable assumptions about our quantum theory of
photo detection.

Figure 3.1: An array of single-photon detecting (SPD) pixels. A series of
beamsplitters with low reflectivity ensure that at most one photon is incident
on each single-photon detector.

First, we focus on single-photon detection. The main reason is that number-
resolved photo detection is possible using arrays of SPDs where each “pixel”
receives at most one photon as in Fig. 3.1 (also see [131, 132], or [133] for
the time-reversed process of creating a single photon on demand). So we focus
on an individual pixel here. (See Ref. [134] for a modeling framework for
systems with multiple inputs and Refs. [135, 136, 137] for non-linear S-matrix
treatments of few-photon transport.)

Second, although a general state of a single photon is a function of four
quantum numbers, one related to the spectral degree of freedom, two related to
the two transverse spatial degrees of freedom, and one related to the
polarization or helicity degree of freedom, we will restrict ourselves to the
spectral (or, in the Fourier transformed-picture, the temporal) degree of
freedom. That is, the input state can be defined in terms of frequency-
dependent creation operators $\hat{a}^{\dagger}(\omega)$ acting on the vacuum.
The reason is that the other three degrees of freedom can, in principle, if
not in practice, be sorted before detection. For example, if one wishes to
distinguish between horizontally and vertically polarized photons, one may use
a polarizing beam splitter and put two detectors behind each of the two output
ports. Similarly, efficient sorting of photons by their orbital angular
momentum quantum number [104] and spatial mode are also possible [105, 106].
It is easier to consider sorting as part of the pre-detection process, rather
than a task for the detector itself 111To describe sorting as well within this
framework, we simply write more transmission functions, e.g. $T_{j}(\omega)$
for all the different input continua $j$, each leading to their own output
continuum. Even more complicatedly, we could consider multiple outputs $i$ for
a given input $j$ and write $T_{i|j}(\omega)$. But even in this case we can
focus on a particular $i$ and $j$ as we do this in this thesis.. On the other
hand, the spectral response of a detector cannot be eliminated; the time-
frequency degree of freedom is intrinsic to the resonance-structure of the
photo detecting device.

Third, we are going to assume that the transmission stage of each pixel’s
operation is passive. That is, apart from being turned on at some point, and
being turned off at some later point, it operates in a time-independent manner
with time-independent decay rates, couplings, and resonance structure. Thus,
an incoming photon will interact with a time-independent quantum system. As we
will see, active filtering is not needed for perfect detection provided the
photodetector has no internal losses (couplings to additional continua or side
channels). Furthermore, including a time-dependent amplification trigger
allows for arbitrary wavepacket detection, a result we will show in Chapter V.

We can now describe the interaction of a single photon with an arbitrary
quantum system as follows. The system may be naturally decomposed into
subsystems, each of which may have discrete and/or continuous energy
eigenstates. (For example, the photon may be absorbed by a molecule or atom or
quantum dot or any structure with a discrete transition that is almost
resonant with the incoming photon.) The continua will in general be structured
(for example, containing bands and band gaps in between) [138, 139, 140], but
structured continua can be equivalently described as structureless (flat)
continua coupled to (fictitious) discrete states [109, 110, 111, 89], enabling
a Markovian description of the system independent of an input photon’s
bandwidth. Indeed, it is well known that a non-Markovian open system can
always be made Markovian by expanding the Hilbert space (the converse of the
Stinespring dilation [141]). And so an arbitrary quantum system may be
described by a network of discrete states (some physical, some fictitious),
coupled to flat continua. The latter coupling makes the time evolution
irreversible. Of course, an actual detector is indeed irreversible. In
particular, the amplification process (converting the microscopic input signal
into a classical macroscopic output signal) is intrinsically irreversible.

Figure 3.2: Input and output fields coupled to a two-port black box quantum
network. The input and output operators are continuous mode (as functions of
frequency $\omega$) annihilation operators, satisfying canonical commutation
relations [4]. Here the mode $a_{{\rm in}}(\omega)$ carries the single-photon
input state, the reflected mode is described by $a_{{\rm out}}(\omega)$, and
the output mode $b_{{\rm out}}(\omega)$ contains the excitation energy if the
input energy successfully traversed the network (and may serve as input to the
next part of the photo detection process, e.g., amplification, see Ref. [5]).
$b_{{\rm in}}(\omega)$ may contain thermal excitations, but is never occupied
by the photon we wish to detect.

As we will show, this first part of the process can then be fully described in
terms of a complex transmission amplitude $T(\omega)$, which is the
probability amplitude for the component of the input signal at frequency
$\omega$ to enter the next (amplification) process. (By then the energy
$\hbar\omega$ will have been converted into a different type of energy, i.e.,
a different type of excitation, but that plays no particular role here.) It is
straightforward to calculate the transmission function through a network (the
equations are linear!). The point here is that from $T(\omega)$ we can
determine three input-independent quantities of interest to the photo
detection process. (That one complex function $T(\omega)$ is sufficient is due
to our considering only the time/frequency degree of freedom.)

This chapter is organized as follows. After defining our quantities of
interest in section II (short), we start with analyzing the simplest quantum
network to illustrate how we calculate $T(\omega)$ and how our three
quantities of interest behave in section III (also short). We then present a
systematic survey of more complicated networks in the much longer section IV;
for each network class, we analytically calculate $T(\omega)$ and discuss the
behavior of the three quantities of interest. Then in section V, we briefly
discuss extensions to arbitrary quantum networks including the effects of
couplings to additional continua/side channels before summarizing our findings
in the conclusions.

###  3.1 Quantities of Interest

Here, we define three quantities of interest derivable from $T(\omega)$, and
discuss their applicability to photo detecting systems.

First, $|T(\omega)|^{2}$ itself gives an upper bound on the probability for
the frequency component $\omega$ to be detected. (If there were no losses
downstream in the photodetection process, it would equal the probability of
detection of monochromatic light at that frequency.) We are thus particularly
interested in identifying quantum systems for which there is at least one
frequency $\omega_{i}$ for which $|T(\omega_{i})|=1$.

Second, an upper bound to the total detectable frequency range of input light
is given by the spectral bandwidth of the quantum network (not to be confused
with the channel bandwidth in [98] or the range of a single frequency band
[142]). We define this quantity

$\tilde{\Gamma}=\frac{1}{\pi}\int_{0}^{\infty}d\omega|T(\omega)|^{2}.$ (3.1)

(The $\frac{1}{\pi}$ factor gives agreement with a classical Lorentzian
filter; the transmission function is
$T(\omega)=\frac{\Gamma}{\Gamma-i(\omega-\omega_{0})}$ for a filter with
damping factor $\Gamma$ and resonant frequency $\omega_{0}$ and we find
$\tilde{\Gamma}=\Gamma$, see Eq. (3.14) below.) The inverse of this quantity
is also a measure of the time the photon spends in the detector. (Indeed,
$\tilde{\Gamma}^{-1}$ is a lower bound on timing jitter from integrated
detection event when quasi-monochromatic photon states are detected with high
efficiency, see footnote222Timing jitter for a photo detection has three
contributing components [98]. The first comes from the temporal spread of the
mode onto which the measurement projects [143] and is intrinsic to the
resonance structure of the photodetector. The second comes from integrating a
continuously monitored continua to form discrete detection event, which is
necessary for an accurate information-theoretic characterization of photo
detection and influences photo detection efficiency. It is this part of the
jitter that $\tilde{\Gamma}^{-1}$ is a lower bound for; it sets the timescale
for integration where monochromatic states are detected with high efficiency.
The third contribution comes from input signals with long temporal wave-
packets and, since it is input-dependent, will be ignored in this analysis
since here we assume no priors about the single-photon input..)

Third, we can also define a delay (latency) using the polar decomposition of
$T(\omega)$

$T(\omega)=|T(\omega)|\exp(i\phi(\omega))$ (3.2)

and using the standard definition of group delay as

$\tau_{g}(\omega)=-\frac{d\phi(\omega)}{d\omega}.$ (3.3)

We can see how this group delay directly relates to an experimentally-measured
latency by considering a temporal wavepacket $\psi_{\rm in}(t)$ and Fourier
transform $\tilde{\psi}_{\rm in}(\omega)$. We can then write the output
single-photon state

$\displaystyle\left|\psi^{\prime}\right\rangle$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}d\omega\tilde{\psi}_{\rm
in}(\omega)R(\omega)\hat{a}^{\dagger}_{\rm out}(\omega)|{\rm vac}\rangle$
(3.4) $\displaystyle+$ $\displaystyle\int_{0}^{\infty}d\omega\tilde{\psi}_{\rm
in}(\omega)T(\omega)\hat{b}^{\dagger}_{\rm out}(\omega)|{\rm vac}\rangle$

where the first and second terms correspond to the reflected and transmitted
parts of the single-photon state, respectively (for details, see Ref. [143])
and we have introduced a reflection coefficient $R(\omega)$ satisfying
$|T(\omega)|^{2}+|R(\omega)|^{2}=1$ and
$R^{*}(\omega)T(\omega)+R(\omega)T^{*}(\omega)=0$ at every frequency $\omega$.
From (3.4) we note that, after interacting with the network, the transmitted
wavepacket will have the form $\psi_{\rm out,T}(t)=FT^{-1}[\tilde{\psi}_{\rm
in}(\omega)T(\omega)]$. (Note that the wavepacket described by $\psi_{\rm
out,T}(t)$ will be sub-normalized by construction; it only corresponds to the
portion of the full single photon state $\psi^{\prime}(t)\equiv\left\langle
t|\psi^{\prime}\right\rangle$ that is transmitted!) For a long input pulse
with central frequency $\omega^{\prime}$, we find $\psi_{\rm
out,T}(t)\approx|T(\omega^{\prime})|\psi_{\rm
in}(t-\tau)e^{i\omega^{\prime}\tau^{\prime}}$ with $\tau^{\prime}$ the
difference between the group delay $\tau_{g}(\omega^{\prime})$ defined above
and the (here irrelevant) phase delay.

The effect of the group delay on an arbitrary input photon state is to
selectively delay and reshape the transmitted wavepacket. Of course, if an
input photon has a wide spread of frequencies, a differential group delay
$\tau_{g}(\omega)=-\frac{d\phi(\omega)}{d\omega}$ may increase (or decrease)
the temporal spread of the wavepacket, as it will affect each frequency
differently. This increase (or decrease) in the arrival times of different
frequencies is manifestly input-dependent and will not play a role in the
input-independent temporal uncertainty or jitter. We can, however, define an
additional quantity characterizing the input-independent group delay-induced
dispersion

$\mathcal{T}_{g}=\int_{0}^{\infty}d\omega\left|\frac{d\tau_{g}(\omega)}{d\omega}\right||T(\omega)|^{2}$
(3.5)

whose definition agrees with our physical intuition that a constant group
delay over the transmission window will not contribute to dispersion, nor will
frequencies that are not transmitted regardless of how large
$\frac{d\tau_{g}(\omega)}{d\omega}$ may be. We find that, for a flat
transmission function that is unity over some spectral range
$\omega_{0}\pm\delta\omega$ and zero everywhere else and a monotonic group
delay $\tau_{g}(\omega)$, (3.5) gives the difference in group delay between
the minimum and maximumly transmitted frequencies:
$\mathcal{T}_{g}=|\tau_{g}(\omega_{0}-\delta\omega)-\tau_{g}(\omega_{0}+\delta\omega)|$.
This is the maximum dispersion possible for any input to this system.

Finding key conditions that change the transmission function $T(\omega)$ and
frequency-dependent group delay $\tau_{g}(\omega)$ are important for the
design of coupled-resonator optical waveguide (CROW) networks [144] for delay-
lines [145] and spectral filtering [146], where the transmission efficiency,
frequency-dependent group delay, and spectral bandwidth will all affect
performance. These are the three quantities we focus on in the rest of the
chapter.

Knowing $T(\omega)$ for a specific photodetector also allows one to construct
a simplified positive-operator valued measure (POVM) by assuming any
excitation in the output continuum results in a click (for details, see the
last section of this chapter). The simple POVM element corresponding to a
click after the photodetector has been left on for a very long time (in
particular, long compared to the bandwidth $\tau\gg\tilde{\Gamma}^{-1}$) has
the particularly simple form

$\displaystyle\hat{\Pi}=\int_{0}^{\infty}\,d\omega\,|T(\omega)|^{2}\left|\omega\right\rangle\left\langle\omega\right|.$
(3.6)

$\hat{\Pi}$ is defined such that the probability of a photon in an arbitrary
input state $\hat{\rho}$ being detected is given by the Born rule
$\textnormal{Pr}=\textnormal{Tr}\left(\hat{\Pi}\hat{\rho}\right)$. For
example, any photon state
$\hat{\rho}=\sum_{i}\lambda_{i}\,\left|\omega_{i}^{\prime}\right\rangle\left\langle\omega_{i}^{\prime}\right|$
where $|T(\omega_{i}^{\prime})|^{2}=1$ and $\sum_{i}\lambda_{i}=1$ will be
detected with unit probability. (The states the photodetector can detect
perfectly include both pure states [when only one $\lambda_{i}$ is non-zero]
and mixed states comprised entirely of frequencies where
$|T(\omega_{i}^{\prime})|^{2}=1$.) Of course, no photon is truly monochromatic
(or discretely polychromatic), but it can be effectively so if the wave-packet
envelope is long compared to the inverse spectral bandwidth
$\tilde{\Gamma}^{-1}$.

###  3.2 Simple example

Figure 3.3: A simple network comprised of a single discrete state described by
an operator $c$ (the bosonic replacement for the two-level system fermionic
lowering operator $\sigma^{-}$) and decaying to left (input, external) and
right (output, internal) continua $a$ and $b$ at rates $\gamma$ and $\Gamma$,
respectively. That is, here the black box of Fig. 3.2 contains just one two-
level system. This exactly describes filtering of light by a Fabry-Perot
cavity.

The simplest quantum network consists of a single two-level system with a
ground state $\left|g\right\rangle$ and an excited state
$\left|e\right\rangle$, with the incoming photon coupling these states (Fig.
3.3). The two-level system is described by fermionic raising and lowering
operators $\sigma^{+}=\left|e\right\rangle\left\langle g\right|$ and
$\sigma^{-}=\left|g\right\rangle\left\langle e\right|$ (this is also the
simplest model of a photodetector, see [130]). Physically, this excited state
could be any discrete state of a single absorber, e.g. the p-state of an atom
(that is, an atomic excited state with exactly one quanta of angular momentum)
which then decays to a monitored flat continuum. (Indeed, single absorbers
such as atoms [147, 148, 149, 150, 151, 77], single molecules [152], NV-
centers [153], and quantum emitters [154, 155] are able to efficiently couple
to and absorb single photons.)

In the Heisenberg picture, the evolution of these raising and lowering
operators $\sigma^{+}$ and $\sigma^{-}$ will determine whether a photon makes
it from one side of the network to the other. By focusing our analysis on
cases where at most a single photon is in the network, we can use the
equivalence between the two-level system and the simple harmonic oscillator to
simplify our problem ab initio: we replace the fermionic raising and lowering
operators with bosonic creation and annihilation operators $c^{\dagger}$ and
$c$333Since both operators and their expectation values (mode amplitudes) will
satisfy the same systems of equations, we will omit hats throughout this
chapter..

We follow standard input-output theory here [4], separating the full evolution
of both continua $a$ and $b$ into input and output modes (Fig. 3.2). By
formally solving the Heisenberg evolution equations for the two input
continuum mode annihilation operators $b_{{\rm in}}$ and $a_{{\rm in}}$, we
can write the effective system Hamiltonian that governs the evolution of the
system operators $c$ and $c^{\dagger}$

(3.7) $\displaystyle H$
$\displaystyle=\hbar\omega_{0}c^{\dagger}c-\sqrt{\gamma}(c^{\dagger}a_{{\rm
in}}+ca^{\dagger}_{{\rm in}})-\sqrt{\Gamma}(c^{\dagger}b_{{\rm
in}}+cb^{\dagger}_{{\rm in}})$

where we have identified $\omega_{0}$ as the resonance frequency, and $\gamma$
and $\Gamma$ as the left and right side couplings to two continua $a$ and $b$
respectively444In assuming these couplings to be frequency independent, we are
invoking a modified version of the first Markov approximation. Formally, we
define ${\gamma_{i}=2\pi\,\kappa_{i}^{2}(\omega_{i})}$ where $\omega_{i}$ is
the resonance frequency of the $i$th discrete state and $\kappa_{i}(\omega)$
is the coupling between the $i$th discrete state and the left continuum at the
discrete state frequency. This implies that, in an experiment, the decays
$\gamma_{i}$ and resonances $\omega_{i}$ cannot be varied independently, which
is well known in the context of the Thomas-Reiche-Kuhn sum rule for electric
dipole transitions [156].. In the Heisenberg picture, the time evolution of
the discrete state annihilation operator is given by

$\displaystyle\dot{c}=-i\omega_{0}c-\frac{\gamma+\Gamma}{2}c-\sqrt{\gamma}a_{{\rm
in}}-\sqrt{\Gamma}b_{{\rm in}}.$ (3.8)

The input mode operators $a_{{\rm in}}$ and $b_{{\rm in}}$ and output mode
operators $a_{{\rm out}}$ and $b_{{\rm out}}$ are determined by open quantum
system evolution of the discrete state operator $c$ in (3.8) and the two
boundary conditions

$\displaystyle a_{{\rm out}}-a_{{\rm in}}=-\sqrt{\gamma}c$ $\displaystyle
b_{{\rm out}}-b_{{\rm in}}=-\sqrt{\Gamma}c.$ (3.9)

It is easiest to solve the equations by taking the Fourier transform.
Unitarity implies the existence of a transfer matrix relating in and out
fields in the spectral domain

$\displaystyle\begin{bmatrix}a_{{\rm out}}(\omega)\\\ b_{{\rm out}}(\omega)\\\
\end{bmatrix}=\begin{bmatrix}R(\omega)&T(\omega)\\\ T(\omega)&R(\omega)\\\
\end{bmatrix}\,\begin{bmatrix}a_{{\rm in}}(\omega)\\\ b_{{\rm in}}(\omega)\\\
\end{bmatrix}$ (3.10)

where $|T(\omega)|^{2}+|R(\omega)|^{2}=1$ (resulting from our assumption there
are no internal losses) [157]. Defining a detuning $\Delta=\omega-\omega_{0}$,
we can easily solve (3.8) in terms of the Fourier transform of the discrete
state annihilation operator

$\displaystyle c(\omega)=\frac{-\sqrt{\gamma}a_{{\rm
in}}(\omega)-\sqrt{\Gamma}b_{{\rm
in}}(\omega)}{\frac{\gamma+\Gamma}{2}-i\Delta}$ (3.11)

yielding a transmission function

$\displaystyle
T(\omega)=\frac{\sqrt{\gamma\Gamma}}{\frac{\gamma+\Gamma}{2}-i\Delta}.$ (3.12)

We can see from (3.13) that perfect transmission ($|T(\omega)|^{2}=1$) occurs
only when $\gamma=\Gamma$ and $\Delta=0$. These are the well-known conditions
of balanced mirrors and on-resonance required for perfect transmission through
a Fabry-Perot cavity [124].

We can also calculate the frequency dependent group delay

$\displaystyle\tau_{g}(\omega)=\frac{\frac{\Gamma+\gamma}{2}}{\left(\frac{\Gamma+\gamma}{2}\right)^{2}+\Delta^{2}}.$
(3.13)

Like $T(\omega)$, the group delay is also a Lorentzian with width
$\frac{\Gamma+\gamma}{2}$, and that frequencies close to resonance spend the
most time in the network with a maximum group delay of
$\frac{2}{\Gamma+\gamma}$ on resonance. We similarly find the input-
independent group delay-induced dispersion (3.5) to be
$\mathcal{T}_{g}=\frac{8\gamma\Gamma}{(\gamma+\Gamma)^{3}}$.

We can also use $T(\omega)$ to calculate a spectral bandwidth (not to be
confused with the channel bandwidth discussed in [98])

$\displaystyle\tilde{\Gamma}$
$\displaystyle=\frac{1}{\pi}\int_{0}^{\infty}\,d\omega\,|T(\omega)|^{2}$
(3.14) $\displaystyle=\frac{2\Gamma\gamma}{\Gamma+\gamma}$

which is a measure of the number of frequencies that can be efficiently
detected. For this simple case, we note that
$\tau_{g}(\omega)=\tilde{\Gamma}^{-1}|T(\omega)|^{2}$ and thus
$\int\,d\omega\tau_{g}(\omega)=\pi$. While this expression will not be true
for a general network, the shift in phase by $\pi$ as a resonance frequency is
crossed is a universal feature of networks, as we shall see shortly

###  3.3 Quantum Networks

We now set up the general problem of an arbitrary network of discrete states
connecting two continua. The Hamiltonian is a straightforward generalization
of (3.7)

$\displaystyle\hfill H=\sum_{i}\hbar\omega_{i}c^{\dagger}_{i}c_{i}$
$\displaystyle-$
$\displaystyle\sum_{ij}g_{ij}(c_{i}c^{\dagger}_{j}+c^{\dagger}_{i}c_{j})\hfill\phantom{211}$
(3.15) $\displaystyle-\sum_{i}\sqrt{\gamma_{i}}(c^{\dagger}_{i}a_{{\rm
in}}+c_{i}a^{\dagger}_{{\rm in}})$ $\displaystyle-$
$\displaystyle\sum_{i}\sqrt{\Gamma_{i}}(c^{\dagger}_{i}b_{{\rm
in}}+c_{i}b^{\dagger}_{{\rm in}})$

where we’ve now defined a real coherent coupling between discrete states
$g_{ij}$ (we define $g_{ii}=0$ for each state). Some states may not be coupled
to one (or both) continuum, in which case either $\sqrt{\gamma_{i}}$ or
$\sqrt{\Gamma_{i}}$ (or both) will be zero.

We can similarly generalize the operator evolution in (3.8) for an arbitrary
network; moving to the spectral domain, we write the spectral dependence of
the discrete state operators

$\displaystyle-i\Delta_{i}c_{i}(\omega)=$
$\displaystyle-\sum\limits_{j}\,\left(\frac{\sqrt{\gamma_{i}\,\gamma_{j}}+\sqrt{\Gamma_{i}\,\Gamma_{j}}}{2}+ig_{i\,j}\right)c_{j}(\omega)-\sqrt{\gamma_{i}}\,a_{\rm
in}(\omega)-\sqrt{\Gamma_{i}}\,b_{\rm in}(\omega).$ (3.16)

Similarly to (3.2), we can write boundary conditions for the two continua with
an arbitrary network

$\displaystyle a_{{\rm out}}(\omega)-a_{{\rm
in}}(\omega)=-\sum\limits_{i}\sqrt{\gamma_{i}}c_{i}(\omega)$ $\displaystyle
b_{{\rm out}}(\omega)-b_{{\rm
in}}(\omega)=-\sum\limits_{i}\sqrt{\Gamma_{i}}c_{i}(\omega).$ (3.17)

To move from (3.3) to the transfer matrix (4.28) involves solving $N$ systems
of $N$ coupled first-order differential equations555Luckily, there is some
redundancy. Once one has solved for one $c_{i}$ in terms of the input and
output fields, one can permute the labels to generate the remaining
solutions., as each discrete state amplitude $c_{i}(\omega)$ depends on every
other amplitude. Three methods for solving these equations are as follows.

The first is to take the weak-coupling limit
$2\,g_{ij}\ll\sqrt{\gamma_{i}\,\gamma_{j}}+\sqrt{\Gamma_{i}\,\Gamma_{j}}\,\forall
i,j$ and truncate the solutions after some power in $g_{ij}$, making (3.19)
the $0$th order approximate solution with higher order corrections. However,
this method fails if even a single discrete state decouples from both
continua, as is the case for many networks of interest.

The second is to use numerical techniques to diagonalize the systems of
equations and find the transmission function numerically [158]. However, this
rapidly gets harder with large systems, and masks the analytic conditions for
perfect transmission we are interested in identifying.

The third method is to use a variety of mathematical techniques to solve these
$N$ systems of $N$ coupled first-order differential equations analytically,
which is the route taken in this analysis. These tricks include making use of
correlations between discrete states, permuting labels, and using generalized
continued fraction formulae. Here, we will use these techniques to describe
three large classes of systems that can be solved exactly with arbitrary
couplings, decays, and resonant frequencies. These are parallel networks,
series networks, and hybrid networks.

Parallel: each discrete state is directly coupled to both continua
($\gamma_{i}\neq 0$ and $\Gamma_{i}\neq 0\,\forall i$) but not to each other.
Thus there are multiple parallel paths to the same final state and hence we’ll
get interference. Physical photo detection platforms described by parallel
networks include (i) single atoms with multiple p-states that then decay
directly to a continuum (similarly for trapped ions/atoms due to Stark and
Zeeman effects, see Ref. [159] for this in generating single photons), (ii)
quantum dots [160], (iii) structured continua with multiple pseudomodes [161].

Series: each of the two continua are coupled to their own discrete state,
which are in turn coherently coupled by a chain of intermediate single
discrete states. There is just one path from the input continuum to the output
continuum. This model describes (i) an atom in a p-state that first decays to
a d-state or metastable s-state before decaying to a flat continuum (and
analogously for a molecule [26]), (ii) coherently coupled frequency filtering
in front of a photo detecting platform (for instance, using an antireflective
coating [20] or optical cavities [162]), (iii) atomic chains coupled through
their p-states (which could be mediated by, for example, a fiber mode [163]).

Hybrid: a combination of the above cases. For example, two parallel paths of
three steps each or, more generally, layered structured continua (a photon
must pass through one before the other). A physical system that can be modeled
with a hybrid network is a photosynthetic light-harvesting (i.e.
Fenna–Matthews–Olson) complex with multiple paths for coherent transport [164,
165, 166].

In all cases, we will solve the systems of equations for $R(\omega)$ directly
and make use of the identities $|T(\omega)|^{2}=1-|R(\omega)|^{2}$ and
$T^{2}(\omega)=-R^{2}(\omega)\frac{|T(\omega)|^{2}}{|R(\omega)|^{2}}$when we
calculate the transmission efficiency, spectral bandwidth, and group delay.

####  3.3.1 Parallel Networks

Figure 3.4: A parallel network of ${N=5}$ decoupled discrete states, each
described by an operator $c_{i}$ and decaying to left (input, external) and
right (output, internal) continua $a$ and $b$ at rates $\gamma_{i}$ and
$\Gamma_{i}$, respectively.

We now consider a parallel quantum network where each discrete state is
directly coupled to both continua (Fig. 3.4). If the discrete states are
directly coupled to each other, it is helpful to first diagonalize the system
in the basis where they are decoupled. We then consider the new modes to be
the discrete states a photon can occupy, each with a new resonance frequency
and two decay rates to the two continua. (In the strong coupling limit, Rabi
splitting comes into effect and these dressed states are the more physical
description anyways [45].) This decoupled manifold of discrete states, each
decaying at a rate $\gamma_{i}$ to the input continuum and a rate $\Gamma_{i}$
to the output continuum (Fig. 3.4), is the next simplest case to analyze, and
provides a simple model of a photodetector with a simple band structure.

We find the frequency dependence of the discrete state operators

$\displaystyle-i\Delta_{i}c_{i}(\omega)=$
$\displaystyle-\sum\limits_{j}\,\frac{\sqrt{\gamma_{i}\,\gamma_{j}}+\sqrt{\Gamma_{i}\,\Gamma_{j}}}{2}c_{j}(\omega)$
(3.18) $\displaystyle-\sqrt{\gamma_{i}}\,a_{\rm
in}(\omega)-\sqrt{\Gamma_{i}}\,b_{\rm in}(\omega).$

A salient feature of Eq. 3.18 is the decay terms produce purely virtual
coupling between discrete states coupled to the same continua. These _cannot_
be thought of as coupling mediated by continua, as the continua are flat
(Markovian) and perfectly dissipative. If an excitation exits the discrete
states and enters a continuum, there is no possibility of its ever returning
to another discrete; energy transfer from one discrete state is impossible!
Thus, the coupling is purely virtual, and we must instead consider it to be a
purely information-theoretic phenomena666Interestingly, this coupling does
persist for higher numbers of excitations, an effect we intend to explore in
future work..

We use (4.28) to find $R(\omega)$ by considering an input on only one side of
the network (thus setting the expectation value of the other input field
operator to zero). This yields an expression
$a_{out}(\omega)=R(\omega)a_{in}(\omega)$ (or
$b_{out}(\omega)=R(\omega)b_{in}(\omega)$), from which we reconstruct
$T(\omega)$. We analytically find the general form of $R(\omega)$

$\displaystyle
R(\omega)=\frac{\prod\limits_{i}\left(\frac{\Gamma_{i}-\gamma_{i}}{2}-i\Delta_{i}\right)-X_{-}^{(N)}}{\prod\limits_{i}\left(\frac{\Gamma_{i}+\gamma_{i}}{2}-i\Delta_{i}\right)-X_{+}^{(N)}}$
(3.19)

where $X_{\pm}^{(N)}$ is a polynomial of order $N-2$ in the detunings
$\Delta_{i}$. (We find $X_{\pm}^{(1)}=0$.) For $N=2$, we can see that
$X_{\pm}^{(2)}$ is symmetric (anti-symmetric) between $\Gamma_{i}$ and
$\gamma_{i}$

$\displaystyle
X_{\pm}^{(2)}=\left(\frac{\sqrt{\Gamma_{1}\Gamma_{2}}\pm\sqrt{\gamma_{1}\gamma_{2}}}{2}\right)^{2}.$
(3.20)

This (anti-)symmetry is also present in higher-$N$ coefficients. From (3.19)
we can determine a key feature of parallel quantum networks: if some subset of
the discrete states have _balanced_ decay rates such that
$\gamma_{i}=\Gamma_{i}$, for large spacings between discrete states compared
to the other decay rates $|\omega_{i}-\omega_{j}|\gg\gamma_{j},\Gamma_{j}$, we
find $R(\omega_{i})=0$; input monochromatic photons with frequencies on
resonance with those discrete states are transmitted perfectly through the
network777For ${N=2}$, the requirement for large spacing such that
${R(\omega_{i})=0}$ is less stringent, we only need it much larger than
difference in decay rates
${|\omega_{i}-\omega_{j}|\gg(\sqrt{\Gamma_{j}}-\sqrt{\gamma_{j}})^{2}}$..

In general, finding the specific form of $X_{\pm}^{N}$ is a numerical task,
and we will focus on a simpler case where we can utilize another salient
feature of parallel networks: the purely virtual coupling present in (3.18).
Before we consider a network that is uniformly coupled (all decays are the
same), we can consider a network with couplings that are inhomogeneous but
uniformly unbalanced such that $\Gamma_{i}=k\,\gamma_{i}\,\forall i$. We can
then write (3.18) in a simplified form

Figure 3.5: Transmission probability for parallel networks with ${N=4}$
equally spaced discrete states, each with a decay rate to the input continuum
${\gamma_{i}=\gamma_{1}\,(\frac{7}{5})^{i-1}}$. The decays to the monitored
output continuum are uniformly unbalanced
(${\frac{\Gamma_{i}}{\gamma_{i}}=k\,\forall i}$), with ${k=\frac{1}{2}}$ for
the blue (lower) curve and ${k=1}$ for the red (upper) curve. Frequency is
measured w.r.t. the average resonance frequency. The four resonance
frequencies $\omega_{i}$ have maximum transmission probability
${|T(\omega_{i})|^{2}=\frac{4k}{(k+1)^{2}}}$. The three frequencies of perfect
reflection (i.e, ${T(\omega_{i})=0}$) correspond to solutions of
${\sum_{i}\frac{\gamma_{i}}{\Delta_{i}}=0}$. Figure 3.6: Transmission
probability for a parallel network with ${N=5}$ equally spaced discrete states
with balanced decay rates to both continua (${\gamma=\Gamma}$). Frequency is
measured w.r.t. the average resonance frequency. We observe the five resonance
frequencies $\omega_{i}$ each correspond to a perfectly transmitted frequency.
The four frequencies of perfect refection correspond to solutions of
${\sum_{i}\frac{1}{\Delta_{i}}=0}$.  Figure 3.7: Group delay (in units of
$\frac{1}{\gamma}$) for a parallel network with ${N=5}$ equally spaced
discrete states with balanced decay rates to both continua
(${\gamma=\Gamma}$). The group delay is always largest for the highest and
lowest frequency resonances (except in the large spacing limit, where they are
of equal magnitude). Figure 3.8: Transmission function phase (in units of
$2\pi$) for a parallel network with ${N=5}$ equally spaced discrete states
with balanced decay rates to both continua (${\gamma=\Gamma}$). The total
change in phase of the transmission function is directly proportional to the
number of discrete states. This is due to the phase shift in $\pi$ that occurs
when crossing a resonance frequency, as was discussed for the simple example
previously.

$\displaystyle\frac{i\Delta_{i}}{\sqrt{\gamma_{i}}}c_{i}(\omega)$
$\displaystyle=\sum\limits_{j}\frac{\sqrt{\gamma_{j}}(1+k)}{2}c_{j}(\omega)+a_{\rm
in}(\omega)+\sqrt{k}\,b_{\rm in}(\omega)$

so that we can identify strong correlations between discrete state amplitudes

$\displaystyle\frac{i\Delta_{i}}{\sqrt{\gamma_{i}}}c_{i}(\omega)=\frac{i\Delta_{i^{\prime}}}{\sqrt{\gamma_{i^{\prime}}}}c_{i^{\prime}}(\omega).$
(3.22)

For non-degenerate states, Eq. 3.22 means that a photon that is resonant with
one state will only excite that state. (For infinitely-narrow discrete states,
these correlations are satisfied trivially of course; $c_{i^{\prime}}(\omega)$
is only non-zero at $\omega_{i^{\prime}}$.) Eq. 3.22 also indicates that the
relative phase of discrete state amplitudes is frequency dependent; when
$\omega_{i}<\omega<\omega_{i^{\prime}}$, there is a relative phase of $\pi$
between $c_{i}$ and $c_{i^{\prime}}$. (Outside of photo detection, this purely
virtual coupling provides a possible alternative explanation for the
destructive interference present in atoms along a fiber [167, 168] and multi-
mode Fabry-Perot cavities [169].) We can see destructive interference directly
from the form of the reflection coefficient

$\displaystyle
R(\omega)=\frac{i-(k-1)\sum\limits_{i}\frac{\gamma_{i}}{2(\Delta_{i})}}{i-(k+1)\sum\limits_{i}\frac{\gamma_{i}}{2(\Delta_{i})}}$
(3.23)

from which we can determine $|T(\omega)|^{2}$ and the other quantities of
interest.

We can further specialize to the case of homogenous coupling where
$\gamma_{i}=\gamma$ and $\Gamma_{i}=\Gamma$. This case is of interest for
several reasons: most generally, this assumption directly follows from the
first Markov approximation if the spacing between discrete states is small
compared to the decays. It also simplifies the form of the correlations
between discrete states
$\Delta_{i}c_{i}(\omega)=\Delta_{i^{\prime}}c_{i^{\prime}}(\omega)$888We now
can perform a small sanity check. Considering the two continuum fields
together, we define total input and output flux operators
${J_{in}=\sqrt{\gamma}a_{in}+\sqrt{\Gamma}b_{in}}$ and
${J_{out}=\sqrt{\gamma}a_{out}+\sqrt{\Gamma}b_{out}}$. Solving for the total
outputs directly from (3.3), we find
${J_{out}=\frac{i+\frac{\gamma+\Gamma}{2}\,\sum_{i}\frac{1}{\Delta_{i}}}{i-\frac{\gamma+\Gamma}{2}\,\sum_{i}\frac{1}{\Delta_{i}}}\,J_{in}}$.
We can immediately see that ${|J_{out}|=|J_{in}|}$; photon flux is preserved
through the system at every frequency, as it must be since there are no side
channels present., as well as the form of the reflection coefficient

$\displaystyle
R(\omega)=\frac{i-\frac{\Gamma-\gamma}{2}\sum\limits_{i}\frac{1}{\Delta_{i}}}{i-\frac{\Gamma+\gamma}{2}\sum\limits_{i}\frac{1}{\Delta_{i}}}.$
(3.24)

_Perfect transmission.—_ From (3.23) and (3.24) we can see that in both cases
there are $N$ frequencies of maximum transmission corresponding to each
resonant frequency $\Delta_{i}=0$ with transmission probability
$|T(\omega_{i})|^{2}=\frac{4k}{(k+1)^{2}}$ and
$|T(\omega_{i})|^{2}=\frac{4\gamma\Gamma}{(\gamma+\Gamma)^{2}}$, respectively.
We also see $N-1$ frequencies of destructive interference corresponding to the
$N-1$ solutions of $\sum\limits_{i}\frac{\gamma_{i}}{\Delta_{i}}=0$ (Fig. 3.5)
determined solely by the decays and resonances (and notably not by $k$). We
similarly observe for the case of a network with homogenous decay rates $N-1$
frequencies of destructive interference corresponding to the solutions of
$\sum\limits_{i}\frac{1}{\Delta_{i}}=0$ (Fig. 3.6). One might think that, in
principle, a resonant frequency could coincide with a frequency of perfect
reflection when $N>2$, in which case they can annihilate. However, this only
occurs when two discrete states are energetically degenerate. Since the
discrete states would also couple to the same $1$D continuum, this is
forbidden by unitarity. To avoid this, we see a resonant frequency that is
perfectly transmitted and a frequency of destructive interference move closer
together as the spacing between discrete states is decreased. When the
degeneracy becomes exact, a discrete state is forced to decouple from the
system as the frequencies of perfect reflection and transmission annihilate.

In general, the condition for perfect transmission through a parallel network
is that all the couplings be balanced ($\gamma=\Gamma$ or in the inhomogeneous
uniformly balanced case, $k=1$). As we saw in the case for a completely
arbitrary parallel network, we see that perfect transmission at some discrete
state frequency $\omega_{i}$ not only requires balanced coupling
$\gamma_{i}=\Gamma_{i}$, but also that all the other discrete states either be
far away in frequency compared to the their decay rates
($|\omega_{i}-\omega_{j}|\gg\gamma_{j},\Gamma_{j}$), or also be balanced
($\gamma_{j}=\Gamma_{j}$), or a mix of the two.

_Spectral Bandwidth.—_ Once we have the reflection coefficient, we can
calculate the three quantities of interest. For all three cases we’ve
discussed, we find that the spectral bandwidth is purely additive
$\tilde{\Gamma}=\sum_{i}\frac{2\gamma_{i}\Gamma_{i}}{\gamma_{i}+\Gamma_{i}}$
and is completely independent of the spacing between discrete states999The
uncertainty in frequency, as defined in [98] and calculated entropically from
the spectral POVM [143], is in this case directly proportional to the
bandwidth (and hence also independent of discrete state spacing)..

_Group Delay.—_ We note that, unlike the simple model, the sharp peaks in the
group delay (Fig. 3.7) correspond to frequencies of destructive interference
and are greatest for the outermost frequencies of destructive interference
despite all the frequencies of note in (Fig. 3.6) being completely destructive
or constructive. We also observe that the relationship between the three
quantities of interest discussed for the simple model is not present: here
$\tau_{g}(\omega_{i})>\tilde{\Gamma}^{-1}|T(\omega_{i})|^{2}$ for each
resonant frequency. We find that the phase of $T(\omega)$ increases by $\pi$
with each resonant frequency (3.8). This provides a novel application for
single-photon interferometry for resolving tightly-structured resonance
structures and explains why, whereas the spectral bandwidth $\tilde{\Gamma}$
is independent of discrete state spacing, the group delay increases with close
spacing; the same change in phase is occurring in a smaller spectral range so
the magnitude of the group delay
$|\tau_{g}(\omega)|=|\frac{d\phi(\omega)}{d\omega}|$ increases.

####  3.3.2 Series Networks

Figure 3.9: A series network of three coherently coupled discrete states, each
described by an operator $c_{i}$ and coupled to each other at rates $g_{ij}$.
The first and last states decay to left (input, external) and right (output,
internal) continua $a$ and $b$ at rates $\gamma$ and $\Gamma$, respectively.
Figure 3.10: Transmission probability for a quantum network with ${N=5}$
discrete states in series with no relative detuning and balanced decay rates
${\gamma=\Gamma}$ for the special case of homogenous critical coupling
${g_{ij}\rightarrow g=\frac{\sqrt{\gamma\,\Gamma}}{2}}$. For comparison with
the actual transmission probability (blue curve), we also plot a fictitious
transmission probability (red bounding box) that is unity for
${-2g\leq\omega\leq 2g}$ and zero elsewhere. Frequencies are measured w.r.t.
resonance. Meeting both the balanced decay and critical coupling conditions
ensures that the on-resonance transmission is both unity and maximally
broadened, but are not necessary conditions for perfect transmission to occur
at some frequency. Figure 3.11: Transmission probability for a quantum network
with ${N=5}$ discrete states in series with no relative detuning and
unbalanced decay rates ${2\gamma=\Gamma}$ for the special case of homogenous
critical coupling ${g_{ij}\rightarrow g=\frac{\sqrt{\gamma\,\Gamma}}{2}}$.
Again, we observe perfect transmission for certain frequencies. Figure 3.12:
Transmission probability for a quantum network with ${N=4}$ discrete states in
series with no relative detuning and unbalanced decay rates
${2\gamma=\Gamma}$, again for the special case of homogenous critical coupling
${g_{ij}\rightarrow g=\frac{\sqrt{\gamma\,\Gamma}}{2}}$ ensuring perfect
transmission at some frequencies Figure 3.13: Transmission probability for a
quantum network with ${N=4}$ discrete states in series with no relative
detuning and balanced decay rates ${\gamma=\Gamma}$. Here critical coupling
${g_{ij}\rightarrow g=\frac{\sqrt{\gamma\,\Gamma}}{2}}$ ensures both perfect
transmission and maximal broadening on-resonance.

In general, we cannot diagonalize a completely arbitrary network in terms of a
single set of parallel states; there may be a causal relationship embedded in
the network structure; for example, for an atom in an s-state coupled via a
photon to one or many p-states which subsequently decay to multiple d-states,
we would have to diagonalize the p-states and d-states separately. A class of
such systems are series quantum networks (Fig. 3.9), where only one state is
coupled to each continuum, with the other states forming a chain between the
two outer ones ($\gamma_{i>1}=0$, $\Gamma_{i<N}=0$, and $g_{ij}=0$ for $j\neq
i\pm 1$). This provides a natural (and especially simple) model for energy
transport and repeated spectral filtering (a series of Fabry-Perot cavities).
Furthermore, we can analytically determine the transmission function for
arbitrary series networks, as we will now proceed to do.

Figure 3.14: Transmission probability for networks with ${N=5}$ discrete
states in series with no relative detuning for the special case of homogenous
balanced decays ${\gamma=\Gamma}$ (blue curve) and over-coupling
${g=\frac{6}{5}\frac{\sqrt{\gamma\Gamma}}{2}}$. A balanced over-coupled
network will always have more peaks of perfect transmission than an under-
coupled or critically-coupled one, though on-resonance transmission may not be
a local maxima. Figure 3.15: Transmission probability for networks with $N=5$
discrete states in series with no relative detuning for the special case of
homogenous balanced decays ${\gamma=\Gamma}$ (blue curve) and under-coupling
${g=\frac{4}{5}\frac{\sqrt{\gamma\Gamma}}{2}}$.  Figure 3.16: Transmission
probability for networks with ${N=4}$ discrete states in series with no
relative detuning for the special case of homogenous balanced decays
${\gamma=\Gamma}$ (blue curve) and over-coupling
${g=\frac{6}{5}\frac{\sqrt{\gamma\Gamma}}{2}}$. Again, balanced over-coupled
network will always have more peaks of perfect transmission than an under-
coupled or critically-coupled one, though on-resonance transmission may not be
a local maxima. Figure 3.17: Transmission probability for networks with
${N=4}$ discrete states in series with no relative detuning for the special
case of homogenous balanced decays ${\gamma=\Gamma}$ (blue curve) and under-
coupling ${g=\frac{4}{5}\frac{\sqrt{\gamma\Gamma}}{2}}$.

Since each state is only coupled to the two adjacent states, the system of $N$
equations of the form of (3.16) describing evolution of the system is solvable
in a stepladder-type approach. Setting the expectation value of the second
input continuum to zero, we solve for the $N$th state in terms of the $N-1$th.
As we step up the ladder, we find that the expression for the reflection
coefficient has the form of a generalized continued fraction

$\displaystyle R(\omega)$
$\displaystyle=1-\cfrac{\gamma}{\frac{\gamma}{2}-i\Delta_{1}+\cfrac{g_{12}^{2}}{-i\Delta_{2}+\cfrac{g_{23}^{2}}{\dots+\cfrac{g_{N-1N}^{2}}{\frac{\Gamma}{2}-i\Delta_{N}}}}}$
(3.25)

(we have dropped the subscripts on $\gamma_{1}$ and $\Gamma_{N}$). This
equation looks difficult to analyze, but can be greatly simplified using the
Wallis-Euler recursion relations for continued fractions. We define two
function $A_{n}$ and $B_{n}$ given by the following recurrence relations

$\displaystyle B_{-1}=0,B_{0}=1,A_{-1}=1,A_{0}=b_{0}$ $\displaystyle
A_{n}=b_{n}\,A_{n-1}+a_{n}\,A_{n-2}\,(n\geq 1)$ $\displaystyle
B_{n}=b_{n}\,B_{n-1}+a_{n}\,B_{n-2}\,(n\geq 1)$ (3.26)

with coefficients $a_{n}$ and $b_{n}$ given in Tbl. 3.1. The reflection
coefficient is then given $R(\omega)=\frac{A_{N}}{B_{N}}$ for $N$ discrete
states. (For $n<N$, the functions $A_{n}$ and $B_{n}$ have no clear physical
meaning.) From this, we easily solve for conditions where $R(\omega)=0$
(perfect transmission).

$n$ | $a_{n}$ | $b_{n}$
---|---|---
$0$ | $0$ | $1$
$1$ | $-\gamma$ | $\frac{\gamma}{2}-i\,\Delta_{1}$
$2$ | $g_{12}^{2}$ | $-i\,\Delta_{2}$
$\vdots$ | $\vdots$ | $\vdots$
$N-1$ | $g_{N-2N-1}^{2}$ | $-i\,\Delta_{N-1}$
$N$ | $g_{N-1N}^{2}$ | $\frac{\Gamma}{2}-i\,\Delta_{N}$
$>N$ | $0$ | $0$
Table 3.1: Wallis-Euler coefficients for ${N>1}$ discrete states in series.
For a series network with an arbitrarily high number of discrete states $N$,
these can be used to generate transmission functions (that correctly encode
the causal structure that is inherent to the network outside of the strong-
coupling limit) using the Wallis-Euler recursion relations (3.3.2). Figure
3.18: Transmission probability for a series network with ${N=70}$ identical
discrete states (${\omega_{i}=\omega_{j}}$) with both balanced decays
(${\gamma=\Gamma}$) and uniform critical coupling
(${g=\frac{\sqrt{\gamma\Gamma}}{2}}$) conditions met. This results in a
maximally-broadened on-resonance transmission. In the large-$N$ limit, the
amplitude of this ideal (perfect and broadened) transmission function
$|T(\omega)|$ is bounded below by a circle (parabola for $|T(\omega)|^{2}$)
and above by a square (both red) with widths $2g$ and maxima of unity.

_Perfect transmission.—_ We begin by considering the unphysical but
illuminating case of an infinite series of identical discrete states
($g_{i\,i+1}=g\,\forall i$ and $\Delta_{i}=\Delta\,\forall i$) analytically.
We find that the limit $\lim_{N\rightarrow\infty}R(\omega)$ only converges on-
resonance for the special “critical” case

$g=\frac{\sqrt{\gamma\,\Gamma}}{2}.$ (3.27)

Furthermore, the limit only converges to zero on resonance (perfect
transmission) when we also have $\gamma=\Gamma$. This first condition
corresponds to a series of discrete states coupled through their decays (e.g.
Fabry-Perot cavities coupled through their evanescent fields), and the second
condition corresponds to the same requirement of balanced decays we saw for
parallel quantum networks.

While infinite series networks of discrete states are not realistic, these two
conditions play different but important roles in all finite series networks
where the discrete states are identical (though introducing relative detuning
will change the critical values of $g$ and $\Gamma/\gamma$, as we will see).
That there are two conditions can be explained thus; since $R(\omega)$ is in
general complex, the condition $R(\omega)=0$ gives two constraint equations on
the real and imaginary parts of $R(\omega)$. When $N$ is even [odd], the real
part is an order $N$ [$N-1$] polynomial in the detunings $\Delta_{i}$, while
the imaginary part is order $N-1$ [$N$]. In general, this means there are
conditions for a minimum of $N-1$ frequencies of perfect transmission and we
may find $N$ frequencies of perfect transmission only if the lower-order
equation is satisfied trivially for all frequencies. When considering finite
series networks of identical discrete states, these same two conditions appear
in the constraint equations for perfect transmission101010For ${N=1}$, the
lower order equation is just the requirement that ${\gamma=\Gamma}$ and the
higher order equation simply requires ${\Delta_{1}=0}$ (on-resonance). For
${N=2}$, the lower order equation is ${\gamma\Delta_{2}=\Gamma\Delta_{1}}$ and
the higher order equation is
${g^{2}=\Delta_{1}\Delta_{2}+\frac{\gamma\Gamma}{4}}$. The lower order
equation is only frequency independent when ${\gamma=\Gamma}$ and
${\Delta_{2}=\Delta_{1}}$. When the discrete states are degenerate, we find
that ${g\geq\frac{\sqrt{\gamma\,\Gamma}}{2}}$ is the requirement for perfect
transmission at one (${g=\frac{\sqrt{\gamma\,\Gamma}}{2}}$) or two
(${g>\frac{\sqrt{\gamma\,\Gamma}}{2}}$) frequencies..

We now explore in detail the effects of these two conditions on series
networks of identical discrete states with uniform coupling; first, consider
fixing the coupling $g$ to be critical ($g=\frac{\sqrt{\gamma\Gamma}}{2}$).
For odd $N$, we find that when the decays are balanced ($\gamma=\Gamma$), this
ensures $N$ frequencies of perfect transmission (Fig. 3.10) with the resonance
frequency at a local maxima of unity, and when the decays are unbalanced
($\gamma\neq\Gamma$), $N-1$ frequencies of perfect transmission (Fig. 3.11)
with the resonant frequency at a local minima. For even $N$, letting
$g=\frac{\sqrt{\gamma\Gamma}}{2}$ always results in $N-1$ frequencies of
perfect transmission (Fig. 3.12) with the on-resonant frequency at a local
maxima. Here, having balanced decays broadens the the on-resonance maxima
(Fig. 3.13), which will be desirable for detection of non-monochromatic
photons (wavepackets).

Now we instead consider balanced decays ($\gamma=\Gamma$) and observe a switch
in the on-resonance behavior; whereas above we found that for even $N$ the
critical coupling condition was sufficient for on-resonance transmission to be
at a local maxima, we now find that, for odd $N$ that the balanced decay
condition results in $N$ peaks of unity transmission (Fig. 3.14) with on-
resonance transmission at a local maxima. Here three peaks become degenerate
to give $N-2$ frequencies of perfect transmission when
$g\lesssim\frac{\sqrt{\gamma\Gamma}}{2}$ (Fig. 3.15). (This inequality rapidly
becomes exact [$g<\frac{\sqrt{\gamma\Gamma}}{2}$] with increasing $N$.)
Similarly, for even $N$ we find the behavior of $T(\omega)$ depends strongly
on the coupling, flipping between $N$ frequencies of perfect transmission with
on-resonance transmission at a local minima for
$g>\frac{\sqrt{\gamma\Gamma}}{2}$ (Fig. 3.16) and $N-2$ frequencies of perfect
transmission with on-resonance transmission at a non-unity local maxima for
$g<\frac{\sqrt{\gamma\Gamma}}{2}$ (Fig. 3.17).

For both even and odd $N$, the width is increasingly determined by $g$ with
increasing $N$, with the half-width asymptotically approaching $2g$. In the
large-$N$ limit, we observe that the transmission function is asymptotically
bounded between a circle and a square when both conditions for perfect
transmission are met (Fig. 3.18). We also observe that increasing $g$ past
$\frac{\sqrt{\gamma\,\Gamma}}{2}$ while maintaining $\gamma=\Gamma$ induces
Rabi splitting, with the $N$ or $N-1$ frequencies of perfect transmission
spreading outwards (for odd and even $N$, respectively). For the special case
of $N=2$, the presence of a second coupled discrete state resulting in a
splitting and shift of the resonant frequency is in agreement with the effect
seen in Ref. [20] for an antireflective coating; perfect transmission is still
possible provided $\gamma=\Gamma$ but the frequency that is perfectly
transmitted is split in to two.

In the strong coupling (high-$g$) limit, a frequency-comb structure emerges
(Fig. 3.19), with dips approaching perfect reflection. This is due to the
asymptotic irrelevance of the causal ordering (that is, the order in which an
excitation must traverse discrete states to pass from one continuum to the
other): the network becomes approximately diagonalizable as parallel modes
with purely virtual coupling and the $N-1$ frequencies of perfect reflection
from (3.23) manifest.

We now consider the effects of introducing relative detunings between discrete
states as, generally, a quantum network will not be comprised of identical
states. Still there are sufficient degrees of freedom in (3.25) such that, by
tuning the parameters, perfect transmission at _some_ frequencies is always
possible. This is even true when $\gamma\neq\Gamma$ (for $N>1$): considering
only two discrete states in series, we find that perfect transmission occurs
at a frequency
$\omega=\frac{\omega_{1}+\omega_{2}}{2}+\frac{\Gamma\omega_{1}-\gamma\omega_{2}}{\Gamma-\gamma}$
when
$g_{12}=\sqrt{\frac{\gamma\Gamma}{4}+\left(\frac{\Gamma\omega_{1}-\gamma\omega_{2}}{\Gamma-\gamma}\right)^{2}-\left(\frac{\omega_{1}-\omega_{2}}{2}\right)^{2}}$
(Fig. 3.20). For two detuned discrete states in series with balanced decays
($\gamma=\Gamma$), perfect transmission is impossible as the critical value of
$g$ is infinite; the transmission efficiency asymptotically becomes perfect in
the strong-coupling limit. However, this a special case and is not true for
larger numbers ($N>2$) of discrete states. Critically, given a series network
of $N$ discrete states with arbitrary relative detunings, we can _always_ find
at least one set of parameters (couplings and decay rates) such that $N-1$
frequencies are perfectly transmitted.

Figure 3.19: Transmission probability for a series network with ${N=70}$
identical discrete states with balanced decays (${\gamma=\Gamma}$) yielding
perfect transmission in the strong-and-uniform coupling limit
(${g\gg\frac{\sqrt{\gamma\Gamma}}{2}}$). The result is a frequency-comb
structure with $70$ frequencies of unity transmission between $-2g$ and $2g$,
separated by $69$ regions of near-perfect reflection.

_Spectral Bandwidth.—_ Once we have the form of the transmission function, we
can calculate the spectral bandwidth for these systems. The spectral bandwidth
decreases with additional discrete states and is strictly bounded above by the
single discrete state bandwidth
$\tilde{\Gamma}\leq\frac{2\gamma\Gamma}{\gamma+\Gamma}$. For discrete states
without detuning, equality is reached in the strong-coupling limit of
$g\gg\frac{\sqrt{\gamma\Gamma}}{2}$ but independently of whether
$\gamma=\Gamma$ (Fig. LABEL:tildeGammaN1). Introducing detuning between
discrete states lowers the bandwidth, but equality with the upper limit still
occurs for sufficiently strong coupling (Fig.
LABEL:N1RelativeBandwidthDetuning); the strong coupling is able to better mask
the discrepancy between discrete state frequencies as the dressed states
become the more physical description and as the system more strongly resembles
a parallel network. (In this limit, each dressed state is coupled to the
continua at reduced decays so that the total bandwidth is still bounded by
$\tilde{\Gamma}\leq\frac{2\gamma\Gamma}{\gamma+\Gamma}$ with $\gamma$ and
$\Gamma$ the two original decay rates for the series network.)

_Group Delay.—_ Lastly, we consider the group-delay for series networks, which
increases with both $g$ and $N$ but for different reasons. As $g$ increases,
the peaks of the transmission function sharpen so that the phase changes more
rapidly. This results in an increased group delay (Fig.
LABEL:2discretestatesSeriesTrans-d). As $N$ increase, we observe (as we did
for parallel networks) that the peaks in the group delay are not of uniform
magnitude, even when the transmission function itself is rather flat (Fig.
LABEL:20discretestatesSeriesTrans). Instead, the frequencies of maximum delay
are those closest to $\pm 2g$ (Fig. LABEL:20discretestatesSeriesTau) and
increase the most with $N$. This is because the oscillations in the
transmission function are most dense here, resulting in a rapid change in
transmission function phase and thus a larger group delay. On the contrary,
the group delay is relatively flat in the center of the transmission window
(Fig. LABEL:20discretestatesSeriesTau) so that a spectrally-narrow (compared
to $g$) input photon pulse will not be significantly dispersed. (That is, the
maximum dispersion-induced jitter $\mathcal{T}_{g}$ from (3.5) will be very
small.)

Figure 3.20: Transmission probabilities for series networks with ${N=2}$
discrete states with no relative detuning (blue with a single maxima), and
with non-zero relative detuning