# Measuring higher-order photon correlations of faint quantum light: a short
review

K. Laiho<EMAIL_ADDRESS>Institute of Quantum Technologies, German
Aerospace Center (DLR), Wilhelm-Runge-Str. 10, 89081 Ulm, Germany  T. Dirmeier
Max Planck Institute for the Science of Light, Staudtstr. 2, 91058 Erlangen,
Germany  Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), Department
of Physics, Staudtstr. 7/B2, 91058 Erlangen, Germany  M. Schmidt  Technische
Universität Berlin, Institut für Festkörperphysik, Hardenbergstr. 36, 10623
Berlin, Germany  Physikalisch-Technische Bundesanstalt (PTB), Abbestr. 2-12,
10587 Berlin, Germany  S. Reitzenstein  Technische Universität Berlin,
Institut für Festkörperphysik, Hardenbergstr. 36, 10623 Berlin, Germany  C.
Marquardt  Institute of Quantum Technologies, German Aerospace Center (DLR),
Wilhelm-Runge-Str. 10, 89081 Ulm, Germany  Max Planck Institute for the
Science of Light, Staudtstr. 2, 91058 Erlangen, Germany  Friedrich-Alexander-
Universität Erlangen-Nürnberg (FAU), Department of Physics, Staudtstr. 7/B2,
91058 Erlangen, Germany

###### Abstract

Normalized correlation functions provide expedient means for determining the
photon-number properties of light. These higher-order moments, also called the
normalized factorial moments of photon number, can be utilized both in the
fast state classification and in-depth state characterization. Further, non-
classicality criteria have been derived based on their properties. Luckily,
the measurement of the normalized higher-order moments is often loss-
independent making their observation with lossy optical setups and imperfect
detectors experimentally appealing. The normalized higher-order moments can
for example be extracted from the photon-number distribution measured with a
true photon-number-resolving detector or accessed directly via manifold
coincidence counting in the spirit of the Hanbury Brown and Twiss experiment.
Alternatively, they can be inferred via homodyne detection. Here, we provide
an overview of different kind of state classification and characterization
tasks that take use of normalized _higher-order_ moments and consider
different aspects in measuring them with free-traveling light.

##  I Introduction

The definition of light via discrete energy quanta has in the past brought up
a variety of methods for determining its photon-number properties. Nowadays,
accessing the photon-number content of light lies in the core of quantum
optics experiments. The quantum theory of photon correlations, which explores
the higher-order factorial moments of the photon-number distribution, was
seminally formulated in 1963 by Glauber Glauber1963 ; Glauber1963a  and
Sudarshan Sudarshan1963 . A few years earlier in 1956 Hanbury Brown and Twiss
had conducted their famous experiment, in which they were able to discern
between thermal and Poissonian light Brown1956a ; Brown1956 . Additionally,
only shortly after a non-classicality criterium for light was derived in terms
of the normalized higher-order moments Titulaer1965 .

The normalized higher-order moments offer fast and simple tools for the
_quantum state classification_ aiming at recognizing what kind of photon-
number distribution the investigated light source emits. The most often used
classes can directly be accessed via the normalized second-order moment
$g^{(2)}$ and are denoted as the sub-Poissonian $[g^{(2)}<1]$, Poissonian
$[g^{(2)}=1]$ and super-Poissonian $[g^{(2)}>1]$ classes Fox2006 ; Migdall2013
. Indeed, they deliver information of the photon-number variance of the
emitted light when compared with that of a Poissonian light source having the
same mean photon number. Another option is to designate the light source
directly to a specific class of light, when preferably knowing all its
normalized $m$-th order factorial moments $g^{(m)}$. The most famous classes
of quantum light probably are the thermal light $[g^{(m)}=m!]$, the Poissonian
light $[g^{(m)}=1]$ and the single-photon state $[g^{(m\geq 2)}=0]$
S.M.Barnett1997 .

The normalized higher-order moments of single-mode light fields incorporating
only a single frequency, polarization and wavenumber take constant values
Loudon2000 . Therefore, it is often convenient to regard the normalized
higher-order moments in their time-independent form. However, in general the
photon correlations are by definition time dependent and need to be measured
in the reference time frame of the emitted photons. Indeed, the time-dependent
normalized second-order correlation function $\tilde{g}^{(2)}(\tau)$, in which
$\tau$ denotes the relative time difference between the photons emitted from a
given light source, has led to another important figure of merit. Namely,
photon bunching is observed if
$\tilde{g}^{(2)}(\tau=0)>\tilde{g}^{(2)}(\tau\neq 0)$ and anti-bunching if
$\tilde{g}^{(2)}(\tau=0)<\tilde{g}^{(2)}(\tau\neq 0)$ Gerry2005 ; Vogel2006 .
For photons arriving from two completely independent light fields we expect
that $\tilde{g}^{(2)}(\tau)=1$. In an experiment, the temporal resolution of
the used equipment often sets limits to the accuracy of these measurements
Migdall2013 . Therefore, for pulsed light it is usually most convenient to
investigate the time-averaged form of the normalized correlation functions.

The measurement of the normalized second-order moment has become a standard
tool for verifying and examining the properties of different kind of
miniaturized light emitters Migdall2013 . Since the first observation of
strong anti-bunching from resonance fluorescence of an atom Kimble1977 , the
measurement of $\tilde{g}^{(2)}(\tau=0)<1$ has become a well-established
method for determining the quality of single-photon emitters Walls1979 ;
Short1983 ; Leuchs1986 ; Grangier1986 ; Diedrich1987 ; Michler2000 . This has
literally resulted to a race towards a value closest to zero, in order to
prove that the source emits only a single photon at a time Meyer-Scott2020 ;
Rodt2020 . Additionally, the measurement of the normalized second-order moment
has become helpful for recognizing the transition to lasing, i.e. the change
from thermal to Poissonian photon-number distribution, in the so-called
thresholdless nanolasers that no longer display a kink in their input-output
relation as a sign of ongoing lasing Jin1994 ; Khajavikhan2012 ; Kreinberg2019
. Moreover, sources producing photons in pairs, such as the non-linear optical
process of parametric down-conversion (PDC) Burnham1970 , provide an ideal
testbed for investigating the photon correlations. The generated twin
beams—often called signal and idler—obey strict photon-number correlations
Haderka2005 ; Avenhaus2008 . The measurement of the joint normalized higher-
order moments of signal and idler have proven to be beneficial for accessing
the intrinsic properties of the PDC emitters Razavi2009 ; Christ2011 .

Yet, the higher-order moments are also capable for _in-depth quantum state
characterization_ Leuchs2015 . Being connected to a more wider concept of the
_moment generating function_ they allow one to access other fascinating
features of light such as the photon-number parity or even the individual
photon-number contributions S.M.Barnett1997 ; Mandel1996 . The photon-number
parity directly delivers the value of the Wigner function at the origin of the
phase space Cahill1969a ; Cahill1969  that offers an alternative to the
state’s density matrix for investigating its properties. Additionally, the
non-classicality criteria connected to the Glauber-Sudarshan
$P$-representation, which is another quasi-distribution function, can often be
formulated in terms of the normalized higher-order moments and have become
especially useful when searching for quantum features of light Agarwal1992 ;
Miranowicz2010 .

In this review, we explore photon correlations and highlight the use of _the
normalized moments with orders higher than two_ in the quantum state
classification and characterization. We start by summarizing theoretical
aspects of photon correlations in Sec. II, after which we consider
experimental methods required for gaining access to them both in discrete
(Sec. III) and continuous (Sec. IV) variables. Thereafter, in Sec. V we review
state classification tasks, in which the higher-order moments have proven to
been practical. Finally, we regard state characterization tasks varying from
the demonstration of non-classicality in Sec. VI to the reconstruction of the
state’s phase-space properties in Sec. VII before summarizing the findings in
Sec. VIII. This tutorial review is epecially suited for readers who are
already familiar with introductory quantum optics.

##  II  Photon correlations

The time-dependent normalized $m$-th order correlation function for light can
be determined via Glauber1963a

$\displaystyle\tilde{g}^{(m)}(t_{1},t_{2}=t_{1}+\tau,\dots
t_{m})=\frac{\braket{\hat{\textbf{a}}^{\dagger}(t_{1})\hat{\textbf{a}}^{\dagger}(t_{2})\dots\hat{\textbf{a}}^{\dagger}(t_{m})\hat{\textbf{a}}(t_{m})\dots\hat{\textbf{a}}(t_{2})\hat{\textbf{a}}(t_{1})}}{\braket{\hat{\textbf{a}}^{\dagger}(t_{1})\hat{\textbf{a}}(t_{1})}\braket{\hat{\textbf{a}}^{\dagger}(t_{2})\hat{\textbf{a}}(t_{2})}\dots\braket{\hat{\textbf{a}}^{\dagger}(t_{m})\hat{\textbf{a}}(t_{m})}},$
(1)

in which $\hat{\textbf{a}}^{\dagger}(t_{\sigma})$
[$\hat{\textbf{a}}(t_{\sigma})$] denotes the photon creation [annihilation]
operator at time $t_{\sigma}$ ($\sigma=1,2,\dots m$). In the case of
temporally-resolved measurements one is often most interested in the value of
the higher-order correlation at $t_{\sigma}=0$ ordinarily denoting the event
that the photons are emitted simultaneously and delivering information of the
underlying photon-number distribution at that time. The function
$\tilde{g}^{(m)}(t_{1},t_{2}=t_{1}+\tau,\dots t_{m})$ provides access to the
temporal coherence properties of the emitted photons and most importantly, the
dynamics of $\tilde{g}^{(2)}(\tau)$ follow the coherence time of the source
Zhang2005 ; Foertsch2013 ; Guo2017 .

Besides $\tilde{g}^{(2)}(\tau)$ can be measured with a high temporal
resolution if the photo-detection is assisted by the physical phenomenon of
the two-photon absorption Boitier2009 . In other cases, it is often difficult
to measure the time-dependent $\tilde{g}^{(m)}$-function accurately, since the
measured outcome is convolved with the detector’s response function and thus
smeared out, if the time jitter of the used detectors and of the counting
electronics is longer than the temporal coherence of the source Kreinberg2019
; Razavi2009 ; Bettelli2010 ; Blauensteiner2009 . However, sometimes it can be
useful to consider time integration over the measured outcome, which
mathematically delivers the time integration over the original
$\tilde{g}^{(m)}$-function, since the integral over the detector’s response
function is normalized to unity Kreinberg2017 . Physically, it means that the
total amount of the detection events under the convolution stays the same;
just temporal re-binning of the detection events happens due to the jittering.
This way, the values of the original $\tilde{g}^{(2)}(\tau=0)$ function has at
least been reconstructed in Refs Kreinberg2019 ; Kreinberg2017  for the
miniaturized semiconductor light emitters around the lasing transition by
taking the advantage of the modified Siegert relation Blazek2011  and in Ref.
Ivanova2006  for the squeezed vacuum states prepared in PDC.

In contrast, when investigating pulsed light, one often measures the photon
correlations in time-averaged manner. Thus, one measures all photons arriving
at the detector during an integration time, which is chosen to be much longer
than the duration of the optical pulses. Therefore, we replace Eq. (1) with
Christ2011

$\displaystyle{g}^{(m)}=\frac{\int dt_{1}\int dt_{2}\dots\int
dt_{m}\braket{\hat{\textbf{a}}^{\dagger}(t_{1})\hat{\textbf{a}}^{\dagger}(t_{2})\dots\hat{\textbf{a}}^{\dagger}(t_{m})\hat{\textbf{a}}(t_{m})\dots\hat{\textbf{a}}(t_{2})\hat{\textbf{a}}(t_{1})}}{\int
dt_{1}\braket{\hat{\textbf{a}}^{\dagger}(t_{1})\hat{\textbf{a}}(t_{1})}\int
dt_{2}\braket{\hat{\textbf{a}}^{\dagger}(t_{2})\hat{\textbf{a}}(t_{2})}\dots\int
dt_{m}\braket{\hat{\textbf{a}}^{\dagger}(t_{m})\hat{\textbf{a}}(t_{m})}}.$ (2)

Extending the integration limits in Eq. (2) to $(-\infty,\infty)$,
transforming between the time and frequency domains with the Fourier
transform, which can be expressed in the form
$\hat{\textbf{a}}(t)=\frac{1}{\sqrt{2\pi}}\int d\omega\
\hat{\textbf{a}}(\omega)\textrm{e}^{-\imath\omega t}$ with $\omega$ denoting
the angular optical frequency, and utilizing the definition
$\delta(\omega)=\frac{1}{2\pi}\int dt_{\sigma}\textrm{e}^{\imath\omega
t_{\sigma}}$ $(\sigma=1,\dots m)$ of the delta function, we find

$\displaystyle{g}^{(m)}=\frac{\int d\omega_{1}d\omega_{2}\dots d\omega_{m}\
\braket{\hat{\textbf{a}}^{\dagger}(\omega_{1})\hat{\textbf{a}}^{\dagger}(\omega_{2})\dots\hat{\textbf{a}}^{\dagger}(\omega_{m})\hat{\textbf{a}}(\omega_{m})\dots\hat{\textbf{a}}(\omega_{2})\hat{\textbf{a}}(\omega_{1})}}{\int
d\omega_{1}\braket{\hat{\textbf{a}}^{\dagger}(\omega_{1})\hat{\textbf{a}}(\omega_{1})}\int
d\omega_{2}\braket{\hat{\textbf{a}}^{\dagger}(\omega_{2})\hat{\textbf{a}}(\omega_{2})}\dots\int
d\omega_{m}\braket{\hat{\textbf{a}}^{\dagger}(\omega_{m})\hat{\textbf{a}}(\omega_{m})}}.$
(3)

Eq. (3) delivers the value for the time-averaged normalized $m$-th order
correlation function in the frequency domain.

In the framework of broadband and orthonormal spectral modes introduced in
Ref. Rohde2007  we define the broadband photon creation operators
$\hat{\textbf{A}}^{\dagger}_{j}=\int d\omega\ \varphi_{j}(\omega)\
\hat{\textbf{a}}^{\dagger}(\omega)$ with $\varphi_{j}(\omega)$ denoting the
profile of the $j$-th mode, for which it holds

$\displaystyle\int d\omega\ \varphi^{\star}_{j}(\omega)\
\varphi_{j^{\prime}}(\omega)$
$\displaystyle=\left\\{\begin{array}[]{l}1\quad\textrm{if}\quad
j=j^{\prime}\\\ 0\quad\textrm{otherwise}\\\
\end{array}\right.\quad\textrm{and}$ (6)
$\displaystyle\sum_{j}\varphi_{j}(\omega)\
\varphi^{\star}_{j}(\tilde{\omega})$
$\displaystyle=\delta(\omega-\tilde{\omega}).$ (7)

Thus, we can re-write Eq. (3) with the help of the relation
$\hat{\textbf{a}}^{\dagger}(\omega)=\sum_{j}\hat{\textbf{A}}^{\dagger}_{j}\
\varphi^{\star}_{j}(\omega)$. This yields

$\displaystyle{g}^{(m)}=\frac{\braket{:\bigg{(}\sum_{j}\hat{\textbf{A}}^{\dagger}_{j}\hat{\textbf{A}}_{j}\bigg{)}^{m}:}}{\braket{\sum_{j}\hat{\textbf{A}}^{\dagger}_{j}\hat{\textbf{A}}_{j}}^{m}},$
(8)

in which : : denotes the normal operator ordering. Further, Eq. (8) reduces to
the well-known definition of the normalized factorial moments of photon number
given by

$\displaystyle{g}^{(m)}=\frac{\braket{:\hat{N}^{m}:}}{\braket{\hat{N}}^{m}},$
(9)

when written in terms of the multi-mode photon-number operator
$\hat{N}=\sum_{j}\hat{\textbf{A}}^{\dagger}_{j}\hat{\textbf{A}}_{j}$ that is
given as a sum of the photon-number operators of the independent and thus
commuting modes.

A more suitable form for evaluating the normalized higher-order factorial
moments is available, if one has an access to the photon-number content of the
investigated light. Thence, they can be expressed as S.M.Barnett1997

$\displaystyle\varg^{(m)}$
$\displaystyle=\frac{\sum_{n}n(n-1)\dots(n-m+1)\rho_{n}}{\left(\sum_{n}n\rho_{n}\right)^{m}}=\frac{\sum_{n}\frac{n!}{(n-m)!}\
\rho_{n}}{\left(\sum_{n}n\rho_{n}\right)^{m}},$ (10)

in which $\rho_{n}$ denotes the photon-number distribution of the investigated
quantum optical state, or shortly its photon statistics, given in terms of the
photon number $n$. Interestingly, Eqs (9) and (10) are equivalent, as can be
seen, if the photon-number distribution of the investigated state is re-
written as $\rho_{n}=\braket{:\textrm{e}^{-\hat{N}}\frac{\hat{N}^{n}}{n!}:}$
and this expectation value is replaced in Eq. (10). By using
$\textrm{e}^{\hat{N}}=\sum_{n}\frac{\hat{N}^{n}}{n!}$ and by utilizing Baker-
Hausdorff formula for commuting operators we gain

$\displaystyle\varg^{(m)}$
$\displaystyle=\frac{\braket{:\textrm{e}^{-\hat{N}}\frac{\sum_{n}\hat{N}^{n-m}}{(n-m)!}\hat{N}^{m}:}}{\braket{:\textrm{e}^{-\hat{N}}\frac{\sum_{n}\
\hat{N}^{n-1}}{(n-1)!}\hat{N}:}^{m}}=\frac{\braket{:\hat{N}^{m}:}}{\braket{\hat{N}}^{m}}.$
(11)

As suggested by our treatment, the normalized higher-order moments can be
evaluated either from the manifold coincidence discrimination or via the
photon statistics measured with a true photon-number-resolving detector. Next,
we shortly review important aspects of both these measurement schemes in
discrete variables.

##  III Measurement of the normalized higher-order moments in discrete
variables

The normalized higher-order moments have the potential of becoming important
in quantum optical metrology Moreva2019 ; Rebufello2019 . However,
experimental imperfections can introduce artifacts in the observed values
Stensson2018 ; Guo2020  and therefore a careful analysis of the measurement
arrangement is necessary Chesi2019  in order to validate the initial
assumptions made. We start by investigating the Hanbury Brown and Twiss (HBT)
experiment that can resolve the normalized second-order moment, after which we
extend the beam splitter network to be able to measure moments of orders
higher than two. Thereafter, we review methods revealing the normalized
higher-order moments from the photon statistics.

###  III.1 Two-fold coincidence discrimination

Figure 1:  Experimental arrangement for an HBT measurement realized by sending
the investigated light beam with $j$ independent modes to a beamsplitter
having a splitting ratio of $T$. Optical losses of the detectors are modeled
with beam splitters having splitting ratios of $\eta_{1}$ and $\eta_{2}$ in
front of the lossless detectors $D_{1}$ and $D_{2}$, respectively. For more
details see text.

In a typical HBT experiment, as sketched in Fig. 1, the investigated light
beam is sent through a beam splitter with an optical vacuum filling its other
input port, while coincidence counts between the two detectors placed at its
outputs as well as the single counts at each one of them are recorded. The
normalized second-order moment is then typically evaluated via

$\displaystyle{g}^{(2)}=\frac{\braket{\hat{N}_{1}\hat{N}_{2}}}{\braket{\hat{N}_{1}}\braket{\hat{N}_{2}}}\approx\frac{P_{c}}{P_{D_{1}}P_{D_{2}}},$
(12)

in which $\braket{\hat{N}_{1}\hat{N}_{2}}$ denotes the expectation value for
the coincidences and $\braket{\hat{N}_{1}}$ and $\braket{\hat{N}_{2}}$ are the
average photon numbers in the two output arms. We replace the expectation
values on the right-hand side in Eq. (12) with $P_{c}$ denoting the
probability of a coincidence click measured between the detectors $D_{1}$ and
$D_{2}$ and $P_{D_{1}}$ and $P_{D_{2}}$ representing the single click
probabilities at these detectors, correspondingly. This approximation is
possible, only when the detectors’ responses are linear with respect to the
photon number. For click detectors—sometimes also called bucket detectors—that
are able to resolve only between the optical vacuum and at least one photon
impinging on such a detector, this requirement can only be satisfied for faint
light, that is, highly attenuated light beams click_det .

We estimate the expectation values in Eq. (12) for the scheme in Fig. 1 by
assuming that all optical modes experience the same beam splitter ratio and
the same amount of losses in a given beam splitter output arm. The mean photon
numbers at the beam splitter input and output arms (labelled as
$\varsigma=0,1,2$ in Fig. 1) are evaluated via
$\hat{N}_{\varsigma}=\sum_{j}\hat{c}^{\dagger}_{\varsigma j}\hat{c}_{\varsigma
j}$, in which $j$ denotes the independent input modes, for which it applies
$[\hat{c}_{\varsigma
j},\hat{c}^{\dagger}_{\varsigma^{\prime}j^{\prime}}]=\delta_{j=j^{\prime}}\delta_{\varsigma=\varsigma^{\prime}}$
with $\delta$ being the Kroenecker delta function. By using the beam splitter
transformations

$\displaystyle\hat{c}_{1j}$
$\displaystyle\rightarrow\sqrt{\eta_{1}}\left(\sqrt{T}\hat{c}_{0j}+\sqrt{1-T}\hat{v}_{0j}\right)+\sqrt{1-\eta_{1}}\hat{v}_{1j}\quad\textrm{and}$
$\displaystyle\hat{c}_{2j}$
$\displaystyle\rightarrow\sqrt{\eta_{2}}\left(\sqrt{T}\hat{v}_{0j}-\sqrt{1-T}\hat{c}_{0j}\right)+\sqrt{1-\eta_{2}}\hat{v}_{2j},$
(13)

where $\hat{v}_{ij}$ ($i=0,1,2$) denotes the ports filled with optical vacuum,
we re-write the expectation values in Eq. (12) in terms of the photon creation
and annihilation operators at the beam splitter input port and trace over the
optical vacuum modes. The average mean photon numbers deliver
$\braket{\hat{N}_{1}}=\eta_{1}T\braket{\sum_{j}\hat{c}^{\dagger}_{0j}\hat{c}_{0j}}=\eta_{1}T\braket{\hat{N}_{0}}$
and
$\braket{\hat{N}_{2}}=\eta_{2}(1-T)\braket{\sum_{j}\hat{c}^{\dagger}_{0j}\hat{c}_{0j}}=\eta_{2}(1-T)\braket{\hat{N}_{0}}$.
Due to the commuting modes at the beam splitter output
$\braket{\hat{N}_{1}\hat{N}_{2}}=\braket{\sum_{j,j^{\prime}}\hat{c}^{\dagger}_{1j}\hat{c}_{1j}\hat{c}^{\dagger}_{2j^{\prime}}\hat{c}_{2j^{\prime}}}=\braket{\sum_{j,j^{\prime}}\hat{c}^{\dagger}_{1j}\hat{c}^{\dagger}_{2j^{\prime}}\hat{c}_{2j^{\prime}}\hat{c}_{1j}}=\eta_{1}\eta_{2}T(1-T)\braket{\sum_{j,j^{\prime}}\hat{c}^{\dagger}_{0j}\hat{c}^{\dagger}_{0j^{\prime}}\hat{c}_{0j^{\prime}}\hat{c}_{0j}}=\eta_{1}\eta_{2}T(1-T)\braket{:\hat{N}_{0}^{2}:}$.
Finally, we find

$\displaystyle{g}^{(2)}$
$\displaystyle=\frac{\braket{\hat{N}_{1}\hat{N}_{2}}}{\braket{\hat{N}_{1}}\braket{\hat{N}_{2}}}=\frac{\braket{:\hat{N}^{2}_{0}:}}{\braket{\hat{N}_{0}}^{2}}.$
(14)

As indicated in Eq. (14), the evaluation of the normalized second-order moment
in the HBT arrangement is in most cases independent of the beam splitter ratio
and optical losses of the employed detectors. This is often justified, since
the optical components are typically selected such that they exhibit
spectrally only slowly varying characteristics over the bandwidth of the
investigated light emitters. However, more sophisticated treatment is
required, if the different modes suffer from different amount of optical
losses, like for example in the case of spectrally multimodal PDC emission,
which is filtered to a desired bandwidth Tapster1998 ; Branczyk2010 .

###  III.2 Manifold coincidence discrimination

The normalized moments with orders higher than two can be accessed by
extending the HBT setup to a larger beam splitter network. Such a network can
be constructed for example via spatial multiplexing as illustrated in Fig. 2.
The evaluation of the normalized $m$-th order moment can be carried out in
analog to the HBT experiment by counting manifold coincidences. One then
arrives at Avenhaus2010

$\displaystyle{g}^{(m)}$
$\displaystyle=\frac{\braket{\hat{N}_{1}\hat{N}_{2}\dots\hat{N}_{m}}}{\braket{\hat{N}_{1}}\braket{\hat{N}_{2}}\dots\braket{\hat{N}_{m}}}=\frac{\braket{:\hat{N}_{0}^{m}:}}{\braket{\hat{N}_{0}}^{m}},$
(15)

in which $\hat{N}_{o}$ ($o=1,2,\dots m$) denotes the mean-photon number
operator at the output ports of the beam splitter network and $\hat{N}_{0}$
that at the input port.

Figure 2:  A sketch of a spatially extended beam splitter network for
extracting the values of the normalized higher-order factorial moments via
m-fold coincidence counting at the $m$ chosen detectors. The inset shows the
more often used implementation for pulsed light via time-multiplexed
detection. For more details see text.

We exemplify this method on the basis of the beam splitter network in Fig. 2
by following Ref. Avenhaus2010 . With $M$ labeling the depth of the beam
splitter network, we denote the stages of the beam splitter network with
$s=\\{0,\dots M\\}$. Thus, the total number of beam splitters at a given stage
is $l=\\{1,\dots 2^{s}\\}$. Altogether, there are $2l$ outputs at each stage
and at the ouput stage they are labeled with $o=\\{1,\dots 2^{M+1}\\}$. Thus,
the highest measurable moment of the network is $m=2^{M+1}$. At each stage, we
denote the beam splitters with their transmission coefficient $T_{s,l}$ and
the vacuum modes with $\hat{v}_{s,l}$ (not shown in Fig. 2). The beam splitter
input-output relations can be expressed with

$\displaystyle\hat{c}_{s,2l-1}$ $\displaystyle=\sqrt{T_{s-1,l}}\
\hat{c}_{s-1,l}+\sqrt{1-T_{s-1,l}}\ \hat{v}_{s-1,l}\quad\textrm{and}$
$\displaystyle\hat{c}_{s,2l}$ $\displaystyle=-\sqrt{1-T_{s-1,l}}\
\hat{c}_{s-1,l}+\sqrt{T_{s-1,l}}\ \hat{v}_{s-1,l},$ (16)

in which the odd values $(2l-1)$ denote the transmission and even values
$(2l)$ reflection at a beam splitter. In order to evaluate Eq. (15) we require
the photon-number operators
$\hat{N}_{o}=\hat{c}^{\dagger}_{M+1,o}\hat{c}^{\phantom{\dagger}}_{M+1,o}$ at
the output stage $M+1$. When evaluating the expectation values in Eq. (15),
similarly to our treatment in Sec. III.1 one recognizes that after tracing
over the vacuum modes the photon creation and annihilation operators are
linearly proportional to those at the input of the network, that is,
$\hat{c}^{\dagger}_{s=M+1,o}\propto\mathcal{C}^{\star}_{M+1,o}\hat{c}^{\dagger}_{0,1}$
and $\hat{c}_{s=M+1,o}\propto\mathcal{C}_{M+1,o}\hat{c}_{0,1}$, respectively.
Each path from the input of the beamsplitter network to its ouputs has its
individual prefactor, and we denote the ones for $\mathcal{C}_{3,4}$ and
$\mathcal{C}_{3,7}$ in Fig. 2 as an example. Clearly, they cancel out when
evaluating Eq. (15).

In an experiment, Eq. (15) is approximated by dividing an $m$-fold coincidence
probability between $m$ chosen detectors by the product the probabilities of
detecting a single click in each of the chosen $m$ detectors. Similar to the
HBT setup, it is also here necessary to approximate the used detectors as
linear detectors, for which $P_{D_{i}}\propto\eta\braket{\hat{N}_{i}}$, such
that the detected count rates are linear with respect to the mean photon
number of the light impinging on the detector $D_{i}$. Since for click
detectors this can be achieved only at low mean photon numbers, the incident
beam can be attenuated prior to measurement, if necessary. Regarding the
network in Fig. 2, as the used detectors are binary detectors, there are 256
different measurement outcomes ranging from a no-click event to an 8-fold
coincidence-click event. Such extended HBT measurements have been implemented
via spatially or temporally multiplexing beam-splitter networks connected to
avalanche-photo diodes or photo-multiplier tubes Avenhaus2010 ; Jezek2011 ;
Oppel2012 ; Qi2018 , interleaved click detectors made of superconducting
nanowires Stevens2010 ; Goltsman2001 ; Kitaygorsky2009  and with the help of
fast multipixel cameras Assmann2009 ; Wiersig2009 ; Gorobtsov2017 .

###  III.3 Measurement of photon statistics

One can access the normalized higher-order moments also directly from the
measured photon-number distribution by utilizing Eq. (10). The well-known
photo-multiplier tubes are capable of measuring the photon statistics Hong1986
; Ramilli2010 , although they typically suffer from rather low detection
efficiencies. In the past, the photon statistics of continuous wave light has
also been measured in good approximation with avalanche photo diodes being
click detectors Banaszek1999 . More recently, both the most popularly used
click detectors—the avalanche photo diodes and the superconducting nanowire
detectors— have also been shown to have an intrinsic photon-number resolution,
however being currently only able to count a small number of photons
Thomas2010 ; Cahall2017 .

Quasi photon-number-resolving detectors based on spatial or temporal
multiplexing and click detection also give access to the photon statistics
after considering the probabilistic splitting of photons to different spatial
or temporal bins Achilles2004 ; Afek2009 . The effect of this convolution
procedure can be formulated in an ensemble measurement in a matrix form. Apart
from that the loss-tolerance can also be achieved by modeling the losses with
an invertable loss-matrix. This method has been applied for revealing the
loss-inverted photon statistics from the recorded loss-degraded click
statistics Avenhaus2008 ; Achilles2005 . Also other methods such as different
kind of optimization routines can be used to access the photon statistics from
the measured clicks Allevi2009 ; Harder2014 ; Perina2018 . Regarding the
evaluation of the normalized higher-order moments with Eq. (10), much care has
to be taken that it is evaluated from the lossy photon statistics, i.e. from
the backconvoluted one, and not directly from the measured raw click
statistics. If the measured raw click statistics, which is lossy and
convoluted, is used for evaluating the normalized higher-order moments via Eq.
(10) corrections on that equation need to be applied Sperling2012 ;
Bartley2013 .

Better adapted for the measurement of photon statistics are the _true_ non-
commercial photon-number-resolving detectors, like the visible-light-photon-
counter VLPC Waks2004 ; Waks2006  and the transition-edge sensor (TES)
Rosenberg2005 ; Brida2012 . Then, the evaluation of the normalized higher-
order moments via photon statistics with Eq. (10) is a straightforward task.
However, the accuracy, at which the $m$-th order moment can be extracted,
depends both on the underlying photon statistics and also on the amount of
statistics taken. We exemplify this by evaluating the higher-order moments for
Poissonian light measured with a TES. Operating between the superconducting
phase and the normal resistive phase close to its critical temperature, the
number of the photons absorbed in a TES can be read out since the heat from
the absorption of a photon causes a gradual transition in the temperature of
the TES towards the normal resistive phase. Recently, many experiments have
taken the advantage of their close-to-unity efficiency Lita2008 , excellent
energy resolution Schmidt2018  and the capability to count tens of photons
Schmidt2021  for reconstruction photon statistics with TESs. Currently, on the
down-side is their rather long temporal resolution of typically
$>$1\text{\,}\mathrm{\SIUnitSymbolMicro s}$$.

Figure 3:  (a-c) Reconstruction of the photon statistics for a telecom laser
with $1542\text{\,}\mathrm{nm}$ wavelength from $2\cdot 10^{4}$ TES’s traces
(shown in a) measured at $77.5\text{\,}\mathrm{kHz}$. The black solid line in
(b), which represents function in the form
$\sum_{n}\gamma_{n}\exp{[-\alpha_{n}(A-\beta_{n})^{2}]}$ with $\alpha_{n}$,
$\beta_{n}$ and $\gamma_{n}$ being fitting parameters for each photon number
$n$ and given in terms of the trace area $A$, is fitted to the histogram of
TES’s trace areas (yellow bars). The individual photon number contributions
illustrated with blue bars in (c) are extracted from the histogram of the
trace areas via $\frac{A_{n}}{\sum_{A_{n}}}$ with $A_{n}$ denoting the area
under the peak allocated with the photon number $n$, which can be extracted
for example via $A_{n}=\gamma_{n}\sqrt{\pi/\alpha_{n}}$. The dashed lines in
(a) show the temporal acceptance window for the calculation of the trace
areas. The black bars in (c) illustrate a Poissonian distribution with
approximately the same mean photon number as the measured photon statistics
has.

The area of an individual TES trace is proportional to the measured photon
number Schmidt2018 ; Schmidt2021 . In Fig. 3(a) we illustrate typical TES
responses in an ensemble measurement of a coherent laser source. The photon
statistics can be reconstructed by forming a histogram of the trace areas in
Fig. 3(b), and by allocating a photon number for each peak in it. The energy
resolution of the detector is determined as the ratio of the full-width-half-
maximum of a single histogram peak ($\Delta E$) to the gap between two
histogram peaks corresponding to the energy of a quantum ($E=\hbar\omega$).
For the used detector we achieve the ratio $\frac{\Delta E}{E}\approx 0.4$ at
the single-photon level. Further, the used detector has an efficiency $>70\%$.
In Fig. 3(c) we illustrate the extracted photon statistics (blue bars) and
compare it with a Poissonian photon-number distribution (black bars) having
the same mean photon number of approximately $2.7$ as the experimentally
reconstructed photon statistics has.

Now, applying Eq. (10) we can extract the normalized moments up to the fourth
order. In order to extract the accuracy, at which these higher-order moments
are retrieved from the measured photon statistics, we make a Monte-Carlo
simulation with maximally $10^{4}$ trials. For this purpose we let each
photon-number contribution in Fig. 3(c) vary within its extracted error and
calculate the value of Eq. (10) for each in the ensemble simulated photon
statistics. This way we gain the accuracy, at which the normalized higher-
order moments are recorded, as shown in Fig. 4. Clearly, the normalized
second-order moment can be extracted very accurately, while the normalized
moments with orders higher than that cannot be extracted as precisely and the
inaccuracy increases with the growing order of $m$. Moreover, the truncation
of the photon number to ten in the measurement resulting from the limited
acquisition time causes a slight but clearly visible systematic error, which
diminishes the values gained for $g^{(m)}$. Therefore, in Fig. 4 we restrict
ourselves to $g^{(m\leq 4)}$. Altogether, our very simple example with
Poissonian photon statistics illustrates that with increasing order it becomes
more challenging to measure the normalized higher-order moments accurately and
that a much more effort is needed to do so.

Figure 4:  Distributions for the accuracy of the normalized factorial moments
extracted from experimental photon statistics in Fig. 3(c) and delivering the
values (a) $g^{(2)}=0.996(5)$, (b) $g^{(3)}=0.988(15)$ and (c)
$g^{(4)}=0.975(30)$. The errors correspond to the standard deviation within
the simulated ensemble.

##  IV Accessing the normalized higher-order moments via homodyne detection

Figure 5:  Typical realization of a homodyne detection scheme. Signal and
local oscillator fields are interfered on a symmetric beam splitter and
detected on a pair of PIN photo diodes $D_{1}$ and $D_{2}$. The relative phase
between those two fields determines which of the field quadratures will be
detected.

As an alternative for photon counting, it is also possible to use homodyne
detection for accessing the normalized higher-order moments Leuchs2017 . In
homodyne detection, the signal state of interest with an annihilation operator
$\hat{a}_{S}$ is interfered with a bright local oscillator field having the
annihilation operator $\hat{a}_{L}$ at a symmetric beam splitter and then
detected on a pair of PIN photodiodes as shown in Fig. 5. The observable of
our choice is the difference current from both diodes, which is proportional
to the difference in the detected mean photon number. This is expressed as

$\displaystyle\hat{n}_{-}=\hat{a}^{\dagger}_{S}\hat{a}_{L}+\hat{a}^{\dagger}_{L}\hat{a}_{S}.$
(17)

In the limit of a bright local oscillator field, the field operators
$\hat{a}_{L}$ can be replaced by a field amplitude $\alpha_{L}$ with a
relative phase $\theta$ to the signal field. Thus, Eq. (17) can then be
written as

$\displaystyle\hat{n}_{-}=\alpha^{\ast}_{L}\hat{a}_{S}~{}e^{-i\theta}+\alpha_{L}\hat{a}_{S}^{\dagger}~{}e^{i\theta}=|\alpha_{L}|\cdot\hat{X}_{\theta},$
(18)

where
$\hat{X}_{\theta}=\hat{a}_{S}^{\dagger}e^{i\theta}+\hat{a}_{S}e^{-i\theta}$ is
the generalized quadrature operator for arbitrary phases $\theta$. We note
that this definition of the quadrature operator is chosen such that the
quadrature variance of the vacuum state, also called the shot noise, equals to
unity, which is convient for the homodyne detection. We emphasize that other
definitions like the one used in Sec. VII, for which the shot noise takes
another value, are possible and, at times, useful. Additionally, we are only
considering a single-mode picture, which is supported by the fact, that the
signal field is interfered with the bright local oscillator. This interference
realizes an effective modal filter so that the PIN diodes of the homodyne
setup only register the overlapping parts of the signal mode Raymer1995 ;
Dirmeier2020 .

Limiting ourselves to the normalized second-order moment, we substitute the
annihilation and creation operators using the expressions
$\hat{a}=1/2~{}(\hat{X}_{0}+\imath\hat{X}_{\frac{\pi}{2}})$ and
$\hat{a}^{\dagger}=1/2~{}(\hat{X}_{0}-\imath\hat{X}_{\frac{\pi}{2}})$. Thus,
the normalized second-order moment is expressed as McAlister1997 ; Grosse2007

$\displaystyle
g^{(2)}=\frac{\braket{\sum_{\iotaup,\kappaup}{\hat{X}_{\iotaup}^{2}\hat{X}_{\kappaup}^{2}}-4\sum_{\iotaup}{\hat{X}_{\iotaup}^{2}}+12}}{\braket{\sum_{\iotaup}{\hat{X}_{\iotaup}^{2}}-2}^{2}}$
(19)

with $\iotaup,\kappaup=\\{0,\frac{\pi}{2}\\}$. To acquire the necessary
quadrature data, one has to either measure the signal state in phase-
randomized manner McAlister1997 ; Roumpos2013 ; Lueders2018  or to detect both
quadratures, $\hat{X}_{0}$ and $\hat{X}_{\frac{\pi}{2}}$, simultaneously
Grosse2007 ; Qi2020 . For the former, one randomly varies the local oscillator
phase in Fig. 5. To do the latter, it is required to split the input state at
a symmetric beam splitter and send the result onto two homodyne detectors with
a phase difference of $\frac{\pi}{2}$. This method can also be extended to the
normalized higher-order moments. Using the same substitution as before, we
receive an alternative expression for Eq. (9)

$\displaystyle
g^{(m)}=\frac{\braket{:\hat{N}^{m}:}}{\braket{:\hat{N}:}^{m}}=\frac{\braket{(\hat{X}_{0}-\imath\hat{X}_{\frac{\pi}{2}})^{m}(\hat{X}_{0}+\imath\hat{X}_{\frac{\pi}{2}})^{m}}}{\braket{(\hat{X}_{0}-\imath\hat{X}_{\frac{\pi}{2}})(\hat{X}_{0}+\imath\hat{X}_{\frac{\pi}{2}})}^{m}}.$
(20)

To illustrate the principle, we simulate the phase-randomized detection of
coherent and thermal states having a mean photon number of unity and repeat 20
times the simulation with $18\cdot 10^{6}$ samples. The theoretically
calculated Wigner functions and a qualitative depiction of the marginal
distributions, which determine the simulated data are depicted in Fig. 6. The
results of calculating the normalized higher-order moments $g^{(m)}$ for our
simulated data are shown in table 1. We calculate the different moments by
normalizing the measurement data to the shot noise of the local oscillator
$\hat{q}=\hat{n}_{-}/\alpha_{L}$ with $\hat{n}_{-}$ being defined in Eq. (17).
Assuming a random phase for each measurement, we can calculate the normalized
second-order moment as

$\displaystyle
g^{(2)}=\frac{4\cdot\left(\langle\hat{q}^{4}\rangle-6\langle\hat{q}^{2}\rangle+3\right)}{6\left(\langle\hat{q}^{2}\rangle-1\right)^{2}}.$
(21)

In analogy, the moments of order higher than two can be calculated in a
similar manner. The details on deriving these expressions can be found in
Roumpos2013 .

Figure 6:  Simulation of the homodyne detection for extracting the normalized
higher-order moments. We illustrate the Wigner functions of (a) a coherent
state with its phase aligned along the real axis and (c) a thermal state that
both have the mean photon number of unity. The former is given by
$W(\alpha)=\frac{2}{\pi}\exp\left(-2|1-\alpha|^{2}\right)$ and the latter
takes the form $W(\alpha)=\frac{2}{3\pi}\exp\left(-2/3|\alpha|^{2}\right)$
that are both presented here in terms of the complex amplitude $\alpha$ being
consistent with the definition of the Wigner function in Sec. VII. The blue
solid lines indicate the probabilities of the marginal distributions, when the
Wigner functions are projected on the real $\mathrm{pr}(\textrm{Re}(\alpha))$
and imaginary $\mathrm{pr}(\textrm{Im}(\alpha))$ quadratures. We emphasize
that these quadratures are in analog to $\hat{X}_{0}$ and
$\hat{X}_{\frac{\pi}{2}}$ but incorporate another shot noise normalization.
Time traces of the simulated phase-averaged quadrature measurements are
presented in (b) for a coherent state like the one in (a) and in (d) for a
thermal state like the one in (c) with 5000 samples shown. We notice that in
(b) and (d) the shot noise variance is normalized to unity as treated in the
main text.

The values for the coherent state stay close to unity as one would expect for
a state with a Poissonian photon number distribution. In case of the thermal
state, we directly notice the factorial increase in $g^{(m)}$ starting from
the value of $g^{(2)}=2$. We again emphasize that it is important to evaluate
the errors, at which the normalized higher-order moments can be extracted. The
errors increase with growing order of $m$ even in an ideal case of no
experimental imperfections taken into account. This is a statistical effect
due to the limited size of the measured data set and can be remedied by
sampling more data, that is, in an experiment measuring longer. However, the
increment of the measurement time is only possible if the system under study
is stable enough. Often the acquisition time cannot be limitless extended and
a trade-off of between statistical errors and other system fluctuations needs
to be met.

Table 1: Simulated results for the normalized higher-order moments up to the 5th order retrieved from $18\cdot 10^{6}$ samples simulated 20 times for a coherent and thermal state with mean photon number of unity. | $g^{(2)}$ | $g^{(3)}$ | $g^{(4)}$ | $g^{(5)}$
---|---|---|---|---
coherent state | $1.0000(6)$ | $0.998(4)$ | $1.00(3)$ | $1.05(15)$
thermal state | $1.9991(5)$ | $5.993(5)$ | $23.97(5)$ | $120.3(7)$

This method has been demonstrated in a variety of optical applications. It was
used to monitor the change in coherence properties of the light emitted by a
diode laser when passing the lasing threshold Lueders2018 . Also, in the
context of the generation and characterization of squeezed states, it has been
used to probe the higher-order moments of displaced squeezed states Grosse2007
and to reconstruct the wave function of the state Han2020 . In the particular
case of measuring Gaussian states, it is possible to access all
characteristics of a quantum state via the quadrature covariance matrix and
the mean of the quadrature distributions as described for example in Ref.
Olivares2012 . As a consequence, it is also possible to calculate the
normalized higher-order moments using the covariance matrix of such Gaussian
quantum states Olivares2018 . The homodyne detection can also be extended to
systems beyond the optical domain and has been demonstrated in the context of
characterizing microwave photons in superconducting quantum circuits
Eichler2011 .

##  V Classification of quantum states of light

The normalized second-order moment is already a standard tool in analyzing the
photon-number content of quantum light emitters and is most popularly used for
their classification. Beyond that the normalized moments with orders higher
than two are also becoming essential when gathering information about the
nature of the emitted light. Next, we inspect different kind of state
classification tasks, in which the normalized higher-order moments have been
beneficial.

###  V.1 Poissonian and thermal light sources

Poissonian and thermal photon statistics play historically a crucial role and
are still today employed in many different quantum optics applications. The
Poissonian photon statistics takes the form

$\displaystyle\rho^{\textrm{Poissonian}}_{n}=\exp{(-\tilde{n})}\frac{\tilde{n}^{n}}{n!}$
(22)

with $\varg^{(m)}=1$ as is easily extraced via Eq. (10), whereas the thermal
one can be expressed as

$\displaystyle\rho^{\textrm{thermal}}_{n}=\frac{\tilde{n}^{n}}{(1+\tilde{n})^{n+1}}=|\xi|^{2n}(1-|\xi|^{2})$
(23)

with $\varg^{(m)}=m!$ delivered by Eq. (10). In Eq. (23) we have presented the
two most often used forms for thermal photon statistics connected via
$|\xi|^{2}=\frac{\tilde{n}}{1+\tilde{n}}$, which is usually called the
strength of the thermal state. In Eqs (22) and (23) $\tilde{n}$ denotes the
mean photon number of the state. In Fig. 7(a–b) we illustrate the Poissonian
and thermal photon statistics having the mean photon number of unity. Clearly,
there are distinct differences between the statistics that are also reflected
in their respective higher-order moments. The thermal statistics always has a
non-zero vacuum contribution, which takes the highest value in the
distribution, whereas that of the Poissonian one gradually vanishes when
increasing mean photon number. Thermal photon statistics thus is wider, which
reflected by the variance, and has a stronger tilt (skewness) and longer tail
(kurtosis) that can mathematically be associated with the second-, third- and
fourth-order moments, respectively Mandel1996 . Therefore, the higher-order
moments can directly reveal important figures of merit of the shape of the
studied photon-number distribution.

Figure 7: Photon statistics of (a) Poissonian and (b) thermal states having
the mean photon number of unity. The different character of their photon
statistics can easily be visually certified.

The measurements of the normalized moments have become a powerful tool in
understanding the origin of the lasing in nano- and microstructured devices by
revealing the difference between thermal statistics below the laser’s
threshold and Poissonian one above it Assmann2009 . Through such
investigations one has gained access to the importance of different parameters
in lasing structures, such as the light confinement in a cavity with a high
quality factor or the enhancement experienced via small mode-volumes
Strauf2006 ; Ulrich2007 . In thresholdless lasers that have a high rate of
spontaneous emission into the lasing mode, the pronounced kink in the laser’s
input-output curve disappears Noda2006 . In that case one can classify the
emitted radiation via its photon statistics, while also observing the effect
of the cavity parameters on the behavior of the conventional input-output
characteristics Chow2018 ; Reeves2018 .

More insight of the emitted light can be gained by measuring also the
normalized moments with orders higher than two, for example, in order to
extract the degree, to which the investigated light resembles that from a
chaotic source i.e., thermal source or from a laser source Elvira2011 ;
Rundquist2014 ; Schlottmann2017 . The measurements of normalized higher-order
moments have also become expedient for exploring the internal dynamics of
nanolasers Wiersig2009  or more sophisticated photon-number effects like
superradiance accompanied with photon bunching that is stronger than for the
thermal light Jahnke2016 ; Wang2020 . Altogether, these moments are expedient
in resolving the quality of the light emitted from the miniaturized laser
sources Dynes2011 .

Thermal light plays a crucial role both in the optical Zhang2005 ; Boitier2009
; Paleari2004  and microwave regimes Goetz2017 . Pseudo-thermal light is
usually produced by sending laser light through a rotating ground disk, after
which a single speckle is collected via a pinhole Arecchi1965 . The action of
this experimental arrangement can be generalized to other light sources also
Li2020 . Interestingly, the properties of photon-number subtracted thermal
states show a gradual transition towards Poissonian statistics indicating a
drop in the photon-number correlations Y.Zhai2013 ; Katamadze2020 .
Undoubtedly, both true- and pseudo-thermal light sources have also become
important in the large field of quantum imaging that is based on the
measurements of photon correlations Zhang2005 ; Valencia2005 . In fact, the
utilization of the higher-order correlations of thermal light Chan2009  can
result in the increased quality of the ghost images Chen2010 ; Chan2010 ;
Iskhakov2011 . Overall, the dissimilar behavior of the normalized higher-order
moments in the different classes of light allow one even to recognize and
resolve the mode occupation in complex optical states Goldschmidt2013 .
Moreover, the higher-order moments of other exotic light sources such as the
free-electron laser emitting at the extreme ultraviolet regime have been
measured to understand its spatio-temporal properties and photon-number
characteristics in terms of spectral filtering Singer2013 .

###  V.2 Characteristics of single-photon emitters

A perfect $k$-photon emitter incorporates only the $k$-th photon contribution
and has the photon statistics

$\displaystyle\rho^{\ket{k}}_{n}=\left\\{\begin{array}[]{cc}1&\textrm{if
\quad}n=k\\\ 0&\textrm{otherwise}\end{array}\right.,$ (26)

where $k$ is belonging to the natural numbers. Using Eq. (10) the $m$-th order
factorial moment of the photon statistic in Eq. (26) takes the form

$\displaystyle\varg^{(m)}=\frac{m(m-1)\dots(m-k+1)}{k^{m}}.$ (27)

Thus, for the _ideal single-photon emitter_ that has $k=1$ all the values
$\varg^{(m\geq 2)}=0$. Therefore, the vanishing normalized second-order moment
is well adapted for evidencing that all photon-number contributions higher
than or equal to two disappear. Due to the experimental imperfections it is
very challenging to reach this value, and therefore, often a value
$\varg^{(2)}=0.5$, which is valid for an ideal two-photon state
[$\varg^{(2)}=1-\frac{1}{k}$ with $k=2$ in Eq. (27)], is regarded as the upper
limit for the single-photon emitters Chavez-Mackay2020 .

Figure 8: The behavior of the normalized moments $\varg^{(m)}$ up to the 5th-
order ($m=2\dots 5$) for photon number states in terms of the photon number
$k$.

Nevertheless, all values $g^{(2)}<1$ hint at a sub-Poissonian light source and
verify its non-classicality. However, we emphasize that the normalized second-
order moment cannot distinguish the vacuum contribution of the light source.
Therefore, the extraction efficiency or the state fidelity with respect to the
one-photon state is often reported alongside Gruenwald2019 . This way, the
single photon characteristics have been demonstrated from the emission from
atoms and ions Kimble1977 ; Grangier1986 ; Diedrich1987  and from the solid
state including at least quantum dots Michler2000 ; Dauler2008 ; Arakawa2020 ,
silicon- and nitrogen-vacancy centers in diamond Jelezko2006 ; Neu2011 ;
Steudle2012 , cold and room-temperature molecules Lounis2000 ; Toninelli2020
and microwave circuits that can be regarded as artificial atoms Bozyigit2011 ;
Gu2017 . Further heralding of single photons has been demonstrated via the
non-linear optical processes of four-wave mixing McMillan2013 ; Faruque2019
and PDC [see Sec. V.4].

Due to the fact that true single photons are difficult to prepare in the
laboratory and that the prepared states most often incorporate experimental
imperfections, such as higher-photon-number contributions, it is also
worthwhile to examine their normalized moments with orders higher than two.
Such measurements have been employed in several experiments in order to more
deeply understand their characteristics. In order to estimate the ratio of the
single-photon emission to the parasitic background light such investigations
have been conducted at least up to the third order with quantum dots
Stevens2014  and up to the second order with ensembles of emitters based on
nitrogen-vacancy centers Moreva2017 . Further, the normalized moments of the
emission from colloidal two-dimensional nanocrystals have been investigated
upto the fourth order in order to analyze their behavior in terms of the size
and emission wavelength of the emitters Amgar2019 . Additionally, measurements
of normalized moments up to the fourth order with light emitted from ensembles
of colloidal quantum dots acting as single-photon emitters have been
investigated to explain the effect of the cluster size on the emitted light
characteristics and its non-classicality Qi2018 .

###  V.3 Photon-pair states and other multi-photon states

Light sources producing photons in pairs incorporate strong photon-number
correlation between the created photons. The most famous state including only
even photon-number contributions probably is the squeezed vacuum, usually
created by non-linear optics via either PDC or four-wave mixing Slusher1985 ;
Wu1986 . The strong correlation between the created photon pairs can be
verified by measuring the normalized second-order moment Ivanova2006 ;
Kuga1993 ; Boitier2011 . Displacing the squeezed vacuum in phase-space
produces quadrature squeezed states of light and results in interesting
features in the state’s characteristics like for example on sub-Poissonian
behaviour and clear non-classical features that have been probed by measuring
the second-order normalized moment Grosse2007 ; Han2020 ; Olivares2018 ;
Koashi1993 . However, twin-photon states can also be created from the emission
from solid state. Bunching higher than that in thermal light has been reported
by measuring the normalized second-order moment from a twin-photon cascade
emitted from quantum dots Heindel2017 . Also correlations between photon
triplets emitted from a solid state source have been verified Khoshnegar2017 .
Moreover, quadrature squeezed light can also be prepared in quantum-dot
systems Schulte2015 .

The generation of more sophisticated states like path-entangled NOON-states,
describing a non-separable superposition
$\propto\ket{\mathcal{N}0}+\ket{0\mathcal{N}}$ with $\mathcal{N}$ photons
distributed between two independent paths, often incorporates preparation of
photon-pair states and their measurements with correlation functions
Mitchell2004 ; Afek2010 . Also other exotic states such as Schrödinger Kitten
states defined as a superposition of two coherent states with opposite phases
can be created with photon subtraction from squeezed vacuum states Neergaard-
Nielsen2006 ; Ourjoumtsev2007 ; Wakui2007 . The generation of other
superposition states, such as that of the vacuum and single-photon displaced
in the phase space also often relies on the squeezed state generation and
photon-number subtraction Zavatta2004 ; Coelho2016 .

###  V.4 Characteristics of bimodal states of light

Especially interesting are bimodal states of light, in which photon
correlations are generated between two distinct modes of light. The
characteristics of such states can be measured for example by accessing the
joint photon-number distribution of the state with two true photon-number
resolving detectors. Detection with two TESs has been employed investigating
the characteristics of such states created at least in bimodal microlasers
that incorporate strong anti-correlation between the created modes of light
Schmidt2021  and via twin-beam production from PDC emission, which in contrast
results in strong photon-number correlation between the created modes
Harder2016 ; Sperling2017 ; Magana-Loaiza2019 ; Laiho2019 ; Tiedau2019 .

Figure 9:  Illustration of typical PDC characteristics in a collinear
configuration. In (a) a laser beam is coupled in the non-linear optical
material with a second-order non-linearity $\chi^{(2)}$, through which the
pump ($p$) photons split in signal ($s$) and idler ($i$) photons propagating
parallel to each other. Such a scheme can be realized both in waveguided and
bulk configurations. If signal and idler are created in separable modes in the
spectral, spatial or polarization degree of freedom, twin beams are generated.
In (b) a Gaussian approximation of typical joint spectral properties, that is
$|f(\omega_{s},\omega_{i})|$ from Eq. (30), is demonstrated, which
incorporates a strong anti-correlation in the frequency space. The exact shape
of the joint spectral correlations of signal and idler ordinarily stems from
the dispersion properties of the used material W.P.Grice1997 ; Zielnicki2018 ;
Gianani2020 . Such strong spectral correlations result in multimodal PDC
emission. The insets show the conservation laws for energy and momentum, the
latter of which is described in terms of the wavenumber
$k_{\zeta}=\frac{\omega_{\zeta}}{c}n_{\zeta}$ ($\zeta=s,i,p$) with $c$ being
the speed of light and $n_{\zeta}$ the refractive index of the corresponding
mode. Natural phasematching is achieved if $k_{p}=k_{s}-k_{i}$, which is
possible for example in the semiconductor Bragg-reflection waveguides basing
on modal phasematching Laiho2016 . In periodically poled waveguides and
crystals a wavevector mismatch ($k_{p}-k_{s}-k_{i}\neq 0$) can be compensated
with an artificial wavevector component that can be used for example for
frequency tuning Fiorentino2007 ; Eigner2020 .

In the following, we focus on the properties of twin beams generated from PDC
emission as illustrated in Fig. 9 and concentrate on the conventional case, in
which the PDC process is pumped with a coherent laser source. We investigate
the properties of the joint higher-order normalized moments of the signal and
idler beams that in general provide several means for accessing the intrinsic
PDC process parameters. Recently, it was shown that the characteristics of the
twin beams change if the pump light has other than Poissonian characteristics
Meher2020 . However, until today only very few experiments exploit other kind
of pump light than a laser like for example heralded single photons from PDC
Huebel2010  or four-wave-mixing Ding2015 .

The joint higher-order correlations can be investigated in analog to Eq. (8)
as Christ2011

$\displaystyle{g}^{(w,\upsilon)}=\frac{\braket{:(\sum_{q}\hat{A}^{\dagger}_{q}\hat{A}_{q})^{w}(\sum_{q^{\prime}}\hat{B}^{\dagger}_{q^{\prime}}\hat{B}_{q^{\prime}})^{\upsilon}:}}{\braket{\sum_{q}\hat{A}^{\dagger}_{q}\hat{A}_{q}}^{w}\braket{\sum_{q^{\prime}}\hat{B}^{\dagger}_{q^{\prime}}\hat{B}_{q^{\prime}}}^{\upsilon}},$
(28)

in which $A^{\dagger}_{q}$ ($A_{q}$) and $B^{\dagger}_{q^{\prime}}$
($B_{q^{\prime}}$) denote the broadband photon creation (annihilation)
operators in signal ($s$) and idler ($i$) modes labelled with $q$ and
$q^{\prime}$, respectively, and the indices $w$ and $\upsilon$ label the order
of the joint moment.

The two-mode squeezed state $\ket{\Phi}_{s,i}$ of signal and idler, can be
expressed in terms of the unitary squeezing operator $\hat{S}_{s,i}$ as
Christ2011

$\displaystyle\ket{\Phi}_{s,i}=\hat{S}_{s,i}\ket{0}=\textrm{e}^{\sum_{q}r_{q}\hat{A}_{q}\hat{B}_{q}-h.c.}\ket{0},$
(29)

in which the parameter $r_{q}=\mathcal{B}\lambda_{q}$ is defined in terms of
the process strength $\mathcal{B}$ and the real-valued weight of the $q$-th
mode $\lambda_{q}$ and $h.c.$ denotes the hermitian conjugate. These weights
obey the normalization $\sum_{q}\lambda^{2}_{q}=1$. The effective mode number
can be extracted via $\mathcal{K}=1/\sum_{q}\lambda^{4}_{q}$ and the value
$\mathcal{K}=1$ corresponds to the single-mode (SM) PDC emission, while
$\mathcal{K}\gg 1$ denotes multi-mode (MM) PDC emission Eberly2006 . To this
end, the first SM PDC source was experimentally demonstrated in Ref.
P.J.Mosley2008 .

The joint spectral properties of signal and idler are determined by their
joint spectral amplitude, which is connected to these weights via

$\displaystyle f(\omega_{s},\omega_{i})=\sum_{q}\lambda_{q}\
\varphi_{q}(\omega_{s})\ \phi_{q}(\omega_{i}).$ (30)

The frequency-dependent sets of the basis functions
$\\{\varphi_{q}(\omega_{s})\\}$ and $\\{\phi_{q}(\omega_{i})\\}$ for signal
and idler, respectively, both obey the conditions in Eq. (7) and can be
accessed via the Schmidt-mode decomposition accomplished as the singular-value
decomposition.

In order to evaluate the joint normalized moments of signal and idler, we
employ the transformations S.M.Barnett1997

$\displaystyle\hat{S}^{\dagger}_{s,i}\hat{A}_{q}\hat{S}_{s,i}=\textrm{cosh}(r_{q})\hat{A}_{q}+\textrm{sinh}(r_{q})\hat{B}_{q}\quad{\textrm{and}}$
(31)
$\displaystyle\hat{S}^{\dagger}_{s,i}\hat{B}_{q}\hat{S}_{s,i}=\textrm{cosh}(r_{q})\hat{B}_{q}+\textrm{sinh}(r_{q})\hat{A}_{q},$
(32)

that describe the optical squeezing process. The mean photon number of the PDC
emission can easily be evaluated via
$\braket{n}=\sum_{q^{\prime}}\braket{\hat{B}^{\dagger}_{q^{\prime}}\hat{B}_{q^{\prime}}}=\sum_{q}\braket{\hat{A}^{\dagger}_{q}\hat{A}_{q}}=\sum_{q}\textrm{sinh}^{2}(r_{q})\approx\mathcal{B}^{2}$.
The approximation in the last step is valid for MM PDC in the limit of weak
squeezing, for which $\textrm{sinh}(r_{q})\approx\mathcal{B}\lambda_{q}$
applies. Additionally, in that region, that is, at low pump powers, the
evaluation of the lowest orders of the joint normalized moments delivers
Christ2011

${g}^{(1,1)}\approx\frac{1}{\mathcal{B}^{2}}+\sum_{q}\lambda^{4}_{q}+1=\frac{1}{\braket{n}}+\frac{1}{\mathcal{K}}+1,$
(33)
${g}^{(2,0)}=1+\sum_{q}\lambda^{4}_{q}=1+\frac{1}{\mathcal{K}}\quad\textrm{and}$
(34)
${g}^{(2,1)}=\bigg{[}1+\frac{2}{\braket{n}}\bigg{]}{g}^{(2,0)}+2\bigg{[}\frac{1}{\mathcal{K}}+\sum_{q}\lambda_{q}^{6}\bigg{]}.$
(35)

All the joint normalized moments provide useful insight in to the
characteristics of the signal and idler beams. The most often used joint
normalized moment is probably $g^{(1,1)}$ in Eq. (33), which corresponds to
the well-known coincidences-to-accidentals ratio (CAR) Friberg1985 ; Kono1996
. It can directly and loss-independently be employed for estimating the mean
photon number of the PDC state as shown in Fig. 10(a) after the effective
number of modes is extracted for example via Eq. (34). The inverse
proportionality between $g^{(1,1)}$ and the mean photon number is evident
Tanzilli2002 ; Zhu2013 . In case of MM PDC the estimation of the mean photon
number from CAR is straightforward Guenthner2015 . High values of CAR are
desired in order to prove a strong photon-pair correlation between signal and
idler, such that spurious counts are successfully suppressed Bocquillon2009 ;
Krapick2013 ; He2015 ; Kultavewuti2016 ; Chen2018 .

Figure 10:  Characteristics of the joint normalized moments of twin beams with
the lowest orders. In (a) we illustrate the inverse proportionality between
the CAR represented by $g^{(1,1)}$ and the PDC mean photon number. In (b) we
present the linear dependence between $g^{(2,1)}$ and $g^{(1,1)}$. The solid
black lines represent the characteristics of SM PDC process, whereas the
properties of highly MM PDC calucalated with $\mathcal{K}\rightarrow\infty$
are illustrated with gray dashed lines. The inset in (a) shows the behavior of
$g^{(2,0)}$ with respect to the PDC mean photon number.

The normalized second-order moment of an individual twin beam, that is,
$g^{(2,0)}$ in Eq. (34), provides a direct access to the _effective number_ of
the spectral modes excited. Most interestingly, the value of $g^{(2,0)}$ for a
marginal beam i.e., signal or idler, is in general independent of the pump
power, especially in waveguided PDC. When investigating degenerate PDC, that
is, signal and idler lie at the same wavelength, it is also utterly important
to experimentally verify the independence of $g^{(2,0)}$ from the pump power
Laiho2011 . Namely, even a slight leakage of the signal beam to the idler beam
path or vice versa produces a background of single-beam squeezing, which
causes strongly pump power dependent values Christ2011 ; Wakui2014  and can
result in a wrong interpretation of the effective mode number. A more
generalized analysis of the PDC process has shown that the photon-number
properties of the marginal beams can be manipulated by the photon-number
properties of the pump. In general, $g^{(2,0)}$ can be twice as high than the
normalized second-order moment of the pump Meher2020 .

The number of the excited modes in the PDC emission plays a crucial role when
deciding, to which quantum optics tasks the PDC source in question is suitable
for. Ideally, if the PDC is emitted into a single spectral mode, which is
suitable for heralding photon-number Fock states from the PDC emission, the
photon-number properties of the marginal beams show thermal behavior
Paleari2004 ; Vasilyev2000 . In the case of a MM PDC process, which is
suitable for example for entanglement generation or spectrally multiplexed
states Alibart2016 ; Kues2017 , the photon-number properties of the marginal
beams follow that of Poissonian light Haderka2005 ; Avenhaus2008 . Several
different PDC process parameters can be used to tune the effective mode number
such as the pump bandwidth Eckstein2011 ; Harder2013 ; Bruno2014 . Also the
pump depletion has been shown to affect the spectral mode profiles and thus
dramatically modify the multimode structure of twin beams Allevi2014 . In
particular systems, the number of exited modes can be engineered and a direct
selection of the desired spatial or temporal modes has become possible
Perez2014 ; Foertsch2015 ; Ansari2018 ; Graffitti2020 . Apart from that
spectral filtering provides simple means for reducing the mode number of the
twin beams Tapster1998 .

The joint normalized moments with other orders such as $g^{(1,2)}$ in Eq. (35)
also play a crucial role in understanding the intrinsic behavior of the PDC
emission. The measurement of the joint normalized moments $g^{(w,\upsilon)}$
is possible directly via the measured joint photon-number distribution of
signal and idler Harder2016 ; Magana-Loaiza2019 ; Laiho2019  or via manifold
coincidence counting after sending both signal and idler beams in a spatially
or temporally resolving beam splitter networks Avenhaus2010 ; Laiho2011 . In
the latter case, the joint normalized moments are evaluated by dividing the
probability of a coincidence click between $w$ selected detectors in signal
arm and $\upsilon$ ones in idler arm with a product of the probabilities of
detecting a single click in each of these selected detectors. The joint
normalized photon correlations of the PDC emission grow with the increasing
orders of $w$ and $\upsilon$ as illustrated in Fig. 10(b) Avenhaus2010 ;
Harder2016 ; Laiho2011 ; Allevi2012 . The normalized moments of higher orders
better reveal the interplay of the effective modes in the PDC emission
Laiho2011 ; Wakui2014 , and apart from providing information of the strength
of the correlations between the higher photon numbers, they have been used to
verify the non-classicality of the twin beams Avenhaus2010 ; Allevi2012 .

Figure 11:  The properties of ${g}^{(2)}_{h}$ in terms of CAR and the
efficiency of the heralding detector $\eta$ for both SM (orange) and MM
(magenta) PDC when (a) a click detector and (b) true photon-number-resolving
detector is used for heralding. For the calculation we start with the joint
signal-idler state $\ket{\Psi}=\sum_{n}\Lambda_{n}\ket{n,n}_{si}$, in which
$|\Lambda_{n}|^{2}$ describes the photon statistics of the strictly photon-
number correlated signal and idler and corresponds to thermal statistics in SM
PDC and Poissoinian one in MM PDC. The density matrix of the heralded state is
estimated via
$\varrho_{s}=\frac{1}{\Upsilon}\textrm{Tr}_{i}\left\\{(\hat{1}_{s}\otimes\hat{\Pi}^{\mathcal{O}}_{i})\ket{\Psi}\bra{\Psi}\right\\}$,
in which $\Upsilon$ accounts for the normalization of the heralded state and
$\hat{\Pi}_{i}^{\mathcal{O}}$ denotes the detection operation at the heralding
with $\mathcal{O}$ labeling the detector type ph_det . The second-order
normalized moments are thus evaluated from the heralded state photon
statistics given by
$\rho_{n}=\textrm{Tr}_{s}\left\\{\varrho_{s}\ket{n}_{s,s}\bra{n}\right\\}$ via
Eq. (10).

Altogether, the twin beams are ideal for state manipulation and often employed
for example in the preparation of heralded number states A.I.Lvovsky2001 ;
A.Ourjoumtsev2006 . However, the quality of the heralded state is strongly
affected e.g. by the losses in the herald and the photon-number resolution of
the heralding detector Branczyk2010 ; Laiho2019 ; Tiedau2019 ; DAuria2012 ;
Rohde2015 . Therefore, most often the target states are restricted to the
heralded single photons. The normalized second-order moment conditioned on the
detection of the herald, often labelled with ${g}^{(2)}_{h}$ suits well for
the classification of the these states. Additionally, the heralded state
characteristics is connected to the underlying joint photon-number
distribution of signal and idler, and the values reached for ${g}^{(2)}_{h}$
depend on the higher photon-number contributions of the joint signal-idler
state. Following Ref. Laiho2019  we illustrate in Fig. 11 the achievable
values for ${g}^{(2)}_{h}$ in terms of CAR aka $g^{(1,1)}$ of the PDC emission
and Klyshko efficiency Klyshko1996  of the herald for both SM and MM PDC when
detecting the herald with either a click detector or a true photon-number-
resolving detector. One benefits of the heralding with true photon-number-
resolving detection only at high detection efficiencies. Moreover, the
intrinsic PDC process parameters strongly affect the residual higher photon-
number contributions in the heralded state Laiho2011 . Apart from the
platforms on bulk crystals Hong1986 , recently several integrated optics
devices have shown close-to-zero values for ${g}^{(2)}_{h}$ ranging from
ferro-electric platforms URen2005 ; Ngah2015  to semiconductor nanowires and
waveguides Guo2017 ; Belhassen2018 . Heralding of higher photon-number states
similarly greatly suffers from the losses experienced in the heralding arm
Waks2006 ; Tiedau2019 ; Sperling2015 ; Hlousek2019 . For those states, one no
longer can just decrease the pump power in order to get to higher CAR values
since the yield then dramatically drops.

##  VI Nonclassicality measures and higher-order moments

The normalized higher-order moments can reveal information of the quantum
state’s characteristics in phase-space even without having the complete access
to its density matrix or to its alternative representations, the quasi-
distributions functions. One of these quasi-distribution functions, the
$P$-representation, in other words the normally-ordered expansion of the
density matrix, is given for a single-mode state by

$\displaystyle\varrho(\alpha)=\int
d\alpha^{2}P(\alpha)\ket{\alpha}\bra{\alpha}$ (36)

and defined in terms of coherent states having the amplitude $\alpha$. For a
classical state the $P$-representation is well-behaving and takes only non-
negative values, in other words, it can be interpreted as a classical
probability distribution. Singularities in this function or negative values
are signs of _non-classicality_.

Several non-classicality criteria based on the $P$-representation aim at
revealing cases, in which it cannot anymore be interpreted as a classical
probability distribution. Luckily, these criteria can often be formulated and
observed via the higher-order moments through the definition Lee1990 ;
Klauder2006

$\displaystyle\braket{\hat{\textbf{a}}^{\dagger
m}\hat{\textbf{a}}^{m^{\prime}}}=\int
d\alpha^{2}P(\alpha)\alpha^{*m}\alpha^{m^{\prime}}.$ (37)

In a compact way, one can formulate the non-classicalities with matrices of
these moments Agarwal1992 ; Miranowicz2010 ; Sperling2013 . In their simplest
form such a matrix can be defined as

$\displaystyle\mathcal{M}=\left[\begin{array}[]{ccccc}g^{(0)}&g^{(1)}&g^{(2)}&\cdots&g^{(m-1)}\\\
g^{(1)}&g^{(2)}&g^{(3)}&\cdots&g^{(m)}\\\
g^{(2)}&g^{(3)}&g^{(4)}&\cdots&g^{(m+1)}\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\
g^{(m-1)}&g^{(m)}&g^{(m+1)}&\cdots&g^{(2m-2)}\\\ \end{array}\right].$ (43)

For a classical state, whose $P$-representation does not take any negative
values, all the eigenvalues of the matrix in Eq. (43) are positive. Thus,
searching for negative values within the eigenvalues or finding a negative
determinant for it is a sign of the non-classicality. Moreover, for Poissonian
photon statistics the determinant of such a matrix takes the value of zero,
which can be used as a reference value. Several experiments have verified in
such manner the non-classicality of the joint signal-idler states Magana-
Loaiza2019 ; Sperling2015 ; Arkhipov2016  as well as that of the heralded
photon-number states Sperling2017  and clusters of single-photon emitters
Bohmann2019 .

Already, the well-known condition $g^{(2)}<1$, is one of such non-classicality
criteria directly probing the non-classicality of the $P$-representation of a
single-mode state Titulaer1965 ; Klauder2006 ; Agarwal1992 . However, quantum
states may show a transition between sub- and super-Poissonian photon
statistics. Such effects have been measured in the phase-space at least for
the displaced squeezed states Grosse2007 , displaced single-photon states
Laiho2012  and photon-catalyzed optical coherent states Bartley2012 .
Moreover, when looking more closely for example on the single photons
displaced in phase space, at higher displacements their $g^{(2)}>1$ and the
normalized second-order moment is no longer adequate for proving their non-
classicality Laiho2012 . Indeed, in some cases a more general form of this
non-classicality criteria given by

$\displaystyle g^{(m+1)}<g^{(m)}$ (44)

Titulaer1965 ; Klauder2006  can be useful for proving the non-classicality.

Another interesting non-classicality criterion is based on Schwarz inequality
for the $P$-representation, and it can be expressed in the form Titulaer1965 ;
Klyshko1996a

$\displaystyle{g}^{(h-m)}{g}^{(h+m)}\geq\left[{g}^{(h)}\right]^{2}$ (45)

with $h$ and $m$ used for denoting the required orders for the moments. This
criterion has also been investigated both with heralded single photons
Waks2006a ; Allevi2012  and single photons from solid state emitters Qi2018
to find non-classical features in spite of experimental imperfections in the
studied states.

Many other non-classicality criteria basing on the $P$-representation for
multi-mode states such as the twin beams have also been derived in terms of
the joint normalized higher-order moments $g^{(w,\upsilon)}$ Miranowicz2010 ;
Vogel2008 . The PDC emission producing twin beams have been experimentally
shown to obey such conditions Avenhaus2010 . One of their simplest form is
$g^{(1,1)}>\sqrt{g^{(2,0)}\ g^{(0,2)}}$ resulting in the condition
$\textrm{CAR}>2$ for the twin beams generated from SM PDC, which is pumped
with well-behaving laser Loudon2000 ; Miranowicz2010 ; Vogel2008 ; Reid1986 .
Many quantum optical states exhibit also asymmetric moments as described in
Eq. (37) with $m\neq m^{\prime}$ and providing information of the phase
characteristics of the investigated state. Until, today such asymmetric
moments have been extracted only with homodyne tomography Eichler2011 .
Nevertheless, it is possible to access them directly by measuring the higher-
order moments of states displaced in phase space by varying only the phase of
the displacement, while keeping its amplitude fixed Shchukin2005 .

##  VII Quantum state characterization via higher-order moments

The concept of higher-order moments is connected to the moment generating
function S.M.Barnett1997 ; Barnett2018  taking for single-mode state the form

$\displaystyle
M(\mu)=\sum_{n}(1-\mu)^{n}\rho_{n}=\sum_{m}\frac{{g}^{(m)}}{m!}(-\mu\braket{n})^{m},$
(46)

which is defined in terms of the variable $\mu$ ($0\leq\mu\leq 2$), mean
photon number $\braket{n}$ and the photon statistics $\rho_{n}$. The moment
generating function offers a more versatile access to the quantum state
characterization and can deliver, for example, parameters such as the photon
statistics and all orders of the factorial moments of photon number Perina2005
; Wasilewski2008 ; Beenakker2001 . Additionally, there is a direct connection
to the photon-number parity, delivering the value of the Wigner function, the
symmetric-ordered quasi-distribution function at the origin of the phase space
Cahill1969a ; Royer1977 ; Englert1993 . It can be accessed directly via

$\displaystyle
M(\mu=2)=\sum_{n}(-1)^{n}\rho_{n}=\sum_{m}\frac{{g}^{(m)}}{m!}(-2\braket{n})^{m}.$
(47)

The moment generating function also allows one to re-express the photon-number
parity in terms of the normalized higher-order moments and the loss-corrected
mean photon number. The correction for losses is straightforward, since the
measured loss-degraded mean photon number is just divided by the efficiency,
with which it is being detected S.M.Barnett1997 . Such a method has
experimentally been demonstrated for reconstructing the photon-number parity
of heralded single photons generated via PDC emission from a semiconductor
waveguide Laiho2019 .

Nonetheless, for characterization tasks based on the measurement of higher-
order moments, care has to be taken that the summation ofn the right hand side
of Eq. (46) converges. If that is the case, the higher-order moments offer
expedient means for the direct probing of the Wigner function by
reconstructing the photon-number parity of the states displaced in the phase-
space similar to Refs Banaszek1996 ; S.Wallentowitz1996 ; Leibfried1996 ;
P.Bertet2002 . These methods rely on displacing the single-mode state in the
phase space with a complex amplitude
$\alpha=\textrm{Re}(\alpha)+\imath\textrm{Im}(\alpha)$ corresponding to the
transformation
$\varrho\rightarrow\hat{D}^{\dagger}(\alpha)\varrho\hat{D}(\alpha)$, in which
$\hat{D}(\alpha)$ is the displacement operator in phase space. Experimentally,
the displacement operator is easy to realize in an all-optical manner by
overlapping the studied state with a coherent displacement beam at an
asymmetric beam splitter DAriano1995 ; Paris1996 .

Figure 12: Characteristics of the displaced single-photon state,
$\hat{D}(\alpha)\ket{1}$, extracted via higher-order normalized moments. In
(a) we illustrate the normalized moments of the displaced single photons in
terms of the absolute value of the displacement $|\alpha|$ upto the 6th order
given by
$g^{(m)}_{\alpha}=\frac{|\alpha|^{2(m-1)[m^{2}+|\alpha|^{2}]}}{(1+|\alpha|^{2})^{m}}$
Laiho2012 ; Mahran1986 . In (b) we show the single photon Wigner function
reconstructed by estimating the photon-number parity of the displaced single
photon at the displacement $|\alpha|$ via the alternating sum in Eq. (48). The
summation is truncated upto the 6th (dotted line), 11th (dash-dotted line),
16th (dashed line) and 21st (solid line) moment. In (c) we visualize the three
dimensional Wigner function corresponding to the solid line in (b). The mean
photon number of the displaced single photon required for the reconstruction
is given by $\braket{n}_{\alpha}=1+|\alpha|^{2}$. The horizontal dash-dotted
lines provide guides for the eye.

The complete Wigner function can be probed via the alternating sum of the
displaced state’s photon statistics $\rho_{n}(\alpha)$ via

$\displaystyle W(\alpha)$
$\displaystyle=\frac{2}{\pi}\sum_{n}(-1)^{n}\braket{n}{\hat{D}^{\dagger}(\alpha)\varrho\hat{D}(\alpha)}{n}$
$\displaystyle=\frac{2}{\pi}\sum_{n}(-1)^{n}\rho_{n}(\alpha)$
$\displaystyle=\frac{2}{\pi}\sum_{m}\frac{(-2)^{m}}{m!}{g}^{m}_{\alpha}\braket{n}_{\alpha}^{m}.$
(48)

However, as shown in the last line of Eq. (48) it can also easily be extracted
via the higher-order moments, in other words, via the normalized $m$-th order
moments of the displaced state $g^{(m)}_{\alpha}$ and its mean photon number
$\braket{n}_{\alpha}$. This definition of the phase-space properties is most
suitable, when reconstructing the properties of the Wigner function in
discrete variables. We note that it is different from the one used for the
phase quadratures in Sec. IV, which again is optimally suited for the
detection in continous variables. In Fig. 12 we theoretically reconstruct the
single-photon Wigner function, whose characteristics are phase independent
Nehra2019 . Fig. 12(a) shows the characteristics of the normalized higher-
order moments of single photons displaced in phase space until the 6th order
in terms of the magnitude $|\alpha|$ of the displacement, whereas Fig. 12(b)
presents the values of the alternating sum in Eq. (48) by taking into account
different amount of moments. The truncation of the summation causes
distortions and limits the useful state reconstruction range in the phase
space. Clearly, even in an ideal case without experimental imperfections
moments up to $m\approx 21$ are required for reconstructing the interesting
range of the single-photon Wigner function depicted wholly in Fig. 12(c).

Figure 13:  (a) Q-function as well as (b) its three dimensional counterpart in
case $m\leq 21$ and (c) the absolute value squared of the characteristic
function of the single-photon state as well as (d) its three dimensional
counterpart in case $m\leq 21$ reconstructed from the normalized higher-order
moments of the displaced states and their mean photon number. The range of the
reconstruction is crucially influenced by the truncation of the summations in
Eqs (49) and (50) upto the 6th (dotted line), 11th (dash-dotted line), 16th
(dashed line) and 21st (solid line) moment in (a) and (b), correspondingly.

This way also the Husimi Q-function corresponding to the vacuum contribution
of the displaced state and the magnitude of the symmetric-ordered
characteristic function has been probed at least for free-propagating single
photons Laiho2012  and for the lowest number states in trapped ions Ziesel2013
; Wolf2019 . The Q-function that can be extracted via the vacuum contribution
of the displaced state $\rho_{0}(\alpha)$, can be very reliably reconstructed,
since all the measured orders of the higher-order moments contribute to the
summation, which is given by

$\displaystyle Q(\alpha)$ $\displaystyle=\frac{1}{\pi}\braket{\alpha}{\
\varrho\
}{\alpha}=\frac{1}{\pi}\braket{0}{\hat{D}^{\dagger}(\alpha)\varrho\hat{D}(\alpha)}{0}=\frac{1}{\pi}\rho_{0}(\alpha)$
$\displaystyle=\frac{1}{\pi}\sum_{m}\frac{(-1)^{m}}{m!}g^{(m)}_{\alpha}\braket{n}_{\alpha}^{m}.$
(49)

The characteristic function usually cannot be experimentally accessed via
photon counting. In the symmetric-order its given
$\chi(\alpha)=\textrm{Tr}\\{\hat{D}(\alpha)\varrho\\}$. With this respect the
number states $\ket{n}$ form an exception. The absolute value square of their
symmetric-ordered characteristic function
$\chi(\alpha)_{\ket{n}}=\braket{n}{\hat{D}(\alpha)}{n}$ is given by

$\displaystyle|\chi(\alpha)|^{2}_{\ket{n}}$
$\displaystyle=\braket{n}{\hat{D}^{\dagger}(\alpha)}{n}\braket{n}{\hat{D}(\alpha)}{n}=\rho_{n}(\alpha)_{\ket{n}}$
$\displaystyle=\sum_{m}\frac{(-1)^{m}}{n!m!}g^{(m+n)}_{\alpha}\braket{n}^{m+n}_{\alpha},$
(50)

and corresponds to the probability of the $n$-th photon-number contribution of
the displaced state. Clearly only moments higher than or equal to $n$
contribute to the summation. In Fig. 13 we illustrate the reconstruction of
both the Q-function and the absolute value squared of the characteristic
function for single photons when taking into account different amount of the
higher-order normalized moments. Altogether, the simultaneous access to
several characteristic properties of a quantum state offers an in-depth
insight to the quantum optical state under study.

##  VIII Conclusions

The characteristics of the photon statistics can be investigated via photon
correlations in a straightforward fashion. Moreover, the normalized higher-
order moments are especially expedient in such state characterization tasks,
in which the high optical losses wash out quantum features and prohibit their
direct evaluation from the photon-number contributions. Depending on the
temporal coherences of the investigated light, the photon correlations can be
investigated in time-dependent or time-averaged form. While for continuous-
wave light usually the time-dependent form provides most detailed information,
the time-averaged form is most suitable for pulsed light. Apart from
delivering information of the photon-number content of the emitted light, the
normalized higher-order moments are routinely employed for accessing the
intrinsic properties of the photonic sources. Additionally, they are feasible
for evaluating the state’s non-classicality and can even allow for
reconstructing the state’s phase-space characteristics. Certainly, care has to
be taken when measuring the normalized higher-order moments both in the
discrete and continuous variables, such that enough statistics is taken, which
can experimentally turn challenging. Moreover, the highest reliably resolved
moment restricts the accuracy, at which the state characteristics can be
extracted affecting also their reliable reconstruction range. We believe that
the photon correlations remain important in quantum optical state
classification and characterization tasks and their usage continues evolving
along the development of optical detection techniques at the single- and few
photon level both with and without true photon-number resolution.

## Acknowledgements

We thank M. von Helversen for laboratory support, J. Beyer at PTB for caring
for the TES detector, and A. E. Lita and S. W. Nam from NIST, USA, for
providing the transition-edge-sensor chips.

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