Section 3.1, the state evolves until time $t$ through the action of the
evolution operator $V(t)=U_{0}^{\dagger}(t)U(t)$ as
$\rho(t)=V(t)\rho(0)V^{\dagger}(t)$. Measuring the detector in its excited
state corresponds to the observable

$\outerproduct{e}{e}\otimes\mathds{1}=\outerproduct{e}{e}\otimes\sum_{f}\outerproduct{f}{f}=\sum_{f}\outerproduct{e,f}{e,f}\,,$
(89)

where $f$ indexes Fock states, and we decomposed the identity operator onto
the Fock states, $\mathds{1}=\sum_{f}\outerproduct{f}{f}$. Now the observable
(89), in the interaction picture, evolves as

$U_{0}^{\dagger}(t)\sum_{f}\outerproduct{e,f}{e,f}U_{0}(t).$

We are now ready to compute the probability $p_{\omega_{eg}}(\mathbf{r},t)$ of
the detector being measured in its excited state at time $t$. By Born’s rule
(see Section 3.1), this probability is the average value of the evolved
observable onto the evolved state $\rho(t)$, namely

$\displaystyle p_{\omega_{eg}}(\mathbf{r},t)$
$\displaystyle=\mathrm{Tr}\,\left(U_{0}^{\dagger}(t)\left[\sum_{f}\outerproduct{e,f}{e,f}\right]U_{0}(t)V(t)\outerproduct{g,i}{g,i}V^{\dagger}(t)\right)$
(90a)
$\displaystyle=\sum_{f}\mathrm{Tr}\,\left(\outerproduct{e,f}{e,f}U(t)\outerproduct{g,i}{g,i}U(-t)\right)$
(90b) $\displaystyle=\sum_{f}\left|\bra{e,f}U(t)\ket{g,i}\right|^{2}$ (90c)
$\displaystyle=\sum_{f}\left|\bra{e,f}V(t)\ket{g,i}\right|^{2},$ (90d)

where the last line comes from the definition $V(t)=U_{0}^{\dagger}(t)U(t)$
and the fact that $\ket{e,f}$ is an eigenvector of $H_{0}(t)$, so that
applying $U_{0}(t)$ to $\ket{e,f}$ simply multiplies by a complex number of
modulus $1$. Note also that the dependence in $\mathbf{r}$ is hidden in
$V(t)$.

Up until now, all the discussion is exact. We now introduce a physical
assumption that allows for a simple expression of $V(t)$.

######  Assumption 5 (Weak Coupling).

The interaction between the detector and the field is assumed to be weak,
allowing us to treat the dynamics using perturbation theory. Formally, this
means that we assume the existence of a series expansion of the evolution
operator in terms of the interaction Hamiltonian; see the discussion around
(32). To first order, this amounts to

$\displaystyle
V(t)\simeq\mathds{1}-\frac{\mathrm{i}}{\hbar}\int_{0}^{t}H_{I}(t^{\prime}){\mathrm{d}}t^{\prime}\,.$
(91)

Using (91) and
$\langle\mathrm{e}|\sigma_{x}(t)|g\rangle=\mathrm{e}^{\mathrm{i}\omega_{eg}t}$,
we find

$\displaystyle\langle
e,f|V(t)|g,i\rangle=\frac{\mathrm{i}q}{\hbar}\int_{0}^{t}\mathrm{e}^{\mathrm{i}\omega_{eg}t^{\prime}}\langle
f|E(\mathbf{r},t^{\prime})|i\rangle{\mathrm{d}}t^{\prime}\,.$ (92)

Plugging this into (94), and remembering that the Fock states $\ket{f}$ form a
basis of the Fock space, we obtain

$\displaystyle p_{\omega_{eg}}(\mathbf{r},t)$
$\displaystyle=\left(\frac{q}{\hbar}\right)^{2}\int_{[0,t]^{2}}\mathrm{e}^{\mathrm{i}\omega_{eg}(t^{\prime}-t^{\prime\prime})}\langle
i|E(\mathbf{r},t^{\prime\prime})E(\mathbf{r},t^{\prime})|i\rangle\,{\mathrm{d}}t^{\prime}{\mathrm{d}}t^{\prime\prime}\,.$
(93)

This equation is straightforwardly generalized to the case where the initial
state of the field is not a pure state but a mixed state, described by a
density matrix $\rho=\sum_{i}p_{i}\ket{i}\\!\\!\bra{i}$ in some basis
$(\ket{i})$ of initial states; see Section 3.1. The probability of a detection
event between $0$ and $t$ then becomes

$\displaystyle p_{\omega_{eg}}(\mathbf{r},t)$
$\displaystyle=\left(\frac{q}{\hbar}\right)^{2}\int_{[0,t]^{2}}\mathrm{e}^{\mathrm{i}\omega_{eg}(t^{\prime}-t^{\prime\prime})}\langle
E(\mathbf{r},t^{\prime\prime})E(\mathbf{r},t^{\prime})\rangle_{\rho}\,{\mathrm{d}}t^{\prime}{\mathrm{d}}t^{\prime\prime}\;.\,$
(94)

Equation (94) underlies most of photodetection theory. It relates properties
of the field, encoded in a _coherence function_ $\langle
E(\mathbf{r},t^{\prime\prime})E(\mathbf{r},t^{\prime})\rangle_{\rho}$, to the
excitation probability of the detector.

The second important approximation usually made in photodetection theory is
the _rotating wave approximation_ (RWA), which translates the intuition that a
proper photodetection event is an absorption of an excitation. To motivate and
illustrate it, we consider first a simple example where the field is assumed
to always be in a coherent state.

######  Example 6 (Monochromatic classical wave).

We assume here that the field always remains in a coherent state as time
evolves and is factorized with respect to the state of the qubit; see Section
3.6. Since the free part of the Hamiltonian preserves coherent states up to a
phase, what we are assuming is that the evolution under the interaction
Hamiltonian approximately preserves coherent states. Physically, this
assumption corresponds to, for instance, a macroscopic source emitting a
coherent state or a statistical mixture thereof, like a common lightbulb: the
fact that one observes the emitted light does not change its classical
character. We also assume, for simplicity, that we have an incoming
monochromatic coherent state at frequency $\omega$. Formally, we reduce the
sum in (85) to a single term $\mathbf{k}$ such that
$\omega_{\mathbf{k}}=\omega$, and denote by
$\mathcal{E}(t)=\mathcal{E}(\mathbf{r},t)$ the average value of this field at
the fixed position $\mathbf{r}$ of the detector. The terms in $\mathbf{r}$ can
be omitted by changing the time origin. More explicitly, we set
$\mathcal{E}(t)=\mathcal{E}\sin\omega t$. We want a coherent state
$\ket{\alpha}$ such that the average value of the electric field operator

$\expectationvalue{E(t)}{\alpha}=\expectationvalue{a_{\mathbf{k}}\mathrm{e}^{-\mathrm{i}\omega
t}+a_{\mathbf{k}}^{\dagger}\mathrm{e}^{\mathrm{i}\omega
t}}{\alpha}=\mathcal{N}_{\mathbf{k}}(\alpha\mathrm{e}^{-\mathrm{i}\omega
t}+\alpha^{*}\mathrm{e}^{\mathrm{i}\omega t})$

is equal to $\mathcal{E}(t)$. We thus choose the coherent state parameter
$\alpha$ to be $-\mathcal{E}/2\mathrm{i}\mathcal{N}_{\mathbf{k}}$.303030 The
normalization constant $\mathcal{N}_{k}$ is reinserted into the coherent state
parameter in this derivation to simplify the notation.  Furthermore, assume
that we initially prepare the detector in its ground state $\ket{g}$. In the
interaction picture, our approximation that the state of the field remains the
same coherent state313131Strictly speaking, we should allow $\alpha$ to depend
on time, but the argument below does not change. and factorized allows to
write

$V(t)\ket{g}\otimes\ket{\alpha}\approx(V_{\text{sc}}(t)\ket{g})\otimes\ket{\alpha}\;.$
(95)

The operator $V_{\text{sc}}(t)$ in (95) is defined as the interaction picture
evolution operator of the so-called _semi-classical_ Hamiltonian
$H_{\text{sc}}(t)=-q\mathcal{E}(t)\,\sigma_{x}(t)$. Using the weak coupling
approximation to evaluate the time evolution, we obtain

$\displaystyle p_{\omega_{eg}}(\mathbf{r},t)$
$\displaystyle=\mathrm{Tr}\,\left(U_{0}^{\dagger}(t)\big{[}\outerproduct{e}{e}\otimes\mathds{1}\big{]}U_{0}(t)\,V(t)\big{[}\outerproduct{g}{g}\otimes\outerproduct{\alpha}{\alpha}\big{]}V^{\dagger}(t)\right)$
(96a)
$\displaystyle=\mathrm{Tr}\,\left(\big{[}\outerproduct{e}{e}\otimes\mathds{1}\big{]}\,V(t)\big{[}\outerproduct{g}{g}\otimes\outerproduct{\alpha}{\alpha}\big{]}V^{\dagger}(t)\right)$
(96b)
$\displaystyle=\mathrm{Tr}\,\big{(}V_{\text{sc}}(t)\outerproduct{g}{g}V^{\dagger}_{\text{sc}}(t)\outerproduct{e}{e}\big{)}$
(96c)
$\displaystyle=\left|\frac{\mathrm{i}q}{\hbar}\int_{0}^{t}\mathcal{E}(t^{\prime})\,\mathrm{e}^{\mathrm{i}\omega_{eg}t^{\prime}}\,{\mathrm{d}}t^{\prime}\right|^{2}$
(96d)
$\displaystyle=\left|\frac{q\mathcal{E}}{2\hbar\mathrm{i}}\left(\frac{1-\mathrm{e}^{\mathrm{i}(\omega_{eg}-\omega)t}}{\omega_{eg}-\omega}-\frac{1-\mathrm{e}^{\mathrm{i}(\omega_{eg}+\omega)t}}{\omega_{eg}+\omega}\right)\right|^{2}\,,$
(96e)

where the second line is obtained because the state $\ket{e}$ is an eigenvalue
of the free evolution operator $U_{0}(t)$, the third follows from the semi-
classical assumption that the state of the field always stays in a coherent
state, the fourth from the weak coupling assumption and the form of
$H_{\text{sc}}$, and the last from the monochromatic assumption.

At this point, we make an extra assumption called the _rotating wave
approximation_ (RWA). In our special case, the RWA amounts to neglecting the
second term of the right-hand side of (96e). Because both $\omega$ and
$\omega_{eg}$ are strictly positive, the denominator of the second term never
vanishes, unlike the denominator of the first term. Around resonance
$\omega\approx\omega_{eg}$, the first term will largely dominate the second. ∎

We make the RWA from Example 6 a general principle that applies to arbitrary
quantum fields. It echoes the use of the analytic signal in representing a
real classical field; see Section 2.

######  Assumption 6 (Rotating wave approximation).

The photodetection response is a function of the positive frequency part of
the quantum field only.

With Assumption 6, the transition probability (94) now reads

$\displaystyle
p_{\omega_{eg}}(\mathbf{r},t)=\left(\frac{e\,q}{\hbar}\right)^{2}\int_{0}^{t}\mathrm{e}^{\mathrm{i}\omega_{eg}(t^{\prime}-t^{\prime\prime})}G^{(1)}_{\rho}\big{(}\mathbf{r}\,t^{\prime},\mathbf{r}\,t^{\prime\prime}\big{)}\,{\mathrm{d}}t^{\prime}{\mathrm{d}}t^{\prime\prime}\,,$
(97)

where323232 The missing commas in the arguments are a convention that will
become handy when we consider higher-order coherence functions.  we introduced
the _first-order coherence function_

$\displaystyle
G^{(1)}_{\rho}\big{(}\mathbf{r}^{\prime}\,t^{\prime},\mathbf{r}\,t\big{)}=\langle
E^{-}(\mathbf{r}^{\prime},t^{\prime})E^{+}(\mathbf{r},t)\rangle_{\rho}\,,$
(98)

where $E^{+}$ is associated to the positive part of the spectrum (annihilation
operators) and $E^{-}$ is associated to the negative part (creation
operators). The coherence function (98) was originally introduced by Glauber
(1963).333333 As a side remark, remember that this coherence function is
defined after choosing a specific polarization of the field. In the general
case, the electric field should be expanded along a basis of the three-
dimensional polarization space and we should consider all combinations of
coherence functions  $G^{(1)}_{ij}(t^{\prime},t)=\langle
E_{j}^{-}(t^{\prime})E_{i}^{+}(t)\rangle_{\rho}$ where $i,j$ label the basis
elements.

At least informally, it is not difficult to generalize the computations
leading to (94) to more sophisticated detectors. In experiments, for instance,
photodetection is often performed through ionization: an electron of an atom
is extracted by the incident light and this event is amplified and detected.
This detector is best described by a system having, on top of its ground
$\ket{g}$ state, a countable set of excited states $\ket{e_{i}}$, and a
continuum of so-called _diffusive_ states $\ket{p},p\in\mathbb{R}$. This leads
to non-compact Hamiltonian operators, having a spectrum that is partly
discrete, partly continuous. Nonetheless, the computation of the transition
probability $p(\mathbf{r},t)$ follows from the same lines as above, with the
new observation corresponding to projecting on all states that are not the
ground state, that is,
$(\sum_{n}\outerproduct{e_{n}}{e_{n}}+\int\outerproduct{p}{p}{\mathrm{d}}p)\otimes\mathds{1}$
instead of $\outerproduct{e}{e}\otimes\mathds{1}$. Because we are working with
a first-order approximation in Assumption 5, this leads to a probability of
transition that is an expectation of the single-state probability of
transition (94), namely

$\displaystyle p(\mathbf{r},t)=\int
p_{\omega_{eg}}(\mathbf{r},t){\mathrm{d}}\xi(\omega_{eg})=\int_{[0,t]^{2}}\kappa(t^{\prime}-t^{\prime\prime})G^{(1)}_{\rho}\big{(}\mathbf{r}\,t^{\prime},\mathbf{r}\,t^{\prime\prime}\big{)}\,{\mathrm{d}}t^{\prime}{\mathrm{d}}t^{\prime\prime}\,,$
(99)

where $\xi$ has a support that is the union of a discrete set, corresponding
to the states $\ket{e_{n}}$, and an uncountable set, corresponding to
$\ket{p},p\in\mathbb{R}$, and the structure function is defined as

$\kappa(\tau):=\left(\frac{e}{\hbar}\right)^{2}\int\mathrm{e}^{-\mathrm{i}\omega\tau}{\mathrm{d}}\xi(\omega).$

We can think of either $\xi$ or $\kappa$ as being a characteristic of the
detector, describing its time or frequency efficiency. For instance, if $\xi$
is supported on a single point or closely around a single point, we only
collect photons at a very specific energy, and call the detector _narrow-
band_. If $\xi$ is supported on a large interval, we collect all photons
equally, and call the detector _broad-band_.

For completeness, we close this section by recovering the semi-classical
treatment of Section 2.3 through our quantum treatment, using the coherent
state formalism of Section 3.6.

######  Example 7 (General rotating wave).

By linearity arguments, the monochromatic light case of Example 6 can be used
to derive the excitation probability in the case of a more general field.
Thinking of the field being enclosed in a large box, so as to be able to work
with Fourier series instead of Fourier transforms for simplicity, let us
decompose the electric field $\mathcal{E}_{\boldsymbol{\alpha}}(\mathbf{r},t)$
on a set of monochromatic plane waves

$\displaystyle\mathcal{E}_{\boldsymbol{\alpha}}(\mathbf{r},t)=\sum_{\mathbf{k}}\mathcal{N}_{\mathbf{k}}\left[\alpha_{\mathbf{k}}\mathrm{e}^{\mathrm{i}(\mathbf{k}\cdot\mathbf{r}-\omega_{\mathbf{k}}t)}+\alpha^{*}_{\mathbf{k}}\mathrm{e}^{-\mathrm{i}(\mathbf{k}\cdot\mathbf{r}-\omega_{\mathbf{k}}t)}\right]\,,$
(100)

with $\mathcal{N}_{\mathbf{k}}$ a normalization coefficient and
$\alpha_{\mathbf{k}}\in\mathbb{C}$ the amplitude of the mode $\mathbf{k}$.
Note that $\mathcal{E}$ is a (real-valued) function, not an operator: we are
still treating the field as a classical field. The first part of the sum
corresponds to the positive frequency part of the field (the “analytic”
field), while the second part corresponds to the negative frequency part
$\mathcal{E}^{-}$.

An important observable of the electromagnetic field is its intensity. A
general result of Maxwell’s equations (Mandel and Wolf, 1995) is that the
energy carried by the field is given by the squared modulus of the field.
Under our rotating wave Assumption 6, the intensity $I(\mathbf{r},t)$ of the
field reads

$\displaystyle
I(\mathbf{r},t)=\mathcal{E}_{\boldsymbol{\alpha}}^{-}(\mathbf{r},t)\mathcal{E}_{\boldsymbol{\alpha}}^{+}(\mathbf{r},t)=|\mathcal{E}_{\boldsymbol{\alpha}}^{+}(\mathbf{r},t)|^{2}\,.$
(101)

Investigating this intensity, and especially its correlation structure in the
case of random sources, has led to the major discoveries of photon bunching
and in the end of the general theory of optical coherence in quantum theory
(Mandel and Wolf, 1995). More precisely, say the field (100) is not perfectly
known; it is then reasonable to model it as a stochastic process, i.e., a
random function. A very common assumption is to assume the process to be a
stationary Gaussian process. One partial justification is that the field at a
given spacetime point is the sum of the contributions of many independent
sources, and is well approximated by a Gaussian. Formally, the field state
becomes a statistical mixture of coherent states, a _mixed state_ in the
vocabulary of Section 3.1. The probability of excitation remains of the form
(94), with first-order coherence function

$\displaystyle
G^{(1)}_{\rho}\big{(}\mathbf{r}\,t^{\prime},\mathbf{r}\,t^{\prime}\big{)}$
$\displaystyle=\mathbb{E}_{\boldsymbol{\alpha}}\expectationvalue{E^{-}(\mathbf{r}^{\prime},t^{\prime})E^{+}(\mathbf{r},t)}{\boldsymbol{\alpha}}$
(102)
$\displaystyle=\mathbb{E}_{\boldsymbol{\alpha}}\mathcal{E}_{\boldsymbol{\alpha}}^{-}(\mathbf{r}^{\prime},t^{\prime})\mathcal{E}_{\boldsymbol{\alpha}}^{+}(\mathbf{r},t)$
(103) $\displaystyle=\int
p_{\text{classical}}(\boldsymbol{\alpha})\mathcal{E}_{\boldsymbol{\alpha}}^{-}(\mathbf{r}^{\prime},t^{\prime})\mathcal{E}_{\boldsymbol{\alpha}}^{+}(\mathbf{r},t)\,{\mathrm{d}}\boldsymbol{\alpha}\,.$
(104)

The expectation is over the stochastic process $\mathcal{E}$. Physicists
prefer to denote it informally using a probability density
$p_{\text{classical}}$ over the Fourier coefficients
$\boldsymbol{\alpha}=(\alpha_{\mathbf{k}})$ of the field in (100), and we use
that notation here for future notational compatibility with Glauber-Sudarshan
decompositions. In Section 4.2, we shall use Wick’s theorem to derive higher-
order coherence functions, and recognize the set of correlation functions of a
permanental point process.

∎

####  4.1.3 Role of the detector structure function

In this section, we consider the influence of the detector response $\kappa$
on the probability of excitation (99).

##### Broad-band detector.

Consider first a broad-band detector, i.e. a detector with structure function
that is sharply peaked in time or, equivalently, does not depend much on the
frequency. Ideally, this corresponds to
$\kappa(t-t^{\prime})=\kappa_{0}\,\delta(t-t^{\prime})$ with $\kappa_{0}$ a
real constant. The probability of excitation (99) becomes

$\displaystyle
p(\mathbf{r},t)=\kappa_{0}\int_{0}^{t}G^{(1)}_{\rho}\big{(}\mathbf{r}\,t^{\prime},\mathbf{r}\,t^{\prime}\big{)}\,{\mathrm{d}}t^{\prime}\,.$
(105)

Anticipating over Section 4.3, we would like to map this probability to a
point process on the positive half-line $\mathbb{R}_{+}$. One way to associate
the two notions is to imagine $t>0$ infinitely small in (105), and “re-
setting” the detector in its ground state at each time multiple of $t$,
keeping track of whether the detector was found in its excited state, say in
the middle of each small time interval. Loosely speaking, in the limit
$t\rightarrow 0$, these detection times would form a point process on the
positive half of the real line, with first correlation function
$G^{(1)}_{\rho}\big{(}\mathbf{r}\,t^{\prime},\mathbf{r}\,t^{\prime}\big{)}$.
To obtain a more formal correspondence, one needs to further model the
detection process, for instance including de-excitation of the detector, see
e.g. Bouten et al. (2007).

##### Narrow-band detector.

A narrow-band detector is a detector that is well-resolved in frequency, like
the two-level atom we used in our derivation of the photodetection signal. At
the extreme, we think of the structure function as a delta distribution in
Fourier space at a frequency $\omega_{0}$, so that
$\mathcal{F}\kappa(\omega)=\kappa_{0}\delta(\omega-\omega_{0})$, and thus
$\kappa(t)=\kappa_{0}/\sqrt{2\pi}\mathrm{e}^{\mathrm{i}\omega_{0}t}$. The
excitation probability (99) now reads

$\displaystyle p(\mathbf{r},t)$
$\displaystyle=\frac{\kappa_{0}}{\sqrt{2\pi}}\int_{[0,t]^{2}}G^{(1)}_{\rho}\big{(}\mathbf{r}\,t^{\prime},\mathbf{r}\,t^{\prime\prime}\big{)}\mathrm{e}^{\mathrm{i}\omega_{0}(t^{\prime}-t^{\prime\prime})}\,{\mathrm{d}}t^{\prime}{\mathrm{d}}t^{\prime\prime}\;.\,,$
(106)

The meaning of this equation is even more transparent when we switch to the
frequency representation of the coherence function. Indeed, let us define the
Fourier representation of the first-order coherence function through the
Fourier transform(-like) equation

$\displaystyle
G^{(1)}_{\rho}(\mathbf{r}^{\prime}\,t^{\prime},\mathbf{r}^{\prime\prime}\,t^{\prime\prime})=\int_{\mathbb{R}^{2}}G^{(1)}_{\rho}(\mathbf{r}^{\prime}\,\omega^{\prime},\mathbf{r}^{\prime\prime}\,\omega^{\prime\prime})\mathrm{e}^{-\mathrm{i}\omega^{\prime}t^{\prime}}\mathrm{e}^{\mathrm{i}\omega^{\prime\prime}t^{\prime\prime}}\,\frac{{\mathrm{d}}\omega^{\prime}{\mathrm{d}}\omega^{\prime\prime}}{2\pi}\,,$
(107)

where we follow the physicists’ convention to denote both the function and its
Fourier transform by the same notation and index the function by the name of
its variable. Putting everything together, we arrive at

$\displaystyle
p(\mathbf{r},t)=\frac{\kappa_{0}}{\sqrt{2\pi}}\int_{\mathbb{R}^{2}}G^{(1)}_{\rho}(\mathbf{r}\,\omega,\mathbf{r}^{\prime}\,\omega^{\prime})\,2t\frac{\sin(t(\omega-\omega_{0}))}{(\omega-\omega_{0})}\,\frac{{\mathrm{d}}\omega{\mathrm{d}}\omega^{\prime}}{2\pi}\;.$
(108)

In the long time limit343434 usually the coherence function varies over some
timescale, so large time means $t$ sufficiently large compared to that
timescale , the integration window is approximately equal to
$t\delta(\omega-\omega_{0})$. We end up with the simple expression

$\displaystyle
p(\mathbf{r},t)\approx\frac{\kappa_{0}t}{\sqrt{2\pi}}G^{(1)}_{\rho}(\mathbf{r}\,\omega_{0},\mathbf{r}\,\omega_{0})\,.$
(109)

The interpretation of this expression is easier now. For a narrow-band
detector in frequency space centered around a frequency $\omega_{0}$, the
excitation probability is proportional to the diagonal $\omega_{0}$ part of
the first-order coherence function which is none other than the average
intensity of the mode $\omega_{0}$.

###  4.2 Correlation between photodetection events

We now examine the joint probability of detecting two, and then more, photons.

####  4.2.1 Second-order coherence

Consider two detectors labelled $1$ and $2$ and placed at two distinct
positions $\mathbf{r}_{1}$ and $\mathbf{r}_{2}$. They are excited by an
incident radiation, and we ask for the probability that detector $1$ gets
excited at time $t_{1}$ _and_ detector $2$ gets excited at time $t_{2}$.
Actually, the derivation of the joint probability of detection is quite
similar to the derivation for the one photodection event in Section 4.1.2, and
we will only give the general principles of the method here again.

The dynamics are once again modeled through a Hamiltonian, whose interaction
term is given by the following generalization of (88),

$\displaystyle H_{I}(t)=q_{1}\sigma^{(1)}_{x}(t)\otimes\mathds{1}\otimes
E(\mathbf{r}_{1},t)+q_{2}\mathds{1}\otimes\sigma^{(2)}_{x}(t)\otimes
E(\mathbf{r}_{2},t)\,.$ (110)

Such a form is natural to physicists, in the sense that it models the local
interaction of each of two detectors with the field at their respective
position.

Computing the joint probability of excitation requires to compute first a
matrix element of the evolution operator in the interaction picture, which is
again done assuming weak coupling. We will see that in order to have a non-
trivial result, we need to push the expansion in 5 up to second order.

######  Assumption 7 (Weak Coupling; second order).

The interaction between the detectors and the field is assumed to be weak,
allowing to treat the dynamics using perturbation theory. To second order,
this amounts to

$\displaystyle
V(t)\simeq\mathds{1}-\frac{\mathrm{i}}{\hbar}\int_{0}^{t}H_{I}(t_{1}){\mathrm{d}}t_{1}-\frac{1}{\hbar^{2}}\int_{0}^{t}\int_{0}^{t_{1}}H_{I}(t_{1})H_{I}(t_{2})\,{\mathrm{d}}t_{2}{\mathrm{d}}t_{1}\,.$
(111)

Being interested in the joint probability of excitation after a time $t$, we
have to compute the matrix element $\langle
e_{1},e_{2},f|V(t)|g_{1},g_{2},i\rangle$ where $(\ket{g_{1}},\ket{g_{2}})$ and
$(\ket{e_{1}},\ket{e_{2}})$ are the ground and excited states of both
detectors. Since they are all pairwise orthogonal, the $\mathds{1}$ term in
(111) will give a zero contribution. As for the linear term, and given the
form of the interaction (110) as a sum of two contributions from each
detector, it will split into two terms, each acting trivially on one of the
detectors, again yielding a zero result. For instance

$\displaystyle\langle
e_{1},e_{2},f|\sigma^{(1)}_{x}(t)\otimes\mathds{1}\otimes
E(\mathbf{r}_{1},t)|g_{1},g_{2},i\rangle=\matrixelement{e_{1}}{\sigma^{(1)}_{x}(t)}{g_{1}}\innerproduct{e_{2}}{g_{2}}\matrixelement{f}{E(\mathbf{r}_{1},t)}{i}=0$

by orthogonality of $\ket{e_{2}}$ and $\ket{g_{2}}$. This is the reason why we
have to go to the second order in perturbation theory to compute the joint
probability distribution. In general, to compute the joint probability
distribution of $N$ detectors clicking, we will need to push the expansion of
7 to the order $N$, since none of the lower-order terms contributes. Going
back to $N=2$, the quadratic term also simplifies with only the cross terms
$E(\mathbf{r}_{2},t_{2})E(\mathbf{r}_{1},t_{1})$ giving a non-zero
contribution. Using again the RWA of 6, we end up with

$\displaystyle\langle e_{1},e_{2},f|$ $\displaystyle
V(t)|g_{1},g_{2},i\rangle$ (112)
$\displaystyle=-\frac{1}{\hbar^{2}}\int_{0}^{t}\int_{t_{1}}^{t}\mathrm{e}^{\mathrm{i}\omega_{eg_{1}}t_{1}}\mathrm{e}^{\mathrm{i}\omega_{eg_{2}}t_{2}}\langle
f|E^{+}(\mathbf{r}_{2},t_{2})E^{+}(\mathbf{r}_{1},t_{1})|i\rangle\,{\mathrm{d}}t_{2}{\mathrm{d}}t_{1}+1\leftrightarrow
2$
$\displaystyle=-\frac{1}{\hbar^{2}}\int_{0}^{t}\int_{0}^{t}\mathrm{e}^{\mathrm{i}\omega_{eg_{1}}t_{1}}\mathrm{e}^{\mathrm{i}\omega_{eg_{2}}t_{2}}\langle
f|E^{+}(\mathbf{r}_{2},t_{2})E^{+}(\mathbf{r}_{1},t_{1})|i\rangle\,{\mathrm{d}}t_{2}{\mathrm{d}}t_{1}\,.$
(113)

The notation $1\leftrightarrow 2$ is a shorthand for the same term with the
two indices transposed. The second line is obtained by grouping the two terms
into one and changing the integration range, noting that the involved
operators commute. Just like we did in Section 4.1.2, we use Born’s rule and
sum over the final state of the field, since we are only interested in the
state of the detectors, to obtain the joint excitation probability. Once
again, we express it as the Fourier transform of a second-order coherence
function. Assuming a fixed polarization for the field for simplicity, we have

$\displaystyle
G^{(2)}_{\rho}(\mathbf{r}_{1}^{\prime}\,t_{1}^{\prime},\mathbf{r}_{2}^{\prime}\,t^{\prime}_{2},\mathbf{r}_{2}\,t_{2},\mathbf{r}_{1}\,t_{1}):=\langle
E^{-}(\mathbf{r}_{1}^{\prime},t^{\prime}_{1})E^{-}(\mathbf{r}_{2}^{\prime},t^{\prime}_{2})E^{+}(\mathbf{r}_{2},t_{2})E^{+}(\mathbf{r}_{1},t_{1})\rangle_{\rho}\,.$
(114)

By once again introducing structure functions $\kappa_{1}$ and $\kappa_{2}$
for both detectors, we obtain our final expression for the joint excitation
probability

$\displaystyle
p(\mathbf{r}_{2}\,t,\mathbf{r}_{1}\,t)=\int_{0}^{t}\kappa_{1}(t^{\prime}_{1}-t^{\prime\prime}_{1})\kappa_{2}(t^{\prime}_{2}-t^{\prime\prime}_{2})G^{(2)}_{\rho}(\mathbf{r}_{1}\,t^{\prime\prime}_{1},\mathbf{r}_{2}\,t_{2}^{\prime\prime},\mathbf{r}_{2}\,t_{2}^{\prime},\mathbf{r}_{1}\,t_{1}^{\prime}))\,{\mathrm{d}}t_{1}^{\prime}{\mathrm{d}}t_{1}^{\prime\prime}{\mathrm{d}}t_{2}^{\prime}{\mathrm{d}}t_{2}^{\prime\prime}\,,$
(115)

##### Intensity correlations.

Of particular experimental relevance is the diagonal of the second-order
coherence function, namely

$\displaystyle
G^{(2)}_{\rho}(\mathbf{r}_{1}\,t_{1},\mathbf{r}_{2}\,t_{2},\mathbf{r}_{2}\,t_{2},\mathbf{r}_{1}\,t_{1})=\langle
E^{-}(\mathbf{r}_{1},t_{1})E^{-}(\mathbf{r}_{2},t_{2})E^{+}(\mathbf{r}_{2},t_{2})E^{+}(\mathbf{r}_{1},t_{1})\rangle_{\rho}\,.$
(116)

For reference, if we were treating a classical field (i.e., modeled as a real-
or complex-valued stochastic process rather than a collection of noncommuting
operators), all the fields would commute and from the definition of the field
intensity, we would obtain

$\displaystyle
G^{(2)}_{\rho}(\mathbf{r}_{1}\,t_{1},\mathbf{r}_{2}\,t_{2},\mathbf{r}_{2}\,t_{2},\mathbf{r}_{1}\,t_{1})=\mathbb{E}I(\mathbf{r}_{2},t_{2})I(\mathbf{r}_{1},t_{1})\,.$
(117)

Equation 116 is the non-commuting counterpart to (117), which gives us
information about the intensity correlations of the field. The order of the
operators in (116), called _normal order_ , is crucial for the interpretation
of the photodetection signal as the absorption of excitations by the detector,
but also to take properly into account the fundamental quantum fluctuations
that affect the detection signal.

Intensity correlations can be probed experimentally. For instance, assume a
perfect temporal resolution for each detector, and that they are functioning
in the time intervals $[0,t_{i}]$, $i=1,2$. This is modeled by choosing for
structure functions

$\kappa(t_{i}-t_{i}^{\prime})=\kappa_{i}\,\Theta_{[0,t_{i}]}\delta(t_{i}-t_{i}^{\prime}),i=1,2,$

with $\kappa_{1},\kappa_{2}>0$. The probability of joint excitation is then
given by

$\displaystyle
p(\mathbf{r}_{2}\,t_{2},\mathbf{r}_{1}\,t_{1})=\kappa_{1}\kappa_{2}\int_{0}^{t_{2}}\int_{0}^{t_{1}}G^{(2)}_{\rho}(\mathbf{r}_{1}\,t^{\prime},\mathbf{r}_{2}\,t^{\prime\prime},\mathbf{r}_{2}\,t^{\prime\prime},\mathbf{r}_{1}\,t^{\prime}))\,{\mathrm{d}}t^{\prime}{\mathrm{d}}t^{\prime\prime}\,.$
(118)

In particular, taking the derivative in $t_{1}$ and $t_{2}$ yields the so-
called transition rate,

$\displaystyle\Gamma(\mathbf{r}_{2}\,t_{2},\mathbf{r}_{1}\,t_{1})=\kappa_{1}\kappa_{2}G^{(2)}_{\rho}(\mathbf{r}_{1}\,t_{1},\mathbf{r}_{2}\,t_{2},\mathbf{r}_{2}\,t_{2},\mathbf{r}_{1}\,t_{1})\,,$
(119)

which is proportional to (116), and reduces to the intensity correlation
function in the classical setting.

####  4.2.2 Higher-order coherences

The previous derivation can be generalized straightforwardly to obtain the
joint probability of $n$ detection events. Denoting the spacetime coordinates
by $x_{i}=(\mathbf{r}_{i},t_{i})$, we define the $n$th order coherence
function by

$\displaystyle G^{(n)}_{\rho}(x_{1},\cdots,x_{2n})=\langle E^{-}(x_{1})\cdots
E^{-}(x_{n})E^{+}(x_{n+1})\cdots E^{+}(x_{2n})\rangle_{\rho}\,,\quad n\geq 1.$
(120)

This definition generalizes the first- and second-order coherence functions.
Upon noting that $\mathrm{Tr}\,(\rho O^{\dagger}O)\geq 0$ for any operator
$O$, we can derive general inequalities for the coherence functions (120). For
instance, consider the operator $O=\sum_{k=1}^{m}\lambda_{k}E^{+}(x_{k})$, for
arbitrary $\lambda_{k}$’s. The nonnegativity of the quadratic form
$\mathrm{Tr}\,(\rho O^{\dagger}O)$ implies a contraint in the underlying
determinant. For $m=2$, we obtain

$G^{(1)}_{\rho}(x,x)G^{(1)}_{\rho}(y,y)\geq|G^{(1)}_{\rho}(x,y)|^{2},$

which we interpret as a sign of bunching.

Finally, we note that for general states, higher-order coherence functions are
typically hard to compute. One special case is for a Gaussian state (61),
where Wick’s Theorem 3 applies. The $n$th-order coherence function can then be
expanded as

$\displaystyle
G^{(n)}_{\rho}(x_{1},\cdots,x_{2n})=\sum_{\sigma\in\mathfrak{S}_{n}}\prod_{i=1}^{n}\langle
E^{-}(x_{i})E^{+}(x_{n+\sigma(i)})\rangle_{\rho}\,.$ (121)

We now have all the tools to identify the point processes behind the detection
of photons.

###  4.3 Turning coherence functions into correlation functions

There is a specific set of states for which the coherence functions are
particularly simple, namely the coherent states of Section 3.6. Specifically,
take a multi-mode coherent state
$\ket{\boldsymbol{\alpha}}=\otimes_{\mathbf{k}}\ket{\alpha_{\mathbf{k}}}$.
These states are eigenstates of the electric field (85) itself, with
eigenvalue the classical field from (100) , i.e.,

$\displaystyle
E^{+}({\mathbf{r}}_{1},t_{1})\ket{\boldsymbol{\alpha}}=\mathcal{E}^{+}_{\boldsymbol{\alpha}}({\mathbf{r}}_{1},t_{1})\ket{\boldsymbol{\alpha}}=\left(\sum_{\mathbf{k}}\mathcal{E}^{+}_{\alpha_{\mathbf{k}}}({\mathbf{r}}_{1},t_{1})\right)\otimes_{\mathbf{k}}\ket{\alpha_{\mathbf{k}}}\,.$
(122)

Physically, coherent states correspond to ideal lasers. Now, by construction,
the quantum coherence functions (120) are reduced to the classical coherence
functions when the field is prepared in such a state, i.e., writing
$x_{i}=(\mathbf{r}_{i},t_{i})$,

$\displaystyle
G^{(n)}_{\outerproduct{\boldsymbol{\alpha}}{\boldsymbol{\alpha}}}(x_{1},\cdots,x_{2n})$
$\displaystyle=\expectationvalue{E^{-}(x_{1})\cdots
E^{-}(x_{n})E^{+}(x_{n+1})\cdots E^{+}(x_{2n})}{\boldsymbol{\alpha}}$
$\displaystyle=\prod_{i=1}^{n}\overline{\mathcal{E}^{+}_{\boldsymbol{\alpha}}(x_{i})}\mathcal{E}^{+}_{\boldsymbol{\alpha}}(x_{n+i}).$
(123)

This factorization across coherent states is what allows us recovering the
correlation functions of a Poisson point process.

######  Example 8 (Ideal lasers yield Poisson processes).

Take $x_{i}=x_{n+i}=(\mathbf{r},t_{i})$ for some fixed position
$\mathbf{r}\in\mathbb{R}^{3}$ in (123), and assume a broad-band detector
placed at $\mathbf{r}$, see Section 4.1.3. The joint detection probabilities
yield measurement times that follow a point process with correlation functions

$\rho_{n}(t_{1},\dots,t_{n})=|\mathcal{E}^{+}_{\boldsymbol{\alpha}}(t_{1})\cdots\mathcal{E}^{+}_{\boldsymbol{\alpha}}(t_{n})|^{2}.$

Such a point process indeed exists: it is the Poisson point process of first
correlation function $|\mathcal{E}_{\boldsymbol{\alpha}}^{+}|^{2}$; see
Section 2.2. ∎

Now, consider a slightly more general state $\rho$ that can be written as a
statistical mixture $\rho=\mathbb{E}_{\boldsymbol{\alpha}\sim
p_{\text{classical}}}\outerproduct{\boldsymbol{\alpha}}{\boldsymbol{\alpha}}$
of coherent states. Concretely, this state describes non-ideal laser light,
e.g. where there is uncertainty on which light amplitude is emitted. For
instance, a single-mode randomly phased laser falls into this category, where
$p_{\text{classical}}$ is uniform over a circle in the complex plane: the
mixture is over all possible phases with a fixed intensity $|\alpha|^{2}$. As
for the coherence functions, by linearity, we obtain

$\displaystyle G^{(n)}_{\rho}(x_{1},\cdots,x_{2n})$
$\displaystyle=\mathbb{E}_{\boldsymbol{\alpha}\sim
p_{\text{classical}}}\prod_{i=1}^{n}\overline{\mathcal{E}^{+}_{\boldsymbol{\alpha}}(x_{i})}\mathcal{E}^{+}_{\boldsymbol{\alpha}}(x_{n+i}).$

######  Example 9 (Mixture states in phase space yield Cox processes).

Taking again $x_{i}=x_{n+i}=(\mathbf{r},t_{i})$ for some fixed position
$\mathbf{r}\in\mathbb{R}^{3}$ in (123) and a broad-band detector, the
detection times follow a point process with correlation functions

$\rho_{n}(t_{1},\dots,t_{n})=\mathbb{E}_{\boldsymbol{\alpha}\sim
p_{\text{classical}}}|\mathcal{E}^{+}_{\boldsymbol{\alpha}}(t_{1})\cdots\mathcal{E}^{+}_{\boldsymbol{\alpha}}(t_{n})|^{2}.$

The detection times thus form a Cox point process; see Section 2. In
particular, arrival times tend to exhibit bunching: they are more clustered
together than a Poisson process. Moreover, assuming that each $\alpha_{k}$ in
(100) is Gaussian, $\mathcal{E}_{\alpha}$ is a Gaussian process, and the
detection times then form a permanental point process. Permanental point
processes are the archetype of a point process exhibiting bunching; see
Section 2.3 and a few samples in Figure 1(b). The fact that we obtain a
permanental point process validates _a posteriori_ the semiclassical analysis
carried out in Section 2. ∎

Additionally, note that while we mention ideally resolved detectors in time
for simplicity, non-broad-band detectors also yield standard point processes
when the detector’s characteristic functions can be normalized. Indeed,
consider for simplicity the point process formed by the detection times at
detectors that share the same characteristic function $\kappa$. The
correlation functions then correspond to the point process formed by the marks
in a marked point process with independent marks. In the vocabulary of (Daley
and Vere-Jones, 2003, Chapter 6), the ground process is formed by the ideal
detection times, and for each such time $t_{i}$, an actual detection time is
drawn from the probability density function proportional to $\kappa(\cdot-
t_{i})$, independently from all other ideal detection times and marks.

Now we consider states that are not statistical mixtures of coherent states,
the so-called _non-classical_ or _genuinely quantum_ states. As we saw in
Section 3.6, coherent states form an overcomplete family of states. This
suggests that even nonclassical states admit a decomposition as a linear
combination of coherent states. The decomposition of any mixed state as a
linear combination of projectors onto coherent states is known as the
_Glauber-Sudarshan_ decomposition. However, the linear decomposition is not a
statistical mixture over a _bona fide_ probability density, and needs to be
interpreted in a weak sense. Informally, physicists write

$\rho=\int
P_{\rho}(\boldsymbol{\alpha})\outerproduct{\boldsymbol{\alpha}}{\boldsymbol{\alpha}}{\mathrm{d}}\boldsymbol{\alpha},$
(124)

with $P_{\rho}(\boldsymbol{\alpha})$ a tempered distribution for each
$\boldsymbol{\alpha}$; see (Mandel and Wolf, 1995, Section 11.8) and
references therein. For a general state, there is no reason for $P_{\rho}$ to
be a positive function, or even a function at all. We do not attempt to
formalize the Glauber-Sudarshan decomposition here, but we study the simplest
example of a nonclassical state: a Fock state.

Consider a single-mode Fock state

$\rho=\outerproduct{k}{k}\otimes I\otimes\dots.$

The associated Glauber-Sudarshan distribution, denoted for simplicity
$P_{\ket{k}}$, is the “derivative of a Dirac delta”, namely

$\displaystyle
P_{\ket{k}}(\alpha)=\frac{\mathrm{e}^{|\alpha|^{2}}}{k!}\frac{\partial^{2k}}{\partial^{k}\alpha\partial^{k}\overline{\alpha}}\delta(\alpha)\,;$
(125)

see (Mandel and Wolf, 1995, Section 11.8). This representation of a Fock state
is very singular and comes with no simple statistical interpretation. To get
some intuition about Equation 125, let us fix two arbitrary states $\ket{u}$
and $\ket{v}$. We have in fact the matrix elements

$\displaystyle\matrixelement{u}{\rho}{v}$
$\displaystyle=\int_{\mathbb{C}^{2}}\left(\frac{\partial^{2k}}{\partial^{k}\alpha\partial^{k}\overline{\alpha}}\frac{\mathrm{e}^{|\alpha|^{2}}}{k!}\innerproduct{u}{\alpha}\innerproduct{\alpha}{v}\right)\delta(\alpha)\,{\mathrm{d}}\alpha$
(126)
$\displaystyle=\left(\frac{\partial^{2k}}{\partial^{k}\alpha\partial^{k}\overline{\alpha}}\frac{\mathrm{e}^{|\alpha|^{2}}}{k!}\innerproduct{u}{\alpha}\innerproduct{\alpha}{v}\right)\bigg{|}_{\alpha=\overline{\alpha}=0}\,,$
(127)

as long as $\alpha\mapsto\innerproduct{u}{\alpha}\innerproduct{\alpha}{v}$ is
smooth enough. This indeed corresponds to the informal (124), when computing
matrix elements and with $P_{\rho}$ a specific tempered distribution. In
particular, if $\ket{m}$ and $\ket{n}$ are two single-mode Fock states, and
remembering from Section 3.6 that
$\innerproduct{n}{\alpha}=\mathrm{e}^{-\frac{|\alpha|^{2}}{2}}\alpha^{n}/\sqrt{n!}$,
we verify that (127) yields

$\matrixelement{n}{\rho}{m}=\left(\frac{\partial^{2k}}{\partial^{k}\alpha\partial^{k}\overline{\alpha}}\frac{\mathrm{e}^{|\alpha|^{2}}}{k!}\mathrm{e}^{-|\alpha|^{2}}\frac{\alpha^{n}\overline{\alpha}^{m}}{\sqrt{n!}\sqrt{m!}}\right)\bigg{|}_{\alpha=\overline{\alpha}=0}=\delta_{kn}\,\delta_{km}$

which is $1$ if $n=m=k$ and otherwise $0$, as expected.

As for the coherence functions themselves, we can use this representation to
give a form very similar to the statistical mixture we discussed above.
Indeed, by linearity, we can write

$\displaystyle G_{\rho}^{(n)}(x_{1},\dots,x_{2n})$ $\displaystyle=\int
P_{\rho}(\boldsymbol{\alpha})\prod_{i=1}^{n}\overline{\mathcal{E}^{+}_{\boldsymbol{\alpha}}(x_{i})}\mathcal{E}^{+}_{\boldsymbol{\alpha}}(x_{n+i})\,{\mathrm{d}}\boldsymbol{\alpha}\,.$
(128)

As an example, consider again a single-mode state in a Fock state with $k$
photons. The coherence function can then be obtained directly from Equations
125 and 128

$\displaystyle
G_{\ket{k}}^{(n)}(x_{1},\dots,x_{2n})=\frac{\partial^{2k}}{\partial^{k}\alpha\partial^{k}\overline{\alpha}}\left(\frac{\mathrm{e}^{|\alpha|^{2}}}{k!}\prod_{i=1}^{n}\overline{\mathcal{E}^{+}_{\alpha}(x_{i})}\mathcal{E}^{+}_{\alpha}(x_{n+i})\right)\bigg{|}_{\alpha=\overline{\alpha}=0}\,.$
(129)

######  Example 10 (A candidate point process corresponding to a Fock state.).

Taking again $x_{i}=x_{n+i}$ and a broad-band detector, we obtain candidate
correlation functions to define a point process for a single-mode Fock state
with $k$ photons

$\displaystyle\rho_{n}(t_{1},\dots,t_{n})=\frac{\partial^{2k}}{\partial^{k}\alpha\partial^{k}\overline{\alpha}}\left(\frac{\mathrm{e}^{|\alpha|^{2}}}{k!}|\mathcal{E}^{+}_{\alpha}(t_{1})\cdots\mathcal{E}^{+}_{\alpha}(t_{n})|^{2}\right)\bigg{|}_{\alpha=\overline{\alpha}=0}.$
(130)

Explicitly, we have

$\mathcal{E}^{+}_{\alpha}(t)=\mathcal{N}\alpha\mathrm{e}^{-\mathrm{i}\omega
t}.$ (131)

Then (130) implies $\rho_{n}(t_{1},\dots,t_{n})=0$ for $k<n$. For $k\geq n$,
we obtain

$\displaystyle\rho_{n}(t_{1},\dots,t_{n})$
$\displaystyle=\mathcal{N}^{n}\frac{\partial^{2k}}{\partial^{k}\alpha\partial^{k}\overline{\alpha}}\frac{\mathrm{e}^{|\alpha|^{2}}}{k!}\alpha^{n}\overline{\alpha}^{n}\bigg{|}_{\alpha=\overline{\alpha}=0}$
(132)
$\displaystyle=\frac{\mathcal{N}^{n}}{k!}\frac{\partial^{2k}}{\partial^{k}\alpha\partial^{k}\overline{\alpha}}\frac{(\alpha\overline{\alpha})^{k}}{(n-k)!}\bigg{|}_{\alpha=\overline{\alpha}=0}$
(133) $\displaystyle=\mathcal{N}^{n}\frac{k!}{(k-n)!}.$ (134)

We find an expression reminiscent of the correlation functions of $k$
independent and uniformly distributed times (Johansson, 2006, Example 2.6).
However, these tentative correlation functions are not integrable, a problem
linked with the fact that the plane waves in (85) or (100), while allowing
simple formal derivations and a clear separation of positive and negative
frequencies, are not actually in $L^{2}$.

In fact, it is possible to build a quantum field theory of radiation replacing
plane waves by any orthonormal set of $L^{2}$ solutions
$\varphi_{\ell}(\mathbf{r},t)$ of the classical equation of motion (Fabre and
Treps, 2020). The positive part of the field can then be decomposed on these
modes as
$\mathcal{E}^{+}_{\alpha}(\mathbf{r},t)=\sum_{\ell}\alpha_{\ell}\varphi^{+}_{\ell}(\mathbf{r},t)$,
where the $+$ subscript still corresponds to the analytic signal 5 taken in
the time variable. The whole framework introduced in Section 3 follows, with
ladder operators and coherent states now relative to this new basis.
Similarly, Equation 130 stills holds with $\boldsymbol{\alpha}$ corresponding
to the coherent state parameters in the chosen modes. However, in the current
example of a single-mode Fock state with $k$ photons, the electric field (131)
becomes $\mathcal{E}^{+}_{\alpha}(t)=\mathcal{N}\alpha\varphi^{+}(t)$, where
we have dropped the dependence on $\mathbf{r}$ as in (131) since the detector
remains at a fixed position. We thus obtain, in lieu of (135),

$\displaystyle\rho_{n}(t_{1},\dots,t_{n})=\mathcal{N}^{n}\frac{k!}{(k-n)!}|\varphi^{+}(t_{1})|^{2}\dots|\varphi^{+}(t_{n})|^{2},\quad
n\geq 1.$ (135)

Since $\varphi^{+}\in L^{2}$, these are valid correlation functions, namely of
the point process formed by $k$ times $t_{1},\dots,t_{k}$ drawn
independepently with probability density function proportional to
$t\mapsto|\varphi^{+}(t)|^{2}$. ∎

###  4.4 Single-photon sources can lead to anti-bunching

The fact that photons emitted by a Gaussian classical field yield bunched
detection times is known as the Hanbury-Brown & Twiss (HBT) effect, and was
first evidenced by Hanbury Brown and Twiss (1958). While we shall come back to
HBT-type experiments in Part II, we pause here to insist on the fact that
bunching is a consequence of considering multiple independent sources. It is
perfectly possible to obtain point processes that anti-bunch with photons, in
the sense that the resulting second correlation function $\rho_{2}(x,y)$ of
the point process is small close to the diagonal $\\{(x,y):x=y\\}$.

To gain intuition without the quantum overhead, we first recast the HBT effect
as a consequence of interference between classical sources. Consider $S$
independent353535For instance, we neglect interatomic coupling through the
radiation for atomic sources. classical sources, represented by $S$
independent, zero-mean stochastic processes $\mathcal{E}_{s}(\mathbf{r},t)$.
The total field is
$\mathcal{E}(\mathbf{r},t)=\sum_{s}\mathcal{E}_{s}(\mathbf{r},t)$.

As derived in Section 4.1, the first-order coherence function is

$\displaystyle
G^{(1)}(\mathbf{r}^{\prime}\,t^{\prime},\mathbf{r}\,t)=\mathbb{E}\,\left(\sum_{s}\mathcal{E}^{-}_{s}(\mathbf{r}^{\prime},t^{\prime})\right)\left(\sum_{s}\mathcal{E}^{+}_{s}(\mathbf{r},t)\right)\,.$
(136)

By assumption, all cross-terms in the product have expectation zero, so that

$\displaystyle
G^{(1)}(\mathbf{r}^{\prime}\,t^{\prime},\mathbf{r}\,t)=\sum_{s}\mathbb{E}\,\mathcal{E}^{-}_{s}(\mathbf{r}^{\prime},t^{\prime})\mathcal{E}^{+}_{s}(\mathbf{r},t):=\sum_{s}G^{(1)}_{s}(\mathbf{r}^{\prime}\,t^{\prime},\mathbf{r}\,t)\,.$
(137)

The total first-order coherence is the sum of the first-order coherence of
each source. Quite generically, the sum will tend to zero when
$|\mathbf{r}-\mathbf{r}^{\prime}|$ or $|t-t^{\prime}|$ are large compared to
the so-called _spatial coherence scale_ and _correlation time_ , respectively.

We can also compute physical quantities like the average field intensity,
which, in this simple model, is the sum of the average field intensities of
each source,

$\displaystyle\mathbb{E}\,I(\mathbf{r},t)=\mathbb{E}\,\mathcal{E}^{*}(\mathbf{r},t)\mathcal{E}(\mathbf{r},t)=G^{(1)}(\mathbf{r}\,t,\mathbf{r}\,t)=\sum_{s}\mathbb{E}\,(|\mathcal{E}_{s}(\mathbf{r},t)|^{2})=\sum_{s}\mathbb{E}\,I_{s}(\mathbf{r},t)\,.$
(138)

To go further and be able to discuss bunching or anti-bunching, we need to
consider the second-order coherence function, which characterizes the density
of pairs of detection events; see (120). It writes

$\displaystyle G^{(2)}$
$\displaystyle(\mathbf{r}_{1}^{\prime}\,t_{1}^{\prime},\mathbf{r}_{2}^{\prime}\,t_{2}^{\prime},\mathbf{r}_{2}\,t_{2},\mathbf{r}_{1}\,t_{1})$
(139)
$\displaystyle=\mathbb{E}\,\left(\sum_{s}\mathcal{E}^{-}_{s}(\mathbf{r}^{\prime}_{1},t^{\prime}_{1})\right)\left(\sum_{s}\mathcal{E}^{-}_{s}(\mathbf{r}^{\prime}_{2},t^{\prime}_{2})\right)\left(\sum_{s}\mathcal{E}^{+}_{s}(\mathbf{r}_{2},_{2}t)\right)\left(\sum_{s}\mathcal{E}^{+}_{s}(\mathbf{r}_{1},t_{1})\right)\,.$
(140)

Using our assumptions of independence and zero mean, this coherence function
is a sum of one-source and two-source terms. We focus on two-source terms.
Developing (140), the contribution of two-source terms to
$\mathbb{E}I(\mathbf{r},t)I(\mathbf{r}^{\prime},t^{\prime})$ is

$\displaystyle\sum_{s\neq
s^{\prime}}\mathbb{E}\,(|\mathcal{E}^{+}_{s}(\mathbf{r},t)\mathcal{E}^{+}_{s^{\prime}}(\mathbf{r}^{\prime},t^{\prime})+\mathcal{E}^{+}_{s}(\mathbf{r}^{\prime},t^{\prime})\mathcal{E}^{+}_{s^{\prime}}(\mathbf{r},t)|^{2})\,.$
(141)

At coincidence $(\mathbf{r},t)=(\mathbf{r}^{\prime},t^{\prime})$, both
amplitudes are the same and give a contribution
$4|\mathcal{E}^{+}_{s}(\mathbf{r},t)\mathcal{E}^{+}_{s^{\prime}}(\mathbf{r},t)|^{2}$.
However, when $(\mathbf{r},t)$ and $(\mathbf{r}^{\prime},t^{\prime})$ differ,
a phase appears between the two contributions and reduces the modulus of the
sum. We recover here the bunching effect of light as an interference effect
between pairs of complex amplitudes. Note that $S=2$ sources are enough to
create this interference.

If we want to observe the opposite effect, a _reduction_ in the rate at
coincidence, we need a single-photon source such as a single two-level atom
described by a Hilbert space $\mathbb{C}^{2}$ excited by a pumping laser
field. A natural analogy is that of a gun (the atom) that needs to be reloaded
(by the laser pump) before firing a new bullet (the photon). We shall see that
the need to reload implies anti-bunching for the fired photons.

What we want to compute is once again the second-order coherence between two
detections at the same photodetection location $r$, but delayed by a time
$\tau$, and abusively denoted by $G^{(2)}_{\rho}(r,r,\tau)$. The whole state
space of the source, the field and the detector is then
$\mathbb{C}^{2}\otimes\mathscr{H}^{\infty}\otimes\mathbb{C}^{2}$. An accurate
model of the situation would require a proper description of the interaction
between the field, the source and the detector but we only give some elements
of the general computation; see (Cohen-Tannoudji, 1979) for details. The new
element here is the interaction of the field and the source and the main
technical problem is to relate the field $E^{+}(t)$ emitted by the source to
some of its physical properties. We ask the reader to admit that
$E^{+}(t)\propto\sigma_{-}(t)$, where
$\sigma_{-}(t)=U^{\dagger}(t)(\sigma_{-}(0)\otimes I\otimes I)U(t)$.
Physically, this expression means that when the source is excited and then
relaxes to its ground state, it emits light. The second-order coherence of the
field can then be written as a function of the time evolution of a source
operator as

$\displaystyle G^{(2)}_{\rho}(r,r,\tau)$
$\displaystyle\propto\langle\sigma_{+}(0)\sigma_{+}(\tau)\sigma_{-}(\tau)\sigma_{-}(0)\rangle_{\rho}$
$\displaystyle\propto\langle\sigma_{+}(0)U^{\dagger}(\tau)\sigma_{+}(0)U(\tau)U^{\dagger}(\tau)\sigma_{-}(0)U(\tau)\sigma_{-}(0)\rangle_{\rho}\,.$
(142)

Naturally, we assume that the initial state of the atom is excited and the
field is in the vacuum so that
$\rho=\ket{e}\\!\\!\bra{e}\otimes\ket{0}\\!\\!\bra{0}$. With
$\sigma_{-}(0)=\ket{g}\\!\\!\bra{e}$ we end up with the very simple form:

$\displaystyle G^{(2)}_{\rho}(r,r,\tau)$
$\displaystyle\propto\mathrm{Tr}\,\left(\big{(}\outerproduct{g}{g}\otimes\outerproduct{0}{0}\big{)}U^{\dagger}(\tau)\outerproduct{e}{e}U(\tau)\right)$
$\displaystyle\propto\langle
e|U(\tau)\big{(}\outerproduct{g}{g}\otimes\outerproduct{0}{0}\big{)}U^{\dagger}(\tau)|e\rangle\,.$
(143)

Two comments are in order. First, we could have also assumed a more general
form for the atomic density matrix which would amount to multiply the above
expression by the probability $p(e)$ for the atom to be initially excited. Its
appearance is natural since the atom has to be excited initially. Second, the
physical interpretation of the second-order coherence (143) is as follows.
Starting from a factorized state in which the atom has already emitted a
photon and is thus in its ground state, the system evolves until time $\tau$
where we compute the probability for the atom to be excited again. Equation
143 is the conditional probability that, knowing that the atom is in its
ground state initially, it gets excited at time $\tau$.

From there, predicting photon anti-bunching at coincidence is straightforward:
near coincidence $\tau=0$, the evolution operator is $U(0)=\mathds{1}$. Since
$\ket{e}$ and $\ket{g}$ are orthogonal, this implies

$\displaystyle G^{(2)}_{\rho}(r,r,0)=0\,.$ (144)

In other words, at coincidence, the probability to have a second emission is
zero. This result is in fact physically quite intuitive if we think at the
single-source level. Indeed, after the atom emits one photon, it is surely in
its ground state and cannot emit a second photon right away. It has to be re-
excited first by the laser source before emitting the second photon. A typical
experimental signal is shown in (3) where we clearly see the coherence signal
going to zero at coincidence. The oscillation pattern is also nicely
understood as Rabi oscillations between the two atomic levels (Degiovanni et
al., 2021). Nowadays, this anti-bunching is used as signature of a good single
photon source, along with a second major interference effect called Hong-Ou
Mandel; see e.g. (Degiovanni et al., 2021).

Figure 3:  Experimental measurement of the second-order correlation signal of
light coming from a single quantum dot showing anti-bunching at coincidence
(zero time delay). The traces show different pump laser powers, all of which
are multiples of the lowest power $P_{0}$. The oscillations at larger delay
correspond to so-called _Rabi oscillations_ between the two energy levels; as
the pump power is increased these oscillations increase in frequency. The
solid line is a fit to a model. Figures taken from Flagg et al. (2009).

##  5 Electrodetection and fermionic coherences

While photonic optics experiments have reached a high level of sophistication
(Haroche and Raimond, 2006; Grynberg et al., 2010), it is only recently that
electronic coherences have started to be probed, thanks to experimental
advances in the manipulation, control and measurement of small quantum systems
in condensed matter and atomic physics. Formally, the main difference with
bosons comes from the anti-commutation of the associated ladder operators,
resulting in the Pauli exclusion principle and in determinants appearing in
Wick’s Theorem 3. Moreover, while we limit ourselves to free (i.e., non-
interacting) fermions in this section, in realistic physical models fermions
are usually subject to the effects of interaction. These two aspects
drastically change the physics compared to photons. Photons are noninteracting
objects with a truly empty reference state, the vacuum. Electrons, on the
contrary, are subject to Coulomb interaction and their ground state, called
the Fermi sea and representing a metal at equilibrium, is full of fermions:
their vacuum is not empty.

This Section follows the lines of Section 4.2, explaining how similar
computations are affected by the fermionic character of the particles. In
Section 5.1, we remind what a fermionic field is and discuss how to model the
detection of electrons. In Section 5.2, we discuss the coherence functions of
arbitrary orders, and use the second-order one in Section 5.3 to justify the
default anti-bunching character of fermions. In Section 5.4, we finally
discuss the difficulties in recovering classical currents in the way coherent
states do for bosonic fields.

###  5.1 Modeling electrodetection events

We proceed as for photodetection by first modeling the field and then the
detection, so as to be able to write correlation functions.

####  5.1.1 Modeling the fermionic field

A _fermionic quantum field_ $\psi(\mathbf{r},t)$ is a collection of operators
indexed by space and time and acting on a Fock space
$\mathcal{H}^{\infty}_{\text{Fermions}}$ of fermions; see Section 3.4. The
field models an indefinite number of fermionic particles, like electrons.
Similarly to a bosonic field, the state of a fermionic system is built by
acting on a reference state with operators that create or annihilate fermions
in a given state of a basis of the single-particle Hilbert space. When it is
clear that we speak of fermions,363636 $a^{\dagger}$ and $a$ are usually kept
for photons, or general particles.  these ladder operators are usually denoted
as $c^{\dagger}$ and $c$, and indexed by the label of the single-particle
state. Unlike for bosons, the creation and annihilation operators satisfy the
canonical _anti_ commutation relations (48).

In theory, the reference state can be taken to be the vacuum, meaning the
complete absence of any particle, as we did for bosons. But when modeling
electronic experiments, physicists often have to take another reference state,
labeled as $\ket{F}$, and called the _Fermi sea_. The Fermi sea is meant to
represent, for instance, the metal used in experiments, which is itself full
of electrons. In the simplest case, corresponding to temperature zero, the
Fermi sea is a state with one fermion in every energy level of the considered
system, up to some reference level called the _Fermi level_. For instance, in
a system built using a single-particle Hamiltonian with eigenpairs
$(\varphi_{n},\varepsilon_{n})_{n\in\mathbb{N}}$, so that $c^{\dagger}_{n}$
creates a particle in mode $\varphi_{n}$, the Fermi sea is

$\ket{F}=\prod_{n:\varepsilon_{n}<\varepsilon_{F}}c^{\dagger}_{n}\ket{0},$

with the Fermi level $\varepsilon_{F}$ defined by the experimental setting.
Like the vacuum, the Fermi sea can also be built as the limit of a Gaussian
state when $\beta\rightarrow\infty$ and the chemical potential $\zeta$ is
fixed as a function of $\varepsilon_{F}$; see the computations in Section 6.

######  Remark 8.

Depending on the situation to model, more physical parameters can be
introduced to describe the field in addition to the spacetime coordinates. For
instance, the spin of the fermionic excitation could contribute, or more
fermionic fields can be used to describe different channels, as in the
description of the quantum Hall effect (Ezawa, 2008).

####  5.1.2 First-order electronic coherence

As we did for photons, we consider a two-level detector with Hilbert space
$\mathbb{C}^{2}$ probing a fermionic field $\psi(\mathbf{r})$ living in the
Hilbert space $\mathcal{H}^{\infty}_{\text{Fermions}}$.373737 Actual
experimental settings are more intricate to describe than for photons exciting
a two-level atom; Part II will contain examples.  The dynamics of the detector
is described here by a fermionic operator $d$ that satisfies the
anticommutation rule $\\{d,d^{\dagger}\\}=0$. Their interaction, again assumed
to be weak, is modeled by Hamiltonian (in the interaction picture)

$\displaystyle
H_{I}(t)=d^{\dagger}(t)\psi(\mathbf{r},t)+d(t)\psi^{\dagger}(\mathbf{r},t)\,,$
(145)

Note that we place ourselves in the interaction picture, so that the time
dependence in (145) results from the evolution of the free part of the
Hamiltonian. The first term of the interaction Hamiltonian describes the
absorption process of an electron that excites the detector while the second
term describes an electron being emitted by the detector. With the same
assumptions as in the photodetection problem, we can compute the excitation
probability of the detector as a function of an electronic correlation
function, naturally called in this context the _first-order electronic
coherence function_ , and defined as

$\displaystyle
G^{(1e)}_{\rho}(\mathbf{r}^{\prime}\,t^{\prime},\mathbf{r}\,t)=\langle\psi^{\dagger}(\mathbf{r}^{\prime},t^{\prime})\psi(\mathbf{r},t)\rangle_{\rho}\,.$
(146)

######  Remark 9.

In concrete condensed matter systems, we argued above that the natural ground
state is in fact a Fermi sea $\ket{F}$, contrary to the photonic case where
the natural ground state is the true vacuum $\ket{0}$. In such a context, it
is then also interesting to consider the de-excitation probability of the
detector, sending back the electron into the system, or equivalently creating
a _hole_ in the Fermi sea. This transition probability is controlled by the
_first-order hole coherence function_

$\displaystyle
G^{(1h)}_{\rho}(\mathbf{r}^{\prime}\,t^{\prime},\mathbf{r}\,t)=\langle\psi(\mathbf{r}^{\prime},t^{\prime})\psi^{\dagger}(\mathbf{r},t)\rangle_{\rho}\,.$
(147)

Note however that the two types of coherence functions are not independent.
Indeed, thanks to the canonical anticommutation relations, they satisfy the
equal time relation

$G^{(1e)}_{\rho}(\mathbf{r}^{\prime}\,t,\mathbf{r}\,t)+G^{(1h)}_{\rho}(\mathbf{r}^{\prime}\,t,\mathbf{r}\,t)=\delta(\mathbf{r}^{\prime}-\mathbf{r}).$

In the following, we work with one of the two coherence functions, and we pick
$G^{(1e)}_{\rho}$ by convention.

######  Example 11 (Free fermions at non-zero temperature $T$).

The simplest example we can consider is a set of free fermions. In particular,
we consider the free field Hamiltonian
$H_{\text{FF}}=\sum_{\mathbf{k}}\nu_{\mathbf{k}}c_{\mathbf{k}}^{\dagger}c_{\mathbf{k}}$,
and further assume that the energy level $\nu_{\mathbf{k}}=\nu_{k}$ only
depends on $k=\|\mathbf{k}\|$. We prepare the state in the grand-canonical
ensemble (62), which we denote here as $\rho=\rho_{\beta,\zeta}$. Note that
$\rho$ commutes with $H_{\text{FF}}$, so that Schrödinger’s equation yields
that $\rho$ does not change in time: it is an equilibrium state. In
particular, the coherence function
$G^{(1e)}_{\rho}(\mathbf{r}^{\prime}t^{\prime},\mathbf{r}t)$ depends only on
$t-t^{\prime}$. We arbitrarily set $t=t^{\prime}$, and write
$G^{(1e)}_{\rho}(\mathbf{r}^{\prime},\mathbf{r})$. Introducing the Fourier
transform of the annihilation operator

$\psi(\mathbf{r})=\int_{\mathbb{R}^{3}}c_{\mathbf{k}}\,\mathrm{e}^{\mathrm{i}(\mathbf{k}\cdot\mathbf{r})}\,{\mathrm{d}}\mathbf{k}/(2\pi)^{3},$

we can write the first electronic coherence function as

$\displaystyle
G^{(1e)}_{\rho}(\mathbf{r}^{\prime},\mathbf{r})=\langle\psi^{\dagger}(\mathbf{r}^{\prime})\psi(\mathbf{r})\rangle_{\rho}=\int_{\mathbb{R}^{6}}\langle
c^{\dagger}_{\mathbf{k}^{\prime}}c_{\mathbf{k}}\rangle_{\rho}\,\mathrm{e}^{\mathrm{i}(\mathbf{r}\cdot\mathbf{k}-\mathbf{r}^{\prime}\cdot\mathbf{k}^{\prime})}\,\frac{{\mathrm{d}}\mathbf{k}}{(2\pi)^{3}}\frac{{\mathrm{d}}\mathbf{k}^{\prime}}{(2\pi)^{3}}\,.$
(148)

Now note that383838 This computation is standard in physics. At this stage, we
take it for granted; but see Equation 183 for an explicit derivation.

$\langle
c^{\dagger}_{\mathbf{k}^{\prime}}c_{\mathbf{k}}\rangle_{\rho}=f_{\beta,\zeta}(k)\delta(\mathbf{k}-\mathbf{k}^{\prime})$

with $k=\|\mathbf{k}\|$ where $f_{\beta,\zeta}$ is the Fermi-Dirac
distribution

$\displaystyle
f_{\beta,\zeta}(k)=\frac{1}{\mathrm{e}^{\beta(\nu_{k}-\zeta)}+1}\,.$ (149)

We obtain

$\displaystyle
G^{(1e)}_{\rho}(\mathbf{r}^{\prime},\mathbf{r})=\int_{\mathbb{R}^{3}}f_{\beta,\zeta}(k)\,\mathrm{e}^{\mathrm{i}(\mathbf{r}-\mathbf{r}^{\prime})\cdot\mathbf{k}}\,\frac{{\mathrm{d}}\mathbf{k}}{(2\pi)^{3}}\,.$
(150)

A practically relevant case is the zero-temperature limit, where the grand-
canonical ensemble state is replaced by the Fermi sea itself
$\outerproduct{F}{F}$. The Fermi-Dirac distribution then simplifies to the
indicator $\mathbbm{1}_{B(k_{F})}$ of the centered ball $B(k_{F})$ of radius
$k_{F}$ known as the Fermi momentum, which is an input of the model393939
Equivalently, we could have specified the chemical potential $\zeta$ to a
certain value $\zeta_{F}$, called the Fermi chemical potential. Moreover,
$k_{F}$ is related to the Fermi velocity $v_{F}$ or the Fermi energy $\nu_{F}$
by the free Hamiltonian so these too can be used as inputs of the model.. The
first-order electronic coherence function is then

$\displaystyle G^{(1e)}_{\ket{F}}(\mathbf{r}^{\prime},\mathbf{r})$
$\displaystyle=\frac{1}{\pi^{2}\|\mathbf{r}-\mathbf{r}^{\prime}\|}\int_{0}^{k_{F}}k\sin(k\|\mathbf{r}-\mathbf{r}^{\prime}\|)\,{\mathrm{d}}k$
$\displaystyle=\frac{1}{\pi^{2}\|\mathbf{r}-\mathbf{r}^{\prime}\|}\left(\frac{\sin(k_{F}\|\mathbf{r}-\mathbf{r}^{\prime}\|)}{\|\mathbf{r}-\mathbf{r}^{\prime}\|^{2}}-\frac{k_{F}}{\|\mathbf{r}-\mathbf{r}^{\prime}\|}\cos(k_{F}\|\mathbf{r}-\mathbf{r}^{\prime}\|)\right)\,.$
(151)

∎

######  Example 12 (Chiral free fermions at non-zero temperature $T$).

A second example appears naturally when discussing electronic coherences. This
time, we consider a simple model of a _chiral quantum wire_ , that is,
electrons moving in one dimension and only in a given direction (e.g. to the
right). The Hilbert space is similar to Example 11,
$\mathbb{C}^{2}\times\mathcal{H}_{\text{Fermions}}^{\infty}$, with
$\mathcal{H}=L^{2}(\mathbb{R})$ as a single-particle Hilbert space. Rather
than describing the Hamiltonian, a typical shortcut to describe the model is
to directly discuss the solutions to Schrödinger’s equation, or, as we do here
for the field operators, how operators evolve in the Heisenberg picture.

Compared to Example 11, we are not at equilibrium, and thus need to keep track
of the time dependence of the field $\psi(x,t)$. The Fourier variable
conjugated to the time $t$ is denoted by $\omega$, just like the momentum
$\mathbf{k}$ is conjugated to the position $\mathbf{r}$. Being in one
dimension in this example, the vectors $\mathbf{r}$ and $\mathbf{k}$ are in
fact scalar quantities that we will denote $x$ and $k$ respectively. To
further constrain the model, we posit an equation of motion for the field,
i.e., a relation between its time- and space derivatives404040 Again, this is
a shortcut to avoid describing the Hamiltonian and Schrödinger’s equation. .
The non-zero solutions of this equation are described by a relation between
all the Fourier variables, called _a dispersion relation_. In our quantum wire
model, the dispersion relation is assumed to be linear, meaning that the
eigenvalues of the free field Hamiltonian are given by $\nu_{k}=v_{F}|k|$,
with $v_{F}$ the so-called Fermi velocity of the electron in the quantum wire.
Finally, we again assume that the electrons coming into the wire come from a
reservoir of particles, whose state is the grand-canonical ensemble
$\rho=\rho_{\beta,\zeta}$ at temperature $\beta=1/k_{B}T$ and chemical
potential $\zeta$.

Figure 4: Representation of a simple model of a quantum wire where fermions
are sent from a reservoir at equilibrium (grand-canonical ensemble) into a
one-dimensional wire where they can travel and are probed by an electro-
detection device. Detection probabilities are given by electronic coherence
functions.

Now that we have all the additional elements for this model, we can proceed to
compute the first-order coherence function. Assuming that the field lives in a
large box, from the Fourier decomposition
$\psi(x,t)=\sum_{k}c_{k}\,\mathrm{e}^{\mathrm{i}(kx-\nu_{k}t)}$, the first-
order coherence expands as

$\displaystyle
G^{(1e)}_{\rho}(x^{\prime}\,t^{\prime},x\,t)=\langle\psi^{\dagger}(x^{\prime},t^{\prime})\psi(x,t)\rangle_{\rho}=\sum_{k,k^{\prime}}\langle
c^{\dagger}_{k^{\prime}}c_{k}\rangle_{\rho}\,\mathrm{e}^{\mathrm{i}(kx-k^{\prime}x^{\prime}-(\nu_{k}t-\nu_{k^{\prime}}t^{\prime}))}\,.$
(152)

The average value of $c^{\dagger}_{k^{\prime}}c_{k}$ on the grand canonical
state $\rho$ can be shown to be $\langle
c^{\dagger}_{k^{\prime}}c_{k}\rangle=\delta(k-k^{\prime})f_{\beta,\zeta}(k)$.
In short, the first-order electronic coherence is simply a Fourier transform
of the Fermi-Dirac distribution. Doing the change of variable to $\omega$, we
have

$\displaystyle
G^{(1e)}_{\rho}(x^{\prime}\,t^{\prime},x\,t)=\int_{\mathbb{R}}\frac{1}{\mathrm{e}^{\beta(\hbar\omega-\zeta)}+1}\,\mathrm{e}^{\mathrm{i}(x-v_{F}t-(x^{\prime}-v_{F}t^{\prime}))\frac{\omega}{v_{F}}}\,\frac{{\mathrm{d}}\omega}{2\pi
v_{F}}\,.$ (153)

As it is traditionally done in this field, we will use a rescaled time
variable $t-x/v_{F}$ as $t$. This simplification between time and space is
very special to the one-dimensional case thanks to the ballistic propagation
of the electron. We then write the simpler expression

$\displaystyle
G^{(1e)}_{\rho}(t^{\prime},t)=\int_{\mathbb{R}}\frac{1}{\mathrm{e}^{\beta(\hbar\omega-\zeta)}+1}\,\mathrm{e}^{\mathrm{i}(t^{\prime}-t)\omega}\,\frac{{\mathrm{d}}\omega}{2\pi
v_{F}}\,.$ (154)

As such, the integral is divergent when $t=t^{\prime}$ and has to be
regularized. This divergence has a physical origin: the linear dispersion
relation posits a Fermi sea with an infinite number of fermions with negative
energy. In practice however, the system has a finite size and the Fermi sea
has a finite bandwidth. The strategy is then to introduce a small parameter
$\varepsilon>0$, which corresponds to the inverse of that bandwidth, and
compute the integral with the substitution$t-t^{\prime}\to
t-t^{\prime}+\mathrm{i}\varepsilon$. With these modifications, the residue
theorem gives

$\displaystyle G^{(1e)}_{F}(t^{\prime},t)=\frac{\mathrm{i}}{2\pi
v_{F}\tau_{\text{th}}}\frac{\mathrm{e}^{-\mathrm{i}\frac{\zeta}{\hbar}(t-t^{\prime})}}{\sinh\left(\frac{t-t^{\prime}+\mathrm{i}\varepsilon}{\tau_{\text{th}}}\right)},$
(155)

where $\tau_{\text{th}}=\hbar\beta/\pi$ is called the _thermal coherence
time_. A particular case to have in mind is the zero-temperature limit of the
coherence function. It can in fact be computed directly as the Fourier
transform (in the sense of distributions) of the zero-temperature Fermi-Dirac
energy density distribution $f_{0,\zeta}(\omega)=\Theta(\zeta-\omega)$, where
$\Theta$ is the Heavyside function. It reads414141The $+\mathrm{i}\varepsilon$
notation is a shorthand notation for the more rigorous distribution
$\displaystyle\lim\limits_{\varepsilon\to
0}\frac{1}{x+\mathrm{i}\varepsilon}=\text{P}\left(\frac{1}{x}\right)-\mathrm{i}\pi\delta(x)\,.$
(156)

$\displaystyle G^{(1e)}_{F}(t^{\prime},t)=\frac{\mathrm{i}}{2\pi
v_{F}}\frac{\mathrm{e}^{-\mathrm{i}\frac{\zeta}{\hbar}(t-t^{\prime})}}{t-t^{\prime}+\mathrm{i}\varepsilon}\,.$
(157)

∎

###  5.2 Correlation between electrodetection events

The first-order coherence gives information about the system at the one-
particle level. If we want to probe two-particle or higher levels of
information like the correlations between electrodetection events, we have to
study the coherence functions of higher order. They are defined similarly as
the photonic coherences of Section 4. For instance, the second-order
electronic and hole coherence functions are defined as

$\displaystyle
G^{(2e)}_{\rho}(\mathbf{r}^{\prime}_{1}\,t^{\prime}_{1},\mathbf{r}^{\prime}_{2}\,t_{2}^{\prime},\mathbf{r}_{2}\,t_{2},\mathbf{r}_{1}\,t_{1})$
$\displaystyle=\langle\psi^{\dagger}(\mathbf{r}^{\prime}_{1},t^{\prime}_{1})\psi^{\dagger}(\mathbf{r}^{\prime}_{2},t^{\prime}_{2})\psi(\mathbf{r}_{2},t_{2})\psi(\mathbf{r}_{1},t_{1})\rangle_{\rho}$
(158a) $\displaystyle
G^{(2h)}_{\rho}(\mathbf{r}^{\prime}_{1}\,t^{\prime}_{1},\mathbf{r}^{\prime}_{2}\,t_{2}^{\prime},\mathbf{r}_{2}\,t_{2},\mathbf{r}_{1}\,t_{1})$
$\displaystyle=\langle\psi(\mathbf{r}^{\prime}_{1},t^{\prime}_{1})\psi(\mathbf{r}^{\prime}_{2},t^{\prime}_{2})\psi^{\dagger}(\mathbf{r}_{2},t_{2})\psi^{\dagger}(\mathbf{r}_{1},t_{1})\rangle_{\rho}\,.$
(158b)

This straightforwardly generalizes to higher orders. Using the shorthand
notation $x=(\mathbf{r},t)$ for the spatio-temporal coordinates, the $n-$th-
order electronic and hole coherences are

$\displaystyle
G^{(ne)}_{\rho}(x_{1}^{\prime},\cdots,x_{n}^{\prime},x_{n},\cdots,x_{1})$
$\displaystyle=\langle\psi^{\dagger}(x_{1}^{\prime})\cdots\psi^{\dagger}(x_{n}^{\prime})\psi(x_{n})\cdots\psi(x_{1})\rangle_{\rho}$
(159a) $\displaystyle
G^{(nh)}_{\rho}(x_{1}^{\prime},\cdots,x_{n}^{\prime},x_{n},\cdots,x_{1})$
$\displaystyle=\langle\psi(x_{1}^{\prime})\cdots\psi(x_{n}^{\prime})\psi^{\dagger}(x_{n})\cdots\psi^{\dagger}(x_{1})\rangle_{\rho}\,.$
(159b)

######  Example 13 (Continuation of 11).

The grand canonical ensemble $\rho=\rho_{\beta,\zeta}$ for free fermions is
Gaussian, so that Theorem 3 (Wick’s) applies and we obtain a $2\times 2$
determinant

$\displaystyle
G^{(2e)}_{\rho}(\mathbf{r}^{\prime}_{1},\mathbf{r}^{\prime}_{2},\mathbf{r}_{2},\mathbf{r}_{2})$
$\displaystyle=\langle\psi^{\dagger}(\mathbf{r}^{\prime}_{1})\psi^{\dagger}(\mathbf{r}^{\prime}_{2})\psi(\mathbf{r}_{2})\psi(\mathbf{r}_{1})\rangle_{\rho}$
(160a)
$\displaystyle\underset{\text{Wick}}{=}G^{(1e)}_{\rho}(\mathbf{r}^{\prime}_{1},\mathbf{r}_{1})G^{(1e)}_{\rho}(\mathbf{r}^{\prime}_{1},\mathbf{r}_{1})-G^{(1e)}_{\rho}(\mathbf{r}^{\prime}_{1},\mathbf{r}_{2})G^{(1e)}_{\rho}(\mathbf{r}^{\prime}_{2},\mathbf{r}_{1})\,.$
(160b)

The minus sign is of course a consequence of the fermionic character of the
electronic field. In fact, we can already see at this level the repulsion
between different fermions. Indeed, if we look for instance at the diagonal
part $\mathbf{r}^{\prime}_{1}=\mathbf{r}_{1}$ and
$\mathbf{r}^{\prime}_{2}=\mathbf{r}_{2}$, in the zero temperature limit, we
have

$\displaystyle
G^{(2e)}_{\ket{F}}(\mathbf{r}_{1},\mathbf{r}_{2},\mathbf{r}_{2},\mathbf{r}_{1})\propto\left(1-\left|g^{(1e)}_{\ket{F}}(\|\mathbf{r}_{1}-\mathbf{r}_{2}\|)\right|^{2}\right)\,,$
(161)

where $g^{(1e)}_{\ket{F}}(\|\mathbf{r}_{1}-\mathbf{r}_{2}\|)$ is the first-
order electronic coherence function (151) normalized to unity. We clearly see
that when $\mathbf{r}_{1}=\mathbf{r}_{2}$, the second-order coherence function
is exactly zero: two fermions cannot be at the same place at the same time.
This is another avatar of the general anti-bunching effect of fermionic
systems. ∎

######  Example 14 (Second-order coherence of $N$ excitations).

To gain some intuitive understanding of the information content of the second-
order coherence function, let’s continue 12 of a quantum wire. Remember that
in this simple one-dimensional model, time and space coordinates are
identified thanks to the linear dispersion relation and that we are looking
for the coherence functions at the fixed position of the detector (position
that we omit in the arguments to simplify the notations). Consider a state
containing two electrons above the true vacuum $\ket{0}$ (not the Fermi sea)
in two orthogonal wavefunctions
$\ket{\varphi_{1},\varphi_{2}}=\psi^{\dagger}[\varphi_{1}]\psi^{\dagger}[\varphi_{2}]\ket{0}$,
where $\psi[\varphi]=\int\psi(t)\varphi(t){\mathrm{d}}t$. Note again that
wavefunctions, strictly speaking functions of $x$ and $t$, are here functions
of $x-v_{F}t$ only because of the linear dispersion relation. Since we further
consider a fixed position here (that of the detector), we simply write
$t\mapsto\psi(t)$ and $t\mapsto\varphi(t)$.

Since the true vacuum is a Gaussian state, we can apply Wick’s Theorem 3 to
the second-order coherence function

$\displaystyle
G^{(2e)}_{\ket{\varphi_{1},\varphi_{2}}}(t^{\prime}_{1},t^{\prime}_{2},t_{2},t_{1})$
$\displaystyle=\int_{\mathbb{R}^{4}}\langle
0|\psi(y^{\prime}_{1})\psi(y^{\prime}_{2})\psi^{\dagger}(t^{\prime}_{1})\psi^{\dagger}(t^{\prime}_{2})\psi(t_{2})\psi(t_{1})\psi^{\dagger}(y_{2})\psi^{\dagger}(y_{1})|0\rangle\,$
$\displaystyle\varphi_{1}^{*}(y^{\prime}_{1})\varphi_{2}^{*}(y^{\prime}_{2})\varphi_{2}(y_{2})\varphi_{1}(y_{1})\,{\mathrm{d}}y_{1}{\mathrm{d}}y^{\prime}_{1}{\mathrm{d}}y_{2}{\mathrm{d}}y^{\prime}_{2}$
$\displaystyle=\varphi^{*}_{1}(t_{1}^{\prime})\varphi^{*}_{2}(t_{2}^{\prime})\varphi_{1}(t_{1})\varphi_{2}(t_{2})+\varphi^{*}_{1}(t_{2}^{\prime})\varphi^{*}_{2}(t_{1}^{\prime})\varphi_{1}(t_{2})\varphi_{2}(t_{1})$
$\displaystyle-\varphi^{*}_{1}(t_{2}^{\prime})\varphi^{*}_{2}(t_{1}^{\prime})\varphi_{1}(t_{1})\varphi_{2}(t_{2})-\varphi^{*}_{1}(t_{1}^{\prime})\varphi^{*}_{2}(t_{2}^{\prime})\varphi_{1}(t_{2})\varphi_{2}(t_{1})\,.$
(162)

We see that the form of this coherence contains the expected anti-symmetries
coming with fermionic statistics. It can be written even more explicitly by
introducing the antisymmmetrized wave-function

$\displaystyle\Phi_{12}(t_{1},t_{2})=\det\left[\begin{matrix}\varphi_{1}(t_{1})&\varphi_{2}(t_{1})\\\
\varphi_{1}(t_{2})&\varphi_{2}(t_{2})\end{matrix}\right]\,.$ (163)

The second-order electronic coherence function then reads

$\displaystyle
G^{(2e)}_{\ket{\varphi_{1},\varphi_{2}}}(t^{\prime}_{1},t^{\prime}_{2},t_{2},t_{1})=\Phi^{*}_{12}(t^{\prime}_{1},t^{\prime}_{2})\Phi_{12}(t_{1},t_{2})\,.$
(164)

The second-order coherence function is thus essentially the full many-body
wavefunction of a two-particle state.

When more than two fermions are present, the second-order coherence function
only keeps track of two-particle terms. To see this, consider a state
containing $N$ electrons above the vacuum in mutually orthogonal wavepackets
$\ket{\psi_{N}}=\psi^{\dagger}[\varphi_{1}]\dots\psi^{\dagger}[\varphi_{N}]\ket{0}$.
A straightforward, but cumbersome, application of Wick’s theorem gives

$\displaystyle
G^{(2e)}_{\ket{\psi_{N}}}(t^{\prime}_{1},t^{\prime}_{2},t_{2},t_{1})=\sum_{k<l}\Phi^{*}_{kl}(t^{\prime}_{1},t^{\prime}_{2})\Phi_{kl}(t_{1},t_{2})\,.$
(165)

Informally, the second-order coherence function contains only the “two-
particle physics”.

∎

######  Example 15 ($k$th-order coherence of $N$ excitations).

Continuing the quantum wire example, we can understand its $k-$particle
physics by computing the $k$th-order coherence function. Consider again $N$
fermions prepared in mutually orthogonal wavepackets
$\ket{\psi_{N}}=\psi^{\dagger}[\varphi_{1}]\dots\psi^{\dagger}[\varphi_{N}]\ket{0}$.
By Wick’s theorem, the $k$th-order coherence function is

$\displaystyle
G^{(ke)}_{\ket{\psi_{N}}}(t^{\prime}_{1},\cdots,t^{\prime}_{k},t_{1},\cdots,t_{k})=\sum_{i_{1}<\cdots<i_{k}}\Phi^{*}_{i_{1},\dots,i_{k}}(t^{\prime}_{1},\cdots,t^{\prime}_{k})\Phi_{i_{1},\dots,i_{k}}(t_{1},\cdots,t_{k})\,,$
(166)

with

$\displaystyle\Phi_{i_{1},\dots,i_{k}}(t_{1},\cdots,t_{k})=\det\left[\begin{matrix}\varphi_{i_{1}}(t_{1})&\cdots&\varphi_{i_{k}}(t_{1})\\\
\vdots&\ddots&\vdots\\\
\varphi_{i_{1}}(t_{k})&\cdots&\varphi_{i_{k}}(t_{k})\end{matrix}\right]\,.$
(167)

In particular, the $N$-th coherence function is

$\displaystyle
G^{(Ne)}_{\ket{\psi_{N}}}(t^{\prime}_{1},\cdots,t^{\prime}_{N},t_{1},\cdots,t_{N})=\Phi^{*}_{1,\dots,N}(t^{\prime}_{1},\cdots,t^{\prime}_{N})\Phi_{1,\dots,N}(t_{1},\cdots,t_{N})\,.$
(168)

As expected, the $N$-th coherence function is essentially the full many-body
wavefunction of a quantum state with $N$ excitations in mutually orthogonal
wavefunctions. ∎

| $\varphi_{e_{1}}$ | $\cdots$ | $\varphi_{e_{k}}$ | $\cdots$ | $\varphi_{e_{N}}$
---|---|---|---|---|---
$G^{(1e)}$ | $\varphi^{*}_{e_{1}}\varphi_{e_{1}}$ | $\cdots$ | $\sum_{p=1}^{k}\varphi^{*}_{e_{p}}\varphi_{e_{p}}$ | $\cdots$ | $\sum_{p=1}^{N}\varphi^{*}_{e_{p}}\varphi_{e_{p}}$
$\vdots$ | $\vdots$ | $\ddots$ |  |  | $\vdots$
$G^{(ke)}$ | 0 |  | $\Phi^{*}_{1,\dots,k}\Phi_{1,\dots,k}$ |  | $\sum_{i_{1}<\cdots<i_{k}}\Phi^{*}_{i_{1},\dots,i_{k}}\Phi_{i_{1},\dots,i_{k}}$
$\vdots$ | $\vdots$ |  |  | $\ddots$ | $\vdots$
$G^{(Ne)}$ | 0 | $\cdots$ | 0 | $\cdots$ | $\Phi^{*}_{1,\dots,N}\Phi_{1,\dots,N}$
Figure 5: Table showing the structure of the electronic coherence functions of
order up to $N$ in the presence of one up to $N$ fermions in some given
wavepackets.

To see the correlation functions of determinantal point processes appear, one
can use an ideal broad-band detector as in Section 5. In particular, imagine
that we could prepare $N$ electrons in mutually orthogonal wavepackets
$\varphi_{1},\dots,\varphi_{N}$, and set up an ideal detector as we did for
photons in Section 4, resulting in a point process with correlation functions
given by the diagonal of $G^{(ne)}_{\ket{\psi_{N}}}$,

$\displaystyle\rho_{n}(t_{1},\dots,t_{n})$
$\displaystyle=G^{(ne)}_{\ket{\psi_{N}}}(t_{1},\cdots,t_{n},t_{1},\cdots,t_{n}),$
(169) $\displaystyle=\left|\Phi_{1,\dots,N}(t_{1},\cdots,t_{N})\right|^{2}$
(170) $\displaystyle=\det\left[K(t_{i},t_{j})\right]_{i,j=1}^{n},$ (171)

where

$K(t,s)=\sum_{i=1}^{N}\varphi_{e_{i}}(t)\varphi_{e_{i}}^{*}(s).$ (172)

We recognize a DPP with (projection) kernel $K$, as introduced in Section 2.4.

######  Remark 10.

To illustrate what reference computations different scientists may have in
mind, note that, following a physics viewpoint, we computed in Examples 14 and
15 coherence functions applying Wick’s theorem, one after the other, starting
from the lower ones. Probabilists and statisticians are more used to derive
correlation functions starting from the definition of the point process. Had
we started by defining the DPP with kernel $K$ in (172), and asked for the
diagonal of, say, the second coherence function in (14), it would have been
enough to notice, from the definition (186) of the correlation functions of a
DPP, that

$\displaystyle\rho_{2}(t_{1},t_{2})$
$\displaystyle=\det\left[\begin{matrix}K(t_{1},t_{1})&K(t_{1},t_{2})\\\
K(t_{2},t_{1})&K(t_{2},t_{2})\end{matrix}\right].$ (173)

By definition (172) of $K$,

$\displaystyle\rho_{2}(t_{1},t_{2})$
$\displaystyle=\det\left[\begin{matrix}\varphi_{1}(t_{1})&\dots&\varphi_{N}(t_{1})\\\
\varphi_{1}(t_{2})&\dots&\varphi_{N}(t_{N})\end{matrix}\right]\left[\begin{matrix}\varphi_{1}^{*}(t_{1})&\varphi_{1}^{*}(t_{2})\\\
\vdots&\vdots\\\
\varphi_{N}^{*}(t_{1})&\varphi_{N}^{*}(t_{2})\end{matrix}\right]\,.$ (174)

The diagonal version of (165) then results from the Cauchy-Binet formula,
which allows rewriting $\rho_{2}(t_{1},t_{2})$ as a sum of $2\times 2$
determinants; see e.g. (Kulesza and Taskar, 2012).

###  5.3 Electron anti-bunching

For photons, we saw that HBT-type experiments can reveal both a bunching
effect (for classical light beams) and an anti-bunching effect (for quantum
beams), depending on the source.

A similar experimental setup as the one pictured for photons in Section 4 can
be imagined for fermionic excitations. We defer the discussions on how to
properly do this experimentally with fermionic atoms or single electronic
excitations to Part II, but we can already foresee from either DPP
constructions like the one with kernel (172), or directly from the coherence
functions of Section 5.2 that fermions will exhibit anti-bunching detection
times. Unlike bosons, this is a direct consequence of the anti-symmetric
statistic of fermions, and does not qualitatively change when introducing
source models.

If the arrival times follow a DPP with an Hermitian kernel like (172), we saw
indeed in Section 2.4 that samples tend to spread regularly, and form less
clusters than a Poisson point process. Alternately, this statistical anti-
bunching effect can be directly seen from the coherence funtions. Indeed, as
for photons, the HBT signal is controlled by the second-order coherence
function at the position $\mathbf{r}$ of the detector,

$\displaystyle
G^{(2e)}_{\rho}(\mathbf{r},t_{2},t_{1})=\langle\psi^{\dagger}(\mathbf{r},t_{1})\psi^{\dagger}(\mathbf{r},t_{2})\psi(\mathbf{r},t_{2})\psi(\mathbf{r},t_{1})\rangle_{\rho}\,.$
(175)

At coincidence $\tau=t_{2}-t_{1}=0$, because the CAR (48) impose
$\psi(\mathbf{r},t_{1})^{2}=0$, we obtain

$\displaystyle G^{(2e)}_{\rho}(\mathbf{r},t,t)=0\,.$ (176)

If $G^{(2e)}_{\rho}$ is smooth, then we expect few coincidences at small time
differences.

As a final remark, we insist that bunching or anti-bunching of quantum
excitations is not a signature of the statistics of the elementary
excitations. While it is a consequence of the statistics for fermions, bosons
also anti-bunch when we add to the model a single-excitation source. Hence, it
is better to think of anti-bunching as a signature of the quantum nature of
the fundamental excitation.

###  5.4 Recovering a classical current is not as easy as for bosons

For photons, coherent states allowed for relating the quantum formalism to
classical fields; see Section 4. Describing a classical electric current or,
more generally, a classical theory of fermionic fields, is more difficult. In
this section, we examine reasons why it is not possible to build a set of
states having all the properties that make bosonic coherent states handy.
First, even considering a single mode, bosonic coherent states are
superpositions of arbitrary numbers of bosons, which allows for having very
small fluctuations around the average boson number. Indeed the number of
photons in the mode is a Poisson variable with arbitrarily large parameter,
and thus arbitrarily small relative variance. Such small fluctuations are not
achievable with fermions, because it is impossible to have more than one
fermion per mode.

A second difference is that the eigenvalue of a tentative fermionic coherent
state of the fermionic annihilation operator cannot be a complex number.
Indeed, were $\ket{\alpha}$ an eigenstate of the fermionic annihilation
operator $a$, then the CAR (48) would imply that its eigenvalue $\alpha$
satifies $\alpha^{2}=0$. Moreover, in the many-mode case, a two-mode coherent
state $\ket{\alpha_{1},\alpha_{2}}$ would satisfy the anti-commutation
relation
$a_{1}a_{2}\ket{\alpha_{1},\alpha_{2}}=\alpha_{1}\alpha_{2}\ket{\alpha_{1},\alpha_{2}}=-a_{2}a_{1}\ket{\alpha_{1},\alpha_{2}}=-\alpha_{2}\alpha_{1}\ket{\alpha_{1},\alpha_{2}}$.424242This
result is independent on whether one assumes that $\alpha_{i}$ commutes or
anti-commutes with $a_{j}$. The eigenvalues of a fermionic coherent state thus
cannot be complex numbers.434343 It is however possible to define fermionic
coherent states in terms of anti-commuting variables (elements of an exterior
algebra) by the action of the analogue of a displacement operator (Berezin,
1966, Chapter 1). While they are very useful objects to define and use
(especially to write path integrals), they do not possess the nice set of
properties of bosonic coherent states that allow for deriving coherence
functions.

A more subtle difference is that fermionic fields cannot have an observable
average amplitude,444444 Just like $a$ for photons, $\psi$ is not Hermitian,
and thus not an observable, _stricto sensu_. Yet, for $a$, the average value
corresponds to the intensity of the field, and is thus experimentally
accessible as a statistical average. It thus makes sense to wonder whether the
average value of $\psi$ can be non-zero.  unlike bosonic fields prepared in a
coherent state.454545 This property can actually be used as a definition of
bosonic coherent states (Cohen-Tannoudji et al., 2019a).  This impossibility
is fundamental and is related to _superselection rules_ (Wick et al., 1952).
For concreteness, consider a general fermionic state
$\ket{\varphi}=\alpha\ket{0}+\beta\ket{1}$ with
$\ket{1}=\psi^{\dagger}\ket{0}$, where $\psi^{\dagger}$ is any fermionic
creation operator. Now, physicists usually impose symmetries on their models,
which implies that some states should yield the same measurements. In
particular, one would like to model the fact that rotating the physical system
around any axis by an angle of $2\pi$ should not modify the law of
measurements. For fermions, the so-called _spin-statistics theorem_ of
relativistic quantum field theory tells us that this kind of rotations should
be modelled by the action of the half-integer representations of the group
$SU(2)$. For instance, in the representation of order $1/2$, a rotation of
angle $2\pi$ corresponds to multiplying the state by
$\mathrm{e}^{\mathrm{i}\pi N}$, where $N$ is the number operator. Applied to
$\ket{\varphi}=\alpha\ket{0}+\beta\ket{1}$, this yields
$\ket{\varphi^{\prime}}:=\mathrm{e}^{\mathrm{i}\pi
N}\ket{\varphi}=\alpha\ket{0}-\beta\ket{1}$, where the minus sign comes from
the fact that $1$ is an odd number. However, since such a rotation should not
correspond to any change of the system, we impose that
$\ket{\varphi^{\prime}}=\ket{\varphi}$ up to a phase, and consequently either
$\alpha$ or $\beta$ is zero. Thus $\ket{\varphi}$ is in fact not a
superposition, implying that $\expectationvalue{\psi}_{\ket{\varphi}}=0$. More
generally, the _parity superselection rule_ imposes that a given state can
only consist of superpositions of states with the same parity of the number of
fermions. Therefore, a single fermionic annihilation operator can only have a
vanishing average.464646 This generalises beyond physically realistic fermions
of half-integer spins, and applies to non-relativistic quantum mechanics, see
for instance (Johansson, 2016; Szalay et al., 2021).

For all these reasons, defining a classical regime with fermionic fields is an
open question. This actually raises deep conceptual questions when fermions
are involved. For instance, it is not clear what is meant by a _classical
electronic current_ if we start from a pure quantum description of the
electronic current.

##  6 Wrapping up and open questions

To conclude this Part I, we first show how to use the formalism of Sections 2
to 5 to write a generic permanental or determinantal point process as the
detection process of a system of free bosons or fermions, respectively.
Second, we propose a list of questions, from simple ones to open problems,
that are raised by the connections between point processes and physical
measurements in quantum field theory. Third, we announce the sections that are
to appear in Part II of this document.

###  6.1 From a point process to free particles

In Sections 4 and 5, we obtained some point processes, including permanental
and determinantal point processes, by modeling the detection of physical
particles. A natural question is whether all permanental and determinantal
point processes arise in this form. We now show that, at least formally and
under weak assumptions on the kernel, the answer is yes. More precisely, let
$\mu$ be a Borel measure on a complete metric space $\mathbb{X}$, and
$(\varphi_{i})$ an orthonormal family in $L^{2}(\mu)$. Consider the kernel

$K(x,y):=\sum_{i}\lambda_{i}\overline{\varphi_{i}(x)}\varphi_{i}(y),$ (177)

where, for all $i$, $\lambda_{i}\in[0,1]$ if $\eta=-1$, and
$\lambda_{i}\in\mathbb{R}_{+}$ if $\eta=+1$. We now build a Fock space, a
quantum state, and a measurement model that together lead to a permanental or
determinantal point process with $K$ as its kernel. Let $\eta=\pm 1$ depending
on whether one wants a permanental or a determinantal point process.

##### Building a Fock space.

Build the Fock space like in Section 3.4, using the basis $(\varphi_{i})$ as
single-particle basis and the symmetrization property dictated by $\eta$.
Consider the operator

$H=\sum_{i}\nu_{i}a_{i}^{\dagger}a_{i},$

where $(\nu_{i})$ is left as a free parameter for now, and will later be
chosen in relation to the spectrum of $K$. We think of $H$ as a Hamiltonian.

##### A Gaussian density matrix.

Let $\zeta>0$, and define the so-called grand canonical ensemble as in (62) by
the Gaussian density matrix

$\rho=\frac{\mathrm{e}^{-\beta\left({H}-\zeta\sum_{i}a_{i}^{\dagger}a_{i}\right)}}{Z_{\eta}}=\frac{1}{Z_{\eta}}\mathrm{e}^{-\beta\sum_{i}(\nu_{i}-\zeta)a_{i}^{\dagger}a_{i}}.$

We explicitly write the dependence of $Z$ to $\eta$, as bosons and fermions
lead to different normalization constants. More precisely, since the Fock
states $\ket{\mathbf{n}}=\ket{n_{1},n_{2},\dots}$ are eigenvectors of all the
number operators $a_{i}^{\dagger}a_{i}$ in the exponential, the normalization
constant is

$Z_{\eta}=\mathrm{Tr}\,\mathrm{e}^{-\beta\sum_{i}(\nu_{i}-\zeta)a_{i}^{\dagger}a_{i}}=\sum_{\mathbf{n}}\expectationvalue{\mathrm{e}^{-\beta\sum_{i}(\nu_{i}-\zeta)a_{i}^{\dagger}a_{i}}}{\mathbf{n}}=\sum_{\mathbf{n}}\mathrm{e}^{-\beta\sum_{i}(\nu_{i}-\zeta)n_{i}}.$
(178)

For bosons, we sum over all sequences of integers with only $p$ non-zero
components, and this for all $p\geq 1$. In particular,

$Z_{1}=\prod_{p\in\mathbb{N}}\sum_{n\in\mathbb{N}}e^{-\beta(\nu_{p}-\zeta)n}=\prod_{p\in\mathbb{N}}\frac{1}{1-\mathrm{e}^{-\beta(\nu_{p}-\zeta)}},$
(179)

where we implicitly assumed $\nu_{p}-\zeta>0$ for all $p$, for the geometric
sums in (179) to converge.474747 Note that for a given $p$, the limit
$\nu_{p}\to\zeta$ corresponds to the onset of Bose-Einstein condensation,
where a macroscopic number of bosons start to occupy the state $p$.  For
$Z_{1}$ to be finite, we further need a condition on the $(\nu_{p})$, such as

$\sum_{p}\log\left(\frac{1}{\mathrm{e}^{\beta(\nu_{p}-\zeta)}-1}\right)<\infty.$
(180)

For fermions, we only sum in (178) over all sequences in
$\\{0,1\\}^{\mathbb{N}}$ that have $p$ non-zeros components, for all sizes
$p\geq 1$, so that

$Z_{-1}=\prod_{p\in\mathbb{N}}\sum_{n\in\\{0,1\\}}\mathrm{e}^{-\beta(\nu_{p}-\zeta)n}=\prod_{p\in\mathbb{N}}\left(1+\mathrm{e}^{-\beta(\nu_{p}-\zeta)}\right).$
(181)

Guaranteeing convergence is easier than for bosons, e.g. assuming that
$\sum_{p}(\nu_{p}-\zeta)$ converges.

Now that we have simple expressions for $Z_{\eta}$, $\eta=\pm 1$, we can use
the partition function trick484848Statisticians might say the _score function_
trick. to compute the expected number of particles in mode $i$,

$\displaystyle\expectationvalue{a_{i}^{\dagger}a_{i}}_{\rho}$
$\displaystyle=\frac{1}{Z_{\eta}}\mathrm{Tr}\,a_{i}^{\dagger}a_{i}\rho$ (182a)
$\displaystyle=\frac{\sum_{n}n_{i}\mathrm{e}^{-\beta\sum_{j}(\nu_{j}-\zeta)n_{j}}}{\sum_{n}\mathrm{e}^{-\beta\sum_{j}(\nu_{j}-\zeta)n_{j}}}$
(182b) $\displaystyle=-\frac{1}{\beta}\frac{\partial}{\partial\nu_{i}}\log
Z_{\eta}(\nu).$ (182c)

Using (179) and (181) in turn, this yields

$\expectationvalue{a_{i}^{\dagger}a_{i}}_{\rho}=\frac{1}{\mathrm{e}^{\beta(\nu_{i}-\zeta)}-\eta},$
(183)

which is the so-called _Fermi-Dirac_ distribution494949The word _distribution_
is used here in a loose sense. for $\eta=-1$ and the _Bose-Einstein_
distribution for $\eta=1$. Note that along the lines of Lemma 2, one can show
that $\langle a_{i}^{\dagger}a_{j}\rangle_{\rho}=0$ if $i\neq j$.

##### A measurement basis.

Now, consider the alternative basis $\ket{x}$, $x\in\mathbb{X}$, with
annihilation operator

$\psi(x)=\sum_{i}\langle x,\varphi_{i}\rangle a_{i};$

see (54). We think of $\ket{x}$ as the position basis, so that we write
$\langle x,\varphi_{i}\rangle=\varphi_{i}(x)$, and we actually mean a
generalized basis if necessary, e.g., if $X=\mathbb{R}^{d}$; see Section 3.1.
In the basis $\ket{x}$, the state $\rho$ rewrites

$\rho=\frac{1}{Z}\exp\left(-\beta\sum_{i}(\nu_{i}-\zeta)\int\varphi_{i}(x)\overline{\varphi_{i}(y)}\psi^{\dagger}(x)\psi(y){\mathrm{d}}\mu(x){\mathrm{d}}\mu(y)\right).$

##### Wick’s theorem applies.

By Theorem 3, see also Example 5, for all $n\geq 1$, we obtain the coherence
functions

$\expectationvalue{\psi^{\dagger}(x_{1})\psi(x_{1})\dots\psi^{\dagger}(x_{n})\psi(x_{n})}=f_{\eta}(\mathbf{K}),$

where

$\mathbf{K}=\left(\expectationvalue{\psi(x_{i})^{\dagger}\psi(x_{j})}_{\rho}\right)_{1\leq
i,j\leq n}$

and $f_{\eta}$ is the determinant if $\eta=-1$, and the permanent if $\eta=1$.
If the framework is consistent, an ideal detection experiment that measures
all particles in $\rho$ in the basis $\ket{x}$ should thus correspond to a
permanental/determinantal point process on $\mathbb{X}$, with kernel

$\displaystyle\expectationvalue{\psi^{\dagger}(x)\psi(y)}_{\rho}$
$\displaystyle=\sum_{i,j}\overline{\varphi_{i}(x)}\varphi_{j}(y)\expectationvalue{a_{i}^{\dagger}a_{j}}_{\rho}.$
(184)

This is precisely $K$ in (177), provided that we choose $\nu_{i}$ so that

$\lambda_{i}=(\mathrm{e}^{\beta(\nu_{i}-\zeta)}-\eta)^{-1}.$ (185)

This is achieved by setting

$\beta(\nu_{i}-\zeta)=\log(\frac{1+\eta\lambda_{i}}{\lambda_{i}}).$

Note that when $\eta=1$, one can check that the corresponding permanental
point process exists by an adaptation of Macchi’s Cox process construction of
Section 2.3. When $\eta=-1$, the determinantal point process exists by the
Macchi-Soshikov theorem; see Section 2.4.

##### Obtaining any kernel.

We conclude with several comments. First, inverting (183) allows us to obtain
any spectrum $(\lambda_{i})$ with $\lambda_{i}\in(0,1)$ for fermions, and
$\lambda_{i}>0$ for bosons. With the right assumptions, projection kernels for
DPPs can also be obtained, but as a limit as $\beta\rightarrow+\infty$, i.e.
when the temperature $1/\beta$ goes to zero. The corresponding projection
operator is onto the span of the $\varphi_{i}$s for which $\nu_{i}<\zeta$. In
other words, the chemical potential controls the rank of the limiting
projection operator. Note that for DPPs, uniform convergence of the kernel on
compact subsets of $\mathbb{X}$ implies convergence of the point process in a
natural sense; see e.g. (Anderson et al., 2010, Section 4.2.8). Another way to
formally obtain projection DPPs is as a ground state; see Section 3.3.2.

Second, the construction in this section, with a state $\rho$ and measuring
correlators, is as close as one can hope for a Cox process-like decomposition
of determinantal point processes. Loosely speaking, there is a Gaussian object
above determinantal point processes with self-adjoint kernels, but it is a
Gaussian density matrix, i.e., an operator, not a functional process.505050
Alternatively, the formulation in terms of operators can be replaced by a
“Gaussian functional integral” over complex commuting (anti-commuting) fields
for permanents (determinants). The connection with Gaussian measures is then
more direct, at the price of subtle mathematical difficulties. Furthermore, in
the case of fermions, this implies generalising the notion of integration to
variables that anti-commute, see for instance (Berezin, 1966, Chapter 1).
Third, while we restricted here to Hermitian kernels, thus corresponding to an
observable quantity, the Hermitian assumption does not play a major role in
the mathematical framework. Furthermore, non-Hermitian free fermions may not
be so removed from physics, see e.g. (Ashida et al., 2020) for a recent survey
and (Guo et al., 2021) for a formal example.

##### On physical realizability.

Finally, we have seen in Section 5 examples of experimentally realizable DPPs,
in the sense that the corresponding idealized experimental setups can, in
principle, be built in a physics laboratory. We will see in Part II concrete
examples of DPP samples obtained from lab experiments. Yet, the generic
construction given in Section 6.1 remains rather theoretical, as it is not
clear how to build an experimental setup leading to _any_ given DPP, that is,
any choice of underlying space $(\mathbb{X},\mu)$ and any valid kernel $K$.
One might be able to treat $\mathbb{X}=\mathbb{R}^{d}$ for $d$ up to $3$ by
associating mathematical coordinates to space-coordinates. Going slightly
beyond $3$ might be doable using either time or additional physical degrees of
freedom (“pseudo-dimensions”) as the next dimensions, but it is unclear to
physicists where fermions in arbitrary dimension would be needed as a model.
Outside the problem of dimensionality, fixing the spectrum of the Hamiltonian
to an arbitrary sequence can also be experimentally difficult.

###  6.2 Going further: from simple to open questions

Many natural questions come to mind when one looks back at the correspondence
between permanental point processes and free bosons, or between DPPs and free
fermions.

####  6.2.1 A quantum state is more than a point process

A point process is the result of observing a state $\rho$ with a detector, but
the state itself contains a lot more information. For instance, the
correlation functions that we identified in Sections 4 and 5 are only the
diagonals of the corresponding coherence functions. Relatedly, depending on
the measurement basis, a single state $\rho$ can yield many point processes.

As a concrete example, consider for simplicity fermions living in
$\mathbb{X}=\\{1,\dots,N\\}$, $\mu$ the counting measure on $\mathbb{X}$, and
the Hamiltonian

$H=\sum_{i=1}^{N}\nu_{i}a^{\dagger}(\varphi_{i})a(\varphi_{i})$

acting on the corresponding Fock space. By (184), measuring in an orthonormal
basis $(v_{i})$ of $\mathbb{C}^{N}$, with
$\mathbf{V}=(\innerproduct{\varphi_{i}}{v_{j}})_{i,j}$, we obtain the DPP with
kernel matrix

$\mathbf{K}=\left(\sum_{k}\lambda_{k}\innerproduct{\varphi_{k}}{v_{i}}\innerproduct{v_{j}}{\varphi_{k}}\right)_{1\leq
i,j\leq N}=\mathbf{V}\text{diag}(\lambda_{k})\mathbf{V}^{\dagger},$

where $(\lambda_{i})$ is given by (185). All Hermitian kernels with spectrum
$(\lambda_{k})$ can thus be obtained from the same Hamiltonian, just changing
the measurement basis.

At one extreme, measuring in the original basis used to build the Hamiltonian,
i.e. taking $\mathbf{V}=\mathbf{I}$, leads to a diagonal kernel matrix. This
means that the corresponding point process is a set of independent Bernoulli
samples for fermions, and independent geometric samples for bosons (Hough et
al., 2006). Correlation appears when the observation basis $(v_{k})$ and the
creation basis $(\varphi_{k})$ do not match. For instance, let
$v_{k}=\varphi_{k}$ for $k=1,\dots,N-2$, but apply a special unitary
transformation to the last two vectors. That is, for
$\alpha,\beta\in\mathbb{C}$ such that $|\alpha|^{2}+|\beta|^{2}=1$, complete
the observation basis with

$v_{N-1}=\alpha\varphi_{N-1}-\overline{\beta}\varphi_{N}\quad\text{ and }\quad
v_{N}=\beta\varphi_{N-1}+\overline{\alpha}\varphi_{N}.$

The resulting permanental or determinantal point process, call it
$\gamma^{\alpha,\beta}$, has a block diagonal kernel

$\mathbf{K}^{\alpha,\beta}=\begin{pmatrix}\lambda_{1}&&&&\\\ &\ddots&&&\\\
&&\lambda_{N-2}&&\\\
&&&\lambda_{N-1}|\alpha|^{2}+\lambda_{N}|\beta|^{2}&(\lambda_{N-1}-\lambda_{N})\overline{\alpha}\beta\\\
&&&(\lambda_{N-1}-\lambda_{N})\alpha\overline{\beta}&\lambda_{N-1}|\beta|^{2}+\lambda_{N}|\alpha|^{2}\end{pmatrix}.$

Note that since the diagonal of $\mathbf{K}^{\alpha,\beta}$ varies with
$(\alpha,\beta)$, the first correlation function of the point process is also
altered. For instance, the marginal probability that item $N-1$ belongs to the
point process is $\lambda_{N-1}|\alpha|^{2}+\lambda_{N}|\beta|^{2}$. If
$\lambda_{N-1}\geq\lambda_{N}$, then as soon as $\beta\neq 0$, the marginal
probability of item $N-1$ occurring decreases. Note that for $\eta=-1$,
though, the average number of points in $\gamma^{\alpha,\beta}$ remains
constant, as the trace of $\mathbf{K}^{\alpha,\beta}$ does not depend on
$(\alpha,\beta)$. Meanwhile, the second correlation function
$\rho_{2,\theta}(N-1,N)$ is the permanent/determinant of the trailing block of
$\mathbf{K}^{\alpha,\beta}$, namely

$\rho_{2}^{\alpha,\beta}(N-1,N)=|\alpha|^{2}|\beta|^{2}\left(\lambda_{N-1}^{2}+\lambda_{N}^{2}+\eta(\lambda_{N-1}-\lambda_{N})^{2}\right)+\left(|\alpha|^{4}+|\beta|^{4}\right)\lambda_{N-1}\lambda_{N}.$

For the sake of illustration, we focus now on $\eta=-1$, in which case

$\rho_{2}^{\alpha,\beta}(N-1,N)=\lambda_{N-1}\lambda_{N}\left(|\alpha|^{2}+|\beta|^{2}\right)^{2}=\lambda_{N-1}\lambda_{N}.$

The probability of co-occurrence of items $N-1$ and $N$ thus does not depend
on $\alpha$ and $\beta$, while the product of their marginal probabilities of
occurrence does change: we have thus introduced correlation between the events
$\\{N-1\in\gamma^{\alpha,\beta}\\}$ and $\\{N\in\gamma^{\alpha,\beta}\\}$.
Because $\gamma^{\alpha,\beta}$ is a DPP with Hermitian kernel, this
correlation is nonpositive.

To go further, an interesting question would be to relate the different ways
to quantify repulsiveness in a DPP (Biscio and Lavancier, 2016; Møller and
O’Reilly, 2021) to properties of the bases used in the construction of free
fermions.

####  6.2.2 A DPP from bosons in a non-Gaussian state

Another example that illustrates the subtleties of associating correlation
functions to states is that (non-Gaussian) density matrices corresponding to
interacting (i.e., non-free) bosons can give rise to a DPP for some
observables.

One example is the Tonks-Girardeau gas (Girardeau, 1960), which consists of
impenetrable (statisticians would say _hardcore_) bosons in
$\mathbb{X}=\mathbb{R}$. The very strong interactions prevent two bosons to be
at the same position, which is reminiscent of the Pauli exclusion principle
for fermions. In particular, one considers the usual bosonic field operators
$a(x),a^{\dagger}(y)$, satisfying the canonical commutation relations as long
as $x\neq y$, but the hardcore constraint is enforced by requiring that
$a(x)^{2}=a^{\dagger}(x)^{2}=0$ and $[a(x),a^{\dagger}(x)]_{+}=1$. These
additional constraints are the results of adding a large penalty term to the
Hamiltonian at coincidence.

Now, there actually is a transformation, called the _Jordan-Wigner
transformation_ , that maps these hardcore bosons onto free fermions (Lieb et
al., 1961). Concretely, one can write $a(x)=S(x)\psi(x)$ and
$a^{\dagger}(x)=\psi^{\dagger}(x)S^{\dagger}(x)$, where
$\psi(x),\psi^{{\dagger}}(x)$ obey the standard canonical anti-commutation
relations, and $S(x)$ is a unitary operator called a _string operator_ , which
allows for preserving the commutation relations of the bosonic
operators.515151 In particular, it satisfies $[S(x),S(y)]=0$ and
$S(x)\psi(y)+{\rm sign}(x-y)\psi(y)S(x)=0$.

By construction, all observables that are built from number
operators525252Usually rather called _density operators_ when we use a
generalized basis like here. are the same as for free fermions, since
$a^{\dagger}(x)a(x)=\psi^{\dagger}(x)\psi(x)$, and the correlation functions
of an ideal detection experiment are those of a DPP with kernel
$\langle\psi^{\dagger}(x)\psi(y)\rangle$. However, the coherence function
$\langle
a^{\dagger}(x)a(y)\rangle=\langle\psi^{\dagger}(x)S^{\dagger}(x)S(y)\psi(y)\rangle$
is very different from that of the free fermions. For instance, the largest
eigenvalue of the operator with kernel $\langle a^{\dagger}(x)a(y)\rangle$
evaluated in a pure state typically scales like the square root of the number
of particles; see e.g. Forrester et al. (2003) for investigations on the
ground state properties of a Tonks gas in a harmonic trap.

####  6.2.3 Interacting field theories and point processes

The relationship between, on one side, permanental and determinantal point
processes, and, on the other side, bosonic and fermionic quantum field theory,
is based on free models, i.e., with a quadratic Hamiltonian in the fields.
However, in physical models, Hamiltonians are rarely quadratic, but take into
account interactions between the particles. For a given Hamiltonian, success
is achieved when it is possible to approximate the coherence functions of the
field, which correspond to concrete experimental measurements. The main issue
is that Wick’s theorem does not apply.

There are interacting Hamiltonians that still yield closed-form correlation
functions for the underlying ideal detection experiments. Such systems are
usually called _integrable_ by physicists. For instance, the Calogero-
Sutherland model is a one-dimensional system of interacting fermions, for
which ideal detection leads to well-known point processes, called
$\beta$-ensembles in random matrix theory; see e.g. (Forrester, 2010, Chapter
11) as well as (Stéphan, 2019; Smith et al., 2021).

Yet, most interacting systems are not integrable. In Sections 4 and 5, we have
used simple perturbation-theoretic arguments like Assumption 5 to work with
the interaction Hamiltonian. There is now a significant and sophisticated
toolbox to approximate coherence functions in the presence of interaction. Key
tools include mean-field methods (Goldenfeld, 2018, Chapter 3), Feynman
diagrams (Folland, 2008, Chapter 6) and renormalisation (Folland, 2008,
Chapter 7). At a high level, it would be interesting to investigate what these
methods say about point processes. In words, one of the lessons of quantum
field theory is that it is possible, even in the presence of interaction, to
get “close” to a free situation in some regimes. Does this imply approximation
results for the sophisticated point processes behind models with interaction?
Conversely, can we use the mathematical technology of DPPs to perform non-
trivial calculations for many-body problems?

Moreover, a central concept in quantum field theory is that of universality,
i.e., the fact that many different interacting models behave, in a certain
regime usually qualified as the _low-energy regime_ , essentially in the same
way. Put differently, the detailed form of the interactions are irrelevant to
understand this low-energy regime, only the dimension of the space and the
symmetries of the model usually matter. The archetypal phenomenon illustrating
this is phase transitions (Goldenfeld, 2018). Does this universality connect
with similar results for point processes, such as those obtained for the
eigenvalues of random matrices (Anderson et al., 2010)?

####  6.2.4 Constructive arguments for point processes

The fermionic system “above” a DPP can help address fundamental questions on
DPPs, about their invariance or their construction. For instance, a single DPP
corresponds to many kernels. In particular, for any $f$, the kernel

$x,y\mapsto\frac{f(x)}{f(y)}K(x,y)$

yields the same correlation functions (186) as $K$. Without strong assumptions
on the kernel, it has proven difficult to find all the transformations of a
kernel that leaves a DPP invariant. It would be interesting if the fermionic
framework helped us to understand these invariances; see e.g. recent partial
results by Olshanski (2020).

As another example, generalizing permanental and determinantal point
processes, one can define $\alpha$-DPPs as having correlation functions

$\displaystyle\rho_{k}(x_{1},\ldots,x_{k})={\det}_{\alpha}\Big{[}K(x_{i},x_{j})\Big{]}_{i,j=1}^{k}\,,\quad
k\geq 1,$ (186)

where, for an $n\times n$ matrix $\mathbf{A}=((a_{ij}))$,

${\det}_{\alpha}(\mathbf{A}):=\sum_{\sigma\in\mathfrak{S}_{n}}\alpha^{n-\nu(\sigma)}\prod_{i=1}^{n}a_{i\sigma(i)}.$

Shirai and Takahashi (2003) have studied the existence of $\alpha$-DPPs for
$\alpha\in[-1,1]$, where $\alpha=-1$ corresponds to a DPP, and $+1$ to a
permanental point process. Some $\alpha$-DPPs appear naturally when
marginalizing a projection DPP with a separable kernel over coordinates
(Mazoyer et al., 2020). It is a natural question whether $\alpha$-DPPs
correspond to any physical system of particles. As a partial affirmative
answer, Cunden et al. (2019) give a limit procedure to construct certain
$\alpha$-DPPs out of fermionic processes.

In the same vein, can we build point processes from particles with more exotic
commutation rules for their ladder operators? A natural physical example is
anyons (Ezawa, 2008, Chapter 8), for which a phase factor appears for each
transposition.

The links of fermions to Pfaffian point processes, another generalization of
DPPs are also a promising research direction (Koshida, 2021), as well as the
point processes behind quasi-free states (Bach et al., 1994; Lytvynov, 2002;
Lytvynov and Mei, 2007; Olshanski, 2020), namely (possibly non-Gaussian)
states to which Wick’s theorem still applies.

###  6.3 A teaser for Part II

This manuscript is intended to become Part I of a monograph. In Part II, we
shall present selected topics at the intersection of point processes and
quantum optics, using the vocabulary of Part I. We will describe landmark
experimental measurements of HBT signals, with both photons and (bosonic and
fermionic) atoms. We will discuss the application of determinantal point
processes to the study of non-interacting trapped fermions in statistical
physics. We will show how the formalism of quantum field theory can help to
prove fundamental results on point processes appearing in combinatorics.
Finally, we will discuss electronic quantum optics and its interactions with
signal processing.

### Acknowledgments

We thank all participants to the Lille and Lyon workshops, and all
participants to the Lille workgroup on point processes and applications; the
discussions held in these circles helped a lot in the construction of this
document. We hope that the document, in return, participates to developing
this cross-disciplinary research field. In particular, RB thanks all
physicists with whom he has interacted over the last years for their
enthusiasm and their patience, explaining basic physical concepts over and
over until we converged to a description that spoke to all of us.

We acknowledge support from ERC grant Blackjack (ERC-2019-STG-851866), ANR AI
chair Baccarat (ANR-20-CHIA-0002), ANR grant cl1shot reloaded
(ANR-14-CE32-0017), ANR grant QuSig4QuSense (ANR-21-CE47-0012), the Joint
Research Project SEQUOIA (17FUN04) within the European Metrology Programme for
Innovation and Research (EMPIR), ANR grants QRITiC I-SITE ULNE and
ANR-16-IDEX-0004 ULNE, ANR grant COSQUA (ANR-20-CE47-0001-01), ANR grant MENTA
(ANR-22-QUA2-0008-01), ANR Grant RaMaTraF (ANR-17-CE30-0027-01), ANR grant
Dimers (ANR-18-CE40-0033), ANR grant Combiné (ANR-19-CE48-0011), Labex PALM
(ANR-10-LABX-0039-PALM), and Labex MILYON.

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