Research

Quantum Optics, Multiphoton sources

Elena del Valle

# Two photons everywhere

E. Zubizarreta Casalengua1 F. P. Laussy2 and E. del Valle3 1Walter Schottky
Institute, School of Computation, Information and Technology and MCQST,
Technische Universität München, 85748 Garching, Germany
2Instituto de Ciencia de Materiales de Madrid ICMM-CSIC, 28049 Madrid, Spain
3Departamento de Física Teórica de la Materia Condensada e IFIMAC, Universidad
Autónoma de Madrid, 28049 Madrid, Spain.
Institute for Advanced Study, Technische Universität München, 85748 Garching,
Germany<EMAIL_ADDRESS>
###### Abstract

We discuss two-photon physics, taking for illustration the particular but
topical case of resonance fluorescence. We show that the basic concepts of
interferences and correlations provide at the two-photon level an independent
and drastically different picture than at the one-photon level, with
landscapes of correlations that reveal various processes by spanning over all
the possible frequencies at which the system can emit. Such landscapes
typically present lines of photon bunching and circles of antibunching. The
theoretical edifice to account for these features rests on two pillars: i) a
theory of frequency-resolved photon correlations and ii) admixing classical
and quantum fields. While experimental efforts have been to date concentrated
on correlations between spectral peaks, strong correlations exist between
photons emitted away from the peaks, which are accessible only through
multiphoton observables. These could be exploited for both fundamental
understanding of quantum-optical processes as well as applications by
harnessing these unsuspected resources.

###### keywords:

Squeezing, Antibunching, Correlations, Interferences

##  1 Introduction

Quantum mechanics is notorious for its quantized spectral lines. This is how
the theory was born and the fact after which it is named, following Bohr [1]’s
quantization of angular momentum of the electron orbitals. This produced
Rydberg’s constant as a product of fundamental constants and explained the
lines as somethingyouwanttohide

Figure 1: (a) Old quantum mechanics: transitions between quantized orbitals of
the electron lead to a quantized spectrum. Transitions to $n=2$ form the
visible (Balmer) series. The $2\to 1$ (Lyman) is in the ultraviolet. The
$2s\to 1s$ is furthermore dipole forbidden, but it can occur with a continuum
of two-photon emission, which falls again in the visible range. (b)
Transitions between dressed states $\left|\pm\right\rangle$ lead to a triplet
spectrum (Mollow). A two-photon, so-called _leapfrog_ , transition can also
occur that jumps over an intermediate state. However weak is this transition,
it can be revealed by two-photon correlations. (c) In the low-driving limit,
detuning makes the same physics take another turn, with the triplet now formed
by the Rayleigh scattering as the central peak and (d) breaking the leapfrog
over one real-state transition but keeping the other virtual, accounting for
the two side peaks. This results in a rich two-photon physics beyond the side
peaks, not visible in photoluminescence but revealed in the two-photon
correlation spectrum.

transitions between energy levels (Fig. 1(a)). At about the same time, a much
less resounding finding was recorded in a bulletin of the Lowell Observatory:
a nebula in the Pleiades was observed to have a completely continuous spectrum
of emission [2]. There are many ways to produce a continuous spectrum, from
unbound charges to, as was the case here, scattering from a broad spectrum
(the light from the star Merope). Systematic and careful observations revealed
that emission nebulae (in particular planetary nebulae) which do not reflect
but emit light directly, also feature a continuous spectrum in addition to the
spectral lines of what was by then well-established quantum theory [3]. One
part—the Balmer continuum—could be explained with the known mechanisms from
reflection nebulae, but on the other side of the Balmer limit, sitting with
the quantized spectral lines in the visible spectrum, lied a continuum
spectrum which was of unknown origin [4]. This got simultaneously and
independently resolved in 1950, from both sides of the iron curtain [5, 6], as
a fitting companion to Bohr’s model of quantum jumps in the form of a two-
photon emission from the metastable $2S_{1/2}$ shell of the hydrogen atom down
to the ground state $1S_{1/2}$ (Fig. 1(a)). Would the atom be excited in the
$2P$ level, it would undergo a normal, “Bohrian” transition back to the ground
state, but the $2S$–$1S$ is dipole forbidden so it has to go through the next
(quadrupole) order, which is however very weak, and, in ambient conditions, is
knocked-off by collision back into a radiative state. In the extremely
rarefied cosmic conditions, however, there is plenty of space as well as
enough time to have isolated Hydrogen atoms left stuck in great quantity in
their $2S$ state. For those, the next best route back to $1S$ is the
“Doppelemission” (double-emission or two-photon emission) theorized by
Goeppert Mayer [7] and first applied to Hydrogen by Breit and Teller [8].
Although a higher-order process, it is not dipole forbidden and finds in outer
space the ideal quiet laboratory conditions making possible its direct
observation, indeed giving it the name of “visual continuum”. The Earth-based
laboratory is less auspicious for a direct (visual) observation of two-photon
emission [9] but the process nevertheless opened the multiphoton page of
atomic physics [10]. Such two-photon transitions have, among other things, be
generalized to link any $(n,l,m)$ states of the atom [11] and can be
stimulated with a laser [12]. In the solid state, the opposite regime of the
interstellar one can be realized of high populations in small volumes, e.g.,
in semiconductors, where delocalized electrons in the crystal under high laser
excitation can jump the bandgap in sufficient amount to produce a measurable
and even controllable two-photon continuous spectrum [13]. Another solid-state
approach is to inflate the light-matter coupling by focusing light onto the
emitter, to make higher-order processes “less smaller” and thus lift their
“forbidden” character, possibly making them even comparable to first-order
processes [14].

Here, we present a quantum optical alternative to the QED description. The
latter typically relies on a perturbative treatment of the two-photon
processes of a complex system: the Hydrogen atom in its simplest case, up to
an interstellar ionized gas bathed in the radiation of other astronomical
objects. Instead, we consider the _exact_ treatment of a _simple_ system,
namely, resonance fluorescence, i.e., the photon emission (fluorescence) of a
two-level system driven at the same energy as it is excited (resonance). This
will allow us to focus on the two-photon physics itself, instead of
interesting but secondary problems specific to Hydrogen or to the
thermodynamics of nebulae. While we invoke a variety of our results collected
over the last decade, we try to keep the discussion self-contained with no
need of prior familiarity with our earlier works on, mainly, frequency-
resolved multiphoton correlations [15] or multiphoton interferences of quantum
fields [16]. We furthermore connect these two aspects to provide a new and
fairly comprehensive picture of the phenomenology of two-photon emission from
resonance fluorescence, explaining features hitherto only observed. Section 2
introduces the textbook problem of resonance fluorescence but revisited with
the “sensor formalism” [15], which is an alternative way to compute spectra
that will allow us, in Section 3, to provide a first departure from
conventional treatments, by introducing our concept of frequency-resolved
correlations, which will put the two-photon observables on the same footing as
the one-photon ones, and explain the reason for our title of “two photons
everywhere”. In Section 4, we introduce the other pillar of our edifice:
interferences of quantum fields, as underpinning their statistical properties
[16]. The basic idea, that holds with quantum states, is applied to a
dynamical system in Section 5, which, in spirit, is resonance fluorescence
itself, but we show that the two-photon observables are actually captured by
the simpler case of a squeezed cavity, allowing us to identify what is
specific to quantum states admixtures of Gaussian states (squeezing and
coherent states) and what is to the “more quantum” two-level system, to which
we return in Section 6 with a focus on detuning. We provide a surprisingly
compact expression for two-photon correlations in this case, that is exact to
leading order in the driving and captures its main phenomenology. In Section
7, we show how our approach gives way to countless variations, even if
remaining at the level of resonance fluorescence, although this could and
should be extended to all possible quantum emitters. Specifically, we consider
entanglement and other quantum resources such as two-mode squeezing, which we
merely exhibit but that could be similarly explained and exploited. Our
approach could also usefully revisit closely related problems such as two-
photon absorption [17], two-photon resonance fluorescence [18] or two-photon
gain [19].

##  2 Resonance Fluorescence

Resonance fluorescence is the simplest problem of quantum optics, yet a still
actively investigated one. Its Hamiltonian in the rotating frame of the laser
is (with $\hbar=1$)

$H_{\sigma}\equiv\Delta_{\sigma}\,{\sigma^{\dagger}}\sigma+\Omega_{\sigma}({\sigma^{\dagger}}+\sigma)$
(1)

where $\sigma$ is the annihilation operator of a two-level system driven
coherently by a laser with amplitude $\Omega_{\sigma}\in\mathbb{R}$, and
$\Delta_{\sigma}$ is the detuning from the laser frequency
$\omega_{\mathrm{L}}$ which we shall take as the reference, i.e.
$\omega_{\mathrm{L}}=0$. This problem provides a rich quantum-optical
playground, from essentially two regimes of excitation: at low-driving (so-
called Heitler regime [20]), the spectrum is dominated by the Rayleigh-
scattered light of the laser, while at high-driving (so-called Mollow regime
[21]), there is a splitting of the spectral shape into a triplet (Fig. 1).
While these two regimes are very different in character, we can find a first
unifying theme through large detuning. When the laser drives the system far
from its resonance frequency (although still referring to “resonance
fluorescence”), the spectral response remains a symmetric triplet. This is
shown at the top of Fig. 2 for (a) the Mollow triplet at resonance, (b) the
detuned Mollow triplet with an increasingly bright coherent peak sitting on
top of a dim fluorescent one and (c) the detuned (Heitler regime of) resonance
fluorescence with vanishing fluorescent contributions. The driving
$\Omega_{\sigma}$ has been chosen so that the spectral position of the side
peaks is the same in units of the Mollow splitting
$\Omega_{+}\approx\sqrt{\Delta_{\sigma}^{2}+4\Omega_{\sigma}^{2}}$ (at large
enough detunings). The frequency $\omega$ is normalized to this splitting,
i.e., $\varpi\equiv(\omega-\omega_{\mathrm{L}})/\Omega_{+}$. The central peak
is either (a) a fluorescent (broad) peak, (c) the Rayleigh-scattered coherent
peak (intense and narrow central peak, note that the side peaks are magnified
by a factor $800$) or (b) an admixture of a fluorescent and coherent peaks.

Figure 2: The two-photon physics of resonance fluorescence. Top row:
photoluminescence spectra for (a) the Mollow triplet at resonance, (b) the
detuned Mollow triplet and (c) the detuned Heitler triplet. The coherent
(scattered) light is shown in yellow. Second row: two-photon correlation
spectra $g^{(2)}_{\Gamma}(\varpi_{1},\varpi_{2})$ for the corresponding
spectra, with bunching in red, no-correlation in white and antibunching or no-
coincidence emission in blue. Two sets of features dominate the landscape:
straight lines of bunching (red) and circles of antibunching (blue). Lines
result from multiphoton transitions that involve virtual photons—“leapfrog
processes”—while circles result from self-homodyning destructive
interferences. Three bottom rows: covariance $\mathcal{I}_{0}$ (third row),
anomalous two-photon moments $\mathcal{I}_{1}$ (fourth) and squeezing
$\mathcal{I}_{2}$ (bottom) which sum together with 1 to the two-photon
spectrum. Shaded regions refer to negative quantities. Parameters:
$\gamma_{\sigma}=1$ (setting the unit) and $\Gamma=2$ everywhere, while
$(\Delta_{\sigma},\Omega_{\sigma})=$ (a) $(0,40.05)$, (b) $(60,26.53)$ and (c)
$(80,2)$.

One can conveniently derive such results with our sensor formalism [15], which
is an alternative to formal results that rely on mathematical structures of
the problem—such as the quantum regression theorem or the Wiener-Khinchin
theorem—to obtain instead observables from a physical modeling of what is
being measured. This was developed to extend the physical spectrum of Eberly
and Wódkiewicz [22] to $N$-photon observables, which instead of computing
awkward multi-time, normally-ordered integrals, simply attaches to the system
“sensors” (two of them for two-photon correlations, $N$ sensors in general)
and computes correlations directly from their usual quantum averages.
Technically, this requires to “plug” to the Hamiltonian of the system (in our
case, Eq. (1)), the sensors, themselves most simply described as two-level
systems $\varsigma_{i}$ whose frequencies $\omega_{i}$ define which
frequencies are being “measured”. The name “sensor” was chosen as opposed to
“detector” since correlations are obtained in the limit $\epsilon\to 0$ of
their vanishing coupling to the system, thereby not affecting its dynamics.
This restricts their use to correlations as opposed to signal. Alternatively,
one can also use the “cascading systems” [23, 24], which was shown in Ref.
[25] to be equivalent to the sensor method and to conveniently substitute it
in cases where the signal is needed, e.g., to perform frequency-resolved Monte
Carlo simulations. For resonance fluorescence, the augmented Hamiltonian thus
reads:

$H_{\sigma;\varsigma}\equiv
H_{\sigma}+\Delta_{1}{\varsigma_{1}^{\dagger}}\varsigma_{1}+\Delta_{2}{\varsigma_{2}^{\dagger}}\varsigma_{2}+\epsilon\sum_{i=1,2}({\sigma^{\dagger}}\varsigma_{i}+{\varsigma_{i}^{\dagger}}\sigma)\,.$
(2)

In the rotating frame of the laser, the frequencies
$\Delta_{i}\equiv\omega_{i}-\omega_{\mathrm{L}}$ are simply $\omega_{i}$ since
$\omega_{\mathrm{L}}=0$. As a quantum-optical problem, one should include
dissipation, which can be provided by a master equation in the Lindblad form,
so that the dynamics for the full density matrix $\rho$—of the system itself
as well as its sensors—is governed by the equation

$\partial_{t}{\rho}=-i[H_{\sigma},\rho]+{\gamma_{\sigma}\over
2}\mathcal{L}_{\sigma}\rho+\sum_{j=1,2}\frac{\gamma_{j}}{2}\mathcal{L}_{\varsigma_{j}}\rho,$
(3)

where the Lindblad terms are of the form $\mathcal{L}_{c}\rho\equiv
2c\rho{c^{\dagger}}-{c^{\dagger}}c\rho-\rho{c^{\dagger}}c$ for any operator
$c$. Importantly, in addition to the decay rate $\gamma_{\sigma}$ of the two-
level system, we also bring the decay rate $\gamma_{i}$ of the $i$th sensor,
that describes its frequency bandwidth. For two-photon observables (and higher
$N$), such a parameter is mandatory for a physical description of the system,
unlike one-photon observables like the power spectrum, which can be well
described for a vanishing linewidth of their detector (recovering the Wiener-
Khinchin result). In the following we shall assume the same frequency window
for both sensors, i.e., $\gamma_{1,2}=\Gamma$, which we shall refer to as the
“filter width” as this also corresponds to placing an interference filter
before an ideal detector. The sensor method allows us to obtain the
luminescence spectrum $S_{\Gamma}(\omega)={\Gamma\over
2\pi\epsilon^{2}}\langle{\varsigma^{\dagger}}\varsigma\rangle(\omega)$ (no
index needed since a single sensor is enough) as the population of the sensor
$\varsigma$ placed at the frequency $\omega$, which is a mere parameter in Eq.
(2) as opposed to a physical observable and thus an operator, which would
complicate very much the treatment. In particular, note that we need not
invoke any quantum-regression theorem, Fourier transform, etc. For resonance
fluorescence, the spectral shape has a simple, characteristic structure:

$S_{\Gamma}(\omega)=|\langle\sigma\rangle|^{2}\mathscr{L}_{\Gamma}(\omega)+\big{(}\langle{\sigma^{\dagger}}\sigma\rangle-|\langle\sigma\rangle|^{2}\big{)}\mathscr{M}_{\Gamma}(\omega)$
(4)

where

$\langle\sigma\rangle=\frac{2\Omega_{\sigma}(2\Delta_{\sigma}+i\,\gamma_{\sigma})}{\gamma_{\sigma}^{2}+4\Delta_{\sigma}^{2}+8\Omega_{\sigma}^{2}}\quad\text{and}\quad\langle{\sigma^{\dagger}}\sigma\rangle=\frac{4\Omega_{\sigma}^{2}}{\gamma_{\sigma}^{2}+4\Delta_{\sigma}^{2}+8\Omega_{\sigma}^{2}}$
(5)

are the coherently-scattered field $\langle\sigma\rangle$ (with intensity the
modulus square of this) and the total population
$\langle{\sigma^{\dagger}}\sigma\rangle$, including the coherent part, so the
incoherent part alone is
$\langle{\sigma^{\dagger}}\sigma\rangle-|\langle\sigma\rangle|^{2}=32\Omega_{\sigma}^{2}/(\gamma_{\sigma}^{2}+4\Delta_{\sigma}^{2}+8\Omega_{\sigma}^{2})^{2}$
(this happens to be $2\langle{\sigma^{\dagger}}\sigma\rangle^{2}$). These are
not $\Gamma$ dependent. They weight, respectively, a central Lorentzian peak
$\mathscr{L}_{\Gamma}(\omega)\equiv\frac{1}{\pi}\frac{\Gamma/2}{(\Gamma/2)^{2}+\omega^{2}}$
and a symmetric triplet (both normalized), whose expression is a bit more
involved:

$\mathscr{M}_{\Gamma}(\omega)\equiv\\\
{2\over\pi}{(\gamma_{12}^{2}+4\omega^{2})(\gamma_{11}^{2}\gamma_{12}+4\gamma_{12}\Delta_{\sigma}^{2}+4\gamma_{10}\omega^{2})+8\Omega_{\sigma}^{2}(\gamma_{11}\gamma_{12}\gamma_{35}+4\gamma_{12}\Delta_{\sigma}^{2}-4\gamma_{1\bar{2}}\omega^{2})+128\Omega_{\sigma}^{4}\gamma_{11}\over\begin{multlined}(\gamma_{12}^{2}+4\omega^{2})\big{[}(\gamma_{11}^{2}+4\Delta_{\sigma}^{2})^{2}+8(\gamma_{11}^{2}-4\Delta_{\sigma}^{2})\omega^{2}+16\omega^{4}\big{]}+{}\\\
{}+32\Omega_{\sigma}^{2}\big{[}\gamma_{11}\gamma_{12}(\gamma_{11}^{2}+4\Delta_{\sigma}^{2})+4(\gamma_{01}\gamma_{11}+4\Delta_{\sigma}^{2})\omega^{2}-16\omega^{4}\big{]}+256\Omega_{\sigma}^{4}(\gamma_{11}^{2}+4\omega^{2})\end{multlined}(\gamma_{12}^{2}+4\omega^{2})\big{[}(\gamma_{11}^{2}+4\Delta_{\sigma}^{2})^{2}+8(\gamma_{11}^{2}-4\Delta_{\sigma}^{2})\omega^{2}+16\omega^{4}\big{]}+{}\\\
{}+32\Omega_{\sigma}^{2}\big{[}\gamma_{11}\gamma_{12}(\gamma_{11}^{2}+4\Delta_{\sigma}^{2})+4(\gamma_{01}\gamma_{11}+4\Delta_{\sigma}^{2})\omega^{2}-16\omega^{4}\big{]}+256\Omega_{\sigma}^{4}(\gamma_{11}^{2}+4\omega^{2})}$
(6)

where we use the notation $\gamma_{ij}\equiv i\Gamma+j\gamma_{\sigma}$ for any
integer $i$, $j$, with also $\bar{\jmath}=-j$ so, e.g.,
$\gamma_{35}=3\Gamma+5\gamma_{\sigma}$. The expression is not particularly
enlightening but it is completely general, including the Mollow triplet and
Heitler regime, at and out-of resonance, also with the effect of detection
$\Gamma$. An even more complete (with incoherent pumping and dephasing)
version of this expression for the “physical spectrum” of resonance
fluorescence is available as a self-standing applet [26]. At any rate, its
shape is simple: it is a triplet which maintains a perfect symmetry around the
laser frequency set here at $\omega_{\mathrm{L}}=0$, whose interpretation in
the various regimes of driving is given in Figs. 1(b) and (c). In both cases,
Bohr’s insight provides a discerning picture of the otherwise mysterious
spectral features of resonance fluorescence. At high-driving, the so-called
“dressed atom” picture considers such quantum jumps between the dressed states
$\left|\pm\right\rangle\equiv c_{\pm}\left|\mathrm{g}\right\rangle\pm
c_{\mp}\left|\mathrm{e}\right\rangle$ of a two-level system
$\left|\mathrm{g}\right\rangle$ and $\left|\mathrm{e}\right\rangle$ dressed by
photons from a driving laser, with $c_{\pm}\equiv(1+\xi^{\mp 2})^{-1/2}$ and
$\xi\equiv\Omega_{\sigma}/[\sqrt{\Omega_{\sigma}^{2}+(\Delta_{\sigma}/2)^{2}}+(\Delta_{\sigma}/2)]$.
The central peak, at frequency $\omega_{0}$, is twice as bright than the side
peaks $\omega_{\pm}$ due to two degenerate transitions
$\left|+\right\rangle\to\left|+\right\rangle$ and
$\left|-\right\rangle\to\left|-\right\rangle$. At low driving, now looking at
the system in its bare states, a detuned laser does not make it to the excited
state and so is restrained chiefly to Rayleigh (energy-conserving) scattering.
However, involving two laser photons, one can match the energy
$\omega_{\sigma}$ of the emitter with the excess energy $\omega_{v}\equiv
2\omega_{\mathrm{L}}-\omega_{\sigma}$ (either smaller or larger than
$\omega_{\sigma}$ depending on detuning), forming an exactly symmetric peak.
This perceptive physics is well-known since the 80s [27, 28, 29]. Its
understanding has little evolved since then [30].

Since the photoluminescence spectrum is a single-photon observable, one could
consider a vanishing detector bandwidth $\Gamma\to 0$ (replacing $\gamma_{ij}$
by $j\gamma_{\sigma}$), in which case one sees a tightening of the lines, in
particular, the coherent Lorentzian $\lim_{\Gamma\to
0}\mathscr{L}_{\Gamma}(\omega)$ becomes a Dirac $\delta(\omega)$ function that
reflects the vanishing linewidth of the laser treated as a $c$ number
($\Omega_{\sigma}$) in Eq. (1). The effect of detection is thus a mere (and
expected) broadening of the lines, which does not qualitatively alter the
spectral shape. The case of Heitler resonance
($\Omega_{\sigma}\ll\gamma_{\sigma}$ and $\Delta_{\sigma}=0$) has attracted
deserved but unwary attention. A short discussion will allow us to motivate
the significance of detection, to which the sensor formalism gives utmost
importance. The spectrum (6) reduces in this case to:

$S_{\Gamma}(\omega)={4\Omega_{\sigma}^{2}\over\gamma_{\sigma}^{2}}\left[\left(1-{16\Omega_{\sigma}^{2}\over\gamma_{\sigma}^{2}}\right)\mathscr{L}_{\Gamma}(\omega)+\frac{8\Omega_{\sigma}^{2}}{\gamma_{\sigma}^{2}}\frac{2}{\pi}\frac{\gamma_{12}\gamma_{11}^{2}+4\Gamma\omega^{2}}{(\gamma_{11}^{2}+4\omega^{2})^{2}}\right]$
(7)

and, again, the $\Gamma\to 0$ appears to be a straightforward and good-enough
approximation of the main features

$S_{0}(\omega)={4\Omega_{\sigma}^{2}\over\gamma_{\sigma}^{2}}\left[\left(1-{16\Omega_{\sigma}^{2}\over\gamma_{\sigma}^{2}}\right)\delta(\omega)+{8\Omega_{\sigma}^{2}\over\gamma_{\sigma}^{2}}{\pi\gamma_{\sigma}}\mathscr{L}_{\gamma_{\sigma}}^{2}(\omega)\right]$
(8)

in fact making them particularly transparent, namely, as long as
$\Gamma\leq\gamma_{\sigma}$ (sub-natural linewidth resolution), given that
$\Omega_{\sigma}^{2}\ll\gamma_{\sigma}^{2}$, the spectrum is essentially a
very narrow central line sitting on a weak and broad ($\gamma_{\sigma}$)
spectrum. This scenario can be seen for the central peak in panel (b) of Fig.
2. The fluorescent spectrum is, to leading order, the _square_ of a
Lorentzian, exposing its two-photon origin. This expression corrects the
erroneous Eq. (3) in Ref. [35]. The impact of the detector in Eq. (8) is
merely to contribute a quite innocuous broadening: one can take better and
better detectors and converge to the ideal limit (8) which is essentially
${4\Omega_{\sigma}^{2}\over\gamma_{\sigma}^{2}}\delta(\omega)$, so it would
appear that we are merely discussing technical issues. That is so, at the one-
photon level.

Two-photon observables, however, behave differently, at least, if taken in
their entirety. The complete quantity should retain both the frequencies
$\omega_{1}$ and $\omega_{2}$ as well as the times of detection $t_{1}$ and
$t_{2}$ of both photons, or, in a steady state, the time difference
$\tau\equiv t_{2}-t_{1}$. We will introduce it in the next Section, but for
now, to compare with the results found in the literature, we assume both
frequencies to be that of the emitter $\omega_{1}=\omega_{2}=\omega_{\sigma}$,
itself at resonance with the laser frequency $\omega_{\mathrm{L}}$, in which
case the sensor formalism provides us with
$g^{(2}_{\Gamma}(\tau)=\big{\langle}{\varsigma_{1}^{\dagger}}(0)({\varsigma_{2}^{\dagger}}\varsigma_{2})(\tau)\varsigma_{1}(0)\big{\rangle}/[\langle{\varsigma_{1}^{\dagger}}\varsigma_{1}\rangle\langle{\varsigma_{2}^{\dagger}}\varsigma_{2}\rangle]$
which evaluates to

$g^{(2)}_{\Gamma}(\tau)=\left(e^{-(\Gamma+\gamma_{\sigma})\tau/2}+{\Gamma\gamma_{\sigma}\over\Gamma^{2}-\gamma_{\sigma}^{2}}e^{-\Gamma\tau/2}-{\Gamma^{2}\over\Gamma^{2}-\gamma_{\sigma}^{2}}e^{-\gamma_{\sigma}\tau/2}\right)^{2}\,,$
(9)

(the limit $\Gamma\to\gamma_{\sigma}$ gives
$g^{(2)}_{\gamma_{\sigma}}(\tau)=\big{[}{1+(\gamma_{\sigma}\tau)/2\over
2}e^{-\gamma_{\sigma}\tau/2}-e^{-\gamma_{\sigma}\tau}\big{]}^{2}$). The most
natural way to neglect frequencies is to detect them all, which corresponds to
taking the limit $\Gamma\to\infty$, in which case one recovers the result from
the literature [42]:

$g^{(2)}_{\infty}(\tau)=(1-e^{-\gamma_{\sigma}\tau/2})^{2}\,.$ (10)

This describes an excellent single-photon source (in particular with a
flattening $\approx(\gamma_{\sigma}\tau/2)^{2}$ at small delays $\tau$ around
$g^{(2)}(0)=0$, characteristic of superior single-photon emission [31]). That
is to say, one apparently has a bright very narrow—Eq. (8)—and antibunched—Eq.
(10)—source. This is thanks to various ingredients contributing their
respective benefits: the laser brings the narrow linewidth while the two-level
system brings the single-photon emission. But in this listing of great
properties, which have been experimentally demonstrated [32, 33], one is using
different and in fact incompatible configurations, namely, $\Gamma\to 0$ for
the spectrum and $\Gamma\to\infty$ for the antibunching. In the laboratory,
separate characterizations, overlooking the impact of $\Gamma$, have been
performed of the system, thereby omitting a crucial adversative conjunction in
describing an “ultra-coherent _or_ single photon source” and “subnatural
linewidth _or_ single photons”. This oversight shows that $\Gamma$ is not a
mere technical concern but a central consideration. If working directly with
Eqs. (7) and (9) instead of the textbook limits, then the intrinsic
limitations imposed by detection would be obvious. The difficulty is that one
needs a large enough bandwidth $\Gamma$ so as to have sufficient integration
time to resolve $g^{(2)}(\tau)$ while also having a small enough bandwidth to
have a good frequency resolution and not suffer too much broadening. When
using the wrong limit for each observable, one indeed gets the worst result
possible, namely, $S_{\infty}(\omega)$ vanishes as the spectrum becomes
constant over $(-\infty,+\infty)$ since photons are emitted at all possible
energies, while $g^{(2)}_{0}(\tau)=1$ and all correlations are lost. These are
in fact those of an ideal laser; a physical laser has a linewidth and
filtering below that linewidth would go to the thermal limit
$g^{(2)}_{0}(\tau)=2!$ [34], at any rate, the single-photon correlations are
completely lost. Presented in these terms, it would seem that the
time–frequency incompatibility occurs at the Fourier level and is thus
unavoidable. But this could be easily circumvented merely by replacing the
source with one that is more spectrally narrow, i.e., by reducing
$\gamma_{\sigma}$, keeping the same antibunching with a broader-than-natural
linewidth of the substitute emitter, but still narrower than the original one:
in this way, one can realize arbitrarily narrow and antibunched source,
showing that there is no fundamental limitation, only a technical one of
finding the suitable emitter. In fact, even the sub-natural linewidth can be
realized _simultaneously_ with antibunching, provided a small, but crucial,
variation of the experiments [35]. One first need to understand the nature of
antibunching in this system, which, as we said, comes from the two-level
system. Or does it? Because driving is weak in this regime, the saturation of
the two-level system is not actually needed, in contrast to incoherent
excitation where truncation of the number of excitations is crucial
(maintaining the average population below $1/2$). Here, a weak nonlinearity
would work as well since the antibunching arises in this case from a two-
photon approximation of squeezing antibunching [36] (this is an approximation
because there is also three and higher-order antibunching which a squeezed
state cannot provide). Squeezing and antibunching have a long shared history
of joint appearances, initially with little appreciation of their
interconnectedness [37, 38, 17]. This is in this way that they have been
independently theorized [39, 40], sought [41, 42] and ultimately discovered
[43, 44] in resonance fluorescence. In such a system, they turn out to be two
faces of the same coin, namely, of wave interferences [45]. Interferences are
at the heart of both optics and quantum mechanics, so their importance in
quantum optics can only be understood as momentous [46, 47]. While in
classical optics, the simplest waves are described by two parameters—their
amplitude and phase—quantum fields come with an infinite number of amplitudes
for non-Gaussian states of light, as each multiphoton component
$\left|n\right\rangle$ of the field
$\left|\psi\right\rangle=\sum_{n}c_{n}\left|n\right\rangle$ comes with its own
independent coefficient $c_{n}$. One can produce rich multiphoton correlated
outputs from admixing (or interfering) even the simplest and most popular
Gaussian states [48], namely, the coherent state
$\left|\alpha\right\rangle=\mathscr{D}(\alpha)\left|0\right\rangle$, defined
in terms of the displacement operator
$\mathscr{D}(\alpha)\equiv\exp\big{(}\alpha{a^{\dagger}}-\alpha^{*}a\big{)}$
where $\alpha=|\alpha|e^{i\phi}$ and $a$ the harmonic oscillator annihilation
operator, so that $\left(\mathscr{D}\right)^{\dagger}a\mathscr{D}=a+\alpha$,
and the single-mode (quadrature) squeezed state, defined in term of the
squeezing operator [49, 50]
$\mathcal{S}_{1}\equiv\exp[\frac{1}{2}(\xi^{*}a^{2}-\xi{a^{\dagger}}^{2})]$.
The idea is as simple as for classical optics interferences: by fine-tuning
their respective coefficients, one can realize destructive or constructive
interferences for a given component $\left|n\right\rangle$ and thus suppress
or maximize it in the output. At the Gaussian level, this comes with strong
constrains since few parameters define all the multiphoton weights, but
because such states are easily produced, they have elicited most of the
interest. The idea of admixing squeezing with a coherent state appeared very
early, precisely with the aim to produce antibunching [51] (under the name of
“anticorrelation effect”), and in fact even before the more straightforward
idea of a two-level system being restored into its ground state [52]. It was
then believed that such squeezed anticorrelations might exist in the transient
dynamics only. While antibunching from the two-level system was quickly
observed [43], the one based on squeezing took more time (and with pulsed
excitation) with the signal and pump of a degenerate parametric amplifier,
producing both bunching and antibunching by varying the relative phase [53].
The effect was more clearly understood in terms of two-photon interferences by
Ou _et al._ who also realized it in the stationary regime [54]. Now that we
understand better the origin of antibunching in resonance fluorescence [55]
(as squeezing antibunching), we can return to our hope of realizing it
simultaneously with a subnatural linewidth. The loss of antibunching [56] in
resonance fluorescence is due to the interference between the incoherent and
coherent components being disrupted by the detection, which filters out the
spectrally broad incoherent (squeezed) fraction more than it does the coherent
(narrow) one. One could thus restore antibunching by correcting for this
imbalance, which is an excess of coherent state, therefore, destructive
interferences with an external laser can achieve that, as was indeed shown
theoretically in Ref. [35]. Such techniques that go by the name of
“homodyning” have been demonstrated experimentally to extract the quantum part
of a signal [57]. Theoretically, this simply consists in adding a complex
field $\alpha$ to the homodyned operator, so $\sigma$ in our case, i.e., to
substitute $\sigma\to\alpha+\sigma$ for a tunable $\alpha\in\mathbb{C}$ in the
Hamiltonian $H_{\sigma}$. We will go into the details of this mechanism in
Section 4 where we will generalize it to all frequencies of the system, while
our current discussion is merely its particular case for photons with the same
frequency. For now, it will be enough to note that evidences of such
interferences and their effect on the correlations were reported in
independent and complementary works that tamper with either the incoherent
fraction (and thus loosing antibunching) [58, 55] or on the opposite with the
coherent fraction (producing excess bunching instead) [59]. Restoring
antibunching remains to be demonstrated experimentally. At the theoretical
level, our discussion so far should have established that the detection, which
manifests itself through the parameter $\Gamma$, is at the center of quantum-
optical characterizations and should not be treated lightheartedly. A deeper
concept is concealed in Eq. (9), namely, that what is being measured is truly
the correlations of photons with frequency $\omega_{\mathrm{L}}=0$, i.e., one
is making a _complete_ characterization of both the time _and_ frequency of
the photons, which is why $\Gamma$ is mandatory. What the textbook limit
$\Gamma\to\infty$ really does is to neglect the frequency information. This is
not entirely apparent from Eq. (9) because it seems that no other frequency
than the one at which the system emits would make sense. This is not the case.
Such a complete description for more general multiphoton observables is
discussed in the next Section, which embarks us on another departure with the
bulk of the literature, this time not for a computational technique only, but
at the conceptual level of what it means to detect multiphotons.

##  3 Frequency-resolved photon correlations

When considering the emission from a quantum emitter, and performing, say,
two-photon correlations from its output, there is an irresistible temptation
to correlate photons that are “visible” in the emission spectrum, i.e., that
originate from spectral peaks. If the emission further comes in the form of
several peaks, that invites for cross-correlating them. This is due to the
persistence of the classical picture even to the quantum opticians, despite
now many decades of quantum theory telling us that quantum states at the
multi-particle level are not conditioned by their attributes at the one-
particle level [60]. This makes it conceivable that multi-photon correlations
can be more pronounced or interesting in spectral regions where the intensity
(or population, i.e., a one-photon observable of the type
$\langle{a^{\dagger}}a\rangle$) is itself small or even negligible. One must,
at the quantum level, separate in principle quantity and quality. It is in
fact, beyond conceivable, compulsory to elevate one’s understanding of
multiphoton emission to such situations where the system does not emit at the
one-photon level, but does at the two-photon one, and vice-versa. There can
also be joint intense emission of the two type, or jointly suppressed. All
combinations are possible. They are therefore conceptually disconnected. To
see this, one needs a quantity to visualize two-photon physics, which we now
introduce. Such a quantity should have no such prejudice for the spectral
peaks, and treat every pair of photons the same. It should be represented on a
2D plot as it requires two axes, one for each photon. It should also extend or
generalize already existing quantities as it is unlikely that the essence of
two-photon physics has been missed entirely, although it might well have been
left incomplete. Such a two-photon quantity we call the _two-photon spectrum_
[61, 62] or two-photon correlation spectrum, which we define in terms of the
intensity or population operator $\hat{n}(\omega,t)$ that quantifies the
amount of quantum radiation at frequency $\omega$ at time $t$ [15], leading
to:

$g^{(2)}_{\Gamma}(\omega_{1},\omega_{2},t_{1},t_{2})={\langle{:}\hat{n}(\omega_{1},t_{1})\hat{n}(\omega_{2},t_{2}){:}\rangle\over\langle\hat{n}(\omega_{1},t_{1})\rangle\langle\hat{n}(\omega_{2},t_{2})\rangle}\,.$
(11)

This is a clear generalization of Glauber’s two-photon correlation function,
but retaining the frequency information along with the temporal one. For this
reason, the detector’s bandwidth $\Gamma$ is mandatory. This quantity has been
considered by various Authors [63, 64, 65] but it has presented considerable
difficulties and only partial results could be derived, sometimes with
inconsistencies. From the sensor formalism, however, it is easily and exactly
obtained as

$g^{(2)}_{\Gamma}\left(\omega_{1},\omega_{2},t_{1},t_{2}\right)=\frac{\big{\langle}{{\varsigma_{1}^{\dagger}}(t_{1})({\varsigma_{2}^{\dagger}}\varsigma_{2})(t_{2})\varsigma_{1}(t_{1})}\big{\rangle}}{\big{\langle}{({\varsigma_{1}^{\dagger}}\varsigma_{1})(t_{1})\big{\rangle}}{\big{\langle}({\varsigma_{2}^{\dagger}}\varsigma_{2})(t_{2})}\big{\rangle}}\,.$
(12)

While one can consider both time and frequency (see, e.g., Refs. [66, 67]), it
will be enough for the present text to consider coincidences of a stationary
state, i.e., the case $t_{1}=t_{2}\to\infty$. This will not diminish the main
insight that in so doing, we are making a joint characterization in time
(albeit coincidences only) and frequencies. We now embrace the full picture
and compute numerically two-frequency coincidences (12) through standard
quantum averages of the sensor method. Since frequencies are mere parameters,
this allows a convenient and exact computation of what was previously obtained
at the cost of great efforts and several oversimplifying approximations [68,
69]. The results are shown in Fig. 2’s panels (d–f) [2nd row], below the
corresponding spectral shapes, reproducing three notable cases from the
pioneering work [62]. Unlike the top row, where $S_{\Gamma}(\omega)$ could be
obtained in the limit $\Gamma\to 0$, these two-photon spectra would become
“trivial” in this limit, with $g^{(2)}_{0}(\omega_{1},\omega_{2})=1$, i.e.,
washing out all traces of correlations as the detectors integrate over
infinite times. The other limit is more interesting but brings nothing new as
it recovers Eq. (10), and so, at $\tau=0$, is identically zero. What is of
interest, clearly, is the intermediate case where a landscape of correlations
is revealed. It will be enough to limit ourselves with a description of the
qualitative features, although we repeat that the results are numerically
exact, so one could study cross sections in more quantitative details. In our
qualitative description, red colors correspond to bunching, i.e., to photons
with the respective frequencies arriving together, with antidiagonals of
bunching in cases (a) and (b) that correspond to the two-photon Mollow
triplet. These lines indeed generalize Bohr’s quantization condition for
transitions between two energies $E_{i}-E_{j}=\omega$, to a two-photon jump

$E_{k}-E_{l}=\omega_{1}+\omega_{2}$ (13)

where the energies $E_{k}$ and $E_{l}$ are non-contiguous in the energy ladder
and so the system effectively “jumps over” a real state, for which reason this
has been termed a _leapfrog process_ [61, 62]. This is the direct quantum-
optical counterpart of Göpert Mayer’s Doppelemission with a few but important
variations. In her case, there was nothing to jump over, and the multiphoton
emission was thus in a different frequency range, indeed, in its manifestation
from nebulae emission, this converts UV $2S\to 1S$ Lyman photons into the
visual continuum siding with the visible Balmer series $X\to 2S$. Higher-order
processes fall into still more remote spectral windows, while our leapfrog
processes, even from three or higher-number of photons, are all over the place
along with the single photons that one sees in the spectral shape. The second
variation is our reliance on the quantum-optical quantity
$g^{(2)}_{\Gamma}(\omega_{1},\omega_{2})$ (this would be $g^{(n)}$ for higher
photon numbers), which allows to “reveal” this hitherto unsuspected structure,
whose existence is, again, not conditioned to the amount of signal, and can be
captured by using an adequate two-photon observable, which, as Glauber
clarified for quantum light, is the intensity-intensity correlator [60]. As a
result, one should look for such multiphoton emission _everywhere_ , not only
when it is betrayed, for one reason or another, at the one-photon level. In
most cases, it remains invisible (at the one photon level). The appropriate
photon correlations allow us to extract them from a background of unrelated or
more intense emission. Another variation is that, two-photon light being more
complex than single-photon one, it comes with more characteristics, that we
can capture with colors as opposed to a magnitude. We use white for
uncorrelated emission where the two photons are emitted independently the one
from the other. Interestingly, one can see in Fig. 2 that such uncorrelated
photons form a grid of horizontal and vertical lines defined by the (one
photon) spectral peaks. This means that photons from the spectrum contribute
not only the bulk, indeed, of the emission, but more significantly, the
classical signal which is independent from the rest of the emission. This is
another call for quantum opticians to seek their photons away from the peaks:
quantum photons are emitted where one does not seem them. This is because
photons from the spectral peaks are “real” photons that arise from the Bohrian
jump from one real (here, dressed) state to another, while leapfrog photons
are virtual, in the sense that they involve a virtual intermediate state with
energy $E_{l}^{*}$ which could be any value in between, and thus resulting in
a correlated signal for the entire line satisfying Eq. (13). Virtual does not
mean “non-existing” and would one still complain of no emission outside from
the peaks, we would retort again that this is a one-photon concern. The no-
emission at the two-photon level is unrelated to no-emission at the one-photon
one. No two-photon emission is in fact what we encode with the blue color in
the two-photon spectrum. And one can see in Fig. 2(d) how the strongest no
two-photon emission occurs at $(\varpi_{1},\varpi_{2})=(-1,-1)$ or $(+1,+1)$,
i.e., at the spectral side peaks, where one-photon emission is indeed strong.
Such a scenario is well known, as it corresponds to single-photon emission. In
contrast, at $(\varpi_{1},\varpi_{2})={1\over 2}(-1,-1)$ or ${1\over
2}(+1,+1)$, one has the opposite situation of strong two-photon emission but
small one-photon emission (in fact the smallest emission in the range
$|\varpi|\lessapprox 1.25$), which corresponds to the equally fundamental case
of two-photon leaprog emission, which may be even more important than single-
photon emission, although, because it is innocuous at the one-photon level, it
failed to attract much attention so far. One must highlight in this regard the
exceptional contributions from the beautiful and still unique experiments of
the Muller group [70, 71, 72, 73], who has observed these features in
spectacular agreement with the theory. Finally, a manifestation of the
independent yet tangible existence of the two-photon physics can be based on
arguments of theoretical aesthetics: the robust spectral (one-photon)
symmetry, which appears to be lifted at the two-photon level since the two
peaks behave differently in the two-photon spectrum, is instead “rotated”: the
symmetry is with respect to the two-photon diagonal, so independent from the
one-photon structure, but also present when looked at properly. This needed
change of perspective, we believe, is an evidence of the independent two-
photon picture. If such arguments fail to move one’s sensibility, and although
we are focused on the fundamental aspects in this text, we should then mention
that there are obvious and immediate technological prospects of these results
that turn these correlated virtual photons into quantum emission of a new
type. For instance, placing a cavity at the $N$-photon leapfrog degenerate
frequency (halfway between the peaks for two photons) would Purcell-enhance
their emission and open a bright channel of pure $N$-photon emission [74].
This is clearly another evidence of the “existence” of such processes, that
can power devices of a new type. There are other ways to similarly exploit
this hidden physics, but we wish to return instead to its basic structure, and
now focus on the perplexing blue circle cast between the central and virtual
peaks, as seen in Fig. 2(f). This circle is of a different nature from the
phenomenology that we have discussed so far, since this is a curve as opposed
to straight lines, and suppressing photons instead of having them come
together as joint emission. We now explain here for the first time the
underlying mechanism for this circle.

##  4 Admixing two-mode squeezed and coherent states

Two-photon physics is primarily described by Fock states
$\left|2\right\rangle$ in a given mode, or by non-degenerate pairs
$\left|1_{1}1_{2}\right\rangle$ in two modes. Next comes the so-called
_squeezed states_ , which was popularized by Walls [75] for their ability to
squeeze through the uncertainty principle [76], but were initially (and
possibly, more appropriately) called “two-photon coherent states” by Yuen
[77]. Because two-photon spectra correlate two frequencies (providing two
continuously varying modes), two-mode squeezing can be expected to have some
relevance in presence of coherence, as is the case for resonance fluorescence.
Two-mode squeezing considers two bosonic modes with annihilation operators
$a_{1}$ and $a_{2}$, and a so-called “squeezing matrix” $\zeta_{ij}$ (complex-
valued symmetric). The two-mode squeezing operator [78]

$\mathcal{S}_{2}^{(ij)}=\exp[\zeta_{ij}a_{i}a_{j}-\zeta_{ij}^{*}{a_{i}^{\dagger}}{a_{j}^{\dagger}}],$
(14)

can be seen as a generalization of the already introduced single-mode
squeezing operator

$\mathcal{S}_{1}^{(i)}=\exp[\frac{1}{2}(\xi_{i}^{*}a_{i}^{2}-\xi_{i}{a_{i}^{\dagger}}^{2})]\,,$
(15)

where the squeezing parameters are conveniently defined as
$\xi_{i}=r_{i}e^{i\theta_{i}}$ for $i=1,2$ and
$\zeta_{ij}=t_{ij}e^{i\vartheta_{ij}}$ a matrix with zero diagonal elements,
namely $\zeta_{12}=t_{12}e^{i\vartheta_{12}}$ with $\zeta_{jj}=0$ for $j=1,2$.
One can squeeze the two modes independently, with the product of the single-
mode squeezing operators
$\mathcal{S}_{1}^{(1)}(\xi_{1})\mathcal{S}_{1}^{(2)}(\xi_{2})=\mathcal{S}_{1}^{(2)}(\xi_{2})\mathcal{S}_{1}^{(1)}(\xi_{1})$
(the operators commute), or squeeze them jointly with the two-mode squeezing
operator $\mathcal{S}_{2}^{(12)}(\zeta_{12})$, transforming annihilation
operators as [78]:

$(\mathcal{S}_{2}^{(12)})^{\dagger}a_{i}\mathcal{S}_{2}^{(12)}=\sum_{k=1,2}\big{(}\mathcal{M}_{ik}a_{k}-\mathcal{N}_{ik}{a_{k}^{\dagger}}\big{)}\,,$
(16)

for $1\leq i\leq 2$. The mixing matrix $\mathcal{M}$ is positive while the
second mixing matrix $\mathcal{N}$ can be complex-valued. For single-mode
squeezing, they are diagonal with values $\mathcal{M}_{ii}=\cosh(r_{i})$ and
$\mathcal{N}_{ii}=e^{i\theta_{i}}\sinh(r_{i})$, respectively. On the other
hand, for two-mode squeezing, the non-vanishing elements are
$\mathcal{M}_{11}=\mathcal{M}_{22}=\cosh(t_{12})$ and
$\mathcal{N}_{12}=\mathcal{N}_{21}=e^{i\vartheta_{12}}\sinh(t_{12})$. This
leads to the transformation rules for the one- and two-mode operators:

$(\mathcal{S}_{1}^{(i)})^{\dagger}a_{i}\mathcal{S}_{1}^{(i)}=\
\mu_{i}a_{i}-\nu_{i}{a_{i}^{\dagger}}\quad\text{and}\quad(\mathcal{S}_{2}^{(12)})^{\dagger}a_{i}\mathcal{S}_{2}^{(12)}=\
\mathcal{M}_{11}a_{i}-\mathcal{N}_{12}{a_{\bar{\imath}}^{\dagger}}\,,$ (17)

where $\mu_{i}=\cosh(r_{i})$, $\nu_{i}=e^{i\theta_{i}}\sinh(r_{i})$,
$\mathcal{M}_{11}=\cosh(t_{12})$,
$\mathcal{N}_{12}=e^{i\vartheta_{12}}\sinh(t_{12})$ and $\bar{\imath}\equiv
3-i$ exchanges $1$ and $2$. Alternatively, one could use the general form of
$\mathcal{S}_{2}^{(12)}$, with structurally identical results. The combination
of these squeezing leads to a generic two-mode squeezed state:

$\left|\xi_{1},\xi_{2},\zeta_{12}\right\rangle\equiv\mathcal{S}_{1}^{(1)}\mathcal{S}_{1}^{(2)}\mathcal{S}_{2}^{(12)}\left|0\,0\right\rangle.$
(18)

For this state, it is straightforward to compute any correlator
$\langle{a_{1}^{\dagger}}^{m}a_{1}^{n}{a_{2}^{\dagger}}^{p}a_{2}^{q}\rangle$
for integers $m,n,p,q$ from Eqs. (16). For instance, we find for the
population of mode $i=1,2$:

$\langle{a_{i}^{\dagger}}a_{i}\rangle=\cosh^{2}r_{i}\sinh^{2}t+\sinh^{2}r_{i}\cosh^{2}t\,.$
(19)

One could carry-on like this and compute other correlators but since we shall
be concerned in the following with leading-order processes, we can rescale the
squeezing parameters as $r_{i}\rightarrow\epsilon^{2}\,r_{i}$ and
$t\rightarrow\epsilon^{2}\,t$ for an $\epsilon$ that will be taken close to
zero. In this limit, thanks to normalization, we can give a more comprehensive
list of the correlators, including the second-order correlation functions
$g^{(2)}_{i}\equiv{\langle{a_{i}^{\dagger}}^{2}a_{i}^{2}\rangle}/{\langle{a_{i}^{\dagger}}a_{i}\rangle^{2}}$
for $i=1,2$ and
$g^{(2)}_{12}\equiv{\langle{a_{1}^{\dagger}}{a_{2}^{\dagger}}a_{2}a_{1}\rangle}/{\langle{a_{1}^{\dagger}}a_{1}\rangle\langle{a_{2}^{\dagger}}a_{2}\rangle}$:

$\displaystyle\langle{a_{i}^{\dagger}}a_{i}\rangle\approx\left(r_{i}^{2}+t_{12}^{2}\right)\epsilon^{4}\,,\qquad\langle
a_{1}a_{2}\rangle\approx-\epsilon^{2}\zeta_{12}\,,\qquad\langle
a_{i}^{2}\rangle\approx-\epsilon^{2}\xi_{i}\,,$ (20a) $\displaystyle
g^{(2)}_{i}\approx\frac{r_{1}^{4}}{(r_{1}^{2}+t_{12}^{2})(r_{2}^{2}+t_{12}^{2})\epsilon^{4}}\,,\qquad
g^{(2)}_{12}\approx\frac{t_{12}^{4}}{(r_{1}^{2}+t_{12}^{2})(r_{2}^{2}+t_{12}^{2})\epsilon^{4}}\,.$
(20b)

Since $\epsilon$ is very small, one can see that all the two-particle
fluctuations are bunched. We can, however, produce antibunching by interfering
squeezing with a coherent state, as we already discussed for the Heitler
regime at resonance. Here, we will look more closely at the general case of
interferences that involve two-mode squeezing, and show that this captures
much of the two-photon physics of resonance fluorescence. This extends to two
modes the idea already scrutinized for one mode [35, 16], that admixture of
coherence with squeezing results in rich and qualitatively different
correlations than those available in the squeezed or coherent states alone.
This should similarly allow to generalize the specific technique of tuning
photon statistics with coherent fields [48] to multimode correlations,
especially as multimode coherent states are not correlated, so they can be
tuned independently for each squeezed modes. From the weak driving that
defines resonance fluorescence, it is enough to deal with coherent-squeezed
states in the limit of small squeezing. Since the coherent contribution of the
quantum states is taken of the same order than squeezing, we take
$\alpha_{i}\rightarrow\epsilon\,\alpha_{i}$, meaning, however, that coherence
is stronger since it appears to first order in the admixture while squeezing
is of second order. This brings us to the fundamental object of two-photon
resonance fluorescence:

$\left|\bm{\alpha},\bm{\xi},\zeta_{12}\right\rangle=\mathscr{D}_{1}\mathscr{D}_{2}\,\mathcal{S}_{1}^{(1)}\mathcal{S}_{1}^{(2)}\mathcal{S}_{2}^{(12)}\left|0\,0\right\rangle\,,$
(21)

where $\bm{\alpha}\equiv(\alpha_{1},\alpha_{2})$ and
$\bm{\xi}\equiv(\xi_{1},\xi_{2})$ admixes coherence and squeezing of single
modes [48] along with $\zeta_{12}$ that provides the general two-photon
physics. Similarly as before, we can now compute the key two-photon
correlators for this state, to leading order in $\epsilon$:

$\displaystyle\phantom{a}\hskip-56.9055pt\langle{a_{i}^{\dagger}}a_{i}\rangle\approx$
$\displaystyle\
\epsilon^{2}|\alpha_{i}|^{2}+\epsilon^{4}\big{\\{}t_{12}^{2}+r_{i}^{2}-2|\alpha_{i}|^{2}r_{i}\cos(2\phi_{i}-\theta_{i})-\
2|\alpha_{1}||\alpha_{2}|t_{12}\cos(\phi_{1}+\phi_{2}-\vartheta_{12})\big{\\}}\,,$
(22a) $\displaystyle\langle a_{i}^{2}\rangle\approx$ $\displaystyle\
\epsilon^{2}\left(\alpha_{i}^{2}-\xi_{i}\right)\,,\quad\langle
a_{1}a_{2}\rangle\approx\
\epsilon^{2}\left(\alpha_{1}\alpha_{2}-\zeta_{12}\right)\,,$ (22b)
$\displaystyle g^{(2)}_{i}\approx$ $\displaystyle
1-\frac{2\,r_{i}\cos(2\phi_{i}-\theta_{i})}{|\alpha_{i}|^{2}}+\frac{r_{i}^{2}}{|\alpha_{i}|^{4}}\,,$
(22c) $\displaystyle g^{(2)}_{12}\approx$ $\displaystyle\
1-\frac{2\,t\cos(\phi_{1}+\phi_{2}-\vartheta_{12})}{|\alpha_{1}||\alpha_{2}|}+\frac{t_{12}^{2}}{|\alpha_{1}|^{2}|\alpha_{2}|^{2}}\,,$
(22d)

for $1\leq i,j\leq 2$. As compared to Eqs. (20), one can see that the
correlators $g^{(2)}$ can now take a much wider span of possible values, in
particular they can now be less than unity and even vanish exactly as well as
diverge to leading order. This is the two-mode generalization of our previous
single-mode admixing [48], where it was also the case that the population, Eq.
(22a), is essentially coherent, since $\epsilon^{2}\gg\epsilon^{4}$, but that,
regardless of this preponderance of the coherent state, two-photon observables
are ruled by the squeezing component. Here too, the intensity of the squeezed
photons has to be much smaller than the coherent one. The reason is that in
order to interfere at the two-photon level (to produce antibunching) the
contributions have to be of the same order, what we have achieved thanks to
the parametrization. The coherent field makes the system gain one photon, with
a probability proportional to $|\alpha_{i}|$. To climb up from vacuum to
$\left|2\right\rangle$, the system has to absorb two photons, one each time,
so the chances are proportional to $|\alpha_{i}|^{2}$. On the other hand, the
squeezed field carries the photons two by two. Then, the system can directly
jump from $\left|0\right\rangle$ to $\left|2\right\rangle$ by absorbing two
photons with a probability proportional to $r_{i}$. To ensure that both
processes compete on equal footing, the ratio $|\alpha_{i}|^{2}/r_{i}$ has to
remain finite when the limits $|\alpha_{i}|,r_{i}\rightarrow 0$ are taken. For
the two-mode case, the interference yields the cancellation of the
$\left|1,1\right\rangle$ state and the argument is exactly the same just
exchanging $|\alpha_{i}|^{2}\rightarrow|\alpha_{1}||\alpha_{2}|$ and
$r_{i}\rightarrow t_{12}$. This explains the counter-intuitive fact that a
weak incoherent fraction determines the antibunching of an intense coherent
radiation [25, 55]. A major departure from two-mode squeezing as compared to
single-mode admixing, is the degeneracy for the conditions that realize
interferences producing extreme correlations, such as perfect antibunching
(exact zero) or diverging superbunching. Indeed, perfect anticorrelation for a
single mode is obtained when these conditions are fulfilled:

$\displaystyle r_{i}=|\alpha_{i}|^{2}\,,$ $\displaystyle
2\phi_{i}-\theta_{i}=0\,,$ (23)

while two-mode anticorrelations are maximized when:

$\displaystyle t_{12}=|\alpha_{1}\alpha_{2}|\,,$
$\displaystyle\phi_{1}+\phi_{2}-\vartheta_{12}=0\,.$ (24)

Just like two-photon leapfrog transitions (cf. Sec. 3) lift the strict
conservation of energy for each photon to constrain their sum instead—allowing
each photon in the pair to come with a continuous energy [62, 67]—here, the
wave-interference is realized for a sum of the phases, in contrast to the
single phase for a single mode [16]. This allows, again, for a continuous
range of parameters to realize the quantum interference. Such admixtures can
be tracked in any dynamical two-mode quantum state, possibly mixed with a
density matrix $\rho$. We will consider, explicitly, the case of resonance
fluorescence, which is the simplest one, but of course our discussion is
general, whether the coherence is provided internally (as is the case in
resonance fluorescence where it is inherited from the coherent driving) or is
explicitly added externally, with a supplementary (homodyning) laser, a case
we shall also touch upon. It could also be developed by the system itself
(e.g., when it undergoes lasing). Whatever the origin for the various
contributions, one can separate them from the whole operators $a_{i}$ that act
on the whole state from those $\tilde{a}_{i}$ that act only on the
fluctuations, or incoherent part. They are linked by the relation

$a_{i}=\alpha_{i}+\tilde{a}_{i}$ (25)

where, again, $a_{i}$ and $\tilde{a}_{i}$ are operators while $\alpha_{i}$ is
a $c$-number. This again generalizes our earlier case [48] and while we limit
ourselves here to $i=1,2$, one could further extend it to any number of modes.
By construction, one has $\langle\tilde{a}_{i}\rangle=0$, resulting in no
interferences for the intensities:
$\langle{a_{i}^{\dagger}}a_{i}\rangle=|\alpha_{i}|^{2}+\langle{\tilde{a}_{i}^{\dagger}}\tilde{a}_{i}\rangle$.
In stark contrast, the cross-correlator
$\langle{a_{1}^{\dagger}}{a_{2}^{\dagger}}a_{2}a_{1}\rangle$ becomes
$\langle({\tilde{a}_{1}^{\dagger}}+\alpha_{1}^{*})({\tilde{a}_{2}^{\dagger}}+\alpha_{2}^{*})(a_{2}+\alpha_{2})(a_{1}+\alpha_{1})\rangle$.
Expanding this, we get:

$\langle{a_{1}^{\dagger}}{a_{2}^{\dagger}}a_{2}a_{1}\rangle=\langle{\tilde{a}_{1}^{\dagger}}{\tilde{a}_{2}^{\dagger}}\tilde{a}_{2}\tilde{a}_{1}\rangle+\sum_{i=1,2}\big{(}\alpha_{i}\langle{\tilde{a}_{1}^{\dagger}}{\tilde{a}_{2}^{\dagger}}\tilde{a}_{\bar{\imath}}\rangle+\mathrm{cc}\big{)}+{}\\\
+\big{(}\alpha_{1}\alpha_{2}\langle{\tilde{a}_{1}^{\dagger}}{\tilde{a}_{2}^{\dagger}}\rangle+\alpha_{1}\alpha_{2}^{*}\langle{\tilde{a}_{1}^{\dagger}}{\tilde{a}_{2}}\rangle+\mathrm{cc}\big{)}+\sum_{i=1,2}|\alpha_{i}|^{2}\langle{\tilde{a}_{\bar{\imath}}^{\dagger}}\tilde{a}_{\bar{\imath}}\rangle$
(26)

where $\bar{\imath}\equiv 3-i$, i.e., toggles the index in $\\{1,2\\}$. This
shows that, unlike intensities, the second-order correlation functions are the
result of interferences, namely, of four terms:

$g^{(2)}_{12}=1+\mathcal{I}_{0}+\mathcal{I}_{1}+\mathcal{I}_{2}\,,$ (27)

with

$\displaystyle\mathcal{I}_{0}$ $\displaystyle\equiv{1\over
n_{1}n_{2}}\left\\{\langle{\tilde{a}_{1}^{\dagger}}{\tilde{a}_{2}^{\dagger}}\tilde{a}_{2}\tilde{a}_{1}\rangle-\langle{\tilde{a}_{1}^{\dagger}}\tilde{a}_{1}\rangle\langle{\tilde{a}_{2}^{\dagger}}\tilde{a}_{2}\rangle\right\\}\,,$
(28a) $\displaystyle\mathcal{I}_{1}$ $\displaystyle\equiv{2\over
n_{1}n_{2}}\operatorname{Re}\left\\{{\alpha_{1}\langle{\tilde{a}_{1}^{\dagger}}{\tilde{a}_{2}^{\dagger}}\tilde{a}_{2}\rangle+\alpha_{2}\langle{\tilde{a}_{1}^{\dagger}}{\tilde{a}_{2}^{\dagger}}\tilde{a}_{1}\rangle}\right\\}\,,$
(28b) $\displaystyle\mathcal{I}_{2}$ $\displaystyle\equiv{2\over
n_{1}n_{2}}\operatorname{Re}\left\\{\alpha_{1}\alpha_{2}\langle{\tilde{a}_{1}^{\dagger}}{\tilde{a}_{2}^{\dagger}}\rangle+\alpha_{1}\alpha_{2}^{*}\langle{\tilde{a}_{1}^{\dagger}}\tilde{a}_{2}\rangle\right\\}\,,$
(28c)

with $n_{i}\equiv\langle{a_{i}^{\dagger}}a_{i}\rangle$ so that in each case,
$\mathcal{I}_{i}n_{1}n_{2}$ is of order $|\alpha_{1}|^{p}|\alpha_{2}|^{q}$
with $p+q=i$. We now have recovered the general version of the phenomenon
discussed in Section 2 of antibunching produced by an interference of various
contributions, which are the single-mode $\mathcal{I}_{i}$, i.e., with
$a_{1}=a_{2}=a$, in which case $\mathcal{I}_{0}$ is related to the $g^{(2)}$
of the incoherent fraction, being $-1$ for non-Gaussian (quantum) antibunching
and $+1$ for chaotic, bunched states, while $\mathcal{I}_{2}$ is related to
squeezing, being $-2$ in presence of squeezing. $\mathcal{I}_{1}$ is a measure
of anomalous moments which is negligible in the two regimes of interest,
namely

  * •

Heitler regime, where
$g^{(2)}=0=1+(\mathcal{I}_{0}=1)+(\mathcal{I}_{1}=0)+(\mathcal{I}_{2}=-2)$,

  * •

Mollow regime, where
$g^{(2)}=0=1+(\mathcal{I}_{0}=-1)+(\mathcal{I}_{1}=0)+(\mathcal{I}_{2}=0)$,

with, therefore, a completely different interpretation for how $g^{(2)}=0$ is
obtained. In the general case that we consider here, the numerator of
$\mathcal{I}_{1}$ has the physical meaning of a covariance between the
incoherent fractions, being positive when both modes are occupied or depleted
together, and negative when one is largely occupied and the other depleted. As
for the single-mode, the normalization is to the _total_ population, not to
that of the incoherent fraction alone. $\mathcal{I}_{2}$ also has some
interpretation in terms of two-mode squeezing, in particular including the
coherent fractions $\alpha_{1}$ and $\alpha_{2}$. We shall similarly disregard
the exact meaning of $\mathcal{I}_{1}$ that interpolates between the two cases
by involving anomalous correlators of the population and order parameter of
the type $\langle a_{i}\hat{n}_{\bar{\imath}}\rangle$, also weighted by the
coherent fraction $\alpha_{i}^{*}$. We can also write these expressions not
for the quantum fields only (fluctuations), but for the full-state
correlators, which might be more accessible either experimentally or
theoretically. To do so, we can use the backward relation:

$\langle\tilde{a}_{1}^{\dagger i}\tilde{a}_{2}^{\dagger
j}\tilde{a}_{2}^{k}\tilde{a}_{1}^{l}\rangle=\sum_{i^{\prime},j^{\prime},k^{\prime},l^{\prime}=0}^{i,j,k,l}(-1)^{i^{\prime}+j^{\prime}+k^{\prime}+l^{\prime}}\binom{i}{i^{\prime}}\binom{j}{j^{\prime}}\binom{k}{k^{\prime}}\binom{l}{l^{\prime}}(\alpha_{1}^{*})^{i-i^{\prime}}(\alpha_{2}^{*})^{j-j^{\prime}}\alpha_{2}^{k^{\prime}}\alpha_{1}^{l^{\prime}}\times\\\
\langle
a_{1}^{\dagger(i-i^{\prime})}a_{2}^{\dagger(j-j^{\prime})}a_{1}^{k-k^{\prime}}a_{2}^{l-l^{\prime}}\rangle,$
(29)

which applied to $\mathcal{I}_{i}$ leads to:

$\displaystyle\mathcal{I}_{0}=\,$
$\displaystyle\big{(}\langle{a_{1}^{\dagger}}{a_{2}^{\dagger}}a_{2}a_{1}\rangle-\langle{a_{1}^{\dagger}}a_{1}\rangle\langle{a_{2}^{\dagger}}a_{2}\rangle-4\,|\alpha_{1}|^{2}|\alpha_{2}|^{2}+2\,|\alpha_{1}|^{2}\langle{a_{2}^{\dagger}}a_{2}\rangle+$
$\displaystyle
2\,|\alpha_{2}|^{2}\langle{a_{1}^{\dagger}}a_{1}\rangle\,+2\,\operatorname{Re}\\{\alpha_{1}\alpha_{2}\langle{a_{1}^{\dagger}}{a_{2}^{\dagger}}\rangle+\alpha_{1}\alpha_{2}^{*}\langle{a_{1}^{\dagger}}a_{2}\rangle-\alpha_{1}\langle{a_{1}^{\dagger}}{a_{2}^{\dagger}}a_{2}\rangle-\alpha_{2}\langle{a_{1}^{\dagger}}{a_{2}^{\dagger}}a_{1}\rangle\\}\big{)}/n_{1}n_{2},$
(30a) $\displaystyle\mathcal{I}_{1}=$ $\displaystyle
2\,\operatorname{Re}\big{(}{\alpha_{1}\langle{a_{1}^{\dagger}}{a_{2}^{\dagger}}a_{2}\rangle+\alpha_{2}\langle{a_{1}^{\dagger}}{b^{\dagger}}a_{1}\rangle-2\,\alpha_{1}\alpha_{2}\langle{a_{1}^{\dagger}}{a_{2}^{\dagger}}\rangle-2\,\alpha_{1}\alpha_{2}^{*}\langle{a_{1}^{\dagger}}a_{2}\rangle}+$
$\displaystyle\
2\,|\alpha_{1}|^{2}|\alpha_{2}|^{2}-|\alpha_{1}|^{2}\langle{a_{2}^{\dagger}}a_{2}\rangle-|\alpha_{2}|^{2}\langle{a_{1}^{\dagger}}a_{1}\rangle\big{)}/n_{1}n_{2},$
(30b) $\displaystyle\mathcal{I}_{2}=$ $\displaystyle
2\,\operatorname{Re}\big{(}{\alpha_{1}\alpha_{2}\,\langle{a_{1}^{\dagger}}{a_{2}^{\dagger}}\rangle+\alpha_{1}\alpha_{2}^{*}\langle{a_{1}^{\dagger}}a_{2}\rangle}-2\,|\alpha_{1}|^{2}|\alpha_{2}|^{2}\big{)}/n_{1}n_{2}\,.$
(30c)

One can recover from this general result the particular case (21) of a
coherent squeezed state by checking that these quantities for
$\left|\bm{\alpha},\bm{\xi},\zeta_{12}\right\rangle$ with vanishing amplitudes
($\alpha_{i}\rightarrow\epsilon\alpha_{i}$ $r_{i}\rightarrow\epsilon^{2}r_{i}$
and $t_{12}\rightarrow\epsilon^{2}t_{12}$, with $\epsilon\rightarrow 0$), read

$\mathcal{I}_{0}\approx\frac{t_{12}^{2}}{|\alpha_{1}|^{2}|\alpha_{2}|^{2}}\,,\qquad\mathcal{I}_{2}\approx-\frac{2t_{12}\cos(\phi_{1}+\phi_{2}-\vartheta_{12})}{|\alpha_{1}||\alpha_{2}|}\,,$
(31)

while the anomalous $\mathcal{I}_{1}$ exactly zero for any set of parameters.
This recovers Eq. (22d) from Eq. (27) and, with the conditions for the
cancellation of $g^{(2)}_{12}$ in Eq. (24), this gives

$\mathcal{I}_{0}\approx 1\,,\qquad\mathcal{I}_{2}\approx-2\,,$ (32)

i.e., the compensation between the mean field and the squeezed two-mode
fluctuations that leads to $g^{(2)}_{12}=0$ at first order in the parameters
and that we also find in the single-mode antibunching [48, 55]. All the
previous considerations would be interesting, but formal results only, would
it not be for the link that photodetection offers to the theory of frequency-
resolved photon correlations, as discussed in Section 3. Namely, retaining the
frequency information of the photons that one correlates, brings to the fore
the decomposition Eq. (27) but now for the continuously varying frequencies
$\omega_{i}$ as opposed to independent modes $a_{i}$. We now turn to such a
dynamical setting.

##  5 Squeezed cavity

We now return to the two-photon spectrum, to reveal how the concepts of
squeezed and coherent admixtures introduced in Section 4 explain the results
we reported for resonance fluorescence, as shown in Fig. 2. We can best show
that this is the case by bringing directly all these components externally to
a cavity $a$ and collect the re-emitted admixed light, instead of a two-level
system producing them internally. This approach also invites closer inspection
of such excitation schemes [79]. The sensor technique for this scenario
produces the Hamiltonian:

$H_{a}=\Delta_{a}\,{a^{\dagger}}a+i\frac{\Lambda_{a}}{2}(a^{\dagger
2}-a^{2})+\Omega_{a}(e^{i\vartheta}{a^{\dagger}}+e^{-i\vartheta}a)+\Delta_{1}{\varsigma_{1}^{\dagger}}\varsigma_{1}+\Delta_{2}{\varsigma_{2}^{\dagger}}\varsigma_{2}+\epsilon\sum_{i=1,2}({a^{\dagger}}\varsigma_{i}+{\varsigma_{i}^{\dagger}}a)\,.$
(33)

Here we have a passive, linear cavity $a$, driven by both a squeezed source
with strength $\Lambda_{a}$ and a laser with amplitude $\Omega_{a}$, with
$\vartheta$ is the phase difference between the coherent and squeezed sources.
In resonance fluorescence, only the coherent state is provided externally
while the weak nonlinearity, given the weak driving, produces the squeezing.
In other cases, everything could be produced by the system itself (e.g., like
the Mollow triplet produced under incoherent pumping when placed in a cavity
that undergoes lasing [80]). While the 2LS is inherently antibunched
($g^{(2)}_{\sigma}(0)=0$), independent of the nature and strength of the
driving, a squeezed coherent cavity may modulate the photon statistics from
bunched to antibunched as the parameters change. Since we are interested in
showing the similarity of this system with resonance fluorescence, we first
seek to find the conditions that minimize the cavity’s two-photon correlator
$g^{(2)}_{a}$. In the steady-state, the bare correlations of the squeezed
cavity are

$\langle
a\rangle=-\frac{2i\Omega_{a}[e^{i\vartheta}(\gamma_{a}-2i\Delta_{a})-2e^{-i\vartheta}\Lambda_{a}]}{\gamma_{a}^{2}+4\Delta_{a}^{2}-4\Lambda_{a}^{2}}\quad\text{and}\quad\langle{a^{\dagger}}a\rangle=\frac{2\Lambda_{a}^{2}}{\gamma_{a}^{2}+4\Delta_{a}^{2}-4\Lambda_{a}^{2}}+|\langle
a\rangle|^{2}\,.$ (34)

We do not provide the expression for the general $g^{(2)}_{a}$ as it is too
voluminous and not immediately needed for our discussion, unlike the variance
$\langle a^{2}\rangle-\langle
a\rangle^{2}=\Lambda_{a}(\gamma_{a}-2i\Delta_{a})/[\gamma_{a}^{2}+4\Delta_{a}^{2}-4\Lambda_{a}^{2}]$
that is required for the phase matching $\theta=2\phi$ between
$\phi\equiv\arg(\langle a\rangle)$ and $\theta\equiv\arg(\langle
a^{2}\rangle-\langle a\rangle^{2})$ needed to minimize $g^{(2)}_{a}$. This is
satisfied for $\tan(2\vartheta)={2\Delta_{a}/\gamma_{a}}$. There is an upper-
bound for the squeezing amplitude $\Lambda_{a}$ beyond which the system is
unstable and results in unphysical steady states, namely, it must be that
$\Lambda_{a}<\sqrt{\gamma_{a}^{2}+4\Delta_{a}^{2}}\,/2$. Reparameterizing
$\Lambda_{a}$ as $\Lambda_{a}=\Gamma_{a}\lambda/2$ where
$\Gamma_{a}\equiv\sqrt{\gamma_{a}^{2}+4\Delta_{a}^{2}}$, expresses this
stability condition to $\lambda<1$. The cavity phase and its population in the
phase-matching, i.e., optimum-antibunching configuration, are then found as

$\langle
a\rangle=\frac{2i\Omega_{a}\sqrt{\gamma_{a}-2i\Delta_{a}}}{(1+\lambda)\Gamma_{a}^{3}}\quad\text{and}\quad\langle{a^{\dagger}}a\rangle=(1+\lambda)^{-2}\left[\frac{4\Omega_{a}^{2}}{\Gamma_{a}^{2}}+\frac{\lambda^{2}(1+\lambda)}{2(1-\lambda)}\right]\,,$
(35)

with now a tractable two-photon coincidences phase-matched Glauber correlator
(in the $\Gamma\to\infty$ limit):

$g^{(2)}a(0)=\frac{(1-\lambda)^{2}}{[\Gamma_{a}^{2}\lambda^{2}(\lambda^{2}-1)-8\Omega_{a}^{2}(1-\lambda^{2})]^{2}}\times\\\
\big{[}\Gamma_{a}^{4}\lambda^{2}(1+\lambda)^{2}(1+2\lambda^{2})+16\Omega_{2}^{2}\Gamma_{a}^{2}\lambda(\lambda^{2}-1)(1-2\lambda)+64(1-\lambda)^{2}\Omega_{a}^{4}\big{]}\,.$
(36)

Deriving Eq. (36) with respect to $\Omega_{a}$, we find the minimum value
possible for $g^{(2)}_{a}$ of a squeezed cavity admixed to a coherent state:

$g^{(2)}_{a,\mathrm{min}}=2\frac{\lambda(2-\lambda)}{1+2\lambda-\lambda^{2}}\overset{\lambda\rightarrow
0}{\approx}0+4\lambda\,,$ (37)

which is obtained for the “optimum” driving $\Omega_{a,\mathrm{optmin}}$:

$\Omega_{a,\mathrm{optmin}}=\frac{\Gamma_{a}\sqrt{\lambda/2}(1+\lambda)}{2(1-\lambda)}\overset{\lambda\rightarrow
0}{\approx}\frac{\Gamma_{a}}{2}\sqrt{\lambda/2}\,.$ (38)

This produces antibunching for the whole range $0\leq\lambda<1$ and reaches
zero as $\lambda$ gets smaller (for both $\Lambda_{a}\rightarrow 0$ and
$\Delta_{a}\rightarrow\infty$, which is the case of interest). Applying the
optimum antibunching conditions in the limiting case
$\Delta_{a}\rightarrow\infty$ and, therefore $\lambda\rightarrow 0$, we get
for the spectrum

$\displaystyle S_{a,\Gamma}(\omega)\approx|\langle
a\rangle|^{2}\mathscr{L}_{\Gamma}(\omega)+(\langle{a^{\dagger}}a\rangle-|\langle
a\rangle|^{2})\frac{2}{\pi}\frac{\gamma_{12}(\gamma_{11}^{2}+4\Delta_{a}^{2})+4\Gamma\omega^{2}}{(\gamma_{11}^{2}+4\omega^{2})^{2}+8\Delta_{a}^{2}(\gamma_{11}^{2}-4\omega^{2})+16\Delta_{a}^{4}}$
(39)

where we also used the shortcut $\gamma_{ij}\equiv i\Gamma+j\gamma_{a}$ but
this time for $\gamma_{a}$. We will now show that this matches exactly the
two-level system result. The same occurs for the two-photon correlator for the
squeezed cavity,

$g^{(2)}_{a}(\tau)\approx
1+e^{-\gamma_{a}\tau}-2\,e^{-\gamma_{a}/2\,\tau}\cos(2\Delta_{a}\tau)\,,$ (40)

which is identical at resonance to the expression for the unfiltered two-level
system, cf. Eq. (10). In the Next Section, we show that the agreement holds
also in presence of detuning. We conclude this preliminary Section with the
numerically exact two-photon spectrum for the squeezed cavity in the optimum
phase-matching, $\theta=2\phi$, and pumping, Eq. (38), conditions, which is
shown in Fig. 3(a). Its excellent qualitative agreement for most of the
features with resonance fluorescence involving the two-level system, is
obvious (cf. Fig. 2(f)). This confirms the central importance of quantum state
interferences in general and of coherent and squeezed states in particular, to
account for the phenomenology of two-photon physics of detuned resonance
fluorescence. This is fully established in the next Section.

Figure 3: Three manifestations of the origin of the circles of antibunching:
(a) The numerical two-photon spectrum for the squeezed cavity reproduces an
almost identical structure to that of exact resonance fluorescence (cf. Fig.
2(f)). (b) the circle is removed when the coherent fraction itself is removed
(by homodyning), leaving only the strong superbunching leapfrog processes. (c)
The analytical expression also provides exact straight lines and a circle
appearing as Eqs. (45–46) and Eq. (44), respectively. The mesh of thin lines
are those observed in some two-photon landscapes that go beyond the physics
covered in this text. Parameters: (a) $\gamma_{a}=1$, $\Delta_{a}=80.1$,
$\lambda_{a}=0.001$ and $\Gamma=2$, optimizing antibunching. Panel (b) is the
same as Fig. 2(f) but with an external homodyning field
$\alpha=-\langle\sigma\rangle$.

##  6 Detuned resonance fluorescence

We can now complete our argument by returning to the two-photon spectrum of
resonance fluorescence (i.e., for the two-level system $\sigma$). We will
focus on detuning, since this highlights various features of interest,
starting with the spectral separation of the coherent peak, which stays pinned
at the center, from the incoherent emission, that splits into the symmetric
doublet of side peaks, with a neat interpretation in terms of two-photon
scattering as sketched in Fig. 1(c) [29]. Our goal now is to show that the one
and two-photon spectra of the coherently-squeezed cavity derived in the
previous Section, are, beyond their qualitative agreement with the two-level
system, in fact in quantitative agreement to leading order in the driving.
This will confirm that in this detuned Heitler regime, and at the two-photon
level, the main role of the two-level system is to bring in a component of
squeezing, and that the observed phenomenology then follows from two-mode
interferences between squeezed and coherent states, with frequencies spanning
over all possible modes of the system. The one-photon detuned Heitler spectrum
$S_{\Gamma}(\omega)$ from Eq. (4), with $\Delta_{\sigma}\gg\Omega_{\sigma}$,
indeed recovers the squeezed-coherent cavity spectrum (39) with the
correspondence
${\Omega_{\sigma}^{2}\over\Delta_{\sigma}^{2}}\to{\Lambda_{a}\over
2|\Delta_{a}|}$ (which are the populations
$\langle{\sigma^{\dagger}}\sigma\rangle$ and $\langle{a^{\dagger}}a\rangle$,
respectively). The same occurs with two-photon correlations, although the
general expression being awkward, it is not practical to display here their
mathematical identity, which we have, however, checked. We can, instead,
highlight particular cases of interest, including the insightful tampering of
the photon interferences by filtering out the central peak with a notch filter
[59]. In this case, instead of filtering part of the spectrum and correlating
it, one filters out another part and correlate what remains. Such an
observable could be derived using the frequency-filtered approach described
previously, but filtering out the coherent peak is more expediently achieved
by homodyning it out, admixing the total luminescence with a laser of matching
amplitude $\alpha$ but with opposite phase. If we parameterize this field as
$\alpha=-\mathcal{F}\langle\sigma\rangle$ in terms of $0\leq\mathcal{F}\leq
1$, the two-photon correlations for the homodyned field are found as:

$g^{(2)}_{\mathcal{F}}(\tau)=1+\frac{1}{(1-\mathcal{F})^{4}}e^{-\gamma_{\sigma}\tau}-\frac{2}{(1-\mathcal{F})^{2}}e^{-(\gamma_{\sigma}/2)\tau}\cos(\Delta_{\sigma}\tau)$
(41)

where the homodyning is varied from $\mathcal{F}=0$ (no correction, so the
full emission is taken) in which case Eq. (40) is recovered, to
$\mathcal{F}=1$, where the coherent peak is removed completely (so only the
incoherent emission is taken), in which case one observes a simple and smooth
bunching
$g^{(2)}_{\mathcal{F}=1}(\tau)=1+\frac{\Delta_{\sigma}^{4}}{4\Omega_{\sigma}^{4}}e^{-\gamma_{\sigma}\tau}$,
that can be understood as the two photons $\omega_{v}$ and $\omega_{\sigma}$
arriving together. The presence of the laser thus makes them arrive at
different times so that their coincidence is suppressed, but oscillations
ensue as a result of this time redistribution. The effect can also be
understood as beating due to detuning. More general spectral filtering brings
us directly to the full two-photon spectrum. In presence of homodyning, i.e.,
the two-photon spectrum of the incoherent part alone, transforms the landscape
of correlations from Fig. 2(f) to 3(b), i.e., the circle disappears. This is
another proof of its origin from interferences between the squeezing, that
remains and produces extreme bunching (cf. the scale), and coherence. The
result is similar for the squeezed cavity, i.e., with $\Omega_{a}=0$, but
without the side leapfrogs, the central cross and the antibuching around the
real peak (not shown). The general two-photon spectrum can be obtained
analytically (for both the two-level system and the squeezed cavity, both at
and out-of resonance, with or without homodyning), but it is too voluminous to
be replicated here. Excellent approximations can, however, be derived in the
the highly-detuned regime
$\Delta_{\sigma}\gg\Omega_{\sigma}\gg\gamma_{\sigma}$, where emission drops
due to the inefficient excitation of the system and thus enters in the Heitler
regime, although the driving is taken much larger than decay. As a
consequence, the main contribution for the emitted light is the coherent one.
To leading order, the population of each sensor
$\langle{\varsigma_{i}^{\dagger}}\varsigma_{i}\rangle$—that both provide, from
the sensor method, the spectral shape $S_{\Gamma}(\varpi)$—recovers the
spectral shape Eq. (39), featuring a diverging contribution at the origin that
corresponds to the coherent $\delta$ peak, in addition to the symmetric side
peaks, which are also revealed as not Lorentzian:

$S_{\Gamma}(\varpi)=\frac{\epsilon^{2}\Omega_{\sigma}^{2}}{\Delta_{\sigma}^{4}}{1\over\varpi^{2}}+\frac{2\epsilon^{2}\Omega_{\sigma}^{4}}{\Gamma\Delta_{\sigma}^{6}}{\Gamma+2\gamma_{\sigma}+\Gamma\varpi^{2}\over(1-\varpi^{2})^{2}}+\text{
higher orders}\,.$ (42)

The two-photon correlation spectrum is similarly obtained in both cases as:

$g^{(2)}_{\Gamma}(\varpi_{1},\varpi_{2})=\frac{({\varpi}_{1}^{2}+{\varpi}_{1}+{\varpi}_{2}^{2}+{\varpi}_{2})^{2}}{({\varpi}_{1}+{\varpi}_{2})^{2}({\varpi}_{1}+1)^{2}({\varpi}_{2}+1)^{2}}+\text{higher
orders.}$ (43)

This surprisingly compact expression gives an excellent qualitative account of
the detuned resonance fluorescence landscape shown in Fig. 2(f), with
divergences and exact zeros that capture the resonances, which get tamed by
the higher-order corrections as is the case for populations with Eq. (42).
From the analysis of the zeros of both the numerator and denominator of this
expression, we can thus locate the main features of the two-photon spectrum.
Minima correspond to antibunching. They are found by setting the numerator of
Eq. (43) to zero, which, rewriting the equation as

$\Big{(}\varpi_{1}+{1\over 2}\Big{)}^{2}+\Big{(}\varpi_{2}+{1\over
2}\Big{)}^{2}={1\over 2}$ (44)

is that of a circle centered at $\varpi_{1}=\varpi_{2}=-1/2$ and with radius
$\sqrt{2}/2$. This not only gives the most compelling explanation for the
circle of antibunching—as a two-mode interference between squeezing and
coherence—it also shows that it is, indeed, an exact circle. Bunching maxima
are actually divergences to leading order, and located at the condition that
vanish the denominator, i.e.,

${\varpi}_{1}=-1\,,\quad{\varpi}_{2}=-1\quad\text{and}\quad{\varpi}_{1}+{\varpi}_{2}=0$
(45)

producing the three main bunching lines of detuned resonance fluorescence,
which make a right-angled triangle. Additional maxima can be found looking for
the zeros of higher orders (less pronounced in the plot). These are at

${\varpi}_{1}=1\,,\quad\varpi_{2}=1\quad\text{and}\quad\varpi_{1}+\varpi_{2}=\pm
1\,.$ (46)

The horizontal and vertical correlations now concern the real peak, which is
much less correlated than its virtual counterpart. The antidiagonal lines are
the other leapfrog processes. All these lines, along with the circle, are
plotted in Fig. 2(c) as a backbone for the two-photon correlation spectrum.
The decomposition of the 2PS into its interference terms, using Eqs. (30),
leads to:

$\displaystyle\mathcal{I}_{0}$
$\displaystyle\approx\frac{\varpi_{1}^{2}\varpi_{2}^{2}(2+\varpi_{1}+\varpi_{2})^{2}}{(\varpi_{1}+\varpi_{2})^{2}(\varpi_{1}+1)^{2}(\varpi_{2}+1)^{2}}\,,$
(47a) $\displaystyle\mathcal{I}_{1}$ $\displaystyle\approx 0\,,$ (47b)
$\displaystyle\mathcal{I}_{2}$
$\displaystyle\approx-\frac{2\varpi_{1}\varpi_{2}(2+\varpi_{1}+\varpi_{2})}{(\varpi_{1}+\varpi_{2})(\varpi_{1}+1)(\varpi_{2}+1)}\,.$
(47c)

This reveals that, furthermore

$\mathcal{I}_{0}=(\mathcal{I}_{2}/2)^{2}$ (48)

which, from Eq. (27), provides $g^{(2)}$ as a function of the squeezing
component alone:

$g^{(2)}_{\Gamma}=\left(1+{\mathcal{I}_{2}\over 2}\right)^{2}\,.$ (49)

This relationship holds when the coherent fraction dominates over the
incoherent, or quantum, fraction, with also phase-matching between them, in
which case, squeezing can be related to photon fluctuations [37, 83]. This is
the case for instance in Eqs. (31) when $\phi_{1}+\phi_{2}-\vartheta_{12}=0$
or $\pi$ This is also the case in the Heitler regime where the coherent
fraction overtakes the signal [35, 16], including in presence of filtering
[55]. This relationship is the chief reason why squeezing has been more
difficult to observe than antibunching [84]. In more general situations, in
particular when $\mathcal{I}_{1}$ is nonzero, there are deviations from this
ideal. The exact numerical results of $\mathcal{I}_{i}$ for resonance
fluorescence are shown in Fig. 2(g–o). The rightmost column of the detuned
Heitler regime is well approximated by Eqs. (47) with, in particular,
$\mathcal{I}_{1}\approx 0$ (the darker area means a negative sign for the
plotted quantity) and $\mathcal{I}_{0}$ and $\mathcal{I}_{2}$ having the
redundant appearance from Eq. (48). The squeezing component changes sign on
its various domains, as shown in the figure. The condition that produces the
circle of exact antibunching leads to

$\mathcal{I}_{0}\approx 1\,,\quad\mathcal{I}_{1}\approx
0\,,\quad\mathcal{I}_{2}\approx-2\,,$ (50)

which are the same as those for two-photon suppression from destructive
interferences between the squeezing and coherent component, but here occurring
at the two-photon (with two different frequencies) level. Since
$\mathcal{I}_{2}<0$ for this to occur, this happens in the four triangles that
corner the circle in Fig. 2(o). The right-angled triangle is also interesting.
Its lines originate from $\mathcal{I}_{0}$, like the leapfrog processes at
resonance. They indeed lie at the frontier of $\mathcal{I}_{2}$ changing sign,
as can be seen in Fig. 2(o) where they match the dotted lines where
$\mathcal{I}_{2}=0$, although it is large (in absolute value) on both sides.
The antidiagonal line corresponds to the central leapfrog line. The two new
lines that appear, horizontal and vertical, betray the virtual character of
the peak, which remains correlated, at this _pinned_ frequency (this is the
novelty as compared to other leapfrog photons), with all the other photons, at
all the frequencies, including the real photon from the other peak, in which
case one realizes the heralded two-photon scheme [68, 81, 82]. It is
noteworthy that not only the circle, but also the vertical and horizontal
lines disappear with homodyning (cf. Fig. 3(b)), which suggests that the
anomalous correlator $\mathcal{I}_{1}$ plays an important role in the
constitution of these “anomalous” lines, which therefore demand further
attention, that is beyond the scope of our current discussion. In contrast, in
the first column of Fig. 2, i.e., at resonance, one has the opposite situation
where $\mathcal{I}_{0}\gg\mathcal{I}_{2}$ and, furthermore,
$\mathcal{I}_{0}<0$ in the two diagonal quadrants, which contain the spectral
peaks (with a meticulous exclusion of the leapfrogs). This signals the non-
Gaussian, i.e., Fock, or multiphoton emission, character of both the leapfrogs
which have no squeezing associated to their constitution, and of the side
peaks, which have the characteristic $\mathcal{I}_{0}$ of antibunching
dominated by the fluctuations of the incoherent emission. The second column as
the system transits from one case to the other shows the “evaporation” of the
negative domain in $\mathcal{I}_{0}$, which reduces to a tiny islet around the
real peak only at high detuning, as well as the emergence and shaping of
squeezing in $\mathcal{I}_{2}$. It also shows the role of the anomalous
correlator $\mathcal{I}_{1}$ in bridging between these two cases, being close
to negligible in the other limits.

This achieves our description of the two-photon landscape of resonance
fluorescence. All the features have been explained. There is a rich
combination of various mechanisms, from multiphoton emission to multiphoton
interferences, being realized in different regions of phase space where they
can be isolated and exploited. We have largely focused on zero time-delay, but
one could similarly consider the time dynamics of this physics, as we did in
Ref. [82] where, cross-correlating the side peaks of detuned resonance
fluorescence with or without homodyning the coherent peak, we observed a
considerable transformation and enhancement of the correlation functions.
Clearly, the problem is far from being exhausted. Also it should be clarified
that although at the two-photon level, the physics is captured by two-mode
squeezing, the broader problem including higher photon numbers goes much
beyond this vantage point. One should not make the same mistake with two-
photon spectra as has been made with single-photon ones, i.e., assuming that
the picture is now complete. This might be the case for the squeezed cavity,
since it only involves Gaussian states and thus is fully described through its
two-photon correlators, higher-order ones being functions of those. But the
two-level system gives rise to non-Gaussian states, and so, the three-photon
spectrum (and others of higher orders) are most certainly providing as radical
changes as compared to the two-photon case than the two-photon case does to
the single-photon one. Some of this higher-order physics does in fact indeed
transpire in two-photon observables. In the next Section, we give a brief
overview of a few alternative quantifiers of two-photon emission. There, one
will get a chance to sight unaccounted-for phenomena.

##  7 Violation of Inequalities

The above results invite one to seek new regimes of strongly-correlated
quantum emission, away from the spectral peaks. There is much to prospect for,
and various regions (photons of different frequencies) provide us with
different types of light. To highlight this point, we compute various
quantifiers of nonclassical correlations, which can be applied to frequency-
filtered photons [85]. We consider here the case of highly-detuned resonance
fluorescence, since it was not previously considered in this context, and for
being both a clean and fruitful source of correlations. We also add a new
quantifier for two-mode squeezing, given the emphasis we have given to its
role, also clarifying that while we have compellingly established the
importance of squeezing in the broad phenomenology, we are far from having
exhausted the topic. Figure 4(a) shows the violation of Cauchy-Schwarz
inequalities when $R>1$ where

$R\equiv\big{[}g^{(2)}_{12}\big{]}^{2}/\big{[}g^{(2)}_{11}g^{(2)}_{22}\big{]}\,,$
(51)

while panel (b) shows the violation of Bell’s inequalities when $B>2$ where,
this time:

$B\equiv\sqrt{2}\bigg{|}\frac{\langle\varsigma_{1}^{\dagger
2}\varsigma^{2}_{1}\rangle+\langle\varsigma_{2}^{\dagger
2}\varsigma^{2}_{2}\rangle-4\langle\varsigma_{1}^{\dagger}\varsigma_{2}^{\dagger}\varsigma_{2}\varsigma_{1}\rangle-\langle\varsigma_{1}^{\dagger
2}\varsigma^{2}_{2}\rangle-\langle\varsigma_{2}^{\dagger
2}\varsigma^{2}_{1}\rangle}{\langle\varsigma_{1}^{\dagger
2}\varsigma^{2}_{1}\rangle+\langle\varsigma_{2}^{\dagger
2}\varsigma^{2}_{2}\rangle+2\langle\varsigma_{1}^{\dagger}\varsigma_{2}^{\dagger}\varsigma_{2}\varsigma_{1}\rangle}\bigg{|}\,.$
(52)

Panel (c) shows two-mode squeezing, which is present if and only if $S>1$,
where [86]:

$S\equiv\frac{|\langle\varsigma_{1}^{2}\varsigma_{2}^{2}\rangle-\langle\varsigma_{1}\varsigma_{2}\rangle^{2}|}{\langle\varsigma_{1}^{\dagger}\varsigma_{2}^{\dagger}\varsigma_{2}\varsigma_{1}\rangle-|\langle\varsigma_{1}\rangle\langle\varsigma_{2}\rangle|^{2}}\,.$
(53)

All these quantities are plotted in Fig. 4. From the Cauchy-Schwarz
inequality, one can see again the region of no-coincidence emission, where the
system stops emitting at the two-photon level, which is the black circle. This
quantity might be even more suitable to identify the region of no two-photon
emission, being a more properly normalized version of two-photon correlations.
It generalizes to two-photon emission the spectral line elimination via
quantum interferences in spontaneous emission [87]. The green regions show
nonclassical emission. Since the main diagonal is white, it means that the
emission at any frequency from the spectrum is classical, which is also the
case at resonance [85]. Both the main leapfrog and photons involving the
virtual peak are strongly quantum correlated, and are likely valuable
resources for quantum emitters of a new type. The Bell inequality is,
interestingly, of a quite different character, being mainly attached to the
main leapfrog (a bit as well to the circle of no-emission, surprisingly)
where, unlike other quantifiers that behave more like resonances, it spreads
and increases as one gets away from the triplet. Those are features that we
cannot currently explain. Finally, Panel (c) which indicates two-mode
squeezing is probably the most interesting as it first confirms that this is a
feature to be found almost everywhere in detuned resonance fluorescence,
except in the vicinity of the real peaks. More interestingly, however, new
lines whose slope betray other photon combinations ($2\varpi_{1}+\varpi_{2}=k$
and $\varpi_{1}+2\varpi_{2}=k$ where $k=-1$ or $0$) reveal that there is rich
additional physics that we have not yet touched upon, possibly involving
squeezed correlations of higher photon numbers. Even at the qualitative level
of explaining the basic but strong features that shape the landscape, we find
evidence of more complicated processes inherited from beyond two-photon
physics. Some of them have been discussed through $g^{(n)}$ correlators at
resonance [67]. The features not yet accounted for from the quantifiers
discussed in this Section are reported in Fig. 3(c) as thin lines, requiring
new physics. In its pursuit, we could indeed, using the same approach and
techniques, but computing other observables, characterize other types of
correlations, such as three-mode squeezing, other types of entanglement,
quadrature-based tripartite inseparability [88, 89], etc. It is clear that
there remains much to explore, understand and turn into devices, even with the
simplest problem of quantum optics, while the same approach can be applied to
more complicated systems, from cosmic radiation to atoms and molecules passing
by condensed matter and other types of quantum emitters.

Figure 4: Violations of (a) Cauchy and (b) Bell inequalities in highly-detuned
resonance fluorescence, as well as presence of (c) two-mode squeezing.
Filtering harvests correlations. Sight of new physics beyond that discussed in
the text is also apparent. Parameters are the same as for Fig. 2(f).

##  8 Conclusion and Outlook

We have provided a detailed, and essentially analytical, picture of the two-
photon physics of resonance fluorescence. Maybe the most important message is
that this should be contemplated on its own, independently from one-photon
observables, however ingrained is one’s attachment to intensity, signal and
photoluminescence spectra. This text was written at the invitation of the
Royal Society to commemorate the Newton International Fellowship awarded to
one of us (EdV, in 2009). It brings together various of EdV’s research lines
which started at this occasion with a theory of lasing that produced the
Mollow triplet with coherence provided by the emitter itself as opposed to
being brought from outside [80]. This led her to study in more details the
coupling (and decoupling) of two-level systems [90] and how they develop and
maintain correlations in the steady state [91], sustaining a rich span of
different regimes interpolating between incoherent and coherent [92] from the
interplay of quantum and dissipative light-matter interactions. The need to
scrutinize as completely as possible photon correlations from such platforms
led to the introduction of the frequency degree of freedom, this time as part
of a Humboldt Fellowship. This provides the first pillar on which stands the
edifice that explains the two-photon physics of resonance fluorescence (and a
wealth of other problems in its wake): the theory of frequency-resolved
multiphoton correlations [15, 61]. The other pillar is the theory of quantum
field admixtures [16]. Remarkably, while these have been pursued
independently, they turn out to be naturally and deeply interconnected,
besides, on what is possibly the simplest-possible nontrivial problem of
quantum optics: the coherently driven two-level system. We believe that this
betrays the extremely fundamental and universal character of these concepts.
The other Authors of this text, as we are sure would also our past
collaborators on these topics, concur that the surprisingly complex and rich
phenomenology that is being revealed, far from exhausting or completing the
description of the problem, is only making it even more mysterious and
unfathomable. This evokes to us the aphorism of Newton himself:

> «I seem to have been only like a boy playing on the sea-shore, and diverting
> myself in now and then finding a smoother pebble or a prettier shell than
> ordinary, whilst the great ocean of truth lay all undiscovered before me.»

While we have for brevity, simplicity and current experimental interest,
focused on two-photon physics, the above approach is completely general and
can, indeed inevitably will, be generalized to multiphoton correlations and
$n$-mode squeezing. Spheres of antibunching [67] have already been spotted in
three-photon correlation spectra and it is obvious that resonance fluorescence
abounds with three-mode squeezing that remain to be characterized, measured
and exploited. Among other compelling immediate continuations of this work, we
can mention i) seizing additional control and tuneability of the correlations
by externally adjusting the interferences with a laser [25], ii) using
strongly-correlated two-photon spectral locations for quantum-spectroscopic
applications [93] or iii) Purcell-enhancing them to realize new types of
devices with high-purity and strong signal [74]. There are definitely still
other and possibly even more exciting prospects. At any rate, it is clear
that, although inconspicuous at the one-photon level, there is a rich and
varied two-photon physics, which takes place everywhere.

We acknowledge discussions with and interest from colleagues throughout the
years, in particular Juan Camilo López Carreño, Carlos Sánchez Montenegro,
Alejandro González Tudela, Carlos Antón Solanas, Carlos Tejedor, Sang Kyu Kim
and Kai Müller. EdV acknowledges support from the CAM Pricit Plan (Ayudas de
Excelencia del Profesorado Universitario), TUM-IAS Hans Fischer Fellowship and
projects AEI/10.13039/501100011033 (2DEnLight) and Sinérgico CAM 2020
Y2020/TCS-6545 (NanoQuCo-CM). FPL acknowledges the HORIZON
EIC-2022-PATHFINDERCHALLENGES-01 HEISINGBERG project 101114978. This work was
written at the invitation of EdV’s Newton Fellowship, which is also most
gratefully acknowledged.

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