# Photon-photon Correlations from a Pair of Strongly Coupled Two-Level
Emitters

Elnaz Darsheshdar<EMAIL_ADDRESS>Departamento de Física, Universidade
Federal de São Carlos, P.O. Box 676, 13565-905, São Carlos, São Paulo, Brazil
Mathilde Hugbart<EMAIL_ADDRESS>Université Côte d’Azur,
CNRS, INPHYNI, France  Romain Bachelard<EMAIL_ADDRESS>Departamento de Física, Universidade Federal de São Carlos, P.O. Box 676,
13565-905, São Carlos, São Paulo, Brazil  Celso Jorge Villas-Boas
<EMAIL_ADDRESS>Departamento de Física, Universidade Federal de São
Carlos, P.O. Box 676, 13565-905, São Carlos, São Paulo, Brazil

###### Abstract

We investigate two-color photon correlations in the light emitted by strongly
coupled two-level emitters. Spectral filtering allows us to manipulate the
collected light statistics and we show that the resonances induced by dipole-
dipole interactions give rise to specific correlations, where the time-
symmetry of the correlations is broken. Based on the collective dressed
states, our study encompasses both the case of real processes, where the
photons are associated with specific resonances and classical correlations
between each other, and virtual processes, where pairs of photons are emitted
with non-classical correlations.

##  I Introduction

Two-level emitters essentially behave as classical oscillators when they are
weakly driven, and the light elastically scattered by such systems presents a
range of optical phenomena that can be understood using the tools of linear
optics: dispersion, Rayleigh scattering, but also cooperative phenomena such
as superradiance Agarwal (1974a). Differently, for a single emitter, a strong
drive results in a strong inelastic component in the scattering, with the
emergence of a “Mollow triplet” composed of a carrier centered at the laser
frequency, and two symmetric sidebands shifted away by the Rabi frequency of
the driving field Mollow (1969). Of particular interest are the strong
correlations between the photons emitted from the two sidebands: originally
measured in atomic systems Aspect  _et al._ (1980) and extensively studied
theoretically Cohen-Tannoudji  _et al._ (1979); Dalibard and Reynaud (1983);
Schrama  _et al._ (1992), the field has known a resurgence of attention in the
context of quantum dots, with a recent measurement of such photon
(anti)correlations between sidebands Ulhaq  _et al._ (2012), thus
demonstrating the potential of artificial atoms as sources of heralded
photons.

In this context, the coupling of emitters gives access to new control
parameters, as interaction-induced resonances rise, but also interferences
phenomena Wolf  _et al._ (2020). Indeed, the coupling of the emitters through
common radiation modes result in the commonly called dipole-dipole
interactions, which manifests in both the exchange of excitations and
cooperative decay processes Lehmberg (1970a, b). As a consequence, the
fluorescence spectrum of strongly driven atoms presents new sidebands Senitzky
(1978); Agarwal  _et al._ (1980); Ben-Aryeh and Bowden (1988) which, for a
weak interaction, appear at twice the Rabi frequency from the carrier. Such
effects are expected to show up, for instance, in many-atom extended cloud,
with the resonant optical thickness playing the role of cooperativity
parameter Pucci  _et al._ (2017).

Nevertheless, the diversity of photon-photon correlations emitted from
strongly-interacting systems has barely been scratched. The recent development
of the so-called sensor method del Valle  _et al._ (2012), where the photons
emitted in a given mode are monitored by introducing an artificial two-level
system resonant with its frequency (analogously to a Fabry-Perot cavity), has
allowed to explore extensively multi-photon correlations for single emitters
Carreño  _et al._ (2017). In particular, the potential of virtual transitions,
where photons are emitted in bundles, as a source of quantum correlations has
been pointed out. As for interacting emitters, the quantum correlations which
emerge for two weakly interacting emitters have been investigated in the
specific configuration of a pump driving a single emitter, although the
fluorescence spectrum is not substantially affected by the interaction in this
configuration an Peng  _et al._ (2019).

In this work, we investigate two-color correlations in the light emitted by
two strongly-driven strongly-interacting emitters. The correlations are
interpreted introducing the collective dressed states picture, which allows us
to describe both bunching and anti-bunching based on the transitions between
them. At odds from weakly-interacting emitters, the strong interaction lifts
the degeneracy of the energy differences between the different states leading
to a temporal breaking of symmetry for the correlations: photons of different
frequencies may not be emitted in any order. Finally we show that most of the
virtual processes, which involve pairs of photons, yield non-classical
correlations when the sum of their energies fits any of the interaction-
induced sidebands.

##  II Modeling and detection scheme

In our work, we consider two identical two-level systems. Experimentally, this
can be either two atoms, two molecules or two quantum dots. For this last
example, although quantum dots are promising single-emitter platforms, it
remains very challenging to produce very similar dots, i.e., with very similar
transition frequencies and linewidths. On the other hand, laser cooling has
allowed to bring the interactions between cold atoms under a very high degree
of control. In this paper, the two-level system is considered to be a
motionless atom.

The system under study is thus composed of two identical driven two-level
systems (TLS) at positions ${{\textbf{r}}_{i}}$, with transition frequency
$\omega_{a}$ and linewidth $\Gamma$. Each atom is described by the spin-half
angular momentum algebra, with $\sigma_{i}^{-}$ ($\sigma_{i}^{+}$) the
lowering (rising) operator of the $i$th atom ($i=1,2$). In the Born, Markov
and rotating-wave approximations, the master equation which describes the
dynamics of its density matrix $\rho$, in the laser reference frame, is given
by Agarwal (1974b) (we set $\hbar=1$ along the paper):

$\frac{\partial\rho}{\partial t}=i\left[\rho,H\right]+{\mathcal{L}}\rho,$ (1)

where the coherent and incoherent parts are encoded in the following at
resonance Hamiltonian and Lindblad super-operator, respectively:

$\displaystyle H$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum\limits_{i}[\Omega^{*}(\mathbf{r}_{j})\sigma_{i}^{-}+\Omega(\mathbf{r}_{j})\sigma_{i}^{+}]+\Gamma\sum\limits_{i,j\neq
i}{{\delta}_{ij}}\sigma_{i}^{+}\sigma_{j}^{-},$ (2)
$\displaystyle\mathcal{L}\rho$ $\displaystyle=$
$\displaystyle\frac{\Gamma}{2}\sum\limits_{i}(2\sigma_{i}^{-}\rho\sigma_{i}^{+}-\sigma_{i}^{+}\sigma_{i}^{-}\rho-\rho\sigma_{i}^{+}\sigma_{i}^{-})$
(3) $\displaystyle+\frac{\Gamma}{2}\sum\limits_{i,j\neq
i}{{\gamma}_{ij}}(2\sigma_{j}^{-}\rho\sigma_{i}^{+}-\sigma_{i}^{+}\sigma_{j}^{-}\rho-\rho\sigma_{i}^{+}\sigma_{j}^{-}).$

The atoms are here resonantly driven by a monochromatic plane–wave
$\Omega(\mathbf{r})=\Omega e^{i\mathbf{k}_{L}.\mathbf{r}}$, where $\Omega$
stands for the Rabi frequency and $\mathbf{k}_{L}$ the light wavevector. The
dipole-dipole interactions give rise to both a coherent and incoherent
coupling

$\displaystyle{{\delta}_{ij}}=-\frac{3}{4}\left(1-\cos^{2}{{\theta_{ij}}}\right)\frac{\cos{(k{r}_{ij})}}{{k{r}_{ij}}}$
(4)
$\displaystyle+\frac{3}{4}\left(1-3\cos^{2}{{\theta_{ij}}}\right)\left[\frac{\sin{(k{r}_{ij})}}{(kr_{ij})^{2}}+\frac{\cos{(k{r}_{ij})}}{(kr_{ij})^{3}}\right],$
$\displaystyle{{\gamma}_{ij}}=\frac{3}{2}\left(1-\cos^{2}{{\theta_{ij}}}\right)\frac{\sin{(k{r}_{ij})}}{{k{r}_{ij}}}$
$\displaystyle+\frac{3}{2}\left(1-3\cos^{2}{{\theta_{ij}}}\right)\left[\frac{\cos{(k{r}_{ij})}}{(kr_{ij})^{2}}-\frac{\sin{(k{r}_{ij})}}{(kr_{ij})^{3}}\right],$

with $\lambda=2\pi/k$ the wavelength transition ($k\approx k_{L}$), $r_{ij}$
the distance between the atoms, and $\theta_{ij}$ the angle between their
dipole moments and the vector joining them,
$\mathbf{r}_{ij}=\mathbf{r}_{j}-\mathbf{r}_{i}$.

Solving the master equation provides the scattered electric field, which is
given, in the far field limit and in direction $\hat{\mathbf{n}}$, by

${{E}^{\dagger}}\left(t\right)=\sum\limits_{j=1}^{2}{\sigma_{j}^{-}\left(t\right){{e}^{-ik\mathbf{\hat{n}}.{{\mathbf{r}}_{j}}}}}.$
(5)

The dependence on $\hat{\mathbf{n}}$ is hereafter kept implicit. Its temporal
coherence is captured by the first-order and second-order two-time correlation
functions:

$\displaystyle{g}^{(1)}\left(\tau\right)$ $\displaystyle=$
$\displaystyle\lim_{t\to\infty}\frac{\left\langle
E\left(t\right){{E}^{\dagger}}\left(t+\tau\right)\right\rangle}{\left\langle
E\left(t\right){{E}^{\dagger}}\left(t\right)\right\rangle},$ (6)
$\displaystyle{g}^{(2)}\left(\tau\right)$ $\displaystyle=$
$\displaystyle\lim_{t\to\infty}\frac{\left\langle
E\left(t\right)E\left(t+\tau\right){{E}^{\dagger}}\left(t+\tau\right){{E}^{\dagger}}\left(t\right)\right\rangle}{\left\langle
E\left(t\right){{E}^{\dagger}}\left(t\right)\right\rangle^{2}},$ (7)

here computed in the steady state. In particular, the fluorescence spectrum,
sometimes referred to one-photon spectrum (1PS), is obtained from the Fourier
transform of the first-order correlation function

$S\left(\omega\right)=\underset{T\to\infty}{\mathop{\lim}}\,\int_{-T}^{T}{{{g}^{(1)}}\left(\tau\right){{e}^{-i\omega\tau}}d\tau}.$
(8)

The 1PS gives the spectral energy distribution of the light scattered
elastically and inelastically, whereas the second-order correlation function
$g^{(2)}$ contains details on the correlations between the emitted photons,
with the antibunching in the trains of photons emitted by a single emitter as
a hallmark of the non-classicality of this emission Kimble  _et al._ (1977).

The problem of time-resolved observables is however more challenging. Indeed,
as one introduces the field operator in the reciprocal space
$\tilde{E}(\omega)=\int_{t=-\infty}^{\infty}e^{-i\omega t}E(t)dt$, the problem
of studying two-color photon-photon correlations brings in the calculation of
a four-time correlator, $\left\langle
E(t_{1})E(t_{2}){E}^{\dagger}(t_{3}){E}^{\dagger}(t_{4})\right\rangle$. Then,
the use of the quantum regression theorem, commonly used for two-time
observables, may become a daunting task Gisin (1993); Brun and Gisin (1996);
Breuer  _et al._ (1997). This is a strong restriction to the study of photon-
photon correlations, which has long limited rigorous results to single-emitter
physics Bel and Brown (2009).

An elegant solution was found in the “sensor method” that allows one to
investigate theoretically frequency-resolved correlations in greater details
del Valle  _et al._ (2012). It relies on the introduction in the system of
extra two-level systems which behave as sensors, as described by the
Hamiltonian

${{H}_{S}}=\sum_{s}\omega_{s}\xi_{s}^{\dagger}{{\xi}_{s}^{-}}+\varepsilon\sum_{s}\left(E\xi_{s}^{-}+E^{\dagger}\xi_{s}^{\dagger}\right),$
(9)

with $\xi_{s}$ ($\xi_{s}^{\dagger}$) the lowering (rising) operator for sensor
$s$, and $\omega_{s}$ its resonant frequency, in the rotating frame at the
laser frequency 111We here consider sensors which all couple to the field
radiated in the same direction, but a generalization to two-direction photon-
photon correlations can be obtained by introducing sensors which couple to the
field (5) emitted in different directions.. The $\varepsilon$ parameter
corresponds to the coupling strength between the sensors and the atomic
system, which must be made sufficiently small to not perturb significantly the
dynamics of the latter and to avoid the saturation of the sensor
($\varepsilon={{10}^{-4}}$ throughout the paper). The sensors are also
characterized by their linewidth $\Gamma_{s}$, a parameter of importance as we
shall see later, which manifests in an extra Lindblad term:

$\mathcal{L}_{S}\rho=\frac{\Gamma_{s}}{2}\sum_{s}\left(2\xi_{s}^{-}\rho\xi_{s}^{\dagger}-\xi_{s}^{\dagger}\xi_{s}^{-}\rho-\rho\xi_{s}^{\dagger}\xi_{s}^{-}\right).$
(10)

The sensor contributions (9-10) are then added to the master equation (1),
with $\rho$ now describing the density matrix of the whole system (atoms plus
sensors).

The steady-state two-photon time- and frequency-resolved correlation is then
obtained from the second-order correlation function from the sensors
operators:

$\displaystyle g_{s}^{(2)}(\omega_{1},\omega_{2},\tau)=$ (11)
$\displaystyle\lim_{t\to\infty}\frac{\left\langle\xi_{1}^{\dagger}(\omega_{1},t)\xi_{2}^{\dagger}(\omega_{2},t+\tau){{\xi}_{2}}(\omega_{2},t+\tau){{\xi}_{1}}(\omega_{1},t)\right\rangle}{\left\langle\xi_{1}^{\dagger}(\omega_{1},t){{\xi}_{1}}(\omega_{1},t)\right\rangle\left\langle\xi_{2}^{\dagger}(\omega_{2},t){{\xi}_{2}}(\omega_{2},t)\right\rangle}.$

Equal-time correlations ($\tau=0$) characterize the simultaneous emission of
photons of frequencies $\omega_{1}$ and $\omega_{2}$, they are hereafter
called 2PFC (two photon frequency resolved correlations) and noted by
$g_{s}^{(2)}(\omega_{1},\omega_{2})$, for simplicity. Thus, at the expense of
two extra degrees of freedom, equal-time frequency-resolved correlations
$g_{s}^{(2)}(\omega_{1},\omega_{2})$ are contained in the steady-state values
of the density matrix, while time- and frequency-resolved ones (i.e.,
$g_{s}^{(2)}(\omega_{1},\omega_{2},\tau)$) are obtained as two-time
correlators, using the “standard” (two-time) quantum regression theorem
Gardiner and Zoller (2014).

Experimentally, frequency-resolved signals can be obtained using frequency
filters such as a Fabry-Perot cavities whose resonance frequency and linewidth
correspond to the sensor ones, $\omega_{s}$ and $\Gamma_{s}$, but also from
time-resolved measurements using beatnote techniques for the $g^{(1)}(\tau)$
function Ortiz-Gutiérrez  _et al._ (2019); Ferreira  _et al._ (2020), for
example. Throughout this work, the different correlation functions were
computed using the Qutip toolbox Johansson  _et al._ (2012, 2013) and the
Matlab® software, using a solver to reach the steady-state.

##  III Strongly interacting atoms

###  III.1 Fluorescence spectrum

The radiation spectrum of a strongly driven two-level emitter has a rather
intuitive interpretation in the dressed state picture: after the light modes
were traced out to obtain Eqs.(2-3) Lehmberg (1970a, b), in this picture the
photon number is restored to obtain hybrid atom-field states. The resulting
atom-field eigenstates have been discussed extensively for single emitters
Compagno  _et al._ (1995), and the coupling of light to atom leads to the
following eigenstates for the Hamiltonian at resonance

$\left|\pm\right\rangle=\frac{1}{\sqrt{2}}\left(\left|\uparrow,n-1\right\rangle\pm\left|\downarrow,n\right\rangle\right),$
(12)

where $\left|\downarrow\right\rangle$ and $\left|\uparrow\right\rangle$ denote
the single-atom ground and excited states, respectively, and $n$ is the photon
number in the driving field (i.e., the laser). This pair of eigenstates forms
the $n$-excitation manifold: in each manifold the eigenstates are split by the
Rabi frequency of the driving field (unless the Cavity Quantum Electrodynamics
regime is reached Jaynes and Cummings (1963); Brune  _et al._ (1996)).

For a pair of atoms, the dipole-dipole interaction in Eqs. (2-3) generates two
collective single-excitation eigenstates, labelled symmetric and anti-
symmetric:

$\displaystyle\ket{S}$ $\displaystyle=$
$\displaystyle(\ket{\uparrow\downarrow}+\ket{\downarrow\uparrow})/\sqrt{2},$
$\displaystyle\ket{A}$ $\displaystyle=$
$\displaystyle(\ket{\uparrow\downarrow}-\ket{\downarrow\uparrow})/\sqrt{2},$
(13)

which present linewidths $\Gamma_{S}=\Gamma(1+\gamma_{12})$ and
$\Gamma_{A}=\Gamma(1-\gamma_{12})$, and energy shifts
$\Delta_{S}=\Gamma\delta_{12}$ and $\Delta_{A}=-\Gamma\delta_{12}$. Throughout
this work we have fixed $\cos(\theta_{12})=1/\sqrt{3}$, which implies
$\delta_{12}<0$ for the interatomic distances considered and, consequently,
$\Delta_{S}<\Delta_{A}$, as shown in Fig. 1.

We here consider the case of two very close, strongly interacting atoms
($kr_{12}\ll 1$ or, more specifically, $\left|{{\delta}_{12}}\right|\gg 1$ and
${{\gamma}_{12}}\approx 1$), in the presence of a strong resonant driving,
characterized by $\Omega^{2}>\Gamma^{2}+4|\Gamma\delta_{12}|^{2}$. Following
the approach of Ref. Compagno  _et al._ (1995), we consider the following
basis

$\displaystyle\left|\phi_{n}^{1}\right\rangle$
$\displaystyle=\left|\uparrow\uparrow,n-2\right\rangle,$ (14a)
$\displaystyle\left|\phi_{n}^{2}\right\rangle$
$\displaystyle=\left|S,n-1\right\rangle,$ (14b)
$\displaystyle\left|\phi_{n}^{3}\right\rangle$
$\displaystyle=\left|A,n-1\right\rangle,$ (14c)
$\displaystyle\left|\phi_{n}^{4}\right\rangle$
$\displaystyle=\left|\downarrow\downarrow,n\right\rangle.$ (14d)

which is the four dimensional subspace of the eigenvectors of the operator
${{N}_{T}}=N_{\nu}+1/2+\sum\limits_{i=1,2}(1+\sigma_{i}^{+}\sigma_{i}^{-})/2$,
where $N_{\nu}$ is the photon number operator for eigenvalue $n$. In this
basis, the eigenstates of the atom-light system are composed by the collective
dressed states incorporating the eigenstates of Hamiltonian (2) for two atoms
with light-mediated interactions, and the photon number states of the light
field i.e., the $n$-excitation manifold for our system is given by:

$\displaystyle\left|u_{n}^{1}\right\rangle=a_{1}\left|\uparrow\uparrow,n-2\right\rangle+a_{2}\sqrt{2}\left|S,n-1\right\rangle+a_{1}\left|\downarrow\downarrow,n\right\rangle,$
(15a) $\displaystyle\left|u_{n}^{a}\right\rangle=\left|A,n-1\right\rangle,$
(15b)
$\displaystyle\left|u_{n}^{2}\right\rangle=-\frac{1}{\sqrt{2}}\left|\uparrow\uparrow,n-2\right\rangle+\frac{1}{\sqrt{2}}\left|\downarrow\downarrow,n\right\rangle,$
(15c)
$\displaystyle\left|u_{n}^{3}\right\rangle=a_{2}\left|\uparrow\uparrow,n-2\right\rangle-
a_{1}\sqrt{2}\left|S,n-1\right\rangle+a_{2}\left|\downarrow\downarrow,n\right\rangle,$
(15d)

with $a_{1}$ and $a_{2}$ two coefficients obtained by diagonalization of the
Hamiltonian (the lengthy expressions for $a_{1}$ and $a_{2}$ are not shown
here, and the normalization of the eigenstates imposes
$a_{1}^{2}+a_{2}^{2}=1/2$).

These dressed states are characterized by an entanglement between atomic and
field states, apart from the one containing the antisymmetric atomic state,
$\left|u_{n}^{a}\right\rangle$. Since the latter state is not entangled with
the field states, nor is it coupled to the other states through the
Hamiltonian, it does not participate to the dressing. Furthermore, in the
limit of strong coupling of the atoms considered here, it can be shown that
this anti-symmetric state does not participate substantially to the steady-
state fluorescence spectrum. Although it is not driven directly by the laser
($kr_{12}\ll 1$ leads to a rather homogeneous phase profile of the laser on
the atoms, thus addressing the symmetric state), it gets substantially
populated by decay from the atomic state $\ket{\uparrow\uparrow}$ and its long
lifetime allows it to hold a substantial population Cipris  _et al._ (2020).
Nevertheless, the weak linewidth also translates in a low number of emitted
photons. Thus, unless specified (see Sec.III.2), we hereafter neglect this
state in our analysis.

This leads us to introduce the collective operator
$\sigma_{S}^{+-}=(\sigma_{1}^{+-}+\sigma_{2}^{+-})/\sqrt{2}$ and simplify the
Linbladian (3) into
${\mathcal{L}_{\sigma_{S}}}\rho={\Gamma_{S}}(2\sigma_{S}^{-}\rho\sigma_{S}^{+}-\sigma_{S}^{+}\sigma_{S}^{-}\rho-\rho\sigma_{S}^{+}\sigma_{S}^{-})$
in the strong interaction regime. As a consequence the $n$-excitation manifold
reduces to the triplet of $(\ket{u_{n}^{1}},\ket{u_{n}^{2}},\ket{u_{n}^{3}}$),
with the frequency difference hereafter called:

$\Delta_{ij}=-\Delta_{ji}\equiv E_{n}^{i}-E_{n}^{j}.$ (16)

The dressed energy levels are then composed of the $n$-excitation manifolds,
each composed of the above triplet, and with successive manifolds separated by
energy $\omega_{L}$: the dressed states and the equivalent bare collective
energy levels for two interacting atoms are presented in Fig. 1(a).

(a)-0.1in0in (b)0.1in0.1in

Figure 1: (a) Collective dressed (left) and bare (right) states for two
strongly-interacting atoms, in the rotating frame of the laser and in the lab
frame, respectively. (b) 1PS of two interacting atoms. The colored arrows
above the peaks correspond to the transitions depicted in (a). Simulations
carried out for two atoms driven with a field of Rabi frequency
$\Omega=30\Gamma$, separated by a distance $kr_{12}=0.05$, with dipole
orientation $\theta_{12}=\cos^{-1}(1/\sqrt{3})$.

The 1PS is obtained by solving the master equation from Eqs. (1-3) combined
with the quantum regression theorem or, equivalently, monitoring the
population of a sensor whose resonant frequency $\omega_{s}$ is tuned. The
fluorescence spectrum for two strongly interacting atoms is depicted in
Fig.1(b), where the different peaks can be interpreted in the dressed state
picture. Similarly to the single-atom case, the central peak originates in the
$\ket{u_{n}^{i}}\to\ket{u_{n-1}^{i}}$ transitions ($i=1,\ 2,\ 3$), which do
not alter the atomic state and are characterized by the emission of a photon
at the laser frequency ${{\omega}_{L}}$.

The transformation of the doublet of states into a triplet of states for the
$n$-excitation manifold, due to the interactions, leads to a seven-peak 1PS.
The six sidebands are collective, corresponding to resonant frequencies
$\pm\Delta_{ij}$ not present for single atoms (all transitions are hereafter
given in the laser frame), and the corresponding transitions
$\ket{u_{n}^{i}}\to\ket{u_{n-1}^{j\neq i}}$ are presented schematically in the
dressed state picture of Fig.1(a).

###  III.2 Photon-photon correlations

Let us now study the specific correlations which occur between these different
emission processes. While the transitions from one manifold to the next one
are the origin of the 1PS, the correlations in the emitted photons are the
essence of the 2PFC, $g_{s}^{(2)}\left(\omega_{1},\omega_{2}\right)$, computed
using Eq.(11). The 2PFC corresponding to the situation of Fig.1 is presented
in Fig. 2(a), the complexity of which reflects the diversity in photon-photon
correlations.

(a)0in0in (b)0in-0.1in(c)0in0in

Figure 2: (a) Steady-state photon-photon correlations
$g^{(2)}_{s}(\omega_{1},\omega_{2})$ for a pair of strongly-interacting
strongly-driven atoms. (b) Cascade processes involving the emission of two
photons, according to the energy levels of the dressed states picture (allowed
cascades with plain lines, forbidden processes with dashed/dotted ones, see
text). The associated $g^{(2)}_{s}(\omega_{1},\omega_{2})$ is given by the
same symbols in (a). (c) Time-resolved
$g_{s}^{(2)}(\omega_{1},\omega_{2},\tau)$ for the transitions involving
photons of frequency $(\omega_{1},\omega_{2})=(\Delta_{13},-\Delta_{23})$ (
$+$ symbol in (a) and (b)) with $\Gamma_{s}=\Gamma$. Inset: same curves, for a
broader time window. Simulations realized for two atoms with $kr_{12}=0.05$,
$\theta_{12}=\cos^{-1}(1/\sqrt{3})$, $\Omega=30\Gamma$, and
$\Gamma_{s}=\Gamma$ for (a). $kr=0.006$, $\theta=\cos^{-1}(1/\sqrt{3})$,
$\Omega=250\Gamma$ and $\Gamma_{s}=5\Gamma$ for (c).

_Opposite-sideband correlations –_ We first discuss the correlation between
opposite sidebands, i.e., with frequency $+{{\Delta}_{ij}}$ and
$-{{\Delta}_{ij}}$, as shown by the $\bullet$ symbol for $(i,j)=(1,2)$ in Fig.
2(a): it corresponds to the two-photon cascade
$\left|u_{n}^{1}\right\rangle\to\left|u_{n-1}^{2}\right\rangle\to\left|u_{n-2}^{1}\right\rangle$,
shown in Fig. 2(b). Being an allowed path of relaxation, it leads to photon-
photon bunching,
$g_{s}^{\left(2\right)}\left({{\Delta}_{12}},-{{\Delta}_{12}}\right)>1$: this
case is similar to the opposite-sideband bunching effect reported for single
emitters Schrama  _et al._ (1992); Ulhaq  _et al._ (2012). The same holds true
for other transitions of the form
$\left|u_{n}^{i}\right\rangle\to|u_{n-1}^{j}\rangle\to|u_{n-2}^{i}\rangle$,
corresponding to the other sidebands.

_Equal-sideband correlations –_ Differently, photons emitted from the same
sidebands come antibunched, as in all cases the associated relaxation path is
blocked (as long as there is no degeneracy, i.e.,
$\Delta_{12}\neq\Delta_{23}$). An analogous effect is observed for single
atoms. For instance, a photon of frequency ${{\Delta}_{13}}$ automatically
leads the system to state $\left|u_{n-1}^{3}\right\rangle$, so the next photon
cannot be emitted at the same frequency as it requires for the system to be in
a state $\left|u_{n-1}^{1}\right\rangle$ (states
$\left|u_{n-1}^{1}\right\rangle$ and $\left|u_{n-1}^{3}\right\rangle$ are
orthogonal). For this reason, the associated path of relaxation is considered
blocked (see $\circ$ symbol cascade in Fig. 2(b)), and it is characterized by
antibunched photons:
$g_{s}^{\left(2\right)}\left({{\Delta}_{13}},{{\Delta}_{13}}\right)<1$
($\circ$ symbol in Fig. 2(a)).

Nevertheless, as it can be seen in Fig. 2(a), photons from the same sidebands
suffer from being in the “indistinguishability bunching line” of the 2PFC.
Indeed, two photons with the same frequency cannot be distinguished by the
sensor, which in turn leads to bunching effects. This manifests in the
“overbunched” diagonal line in Fig.2(a).

_Cross-sideband correlations –_ Let us now discuss processes which involve
photons from two different sidebands, corresponding to
$g_{s}^{\left(2\right)}\left(\pm{{\Delta}_{ij}},\pm{{\Delta}_{i^{\prime}j^{\prime}}}\right)$
with $(i,j)\neq(i^{\prime},j^{\prime})$. For these processes which involve the
three atomic states, a more careful analysis is needed, as photons of
different frequencies may be emitted in a specific order. For instance, the
double transition
$\left|u_{n}^{1}\right\rangle\to\left|u_{n-1}^{3}\right\rangle\to\left|u_{n-2}^{2}\right\rangle$,
indicated by a + symbol in the dressed state representation of Fig. 2(b), is
allowed, and thus permits the successive emission of photons of frequency
$\Delta_{13}$ and $-\Delta_{23}$, in that order. Differently, a photon of
frequency $-\Delta_{23}$ cannot be followed by one of frequency $\Delta_{13}$,
since this would correspond to the successive
$\left|u_{n}^{3}\right\rangle\to\left|u_{n-1}^{2}\right\rangle$ and
$\left|u_{n-1}^{1}\right\rangle\to\left|u_{n-2}^{3}\right\rangle$ (see
$\otimes$ symbol in Fig. 2(b)), which is a blocked path since
$\left|u_{n-1}^{2}\right\rangle$ and $\left|u_{n-1}^{1}\right\rangle$ are
orthogonal.

Monitoring the zero-delay photon-photon correlations
$g_{s}^{(2)}(\Delta_{ij},\Delta_{i^{\prime}j^{\prime}})$ does not allow to
distinguish the two processes, but its time-resolved version does. As
illustrated by the computation of $g_{s}^{(2)}(\Delta_{13},-\Delta_{23},\tau)$
in Fig. 2(c), we observe a strong bunching at delays $\tau\sim+1/\Gamma$, but
a below-unity $g_{s}^{(2)}$ for $\tau\sim-1/\Gamma$ (negative times
corresponds to the reverse order, since
$g_{s}^{(2)}(\Delta_{13},-\Delta_{23},\tau)=g_{s}^{(2)}(-\Delta_{23},\Delta_{13},-\tau))$.
The same phenomenon is observed for transitions with photon pairs of frequency
$(\pm\Delta_{12},\mp\Delta_{13})$ and $(\pm\Delta_{12},\pm\Delta_{23})$. Thus,
these double transitions, which involve the three different atomic states,
present a time-symmetry breaking for the $g_{s}^{(2)}$ function, which
corresponds to a specific ordering of the emitted photons. It is due to the
interaction between the emitters, which leads to a splitting of the energy
levels of the atomic system.

It is interesting to note that on timescales $\tau$ of several single-atom
excited state lifetime, the correlator
$g_{s}^{(2)}(\Delta_{13},-\Delta_{23},\tau)$ does not go to $1$ as one would
expect: this is the signature of the anti-symmetric state holding a
substantial part of the atomic excitations, yet these are released on the mode
timescale Cipris  _et al._ (2020) (see inset of Fig.2(c)).

Finally, the case of double processes that depart twice from the same atomic
state, but goes to the two others atomic states (i.e.,
$\left|u_{n}^{i}\right\rangle\to|u_{n-1}^{j}\rangle$ and
$\left|u_{n-1}^{i}\right\rangle\to|u_{n-2}^{l}\rangle$, with $i$, $j$ and $l$
all different), are naturally anti-bunched. Indeed both possible orders for
the double transition are blocked. Consequently,
$g_{s}^{2}(\Delta_{ij},\Delta_{il})$ is below unity, as can be observed in
Fig.2(a).

_Sideband-central peak correlations –_ Finally, cascades which involve one
sideband photon plus a central peak photon can in principle occur
successively, since the latter does not involve a change in the atomic state
(i.e., $\left|u_{n}^{i}\right\rangle\to\left|u_{n-1}^{i}\right\rangle$).
Furthermore, both orders of emission for the photons could equally occur.
Nonetheless, these pairs of photons come anti-bunched. As discussed for the
case of single emitters Arnoldus and Nienhuis (1984), this effect originates
from a destructive interference, due to the fact that the state of the system
is not modified by Rayleigh emission. Thus, despite the two cascades involving
photons of frequency ${\Delta_{ij}}$ and $0$ are degenerate (they have the
same initial and final states), the interference between the amplitude of
their transition probability prevents the process instead of favoring it (see
Fig.2(a)).

Finally, we point out that a clear observation of antibunching, and other
kinds of photon-photon correlations as we investigate in this paper, requires
the use of sensors of linewidth at least comparable to the atomic linewidth
(they are here taken equal: $\Gamma_{s}=\Gamma$). Indeed, it has recently been
shown that antibunching (and other photon-photon correlations), whether in the
temporal domain Carreño  _et al._ (2018); Hanschke  _et al._ (2020); Phillips
_et al._ (2020) (as given by $g^{(2)}(\tau)$) is strongly reduced in case of a
sublinewidth filtering, since it results in long integration time, which in
turn averages out the correlations Muñoz  _et al._ (2014).

(a)0in0in(b)0in0in (c)0in-0.05in

Figure 3: (a) Two-photon “leapfrog” processes, with the system transiting
through virtual dressed levels. (b) Ratio $R_{s}$ from the Cauchy-Schwartz
inequality, and (c) $B_{s}$ from the Bell inequality, when tuning the
frequency of each sensor. Simulations realized for $kr=0.05$,
$\theta=\cos^{-1}(1/\sqrt{3})$, $\Omega=30\Gamma$ and $\Gamma_{s}=\Gamma$.

###  III.3 Leapfrog processes

The cascades described above involve two-photon emission processes that
encompass real transitions through intermediate states, where the system state
is described by the manifolds from the dressed atom picture. There exist other
kinds of two-photon transitions, where the system does not transit through one
of these intermediate states, but rather through a “virtual” manifold,
labelled “leapfrog processes” Gonzalez-Tudela  _et al._ (2013). These
transitions are characterized by the joint emission of two photons, and have
recently been observed in single quantum dots Peiris  _et al._ (2015). Most of
these two-photon collective processes yield correlations much stronger than
real transition ones, and their quantum nature has been demonstrated for
single emitters using Cauchy-Schwartz and Bell inequalities Muñoz  _et al._
(2014).

For these leapfrog processes, the energy of each photon does not need to be
related to a specific level transition energy, only their sum needs to obey
the following relation:

$\omega_{1}+\omega_{2}=0,\ \pm{{\Delta}_{ij}},$ (17)

where the frequency in the laboratory frame is obtained by adding
$2\omega_{L}$. The leapfrog transitions correspond to the anti-diagonal lines
marked by color arrows in Fig. 3(b) and (c), and the associated (virtual)
transitions are depicted in Fig. 3(a). Note that if, in addition to condition
(17), the energy of each photon belongs to the allowed real transitions, the
photon emission process is that described in the previous section, and the
correlations between the photons are classical.

To characterize the non-classicality of these correlations, we use the Cauchy-
Schwartz inequality (CSI) for the second-order correlation functions at zero
delay $g^{(2)}_{kl}=g^{(2)}_{s}(\omega_{k},\omega_{l})$:

$\left[g^{(2)}_{12}\right]^{2}\leq g^{(2)}_{11}g^{(2)}_{22},$ (18)

which we monitor by studying the ratio

$R_{s}=\frac{\left[g^{(2)}_{12}\right]^{2}}{g^{(2)}_{11}g^{(2)}_{22}}.$ (19)

Values $R_{s}$ larger than unity are the signatures of non-classical
correlations between the two emitted photons Muñoz  _et al._ (2014).

In Fig.3(a), leapfrog processes which involve different initial and final
atomic states are presented: the system does not emit photons from specific
(“real”) transitions, only the sum of the two photon energies corresponds to a
transition between specific states in the dressed states picture. As one can
observe from the anti-diagonal lines in Fig.3(b), which correspond to
$\omega_{1}+\omega_{2}=0,\ \pm{{\Delta}_{ij}}$, the CSI is violated for most
of these joint emission processes ($R_{s}>1$). Nevertheless, the CSI is not
violated for antibunched real transitions, i.e., for photon pairs with
frequencies $(0,\pm{{\Delta}_{ij}})$ or $(\pm{{\Delta}_{ij}},0)$ (see, for
example, the $\oslash$ for frequencies $({{\Delta}_{13}},0)$). Neither it is
for pairs of real photons, for frequencies
$(\pm{{\Delta}_{12}},\pm{{\Delta}_{23}})$. Also, as we can observe in Fig.
4(a-b), a sublinewidth filtering leads to weaker violations of CSI Muñoz  _et
al._ (2014).

(a)0in0in(b)0in0in (c)0in-0.05in

Figure 4: Quantifier $R_{s}$ for the Cauchy-Schwartz inequalities (with a
violation above the dashed line) for ${{\Gamma}_{s}}=\Gamma$ (solid blue) and
${{\Gamma}_{s}}=\Gamma/10$ (dotted red), for the leapfrog lines (a)
${{\omega}_{1}}+{{\omega}_{2}}={{\Delta}_{12}}$ and (b)
${{\omega}_{1}}+{{\omega}_{2}}=0$. (c) Quantifier $B_{s}$ for the Bell
inequalities (with a violation above the dashed line) for
${{\Gamma}_{s}}=\Gamma$ (solid blue) and ${{\Gamma}_{s}}=\Gamma/10$ (dotted
red) in the leapfrog line of ${{\omega}_{1}}+{{\omega}_{2}}=0$. The vertical
lines with a symbol at the top refer to the processes discussed in Figs.2(a)
and 3(b) and (c). Simulations realized for $kr=0.05$,
$\theta=\cos^{-1}(1/\sqrt{3})$ and $\Omega=30\Gamma$.

Finally, one observes that the inequality is also violated for some emission
processes which involve real transitions, where the correlations between
emitted photons are classical (see $+$ or $\otimes$ symbols, for example, in
Fig.3(b)). In order to properly observe the classicality of these
correlations, as they correspond to photons of real transitions located in the
leapfrog lines, it is necessary to use sensor with a better resolution.
Indeed, the use of a sensor linewidth $\Gamma_{s}=\Gamma$ leads to an
averaging over processes with different kinds of correlations. To illustrate
this point, we show in Fig. 4(a) the change in CSI as the sensor linewidth is
changed from $\Gamma$ to $\Gamma/10$: the above-mentioned transitions for the
real photons do not violate any longer CSI, showing the classical nature of
their correlations Muñoz  _et al._ (2014).

Furthermore, as for single emitters Muñoz  _et al._ (2014), violations of CSI
may appear for transitions of the central antidiagonal line
($\omega_{1}+\omega_{2}=0$), even for real transitions and for well-resolved
frequencies: Fig. 4(b) presents such violations of CSI (see $\bullet$, $\ast$
and $\times$ symbols). This failure of CSI to detect classical correlations
can be addressed by using Bell inequality (BI), as monitored by a quantifier
adapted to the sensor approach Muñoz  _et al._ (2014):

$B_{s}=\sqrt{2}\left|\frac{B_{1111}+B_{2222}-4B_{1221}-B_{1122}-B_{2211}}{B_{1111}+B_{2222}+2B_{1221}}\right|,$
(20)

with
$B_{jklm}=\langle\xi_{1}^{\dagger}\left(\omega_{j}\right)\xi_{2}^{\dagger}\left(\omega_{k}\right)\xi_{2}\left(\omega_{l}\right)\xi_{1}\left(\omega_{m}\right)\rangle$.
Values $B_{s}>2$ are a violation of the BI, which are considered as a true
signature of quantum correlations. As can be seen in Fig. 3(c) for
${{\Gamma}_{s}}=\Gamma$, BI is violated only for specific areas of the central
antidiagonal line, yet not for the real transitions of $\bullet$ and $\ast$.
This behaviour is similar to the single-emitter case Muñoz  _et al._ (2014),
confirming that only transitions involving virtual states hold true quantum
correlations between the emitted photons. Note that BI and CSI are sensitive
to the frequency resolution of the sensors, as narrow linewidth sensors
correspond to long averaging time, which in turn washes out the correlations
Muñoz  _et al._ (2014).

As illustrated for the pair of photons
$(\pm{{\Delta}_{13}},\mp{{\Delta}_{13}})$, a sensor linewidth
$\Gamma_{s}=\Gamma$ presents a violation of BI, yet reducing the sensor
linewidth to $\Gamma/10$ removes the violation of the BI, see Fig. 4(c) where
the pair of $({{\Delta}_{13}},-{{\Delta}_{13}})$ indicated by $\times$. This
highlights again the narrow linewidth sensors correspond to a finer structure
for the quantum quantifiers and the necessity of using sensors with a
linewidth comparable to the atomic transition one, in order to detect the
stronger correlations.

##  IV Conclusion and perspectives

Strong interactions between two two-level emitters give rise to a series of
new sidebands in the fluorescence spectrum, whose shift from the atomic
transition depends on both the interaction strength and the driving field.
Similarly to the single emitters, the leapfrog processes with frequencies that
sums to zero or the frequency of one of the interaction-induced sidebands, are
characterized by strong correlations, which can be either classical or
quantum. This suggests that strongly coupled emitters are potential sources of
heralded photons, with extra control parameters through the interaction, as
compared to single emitters.

Another regime of interest is that of a weak dipole-dipole interaction, i.e.,
when the collective dressed levels are equally spaced in energy
($\Delta_{12}=\Delta_{23}$), which occurs when the distance between the
emitters is comparable or larger than the optical wavelength. In this regime,
the first correction to the single-atom fluorescence spectrum is the emergence
of sidebands shifted from the laser frequency by twice the Rabi frequency
($\omega=\pm 2\Omega$). This phenomenon was predicted to scale with the
resonant optical depth for dilute extended clouds, as a signature of the
raising two-atom quantum correlations Pucci  _et al._ (2017). We have
monitored photon-photon correlations $g^{(2)}(\omega_{1},\omega_{2})$ for a
pair of atoms at a distance of the order of $\lambda$ ($kr=0.4-1$ and
$\theta=\cos^{-1}(1/\sqrt{3})$) and strongly driven, yet photon-photon
correlations appear to be largely dominated by single-atom physics. This is
reminiscent of the anti-bunching phenomenon which vanishes for a large number
of independent emitters, unless specific conditions for their interference is
achieved Grangier  _et al._ (1986). Furthermore, the leapfrog processes
associated with the new sidebands, $\omega_{1}+\omega_{2}=\pm 2\Omega$,
present no violation of the Cauchy-Schwartz inequality (not shown here). This
suggests that despite these sidebands result from correlations between the
quantum fluctuations of the two dipoles Pucci  _et al._ (2017), the photons
associated with these processes may be only classically correlated.

The variety of sidebands and photon-photon correlations encountered for a pair
of atoms calls for a dedicated study for larger systems. Indeed, although the
coherent manipulation of atoms at scales below the diffraction limit is
experimentally challenging, schemes have been proposed to surpass these
limitations, based on the transparency-induced dark states Agarwal and Kapale
(2006); Cho (2007); Yavuz and Proite (2007); Gorshkov  _et al._ (2008), which
have already allowed for the generation of subwavelength cold atom structures
Miles  _et al._ (2013); Wang  _et al._ (2018); Subhankar  _et al._ (2019);
Tsui  _et al._ (2020). In this context, strongly-interacting cold atom
ensembles may be a promising tunable source for entangled pairs of photons,
but also for larger bunches of photons Carreño  _et al._ (2017).

## Acknowledgment

M. H., R. B. and C. J. V.-B. acknowledge funding from the French National
Research Agency (projects QuaCor ANR19-CE47-0014-01). E. D., R. B. and C. J.
V. -B. benefited from Grants from São Paulo Research Foundation (FAPESP,
Grants Nos. 2018/10813-2, 2018/01447-2, 2018/15554-5, 2019/13143-0, and
2019/11999-5) and from the National Council for Scientific and Technological
Development (CNPq, Grant Nos. 302981/2017-9, 409946/2018-4, and
307077/2018-7). M. H. and R. B. received support from the project CAPES-
COFECUB (Ph879-17/CAPES 88887.130197/2017-01).

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