# Entangled biphoton enhanced double quantum coherence signal as a probe for
cavity polariton correlations in presence of phonon induced dephasing

Arunangshu Debnath<EMAIL_ADDRESS>Max Planck Institute for the
Structure and Dynamics of Matter and Center for Free-Electron Laser Science,
Luruper Chaussee 149, 22761 Hamburg, Germany  Angel Rubio
<EMAIL_ADDRESS>Max Planck Institute for the Structure and Dynamics of
Matter and Center for Free-Electron Laser Science, Luruper Chaussee 149, 22761
Hamburg, Germany  Center for Computational Quantum Physics (CCQ), The Flatiron
Institute, 162 Fifth Avenue, New York NY 10010, United States

(August 27, 2024)

###### Abstract

We theoretically propose a biphoton entanglement-enhanced multidimensional
spectroscopic technique as a probe for the dissipative polariton dynamics in
the ultrafast regime. It is applied to the cavity-confined monomeric
photosynthetic complex that represents a prototypical multi-site excitonic
quantum aggregate. The proposed technique is shown to be particularly
sensitive to inter-manifold polariton coherence between the two and one-
excitation subspaces. It is demonstrated to be able to monitor the dynamical
role of cavity-mediated excitonic correlations, and dephasing in the presence
of phonon-induced dissipation. The non-classicality of the entangled biphoton
sources is shown to enhance the ultra-fast and broadband correlation features
of the signal, giving an indication about the underlying state correlations
responsible for long-range cavity-assisted exciton migration.

##  I Introduction

The quantum aggregates consisting of multiple centers of electronic
excitations, e.g., the naturally occurring light-harvesting photosynthetic
complexes, artificially designed molecular light-harvesters offer a uniquely
favorable testing ground for the entangled photon-induced dynamics. These
systems intrinsically host collective excitations, frequently termed as
molecular excitons, extending over several excitation centers/sites and often
within the coherence domain of the spatial(temporal) length(time) scales.
Thus, it provides opportunities for the external modulation of delocalized
excitons and testing the limits of coherent dynamics. Further, the vibrational
motions associated with these structures often give rise to a collective
dephasing mechanism for the delocalized excitons. Additionally, the
vibrational motions modulate the energy gradient of the kinetics, leading to
situations where coherent excitons undergo inter-site transport and eventually
localize. The spectroscopic investigation and control of such exciton kinetics
offers insights into the microscopic nature of the coherent energy transfer
mechanism and prescribes guidelines for bio-mimetic engineering which builds
on the operational equivalence Boulais  _et al._ (2018); Romero  _et al._
(2014); Scholes  _et al._ (2017); Schlau-Cohen  _et al._ (2012); Cao  _et al._
(2020); Scholes (2020). In a separate line of development, there have been a
series of studies that have demonstrated the effective role of external
electromagnetic cavities in manipulating material excitations. These studies
range from the control of electronic excitations Basov  _et al._ (2021);
Ruggenthaler  _et al._ (2018); Mewes  _et al._ (2020); Garcia-Vidal  _et al._
(2021); Cao (2022); Engelhardt and Cao (2021); Groenhof and Toppari (2018);
Latini  _et al._ (2021a); Lengers  _et al._ (2021); Autry  _et al._ (2020),
vibrational modes Sidler  _et al._ (2020); Yang and Cao (2021), collective
mode responses Salij and Tempelaar (2021); Latini  _et al._ (2021b) to the
cavity mode-assisted modulation of dynamical resonances Haugland  _et al._
(2020); Sidler  _et al._ (2021); Schäfer  _et al._ (2021a); Zhang  _et al._
(2019a, b). Cavity mode interacting resonantly with narrow-band of excitons
and off-resonantly with the rest may modulate the excitation dispersal by
tuning the delocalization properties and influence the dephasing properties
via the spectral weight modulationWang  _et al._ (2017); Maser  _et al._
(2016); Campos-Gonzalez-Angulo and Yuen-Zhou (2022). Hence, combining these
two developments to the case of light-harvesting quantum aggregates offers an
opportunity to investigate the role of the cavity in controlling a
prototypical, extended yet aperiodic system that hosts collective excitons.
Previous spectroscopic studies of the cavity-modulated dynamics Debnath and
Rubio (2020); DelPo  _et al._ (2020); Ribeiro  _et al._ (2021); Renken  _et
al._ (2021); Zhang  _et al._ (2021, 2019c, 2018) have increasingly focused
their attention on the ultrafast, nonlinear techniques analogs e.g. pump-
probe.

Figure 1: (a) Schematic description of the dynamical phenomena (in WMEL
diagrammatic convention Lee and Albrecht (1985)) that can be addressed by the
proposed DQC signal measurement technique. The first two interactions (in red)
are common to both the pathways. The last two interactions (in black and gray)
are distinguished by two alternative possibilities indicated by
$\\{k_{s}^{(I)},-k_{3}^{(I)}\\}$, $\\{k_{s}^{(II)},-k_{3}^{(II)}\\}$ pair.
Entangled photons affect both these pair interactions $\\{k_{1},k_{2}\\}$ and
$\\{-k_{3},k_{s}\\}$.(b) Associated Keldysh-Schwinger diagrams that describe
the corresponding dynamical pathways in the multi-polariton basis contributing
to the signal and which are distinct in the last two interactions in time.

In the majority of these studies, participation of the higher-order
excitations remain relatively unexplored. The dynamics involving the higher-
order nonlinear excitations introduce possibilities for cavity-modulated
exciton-exciton annihilation, cavity-assisted exciton fusion, and many more
correlated mechanisms of practical interest. On a fundamental level, the
external dielectric modification of the Coulomb interaction between different
excitons underlies all of these coherent mechanisms. However, the nonlinear
spectroscopic signatures studied so far depend dominantly on the associated
vibrational (or vibronic) processes Tempelaar  _et al._ (2017); Gutiérrez-Meza
_et al._ (2021). Therefore, in order to explore the complexity of cavity-
modulated nonlinear exciton dynamics, it is desirable to combine ultrafast
spectroscopic tools that are sensitive to high-order exciton correlations in
the presence of collective vibrational dephasing. Among the multi-pulse
nonlinear spectroscopic techniques available for mapping out the correlation
between two excitons and probing the two-exciton state-specific dephasing in
the ultrafast regime, double-quantum multidimensional correlation spectra
(DQC) have been proven useful Mukamel (1999); Lomsadze and Cundiff (2020); Gao
_et al._ (2016); Kim  _et al._ (2009). The ultrafast nature of the exciton
dynamics in aggregates results in difficulty in measuring the role of
energetically-distant states even without the presence of the cavity mode.
These delocalized multi-exciton states often participate in dynamics within
the same timescale due to dynamical dephasing properties. Associated with the
normal exciton-number conserving dephasing process, one encounters a dephasing
mechanism that couples the energy-manifolds with the different number of
excitons. The presence of cavity adds additional cavity-exciton hybrid states,
introduces multiple dynamical timescales, and gives rise to novel cavity-
exciton-phonon coupling mechanisms. These excitations are of principal
interest in this communication. The investigation of them requires the
deployment of probes that specifically excite spectrally narrow-band states
(creation of non-linear polarization), allow the associated dynamics to evolve
in time in the presence of dephasing (evolution of the polarization), and
project them to desired frequency components of interest (projection of
polarization to signal components). Robust, correlated state excitation can be
controllably achieved by deploying the entangled photon pairs, namely
entangled biphotons, via a scheme being referred to as entangled two-photon
absorption Svozilík  _et al._ (2018); León-Montiel  _et al._ (2019); Kang  _et
al._ (2020); Schlawin  _et al._ (2018); Saleh  _et al._ (1998). In comparison
to the shaped laser pulses, the entangled biphotons have been shown to improve
the spectral resolution while scaling favorably with the intensity of the
sources. The non-classical correlation properties can also be utilized to
obtain favorable spectral-temporal resolution in the probing via selective
state projections of the nonlinear polarization Richter and Mukamel (2010);
Roslyak and Mukamel (2009); Zhang  _et al._ (2021); Mukamel  _et al._ (2020);
Landes  _et al._ (2021); Debnath and Rubio (2020); Bittner  _et al._ (2020);
Schlawin and Buchleitner (2017); Li  _et al._ (2019); Richter  _et al._
(2018). The latter requires the selection of a few states whose correlation
properties are of particular interest from a manifold. In this regard, a
combination of the aforementioned DQC signal measurement scheme with the
entangled photon sources may provide a technique that studies cavity-modulated
correlated exciton kinetics in the ultrafast regime with higher spectral
resolution.
In what follows, in section II, we introduce the Frenkel exciton Hamiltonian
and describe the underlying model. It will be used to describe the dissipative
exciton-polariton phenomenology and obtain the relevant Green’s functions.
Subsequently, in section II we introduce the DQC signal in a modular manner,
discuss the nature of the signal and present the simulation results in the
relevant parameter regime. Section IV discusses limitations, the scope of the
presented signal within the broader scope of entangled photon-enhanced
spectroscopies, and the outlook.

##  II Dissipative exciton-polariton phenomenology

Here we present the exciton Hamiltonian concurrently interacting with a cavity
mode and the phonon reservoir, which will be used to build up the
phenomenology using a quasi-particle picture. It is given by,

$\displaystyle H$
$\displaystyle=\sum_{m,n=1}^{N_{s}}(E_{m}\delta_{mn}+J_{mn})B_{m}^{\dagger}B_{n}+$
$\displaystyle\sum_{m,n=1}^{N_{s}}\frac{U_{m}^{(2)}}{2}B_{m}^{\dagger}B_{m}^{\dagger}B_{m}B_{m}+\frac{U_{mn}^{(2)}}{2}B_{m}^{\dagger}B_{n}^{\dagger}B_{m}B_{n}$
$\displaystyle+\sum_{\alpha}\omega_{c}(a_{\alpha}^{{\dagger}}a_{\alpha}+1/2)+\sum_{m,\alpha}g_{c,m,\alpha}(a_{\alpha}B_{m}^{\dagger}+a_{\alpha}^{\dagger}B_{m})$
$\displaystyle+\sum_{j}\omega_{j}(b^{\dagger}_{j}b_{j}+\frac{1}{2})+\sum_{m,j}g_{m,j}(b_{j}^{\dagger}+b_{j})B_{m}^{{\dagger}}B_{m}$
(1)

where we set the $\hbar$ to unity. The components of the Hamiltonian are
explained below.

###  II.1 Exciton, Cavity and exciton-cavity interactions

The first three terms constitute the bare exciton Hamiltonian where
$B_{m}(B_{m}^{\dagger})$ are the m-th site exciton creation (annihilation)
operators with the respective commutation relation,
$[\,B_{m},B_{n}\,]=[\,B_{m}^{\dagger},B_{n}^{\dagger}\,]=0$ and
$[\,B_{m},B_{n}^{\dagger}\,]=\delta_{mn}(1-(2-\kappa_{m}^{2}))B^{\dagger}_{m}B^{\dagger}_{m})$
(where $\kappa_{m}=\mu_{21}/\mu_{10}$, is given in terms of ratio of
transition dipoles of $ij$-th level of the $m$-th site) Agranovich and Toshich
(1968); Chernyak  _et al._ (1998); Abramavicius  _et al._ (2009).

Figure 2: The progression of dephasing-broadened bare polariton band with the
cavity frequencies and a fixed value of cavity coupling strength are
presented. For the dephasing a representative set of values (generated from a
random Gaussian distribution of mean and variance of $\mu_{1}=25$,
$\sigma_{1}=10$ and $\mu_{2}=35$ respectively) are chosen for clarity. In the
displayed one-polariton (left panel) and the two-polariton band (right panel),
the increase in the density of states in the latter within a comparable
energy-window (e.g., $\approx 10\times 10^{3}\mathrm{cm^{-1}}$) is noticeable.
The fundamental cavity frequencies along the x-axis are given in terms of
detuning from the central region of the one-polariton band.

$E_{m}^{(1)}$ is the on-site excitation energies and $J_{mn}$ is the inter-
site Coulomb-mediated hopping. The presence of the interacting terms in the
exciton Hamiltonian indicates that the single and two exciton states are
delocalized over sites. The two-excitation manifold which is composed of
permutable composition of pure local and non-local two-exciton, pure two
cavity excitations and joint one exciton-one photon excitation is given by,
$H_{mnkl}^{(2)}=H_{mn}^{(1)}\delta_{kl}+\delta_{mn}H_{kl}^{(1)}$. The terms
$U_{m}^{(1)},U_{mn}^{(2)}$ further dictates the interaction among higher-order
excitonic states which gives rise to static, local energy shifts, exciton-
exciton scattering. The values of $U_{m}^{(1)}$ is taken as
$U_{m}^{(1)}=2(\kappa_{n}^{-2}E_{n}^{(2)}-E_{n}^{(1)})$, with
$\kappa=\mu_{21}/\mu_{10}$ which accounts for the local energy shifts for two-
exciton states and $U_{mn}^{(2)}=0$. Bare two-exciton energy is given by
$E_{m}^{(2)}=2E_{m}^{(1)}+\Delta_{m}$ (where the $\Delta_{m}$ is the two-
exciton state anharmonicity from the Harmonic limit). The parameter values for
the Hamiltonian are obtained from the semi-empirical simulation in
Novoderezhkin  _et al._ (2005, 2004, 2011); van Amerongen and van Grondelle
(2001) where these data-set have been shown to simultaneously fit the spectral
data obtained from the linear absorption, fluorescence, and circular dichroism
spectra. The combined cavity and the cavity-exciton interaction Hamiltonian is
be given by the fourth and fifth term where the $g_{c;m,\alpha}$ is the
cavity-exciton coupling strengths describing the resonant dipolar interaction.
The effects arising from the non-resonant cavity-exciton interaction could be
significant in the static limit if the exciton-cavity coupling strengths are
comparable to the bare Rabi frequencies of the multi-exciton system De
Liberato (2014); Schäfer  _et al._ (2020); Keeling (2007); Vukics  _et al._
(2014). The cavity-exciton parameter considered in this communication
corresponds to moderately weak and within the close to resonant regime which
allows this term being neglected. The cavity photonic modes are described in
the oscillator basis with the corresponding creation (annihilation) operators
denoted as $a_{\alpha}^{\dagger}(a_{\alpha})$, for the $\alpha$-th mode.
Within the scope of this study, a single-mode limit (i.e. $\alpha=1$) and two-
excitation per-mode (with he fundamental frequency denoted by $\omega_{c}$ )
has been considered. Furthermore, we assume an uniform cavity-exciton coupling
strengths for all sites, i.e. $g_{c;m,\alpha}=g_{c}=100\mathrm{cm}^{-1}$ is
independent of index $m$, although the analysis presented is not limited by
that choice. We note that a detailed estimation of the cavity-exciton
couplings requires investigation regarding the mode volume of the cavity and
the quality factor which is beyond the scope of the article. The Coulombic
origin inter-site hopping in the Hamiltonian is of different magnitude,
indicating a different propensity for delocalization in the absence of cavity
interaction. The cavity coupling modulates site-delocalization at different
extents for one-exciton, local and non-local two-excitons. This gives rise to
the possibility of creating a different admixture of cavity-matter
excitations. In addition, it offers cavity mediates coupling between these
configurations, thereby coupling diverse sets of excitations and inter-site
processes. Since the cavity accommodates two-photon excitations, the analysis
is capable of describing resonant two-polariton processes.

###  II.2 Phonon and exciton-phonon interactions

The exciton-phonon interactions originate from the inter and intra-molecular
vibrational motions associated with the relative nuclear motions of the
aggregate. Normal modes of the low-energy vibrational degrees of freedom
related to the collective vibrational coordinates are assigned as phonon modes
and are mapped onto an infinite set of Harmonic oscillators. It is given by
the free phonon Hamiltonian appearing in the sixth term where $\upsilon_{k}$
is the mode frequency associated with the $k$-th normal mode whose
creation(annihilation) operators are denoted via $b_{j}^{\dagger}(b_{j})$. The
phonon operators follow the free-boson commutation relations,
$[\,b_{j},b_{j^{\prime}}\,]=[\,b_{j}^{\dagger},b_{j^{\prime}}^{\dagger}\,]=0$
and $[\,b_{j},b_{j^{\prime}}^{\dagger}\,]=\delta_{jj^{\prime}}$. The seventh
term presenting the exciton-phonon interactions is taken in the site-
uncorrelated, local form and is characterized by the distribution of the
corresponding coupling functions, $\bar{g}_{m,j}$. These phonon modes are
responsible for exciton dephasing, relaxation and exciton-transport phenomena.
The extent of these phenomena are governed by the site-dependent exciton-
phonon coupling strengths $g_{m,j}$ which are taken as $1(1.4)$ for the sites
identified to represent Chl-A (Chl-b). While constructing the two-exciton
phonon interaction Hamiltonian these coupling strengths are taken as
$0.6(g_{m,j}+g_{n,j})$. Due to the differential coupling strengths, the
excitations undergo dephasing at different rates, the one and two-exciton
transport occur to a different extent over the sites leading to time-windowed
interferences at a broader energy window. The inter-band dephasing, a quantity
of particular interest in this article has contributions from the elements of
the relaxation tensor that also participates in the cavity mediated one-
exciton and two-exciton transport. Within this model, the single and the
double excitons are coupled to a common set of phonon modes which can be
characterized by the discrete distribution function,
$J_{0}(\omega)=\pi\sum_{j}|\bar{g}_{j}|^{2}(\delta(\omega-\upsilon_{j})-\delta(\omega+\upsilon_{j}))$
from which the spectral density function is obtained in the continuum
frequency limit. The spectral density function, presented as,
$J(\omega)=2\lambda_{0}(\gamma_{0}\omega)/(\omega^{2}+\gamma_{0}^{2})+\sum_{j=1}^{N_{b}}2\lambda_{j}(\upsilon_{j}^{2}\gamma_{j}\omega)/((\upsilon_{j}^{2}-\omega^{2})^{2}+\omega^{2}\gamma_{j}^{2})$
describes $N_{b}=48$ multi-mode Brownian oscillator modes and one over-damped
oscillator mode. The number of multi-mode Brownian oscillators ($N_{b}$), the
respective spectral shift parameters
($\lambda_{0},\lambda_{j}=\upsilon_{j}\Upsilon_{j}$ where $\upsilon_{j}$ and
$\Upsilon_{j}$ are the $j$-th oscillator frequencies and Huang-Rhys parameters
respectively), and the damping parameter ($\gamma_{0},\gamma_{j}$) which are
required to optimally describe the equilibrium spectral density function have
been obtained from the work of Novoderezhkin et al. Novoderezhkin  _et al._
(2004, 2005, 2011); Zhang  _et al._ (2018). The numerical values corresponding
to those parameters are enlisted in Appendix A.2.

###  II.3 Polariton manifolds

The polariton states are obtained as number-conserving manifolds via exact-
diagonalization of the field-free Hamiltonian subspaces,
$\tilde{H}_{p}^{(n)}=\tilde{H}_{s}+\tilde{H}_{c}+\tilde{H}_{sc}$ by using
$T_{p}^{(n),-1}\tilde{H}_{p}^{(n)}T_{p}^{(n)}=H_{p}^{(n)}=\sum_{n=0,1,2}E_{p^{(n)}}X_{p^{(n)}p^{(n)}}$,
where we define operators $X_{p^{(n)}p^{(n)}}=|p^{(n)}\rangle\langle p^{(n)}|$
as projectors onto the eigenstates of the polariton Hamiltonian $H_{p}^{(n)}$
(denoted by index $n$). The three distinct manifolds thus obtained corresponds
to ground $n=0$, single $n=1$ and double $n=2$ polariton manifolds with
$N_{p^{(0)}}=1$, $N_{p^{(1)}}=15$, $N_{p^{(2)}}=120$ polariton states
respectively. The manifolds are composed of the cluster of states (Fig 2)
which depend on an interplay of the hopping and cavity coupling parameters.
The proposed spectroscopic technique utilizing entangled biphotons, in the
parameter regime of interest, interact only via two-quantum interactions,
which allows the joint-excitation manifold to be truncated at the level of the
double polaritons. For the simulation, we have chosen the cavity mode
frequency and the coupling strength parameters as
$\omega_{c}(\mathrm{cm^{-1}})=15.4\times 10^{3}$ and
$g_{c}(\mathrm{cm^{-1}})=0.1\times 10^{3}$. The cavity frequency is resonant
with the narrow frequency band around the frequency and off-resonant with the
rest of the cluster of states in the bare absorption spectra of the excitonic
system. The cavity coupling results in the polariton states to encompass a
wider range than the cavity-free counterpart with the close-to-resonance
states becoming affected to a greater degree. It can also be noted that the
two-polariton states accommodate more states within the comparable bandwidth
which is determined by the cavity coupling parameter. These two observations,
combined, affirm that the exciton-cavity hybridization dominantly takes around
the resonances, as expected. The spectral weights of the resultant delocalized
polariton states have respective contributions from both the exciton and
cavity modes, as determined by the exciton-cavity coupling matrix elements.

###  II.4 Polariton-laser interactions

The interaction between the external field i.e. biphoton sources and
polaritons are treated within the optical dipolar interaction limit and within
the rotating wave approximation. Corresponding Hamiltonian is given by,

$\displaystyle H_{\mathrm{int}}(t)$
$\displaystyle=\sum_{j}\sum_{n,n^{\prime}=0,1,2;n\neq
n^{\prime}}(\sqrt{2\pi\omega_{j}/V})a_{j}e^{-i\omega_{j}t}d_{p^{(n)}p^{(n^{\prime})}}$
$\displaystyle
X_{p^{(n)}p^{(n^{\prime})}}\exp{(i\omega_{p^{(n^{\prime})}p^{(n)}}t)}+\mathrm{h.c.}$
(2)

where external photon mode creation operators $a_{j}$ and dipole-weighted
inter-manifold polariton transition operators, $X_{p^{(n)}p^{(n^{\prime})}}$
were defined. Also, the mode quantization volume $V$ and the Fourier expansion
frequency $\omega_{j}$ for the photon modes have been introduced. These
operators will be used to derive the signal expressions in the next sections.
It is notable that in the spirit of the weak laser driving limit suitable for
spectroscopy only the one-photon transition operators had been employed.

###  II.5 Phonon induced state broadening and polariton Green’s functions

In this section, we introduce the framework to obtain the state-dependent
dephasing timescales and obtain the polariton Green’s functions required for
the signal expressions. It is carried out by seeking an integral solution of
the generalized master equation (written in the multi-polariton basis)
obtained in the Markovian and the secular limit (often termed as the Redfield
equation). The secular approximation limits the polariton-phonon mode
interactions to be describable within the resonant cases only. The latter
suffices our treatment of acoustic phonons mediate interactions are mediated
by displacive perturbations of exciton states simultaneously dressed by the
cavity interactions that are comparatively much stronger. The kinetic equation
is given by,
$\dot{\sigma}^{(n)}(t)=-iL_{S}^{(n)}\sigma^{(n)}(t)+\int_{0}^{\infty}d\tau\Sigma^{(n)}(t,\tau)\sigma^{(n)}(t)$
where we denote $\sigma^{(n)}$ as number resolved reduced multi-polariton
density operator.We defined the super-operator as,
$L_{s}^{(n)}O=[H_{p}^{(n)},O]$. The polariton-phonon memory kernel capable of
describing the phonon-induced polariton relaxation and dephasing (neglecting
the effects of the driving field on the relaxation) is given by,
$\Sigma^{(n)}(t,\tau)\sigma^{(n)}(t)=-\lambda^{2}\bigg{[}X,D(t)\sigma_{s}(t)-\sigma_{s}(t)D^{\dagger}(t)\bigg{]}$.
The kernel contains the dissipation operator
$D(t)=X(t-t^{\prime})C_{b}(t-t^{\prime})$ where we have the convolution of
polariton-phonon coupling operator
$X(\tau)=\sum_{p^{(n)}}X_{p^{(n)}p^{(n)}}(\tau)$ and time-domain correlation
function given by,

$\displaystyle
C_{b}(\tau)=(\lambda_{0}\gamma_{0}/2)\cot(\beta\gamma_{0}/2)\exp(-\gamma_{0}\tau)+\sum_{j=1}^{N_{b}}\frac{\lambda_{j}}{2\zeta_{j}}$
$\displaystyle\Big{(}\coth(i\beta\phi_{j}^{+}/2)\exp(-\phi_{j}^{+}\tau)-\coth(i\beta\phi_{j}^{-}/2)\exp(-\phi_{j}^{-}\tau)\Big{)}$
$\displaystyle+(-i\lambda_{0}\gamma_{0}/2)\exp(-\gamma_{0}\tau)+\frac{i\lambda_{j}\upsilon_{j}^{2}}{2\zeta_{j}}\Big{(}\exp{(-\phi_{j}^{+}t)}-$
$\displaystyle\exp{(-\phi_{j}^{-}t)}\Big{)}-\sum_{n=1}^{\infty}\Big{(}(4\lambda_{j}\gamma_{j}\upsilon_{j}^{2}/\beta)(\nu_{n}/(\upsilon_{j}^{2}+\nu_{n}^{2})^{2}-\nu_{n}^{2}\gamma_{j}^{2})$
$\displaystyle+(2\lambda_{0}\gamma_{0}/\beta)(\nu_{n}/(\nu_{n}^{2}-\gamma_{0}^{2}))\Big{)}\exp(-i\nu_{n}\tau)$
(3)

In the above we have $\zeta_{j}=\sqrt{(\upsilon_{j}^{2}-\gamma_{j}^{2}/4)}$,
$\phi_{j}^{\pm}=(\gamma_{j}/2)\pm i\zeta_{j}$, and, matsubara frequencies,
$\nu_{n}=n(2\pi/\beta)$ (with $n_{m}=20$). Furthermore, we defined
$\beta=1/\kappa T$ where the Boltzmann constant $\kappa$ and temperature $T$
is denoted. The energy domain version of the relaxation kernel was used to
extract the dephasing parameters. We define the line-broadening functions as,
$\gamma_{p_{1}^{(n)}}(t)=\theta(t)\sum_{p_{2},p_{s}}C_{b}(\omega_{p_{1}p_{2}})T_{p_{s}p_{2}}^{(n)}T_{p_{s}p_{2}}^{(n)}T_{p_{s}p_{1}}^{(n)}T_{p_{s}p_{1}}^{(n)}$
where we have $C_{b}(\Omega)=\int dt\exp{i\Omega t}C^{\pm}(t)$ and
$C_{b}^{\pm}(t)=\int(d\omega/2\pi)(\coth{\beta\omega/2})\cos{\omega t}\mp
i\sin{\omega t})$ which can be evaluated using the previous expressions. With
the help of the line-broadening functions we estimate the inter-manifold
dephasing parameters as,
$\gamma_{p_{1}^{(n)}p_{2}^{(m)}}=(\gamma_{p_{1}^{(n)}}+\gamma_{p_{2}^{(m)}})/2$.
Finally, we obtain the inter-manifold Greens functions relevant for dephasing
as,
$G_{p_{1}^{(n)}p_{2}^{(m)}}(\omega)=i(\omega-\omega_{p_{1}^{(n)}p_{2}^{(m)}}-i\gamma_{p_{1}^{(n)}p_{2}^{(m)}})^{-1}$.
The advanced Green’s functions are defined likewise.

Figure 3: Field correlation functions for different entangled photon states
corresponding to parameter variation of temporal entanglement parameter
$T_{\mathrm{ent}}(\mathrm{fs})$ and pump width $\tau_{p}(\mathrm{fs})$ for
uniform pump frequency $31.0\times 10^{3}\mathrm{cm}^{-1}$ and biphoton
frequencies $\omega_{1}=\omega_{2}=15.5\times 10^{3}\mathrm{cm}^{-1}$. The
bottom right corresponds to the typical case of a classical field two-photon
pulse. (A) corresponds to the $T_{\mathrm{ent}}=10;\tau_{p}=20$, (B)
corresponds to $T_{\mathrm{ent}}=10;\tau_{p}=50$ while (C) assumes values
$T_{\mathrm{ent}}=50;\tau_{p}=20$. The (D) corresponds to a Gaussian pair-
pulse with $\tau_{g}(\mathrm{fs})=10$ with
$\omega_{1,g}=\omega_{2,g}=15.5\times 10^{3}\mathrm{cm}^{-1}$

##  III Multidimensional double-quantum coherence signal with entangled
biphoton sources

The theoretical development mentioned in the preceding sections provides us
the ingredients to introduce the Double Quantum Coherence (DQC) signal. The
DQC signal, as hinted earlier, involves excitation of state-selective (or
narrow-band around the selected state) two-polariton coherence that is
followed by projection of the oscillating polarization components onto two
plausible sets of inter-manifold coherence. Provided that the participant two-
polariton states are sufficiently correlated with the states onto which they
are being projected, the desired nonlinear polarization have dominant
component in the phase-matched direction $k_{s}=k_{1}+k_{2}-k_{3}$. Below we
present a modular derivation of the signal starting from the time-dependent
dual-perturbation scheme, corroborate the derivation to the Keldysh-Schwinger
loop diagrams Fig 1 and introduce the entangled biphoton sources that serves
as probe.

###  III.1 Double quantum coherence signal

The DQC signal is typically generated by inducing four external field-matter
interactions. In the case of time-domain (frequency-domain) classical field
sources, three external fields with controllable delays (relative phases) are
allowed to interact with the matter. The radiation field emitted by the time-
dependent nonlinear polarization is registered, typically, via suitable
heterodyning after another delay (spectrally dispersing) the signal.
Typically, the deployment of quantum fields e.g., the biphoton sources to the
measurement of DQC signals requires considerable care. These field sources are
often parametrically scanned via schemes akin to multi-pulse phase-cycling. An
involved discussion regarding the details of the deployment and measurement
scheme is beyond the scope of the article Dorfman and Mukamel (2014); Raymer
_et al._ (2021). However, we assume that the biphoton generation scheme is
capable of producing two sets of entangled photon pairs that lend themselves
to external manipulation via central frequencies and delays. Before proceeding
further, we introduce the two sets of pathways involved in the DQC signal
generation as,

$\displaystyle\Pi_{a}(\tau_{4},\tau_{3},\tau_{2},\tau_{1})$
$\displaystyle=d_{p_{i}^{(0)}p_{j^{\prime}}^{(1)}}(\tau_{4})d_{p_{j^{\prime}}^{(1)}p_{k}^{(2)}}(\tau_{3})$
$\displaystyle
d_{p_{k}^{(2)}p_{j}^{(1)}}(\tau_{2})d_{p_{j}^{(1)}p^{(0)}}(\tau_{1})$
$\displaystyle\Pi_{b}(\tau_{4},\tau_{3},\tau_{2},\tau_{1})$
$\displaystyle=d_{p_{i}^{(0)}p_{j^{\prime}}^{(1)}}(\tau_{3}^{\prime})d_{p_{j^{\prime}}^{(1)}p_{k}^{(2)}}(\tau_{4}^{\prime})$
$\displaystyle
d_{p_{k}^{(2)}p_{j}^{(1)}}(\tau_{2}^{\prime})d_{p_{j}^{(1)}p^{(0)}}(\tau_{1}^{\prime})$
(4)

where the first and the second one have been written in the Heisenberg
representation and correspond to diagrams in Fig 1). These pathways, notably,
differing in the last two components signify dynamical spectral weights.
Interference features between these pathway components depend on the nature of
the polariton correlation and dephasing properties. We also introduce the
time-domain, four-point external field correlation function as
$D(\tau_{4},\tau_{3},\tau_{2},\tau_{1})$ which is capable of incorporating the
generalized nature of the external field. With the help of these definitions,
we present the signal expression in the time-domain as,
$S=C_{\mathrm{s}}\mathrm{Im}\prod_{i\in
a,b}\int_{\infty}^{\infty}d\tau_{4}^{(i)}d\tau_{3}^{(i)}d\tau_{2}^{(i)}d\tau_{1}^{(i)}\theta(\tau_{43}^{(i)})\theta(\tau_{32}^{(i)})\theta(\tau_{21}^{(i)})\times
D^{(i)}(\tau_{4}^{(i)},\tau_{3}^{(i)},\tau_{2}^{(i)},\tau_{1}^{(i)})\Pi_{i}(\tau_{4}^{(i)},\tau_{3}^{(i)},\tau_{2}^{(i)},\tau_{1}^{(i)})$
where $C_{\mathrm{s}}$ represents the coefficients that arises from the
perturbative expansion and have been taken as a scaling factor. The Heaviside
functions are defined in reference to the interaction times denoted along the
loop diagrams. In going forward, the interaction times i.e. the loop time-
instance variables were transformed to the the loop-delay variables and mapped
onto the real-time parameters that are externally tunable using using,
$\theta(s_{3})\theta(s_{2})\theta(s_{1})\rightarrow\theta(t_{3})\theta(t_{2})\theta(t_{1})$
for the diagram I and
$\theta(s_{3})\theta(s_{2})\theta(s_{1})\rightarrow\theta(s_{3})\theta(s_{2}-s_{3})\theta(s_{1})=\theta(t_{3})\theta(t_{2})\theta(t_{1})$
for the diagram II. These parameters are experimentally realizable. We also
use the time-domain phonon-averaged Green’s functions expanded in the multi-
polariton basis and write the field correlation function in the frequency
domain. Following these exercises, we obtain the generalized signal as,

$\displaystyle S$
$\displaystyle=C_{s}\int_{-\infty}^{\infty}dT_{1}dT_{2}dT_{3}F(\Omega_{3},\Omega_{2},\Omega_{1};T_{3},T_{2},T_{1})$
$\displaystyle\int_{\infty}^{\infty}\frac{d\tilde{\omega}_{3}}{2\pi}\frac{d\tilde{\omega}_{2}}{2\pi}\frac{d\tilde{\omega}_{1}}{2\pi}e^{-i(-\tilde{\omega}_{3}+\tilde{\omega}_{2}+\tilde{\omega}_{1})T_{3}-i(\tilde{\omega}_{1}+\tilde{\omega}_{2})T_{2}-i\tilde{\omega}_{1}T_{1}}$
$\displaystyle\langle
E_{3}^{\dagger}(\tilde{\omega}_{3})E_{4}^{\dagger}(+\tilde{\omega}_{1}+\tilde{\omega}_{2}-\tilde{\omega}_{3})E_{2}(\tilde{\omega}_{2})E_{1}(\tilde{\omega}_{1})\rangle\sum_{p_{j}^{(1)},p_{j^{\prime}}^{(1)},p_{k}^{(2)}}$
$\displaystyle
w^{(1)}G_{p_{k}^{(2)}p_{j^{\prime}}^{(1)}}(\tilde{\omega}_{1}+\tilde{\omega}_{2}-\tilde{\omega}_{3})G_{p_{k}^{(2)}p^{(0)}}(\tilde{\omega}_{2}+\tilde{\omega}_{1})$
$\displaystyle\times G_{p_{j}^{(1)}p^{(0)}}(\tilde{\omega}_{1})+$
$\displaystyle\langle
E_{3}^{\dagger}(\tilde{\omega}_{3})E_{4}^{\dagger}(+\tilde{\omega}_{1}+\tilde{\omega}_{2}-\tilde{\omega}_{3})E_{2}(\tilde{\omega}_{2})E_{1}(\tilde{\omega}_{1})\rangle\sum_{p_{j}^{(1)},p_{j^{\prime}}^{(1)},p_{k}^{(2)}}$
$\displaystyle
w^{(2)}G_{p_{j^{\prime}}^{(1)}p^{(0)}}(-\tilde{\omega}_{3}+\tilde{\omega}_{2}+\tilde{\omega}_{1})G_{p_{k}^{(2)}p^{(0)}}(\tilde{\omega}_{2}+\tilde{\omega}_{1})$
$\displaystyle\times G_{p_{j}^{(1)}p^{(0)}}(\tilde{\omega}_{1})$ (5)

This expression is valid for a general class of DQC signal measurement which
may use different kinds of external field sources beyond biphotons and simple
Gaussian classical fields. The field correlation function acts as a
convolutional probing function for the bare signal. Additional possibilities
for the external manipulation of the field correlation function extend the
applicability of DQC signals to a wide range of scenarios.

Figure 4: Multidimensional correlation plots ((a)-(f)) for signal with
parametric variation of the frequencies and the temporal entanglement
parameters of the entangled biphotons. The short-lived, intermediate one-
polariton resonances along the $\Omega_{3}$ axis can be explored and
associated with the two-polariton resonances along the $\Omega_{2}$ axis. For
parameters corresponding to each plot see text

Further, we aim to introduce a two-dimensional frequency-domain representation
of the signal in order to facilitate a visualization of the correlation
features contained in the matter correlation functions. In order to allow such
representations to be generated via real-time delay-scanning protocols, we
introduce integral transform,
$F(\Omega_{3},\Omega_{2},\Omega_{1};T_{3},T_{2},T_{1})=\theta(T_{3})\theta(T_{2})\theta(T_{1})e^{i\Omega_{3}T_{3}}e^{i\Omega_{2}T_{2}}e^{i\Omega_{1}T_{1}}$.
We also define the following variable mapping of the parameters,
$\tilde{\tau}^{0}_{21}\rightarrow T_{1},\tilde{\tau}^{0}_{32}\rightarrow
T_{2},\tilde{\tau}^{0}_{43}\rightarrow T_{3}$, for the diagram I and
$\tau^{0}_{21}\rightarrow T_{1},\tau^{0}_{42}-\tau^{0}_{43}\rightarrow
T_{2},\tau^{0}_{43}\rightarrow T_{3}$, for the diagram II, (the
$\tau_{ij}^{0}$ is the delays between the centering times). Scanning of these
set of parameters and obtaining joint Fourier transforms w.r.t the delays
generate the desired two-dimensional correlation plots. The final expression
can be presented as the following expression,

$\displaystyle
S(\Omega_{3},\Omega_{2},\Omega_{1})=C_{s}\sum_{p^{(0)}p_{j}^{(1)}p_{j^{\prime}}^{(1)}p_{k}^{(2)}}w^{(1)}\tilde{F}_{\Omega}^{(1)}$
$\displaystyle\langle
E_{4}^{\dagger}(z_{p_{k}^{(2)}p_{j^{\prime}}^{(1)}})E_{3}^{\dagger}(z_{p_{j^{\prime}}^{(1)}p^{(0)}})E_{2}(z_{p_{k}^{(2)}p_{j}^{(1)}})E_{1}(z_{p_{j}^{(1)}p^{(0)}})\rangle$
$\displaystyle+w^{(2)}\tilde{F}_{\Omega}^{(2)}$ $\displaystyle\langle
E_{4}^{\dagger}(z_{p_{k}^{(2)}p_{j^{\prime}}^{(1)}})E_{3}^{\dagger}(z_{p_{j^{\prime}}^{(1)}p^{(0)}})E_{2}(z_{p_{k}^{(2)}p_{j}^{(1)}})E_{1}(z_{p_{j}^{(1)}p^{(0)}})\rangle$
(6)

where the functions are specified as,

$\displaystyle\tilde{F}_{\Omega}^{(1)}$
$\displaystyle=\\{(\Omega_{3}-z_{p_{2}^{(1)}p^{(0)}})(\Omega_{2}-z_{p^{(2)}p^{(0)}})(\Omega_{1}-z_{p_{1}^{(1)}p^{(0)}})\\}^{-1}$
$\displaystyle\tilde{F}_{\Omega}^{(2)}$
$\displaystyle=\\{(\Omega_{3}-z_{p^{(2)}p_{{}^{\prime}}^{(1)}})(\Omega_{2}-z_{p^{(2)}p^{(0)}})(\Omega_{1}-z_{p^{(1)}p^{(0)}})\\}^{-1}$
(7)

These functions encode the polariton dynamical resonances. These resonances
show up, as predicted, during the scan of the Fourier transformed parameter.
The field correlation function encodes the information about the ability to
manipulate the spectral weights of the matter excitations and reveal desired
dynamical resonances.

###  III.2 Entangled biphoton properties

The principal aim of using the entangled biphoton sources is to avail the non-
classical relation between the joint time of arrival and frequency pairs of
the biphotons. This in turn allows one to excite relatively short-lived two-
polariton states (i.e., within an ultra-short time window) that are outside
the excitation energy window of the classical two-photon laser pulses. These
constraints remain difficult to surpass via independent variable manipulation,
even in the case of multiple classical pulses. Below we present some basic
features of the entangled photon source properties that were used in the
simulation and describe their correlation features. The entangled biphoton
field is traditionally generated via the spontaneous parametric down-
conversion (SPDC) process (in the weak down-conversion limit) by pumping the
source material with an ultra-short classical laser. The pump pulse bandwidth
and the central frequency determine the correlation properties and time-
frequency regime of the generated pairs. An effective Hamiltonian procedure
which has been used to derive the correlation properties as outlined
previously Saleh  _et al._ (1998); Cutipa and Chekhova (2022); Andersen  _et
al._ (2016); Arzani  _et al._ (2018) is avoided here for succinctness.
Following a similar derivation, the entangled biphoton field correlation
function can be obtained as, $\langle
E^{{\dagger}}(\omega_{4})E^{{\dagger}}(\omega_{4})E(\omega_{2})E(\omega_{1})\rangle=F_{1}^{*}(\omega_{4},\omega_{3})F_{1}(\omega_{2},\omega_{1})$
where we have,
$F_{1}(\omega_{a},\omega_{b};\omega_{p})=A_{0}(\omega_{a},\omega_{b})\Big{\\{}\mathrm{sinc}[\phi(\omega_{a},\omega_{b})]\exp{i\phi(\omega_{a},\omega_{b})}+a\leftrightarrow
b\Big{\\}}$. The function
$\phi(\omega_{j},\omega_{k})=(\omega_{j}-\omega_{p}/2)\tilde{T}_{1}+(\omega_{k}-\omega_{p}/2)\tilde{T}_{2}$
and the temporal entanglement parameter of the pairs
$\tilde{T}_{1}(\tilde{T}_{2})$, via
$\tilde{T}_{\mathrm{ent}}=\tilde{T}_{2}-\tilde{T}_{1}$, quantifies the
spectral-temporal properties of the entangled biphotons Arzani  _et al._
(2018); Schlawin  _et al._ (2013); Keller and Rubin (1997). The term $A_{0}$
denotes the amplitude of the pump. This form of factorization underlies the
fact that the signal scales linearly with the intensity. The temporal
entanglement parameter can be viewed as an estimator of upper bound of delay
between the time of generation of the entangled photon pairs inside the SPDC
source material. The entanglement time parameter arises from the phase
matching function $\Delta k(\omega_{1},\omega_{2})$ and parametrically depends
on the group velocity of propagation inside the SPDC material. The phase
mismatching function under the linearization approximation around the central
frequencies of the beams, for the collinear case leads to the identification
of parameters, $T_{j}=1/v_{p}-1/v_{j}$ where $j\in\\{1,2\\}$ and $v_{p/j}$
denotes the group velocity of the pump, and two biphotons inside the material.
These values can be estimated from the inverse of the marginal distribution
function of the joint spectral amplitude function in the frequency domain
Landes  _et al._ (2021); Keller and Rubin (1997). In contrast, the classical
field, in a similar weak-field limit factorizes into the product of amplitudes
as, $\langle
E^{\dagger}(\omega_{4})E^{\dagger}(\omega_{3})E(\omega_{2})E(\omega_{1})\rangle=A^{*}_{4}(\omega_{4})A^{*}_{3}(\omega_{3})A_{1}(\omega_{2})A_{1}(\omega_{1})$
and scales quadratically with the intensity. The biphoton field correlation
properties for different typical parameter regimes can be examined by plotting
the joint-spectral amplitude which describes the frequency-dependent
correlation of the same as shown in Fig. 3. The bottom-right plot (i.e.,(d))
in Fig. 3 corresponds to a classical field scenario. The simulation has the
freedom of selecting frequency pairs from the plot region where the function
has finite support.

###  III.3 Simulation

The correlated two-polariton excitations via entangled biphoton sources may
focus on several experimental configurations which will be of particular
interest to the condensed phase spectroscopies. The excitation of specific
two-polariton states via higher-energy sectors of the one-polariton manifold
and contrasting them with those via the lower-energy sectors may give
information about polariton scattering, delocalization, and dephasing. In
other words, the specific two-polariton states may have dominant contributions
from certain one-polariton states which are distant on the site basis but
energetically closer. Alternatively for the same two-polariton excitation,
projecting to the higher and lower-energy one-polariton sector offers insights
into the state compositions. Combining two strategies may provide important
insight into the state resolved polariton correlations. These features can be
probed as shown in the upper and lower panel of Fig. 4. Particularly the
polariton states (e.g., $\omega_{p^{(2)}}=30550\mathrm{cm}^{-1}$) that are
specifically prone to phonon-induced dephasing (higher dephasing induced
broadened) have been excited while the temporal entanglement parameter of the
probe, projected one-polariton sector has been varied.
In the upper panel of the Fig. 4, we present, along the rows, three sets of
results for the variation of the temporal entanglement parameter
$\tilde{T}_{\mathrm{ent}}(\mathrm{fs}$. Each of them correspond to fixed
$\omega_{a_{1}}(\mathrm{cm}^{-1})=15500$ and
$\omega_{b_{1}}(\mathrm{cm}^{-1})=14500$ (therefore exciting the target at
$\omega_{p}^{(2)}=30550$) with pump width $\tau_{p}(\mathrm{fs})=20.0$.
Therefore, the two polariton excitation occurs via the middle sector of the
one-polariton band. It also projects the two-polariton coherence to the mixed-
energy region of the one-polariton band by choosing the corresponding
frequencies as $\omega_{a_{2}}(\mathrm{cm}^{-1})=\omega_{p}^{(2)}-15800$ and
$\omega_{b_{2}}(\mathrm{cm}^{-1})=15800$. It is noticed that the strong
entanglement between the photon pairs, as we move from (A) to (C)
corresponding to $\tilde{T}_{\mathrm{ent}}(\mathrm{fs})$ values $60.0$,
$50.0$, $40.0$ respectively, allows correlated signal features to develop.
These emergent features may not be visible in the signal obtained by using
classical pulse pairs of comparable spectral-temporal properties. Time-
frequency correlated excitation gives the freedom of simultaneously choosing
the narrow-band target while ensuring that system remains less affected by the
high dephasing components of the intermediate states. The latter has the
possibility of making the short-time kinetics in the one-polariton manifold
more accessible. Even an individually controlled classical field two-pulse
scenario may offer less advantage because the temporal and spectral components
are bound.
The bottom row ((d) to (f)) accomplishes the aforementioned goal of exploring
excitation via different energy sectors of the one-polariton manifold. Here
the (d) and (e) allow excitation via middle-sector and (f) lower-sector while
projecting all of them to the same mixed-energy sectors as the above panel.
With the increase in temporal entanglement parameter in going from (d) to
(e)($\tilde{T}_{\mathrm{ent}}(\mathrm{fs})$ values $40.0$, $10.0$,
respectively) we find features shows distinguishable increase. The last panel
(f) whose excitation via lower energy sector (excitation via
$\omega_{a_{1}}(\mathrm{cm}^{-1})=15150$ and
$\omega_{b_{1}}(\mathrm{cm}^{-1})=$) which reveals many more correlation
features. It reveals the higher participation of the particular set of one-
polariton states to chosen target in the two-polariton manifold throughout the
simulation. The short temporal entanglement parameter also certifies the
capability of the biphoton sources to map out correlation involving
energetically distant states. The success of the parameter regime and overall
strategy of the simulation can be traced back to the ability to choose the
frequency pairs from a broader distribution. In addition, the fact that they
are also bound by the temporal constraints allow excitation via fast dephasing
components in the one-polariton band (during the first time delay i.e.,
$T_{1}$) while simultaneously projecting the resultant two-polariton coherence
to short-lived coherences (during the last time delay i.e., $T_{3}$).

##  IV Conclusion and outlook

In this article, we have proposed a theoretical protocol that is suitable for
the investigation of the inter-manifold coherence properties associated with
the two-polariton manifold. We observe that the proposal exclusively focused
on the cavity control of exciton correlation and modulation of exciton-phonon
dephasing via the former. In other words, the cavity affects the exciton
transitions directly and redistributes the excitonic spectral weights. The
extent to which this redistribution occurs is encoded in the action of the
polariton transformation matrices. As a result, the novel hybridized polariton
states interact with the phonons rather differently than that of the cavity-
free case. We demonstrated that biphoton sources are capable of studying the
ultrafast signatures of the related dynamics without losing the state
specificity. In the process, it is capable of mapping out the cavity modulated
exciton correlation. In this direction, two further extensions namely, the
detailed study involving the parametric variation of the cavity-free case and
a comparison employing controlled classical fields are worth looking at. They
will be part of future communication. However, we note that the signal
expressions presented in this communication will be sufficient for such
extended analysis.
The role of cavity coupling has been included non-perturbatively with the same
quasi-particle excitation picture. Here the real-space coupling variations
were neglected for convenience, in the spirit of the first simulation. Also,
the role of the mean number of photons in the cavity has not been investigated
and the role of the cavity has been confined to the coherence created between
the states within the proposed regime of operation.
In comparing and contrasting the present technique to the transmission mode
pump-probe measurements several features distinguish the present technique.
DQC measures one specific component of the nonlinear polarization of the
cavity polariton, unlike the pump-probe analog. The pump-probe technique also
includes the pathways analogous to polaritonic Raman scattering pathways. Thus
the DQC signal is more specific to the purpose of this article. The two-
polariton coherence is explicitly monitored by isolating the signal components
as suggested by the plausible implementation via phase-cycling schemes. It can
be also highlighted that the present technique works by projecting the two-
polariton coherence in two competing coherence components of lower order. The
relevant processes leading to the signal occur during the last two time
intervals. The degree of discrimination of the pathways, reflected in the
associated dynamical spectral weights decides the magnitude of the signal. In
contrast to the pump-probe studies, the off-diagonal spectral signatures in
the correlation plots required to be interpreted differently. In the latter,
the cooperative features appearing in the above-mentioned sector carry less
specific information regarding the physical origin of the polaritonic
correlation due to non-discrimination between pure two-one polariton coherence
and two-polariton-one polariton coherence.
The study can be extended to accommodate the explicit two-polariton transport
phenomenology by studying the fluorescence-detected phase cycling protocols
Malỳ and Mančal (2018); Malỳ and Brixner (2021). The latter is a four-wave
mixing analog but aimed at investigating the longer-time state correlation
properties in the presence of phonon-induced dynamical population
redistribution. The longer-time dynamical information thereby obtained is
complementary to the information provided by the short-time dynamics
investigated in this article. An investigation in this direction is on the
way. Furthermore, one may combine the interferometric detection schemes to
separate the pathways as recently proposed Dorfman  _et al._ (2021); Asban and
Mukamel (2021).

Figure 5: Singular values for the three sets of entangled biphoton sources
corresponding to a similar parameter regime of the ones used in the
simulations of Fig. 3. For the first two plots, (i.e., in (a), (b)) the values
are truncated at $n_{\mathrm{svd}}=50$, and for the third one (i.e., in (c)),
at $n=25$ (after normalization). The classical two-photon pulse yields, as
expected, $n_{\mathrm{svd}}=1$ (i.e., in (d)).

The theoretical description adopted for describing the dissipative polaritonic
matter is formulated at the level of a quasi-particle approach. The quasi-
particle Green’s function was chosen to describe the signal Abrikosov  _et
al._ (2012). Over the last few years, a host of promising methods have been
proposed that can potentially deal with the complexity of the quantum
aggregates in near future. These methods have offered several different
flavors of treating the matter and the cavity modes. We have, namely, quantum
electrodynamics based hybrid (density) functional formulation Ruggenthaler
_et al._ (2014); Schäfer  _et al._ (2021b); Flick and Narang (2018); Yang  _et
al._ (2021), cluster-expansion Haugland  _et al._ (2020), potential-energy
surface-based dynamical calculations combined with the trajectory-based
propagation for the cavity quadrature modes Hoffmann  _et al._ (2020); Zhang
_et al._ (2019d), path integral based unified framework for nuclear modes and
the idealized cavity modes Li  _et al._ (2021); Chowdhury  _et al._ (2021);
Mandal  _et al._ (2022), first-principles simulations Schäfer  _et al._
(2019); Flick  _et al._ (2017, 2019); Welakuh  _et al._ (2022); Svendsen  _et
al._ (2021). Incorporating these methods to describe the phenomenology
described in this article will require a qualified description of exciton
formation, a description of exciton-exciton scattering in the presence of a
dielectric environment, and nuclear propagation.
We also note that the numerical simulation adopted the analytical expressions
which have been derived under the assumption of generalized time-translational
invariance. For systems driven out-of-equilibrium via additional laser pulses,
one may expect to see more correlation features in the signal. The scope to
add additional pulses and using the biphotons as probes, although offers a
more complicated scenario, is a promising avenue for future study of nonlinear
response in correlated quantum materials Rostami  _et al._ (2021);
Parameswaran and Gopalakrishnan (2020); Rostami and Cappelluti (2021); Choi
_et al._ (2020); Grankin  _et al._ (2021).

Figure 6: (a)-(b) The list of the $\upsilon_{j}(\mathrm{cm}^{-1})$ parameters
corresponding to the multi-mode Brownian oscillators..The list of the
$\lambda_{j}(\mathrm{cm}^{-1})$ parameters corresponding to the multi-mode
Brownian oscillators.(c) The spectral function of the structured phonon modes
(corresponding to the multi-mode Brownian oscillators) used in the simulation.

## Appendix A

In this appendix, we provide two sets of supplementary information that aid
the main text. It is composed of a relevant figure of merit for entanglement
in the biphoton source and a description of parameters regarding the phonon
spectral density.

###  A.1 Singular value decomposition of the biphoton spectral function

The singular value decomposition of the joint spectral amplitude of the
biphoton sources provides an estimate for the number of effective modes that
participated in mode-squeezing. It can be analyzed by obtaining the singular
value decomposition of the said function at different parameter regimes. It
can be seen in Fig. 5 that the cases involving the shorter temporal
entanglement parameters increase the number of effectively squeezed modes.
Also, a decrease in the pump bandwidth decreases the modal amplitudes for a
case with comparable temporal entanglement parameters. The classical two-
photon pulse, unsurprisingly, presents a plot reminiscent of the uncorrelated
feature.

###  A.2 Phonon parameters

The site-independent spectral function is composed of discrete frequencies
which primarily induce multiple timescales Markovian dissipation. Here we
enlist the parameter values for the $48$ structured phonon modes used in the
simulation Novoderezhkin  _et al._ (2005, 2004, 2011); van Amerongen and van
Grondelle (2001). Along with these values, the values of other parameters are
given as, $\gamma_{j}(\mathrm{cm}^{-1})=30.0$ for all the multimode Brownian
oscillators. Corresponding Fig. 6(c) illustrates the spectral density
distribution. The parameter values for the overdamped oscillator are given by,
$\lambda_{0}(\mathrm{cm}^{-1})=37.0$ and $\gamma_{0}(\mathrm{cm}^{-1})=30.0$.

###### Acknowledgements.

We acknowledge financial support from the European Research Council
(ERC-2015-AdG-694097), by the Cluster of Excellence “Advanced Imaging of
Matter” (AIM), Grupos Consolidados (IT1249-19) and SFB925 “Light induced
dynamics and control of correlated quantum systems.” The Flatiron Institute is
a division of the Simons Foundation.

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