The first three terms constitute the bare exciton Hamiltonian where B m ( B m † ) are the m-th site exciton creation (annihilation) operators with the respective commutation relation, [ B m , B n ] = [ B m † , B n † ] = 0 and [ B m , B n † ] = δ m n ( 1 − ( 2 − κ m 2 ) ) B m † B m † ) (where κ _m = μ ₂₁/μ ₁₀, is given in terms of ratio of transition dipoles of ij-th level of the m-th site) [56–58]. 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 m n ( 1 ) δ k l + δ m n H k l ( 1 ) . The terms U m ( 1 ) , U m n ( 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 ( κ n − 2 E n ( 2 ) − E n ( 1 ) ) , with κ = μ ₂₁/μ ₁₀ which accounts for the local energy shifts for two-exciton states and U m n ( 2 ) = 0 . Bare two-exciton energy is given by E m ( 2 ) = 2 E m ( 1 ) + Δ m (where the Δ _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 [59–62] 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,α 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 [63–66]. 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 α † ( a α ) , for the α-th mode. Within the scope of this study, a single-mode limit (i.e. α = 1) and two-excitation per-mode (with he fundamental frequency denoted by ω _c ) has been considered. Furthermore, we assume an uniform cavity-exciton coupling strengths for all sites, i.e. g _c;m,α = g _c = 100cm⁻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.

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 υ _k is the mode frequency associated with the k-th normal mode whose creation (annihilation) operators are denoted via b j † ( b j ) . The phonon operators follow the free-boson commutation relations, [ b j , b j ′ ] = [ b j † , b j ′ † ] = 0 and [ b j , b j ′ † ] = δ j j ′ . 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, 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 ( ω ) = π ∑ j | g ̄ j | 2 ( δ ( ω − υ j ) − δ ( ω + υ j ) ) from which the spectral density function is obtained in the continuum frequency limit. The spectral density function, presented as, J ( ω ) = 2 λ 0 ( γ 0 ω ) / ( ω 2 + γ 0 2 ) + ∑ j = 1 N b 2 λ j ( υ j 2 γ j ω ) / ( ( υ j 2 − ω 2 ) 2 + ω 2 γ 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 (λ ₀, λ _j = υ _j ϒ _j where υ _j and ϒ _j are the j-th oscillator frequencies and Huang-Rhys parameters respectively), and the damping parameter (γ ₀, γ _j ) which are required to optimally describe the equilibrium spectral density function have been obtained from the work of [35, 59–61]. The numerical values corresponding to those parameters are enlisted in Appendix A2.

The polariton states are obtained as number-conserving manifolds via exact-diagonalization of the field-free Hamiltonian subspaces, H ̃ p ( n ) = H ̃ s + H ̃ c + H ̃ s c by using T p ( n ) , − 1 H ̃ p ( n ) T p ( n ) = H p ( n ) = ∑ n = 0 , 1 , 2 E p ( n ) X p ( n ) p ( n ) , where we define operators X p ( n ) p ( n ) = p ( n ) 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 (Figure 1). The manifolds are composed of the cluster of states 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 ω _c (cm⁻¹) = 15.4 × 10³ and g _c (cm⁻¹) = 0.1 × 10³. 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.

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, H int t = ∑ j ∑ n , n ′ = 0,1,2 ; n ≠ n ′ 2 π ω j / V a j e − i ω j t d p n p n ′ X p n p n ′ ⁡ exp i ω p n ′ p n t + h . c . (2) where external photon mode creation operators a _j and dipole-weighted inter-manifold polariton transition operators, X p ( n ) p ( n ′ ) were defined. Also, the mode quantization volume V and the Fourier expansion frequency ω _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.

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, σ ̇ ( n ) ( t ) = − i L S ( n ) σ ( n ) ( t ) + ∫ 0 ∞ d τ Σ ( n ) ( t , τ ) σ ( n ) ( t ) where we denote σ ⁽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, Σ ( n ) ( t , τ ) σ ( n ) ( t ) = − λ 2 X , D ( t ) σ s ( t ) − σ s ( t ) D † ( t ) . The kernel contains the dissipation operator D(t) = X (t − t′)C _b (t − t′) where we have the convolution of polariton-phonon coupling operator X ( τ ) = ∑ p ( n ) X p ( n ) p ( n ) ( τ ) and time-domain correlation function given by, C b τ = λ 0 γ 0 / 2 cot β γ 0 / 2 exp − γ 0 τ + ∑ j = 1 N b λ j 2 ζ j coth i β ϕ j + / 2 exp − ϕ j + τ − coth i β ϕ j − / 2 exp − ϕ j − τ + − i λ 0 γ 0 / 2 exp − γ 0 τ + i λ j υ j 2 2 ζ j exp − ϕ j + t − exp − ϕ j − t − ∑ n = 1 ∞ 4 λ j γ j υ j 2 / β ν n / υ j 2 + ν n 2 2 − ν n 2 γ j 2 + 2 λ 0 γ 0 / β ν n / ν n 2 − γ 0 2 exp − i ν n τ (3)

In the above we have ζ j = ( υ j 2 − γ j 2 / 4 ) , ϕ j ± = ( γ j / 2 ) ± i ζ j , and, matsubara frequencies, ν _n = n (2π/β) (with n _m = 20). Furthermore, we defined β = 1/κT where the Boltzmann constant κ 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, γ p 1 ( n ) ( t ) = θ ( t ) ∑ p 2 , p s C b ( ω 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 (Ω) = ∫dt exp iΩtC ^±(t) and C b ± ( t ) = ∫ ( d ω / 2 π ) ( coth ⁡ β ω / 2 ) cos ⁡ ω t ∓ i ⁡ sin ⁡ ω 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, γ p 1 ( n ) p 2 ( m ) = ( γ p 1 ( n ) + γ p 2 ( m ) ) / 2 . Finally, we obtain the inter-manifold Greens functions relevant for dephasing as, G p 1 ( n ) p 2 ( m ) ( ω ) = i ( ω − ω p 1 ( n ) p 2 ( m ) − i γ p 1 ( n ) p 2 ( m ) ) − 1 . The advanced Green’s functions are defined likewise.

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 [67, 68]. 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, Π a τ 4 , τ 3 , τ 2 , τ 1 = d p i 0 p j ′ 1 τ 4 d p j ′ 1 p k 2 τ 3 d p k 2 p j 1 τ 2 d p j 1 p 0 τ 1 Π b τ 4 , τ 3 , τ 2 , τ 1 = d p i 0 p j ′ 1 τ 3 ′ d p j ′ 1 p k 2 τ 4 ′ d p k 2 p j 1 τ 2 ′ d p j 1 p 0 τ 1 ′ (4)

where the first and the second one have been written in the Heisenberg representation and correspond to diagrams in Figure 2). 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 (τ ₄, τ ₃, τ ₂, τ 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 s I m ∏ i ∈ a , b ∫ ∞ ∞ d τ 4 ( i ) d τ 3 ( i ) d τ 2 ( i ) d τ 1 ( i ) θ ( τ 43 ( i ) ) θ ( τ 32 ( i ) ) θ ( τ 21 ( i ) ) × D ( i ) ( τ 4 ( i ) , τ 3 ( i ) , τ 2 ( i ) , τ 1 ( i ) ) Π i ( τ 4 ( i ) , τ 3 ( i ) , τ 2 ( i ) , τ 1 ( i ) ) where C _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 loop-delay variables and mapped onto the real-time parameters that are externally tunable using, θ(s ₃)θ(s ₂)θ(s ₁) → θ(t ₃)θ(t ₂)θ(t ₁) for the diagram I and θ(s ₃)θ(s ₂)θ(s ₁) → θ(s ₃)θ(s ₂ − s ₃)θ(s ₁) = θ(t ₃)θ(t ₂)θ(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, S = C s ∫ − ∞ ∞ d T 1 d T 2 d T 3 F Ω 3 , Ω 2 , Ω 1 ; T 3 , T 2 , T 1 ∫ ∞ ∞ d ω ̃ 3 2 π d ω ̃ 2 2 π d ω ̃ 1 2 π e − i − ω ̃ 3 + ω ̃ 2 + ω ̃ 1 T 3 − i ω ̃ 1 + ω ̃ 2 T 2 − i ω ̃ 1 T 1 ⟨ E 3 † ω ̃ 3 E 4 † + ω ̃ 1 + ω ̃ 2 − ω ̃ 3 E 2 ω ̃ 2 E 1 ω ̃ 1 ⟩ ∑ p j 1 , p j ′ 1 , p k 2 w 1 G p k 2 p j ′ 1 ω ̃ 1 + ω ̃ 2 − ω ̃ 3 G p k 2 p 0 ω ̃ 2 + ω ̃ 1 G p j 1 p 0 ω ̃ 1 + ⟨ E 3 † ω ̃ 3 E 4 † + ω ̃ 1 + ω ̃ 2 − ω ̃ 3 E 2 ω ̃ 2 E 1 ω ̃ 1 ⟩ ∑ p j 1 , p j ′ 1 , p k 2 w 2 G p j ′ 1 p 0 − ω ̃ 3 + ω ̃ 2 + ω ̃ 1 G p k 2 p 0 ω ̃ 2 + ω ̃ 1 G p j 1 p 0 ω ̃ 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. 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 ( Ω 3 , Ω 2 , Ω 1 ; T 3 , T 2 , T 1 ) = θ ( T 3 ) θ ( T 2 ) θ ( T 1 ) e i Ω 3 T 3 e i Ω 2 T 2 e i Ω 1 T 1 . We also define the following variable mapping of the parameters, τ ̃ 21 0 → T 1 , τ ̃ 32 0 → T 2 , τ ̃ 43 0 → T 3 , for the diagram I and τ 21 0 → T 1 , τ 42 0 − τ 43 0 → T 2 , τ 43 0 → T 3 , for the diagram II (the τ i j 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. S Ω 3 , Ω 2 , Ω 1 = C s ∑ p 0 p j 1 p j ′ 1 p k 2 w 1 F ̃ Ω 1 ⟨ E 4 † z p k 2 p j ′ 1 E 3 † z p j ′ 1 p 0 E 2 z p k 2 p j 1 E 1 z p j 1 p 0 ) + w 2 F ̃ Ω 2 ⟨ E 4 † z p k 2 p j ′ 1 E 3 † z p j ′ 1 p 0 E 2 z p k 2 p j 1 E 1 z p j 1 p 0 ) (6) where the functions are specified as, F ̃ Ω 1 = Ω 3 − z p 2 1 p 0 Ω 2 − z p 2 p 0 Ω 1 − z p 1 1 p 0 − 1 F ̃ Ω 2 = Ω 3 − z p 2 p ′ 1 Ω 2 − z p 2 p 0 Ω 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.

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 [47, 69–71] is avoided here for succinctness. Following a similar derivation, the entangled biphoton field correlation function can be obtained as, ⟨ E † ( ω 4 ) E † ( ω 4 ) E ( ω 2 ) E ( ω 1 ) ⟩ = F 1 * ( ω 4 , ω 3 ) F 1 ( ω 2 , ω 1 ) where we have, F 1 ( ω a , ω b ; ω p ) = A 0 ( ω a , ω b ) s i n c [ ϕ ( ω a , ω b ) ] exp ⁡ i ϕ ( ω a , ω b ) + a ↔ b . The function ϕ ( ω j , ω k ) = ( ω j − ω p / 2 ) T ̃ 1 + ( ω k − ω p / 2 ) T ̃ 2 and the temporal entanglement parameter of the pairs T ̃ 1 ( T ̃ 2 ) , via T ̃ ent = T ̃ 2 − T ̃ 1 , quantifies the spectral-temporal properties of the entangled biphotons [71–73]. 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 Δk (ω ₁, ω 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 ∈ {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 [51, 73]. In contrast, the classical field, in a similar weak-field limit factorizes into the product of amplitudes as, ⟨ E † ( ω 4 ) E † ( ω 3 ) E ( ω 2 ) E ( ω 1 ) ⟩ = A 4 * ( ω 4 ) A 3 * ( ω 3 ) A 1 ( ω 2 ) A 1 ( ω 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 Figure 3. The bottom-right plot (i.e., (d)) in Figure 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.

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 Figure 4. Particularly the polariton states (e.g., ω p ( 2 ) = 30550 c m − 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 Figure 4, we present, along the rows, three sets of results for the variation of the temporal entanglement parameter T ̃ ent f s . Each of them correspond to fixed ω a 1 ( c m − 1 ) = 15500 and ω b 1 ( c m − 1 ) = 14500 (therefore exciting the target at ω p ( 2 ) = 30550 ) with pump width τ _p (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 ω a 2 ( c m − 1 ) = ω p ( 2 ) − 15800 and ω b 2 ( c m − 1 ) = 15800 . It is noticed that the strong entanglement between the photon pairs, as we move from (A) to (C) corresponding to T ̃ ent ( f s ) 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) ( T ̃ ent ( f s ) values 40.0, 10.0, respectively) we find features shows distinguishable increase. The last panel (f) whose excitation via lower energy sector (excitation via ω a 1 ( c m − 1 ) = 15150 and ω b 1 ( c m − 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).