Polarization-Entangled Two-Photon Absorption in Inhomogeneously Broadened Ensembles

Entangled photons are promising candidates for a variety of novel spectroscopic applications. In this paper, we simulate two-photon absorption (TPA) of entangled photons in a molecular ensemble with inhomogeneous broadening. We compare our results with a homogeneously broadened case and comment on the consequences for the possible quantum enhancement of TPA cross sections. We find that, while there are differences in the TPA cross section, this difference always remains small and of the order unity. We further consider the impact of the polarization degrees of freedom and carry out the orientational average of a model system Hamiltonian. We find that certain molecular geometries can give rise to a substantial polarization dependence of the entangled TPA rate. This effect can increase the TPA cross section by up to a factor of five.

In theoretical work, recent analyses predict a very small enhancement of the ETPA cross section due to spectral entanglement [48,49]. The key assumption in these publications is that molecular resonances are broadened much more strongly than their atomic counterparts. As a consequence, spectral quantum correlations cannot enhance the absorption probability in the same way as in atomic samples. This would imply that the large absorption cross sections mentioned above cannot be explained by ETPA, and have to be attributed to other processes that remain to be clarified. In contrast, another study by Kan et al. alleges that ETPA takes place into final states with much weaker broadening [50], such that large enhancements due to quantum correlations become possible. Here we investigate whether these two scenarios can give rise to vastly different absorption cross sections. In particular, we consider ETPA in an ensemble of inhomogeneously broadened molecules, where sharp absorption resonances are distributed randomly within a certain frequency distribution (see Figure 2). This enables us to interpolate between these two scenarios mentioned above. We will investigate whether strongly enhanced absorption in a subset of molecules that can be excited resonantly can overcompensate the reduced absorption probabilities in the remaining molecules.
In addition, we will scrutinize the impact of the polarization degrees of freedom on the ETPA cross section. By carrying out an orientational average of the molecular dipoles, we will explore how the relative orientation of the two dipoles affects the ETPA process. A recent study [43] detected no discernible dependence on the photon polarization. Here we show that this is true only for certain molecular dipole orientations.

THE MODEL
We consider the interaction of broadband entangled photons with matter. For a quantum light field propagating in a fixed spatial direction (without loss of generality, we here consider the z-direction), the electric field operator in the interaction picture with respect to the field Hamiltonian reads [51] with the positive frequency component and E (−) its hermitian conjugate. Here, ϵ 0 denotes the vacuum permittivity, and A 0 is the quantization area perpendicular to the propagation direction of the field. The photon annihilation and creation operators obey the usual permutation relations . The polarization vectorê ] distinguishes horizontal or vertical polarization. In the second line, we define the scalar fields E ] . In the following, we will consider a sample, which is placed at the origin of our coordinate system and set z = 0.
We will investigate the ETPA probability of a multilevel system with ground state g, an off-resonant intermediate states e, and a final state f. This is shown in Figure 1. The probability to excite the final state f can be derived with perturbation theory with respect to the light-matter interaction Hamiltonian, which reads within the rotating-wave approximation for a sample placed at the origin of our coordinate system Here, V (−) (t) denotes the negative frequency component of the dipole operator, i.e. it creates a molecular excitation, and we have dropped the spatial dependence of the operators. A detailed account for the derivation of the ETPA probability can be found, e.g., in [49,52]. Here, we only present the final result which, neglecting the polarization degrees of freedom at first, reads with the classical TPA cross section given by Here, we have defined the transition frequency to the final state of the molecule, ω fg , as well as the inverse lifetime γ fg . The intermediate state transition frequency is denoted ω eg , and d ge and d ef are the dipole matrix elements connecting ground and intermediate, or intermediate and final states, respectively. In Eq. 5, we also included the so-called entanglement area A 0 , which could be a function of the entanglement of the two-photon state [44]. Here, we will not consider this effect and treat A 0 as a constant. We further introduced the four-point correlation function of the field in the numerator of the second line of Eq. 5. It is evaluated with respect to the initial state of the light field, which we will specify later in Section 2.3. In writing Eq. 5, we require the intermediate states to be far off-resonant from the entangled photons' centre frequency ω 0 . If this condition is violated the intermediate state resonances cannot be taken out of the frequency integrations and need to be considered separately [2]. In this resonant regime, which will not be considered here, we could already show how quantum correlations can enhance the absorption probability [53][54][55].

Inhomogeneous Broadening
Two-photon transitions in the presence of inhomogeneous broadening were already discussed in [56]. Here we model it by assuming that the resonance of the final state ω fg is distributed around a central value ω (0) fg with a Gaussian distribution with width σ D , i.e.
The TPA probability is then given by the average with respect to this ensemble, where P TPA (ω fg ) is given by Eq. 5 at frequency ω fg . This situation is illustrated is Figure 2, where we plot the two-photon resonance in Eq. 5, i.e., R(γ fg − iω fg + iω) vs. the detuning Δ ω (0) fg − ω. In an inhomogeneously broadened ensemble, the broad resonance is in fact composed of the incoherent mixture of much more narrow resonances of the ensemble constituents. We note that the role of the Lorentzian width γ fg changes between the homogeneously and inhomogeneously broadened samples. In the former case, Eq. 5, γ fg denotes the width that one would measure in a spectroscopic experiment. In the latter case, it is the narrow linewidth of an individual molecule, but experiments would detect the much broader distribution described by Eq. 8. To distinguish the two situations, we will call the Lorentzian width in the inhomogeneously broadened sample the intrinsic width of the resonance, γ inh in the following. In contrast, we will call it the homogeneous width γ hom in the homogeneously broadened case.

Orientational Average
We next discuss how to carry out an orientational average of the molecular absorption. To this end, the dipole elements in Eq. 5 have to be treated as vectors, i.e. d ij → d ij . More specifically, we consider dipole vectors in the molecular reference frame, We further take the polarization of the light fields into account, see Equation 2. We then generalize Eq. 5 to where the TPA tensor is given by Here, we denote 〈. . .,〉 ens the orientational average which accounts for the random orientation of the molecules in the sample with respect to the lab frame. To evaluate this average, we have to transform from the lab frame into the molecular frame. This orientational average was first carried out in [57]. The application to nonlinear spectroscopy can be found, e.g., in [58]. The result is that we can straightforwardly relate the dipole elements in the lab frame with the corresponding dipole vectors in the molecular frame using the transformation where the indices α i = x, y, z label the components of the dipole operator in the molecular frame. The transformation tensor is given by

The Field Correlation Function
We finally have to evaluate the field correlation function in Eqs 5, 11, respectively. For an initial two-photon state, which we will denote |ψ〉, this function factorizes as The gray solid line corresponds to an homogenously broadened ensemble, whereas the small dashed Lorentzians indicate an inhomogeneously broadened ensemble which creates the same broad absorption feature.
Frontiers in Physics | www.frontiersin.org March 2022 | Volume 10 | Article 848624 Here, we have inserted an identity in between a † ]2 (ω 2 ) and a ] 4 (ω 4 ). For a two-photon state, the only non-vanishing contribution stems from the vacuum. In our discussion of the inhomogeneous broadening, where we do not account for the polarization degree of freedom, we simply drop the indices ] i . With the entangled field propagating along the z-direction, see Eq. 2, we consider the horizontal polarization to be along the x-direction and the vertical polarization along the y-direction. In the third line, we defined the function Ψ ],]′ (ω, ω′) = 〈 0|a ] (ω) a ]′ (ω′)|ψ〉, which is often called the two-photon wavefunction. Its absolute square is referred to as the joint spectral intensity. As explained in detail in [52], in the situation described by Eq. 5 where the intermediate states are far off-resonant and can be subsumed into σ (2) , the marginal of the two-photon wavefunction along the anti-diagonal is what is relevant for the TPA process, and we can write Eq. 5 as where the marginal is given by We further defined the detuning Δ = ω fg − 2ω 0 . In the homogeneously broadened case, we simply set this to zero (i.e. we assume that the entangled pair is resonant with the transition), but in the inhomogeneous case, this detuning will be weighted by the ensemble's frequency distribution (7) below. In this paper, we will focus on bi-Gaussian two-photon wavefunctions, i.e. we write where σ N is the spectral width of the individual photon wavepackets and σ B is the width of the sum frequency ω + ω′. Please note the factor 1/ 2 √ in front of the wavefunction, which accounts for the fact that in the case of indistinguishable photons we have the normalization 〈ψ|ψ〉 = 2 dω dω′|Ψ(ω, ω′)| 2 /(2π) 2 = 1. In a strongly entangled state, we will typically have σ N ≫ σ B . This regime is dominated by strong frequency anti-correlations between the entangled photons, as well as strong correlations in their respective arrival times [2]. We further write the full twophoton wavefunction as where we consider specifically the two cases Please not that the factor 2 √ is included in Eq. 20 to ensure normalization with respect to the frequency wavefunction (18). In the first two cases, f 1 and f 2 , the two-photon state is not polarization entangled, as Eq. 18 is symmetric with respect to the exchange of its frequency arguments. In the latter case, the state is polarization entangled. We note that, as pointed out in [52], within the current model the molecular response is symmetric with respect to the exchange of the frequency arguments in Eq. 15, because any photon can excite either the g → e or the e → f transition. Consequently, even if the initial two-photon frequency wavefunction Ψ(ω, ω′) is not symmetric (as is the case, e.g., for type-II downconversion), the correlation function (15) selects the symmetric superposition~Ψ(ω, ω′) + Ψ(ω′, ω). As a consequence, a photonic singlet state of the form (|H s 〉|V i 〉 − |V s 〉|H i 〉)/ 2 √ will not give rise to a TPA signal within the current model due to the destructive interference between the two terms. This situation would change, for instance, in the presence of nearresonant intermediate states or any other process that breaks the abovementioned symmetry between the g → e and e → f transitions.

Inhomogeneous Broadening
We now compare how inhomogeneously broadened ensembles can affect the two-photon absorption probability. In this part of our investigation, we will neglect the polarization degrees of freedom at first. For simplicity, we divide the ETPA probability, Eq. 5, by its constant prefactor σ (2) γ fg /A 2 0 . For each combination of intrinsic linewidth γ inh and inhomogeneous broadening σ D , we calculate the full width at half maximum of the f-state distribution as shown in Figure 3A. We also calculate the ETPA probability from Eq. 8. As shown in Figure 3B, its behaviour inversely follows the broadening of the fstate in the sense that a broader f-distribution in panel (a) results in a reduced ETPA probability. There is however a noticeable asymmetry between the two plots. The ETPA probability appears to be reduced more rapidly with increasing intrinsic linewidth γ inh compared to an increase of the inhomogeneous broadening σ D . This behaviour could be a consequence of the photonic entanglement and in the following we investigate it more closely.
To compare ETPA in homogeneously and inhomogeneously broadened ensembles, we define the ratio between the absorption probability in an inhomogeneously broadened sample, Eq. 8, and in a homogeneously broadened sample, Eq. 5, where we use as homogeneous broadening γ hom the width of the fstate distribution, which we determined in Figure 3A. We note, however, that the convolution of an intrinsic Lorentzian broadening with a Gaussian as in Eq. 8 gives rise to a Voigt line profile. This is Frontiers in Physics | www.frontiersin.org March 2022 | Volume 10 | Article 848624 markedly different from the Lorentzian resonance of the homogeneous case. As a consequence, one should keep in mind that the ratio (23) compares two slightly different situations. We plot the ratio (23) in Figure 4 as a function of both γ inh and σ D for three different two-photon states. In Figure 4A, the entangled photons show very strong frequency anticorrelations, in panel (b) they are separable, and in panel (c) the show strong positive frequency correlations. We see that, even though there are notable differences between these cases, the general trend is identical. The ratio of the two excitation probabilities can be enhanced, and the effect increases with increasing inhomogeneous broadening σ D and with decreasing intrinsic linewidth γ inh . This effect is stronger when the exciting photon pair is entangled with strong frequency anti-correlations as in panel (a). In contrast, positive frequency correlations which are nevertheless associated with an entangled wave function, have the opposite effect and reduce this enhancement.
Still we also find that this enhancement saturates to increasing inhomogeneity of the distribution. This is shown in Figure 5 for different entanglement strengths. The broadening at which the saturation is reached depends strongly on the degree of entanglement. However, the ratio r rel always remains of order unity, as the enhanced resonant excitation probability is balanced by an increase in the number of molecules which can not be excited resonantly and thus become dark. Therefore, inhomogeneous broadening cannot account for the enormous disparity in the reported ETPA cross sections in the literature.

Orientational Average
With our choice of the Gaussian two-photon wavefunction (18), which is symmetric with respect to its frequency arguments, we can separate the orientational average from the frequency integrations. Thus, using the definition of our molecular dipoles (10), their transformation to the laboratory frame in Eqs. 13, 14, we can carry out the index summations ] 1 , . . . , ] 4 and α 1 , . . . , α 4 . Note that this orientational average will always reduce the ETPA rate, since the dipole vectors in Eqs. 11 are chosen normalized. As a consequence, their orientationally  Polarization-Entangled Two-Photon Absorption averaged overlap with any polarization mode will be smaller than unity.
For the polarization state (20), we obtain Here, P TPA iso denotes the isotropic result where the vectorial nature of the dipoles and the photons has been neglected. Likewise, we obtain from Eq. 21 and for the entangled polarization state (22), we arrive at These results are shown in Figure 6. The separable states, Eqs 24, 25, show a weaker dependence on the dipole angle compared to the entangled state (26). The latter shows the largest dependence at an angle ϕ = 0, i.e. for a symmetric superposition state, where it also takes its maximal value. This is consistent with our discussion in Section 2.3, where we pointed out that the molecular response projects onto a symmetric superposition. Hence, such a state is optimal for the ETPA cross section. Conversely, the entangled state shows the smallest variation at ϕ = π/2 (not shown), where it coincides with the product state (25).
The polarization dependence of ETPA in rhodamine 6G molecules was carefully examined in [43], and no discernable dependence on the interphoton polarization angle was reported. This can be recounciled with our simulations provided the dipole angle is around θ ≃ π/3 or 2π/3. Indeed, as we see in Figure 6, in such a case the polarization state should not affect the ETPA probability regardless of the polarization state of the light. This appears broadly consistent with measurements in [59], where an angle of around 40°was reported, thus implying a rather weak ETPA polarization dependence. A further possibility, which we have not discussed here, would be the interference between different excitation pathways. Such a situation could wash out the strong polarization dependence observed here for a single excitation pathway.

CONCLUSION
In summary, we have extended the existing theory of entangled two-photon absorption to describe excitation in inhomogeneously broadened molecular ensembles, and to investigate the influence of the polarization degrees of freedom on ETPA.
In our investigation of inhomogeneously broadened samples, we showed that ETPA can be enhanced compared to homogeneously broadened samples. However, this enhancement always remains relatively small, of order unity. It cannot explain the several orders of magnitudes in difference among the various measured ETPA cross sections in the literature. The presence of near-resonant intermediate states could change this conclusion. As shown in [53][54][55], optimized entangled states of light can substantially enhance the ETPA probability. It remains an open question, however, how these conclusions would be affected by inhomogeneous broadening. Furthermore, entangled states with large spectral tails in their single-photon spectrum also affect the present conclusion. As shown in the context of virtual state spectroscopy [60], these large frequency tails render the key assumption in the current theoretical model -that intermediate states are far detuned from the photonic states (see Eq. 5)problematic and further emphasize the need for the investigation of near-resonant intermediate states in ETPA.

FIGURE 6 |
Polarization dependence of ETPA probability on the angle between the molecular dipoles according to Eq. 10. The gray, dashed line depicts the dependence (24) which is created by the input state (20). The black, dot-dashed line is given by Eq. 25, created by the state (21). The polarization dependence (26), which is created by the state (22), is shown as a solid, yellow line for ϕ = 0, and as a blue, dotted line for ϕ = π/3. absorption probability. It can increase or reduce the absorption probability by as much as a factor of five, provided the molecular dipoles are aligned in parallel or perpendicular. These findings rely on a single excitation pathway, i.e. a single intermediate molecular state. In case there are several, competing excitation pathways, the pronounced angular dependence of the ETPA rate could be washed out by the mixing of these pathways. Furthermore, it is one of the consequences of the employed theoretical model, as observed already by Landes et al. in [60], that the molecular response projects the entangled photon wavefunction onto a symmetrized superposition. As a consequence, the present model predicts a vanishing ETPA probability for a photonic singlet state due to destructive interference. This effect could be tested experimentally, e.g., in atomic ETPA measurements. It could further provide an experimental test in molecular ETPA experiments to gauge the strength of single-photon losses, which should not be suppressed by destructive interference.

DATA AVAILABILITY STATEMENT
The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.