A. Two-photon absorption measurements

In a typical eTPA transmission experiment, pairs of correlated photons, typically produced by spontaneous parametric downconversion (SPDC), interact with an arbitrary sample [see Fig. 1(a)]. The incident photon pairs are expected to drive a two-photon transition so that some pairs can be absorbed and, therefore, removed from the light beam traversing the sample. In most experiments,30,44,4930. J. P. Villabona-Monsalve, O. Calderón-Losada, M. Nuñez Portela, and A. Valencia, “Entangled two photon absorption cross section on the 808 nm region for the common dyes zinc tetraphenylporphyrin and rhodamine B,” J. Phys. Chem. A 121, 7869–7875 (2017). https://doi.org/10.1021/acs.jpca.7b0645044. B. P. Hickam, M. He, N. Harper, S. Szoke, and S. K. Cushing, “Single-photon scattering can account for the discrepancies among entangled two-photon measurement techniques,” J. Phys. Chem. Lett. 13, 4934–4940 (2022). https://doi.org/10.1021/acs.jpclett.2c0086549. S. Corona-Aquino, O. Calderón-Losada, M. Y. Li-Gómez, H. Cruz-Ramírez, V. Álvarez-Venicio, M. D. P. Carreón-Castro, R. de J. León-Montiel, and A. B. U’Ren, “Experimental study of the validity of entangled two-photon absorption measurements in organic compounds,” J. Phys. Chem. A 126, 2185–2195 (2022). https://doi.org/10.1021/acs.jpca.2c00720 the configuration shown in Fig. 1(b) is used to monitor, through coincidence measurements, the reduced photon-pair flux. Although one might be tempted to think that any pair-loss must be due to eTPA, it has been shown that single-photon-loss mechanisms, such as scattering4444. B. P. Hickam, M. He, N. Harper, S. Szoke, and S. K. Cushing, “Single-photon scattering can account for the discrepancies among entangled two-photon measurement techniques,” J. Phys. Chem. Lett. 13, 4934–4940 (2022). https://doi.org/10.1021/acs.jpclett.2c00865 and hot-band absorption,4343. A. Mikhaylov, R. N. Wilson, K. M. Parzuchowski, M. D. Mazurek, C. H. Camp, M. J. Stevens, and R. Jimenez, “Hot-band absorption can mimic entangled two-photon absorption,” J. Phys. Chem. Lett. 13, 1489–1493 (2022). https://doi.org/10.1021/acs.jpclett.1c03751 can lead to a similar behavior as would be expected for eTPA.4949. S. Corona-Aquino, O. Calderón-Losada, M. Y. Li-Gómez, H. Cruz-Ramírez, V. Álvarez-Venicio, M. D. P. Carreón-Castro, R. de J. León-Montiel, and A. B. U’Ren, “Experimental study of the validity of entangled two-photon absorption measurements in organic compounds,” J. Phys. Chem. A 126, 2185–2195 (2022). https://doi.org/10.1021/acs.jpca.2c00720 In view of this, it naturally becomes desirable to find a new, single-photon-loss insensitive photon-pair measurement scheme that enables direct certification of eTPA.In order to assess the possible candidates for eTPA certification, we study and compare three distinct two-photon quantum interferometers, namely single-port and two-port Hong–Ou–Mandel setups and a N00N-state configuration. The three configurations are shown schematically in Figs. 1(b)–1(d), respectively. Note that the N00N state configuration makes use of a superposition of both photons impinging on input port a and both photons impinging on input port b of the BS. Possible experimental implementations for each of these configurations are discussed in detail in the supplementary material. In all configurations, two-photon interference takes place via a beam splitter (BS), which, for the sake of completeness, is assumed for the analysis below to exhibit losses. The input–output transformation of a lossy BS is given by5454. S. M. Barnett, J. Jeffers, A. Gatti, and R. Loudon, “Quantum optics of lossy beam splitters,” Phys. Rev. A 57, 2134–2145 (1998). https://doi.org/10.1103/physreva.57.2134âout=t(ω)âin(ω)+r(ω)b̂in(ω)+F̂a(ω),(1)b̂out=t(ω)b̂in(ω)+r(ω)âin(ω)+F̂b(ω),(2)with âin,out(ω) and b̂in,out(ω) depicting the input and output field modes of the BS, respectively. r(ω) and t(ω) are the BS reflection and transmission coefficients, while F̂a(ω) and F̂b(ω), which commute with the input field operators, are the Langevin noise operators associated with the BS losses.5454. S. M. Barnett, J. Jeffers, A. Gatti, and R. Loudon, “Quantum optics of lossy beam splitters,” Phys. Rev. A 57, 2134–2145 (1998). https://doi.org/10.1103/physreva.57.2134 Note that the Langevin operators lead to reflection and transmission coefficients that do not conserve energy, i.e., r(ω)2+t(ω)2≤1.5454. S. M. Barnett, J. Jeffers, A. Gatti, and R. Loudon, “Quantum optics of lossy beam splitters,” Phys. Rev. A 57, 2134–2145 (1998). https://doi.org/10.1103/physreva.57.2134We can use Eqs. (1) and (2) to monitor the number of photon pairs that impinge on the BS by measuring the photon-coincidence rate at the output of the lossy BS. This can be expressed asR=P(1a,1b)=⟨N̂aN̂b⟩,(3)where ⟨⋯⟩ denotes an expectation value and the continuum number operators5555. S. M. Barnett and P. M. Radmore, Methods in Theoretical Quantum Optics (Oxford University Press, 1997). for the two output ports are given by N̂a=∫dωâout†(ω)âout(ω) and N̂b=∫dωb̂out†(ω)b̂out(ω). Note that Eq. (3) is valid in the case of temporally isolated photon pairs, that is when the probability of two pairs being present within the two-photon correlation time is much lower than one. This is the typical situation in most of the recent eTPA experiments.46–5046. K. M. Parzuchowski, A. Mikhaylov, M. D. Mazurek, R. N. Wilson, D. J. Lum, T. Gerrits, C. H. Camp, M. J. Stevens, and R. Jimenez, “Setting bounds on entangled two-photon absorption cross sections in common fluorophores,” Phys. Rev. Appl. 15, 044012 (2021). https://doi.org/10.1103/physrevapplied.15.04401247. D. Tabakaev, M. Montagnese, G. Haack, L. Bonacina, J.-P. Wolf, H. Zbinden, and R. T. Thew, “Energy-time-entangled two-photon molecular absorption,” Phys. Rev. A 103, 033701 (2021). https://doi.org/10.1103/physreva.103.03370148. T. Landes, M. Allgaier, S. Merkouche, B. J. Smith, A. H. Marcus, and M. G. Raymer, “Experimental feasibility of molecular two-photon absorption with isolated time-frequency-entangled photon pairs,” Phys. Rev. Res. 3, 033154 (2021). https://doi.org/10.1103/physrevresearch.3.03315449. S. Corona-Aquino, O. Calderón-Losada, M. Y. Li-Gómez, H. Cruz-Ramírez, V. Álvarez-Venicio, M. D. P. Carreón-Castro, R. de J. León-Montiel, and A. B. U’Ren, “Experimental study of the validity of entangled two-photon absorption measurements in organic compounds,” J. Phys. Chem. A 126, 2185–2195 (2022). https://doi.org/10.1021/acs.jpca.2c0072050. D. Tabakaev, A. Djorović, L. La Volpe, G. Gaulier, S. Ghosh, L. Bonacina, J.-P. Wolf, H. Zbinden, and R. T. Thew, “Spatial properties of entangled two-photon absorption,” Phys. Rev. Lett. 129, 183601 (2022). https://doi.org/10.1103/physrevlett.129.183601By considering the simplest case of a lossless 50:50 BS, with t = ±ir, where the i represents the π/2 phase difference between the transmitted and reflected beams, and t=r=1/2, we can readily find that the photon-coincidence rate for each of the above configurations (see the supplementary material and Ref. 5656. R.-B. Jin and R. Shimizu, “Extended Wiener–Khinchin theorem for quantum spectral analysis,” Optica 5, 93–98 (2018). https://doi.org/10.1364/optica.5.000093 for details) is given byR±(τ)=14∫dΩsdΩi|ϕ(Ωs,Ωi)|2+|ϕ(Ωi,Ωs)|2±ϕ(Ωs,Ωi)×ϕ*(Ωi,Ωs)e−i(Ωi−Ωs)τ±ϕ*(Ωs,Ωi)ϕ(Ωi,Ωs)×ei(Ωi−Ωs)τ,(4)RN(τ)=14∫dΩsdΩi|ϕ(Ωs,Ωi)|2+|ϕ(Ωi,Ωs)|2+ϕ(Ωs,Ωi)×ϕ*(Ωi,Ωs)+ϕ*(Ωs,Ωi)ϕ(Ωi,Ωs)×,(5)with R+(τ), R−(τ), and RNτ describing the coincidence-count rate for the single-port, two-port, and the N00N-state configurations, respectively. The function ϕ(Ωs, Ωi) represents the signal (s)-idler (i) joint spectral amplitude. The photon-pair state after interacting with the sample can be written, without loss of generality, as |ψ〉=∫dΩsdΩiϕ(Ωs,Ωi)â†(Ωs+ω0)â†(Ωi+ω0)|0〉. In writing Eqs. (4) and (5), we have assumed that the photon pairs are frequency degenerate, with a central frequency ω0. Their frequency deviations from ω0 are, thus, given by Ωj = ωj − ω0 (j = s, i). Moreover, note that we have introduced an external delay in one of the input ports of the BS. This delay allows us to perform a Hong–Ou–Mandel-like measurement on those photon pairs that are not absorbed by the sample. Interestingly, as we will describe below, the coincidence rate as a function of delay carries important information regarding the nature of the photon-pair losses.

B. ETPA as a two-photon spectral filter

Since its conception, eTPA has been described as a process in which correlated photon pairs satisfying the so-called two-photon resonance condition14,28,3714. H.-B. Fei, B. M. Jost, S. Popescu, B. E. A. Saleh, and M. C. Teich, “Entanglement-induced two-photon transparency,” Phys. Rev. Lett. 78, 1679–1682 (1997). https://doi.org/10.1103/physrevlett.78.167928. R. de J. León-Montiel, J. Svozilík, L. J. Salazar-Serrano, and J. P. Torres, “Role of the spectral shape of quantum correlations in two-photon virtual-state spectroscopy,” New J. Phys. 15, 053023 (2013). https://doi.org/10.1088/1367-2630/15/5/05302337. R. de J. León-Montiel, J. Svozilik, J. P. Torres, and A. B. U’Ren, “Temperature-controlled entangled-photon absorption spectroscopy,” Phys. Rev. Lett. 123, 023601 (2019). https://doi.org/10.1103/physrevlett.123.023601 are lost in order to drive a two-photon excitation of the absorbing medium. This means that the sample effectively acts as a frequency filter that removes specific resonance frequencies, thus modifying the joint spectral intensity (JSI) that characterizes the photon pairs. Mathematically, this transformation can be described by5151. F. Triana-Arango, G. Ramos-Ortiz, and R. Ramírez-Alarcón, “Entangled two-photon absorption detection through a Hong-Ou-Mandel interferometer,” http://arxiv:2202.05985 (2022).SΩs,Ωi=ϕΩs,Ωi2=fTPΩs,ΩiΦΩs,Ωi2,(6)where ΦΩs,Ωi and ϕΩs,Ωi stand for the joint amplitude of the photons before and after the interaction with the sample, respectively. The two-photon filter can be readily defined asfTPΩs,Ωi=1−exp−Ωs+Ωi2/2σTP2,(7)with σTP describing the two-photon filter bandwidth. Note that, as pointed out by Schlawin and Buchleitner,2121. F. Schlawin and A. Buchleitner, “Theory of coherent control with quantum light,” New J. Phys. 19, 013009 (2017). https://doi.org/10.1088/1367-2630/aa55ec the diagonal approximation of eTPA is valid, provided that the intermediate states that contribute to the two-photon excitation of the sample are located below the degenerate frequency of the photon pairs—a situation found, for instance, in tetraphenylporphyrin (H2TPP)1313. D.-I. Lee and T. Goodson, “Entangled photon absorption in an organic porphyrin dendrimer,” J. Phys. Chem. B 110, 25582–25585 (2006). https://doi.org/10.1021/jp066767g—and that the lifetime of the doubly excited state is longer than the intermediate states.As previously discussed, in realistic experiments, the two-photon beam may experience single-photon losses (whether at the sample or elsewhere in the setup) that remove, independently, signal or idler photons. These losses can then be accounted for by writing a single-photon filter of the formfs,iΩs,Ωi=1−exp−Ωs,i−Ωs,i02/2σs,i2.(8)Here, Ωs,i0 describes the central frequency deviations of the filter, whereas σs,i represents the single-photon filter bandwidth for the signal and idler modes, respectively. To understand the effects of the single- and two-photon filters, we assume the most general form for the initial joint spectral intensity (JSI) of the photons (see the supplementary material for details),ΦΩs,Ωi=EpΩs,ΩisincLNsΩs+NiΩi/2×exp−iLNsΩs+NiΩi/2.(9)In writing Eq. (9), we have used the definition sincx=sinx/x. EpΩs,Ωi=exp−2Tp2Ωs+Ωi2 corresponds to the Gaussian spectral shape of the classical pulsed pump, with a temporal duration Tp, which pumps the SPDC crystal of length L. Finally, Ns,i=kp′−ks,i′ describes the difference between the inverse group velocity of the pump and the signal and idler photons.Figure 2 shows some examples of filtered JSIs for (a) and (b) symmetric (resulting from type 0 or 1 SPDC with Ns=Ni) and (c) and (d) asymmetric (obtained from type-II SPDC with Ns≠Ni) initial two-photon states. Note that two-photon losses [Fig. 2(a)] are characterized by regions along the anti-diagonal of the JSI, whereas single-photon losses [Figs. 2(b) and 2(c)] correspond to regions along the horizontal or vertical lines for the signal and idler modes.

C. Two-photon coincidence rates in the presence of one- and two-photon losses

Having defined the specific form of the two-photon state following the interaction with the sample—where one- and two-photon-loss processes may be taking place—we are now ready to evaluate the coincidence rates for each of the previously described interferometric schemes. Here, we consider three cases: (i) no filter applied, (ii) a two-photon (eTPA) filter applied, and (iii) a linear filter applied to the idler photon. While other possible cases (including a linear filter applied to both photons and the application of both linear and nonlinear filters) were considered in our analysis, they do not contribute key, additional physical insights and were, thus, omitted from the presented results.

Figure 3 shows the coincidence rate as a function of delay τ for the [(a) and (d)] single-port, [(b) and (e)] two-port, and [(c) and (f)] N00N-state configurations. The top row shows the results for the initially symmetric (Ns=Ni) two-photon states, whereas the bottom row shows the results for the initially asymmetric (Ns≠Ni) states. For the symmetric case in the single- and two-port configurations, note that the no-filter (blue solid line) and eTPA (dashed red line) curves are fully overlapped, with the linear losses curve nearly overlapped with the other two, implying that one would be unable to determine the presence or absence of an eTPA sample from a transmission-based measurement. In striking contrast, for the N00N-state configuration, while the no-filter and linear filter curves are fully overlapped, the eTPA curve clearly deviates from the other two. This means that for symmetric two-photon states, the only scheme that is capable of witnessing eTPA is, indeed, the N00N-state configuration.Let us now turn to the asymmetric case (bottom row of Fig. 3). Note that both the two-photon (eTPA) and single photon filter curves exhibit differences with the no-filter curve. Nevertheless, because the two effects occur together, broadly with a similar behavior, one may be unable to distinguish eTPA from single-photon losses by monitoring changes in the coincidence peak/dip. It is worth mentioning that in Fig. 3, the bandwidths of the one- and two-photon loss filters are assumed to be the same. Of course, if one were to suppress single photon losses, e.g., through a considerable reduction in the single photon filter bandwidth, the resulting curve would more closely follow the no-filter curve, thus allowing one to discern the presence of the eTPA process. This case may, however, not be realizable as, in practice, it is challenging to reliably ensure the absence of linear losses. Remarkably, in the N00N-state configuration once again, the single-photon loss follows the no-filter signal, whereas eTPA clearly shows an altogether different behavior. In Figs. 3(c) and 3(f), the black-dashed line depicts the envelope of the interference pattern for the no-filter and one-photon-loss signals. Note that for the N00N configuration, on the one hand, the central lobe becomes narrower and, on the other hand, additional sidelobes appear. This non-trivial interference pattern is a result of the loss of frequencies in the JSI as a result of the two-photon interaction with the sample.5757. Y. Chen, R. de J. León-Montiel, and L. Chen, “Quantum interferometric two-photon excitation spectroscopy,” New J. Phys. 24, 113014 (2022). https://doi.org/10.1088/1367-2630/ac9d5d An interesting point to note in Figs. 3(d) and 3(e) is that the dip/peak is displaced from τ = 0 due to the asymmetry of the JSI [Eq. (9)]. By numerically analyzing the photon-coincidence rate for different values of Ns and Ni, one can find that the single-port and two-port coincidence rates for asymmetric two-photon states will exhibit a Ns−NiL/2 temporal delay shift.In general, the HOM visibility is unity for a perfectly symmetric JSI (upon the interchange of frequency arguments), while any asymmetry results in a reduction in visibility. Note that the visibility obtained in the single- and double-port configurations, in all of the cases considered above, may be understood in terms of the symmetry properties of the resulting overall JSI, including the effect of any single-photon or two-photon filters. We would like to point out that the above-discussed results are equivalent to those obtained when the BS reflection and transmission coefficients are frequency-dependent (see the supplementary material for details). Finally, we would like to remark that the N00N-state interferometry represents a transmission-based ETPA witness. However, we could expand our ETPA-certification toolbox by investigating equivalent fluorescence-based schemes, such as two-photon Ramsey interferometers.58–6158. G. S. Agarwal and M. O. Scully, “Ramsey spectroscopy with nonclassical light sources,” Phys. Rev. A 53, 467–470 (1996). https://doi.org/10.1103/physreva.53.46759. G. S. Agarwal, P. K. Pathak, and M. O. Scully, “Single-atom and two-atom Ramsey interferometry with quantized fields,” Phys. Rev. A 67, 043807 (2003). https://doi.org/10.1103/physreva.67.04380760. K. Qu and G. S. Agarwal, “Ramsey spectroscopy with squeezed light,” Opt. Lett. 38, 2563–2565 (2013). https://doi.org/10.1364/ol.38.00256361. F. Schlawin and S. Mukamel, “Two-photon spectroscopy of excitons with entangled photons,” J. Chem. Phys. 139, 244110 (2013). https://doi.org/10.1063/1.4848739