Probing exciton dynamics with spectral selectivity through the use of quantum entangled photons

In recent years, quantum light has been recognized as an important resource for the development of quantum metrology, where nonclassical features of light are exploited to enhance the precision and resolution of optical measurements beyond classical techniques Pirandola et al. (2018); Moreau et al. (2019). One of the striking features of quantum light is quantum entanglement. It is a phenomenon where the state of an entire system cannot be described as the product of the quantum states of its individual constituent particles. For instance, the use of photon entanglement has enabled ghost imaging Pittman et al. (1995), quantum imaging with undetected photons Lemos et al. (2014), quantum lithography Boto et al. (2000), cancellation of even-order dispersion Franson (1992), quantum optical coherence tomography Abouraddy et al. (2002); Nasr et al. (2003); Okano et al. (2015), and realization of sub-shot-noise microscopy Ono, Okamoto, and Takeuchi (2013); Triginer Garces et al. (2020); Casacio et al. (2021).

With recent advances in quantum optical technologies, entangled photons have become a promising avenue for the development of new spectroscopic techniques Gea-Banacloche (1989); Javanainen and Gould (1990); Saleh et al. (1998); Oka (2010); Schlawin and Mukamel (2013); de J León-Montiel et al. (2019); Fujihashi, Shimizu, and Ishizaki (2020); Debnath and Rubio (2020); Szoke et al. (2020); Mukamel et al. (2020); Muñoz, Frascella, and Schlawin (2021); Raymer, Landes, and Marcus (2021); Chen and Mukamel (2021); Dorfman et al. (2021); Asban, Dorfman, and Mukamel (2021); Asban and Mukamel (2021); Asban, Chernyak, and Mukamel (2022); Chen and Mukamel (2022); Albarelli et al. (2023); Li et al. (2023). It was experimentally demonstrated that nonclassical correlations between entangled photons had several advantages in spectroscopy, including sub-shot-noise absorption spectroscopy Tapster, Seward, and Rarity (1991); Brida, Genovese, and Berchera (2010); Matsuzaki and Tahara (2022) and increased two-photon absorption signal intensity Georgiades et al. (1995); Dayan et al. (2004); Lee and Goodson, III (2006); Upton et al. (2013); Varnavski and Goodson III (2020). Entanglement-induced two-photon transparency Fei et al. (1997) and suppression of exciton transport Schlawin et al. (2013) controlling the entanglement time, which is the hallmark of the non-classical photon correlation, were also theoretically investigated. In addition to the above advantages, the nonclassical correlations can be used to obtain spectroscopic signals with simpler optical systems compared with conventional methods Yabushita and Kobayashi (2004); Kalashnikov et al. (2016); Mukai et al. (2021); Arahata et al. (2022); Kalashnikov et al. (2017); Eshun et al. (2021). For example, infrared spectroscopy with visible-light source and detector was performed by exploiting entangled visible and infrared photons generated via parametric down-conversion (PDC) Kalashnikov et al. (2016); Mukai et al. (2021); Arahata et al. (2022). Furthermore, the Hong–Ou–Mandel interferometer with entangled photons allows the measurement of the dephasing time of molecules at the femtosecond time scale without the need for ultrashort laser pulses Kalashnikov et al. (2017); Eshun et al. (2021). Inspired by the capabilities of such nonclassical photon correlations, the applications to time-resolved spectroscopic measurements have been theoretically discussed Dorfman, Schlawin, and Mukamel (2014); Schlawin, Dorfman, and Mukamel (2016); Zhang et al. (2022); Fan, Ou, and Zhang (2023). The development of time-resolved spectroscopy that enhances the precision and resolution beyond classical techniques may lead to a better understanding of the mechanism of dynamical processes in complex molecules, such as photosynthetic light-harvesting systems. In contrast to many experimental and theoretical studies on entangled two-photon absorption, only a few theoretical studies have reported the application of entangled photons to time-resolved spectroscopic measurements. There is no comprehensive understanding of which nonclassical states of light are suitable for implementing real-time observation of dynamical processes in condensed phases and which nonclassical photon correlations allow the manipulation of nonlinear signals in a way that cannot be achieved with classical pulses.

In a previous study, we developed a theory of frequency-dispersed transmission measurement using entangled photon pairs generated via PDC pumped with a monochromatic laser Ishizaki (2020a). Especially, it was demonstrated that this measurement scheme enabled time-resolved spectroscopy based on monochromatic pumping when the entanglement time is sufficiently short. Chen et al. demonstrated that a similar scheme could be applied to monitor ultrafast electronic-nuclear motion at conical intersection Chen, Gu, and Mukamel (2022); Gu et al. (2023). Moreover, the simple model calculations in Refs. 53 and 56 suggested that for a finite value of entanglement time, the spectral distribution of the phase-matching function works as a sinc filter in signal processing Owen (2007), which can be used to selectively resolve a specific region of the spectra. Therefore, this spectral filtering mechanism is expected to simplify the interpretation of the spectra in complex molecules.

In this study, we theoretically propose a time-resolved spectroscopy scheme that selectively enhances specific signal contributions by harnessing the nonclassical correlations between entangled photons. We apply our spectroscopic scheme to a photosynthetic pigment-protein complex, and demonstrated that the phase-matching functions of the PDC in nonlinear crystals, such as periodically poled $\mathrm{KTiOPO_{4}}$ crystal and $\beta$-$\mathrm{BaB_{2}O_{4}}$ crystal, allow one to separately measure specific peaks of spectra by tuning the entanglement time and the central frequencies of the entangled photons. Furthermore, we investigated whether the spectral filtering mechanism could be implemented in the range of currently available entangled photon sources.

According to the phase matching conditions, the PDC process can be triggered in different geometries: One distinguishes type-I and type-II, and type-0 down-conversion. For simplicity, we consider a type-II PDC in a birefringent crystal Mandel and Wolf (1995) because the wave vector mismatch can be well approximated to linear order in frequency Grice and Walmsley (1997); Keller and Rubin (1997), as described in Eq. (3). The intensity and normalized spectral envelope of the pump laser are denoted as $I_{\mathrm{p}}$ and $E_{\mathrm{p}}(\omega)$, respectively. A photon of frequency $\omega_{\mathrm{p}}$ in the pump laser is split into a pair of entangled photons whose frequencies $\omega_{\mathrm{H}}$ and $\omega_{\mathrm{V}}$ must satisfy $\omega_{\mathrm{p}}=\omega_{\mathrm{H}}+\omega_{\mathrm{V}}$ because of energy conservation. The polarizations of the generated twins are orthogonal to each other and characterized by horizontal (H) and vertical (V) polarizations. In the weak-down conversion regime, the quantum state of the twin is expressed as Grice and Walmsley (1997); Keller and Rubin (1997)

where the operator $\hat{a}_{\lambda}^{\dagger}(\omega)$ creates a photon of frequency $\omega$ and polarization $\lambda$ and the function $f(\omega_{\mathrm{H}},\omega_{\mathrm{V}})$ is the two-photon amplitude. For simplicity, in Eq. (1), we neglected the spatial variation of the two-photon amplitude and the spatial propagation direction is selected by the collinear configuration Schlawin, Dorfman, and Mukamel (2018). In the following, we consider the electric fields inside a one-dimensional (1D) nonlinear crystal of length $L$. Thus, the two-photon amplitude is given by $f(\omega_{\mathrm{H}},\omega_{\mathrm{V}})=\zeta E_{\mathrm{p}}(\omega_{\mathrm{H}}+\omega_{\mathrm{V}})\mathrm{sinc}[\Delta k(\omega_{\mathrm{H}},\omega_{\mathrm{V}})L/2]$, where $E_{\mathrm{p}}(\omega_{\mathrm{H}}+\omega_{\mathrm{V}})$ is the normalized pump envelope and the sinc function originates from phase-matching Boyd (2003); Graffitti et al. (2018). Note that here $f(\omega_{\mathrm{H}},\omega_{\mathrm{V}})$ is not necessarily normalized, as $\int d\omega_{\mathrm{H}}\int d\omega_{\mathrm{V}}|f(\omega_{\mathrm{H}},\omega_{\mathrm{V}})|^{2}=\zeta^{2}$. Expanding the wave vector mismatch $\Delta k(\omega_{\mathrm{H}},\omega_{\mathrm{V}})$ to first order in the frequencies $\omega_{\mathrm{H}}$ and $\omega_{\mathrm{V}}$ around the center frequencies of the generated beams, $\bar{\omega}_{\mathrm{H}}$ and $\bar{\omega}_{\mathrm{V}}$, we obtain $\Delta k(\omega_{\mathrm{H}},\omega_{\mathrm{V}})=(\omega_{\mathrm{H}}-\bar{\omega}_{\mathrm{H}})T_{\mathrm{H}}+(\omega_{\mathrm{V}}-\bar{\omega}_{\mathrm{V}})T_{\mathrm{V}}$ with $T_{\lambda}=(v_{\mathrm{p}}^{-1}-v_{\lambda}^{-1})L$, where $v_{\mathrm{p}}$ and $v_{\lambda}$ are the group velocities of the pump laser and one of the generated beams at central frequency $\bar{\omega}_{\lambda}$, respectively Keller and Rubin (1997); Rubin et al. (1994). The central frequencies and group velocities are evaluated using the Sellmeier equations Dmitriev, Gurzadyan, and Nikogosyan (2013), which provide empirical relations between the refractive indices of the crystals and the frequencies of the generated beams. In this study, we address the case of monochromatic pumping $E_{\mathrm{p}}(\omega_{\mathrm{H}}+\omega_{\mathrm{V}})=\delta(\omega_{\mathrm{H}}+\omega_{\mathrm{V}}-\omega_{\mathrm{p}})$. Thus, the two-photon amplitude is recast as

The so-called entanglement time $T_{\mathrm{e}}=\lvert T_{\mathrm{H}}-T_{\mathrm{V}}\rvert$ is the maximum time difference between twin photons leaving the crystal Saleh et al. (1998). The positive frequency component of the field operator is given by: $\hat{\mathbf{E}}_{\lambda}^{+}(t)=\mathbf{e}_{\lambda}\int^{\infty}_{0}d\omega\,\mathcal{E}(\omega)\hat{a}_{\lambda}(\omega)e^{-i\omega t},$ where $\mathcal{E}(\omega)\propto i\sqrt{\omega}$ and the negative frequency component is $\hat{\mathbf{E}}_{\lambda}^{-}(t)=[\hat{\mathbf{E}}_{\lambda}^{+}(t)]^{\dagger}$. The unit vectors $\mathbf{e}_{\mathrm{H}}$ and $\mathbf{e}_{\mathrm{V}}$ indicate the directions of the horizontal and vertical polarizations, respectively. We adopt the slowly varying envelope approximation, in which the bandwidth of the field is assumed to be negligibly narrow in comparison with the central frequency Mandel and Wolf (1995). This approximation allows treating the factor $\mathcal{E}(\omega)$ as a constant $\mathcal{E}(\omega)\simeq\mathcal{E}(\bar{\omega}_{\lambda})$. All other constants are merged into a factor $\zeta\propto I_{\mathrm{p}}^{1/2}L\mathcal{E}(\bar{\omega}_{\mathrm{H}})\mathcal{E}(\bar{\omega}_{\mathrm{V}})$, which is regarded as the conversion efficiency of the PDC.

We consider the setup shown in Fig. 1. Twin photons were split using a polarized beam splitter. Although the relative delay between horizontally and vertically polarized photons is innately determined by the entanglement time, the delay interval is further controlled by adjusting the path difference between the beams Hong, Ou, and Mandel (1987); Franson (1989). This controllable delay is denoted by $\Delta t$ in this study. Direct observation of time-frequency duality of biphotons over a delay time of at least a few picoseconds has been experimentally demonstrated Jin, Saito, and Shimizu (2018); MacLean, Donohue, and Resch (2018). The field operator is expressed as $\hat{\mathbf{E}}(t)=\hat{\mathbf{E}}_{\mathrm{H}}(t+\Delta t)+\hat{\mathbf{E}}_{\mathrm{V}}(t)$, indicating that the horizontally and vertically polarized photons act as the pump and probe field for the molecules, respectively. The probe field transmitted through the sample is frequency-dispersed, and the change in the transmitted photon number is registered as a function of the frequency $\omega$, pump frequency $\omega_{\mathrm{p}}$, and external delay $\Delta t$, yielding the signal $S(\omega,\Delta t;\omega_{\mathrm{p}})$.

The Hamiltonian used to describe this pump–probe process is written as $\hat{H}=\hat{H}_{\rm mol}+\hat{H}_{\rm field}+\hat{H}_{\rm mol-field}$. The first term gives the Hamiltonian of the photoactive degrees of freedom (DOFs) in molecules. The second term describes the free electric field. In this work, the electronic ground state $\lvert 0\rangle$, single-excitation manifold $\{\lvert e_{\alpha}\rangle\}$, and double-excitation manifold $\{\lvert f_{\bar{\gamma}}\rangle\}$ are considered as photoactive DOFs. The overline of the subscripts indicates the state in the double-excitation manifold. The optical transitions are described by the dipole operator $\hat{\bm{\mu}}=\hat{\bm{\mu}}_{+}+\hat{\bm{\mu}}_{-}$, where $\hat{\bm{\mu}}_{-}=\sum_{\alpha}\bm{\mu}_{\alpha 0}\lvert 0\rangle\langle e_{\alpha}\rvert+\sum_{\alpha\bar{\gamma}}\bm{\mu}_{\bar{\gamma}\alpha}\lvert e_{\alpha}\rangle\langle f_{\bar{\gamma}}\rvert$ and $\hat{\bm{\mu}}_{+}=[\hat{\bm{\mu}}_{-}]^{\dagger}$. The rotating-wave approximation enables the expression of the molecule-field interaction as $\hat{H}_{\rm mol-field}(t)=-\hat{\bm{\mu}}_{-}\cdot\hat{\mathbf{E}}^{+}(t)-\hat{\bm{\mu}}_{+}\cdot\hat{\mathbf{E}}^{-}(t)$. The signal is expressed as Dorfman, Schlawin, and Mukamel (2016)

where $\hat{\rho}(-\infty)=\lvert 0\rangle\langle 0\rvert\otimes\lvert\psi_{\text{twin}}\rangle\langle\psi_{\text{twin}}\rvert$ and $\hat{\mathbf{E}}_{\lambda}^{-}(\omega)=\int_{0}^{\infty}dt\,\hat{\mathbf{E}}_{\lambda}^{-}(t)e^{-i\omega t}$. We expand the density operator $\hat{\rho}(t)$ with respect to $\hat{H}_{\rm mol-field}$ to the third order, resulting in the sum of eight contributions classified as stimulated emission (SE), ground-state bleaching (GSB), excited-state absorption (ESA), and double quantum coherence (DQC). The DQC signal decays rapidly in comparison with the others when $\Delta t$ is sufficiently long compared to the timescale of environmental reorganization (see Section S1 of Supplementary Material for details); hence, the DQC is disregarded in this work. Each contribution is expressed as follows:

where $x$ indicates rephasing (r) or non-rephasing (nr), and $y$ indicates the GSB, SE, or ESA. Here, $R_{x,y}^{\lambda_{4}\lambda_{3}\lambda_{2}\lambda_{1}}(s_{3},s_{2},s_{1})$ and $C_{x,y}^{\lambda_{4}\lambda_{3}\lambda_{2}\lambda_{1}}(\omega,t;s_{3},s_{2},s_{1})$ are the third-order response functions of the molecules and four-body correlation functions of the field operators, respectively.

For demonstration purposes, we focused on rephasing the SE contribution. Details of the GSB and ESA are provided in Section S2 of Supplementary Material. The rephasing SE contribution is given by $R_{\mathrm{r},\mathrm{SE}}^{\lambda_{4}\lambda_{3}\lambda_{2}\lambda_{1}}(s_{3},s_{2},s_{1})=(i/\hbar)^{3}\sum_{\alpha\beta\gamma\delta}\langle\mu_{\delta 0}^{\lambda_{4}}\mu_{\gamma 0}^{\lambda_{3}}\mu_{\beta 0}^{\lambda_{2}}\mu_{\alpha 0}^{\lambda_{1}}\rangle G_{\gamma 0}(s_{3})G_{\gamma\delta\leftarrow\alpha\beta}(s_{2})G_{0\beta}(s_{1})$. Here, we have defined $\langle\mu_{\epsilon\zeta}^{\lambda_{4}}\mu_{\gamma\delta}^{\lambda_{3}}\mu_{\beta 0}^{\lambda_{2}}\mu_{\alpha 0}^{\lambda_{1}}\rangle=\langle(\bm{\mu}_{\epsilon\zeta}\cdot\mathbf{e}_{\lambda_{4}})(\bm{\mu}_{\gamma\delta}\cdot\mathbf{e}_{\lambda_{3}})(\bm{\mu}_{\beta 0}\cdot\mathbf{e}_{\lambda_{2}})(\bm{\mu}_{\alpha 0}\cdot\mathbf{e}_{\lambda_{1}})\rangle_{\rm ori}$, where the brackets $\langle\dots\rangle_{\rm ori}$ represent the average over molecular orientations Schlau-Cohen, Ishizaki, and Fleming (2011); Schlawin (2022). The matrix element of the time-evolution operator $G_{\gamma\delta\leftarrow\alpha\beta}(t)$ is defined by $\rho_{\gamma\delta}(t)=\sum_{\alpha\beta}G_{\gamma\delta\leftarrow\alpha\beta}(t)\rho_{\alpha\beta}(0)$, and $G_{\alpha\beta}(t)$ is the abbreviation of $G_{\alpha\beta\leftarrow\alpha\beta}(t)$. By substituting the response function and field correlation function Ishizaki (2020a) into Eq. (5), we obtain the following SE signal:

where $\eta=\zeta^{2}\mathcal{E}(\bar{\omega}_{\mathrm{H}})^{2}\mathcal{E}(\bar{\omega}_{\mathrm{V}})^{2}/\hbar^{3}$. The second term in Eq. (6) originates from a field commutator. This term does not depend on $\Delta t$. Therefore, the contribution to the signal can be ignored by considering the difference spectrum:

The Fourier–Laplace transform of $G_{\alpha\beta}(t)$ is written as $G_{\alpha\beta}[\omega]$, and $F_{\gamma\delta\leftarrow\alpha\beta}^{{\lambda_{4}\lambda_{3}\lambda_{2}\lambda_{1}}}(\omega,\Delta t;s_{1})$ is defined as

where $D_{n}(t)=(2\pi)^{-1}\int^{\infty}_{-\infty}d\omega\,e^{-i\omega t}[\Phi(\omega)]^{n}$. As discussed in Ref. 53, Eq. (8) for $\Delta t>T_{\mathrm{e}}/2$ can be simplified as

which is independent of $\omega$. To obtain the information contents of the signal, we assume that the time evolution in the $t_{1}$ and $t_{3}$ periods is described as $G_{\alpha\beta}(t)=e^{-(i\omega_{\alpha\beta}+\epsilon_{+})t}$, thereby leading to the expression of the rephasing SE signal, $G_{\gamma 0}[\omega]F_{\gamma\delta\leftarrow\alpha\beta}^{{\lambda_{4}\lambda_{3}\lambda_{2}\lambda_{1}}}(\omega,\Delta t;0)G_{0\beta}[\omega_{\mathrm{p}}-\omega]\propto\delta(\omega-\omega_{\gamma 0})\delta(\omega_{\mathrm{p}}-\omega-\omega_{\beta 0})G_{\gamma\delta\leftarrow\alpha\beta}(\Delta t)$. It can be understood that the non-classical correlation between the entangled photon pair generated via the PDC pumped with a monochromatic laser restricts the possible optical transitions, ($0\to e_{\beta}$, $0\to e_{\gamma}$), for a given pump frequency. Therefore, Eqs. (6) and (9) indicate that the state-to-state dynamics in the molecules are temporally resolved by sweeping the external delay $\Delta t$ in the time region longer than half of the entanglement time, $T_{\mathrm{e}}/2$. It should be mentioned that phenomena similar to the specific selective excitation described above have been discussed in Ref. 42 in the context of manipulation of two-excitation distributions by the non-classical photon correlation.

Notably, in the limit $T_{\mathrm{e}}\to 0$, the third-order signal in Eq. (4) corresponds to the spectral information along the anti-diagonal line of the absorptive 2D spectrum obtained using the photon-echo technique in the impulsive limit:

except for the $\Delta t$-independent term Ishizaki (2020a), as shown in Fig. 2 (the explicit expression of the 2D photon-echo spectrum, $\mathcal{S}_{\rm 2D}(\omega_{3},\Delta t,\omega_{1})$, is given in Appendix B). It is noted that the sign of the quantum spectrum, $S(\omega,\Delta t;\omega_{\mathrm{p}})$, is the opposite of the sign of the classical 2D photon-echo spectrum, $\mathcal{S}_{\rm 2D}(\omega,\Delta t,\omega_{\mathrm{p}}-\omega)$, as shown in Eq. (10). In the following numerical results, the spectrum $S(\omega,\Delta t;\omega_{\mathrm{p}})$ is plotted multiplied by a minus sign for clarity. Equation (10) also indicates that the pump-probe signal shows no collective two-particle contributions (see Section S3 of Supplementary Material for details). This result is consistent with the arguments in Refs. 73; 74.

Furthermore, the phase matching function $\Phi(\omega-\bar{\omega}_{\mathrm{V}})=\mathrm{sinc}[(\omega-\bar{\omega}_{\mathrm{V}})T_{\mathrm{e}}/2]$ in Eq. (6) can selectively enhance a specific spectral region of the signal by varying the center frequency $\bar{\omega}_{\mathrm{V}}$ of the generated beam. The width at half maximum of $\Phi(\omega-\bar{\omega}_{\mathrm{V}})$ is approximately given by $\lvert\omega-\bar{\omega}_{\mathrm{V}}\rvert\simeq 4/T_{\mathrm{e}}$. Interestingly, this corresponds to a sinc filter in signal processing Owen (2007). Therefore, the entanglement time $T_{\mathrm{e}}$ plays a dual role of the knob for controlling the accessible time region of the dynamics in molecules, $\Delta t>T_{\mathrm{e}}/2$, and the degree of spectral selectivity, $\lvert\omega-\bar{\omega}_{\mathrm{V}}\rvert\simeq 4/T_{\mathrm{e}}$. It is noted that similar spectral filtering can be realized with classical light O’shea et al. (2001), and has been utilized for selective excitation in multidimensional spectra Tollerud, Hall, and Davis (2014).

In the following, we discuss roles of the entanglement time on the temporal resolution and spectral selectivity through numerical investigations of the signals in Eqs. (6)–(8) and Eqs. (S7)–(S11) of the Fenna-Matthews-Olson (FMO) pigment-protein complex in the photosynthetic green sulfur bacterium Chlorobium tepidum Li et al. (1997); Camara-Artigas, Blankenship, and Allen (2003); Tronrud et al. (2009) (Fig. 1C). Due to its relatively small size, it has been widely studied experimentally and theoretically as a prototypical system for discussing photosynthetic energy transfer using nonlinear optical spectroscopy Freiberg et al. (1997); Brixner et al. (2005); Engel et al. (2007); Fujihashi, Fleming, and Ishizaki (2015). Our model includes seven single-excitation states $\{e_{1},\cdots,e_{7}\}$ and 21 double-excitation states $\{f_{\bar{1}},\cdots,f_{\bar{21}}\}$ (for details on the model, see Appendix A). The matrix elements $G_{\alpha\beta}[\omega]$ and $G_{\gamma\delta\leftarrow\alpha\beta}(t)$ in Eq. (6) and Eqs. (S7)–(S11) are calculated using the cumulant expansion for the fluctuations in the electronic energies and the modified Redfield theory Zhang et al. (1998) (see Appendix A).

In this work, we explored the roles of entanglement time for temporal resolution and spectral selectivity through numerical investigations of the entangled photon spectroscopy of FMO complexes. The frequency-dispersed transmission measurement with entangled photons considered in our study exhibits three interesting features. First, nonclassical photon correlation enables time-resolved spectroscopy with monochromatic pumping Ishizaki (2020a). The temporal width of the entangled photon pair is determined by the entanglement time, and hence can be tuned by the crystal length. For example, as illustrated in Fig. 5A, the temporal width of the entangled photon pairs generated with the BBO crystal with a thickness of $0.056\,{\rm mm}$ was a few femtoseconds. Therefore, transmission measurement can temporally resolve the dynamic processes of molecular systems occurring at femtosecond timescales without requiring sophisticated control of the temporal delay between femtosecond laser pulses. Second, the spectral distribution of the phase-matching function can function as a frequency filter, which removes all optical transitions that fall outside the spectral width. The spectral distribution of the phase-matching function can be manipulated by changing the crystal length and phase-matching angle. Thus, specific peaks in crowded 2D spectra can be selectively enhanced or suppressed by controlling the phase-matching function, as in laser spectroscopic experiments using narrow-band pulses. As demonstrated in Fig. 5, the selective enhancement of specific peaks in the congested 2D spectrum of the FMO complex can be achieved using the phase-matching function of the PDC process in BBO crystals with crystal lengths ranging from $0.056\,{\rm mm}$ to $1.68\,{\rm mm}$, which has been used experimentally Branning, Migdall, and Sergienko (2000); Dayan et al. (2004); Yabushita and Kobayashi (2004); Lee and Goodson, III (2006); Eshun et al. (2021). Therefore, spectral filtering is feasible using current quantum optical techniques for generating entangled photons. Third, the spectral distribution of the phase-matching function strongly depends on the properties of the nonlinear crystal. As shown in Figs. 5 and 6, the spectral filter effects can easily be adjusted by changing the nonlinear crystals and/or their properties. Although we considered only BBO and PPKTP crystals in this study, there is a wide range of nonlinear crystals that have been used for PDC in the near-infrared and visible regions Dmitriev, Gurzadyan, and Nikogosyan (2013). Therefore, we anticipate that transmission measurement can be applied not only to FMO complexes but also to other photosynthetic pigment-protein complexes such as the photosystem II reaction center Romero et al. (2014); Fuller et al. (2014); Fujihashi, Higashi, and Ishizaki (2018) by finding an appropriate nonlinear crystal corresponding to the spectral range of the molecular system of interest.

The feasibility of entangled two-photon spectroscopy was recently questioned by several research groups Landes et al. (2021); Parzuchowski et al. (2021). The quantitative estimates using the upper bound on the isolated-entangled-pair cross section indicate that realistic sample concentrations event rates are orders of magnitude below the detection threshold of typical photon-counting systems Raymer, Landes, and Marcus (2021). The same difficulties are expected to be faced in the case of the transmission measurement considered in our study. One solution to overcome vanishingly weak nonlinear signals is to use high-gain squeezed vacuum states Cutipa and Chekhova (2022). As demonstrated in Section S8 of Supplementary Material, at least when the entanglement time is sufficiently short compared with characteristic timescales of the dynamics under investigation, the transmission measurement is capable of temporally resolving the excitation dynamics with high-gain squeezed vacuum states. Whether the measurement can be performed under other parameter conditions such as for long entanglement times is a subject for future research.

In the present work, we did not concentrate on the exploitation of the polarization control of entangled photon pairs. Polarization-controlled measurements have been considered a beneficial technique for separating crowded 2D spectra Hochstrasser (2001); Zanni et al. (2001); Dreyer, Moran, and Mukamel (2003); Schlau-Cohen et al. (2012); Westenhoff et al. (2012). In this regard, exploiting the polarization of entangled photons has the potential to improve the resolution of quantum spectroscopy further. Moreover, hyperentanglement in frequency-time and polarization can be generated by the type-II PDC process Kwiat (1997). Further research on the use of such more elaborately controlled quantum states of light for optical spectroscopy may lead to the development of novel time-resolved spectroscopic measurements with precision and resolution beyond the limits imposed by the laws of classical physics.