Two-dimensional fluorescence spectroscopy with entangled photons and time- and frequency-resolved two-photon coincidence detection

Abstract

Recent theoretical studies highlight how nonclassical photon correlations in entangled photon pairs can selectively address nonlinear optical pathways. However, the resulting signals are typically too weak for practical time-resolved experiments. Here, we propose two-dimensional (2D) time-resolved fluorescence spectroscopy that exploits these correlations and operates with current single-photon detectors. The method provides two advantages over conventional 2D electronic spectroscopy: (i) It yields 2D spectra without phase-stable multipulse control, relying instead on heralded twin-photon correlations, and (ii) it simplifies spectra by isolating the contribution that is spectroscopically equivalent to stimulated emission, thereby suppressing ground-state bleaching and excited-state absorption. Numerical calculations for a natural pigment-protein complex–inspired trimer show that this pathway selectivity enables the extraction of rich information on energy transfer dynamics. These results indicate a feasible route to real-time observation of molecular dynamics using entangled photon pairs.

Quantum entangled photons yield two-dimensional fluorescence spectra without phase-stable multipulse control.

INTRODUCTION

The potential of quantum light to serve as a valuable resource for advancing innovative measurement techniques in spectroscopy has garnered increasing attention (1–19). For instance, entangled photons have the potential to enable substantial advances in the linear scaling of two-photon absorption (2–7), subshot noise absorption spectroscopy (10, 11), and infrared spectroscopy with visible light (12–14). Given these promising capabilities, researchers have begun exploring the feasibility of using entangled photons in time-resolved spectroscopy, including coherent multidimensional optical spectroscopy (20–38). For example, coincidence detection of entangled photon pairs has been shown to enhance the signal-to-noise ratio of pump-probe spectroscopy (22). The nonclassical photon correlation between entangled photons has the potential to enable time-resolved spectroscopy with monochromatic pumping (25–28) and to selectively target specific nonlinear optical processes in a nonlinear optical response (29–35, 38).

Despite this promise, the small magnitude of molecular nonlinear susceptibilities, together with the low conversion efficiency of spontaneous parametric downconversion (SPDC) (10⁻⁶ to 10⁻¹⁰), severely limits the strength of nonlinear signals when molecules are driven directly by entangled photon pairs (39–41). Consequently, time-resolved quantum-light spectroscopies that rely on two-photon irradiation (25, 33) are limited by low signal levels, which has hindered experimental demonstrations.

An alternative approach to overcoming the constraints associated with nonlinear optical signals is the use of entangled photons in time-resolved fluorescence measurements (42–47). Time-resolved fluorescence spectroscopy facilitates the acquisition of data pertaining to third-order nonlinear optical responses by detecting fluorescence signals from the excited states of molecules induced by light irradiation. This method requires irradiating only one of the entangled photon pairs onto the molecules (43); this ensures that the signal intensities are sufficiently strong to be detected using current photon detection technology. Researchers have successfully conducted fluorescence lifetime measurements with monochromatic pumping (44–46) by leveraging the nonclassical correlations of entangled photon pairs. In addition, single-photon absorption events in photosynthetic complexes have been observed (47).

Despite recent progress in fluorescence-based quantum measurements, achieving simultaneous time- and frequency-resolved measurements with entangled photons has been hindered by the absence of time stamping in conventional single-photon detectors such as charge-coupled devices, which leads to prohibitively long acquisition times. In our recent work (48), we developed a time-resolved single-photon spectroscopy method using a delay-line-anode single-photon detector (DLD) that records both the arrival time and position of each photon with subnanosecond precision, thereby enabling time- and frequency-resolved measurements without frequency scanning. For instance, the two-dimensional (2D) frequency distribution of entangled photon pairs generated from a CuCl semiconductor single crystal can be measured within a few minutes—a process that is thousands of times faster than conventional methods.

Building on this foundation, we theoretically propose here a quantum spectroscopic method based on time-resolved fluorescence detection that is fully compatible with existing photon-detection hardware. By exploiting the nonclassical correlations of entangled photons, the method realizes time-resolved spectroscopy—including 2D electronic spectroscopy (2DES) (49–52)—without the need for multiple pulsed lasers. Moreover, we demonstrate that the spectral complexity is substantially reduced because the signal is governed solely by the nonlinear optical process of spontaneous emission. Thus, the proposed method inherits the conceptual advantages of previously developed quantum spectroscopies (25, 33) while achieving practically measurable signal intensities. In particular, implementing the protocol using the detector platform reported in (48) is expected to yield 2D spectral information with experimentally achievable signal intensities and within practical acquisition times (see Materials and Methods or “Note added” in Discussion for details), marking an important step toward experimentally realizable time-resolved quantum-light spectroscopy.

RESULTS

Setup

In this study, we consider the measurement shown in Fig. 1A. A pulsed laser pumps a nonlinear crystal, generating frequency-entangled photon pairs via type II SPDC. A beam sampler reflects a small fraction of the pump to a photodiode, which provides a reference signal to start the timer of a time-correlated single-photon counting (TCSPC) module. The photon pair emerging from the crystal is separated in a polarizing beam splitter (PBS). The idler photon is dispersed by a spectrometer and detected with a single-photon camera [e.g., a DLD (48)], yielding its frequency ω_I and arrival time $t_{\mathrm{I}}$. The signal photon passes through the PBS to excite molecules in a microscope setup; the resulting fluorescence is likewise dispersed by a spectrometer and detected with a second single-photon camera, providing the fluorescence photon’s frequency ω_F and arrival time $t_{\mathrm{F}}$. The TCSPC electronics compute the temporal correlations between the idler and fluorescence photons. By repeating this procedure and accumulating coincidences, we obtain a time-resolved 2D two-photon spectrum. In our proposal, the coincidence counting of the pump pulse, fluorescence, and idler photons serves a role analogous to the ultrashort pulse in the fluorescence upconversion (53). This enables the observation of subpicosecond excited-state dynamics by effectively tagging the timing of the emission event, provided that the detector time resolution is sufficiently short.

Fig. 1. Concept of the proposed quantum spectroscopy and model system.

(A) Schematic of the optical setup. A pulsed laser pumps a nonlinear crystal to generate entangled photon pairs. The signal photon excites the sample, and the resulting fluorescence is analyzed with a spectrometer and a single-photon camera [e.g., DLD (48)]. The idler photon is routed to a reference arm for joint spectral and temporal measurements. Coincidences between the idler and fluorescence channels are recorded. BS, beam splitter. (B) Joint temporal intensity in Eq. 1. By virtue of the Fourier transform relation, the quantum state of light in Eq. 1 exhibits negative time correlation, and its joint temporal amplitude approximately satisfies the condition $t_{\mathrm{S}}=-t_{\mathrm{I}}$. (C) Schematic showing the temporal relationships of the arrival times of the signal, idler, and fluorescence photons in the measurement of (A). Because of the correlation $t_{\mathrm{S}}=-t_{\mathrm{I}}$ from (B), the excitation-fluorescence delay $\mathrm{\Delta }{t}_{\text{FS}}=t_{\mathrm{F}}-t_{\mathrm{S}}$ can be reconstructed from the measured detection times, $\mathrm{\Delta }{t}_{\text{FS}}=t_{\mathrm{F}}+t_{\mathrm{I}}$. Thus, scanning $t_{\mathrm{F}}+t_{\mathrm{I}}$ enables the direct probe of the excited-state dynamics. (D) Monomer subunit of the FMO pigment-protein complex from C. tepidum with seven BChl molecules; BChls 3, 6, and 7 are highlighted in green. The pigments are numbered as in PDB file 3ENI (64).

The essential feature of this spectroscopic scheme is the use of frequency and temporal correlations between the entangled photons. As we show below, the frequency correlation enables us to infer the frequency of the signal photon that interacts with the molecule from the measured frequency of the idler photon. Simultaneously, the temporal correlation allows us to determine the emission time, referenced to the excitation event, from the arrival time of the idler photon, as presented in Fig. 1 (B and C). Exploiting these correlations simplifies the optical setup and yields practical advantages; in particular, they enable time-resolved 2D spectroscopy without the need to control multiple pulsed lasers.

Furthermore, because coincidence detection heralds the arrival of the signal photon, the method enables time-resolved spectroscopy at the single-photon level. This capability is particularly valuable for studies of photosynthetic complexes under very low excitation (47) and for avoiding artifacts associated with multiphoton excitation (34), motivating the development of single-photon-level time-resolved spectroscopic techniques.

Quantum states of entangled twin

For simplicity, we assumed the following: (i) the weak downconversion regime; (ii) degenerate type II SPDC; (iii) the SPDC process is perfectly phase matched at the central frequency ω_p, and the group velocity dispersion through the nonlinear medium is negligible (54); (iv) symmetric group-velocity matching, where the sum of the inverse group velocities of the signal and idler photons is equal to twice the pump inverse group velocity (55, 56); and (v) impulsive pump limit. Under these assumptions, the two-photon state generated by the SPDC, often referred to as the difference-beam state (57), can be written as

$$\mid {\mathrm{\psi }}_{\text{twin}}\rangle =\mathrm{\zeta }\iint d{\mathrm{\omega }}_{\mathrm{S}}d{\mathrm{\omega }}_{\mathrm{I}}\mathrm{\varphi }({\mathrm{\omega }}_{\mathrm{S}}-{\mathrm{\omega }}_{\mathrm{I}}){\hat{a}}_{\mathrm{S}}^{†}\left({\mathrm{\omega }}_{\mathrm{S}}\right){\hat{a}}_{\mathrm{I}}^{†}\left({\mathrm{\omega }}_{\mathrm{I}}\right)\mid \text{vac}\rangle$$ (1)

where ${\hat{a}}_{\mathrm{S}}^{†}\left(\mathrm{\omega }\right)$ and ${\hat{a}}_{\mathrm{I}}^{†}\left(\mathrm{\omega }\right)$ are the creation operators for the signal and idler photons of frequency $\mathrm{\omega }$, respectively, and the prefactor $\mathrm{\zeta }$ represents the conversion efficiency of the SPDC process. The phase matching function is expressed as $\mathrm{\varphi }\left(\mathrm{\omega }\right)=\text{sinc}(\mathrm{\omega }{T}_{\mathrm{e}}/4)$, where the parameter $T_{\mathrm{e}}$ denotes the so-called entanglement time (4). The detailed derivation is presented in section S1.

As illustrated in Fig. 2, the ratio of the inverse entanglement time to the pump bandwidth determines the sign of the frequency correlations between the photon pairs. When the pump bandwidth is much shorter than the inverse entanglement time, the pair exhibits negative frequency correlations. In contrast, when the pump bandwidth is much longer than the inverse entanglement time, the pair exhibits positive frequency correlations, as presented in Fig. 2B. The state in Eq. 1 corresponds to the latter case: For relatively long entanglement times, the joint spectrum in Eq. 1 is concentrated near ${\mathrm{\omega }}_{\mathrm{S}}\simeq {\mathrm{\omega }}_{\mathrm{I}}$.

Fig. 2. Frequency-correlation between the entangled photon pair.

(A) Case of quasi-CW laser. (B) Case of pulsed laser. All panels are computed from eq. S4 in section S1. The panels on the left present the pump envelope ${\mathrm{\alpha }}_{\mathrm{p}}({\mathrm{\omega }}_1+{\mathrm{\omega }}_2)$, the panels on the middle present the phase matching function $\mathrm{\varphi }({\mathrm{\omega }}_1,{\mathrm{\omega }}_2)$, and the panels on the right are the two-photon amplitude $f({\mathrm{\omega }}_1,{\mathrm{\omega }}_2)={\text{ζα}}_{\mathrm{p}}({\mathrm{\omega }}_1+{\mathrm{\omega }}_2)\mathrm{\varphi }({\mathrm{\omega }}_1,{\mathrm{\omega }}_2)$. For clarity, panels (A) and (B) are plotted using different values of the entanglement time $T_{\mathrm{e}}$ to highlight the contrast between positive and negative frequency correlations.

When the entanglement time is sufficiently long compared to the timescales of the dynamics under investigation, that is, $T_{\mathrm{e}}\to \infty$, the phase matching function simplifies to $\mathrm{\varphi }({\mathrm{\omega }}_{\mathrm{S}}-{\mathrm{\omega }}_{\mathrm{I}})\simeq \mathrm{\delta }({\mathrm{\omega }}_{\mathrm{S}}-{\mathrm{\omega }}_{\mathrm{I}})$, leading to a simplified expression for the quantum state of the generated twin

$$\mid {\mathrm{\psi }}_{\text{twin}}\rangle =\mathrm{\zeta }\int d\mathrm{\omega }{\hat{a}}_{\mathrm{S}}^{†}\left(\mathrm{\omega }\right){\hat{a}}_{\mathrm{I}}^{†}\left(\mathrm{\omega }\right)\mid \text{vac}\rangle$$ (2)

This indicates that the frequency of the signal photon can be precisely reconstructed by measuring the frequency of the idler photon. Therefore, this study considered the limit $T_{\mathrm{e}}\to \infty$, as opposed to the continuous wave (CW) pumping case wherein the limit $T_{\mathrm{e}}\to 0$ results in strong frequency correlations between the twin photons. The scenario wherein $T_{\mathrm{e}}$ takes a finite value is discussed in section S3.

The difference-beam state in Eq. 1 is derived under the assumption that the effects of group-velocity dispersion can be neglected. The validity of this assumption in the long entanglement time limit is demonstrated in section S1.

Time- and frequency-resolved two-photon coincidence signal

We consider a system comprising molecules and light fields. The total Hamiltonian is presented in section S2. The free Hamiltonian of the radiation field is expressed as ${\hat{H}}_{\text{field}}={\sum }_{\mathrm{\sigma }=\mathrm{S},\mathrm{F}}\int d\mathrm{\omega } \text{ℏω} {\hat{a}}_{\mathrm{\sigma }}^{†}\left(\mathrm{\omega }\right){\hat{a}}_{\mathrm{\sigma }}\left(\mathrm{\omega }\right)$, where the operator ${\hat{a}}_{\mathrm{F}}^{†}\left(\mathrm{\omega }\right)$ creates a spontaneously emitted photon of frequency $\mathrm{\omega }$. The positive frequency component of the field operator is given by

$${\hat{E}}_{\mathrm{\sigma }}^{+}\left(t\right)=\frac{1}{2\mathrm{\pi }}\int d\mathrm{\omega }{\hat{a}}_{\mathrm{\sigma }}\left(\mathrm{\omega }\right)e^{-i\mathrm{\omega }t}$$ (3)

whereas the negative frequency component is ${\hat{E}}_{\mathrm{\sigma }}^{-}\left(t\right)={\hat{E}}_{\mathrm{\sigma }}^{+}{\left(t\right)}^{†}$. We also assumed that the bandwidth of the fields is negligible compared to the central frequency (58).

We investigated the time- and frequency-resolved two-photon coincidence signal, which facilitates the simultaneous acquisition of spectral and temporal information of photons through the combined use of the spectrometer and the single-photon camera. To obtain this signal, the TCSPC device measures the time correlation between the electric fields of the idler and fluorescence photons, both of which pass through the diffraction grating and, subsequently, the single-photon camera before being output. However, detector dead time and timing jitter in the single-photon camera introduce uncertainty in the arrival time of measured photons, which causes temporal blurring; accordingly, the temporal profile of the detected photon is the convolution of the original electric field with the temporal response of the single-photon camera. Similarly, finite spectral resolution of the single-photon camera leads to spectral blurring; the measured spectrum is the convolution of the true spectrum of photon with the spectral response of the instrument. To incorporate these detector-response effects into the theoretical description, we define the following two functions (59–61)

$$F_{\mathrm{t}}(t,t_a)=\text{exp}[-\frac{1}{2{\mathrm{\sigma }}_{\mathrm{t}}^2}{(t-t_a)}^2]$$ (4)

$$F_{\mathrm{f}}(\mathrm{\omega },{\mathrm{\omega }}_a)=\frac{{\mathrm{\sigma }}_{\mathrm{f}}}{i({\mathrm{\omega }}_a-\mathrm{\omega })+{\mathrm{\sigma }}_{\mathrm{f}}}$$ (5)

where ${\mathrm{\omega }}_a$ and $t_a$ denote the center frequency and arrival time of photon a, respectively, which corresponds to either the idler or the fluorescence photon recorded by the single-photon camera. As illustrated in Fig. 3, Eq. 4 represents the temporal uncertainty in the arrival time $t_a$ of photon a arising from the time response of the single-photon camera. Similarly, Eq. 5 represents the spectral uncertainty in the measured frequency ${\mathrm{\omega }}_a$ of photon a arising from the finite spectral resolution of the single-photon camera. The temporal resolution of the single-photon camera is denoted by ${\mathrm{\sigma }}_{\mathrm{t}}$, while ${\mathrm{\sigma }}_{\mathrm{f}}$ represents the detector’s frequency resolution, which is determined by both the number of grating grooves in the spectrometer and the position resolution of the single-photon camera. Let ${\hat{E}}_a^{+}\left(t\right)$ in Eq. 3 denote the electric field before reaching the single-photon camera. The information obtained at the single-photon cameras can then be expressed as follows (29, 59, 62)

$${\hat{\mathcal{E}}}_a^{+}({\mathrm{\omega }}_a,t_a;t)={\displaystyle \underset{-\infty }{\overset{\infty }{\int }}}\mathit{ds} F_{\mathrm{f}}(t-s,{\mathrm{\omega }}_a)F_{\mathrm{t}}(s,t_a){\hat{E}}_a^{+}\left(s\right)$$ (6)

where $F_{\mathrm{f}}(t,{\mathrm{\omega }}_a)$ is the Fourier transform of Eq. 5, i.e., $F_{\mathrm{f}}(t,{\mathrm{\omega }}_a)=$${\mathrm{\sigma }}_{\mathrm{f}}\mathrm{\theta }\left(t\right)e^{-i{\mathrm{\omega }}_{a}t-{\mathrm{\sigma }}_{\mathrm{f}}t}$ with $\mathrm{\theta }\left(t\right)$ representing the Heaviside step function. The TCSPC device subsequently measures the time correlation between the electric fields detected at the two single-photon cameras, ${\hat{\mathcal{E}}}_{\mathrm{F}}({\mathrm{\omega }}_{\mathrm{F}},t_{\mathrm{F}};t)$ and ${\hat{\mathcal{E}}}_{\mathrm{I}}({\mathrm{\omega }}_{\mathrm{I}},t_{\mathrm{I}};t)$, resulting in the time- and frequency-resolved two-photon coincidence signal (62)

$$\begin{array}{c}S({\mathrm{\omega }}_{\mathrm{F}},t_{\mathrm{F}};{\mathrm{\omega }}_{\mathrm{I}},t_{\mathrm{I}})=\hfill \\ {\displaystyle \underset{-\infty }{\overset{\infty }{\int }}}\mathit{dt}{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}}\mathit{ds} \text{tr}\left[{\hat{\mathcal{E}}}_{\mathrm{F}}^{-}({\mathrm{\omega }}_{\mathrm{F}},t_{\mathrm{F}};t){\hat{\mathcal{E}}}_{\mathrm{F}}^{+}({\mathrm{\omega }}_{\mathrm{F}},t_{\mathrm{F}};t){\hat{\mathcal{E}}}_{\mathrm{I}}^{-}\right.\left.({\mathrm{\omega }}_{\mathrm{I}},t_{\mathrm{I}};s){\hat{\mathcal{E}}}_{\mathrm{I}}^{+}({\mathrm{\omega }}_{\mathrm{I}},t_{\mathrm{I}};s)\hat{\mathrm{\rho }}\left(t\right)\right]\hfill \end{array}$$ (7)

where $\hat{\mathrm{\rho }}\left(t\right)$ denotes the density operator for the total system, with $\hat{\mathrm{\rho }}(-\infty )={\hat{\mathrm{\rho }}}_{\text{mol}}^{\text{eq}}\otimes \mid {\mathrm{\psi }}_{\text{twin}}\rangle \langle {\mathrm{\psi }}_{\text{twin}}\mid$, and ${\hat{\mathrm{\rho }}}_{\text{mol}}^{\text{eq}}$ represents the thermal equilibrium state of the photoactive degrees of freedom of the molecule. The term $\hat{\mathrm{\rho }}\left(t\right)$ in Eq. 7 can be perturbatively expanded with respect to the molecule-field interaction, ${\hat{H}}_{\text{mol}-\text{field}}$, up to the fourth order. As presented in Fig. 4, the signal consists of two contributions classified as rephasing and nonrephasing stimulated emission (SE) (63). Accordingly, Eq. 7 is expressed as follows

$$\begin{array}{c}S({\mathrm{\omega }}_{\mathrm{F}},t_{\mathrm{F}};{\mathrm{\omega }}_{\mathrm{I}},t_{\mathrm{I}})=\frac{{\mathrm{\zeta }}^2}{2}\text{Re}\underset{0}{\overset{\infty }{\int }}d{\mathrm{\tau }}_3\underset{0}{\overset{\infty }{\int }}d{\mathrm{\tau }}_1{F}_{\mathrm{t}}\\ ({\mathrm{\tau }}_3+t_{\mathrm{F}},t_{\mathrm{F}})F_{\mathrm{t}}({\mathrm{\tau }}_1+t_{\mathrm{I}},t_{\mathrm{I}})e^{-({\mathrm{\sigma }}_{\mathrm{f}}-{i\mathrm{\omega }}_{\mathrm{F}}){\mathrm{\tau }}_3}\times \\ [e^{-({\mathrm{\sigma }}_{\mathrm{f}}+{i\mathrm{\omega }}_{\mathrm{I}}){\mathrm{\tau }}_1}{\mathrm{\Phi }}_{\text{SE}}^{\left(\mathrm{r}\right)}({\mathrm{\tau }}_3,t_{\mathrm{F}}+t_{\mathrm{I}},{\mathrm{\tau }}_1)+e^{-({\mathrm{\sigma }}_{\mathrm{f}}-{i\mathrm{\omega }}_{\mathrm{I}}){\mathrm{\tau }}_1}{\mathrm{\Phi }}_{\text{SE}}^{\left(\text{nr}\right)}({\mathrm{\tau }}_3,t_{\mathrm{F}}+t_{\mathrm{I}},{\mathrm{\tau }}_1)]\end{array}$$ (8)

where ${\mathrm{\Phi }}_{\text{SE}}^{\left(\mathrm{r}\right)}({\mathrm{\tau }}_3,{\mathrm{\tau }}_2,{\mathrm{\tau }}_1)$ and ${\mathrm{\Phi }}_{\text{SE}}^{\left(\text{nr}\right)}({\mathrm{\tau }}_3,{\mathrm{\tau }}_2,{\mathrm{\tau }}_1)$ denote the rephasing and nonrephasing response functions, respectively (63). In deriving Eq. 8, we assumed that the time resolution of the detector is sufficiently short compared to the timescale of the system dynamics and the entanglement time, ${\mathrm{\sigma }}_{\mathrm{t}}\to 0$. The derivation of this expression is provided in section S3. Equation 8 demonstrates that the spectroscopic signal as a function of $t_{\mathrm{F}}+t_{\mathrm{I}}$ reflects the excited-state dynamics initiated by the signal photon. This can be explained as follows. By the Fourier transform relation, the state in Eq. 1 exhibits negative time correlation, and its joint temporal amplitude satisfies $t_{\mathrm{I}}=-t_{\mathrm{S}}$, as shown in Fig. 1B. Hence, the excitation-fluorescence delay $\mathrm{\Delta }{t}_{\text{FS}}=t_{\mathrm{F}}-t_{\mathrm{S}}$ can be reconstructed from measured detection times as $\mathrm{\Delta }{t}_{\text{FS}}=t_{\mathrm{F}}+t_{\mathrm{I}}$, as illustrated in Fig. 1C. Scanning $t_{\mathrm{F}}+t_{\mathrm{I}}$ therefore tracks the excited-state dynamics.

Fig. 3. Schematic of the photon detection events.

The light pink–shaded region illustrates the temporal distribution of $F_{\mathrm{t}}(t,t_{\mathrm{I}})$ in Eq. 4. The arrival times of the idler photons are blurred by the finite temporal resolution of the single-photon camera. Although not illustrated here, the fluorescence photons experience the same temporal blurring.

Fig. 4. Double-sided Feynman diagrams.

Double-sided Feynman diagrams representing the fluorescence signals obtained by coincidence detection, as given in Eq. 8. The pathways in (A) and (B) are spectroscopically equivalent to the rephasing and nonrephasing SE components of conventional 2DES, respectively. In the main text, we therefore refer to these fluorescence contributions as the rephasing and nonrephasing signals.

Comparison of quantum and classical spectroscopy

The SE contribution to the absorptive 2D spectrum obtained using heterodyne-detected photon echo (49) and phase-modulated fluorescence–detected 2D electronic spectroscopy (50–52) in the impulsive limit is expressed as

$${\mathcal{S}}_{\text{SE}}({\mathrm{\omega }}_3,{\mathrm{\tau }}_2,{\mathrm{\omega }}_1)={\mathcal{S}}_{\text{SE}}^{\left(\mathrm{r}\right)}({\mathrm{\omega }}_3,{\mathrm{\tau }}_2,{\mathrm{\omega }}_1)+{\mathcal{S}}_{\text{SE}}^{\left(\text{nr}\right)}({\mathrm{\omega }}_3,{\mathrm{\tau }}_2,{\mathrm{\omega }}_1)$$ (9)

where ${\mathcal{S}}_{\text{SE}}^{\left(x\right)}({\mathrm{\omega }}_3,{\mathrm{\tau }}_2,{\mathrm{\omega }}_1)$ represents the real part of the Fourier-Laplace transform of ${\mathrm{\Phi }}_{\text{SE}}^{\left(x\right)}({\mathrm{\tau }}_3,{\mathrm{\tau }}_2,{\mathrm{\tau }}_1)$. The signal in Eq. 8 is analogous to the SE contribution in Eq. 9

$$S({\mathrm{\omega }}_{\mathrm{F}},t_{\mathrm{F}};{\mathrm{\omega }}_{\mathrm{I}},t_{\mathrm{I}})\simeq \frac{{\mathrm{\zeta }}^2}{2}{\mathcal{S}}_{\text{SE}}({\mathrm{\omega }}_{\mathrm{F}},t_{\mathrm{F}};t_{\mathrm{I}},{\mathrm{\omega }}_{\mathrm{I}})$$ (10)

except that it depends on the function $F_{\mathrm{t}}(t,t_a)$ defined in Eq. 4. Restricting detection to the SE contribution reduces the complexity of multidimensional spectral analysis and facilitates the extraction of information about molecular excited-state dynamics, as will be demonstrated by the numerical results.

Notably, a fundamental difference exists between the proposed quantum spectroscopy and conventional 2DES regarding the limits of time and frequency resolution. In 2DES, as described in Eq. 9, when all pulses are of very short duration, the resulting 2D signal directly represents the third-order response function of the molecules. In contrast, in quantum spectroscopy, when the time resolution of the detector is sufficiently short, the signal in Eq. 8 is given by the convolution of the third-order response function and the function in Eq. 4. Under these conditions, for the pathways in Fig. 4, the coherence between the electronic ground and excited states during ${\mathrm{\tau }}_1$ and ${\mathrm{\tau }}_3$ decays more rapidly than the original decay lifetime owing to the time profile of the function in Eq. 4, thereby reducing the frequency resolution. In the case where the detector’s time resolution is infinitely short, $F_{\mathrm{t}}(t,t_a)=\mathrm{\delta }(t-t_a)$, Eq. 8 reduces to a time-resolved signal that has lost frequency resolution

$$S({\mathrm{\omega }}_{\mathrm{F}},t_{\mathrm{F}};{\mathrm{\omega }}_{\mathrm{I}},t_{\mathrm{I}})={\mathrm{\zeta }}^2\text{Re}{\mathrm{\Phi }}_{\text{SE}}^{\left(\mathrm{r}\right)}(0,t_{\mathrm{F}}+t_{\mathrm{I}},0)$$ (11)

Section S4 details the numerical examination of the impact of the time resolution of the detector on quantum spectroscopy.

Numerical results

To demonstrate the advantages of the selective SE contribution, we numerically computed the 2D spectra of a coupled trimer. The model trimer is based on bacteriochlorophylls (BChls) 3, 6, and 7 in the Fenna-Matthews-Olson (FMO) pigment-protein complex from Chlorobaculum tepidum (64), which together form one branch of the two primary energy transfer pathways (49, 65). Their spatial arrangement is shown in Fig. 1D. The site energies and electronic couplings were taken from (66) and are listed in Table 1. The third-order response functions in Eq. 8 are calculated using the second-order cumulant expansion with respect to the electronic energy fluctuations induced by the environment (67), and electronic energy transfer is modeled using the secular Redfield equation (68). The electronic excitation of each pigment is coupled to an environment characterized by the overdamped Brownian oscillator model (63). The timescale of environmental reorganization and the reorganization energy were set to 100 fs and 55 cm⁻¹, respectively. The temperature was set to 77 K. The excitation energy of the pigments and the interaction strength between the pigments are summarized in Table 1. For simplicity, the transition dipole moments of the pigments were assumed to be parallel and were set to the same value for all pigments. As shown below, model reduction and the parallel/equal-dipole approximation can lead to differences from the experimental spectra (49), affecting not only the peak amplitudes but also the number and positions of cross peaks. Our purpose is not to quantitatively reproduce the experimental 2D spectra of the full seven-site FMO complex reported in (49) but rather to provide a qualitative test of SE-selective measurements using a reduced model.

Table 1. Single-excitation Hamiltonian (cm⁻¹) for a coupled trimer modeled on BChls 3, 6, and 7 of the FMO complex.

Site energies and electronic couplings are taken from (66).

BChl	3	6	7
3	12,210	−9.6	6
6	−9.6	12,630	39.7
7	6	39.7	12,440

Figure 5A shows the calculated 2D spectra obtained using the quantum spectroscopy described in Eq. 8. The detection time of idler photons is fixed to $t_{\mathrm{I}}=0$. The time resolution of the detector and the frequency resolution of the spectrometer are set to ${\mathrm{\sigma }}_{\mathrm{t}}=400 \text{fs}$ and ${\mathrm{\sigma }}_{\mathrm{f}}=0$, respectively. The detected times of the fluorescence photons are $t_{\mathrm{F}}=0$ and 4 ps. The absorptive 2D spectrum obtained using the photon echo technique in the impulsive limit is shown in Fig. 5B. Figure 5C depicts the SE contribution to the absorptive 2D spectra presented in Fig. 5B.

Fig. 5. Comparison of quantum and classical 2D spectra.

(A) 2D spectra obtained by the quantum spectroscopy described in Eq. 8. (B) Absorptive 2D spectrum obtained using the classical pulses in the impulsive limit. (C) SE contribution to the absorptive 2D spectra. In (A), the detection time of the idler photon was set as $t_{\mathrm{I}}=0$. The time resolution of the detector and the frequency resolution of the spectrometer were set to ${\mathrm{\sigma }}_{\mathrm{t}}=400 \text{fs}$ and ${\mathrm{\sigma }}_{\mathrm{f}}=0$, respectively. Normalization of contour plots in (A) to (C) is such that the maximum value of each spectrum at $t_{\mathrm{F}}=0$ (${\mathrm{\tau }}_2=0$) is unity, and equally spaced contour levels ($0,\pm 0.1,\pm 0.2,\dots$) are shown.

We first relate the calculated classical 2D spectra to experimental observations in the literature. Reference (49) reported absorptive 2D photon echo spectra of the FMO complex at 77 K, where diagonal peaks follow the main absorption features and cross peaks reveal electronic coupling and energy transfer dynamics as the waiting time increases. In our calculation, the total absorptive 2D spectra in Fig. 5B exhibit differences in peak amplitudes and in the number and positions of cross peaks compared with (49). This is because the measured signal is the superposition of ground-state bleaching (GSB), SE, and excited-state absorption (ESA) contributions, whose overlap and interference can distort the apparent peak pattern. By contrast, the SE contribution in Fig. 5C displays qualitative trends that are consistent with the physical interpretation of the experimental 2D spectra. At ${\mathrm{\tau }}_2=0 \text{ps}$, Fig. 5C shows three diagonal peaks, corresponding to the three one-exciton eigenstates of the (3, 6, 7) subspace; the highest-energy diagonal peak is predominantly associated with BChl 6, the intermediate-energy peak with BChl 7, and the lowest-energy peak with BChl 3. As the waiting time increases, the highest-energy diagonal feature decays and two cross peaks emerge, reflecting downhill population transfer from BChl 6 to BChl 7 and, ultimately, to BChl 3. This behavior is qualitatively consistent with the population dynamics in (65), which suggests that excitation initiated on BChl 6 populates BChl 3 within 1 ps via intermediate pigments including BChls 4, 5, and 7. Moreover, the waiting time–dependent growth of cross-peak features in Fig. 5C is qualitatively analogous to the cross peaks (e.g., those labeled A and B) reported in (49) and attributed there to energy transfer.

We next explain why isolating the SE contribution is advantageous for interpretation. In conventional 2D Fourier transform photon echo measurements, the experimentally accessible absorptive spectrum is the total response (GSB + SE + ESA), and the SE contribution cannot be separated directly. Consequently, these measurements require careful interpretation of complex spectra. For example, the absorptive 2D spectrum in Fig. 5B exhibits negative peaks resulting from the ESA, but the peak positions are shifted from their original ESA positions because of the overlap with nearby positive GSB peaks. This overlap hinders the extraction of accurate information regarding the excited state.

In contrast, as shown in Fig. 5A, the signal obtained from quantum spectroscopy contains spectral information equivalent to the SE contribution, except for the reduced frequency resolution caused by the detector’s time resolution, ${\mathrm{\sigma }}_{\mathrm{t}}$. The impact of the detector’s time resolution is detailed in section S4. Therefore, the proposed quantum spectroscopy presents a promising approach for accurately elucidating energy transfer processes in photosynthetic proteins with multiple pigments, which are often challenging to analyze using conventional 2D photon echo techniques.

However, the time resolution of the DLD is on the order of several hundred picoseconds (48), which is insufficient to observe excitation energy transfer in photosynthetic proteins (49, 69) and organic materials (70) where subpicosecond time resolution is required. This limitation applies not only to the DLD but also to current single-photon imagers such as single-photon avalanche diode arrays (71) and multianode single-photon imagers (72). One potential solution is to combine these detectors with streak tubes, which will allow time resolutions of 200 fs to picoseconds. Here, each DLD captures horizontally dispersed photons on the detector’s surface, although the DLD is circularly shaped and has a vertical dimension assignable to another physical parameter; the streak tube can disperse photons vertically, and subpicosecond information is registered in the direction of the image on DLD.

Practical limits to scaling entangled-photon flux

In this work, toward an experimental realization of time-resolved spectroscopy with entangled photons, we have emphasized the usefulness of fluorescence detection–based measurements using a high-performance single-photon camera as a practical route to achieving sufficiently high signal levels within a realistic acquisition time. In parallel, an alternative approach based on the use of entangled photon pairs at higher flux has also been explored, and several theoretical studies have been reported (28, 73, 74). For our measurement as well, increasing the entangled-photon flux could in principle further shorten the acquisition time.

However, operating at high entangled-photon flux raises several concerns. In particular, there are two major issues. The first concern is multipair generation. As the pump power driving the SPDC process is increased, the probability of generating two or more photon pairs per pump event increases. When multiple entangled pairs are produced, it becomes possible that a signal photon belonging to a different pair than the detected idler photon excites the molecule, and the resulting fluorescence photon is recorded. In such a case, the spectroscopic signal associated with entangled-photon correlations can be masked by contributions originating from effectively uncorrelated photons. As discussed in (73), the two-photon frequency distribution of photon pairs drawn from different SPDC pairs is uncorrelated, and in high-gain PDC, this contribution is comparable to that from genuine entangled pairs. Because our proposed quantum spectroscopy relies on exploiting positive frequency correlations together with negative time correlations, increasing the entangled-photon flux is therefore expected to degrade both the frequency and temporal resolution of the resulting 2D spectra. The second concern is saturation of the single-photon detector. Single-photon detectors and single-photon cameras typically have maximum count rates on the order of a few megahertz; beyond this limit, the detector may saturate and, in severe cases, suffer damage or failure. In the fluorescence-lifetime measurements with entangled photons, the pump power is often attenuated to avoid saturation in the idler channel (44, 46). For example, ref. (44) reduces the laser power using a neutral-density filter.

In the following, we provide a rough estimate of which of these two effects is more likely to become the bottleneck in the measurements proposed here when the pump power is increased. The photon count rate on the idler detector is given by $R=f\overline{n}{\mathrm{\eta }}_{\text{det}}{\mathrm{\eta }}_{\mathrm{I}}$, where f is the repetition rate of the pulsed pump laser, $\overline{n}$ is the mean number of idler photons per pulse, ${\mathrm{\eta }}_{\text{det}}$ is the detector efficiency, and ${\mathrm{\eta }}_{\mathrm{I}}$ is the transmission of the idler optical path, which is included to account for losses due to optical elements. On the basis of the experimental conditions considered in (75), we take f = 76 MHz, a saturation threshold R_max = 1 MHz, and ${\mathrm{\eta }}_{\text{det}}=0.8$. From the saturation condition $R\le R_{\text{max}}$, we obtain an upper bound on the mean photon number per pulse: $\overline{n}\le {\overline{n}}_{\text{sat}}=R_{\text{max}}/\left(f{\mathrm{\eta }}_{\text{det}}{\mathrm{\eta }}_{\mathrm{I}}\right)\simeq 0.0164/{\mathrm{\eta }}_{\mathrm{I}}$. For a representative value ${\mathrm{\eta }}_{\mathrm{I}}=0.2$, this yields ${\mathrm{\eta }}_{\text{sat}}\approx 0.082$. According to (6), even at $\overline{n}\approx 0.1$, the uncorrelated contribution to the entangled two-photon absorption signal is negligibly small compared with the correlated contribution. Therefore, under the present experimental conditions, detector saturation is more likely to become the limiting factor upon increasing the pump power than the contamination from multipair generation. A more detailed and quantitative assessment of multipair effects based on calculations of 2D fluorescence spectra with squeezed-vacuum states is an important topic for future work.

DISCUSSION

We demonstrated a quantum spectroscopy technique using a time-resolved fluorescence approach that is feasible with current photon detection technologies. The proposed quantum spectroscopy offers two advantages over conventional 2DES by leveraging the nonclassical photon correlations of entangled photon pairs. First, the proposed quantum spectroscopy provides 2D spectra without requiring the control of multiple pulsed lasers. Second, whereas conventional 2D spectra contain GSB, SE, and ESA contributions, the proposed spectroscopy selectively detects the SE contribution alone. This substantially reduces spectral complexity, facilitating straightforward extraction of information regarding the excited-state dynamics of molecular systems.

Although our concept is related to earlier theoretical proposals (25, 33), the present scheme is practically superior in two important respects. First, unlike (33), it provides time-resolved spectral information across the entire 2D spectrum. In the optical configuration in (33), the coincidence counting between the signal and idler photons can eliminate the ESA contribution because only one photon is populated in the final state. However, that measurement can include unwanted oscillatory components arising solely from the relative time delay between the signal and idler photons, rather than from molecular dynamics. For nonzero optical delay between the signal and idler photons, such extra contributions can be removed only under the restrictive condition $\mathrm{\omega }={\mathrm{\omega }}_{\mathrm{P}}/2$, where ${\mathrm{\omega }}_{\mathrm{P}}$ is the pump frequency, and ω is the detected photon frequency. As a result, ref. (33) is effectively limited to spectral information near the diagonal. In contrast, our method can extract SE-only information over the full 2D spectral range. Second, the proposed technique stands out for its high experimental feasibility. Our advances in single-photon detection (48) and preliminary measurements (75) demonstrate that the expected signal levels are within the detection capabilities of current devices such as DLDs, making near-term implementation feasible. We anticipate that this framework will open the path to real-time observation of ultrafast molecular dynamics using quantum-entangled photons, bridging the fields of quantum optics and ultrafast spectroscopy.

Last, we comment on the prospect of simultaneously improving frequency and temporal resolution beyond the Fourier-limited trade-off by using entangled photon pairs. Discussions of such potential quantum advantages began nearly a decade ago (59), and several theoretical spectroscopic proposals have suggested this possibility (23, 26, 27, 29, 36, 37); however, a rigorous and quantitative assessment of whether frequency and temporal resolution can be simultaneously enhanced is still lacking. As with conventional 2D spectroscopy using classical light, the quantum spectroscopic scheme proposed in this study does not surpass the Fourier-limited trade-off between frequency and temporal resolution. Nevertheless, by establishing a realistic experimental route to time-resolved spectroscopy with entangled photons, our work is expected to accelerate the development of new quantum light–based time-resolved spectroscopies, potentially paving the way toward resolving this question in future studies.

Note added: During the preparation of this manuscript, we became aware of a related experimental study by Álvarez-Mendoza et al. (76) that demonstrates time- and frequency-resolved fluorescence spectroscopy at the single-photon level using entangled photon pairs from a CW-SPDC source and a Fourier transform spectrometer based on the TWINS (translating wedge–based identical-pulses encoding system). Our work is complementary in two key respects. First, we provide a comprehensive theoretical framework for time-resolved fluorescence spectroscopy with entangled photons, showing how the time-frequency correlations of biphotons enable time-resolved spectroscopy without multipulse control. Second, our approach enables acquisition of 2D spectra within minutes, as validated in (48). In contrast, ref. (76) relies on TWINS scanning and required 120 min even for a single frequency axis; extending that scheme to measure the idler frequency as well would render 2D acquisition times prohibitive.

MATERIALS AND METHODS

Delay-line-anode single-photon detector

Our theoretical framework is based on a DLD combined with a grating spectrometer. The detailed explanation is presented in previous experimental work with the detector’s schematics (48, 75). Here is a brief explanation of the working principle of DLD. The DLD consists of three components: photocathode, microchannel plate (MCP) stack, and delay-line-anode sensor. The photocathode converts incident photons into photoelectrons, and they are amplified via the MCP stack. The material of the photocathode determines the spectral sensitivity. The amplified photoelectrons arrive at a certain point of the meander-wired delay-line-anode sensor, from which the electrical signal is split into opposite directions on the wires. These counterpropagating signals are time stamped at a TCSPC. Given that the photon arrival position corresponds to the arrival time delay of the two pulses, the photon arrival position is specified by calculating the difference. It is important to note that MCP also provides pulses when photons are detected somewhere on the detector. Thus, the DLD can give both temporal and spectral information by combining a grating spectrometer.

In this study, we assume the use of two single-photon cameras with sensitivity in the near-infrared, as suggested by the fluorescence spectrum of the FMO complex. In contrast, the DLD used in our previous work was sensitive only in the visible range (48, 75). This limitation can be alleviated by replacing the red-enhanced photocathode with a high-quantum-efficiency infrared photocathode (77). Alternatively, future proof-of-principle experiments could use molecular systems that emit in the visible range and are detectable with previously reported instrumentation, for example, fluorescent proteins (78) or the photosystem II reaction center (79).

Supplementary Materials

This PDF file includes:

Supplementary Text

Figs. S1 to S5

References