Achieving two-dimensional optical spectroscopy with temporal and spectral resolution using quantum entangled three photons

One of the most widespread techniques for generating these quantum resources is parametric down-conversion (PDC). Mandel and Wolf (1995) In this process, a photon originating from an input laser is converted into an entangled photon pair in a way that satisfies the energy and momentum conservation laws. In this work, we address entangled three photons generated through the cascaded PDC process with two nonlinear crystals, Hübel et al. (2010); Shalm et al. (2013); Hamel et al. (2014); Agne et al. (2017) as shown in Fig. 1. In the primary PDC, the pump photon, which has a frequency of $\omega_{\mathrm{p}}$, passes through the first crystal and is split into a pair of daughter photons (photons 0 and 1) with frequencies of $\omega_{0}$ and $\omega_{1}$. In the second crystal, photon 0 serves as the pump field for the secondary conversion, creating a pair of granddaughter photons (photons 2 and 3) with frequencies of $\omega_{2}$ and $\omega_{3}$. It is noted that the conversion efficiency in the cascaded PDC is typically very low. For instance, in cascaded PDC with a combination of periodically-poled lithium niobate (PPLN) and periodically-poled potassium titanyl phosphate, the detected rate of entangled three photons is approximately 7 counts per hour. Shalm et al. (2013) At this photon count rate, an unrealistically long signal collection time may be required to measure the frequencies of photon for extracting the spectroscopic information. However, the main purpose of this work is to give how the non-classical correlations among entangled three photons can be applied to time-resolved spectroscopy. A discussion on the experimental feasibility of the frequency-dispersed two-photon coincidence counting measurement using the cascaded PDC is beyond the scope of this paper.

For simplicity, we consider the electric fields inside the one-dimensional nonlinear crystals. In the weak down-conversion regime, the state vector of the generated three photons is written as Shalm et al. (2013); Zhang et al. (2018); Ye and Mukamel (2020)

$\displaystyle\lvert\psi_{\mathrm{tri}}\rangle\simeq\iiint d^{3}\omega f(\omega_{1},\omega_{2},\omega_{3})\hat{a}_{1}^{\dagger}(\omega_{1})\hat{a}_{2}^{\dagger}(\omega_{2})\hat{a}_{3}^{\dagger}(\omega_{3})\lvert\mathrm{vac}\rangle.$

(2.1)

The derivation is given in Appendix A. In the equation, $\hat{a}_{\sigma}^{\dagger}(\omega)$ denotes the creation operator of a photon of frequency $\omega$ against the vacuum state $\lvert\mathrm{vac}\rangle$. The operator satisfies the commutation relation $[\hat{a}_{\sigma}(\omega),\hat{a}_{\sigma^{\prime}}^{\dagger}(\omega^{\prime})]=\delta_{\sigma\sigma^{\prime}}\delta(\omega-\omega^{\prime})$. The three-photon amplitude, $f(\omega_{1},\omega_{2},\omega_{3})$, is expressed as

$\displaystyle f(\omega_{1},\omega_{2},\omega_{3})=\eta A_{\mathrm{p}}(\omega_{1}+\omega_{2}+\omega_{3})\phi(\omega_{1},\omega_{2},\omega_{3}),$

(2.2)

where $A_{\mathrm{p}}(\omega)$ is the normalized pump envelope and $\phi(\omega_{1},\omega_{2},\omega_{3})=\mathrm{sinc}[\Delta k_{1}(\omega_{2}+\omega_{3},\omega_{1})L_{1}/2]$ $\times\mathrm{sinc}[\Delta k_{2}(\omega_{2},\omega_{3})L_{2}/2]$ denotes the phase-matching function of the overall cascaded PDC process. The momentum mismatch between the input and output photons in the $n$-th nonlinear crystal is expressed by $\Delta k_{1}(\omega,\omega^{\prime})=k_{\mathrm{p}}(\omega+\omega^{\prime})-k_{0}(\omega)-k_{1}(\omega^{\prime})$ and $\Delta k_{2}(\omega,\omega^{\prime})=k_{0}(\omega+\omega^{\prime})-k_{2}(\omega)-k_{3}(\omega^{\prime})$. The length of the $n$-th crystal is given by $L_{n}$. The momentum mismatches may be linearly approximated around the central frequencies of the generated beams, $\bar{\omega}_{\sigma}$, as in Zhang et al. (2018); Ye and Mukamel (2020)

	$\displaystyle\Delta k_{1}(\omega_{0},\omega_{1})L_{1}$	$\displaystyle=(\omega_{0}-\bar{\omega}_{0})T_{\mathrm{p}0}+(\omega_{1}-\bar{\omega}_{1})T_{\mathrm{p}1},$		(2.3)
	$\displaystyle\Delta k_{2}(\omega_{2},\omega_{3})L_{2}$	$\displaystyle=(\omega_{2}-\bar{\omega}_{2})T_{02}+(\omega_{3}-\bar{\omega}_{3})T_{03},$		(2.4)

where $T_{\mathrm{p}\sigma}=L_{1}/v_{\mathrm{p}}-L_{1}/v_{\sigma}$ and $T_{0\sigma}=L_{2}/v_{0}-L_{2}/v_{\sigma}$. Here, $v_{\mathrm{p}}=\partial k_{\mathrm{p}}/\partial\omega|_{\omega=\omega_{\mathrm{p}}}$ and $v_{\sigma}=\partial k_{\sigma}/\partial\omega|_{\omega=\bar{\omega}_{\sigma}}$ represent the group velocities of the input laser and the generated beam at the frequency $\bar{\omega}_{\sigma}$, respectively. Without loss of generality, we assume that $T_{\mathrm{p}0}\geq T_{\mathrm{p}1}$ and $T_{02}\geq T_{03}$. We merge all other constants into a factor, $\eta$, in Eq. (2.2), which corresponds to the conversion efficiency of the cascaded PDC process.

In this study, we focus on monochromatic pumping with frequency $\omega_{\mathrm{p}}$ for the cascaded PDC process. In this situation, the energy conservation in the two processes is satisfied as $\omega_{\mathrm{p}}=\omega_{1}+\omega_{2}+\omega_{3}$. The three-photon amplitude in Eq. (2.2) can be rewritten as Shalm et al. (2013)

$\displaystyle f(\omega_{1},\omega_{2},\omega_{3})$

$\displaystyle=\eta\delta(\omega_{1}+\omega_{2}+\omega_{3}-\omega_{\mathrm{p}})r(\omega_{1},\omega_{3}),$

(2.5)

where $r(\omega_{1},\omega_{3})=\phi(\omega_{1},\omega_{\mathrm{p}}-\omega_{1}-\omega_{3},\omega_{3})$ is given by

	$\displaystyle r(\omega_{1},\omega_{3})$	$\displaystyle=\mathrm{sinc}\frac{(\omega_{1}-\bar{\omega}_{1})T_{\mathrm{e}}^{(01)}}{2}$
		$\displaystyle\quad\times\mathrm{sinc}\frac{(\omega_{1}-\bar{\omega}_{1})T_{02}+(\omega_{3}-\bar{\omega}_{3})T_{\mathrm{e}}^{(23)}}{2}.$		(2.6)

The difference, $T_{\mathrm{e}}^{(01)}=T_{\mathrm{p}0}-T_{\mathrm{p}1}$, is the entanglement time between photons 0 and 1, Saleh et al. (1998) which represents the maximum relative delay between photons 0 and 1. Similarly, in the secondary PDC, the entanglement time between photons 2 and 3 is defined by $T_{\mathrm{e}}^{(23)}=T_{02}-T_{03}$.

We considered the frequency-dispersed two-photon coincidence counting measurement using the entangled three photons. The delay intervals between two of the three photons are innately determined when generated in the cascaded PDC. For example, the upper bound of the interval between photons 1 and 2 is given by $(T_{\mathrm{e}}^{(01)}+\lvert T_{02}\rvert)/2$. In a similar fashion to the two-photon interference experiments, Hong, Ou, and Mandel (1987); Franson (1989) the delay intervals among the three photons can be controlled by adjusting the path differences between the beams. Agne et al. (2017); Menssen et al. (2017) Therefore, the interval between photons 1 and 2 is experimentally controllable when the external time delay is sufficiently long compared to $T_{\mathrm{e}}^{(01)}$ and $T_{02}$. This external time delay is herein denoted as $\Delta t$. As presented in Fig. 1, photon 2 is employed as the pump field, whereas photon 1 is used for the probe field with the time delay $\Delta t\geq 0$. Photon 3 does not interact with the sample; it serves as a reference for the coincidence measurement. We assume that the efficiency of the photon detectors is perfect. In this situation, the detection of photon 3 makes it possible to verify the generation of entangled three photons. Consequently, the coincidence measurements of photons 1 and 3 enable us to distinguish the genuine spectroscopic signal induced by two of the entangled three photons from undesired accidental photon counts in the photon 1 detector. This is a potential benefit of utilizing two-photon coincidence detection to conduct measurements.

We consider a system comprising molecules and light fields. The positive-frequency component of the electric field operator, which interacts with the molecules, is written as Hong, Ou, and Mandel (1987); Franson (1989); Ishizaki (2020a)

$\displaystyle\hat{E}(t)=\hat{E}_{1}(t)+\hat{E}_{2}(t+\Delta t),$

(2.7)

where $\hat{E}_{\sigma}(t)=(2\pi)^{-1}\int d\omega\,\hat{a}_{\sigma}(\omega)e^{-i\omega t}$. Here, the slowly varying envelope approximation has been adapted with the bandwidth of the fields assumed to be negligible in comparison to the central frequency.Loudon (2000) Under the rotating-wave approximation, the molecule–field interaction can be written as $\hat{H}_{\text{mol--field}}(t)=-\hat{\mu}^{\dagger}\hat{E}(t)-\hat{\mu}\hat{E}^{\dagger}(t)$, where the dipole operator $\hat{\mu}$ is defined by $\hat{\mu}=\sum_{\alpha}\mu_{\alpha 0}\lvert 0\rangle\langle e_{\alpha}\rvert+\sum_{\alpha\bar{\gamma}}\mu_{\bar{\gamma}\alpha}\lvert e_{\alpha}\rangle\langle f_{\bar{\gamma}}\rvert$. In the above, $\lvert 0\rangle$ represents the electronic ground state in the molecules. The summations are performed on indices that run over electronic excited states in the single-excitation manifold $\{\lvert e_{\alpha}\rangle\}$ and double-excitation manifold $\{\lvert f_{\bar{\gamma}}\rangle\}$. The probe fields transmitted through the sample, $\hat{E}_{1}$, and the reference field, $\hat{E}_{3}$, are both frequency-dispersed. Then, changes in the two-photon counting rate, $\mathrm{tr}[\hat{a}_{3}^{\dagger}(\omega_{\rm r})\hat{a}_{1}^{\dagger}(\omega)\hat{a}_{1}(\omega)\hat{a}_{3}(\omega_{\rm r})\hat{\rho}(\infty)]$, are measured. Thus, the frequency-dispersed two-photon counting signal is written as Dorfman, Schlawin, and Mukamel (2016); Schlawin (2017); Ye and Mukamel (2020)

	$\displaystyle S(\omega,\omega_{\rm r};\Delta t)$	$\displaystyle={\rm Im}\int^{\infty}_{-\infty}dt\,e^{i\omega t}$
		$\displaystyle\quad\times\mathrm{tr}[\hat{a}_{3}^{\dagger}(\omega_{\rm r})\hat{a}_{3}(\omega_{\rm r})\hat{E}_{1}^{\dagger}(\omega)\hat{\mu}\hat{\rho}(t)].$		(2.8)

The initial conditions are: $\hat{\rho}(-\infty)=\lvert 0\rangle\langle 0\rvert\otimes\lvert\psi_{\mathrm{tri}}\rangle\langle\psi_{\mathrm{tri}}\rvert$. The lowest-order contribution of Eq. (2.8) only comprises the absorption of photon 1. However, the absorption signal is independent of the frequency of the input laser, $\omega_{\mathrm{p}}$, and the delay time, $\Delta t$, as shown in Appendix C. Moreover, when the entanglement time $T_{\mathrm{e}}^{(23)}$ is much shorter than the characteristic timescales of the dynamics under investigation, the absorption signal is independent of the frequency of the input laser, $\omega_{\mathrm{p}}$, reference frequency, $\omega_{\text{r}}$, and the delay time, $\Delta t$. Thus, in the two-photon coincidence measurement, which improves the signal-to-noise ratio, this process can be separated from the pump-probe-type two-photon process. Consequently, the perturbative expansion of $\hat{\rho}(t)$ with respect to the molecule–field interaction, $\hat{H}_{\text{mol--field}}$, yields the third-order term as the leading order contribution. The resultant signal is expressed as the sum of eight contributions, which are classified into stimulated emission (SE), ground-state bleaching (GSB), excited-state absorption (ESA), and double-quantum coherence (DQC).

To obtain a concrete but simple expression of the signal, here the memory effect straddling different time intervals in the response function is ignored.Ishizaki and Fleming (2012) Hereafter, the reduced Planck constant, $\hbar$, is omitted. Consequently, Eq. (2.8) can be expressed as (see Appendix B)

	$\displaystyle S(\omega,\omega_{\rm r};\Delta t)$	$\displaystyle=S_{\mathrm{SE}}(\omega,\omega_{\rm r};\Delta t)+S_{\mathrm{GSB}}(\omega,\omega_{\rm r};\Delta t)$
		$\displaystyle\quad+S_{\mathrm{ESA}}(\omega,\omega_{\rm r};\Delta t)$
		$\displaystyle\quad+S_{\mathrm{DQC}}(\omega,\omega_{\rm r};\Delta t)$		(2.9)

in terms of the SE, GSB, ESA, and DQC contributions,

	$\displaystyle S_{\mathrm{SE}}(\omega,\omega_{\rm r};\Delta t)$	$\displaystyle=-{\rm Re}\sum_{\alpha\beta\gamma\delta}\mu_{\gamma 0}\mu_{\delta 0}\mu_{\beta 0}\mu_{\alpha 0}$
		$\displaystyle\quad\times\sum_{y={\rm r},{\rm nr}}I_{\gamma 0;\gamma\delta\leftarrow\alpha\beta;\alpha 0}^{(y)}(\omega,\omega_{\rm r};\Delta t)$
		$\displaystyle\quad+\sum_{y=\rm{r},\rm{nr}}\Delta S_{\mathrm{SE}}^{(y)}(\omega,\omega_{\rm r}),$		(2.10)

	$\displaystyle S_{\mathrm{GSB}}(\omega,\omega_{\rm r};\Delta t)$	$\displaystyle=-{\rm Re}\sum_{\alpha\beta}\mu_{\beta 0}^{2}\mu_{\alpha 0}^{2}$
		$\displaystyle\quad\times\sum_{y=\rm{r},\rm{nr}}I_{\beta 0;00\leftarrow 00;\alpha 0}^{(y)}(\omega,\omega_{\rm r};\Delta t)$
		$\displaystyle\quad+\sum_{y={\rm r},{\rm nr}}\Delta S_{\mathrm{GSB}}^{(y)}(\omega,\omega_{\rm r}),$		(2.11)

	$\displaystyle S_{\mathrm{ESA}}(\omega,\omega_{\rm r};\Delta t)$	$\displaystyle=+{\rm Re}\sum_{\alpha\beta\gamma\delta\bar{\epsilon}}\mu_{\bar{\epsilon}\delta}\mu_{\bar{\epsilon}\gamma}\mu_{\beta 0}\mu_{\alpha 0}$
		$\displaystyle\quad\times\sum_{y=\rm{r},\rm{nr}}I_{\bar{\epsilon}\delta;\gamma\delta\leftarrow\alpha\beta;\alpha 0}^{(y)}(\omega,\omega_{\rm r};\Delta t),$		(2.12)

	$\displaystyle S_{\mathrm{DQC}}(\omega,\omega_{\rm r};\Delta t)$	$\displaystyle=\mathrm{Re}\sum_{\alpha\beta\bar{\gamma}}\mu_{\bar{\gamma}\beta}\mu_{\beta 0}\mu_{\bar{\gamma}\alpha}\mu_{\alpha 0}$
		$\displaystyle\quad\times\left\{G_{\bar{\gamma}\beta}[\omega]G_{\bar{\gamma}0}[\omega_{\mathrm{p}}-\omega_{\rm r}]F^{\prime}_{\alpha 0}(\omega,\omega_{\rm r};\Delta t)\right.$
		$\displaystyle\left.\quad-G_{\beta 0}[\omega]G_{\bar{\gamma}0}[\omega_{\mathrm{p}}-\omega_{\rm r}]F^{\prime}_{\alpha 0}(\omega,\omega_{\rm r};\Delta t)\right\},$		(2.13)

where $y$ indicates “rephasing” (r) or “non-rephasing” (nr), and $G_{\alpha\beta}(t)$ describes the time evolution of the $\lvert e_{\alpha}\rangle\langle e_{\beta}\rvert$ coherence. The Fourier-Laplace transform of $G_{\alpha\beta}(t)$ is introduced as $G_{\alpha\beta}[\omega]=\int^{\infty}_{0}dt\,e^{i\omega t}G_{\alpha\beta}(t)$. In Eqs. (2.10) – (2.12), the function $I_{\epsilon\zeta;\gamma\delta\leftarrow\alpha\beta;\alpha 0}^{(y)}(\omega,\omega_{\rm r};\Delta t)$ is defined by

	$\displaystyle I_{\epsilon\zeta;\gamma\delta\leftarrow\alpha\beta;\alpha 0}^{\mathrm{(r)}}(\omega,\omega_{\rm r};\Delta t)$	$\displaystyle=G_{\epsilon\zeta}[\omega]F_{\gamma\delta\leftarrow\alpha\beta}(\omega,\omega_{\rm r};\Delta t,0)$
		$\displaystyle\quad\times G_{\alpha 0}^{\ast}[\omega_{\mathrm{p}}-\omega_{\rm r}-\omega],$		(2.14)

$$I_{\epsilon\zeta;\gamma\delta\leftarrow\alpha\beta;\alpha 0}^{\mathrm{(nr)}}(\omega,\omega_{\rm r};\Delta t)=G_{\epsilon\zeta}[\omega]\int_{0}^{\infty}ds_{1}e^{i(\omega_{\mathrm{p}}-\omega_{\rm r}-\omega)s_{1}}\\ \times F_{\gamma\delta\leftarrow\alpha\beta}(\omega,\omega_{\rm r};\Delta t,s_{1})G_{\alpha 0}(s_{1})$$

(2.15)

in terms of

	$\displaystyle F_{\gamma\delta\leftarrow\alpha\beta}(\omega,\omega_{\rm r};\Delta t,s_{1})$
	$\displaystyle\quad=r(\omega,\omega_{\rm r})\int_{0}^{\infty}ds_{2}G_{\gamma\delta\leftarrow\alpha\beta}(s_{2})$
	$\displaystyle\quad\quad\times e^{-i(\omega-\bar{\omega}_{1})\Delta t}[D_{1}(\omega_{\rm r},s_{2}+s_{1}-\Delta t)e^{i(\omega-\bar{\omega}_{1})(s_{2}+s_{1})}$
	$\displaystyle\quad\quad+D_{1}(\omega_{\rm r},s_{2}+s_{1}+\Delta t)e^{i(\omega+\omega_{\rm r}-\bar{\omega}_{2}-\bar{\omega}_{3})(s_{2}+s_{1})}],$		(2.16)

where $G_{\gamma\delta\leftarrow\alpha\beta}(t)$ is the matrix element of the time-evolution operator defined by $\rho_{\gamma\delta}(t)=\sum_{\alpha\beta}G_{\gamma\delta\leftarrow\alpha\beta}(t-s)\rho_{\alpha\beta}(s)$. In Eq. (2.13), $F^{\prime}_{\alpha\beta}(\omega,\omega_{\rm r};\Delta t)$ is defined by

	$\displaystyle F^{\prime}_{\alpha\beta}$	$\displaystyle(\omega,\omega_{\rm r};\Delta t)$
		$\displaystyle\quad=r(\omega,\omega_{\rm r})\int_{0}^{\infty}ds_{1}G_{\alpha\beta}(s_{1})$
		$\displaystyle\quad\quad\times e^{-i(\omega-\bar{\omega}_{1})\Delta t}[D_{1}(\omega_{\rm r},s_{1}+\Delta t)e^{i\bar{\omega}_{1}s_{1}}$
		$\displaystyle\quad\quad+D_{1}(\omega_{\rm r},s_{1}-\Delta t)e^{i(\bar{\omega}_{2}+\bar{\omega}_{3}-\omega_{\rm r})s_{1}}],$		(2.17)

The function $D_{n}(\omega,t)$ $(n=1,2,\dots)$ is introduced as

$\displaystyle D_{n}(\omega,t)=\int^{\infty}_{-\infty}\frac{d\xi}{2\pi}e^{-i\xi t}r(\xi+\bar{\omega}_{1},\omega)^{n}.$

(2.18)

Note that $D_{1}(\omega_{\rm r},t)$ is non-zero when $\lvert t\rvert\leq(T_{\mathrm{e}}^{(01)}+\lvert T_{02}\rvert)/2$, as illustrated in Fig. 2. The $\Delta t$-independent term in Eqs. (2.10) and (2.11), $\Delta S_{x}^{(y)}(\omega,\omega_{\rm r})$, originates from the field commutator. Details of the $\Delta t$-independent terms are given in Appendix D.

To understand the influence of entanglement times on the spectrum in Eq. (2.9), here we investigate the limiting cases. When $T_{\mathrm{e}}^{(01)}$, $T_{\mathrm{e}}^{(23)}$, and $T_{02}$ are much shorter than the characteristic timescales of the dynamics under investigation,Ishizaki (2020a); Fujihashi, Shimizu, and Ishizaki (2020) we approximately obtain $r(\omega_{1},\omega_{3})=1$ and $D_{n}(\omega,t)=\delta(t)$. Consequently, Eqs. (2.16) and (2.17) can be simplified as

	$\displaystyle F_{\gamma\delta\leftarrow\alpha\beta}(\omega,\omega_{\rm r};\Delta t,s)$	$\displaystyle=G_{\gamma\delta\leftarrow\alpha\beta}(\Delta t-s),$		(2.19)
	$\displaystyle F^{\prime}_{\alpha\beta}(\omega,\omega_{\rm r};\Delta t)$	$\displaystyle=G_{\alpha\beta}(\Delta t)e^{i(\omega_{p}-\omega-\omega_{\rm r})\Delta t},$		(2.20)

and thus, $I_{\epsilon\zeta;\gamma\delta\leftarrow\alpha\beta;\alpha 0}^{(y)}(\omega,\omega_{\rm r};\Delta t)$ is written as

$$I_{\epsilon\zeta;\gamma\delta\leftarrow\alpha\beta;\alpha 0}^{(y)}(\omega,\omega_{\rm r};\Delta t)\\ =G_{\epsilon\zeta}[\omega]G_{\gamma\delta\leftarrow\alpha\beta}(\Delta t)G_{\alpha 0}^{(y)}[\omega_{\mathrm{p}}-\omega_{\rm r}-\omega],$$

(2.21)

where $G_{\alpha 0}^{\mathrm{(r)}}[\omega]=G_{\alpha 0}^{\ast}[\omega]$ and $G_{\alpha 0}^{\mathrm{(nr)}}[\omega]=G_{\alpha 0}[\omega]$ have been introduced. In deriving Eq. (2.21), we assume that $G_{\gamma\delta\leftarrow\alpha\beta}(\Delta t-s_{1})G_{\alpha 0}(s_{1})\simeq G_{\gamma\delta\leftarrow\alpha\beta}(\Delta t)G_{\alpha 0}(s_{1})$ in the non-rephasing case.Cervetto et al. (2004) This approximation is justified when the response function varies slowly as a function of the waiting time, $\Delta t$. As was demonstrated in Ref. Ishizaki, 2020a, the signal $S(\omega,\omega_{\rm r};\Delta t)$ corresponds to the spectral information along the anti-diagonal line, $\omega_{1}+\omega_{3}=\omega_{\mathrm{p}}-\omega_{\rm r}$, on the absorptive two-dimensional (2D) spectrum $\mathcal{S}_{\rm 2D}(\omega_{3},t_{2},\omega_{1})$,

$\displaystyle S(\omega,\omega_{\rm r};\Delta t)\simeq-\mathcal{S}_{\rm 2D}(\omega,\Delta t,\omega_{\mathrm{p}}-\omega_{\rm r}-\omega),$

(2.22)

except for the $\Delta t$-independent terms in Eqs. (D.5) and (D.6), respectively. Equation (2.22) indicates that the two-photon counting signal $S(\omega,\omega_{\rm r};\Delta t)$, is homologous to the 2D spectrum, $\mathcal{S}_{\rm 2D}(\omega_{3},\Delta t,\omega_{1})$. This is true even when the frequency of the input laser, $\omega_{\mathrm{p}}$, is fixed. This correspondence is similar to, but different from, the results reported by Ref. Ishizaki, 2020a, wherein the transmission signal was found to provide the same information as the 2D spectrum only when sweeping the frequency of the input laser, $\omega_{\mathrm{p}}$. In addition, we consider the opposite limit, $T_{\mathrm{e}}^{(01)}\to\infty$ and $T_{\mathrm{e}}^{(23)}\to\infty$. We obtain $r(\omega_{1},\omega_{3})=\delta(\omega_{1}-\bar{\omega}_{1})\delta(\omega_{3}-\bar{\omega}_{3})$. Equations (2.16) and (2.17) can thus be written as

$$F_{\gamma\delta\leftarrow\alpha\beta}(\omega,\omega_{\rm r},\Delta t,s)\\ \propto\delta(\omega-\bar{\omega}_{1})\delta(\omega_{\rm r}-\bar{\omega}_{3})G_{\gamma\delta\leftarrow\alpha\beta}[0],$$

(2.23)

$$F^{\prime}_{\alpha\beta}(\omega,\omega_{\rm r};\Delta t)\\ \propto\delta(\omega-\bar{\omega}_{1})\delta(\omega_{\rm r}-\bar{\omega}_{3})\left(G_{\alpha\beta}[\bar{\omega}_{1}]+G_{\alpha\beta}[\bar{\omega}_{2}]\right),$$

(2.24)

where $G_{\gamma\delta\leftarrow\alpha\beta}[0]=\int_{0}^{\infty}dt\,G_{\gamma\delta\leftarrow\alpha\beta}(t)$ is defined. In this limit, the temporal resolution is eroded, and the spectrum in Eq. (2.9) does not provide any information on the excited-state dynamics.

To demonstrate how the spectrum provides time-resolved information on the state-to-state dynamics, we first investigate the limit of the short entanglement time, $T_{\mathrm{e}}^{(01)}=T_{\mathrm{e}}^{(23)}=T_{02}=0$, although it may be unrealistic for the cascaded PDC process. As shown in Eqs. (2.10) and (2.11), the spectrum contains $\Delta t$-independent contributions. Therefore, we consider the difference spectrum,

$\displaystyle\Delta S(\omega,\omega_{\rm r};\Delta t)=S(\omega,\omega_{\rm r};\Delta t)-S(\omega,\omega_{\rm r};0).$

(3.3)

Figure 4(a) presents the difference spectra of the model dimer for two different waiting times, $\Delta t$, when the frequency of the input laser is $\omega_{\mathrm{p}}=3\Omega_{2}$. The waiting times are $\Delta t=500{}\,\mathrm{fs}$ and $2000{}\,\mathrm{fs}$. Figure 4(a) shows strong signatures of the ESA signal at the location A, and strong signatures of the SE signal at the location labeled B. As was clarified in Eq. (2.21), the possible pairs of optical transitions probed at frequency $\omega=\omega_{\epsilon\zeta}$ are restricted by the resonance condition, $\omega_{\alpha 0}+\omega_{\epsilon\zeta}+\omega_{\rm r}\simeq\omega_{\mathrm{p}}$, which is imposed by the non-classical correlations among the entangled three photons. Hence, the negative peak at A corresponds to the pair of optical transitions ($0\to e_{2}$, $e_{1}\to f_{\bar{3}}$), while the positive peak at B corresponds to the pair of optical transitions ($0\to e_{2}$, $e_{1}\to 0$). The increases in these signal amplitudes during the waiting period $\Delta t$ reflect the excitation relaxation $e_{2}\to e_{1}$, as shown in Fig. 4(b). Therefore, the two-photon counting signal temporally resolves the excitation relaxation $e_{2}\to e_{1}$ through the changes in the amplitudes of peaks A or B during the waiting period, $\Delta t$.

In Fig. 4(a), strong ESA and SE signals can also be observed at locations C and D, respectively. These ESA and SE signals correspond to the pairs of optical transitions ($0\to e_{1}$, $e_{1}\to f_{\bar{3}}$) and ($0\to e_{1}$, $e_{1}\to 0$), respectively. As shown in Fig. 4(b), the difference spectrum exhibited changes in the amplitudes of peaks C and  D occurring within $500{}\,\mathrm{fs}$; these peaks are much faster than the excitation relaxation, $e_{2}\to e_{1}$. Moreover, Fig. 4(b) exhibits the oscillatory transients of peaks A and B, which persisted up to $\Delta t<500{}\,\mathrm{fs}$. However, the electronic coherence in the single-excitation manifold is not considered in this instance. To understand these transient behaviors, we consider the non-rephasing contribution of the ESA signal in Eq. (2.15). For demonstration purposes, we assume that the time evolution in the $t_{1}$ period is denoted by $G_{\alpha 0}(t_{1})=e^{-(i\omega_{\alpha 0}+\Gamma_{\alpha 0})t_{1}}$. With the use of Eqs. (2.19) and (3.2), the expression of $I_{\bar{\epsilon}\beta;\beta\beta\leftarrow\alpha\alpha;\alpha 0}^{\mathrm{(nr)}}(\omega,\omega_{\rm r};\Delta t)$ in Eq. (2.15) can be expressed as

$$I_{\bar{\epsilon}\beta;\beta\beta\leftarrow\alpha\alpha;\alpha 0}^{\mathrm{(nr)}}(\omega,\omega_{\rm r};\Delta t)\\ =-iG_{\bar{\epsilon}\beta}[\omega]\sum_{\xi=1,2}g_{\beta\alpha}^{(\xi)}\frac{e^{i{\Delta\omega_{\alpha 0}}\Delta t-\Gamma_{\alpha 0}\Delta t}-e^{-\lambda_{\xi}\Delta t}}{{\Delta\omega_{\alpha 0}}+i(\Gamma_{\alpha 0}-\lambda_{\xi})},$$

(3.4)

where $\lambda_{1}=0$, $\lambda_{2}=k_{1\leftarrow 2}+k_{2\leftarrow 1}$, and $\Delta\omega_{\alpha 0}=\omega_{\mathrm{p}}-\omega_{\rm r}-\omega-\omega_{\alpha 0}$. Equation (3.4) demonstrates that the amplitude of peak A oscillates at the frequency $\Delta\omega_{20}$. This is the detuning of the $0\to e_{2}$ transition from the frequency of photon 2. Similarly, the transient dynamics in peak C reflect the decay of the $\lvert e_{1}\rangle\langle 0\rvert$ coherence. Therefore, the transient dynamics in peaks A and C are not directly related to the dynamics in the single-excitation manifold during the $t_{2}$ period. The SE contributions to peaks B and D in the short-time region can also be understood in the same manner. If coherence $\lvert e_{\alpha}\rangle\langle e_{\beta}\rvert$ is considered, the time-evolution operator is modeled as $G_{\alpha\beta\leftarrow\alpha\beta}(t_{2})=e^{-(i\omega_{\alpha\beta}+\Gamma_{\alpha\beta})t_{2}}$. Thus, Eq. (2.15) yields

$$I_{\bar{\epsilon}\beta;\alpha\beta\leftarrow\alpha\beta;\alpha 0}^{\mathrm{(nr)}}(\omega,\omega_{\rm r};\Delta t)\\ =-iG_{\bar{\epsilon}\beta}[\omega]\frac{e^{i{\Delta\omega_{\beta 0}}\Delta t-\Gamma_{\alpha 0}\Delta t}-e^{-(i\omega_{\alpha\beta}+\Gamma_{\alpha\beta})\Delta t}}{{\Delta\omega_{\beta 0}}+i(\Gamma_{\alpha 0}-\Gamma_{\alpha\beta})}.$$

(3.5)

Equation (3.5) includes the oscillating component at the detuning frequency $\Delta\omega_{\beta 0}$, as well as the oscillation originating from the $\lvert e_{\alpha}\rangle\langle e_{\beta}\rvert$ coherence. In complex molecular systems such as photosynthetic light-harvesting proteins, the lifetime of the electronic coherence is typically a few hundred femtoseconds. On this time scale, the contribution of the $\lvert e_{\alpha}\rangle\langle 0\rvert$ coherence during the $t_{1}$ period to the signal in Eq. (3.5) cannot be ignored. In this respect, Eq. (3.5) indicates that it is difficult to extract relevant information on the electronic coherence from the oscillatory dynamics in the signal.

We investigate the effects of finite entanglement times on the spectrum. The central frequencies of the entangled three photons that have been generated can be varied by tuning the phase-matching conditions for the two PDC processes. Krapick et al. (2016); Antonosyan, Gevorgyan, and Kryuchkyan (2011) Therefore, we set the three central frequencies of the entangled three photons, that is, $\bar{\omega}_{1}=\bar{\omega}_{2}=\bar{\omega}_{3}=\omega_{\mathrm{p}}/3=\Omega_{2}$, which nearly resonate with the $0\to e_{2}$ transition. It is noted that $T_{\mathrm{e}}^{(01)}$, $T_{\mathrm{e}}^{(23)}$, and $T_{02}$ are not independent because the group velocities $v_{\sigma}=\partial k_{\sigma}/\partial\omega|_{\omega=\bar{\omega}_{\sigma}}$ depend on the central frequencies of the three photons, as presented in Eqs. (2.3) and (2.4). For simplicity, we consider cases that satisfy $T_{\mathrm{e}}\equiv T_{\mathrm{e}}^{(01)}=T_{\mathrm{e}}^{(23)}=2T_{02}$ in what follows. This condition corresponds to the equalities of $\partial k_{1}/\partial\omega|_{\omega=\omega_{\mathrm{p}}/3}=-\partial k_{0}/\partial\omega|_{\omega=\omega_{\mathrm{p}}/3}$ and $\partial k_{2}/\partial\omega|_{\omega=\omega_{\mathrm{p}}/3}=-\partial k_{3}/\partial\omega|_{\omega=\omega_{\mathrm{p}}/3}$. In practical perspective, this requires us to explore the design of quasi-phase-matched crystals that can hold the equalities of the group velocities, which is beyond the scope of the present paper.

Figure 5 presents the difference spectra of the molecular dimer in the cases of (a) $T_{\mathrm{e}}=10{}\,\mathrm{fs}$, (b) $T_{\mathrm{e}}=50{}\,\mathrm{fs}$, (c) $T_{\mathrm{e}}=100{}\,\mathrm{fs}$, and (d) $T_{\mathrm{e}}=500{}\,\mathrm{fs}$. The other parameters are the same as those shown in Fig. 4. The signal in Fig. 5(a) appears to be identical to the signal obtained under the three photon state in the limit of $T_{\mathrm{e}}=0$, illustrated in Fig. 4(a). However, the intensities of peaks B, C, and D decrease with increasing entanglement time, $T_{\mathrm{e}}$, as presented in Figs. 5(b) – (d), respectively. In the case of $T_{\mathrm{e}}=500{}\,\mathrm{fs}$, peaks B, C, and D almost disappear. To understand this dependence on the entanglement time, the rephasing contribution in Eq. (2.16) is considered as an example. Here, we note that $D_{1}(\omega_{\rm r},t)$ is non-zero when $\lvert t\rvert\leq 0.75\,T_{\mathrm{e}}$, as shown in Fig. 2. In the case of $\Delta t>0.75\,T_{\mathrm{e}}$, the expression of $F_{\beta\beta\leftarrow\alpha\alpha}(\omega,\omega_{\rm r};\Delta t,0)$ is obtained as

$$F_{\beta\beta\leftarrow\alpha\alpha}(\omega,\omega_{\rm r};\Delta t,0)\\ =r(\omega,\omega_{\rm r})\sum_{\xi=1,2}r(\omega+i\lambda_{\xi},\omega_{\rm r})g_{\beta\alpha}^{(\xi)}e^{-\lambda_{\xi}\Delta t}.$$

(3.6)

The bandwidth of the phase-matching function in Eq. (2.6) is related to the inverse of the entanglement time, $T_{\mathrm{e}}$. Equation (3.6) indicates that the finite entanglement time acts as a frequency filter through the spectral distribution of the phase-matching function, which limits the accessible spectral range. Figure 6 presents the spectral distribution of the phase-matching function in Eq. (2.6). Comparing Figs. 5 and 6 reveals that all optical transitions that are outside the bandwidth of the phase-matching function are suppressed. Therefore, the finite entanglement times can be used to selectively enhance specific Liouville pathways when the center frequencies of the entangled three photons are tuned to resonate with certain optical transitions. It is noteworthy that a similar property in terms of the finite entanglement time was discussed in the context of entangled two-photon spectroscopy.Munkhbaatar and Myung-Whun (2017)

Further, we investigate the time-evolution of peak A observed in the difference spectra (illustrated in Fig. 5). In the case of $\Delta t>0.75\,T_{\mathrm{e}}$, the contribution of the ESA signal at peak A in Eq. (3.6) is written as

$\displaystyle F_{11\leftarrow 22}(\bar{\omega}_{1},\bar{\omega}_{3};\Delta t,0)=g_{12}^{(1)}+\Lambda g_{12}^{(2)}e^{-\lambda_{2}\Delta t}$

(3.7)

with $\Lambda=8(\lambda_{2}T_{\mathrm{e}})^{-2}\sinh({\lambda_{2}T_{\mathrm{e}}}/{2})\sinh({\lambda_{2}T_{\mathrm{e}}}/{4})$, where the approximations of $\bar{\omega}_{1}\simeq\omega_{\bar{3}1}$ and $\bar{\omega}_{3}\simeq\omega_{\mathrm{p}}-\omega_{\bar{3}1}-\omega_{20}$ are employed. The $\Delta t$-dependence of Eq. (3.7) reflects the monotonous decay governed by the rate of the excitation transfer $\lambda_{2}=k_{1\leftarrow 2}+k_{2\leftarrow 1}$. Therefore, the signal provides information on the dynamics of $e_{2}\to e_{1}$ when $\Delta t>0.75\,T_{\mathrm{e}}$, as shown in Fig. 7(b). When the entanglement time, $T_{\mathrm{e}}$, is sufficiently short compared to the timescales of the excited-state dynamics, Eq. (3.7) becomes $F_{11\leftarrow 22}(\bar{\omega}_{1},\bar{\omega}_{3};\Delta t,0)\simeq G_{11\leftarrow 22}(\Delta t)$, as presented in Fig. 7(a). In contrast, in the case of $\Delta t<0.75\,T_{\mathrm{e}}$, Eq. (2.16) becomes

$$F_{11\leftarrow 22}(\bar{\omega}_{1},\bar{\omega}_{3};\Delta t,0)=-g_{12}^{(1)}-\frac{2g_{12}^{(2)}}{\lambda_{2}^{2}T_{\mathrm{e}}^{2}}e^{-\lambda_{2}T_{\mathrm{e}}}\\ \times[(1+\lambda_{2}\Delta t)e^{-\lambda_{2}\Delta t}+(1-\lambda_{2}\Delta t)e^{\lambda_{2}\Delta t}].$$

(3.8)

Equation (3.8) demonstrates the complicated time-evolution, making it impossible to extract relevant information on the excited-state dynamics from the signal. This can clearly be seen in Fig. 7(b), where it is not possible to temporally resolve the fast oscillatory transients within $0.75\,T_{\mathrm{e}}=375{}\,\mathrm{fs}$. Figures 5 – 7 suggest that the manipulation of the phase-matching function enables filtering out a specific frequency region of the spectra while maintaining ultrafast temporal resolution, resulting in the achievement of the joint temporal and frequency resolution.