Detecting electronic coherences by time-domain high-harmonic spectroscopy

Significance

To generate and monitor the coherent excitation without using X-ray light source, we propose an all-optical method called strong-field Raman scattering which is analogous to the three-step model for the high-order harmonic generation. Once such superposition is prepared, a follow-up probe optical laser can monitor its dynamics by detecting HHG signal which has already been proven to be very sensitive to the small fractions of the excited states. A semiperturbative model combining strong-field and perturbative techniques is developed to describe excitation and measurement of the electronic coherences and populations in multiwave mixing involving strong fields. This opens a route to transfer the powerful techniques of perturbative nonlinear optics to the strong-field regime.

Abstract

Ultrafast spectroscopy is capable of monitoring electronic and vibrational states. For electronic states a few eV apart, an X-ray laser source is required. We propose an alternative method based on the time-domain high-order harmonic spectroscopy where a coherent superposition of the electronic states is first prepared by the strong optical laser pulse. The coherent dynamics can then be probed by the higher-order harmonics generated by the delayed probe pulse. The high nonlinearity typically modeled by the three-step mechanism introduced by Lewenstein and Corkum can serve as a recipe for generation of the coherent excitation with broad bandwidth. The main advantage of the method is that only optical (non–X-ray) lasers are needed. A semiperturbative model based on the Liouville space superoperator approach is developed for the bookkeeping of the different orders of the nonlinear response for the high-order harmonic generation using multiple pulses. Coherence between bound electronic states is monitored in the harmonic spectra from both first- and second-order responses.

Detecting and controlling electron dynamics is one of the goals of the modern ultrafast science. Processes such as long-range coherent electron and energy transfer (1, 2) are significant in many physical, chemical, and biological systems. Monitoring electronic coherences is a challenging task since the coherent superposition of states a few eV apart is difficult to prepare. A broadband X-ray light source, such as high-order harmonic generation (HHG) (3, 4) and free electron laser (5, 6), is needed to create such superpositions, via ionization or the Raman process. Several techniques involving X-ray, such as Raman scattering (7) and attosecond streaking (8, 9), have been applied to measure the ultrafast electronic dynamics. Electronic coherence has been observed in the time-dependent pump-probe HHG signal (10, 11). In these experiments, a coherent excitation is prepared by pumping electrons from the ground state to the excited states directly with the 400-nm pulse or through an impulsive stimulated Raman scattering with the 800-nm pulse. It has been shown that a 0.1$\%$ excitation fraction can be converted into a 20$\%$ signal modulation because of the interference of different ionization and recombination pathways (11, 12). Several well-known models have been developed to calculate the harmonic spectrum from the gaseous media. Strong-field approximation (SFA) (13, 14) or quantum orbits model (15, 16) and the extended quantitative rescattering theory (17) are the main examples. While these methods are suitable to describe HHG process, they do not take into account excited states contributions and dephasing effects, which have been proven to play important roles in the time-domain pump-probe HHG spectroscopy (12, 18).

Here we present a physical model, which combines the SFA and the perturbative techniques to describe the multiwave mixing from a series of pulses in the HHG regime. In particular, we describe free electron dynamics to all orders in field–matter interactions, while bound state contributions can be treated perturbatively. This can be achieved by introducing an effective dipole moment analogous to the Raman polarizability which takes into account field-dependent free electron dynamics in the continuum state. Perturbative treatment allows us to use diagrammatic techniques commonly used in spectroscopy and thus provides an effective tool for bookkeeping of the field–matter interactions. As we show below, this combined treatment is consistent with the parameter regime typically used in HHG experiments and gives a good agreement with the exact solution, while having a significantly lower computational cost.

Pump-Probe HHG Scheme.

Coherence excitation in a multiphoton process is typically weak since the bandwidth of the optical pulse is narrower than the electronic bandgap. Considering the high nonlinearity of the three-step HHG model, ionization, acceleration, and recombination, it is plausible to generate coherent superposition of the residual cation by the strong-field ionization or coherent excitation of the neutral molecule by a recombination with the parent ion. It has been shown that the strong-field ionization by optical pulse can excite coherently the cation states (19, 20) separated more than a few eV apart. The three-step process is indeed similar to the stimulated Raman scattering except that the upper state is replaced by a continuum of states.

Fig. 1A shows the schematics of the proposed experimental setup. Fig. 1 B and C show the comparison between the spontaneous Raman scattering (SPRS) signal (21) and the multipulse high-order harmonics generation. We denote the coherent excitation via continuum states by strong optical pump pulse ($P_1$) as the strong-field Raman scattering (SFRS) process. The bandwidth that is covered by SFRS can be estimated by a Fourier transformation of ${\mu }_{mn}^c\left(t\right)f\left(t\right)$, where $f\left(t\right)$ is the envelope of the electric field and ${\mu }_{mn}^c\left(t\right)$ is the effective dipole moment describing electron transition between the bound states $m$ and $n$ through continuum states (Eq. 5). In principle, the SFRS covers a broad energy range since ${\mu }_{mn}^c\left(t\right)$ involves strong-field ionization. A time-delayed probe pulse ${\mathrm{P}}_2$ is sent to generate the harmonic signal. By taking the Fourier transformation with respect to the time delay, a corresponding spectra containing information about electronic coherences can be obtained (22). The first-order response involving two pulses can be described by the double-sided Feynman diagrams shown in Fig. 1D (see SI Appendix, section C, for details).

Fig. 1.

Illustration of SFRS and pump-probe HHG. (A) Schematic diagram of the multipulse HHG measurement. (B) Sketch of the SPRS signal for detecting the electronic coherences. Coherence between excited states is prepared by SPRS using pulses $P_1$ and $P_2$. $P_3$ is used to excite the system before spontaneous emission of the detected photon, which carries information about coherence. (C) Sketch of the time-domain HHG spectroscopy. Coherent excitation is prepared by a recombination of the ionized electrons, followed by a probe that is used to generate HHG signal. Dependence of the HHG on the time delay between pulses contains information about the electron dynamics. (D) Double-sided Feynman diagram for the first-order response involving two pulses (see SI Appendix, section C, for more details of diagram rules).

Semiperturbative Model.

Interaction between the electric field and the gas media can be described by the Schrödinger equation in the single-electron approximation:

$$i\hslash \frac{\partial \psi \left(t\right)}{\partial t}=\left(\frac{{\mathbf{p}}^2}{2m}+v\left(\mathbf{r}\right)-\mathbf{r}\cdot \mathbf{F}\left(t\right)\right)\psi \left(t\right)$$ [1]

where electron wavefunction

$$\psi \left(t\right)=\sum _m{e}^{-i{E}_{m}t}{C}_m\left(t\right)\left|{\phi }_m\right.⟩+e^{i{I}_{p}t}{\int }_{-\infty }^{+\infty }dv{C}_v\left(t\right)\left|v\right.⟩.$$ [2]

Here $E_m$ is the $m_{\text{th}}$ bound state energy; $I_p$ is the ionization potential of the ground state; and $C_m\left(t\right)$ and $C_v\left(t\right)$ are the bound and continuum state amplitudes, respectively. Using Markovian approximation, we introduce the density matrix for the bound states ${\rho }_{mn}\left(t\right)=C_m\left(t\right)C_n^{*}\left(t\right)$, which satisfies the Liouville–von Neumann equation (see SI Appendix, section A, for details)

$$\frac{\partial {\rho }_{\mathrm{m}\mathrm{n}}\left(t\right)}{\partial t}=-i\sum _k\left[H_{mk}\left(t\right){\rho }_{kn}\left(t\right)-{\rho }_{mk}\left(t\right)H_{nk}^{*}\left(t\right)\right]$$ [3]

where the effective field–matter interaction Hamiltonian is given by $H_{mk}\left(t\right)=-E\left(t\right){\eta }_{mk}\left(t,t\prime \right){|}_{t\prime =t}$, where

$${\eta }_{mk}\left(t,t\prime \right)=\left({\mu }_{mk}^b{e}^{-{\gamma }_{bb}t\prime }+{\mu }_{mk}^c\left(t\right)e^{-{\gamma }_{bc}t\prime }\right)e^{i\left(E_m-E_k\right)t\prime }.$$ [4]

Here ${\mu }_{mk}^b$ and ${\mu }_{mk}^c\left(t\right)$ are the effective transition dipole moments between the bound states $m$ and $k$, and ${\gamma }_{bb}$ (${\gamma }_{bc}$) represents a dephasing for the bound-to-bound (bound-to-continuum) transition, respectively. Note that ${\mu }_{mn}^b=\left.⟨{\phi }_m\right|r\left|{\phi }_n\right.⟩$ is an ordinary transition dipole moment, whereas

$$\begin{array}{ccc}\hfill & {\mu }_{mn}^c\left(t\right)=i{\int }_0^{t}dt\prime \sqrt{\frac{-2\pi \mathrm{i}}{\mathrm{t}-t\prime -i\epsilon }}{\mu }_{m,p_r}E\left(t\prime \right){\mu }_{p_i,n}\hfill & \hfill \\ \hfill & \times e^{-iS\left(t\prime ,t,p_{st}\right)+i{E}_n\left(t-t\prime \right)}\hfill & \hfill \end{array}$$ [5]

is a complex quantity, which is physically analogous to the electronic polarizability, responsible for the three-step process with the initial state n and the final state m (7). Here action is given by

$$S\left(t\prime ,t,p_{st}\right)={\int }_{t\prime }^{t}dt\prime \left[\left(p_{st}-A{\left(t\prime \right)}^2/2\right.\right],$$ [6]

with $p_i=p_{st}-A\left(t\prime \right)$, $p_r=p_{st}-A\left(t\right)$, and $p_{st}=\frac{{\int }_{t\prime }^{t}A\left(t\prime \right)dt\prime }{t-t\prime }$. The above expression for the dipole moment is similar to that used in the SFA model. Since the SFRS process involves the strong-field ionization, the effective bandwidth of the SFRS is much larger than the multiphoton transition energy scale. A half-cycle 1,600-nm pulse is used in simulations shown in Fig. 2. Note that a half-cycle pulse is not a prerequisite to obtain an ultrabroadband excitation. Similar to generating the supercontinuum high-harmonic spectrum, the required bandwidth is determined by $FT\left[{\sum }_{m}f\left(t\right){\mu }_{m1}^c\left(t\right)\right]$ in order to prepare an ultrabroadband coherent excitation. In principle, half-cycle, single-cycle, or few-cycle pulses suffice. If the pulse duration is short, the laser intensity should be strong enough to generate significant excitation probability. If the pulse duration is long, the bandwidth of $FT\left[{\sum }_{m}f\left(t\right){\mu }_{m1}^c\left(t\right)\right]$ is narrow. Thus, the choice for the pulse duration depends on the parameters of the molecule.

Fig. 2.

Energy range covered by a single-photon transition (black line), two-photon transition (red line), three photon-transition (blue line), and SFRS excitation (green line). The driving half-cycle pulse has central frequency $\omega =0.775$ eV.

The corresponding macroscopic polarization yields

$$P\left(t\right)=⟨⟨{\eta }^H\left(\mathrm{t},\mathrm{t}\right)|\rho ⟩⟩,$$ [7]

where

$$\begin{array}{ccc}\hfill & {\eta }^H\left(t,t\prime \right)=\left({\mu }_{mk}^b{e}^{-{\gamma }_{bb}t\prime }+{\mu }_{mk}^c\left(t\right)e^{-{\gamma }_{bc}t\prime }+{\mu }_{km}^{c*}\left(t\right)e^{-{\gamma }_{bc}t\prime }\right)\hfill & \hfill \\ \hfill & \times e^{i\left(E_m-E_k\right)t\prime }\hfill & \hfill \end{array}$$ [8]

is a Hermitian symmetrized form of the $\eta \left(t,t\prime \right)$ (see SI Appendix, Eq. S7, and also ref. 23). Thus, the infinite Liouville space of the strong-field problem involving continuum states is reduced to the finite Liouville space of the bound states only. Note that Eq. 3 is equivalent to equation 3.29 in ref. 24. The polarization Eq. 7 can be then expanded in powers of the electric field, i.e.,

$$P\left(t\right)\equiv P^{\left(0\right)}\left(t\right)+P^{\left(1\right)}\left(t\right)+P^{\left(2\right)}\left(t\right)+\cdots$$ [9]

where the $n_{\text{th}}$-order nonlinear polarization is given by ref. 24.

$$\begin{array}{ll}\hfill & P^{\left(n,n\ge 1\right)}\left(t\right)={\int }_0^{\infty }d{t}_n{\int }_0^{\infty }d{t}_{n-1}\cdots {\int }_0^{\infty }d{t}_1{S}^{\left(n\right)}\hfill \\ \hfill & E\left(t-t_n\right)E\left(t-t_n-t_{n-1}\right)\cdots E\left(t-t_n-t_{n-1}\cdots -t_1\right)\hfill \end{array}$$ [10]

with nonlinear response function defined as

$$\begin{array}{ll}\hfill & S^{\left(n,n\ge 1\right)}=i^n\theta \left(t_1\right)\theta \left(t_2\right)\cdots \theta \left(t_n\right)\hfill \\ \hfill & \langle {\eta }^H\left({t,t}_n+\cdots t_1\right)\left[\eta \left({t-t}_n,t_{n-1}\right),\left[\cdots \left[\eta \left({t-t}_n{-t}_{n-1}\right.\right.\right.\right.\hfill \\ \hfill & \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\left.\left.\left.\left.\cdots -t_2,t_1\right),\left[\eta \left(t-t_n-t_{n-1}\cdots -t_1,0\right),\rho \left(-\infty \right)\right]\right]\cdots \right]\right]\rangle .\hfill \end{array}$$ [11]

Since ${\eta }^H$ depends on the electric field implicitly, the zeroth order polarization yields $P^{\left(0\right)}={\eta }_{gg}\left(t\right)+c.c$, assuming the molecule is initially in its ground state $g$. Thus, it is clear that the $P^{\left(0\right)}$ is equivalent to the SFA model. Higher-order polarization terms $P^{\left(n\right)},n\ge 1$ yield additional molecular information, which is not accessible by the SFA.

Results and Discussion

Formaldehyde molecule fixed at its equilibrium position is taken as sample to illustrate the above method. Only one ground state ($g$), three excited states ($e_1,e_2,e_3$), and continuum states are retained for simplicity. Energy levels and transition dipole moments are presented explicitly in SI Appendix, section B. We first check the validity of the semiperturbative model. Fig. 3A shows the intensity dependence of the high-order harmonic yield for different order harmonics by solving Liouville–von Neumann equation. The scaling law changes from the weak-field (perturbative) to strong-field regime around intensity $1\times 10^{13}$ W/${\mathrm{c}\mathrm{m}}^2$. Fig. 3B reproduces the scaling law of noncollinear harmonic generation as a function of weak field. In our model, the lowest order of the wave mixing is given by Eq. 13 shown below. By doing Fourier transformation of ${\mu }^c\left(t\right)$ and replacing ${-m\omega }_{\mathrm{1,2}}t$ by ${-m\omega }_{\mathrm{1,2}}t+m{\mathbf{k}}_{\mathrm{1,2}}\mathbf{r}$ (25), we can get the sum-frequency wave-mixing signal where the high-harmonic frequency is given by ${\omega }_s=\left(m\pm 1\right){\omega }_1+n{\omega }_2$, and the direction is defined by ${\mathbf{k}}_s=\left(m\pm 1\right){\mathbf{k}}_1+n{\mathbf{k}}_2$. One can also obtain a signal at ${\omega }_s=m{\omega }_1+\left(n\pm 1\right){\omega }_2$ and ${\mathbf{k}}_s=m{\mathbf{k}}_1+\left(n\pm 1\right){\mathbf{k}}_2$. For a system with inversion symmetry, this form can ensure that the total harmonic order ($m+n\pm 1$) is odd. Our theory therefore provides a good fit for the experimental condition in ref. 26. Note that for the case of overlapping pulses, our model underestimates the laser intensity since each strong-field dipole ${\mu }_{mn}^c\left(t\right)$ contains only one electric field corresponding to that particular pulse. This is the reason for the higher laser intensity requirements for the simulated signal compared to the experimental one, as stated in the caption of Fig. 3B. There is no such defect if the pulses are well separated in time as in most of the time-resolved pump-probe experiments. Fig. 3 C and D show the comparison between the calculated spectra from the exact nonperturbative Eq. 2 and perturbative expansion in Eq. 9 for the laser intensities $1\times {10}^{13}$ W/${\mathrm{c}\mathrm{m}}^2$ and $4\times {10}^{13}$ W/${\mathrm{c}\mathrm{m}}^2$. The central wavelength is set to 800 nm. For the weak (Fig. 3C) laser intensity, the simulations which include the zeroth-, the first-, and the second-order responses reproduce the exact solution perfectly. It is apparent that the present model is no longer valid in the high-laser intensity regime, while the general features, such as rapid decay of the lower-order harmonics, long plateau, and the cutoff, are still qualitatively reproducible. The key point is that we treat the strong-field ionization process to all orders in the field, which is embedded in ${\mu }^c\left(t\right)$. At the same time, excited states are included perturbatively as their populations are much lower in comparison with the ground state. Once the laser intensity reaches the strong-field limit, higher-order responses should be included, and the transition between bound states should be treated to higher order. The following two assumptions are crucial for the effectiveness of the present model: 1) driving laser is strong enough to ionize electrons by tunneling in order to generate broadband ${\mu }^c\left(t\right)$ and 2) laser intensity should satisfy the harmonic approximation (exponential dependence of the harmonic yield with the laser intensity). These two conditions can be typically satisfied in the parameter regime where the laser intensity transits from the harmonic (perturbative) to the strong-field regime, as shown in the shaded region in Fig. 3A.

Fig. 3.

Validity of the semiperturbative model. (A) Dependence of the harmonic yield on driving laser intensity. (B) Harmonic yield vs. 400 nm pump intensity for the noncollinear HHG simulation in Eq. 13 (black solid lines) and experimental data taken from ref. 26 for different wave-mixing signals $\left(q_1,q_2\right)$. The system, noncollinear geometry, and laser parameters are the same as in ref. 26. In the simulations, only HOMO orbital and continuum states are included, and the transition dipole moments are taken from ref. 27. The horizontal axis of simulation data is divided by a factor of 10 (see Results and Discussion), which does not affect the scaling law. (C and D) Comparison of the high-order harmonic spectra from the sum of the three lowest-order responses, e.g., zeroth-, first-, and second-order responses (red line) and the numerical solution of Eq. 2 (black line) for three different laser intensities, separately.

We now describe how to use the pump-probe scheme for detection of the electronic coherence. Single-cycle 800 nm/$1\times {10}^{13}$ W/${\mathrm{c}\mathrm{m}}^2$ pump pulse is used in the simulations (28). Pulse $P_2$ is identical to the $P_1$, except that FWHM of $P_2$ is five optical cycles. In the followed simulation, direct transitions between the bound states are ignored since we focus only on the SFRS processes and above-threshold harmonics. Clearly, the HHG signal from the lower-order responses has dominant contribution. The zeroth-order response is independent of the time delay and is regarded as a background. In addition, different order responses can be further separated by the phase matching using noncollinear pulse geometry (26, 29). Here we concentrate on the first- and the second-order responses. The first-order polarization can be written as

$$\begin{array}{c}\hfill P^{\left(1\right)}\left(t\right)={\int }_0^{t}d{t}_1{S}^{\left(1\right)}{E}_1\left(t-t_1\right)\hfill \end{array}$$ [12]

with

$$S^{\left(1\right)}=⟨{\eta }^{H,p_2}\left(t,t_1\right){\eta }^{p_1}\left(t-t_1,0\right)⟩-c.c.$$ [13]

Here the superscript $p_i$ indicates the pulse order. The second-order response includes two components ($j$ = 1,2):

$$P_j^{\left(2\right)}\left(t\right)={\int }_0^{\infty }d{t}_2{\int }_0^{\infty }d{t}_1{S}_j^{\left(2\right)}{E}_j\left(t-t_2\right)E_1\left(t-t_2-t_1\right)$$ [14]

with

$$\begin{array}{ll}\hfill S_j^{\left(2\right)}& =-\left\{\theta \left(t_1\right)\theta \left(t_2\right)\left(Q_1^j-Q_2^j\right)+c.c.\right\}\hfill \\ \hfill Q_1^j& =⟨{\eta }^{H,p_2}\left(t,t_1+t_2\right){\eta }^{p_j}\left(t-t_2,t_1\right){\eta }^{p_1}\left(t-t_2-t_1,0\right)⟩\hfill \\ \hfill Q_2^j& =⟨{\eta }^{p_j}\left(t-t_2,t_1\right){\eta }^{H,p_2}\left(t,t_1+t_2\right){\eta }^{p_1}\left(t-t_2-t_1,0\right)⟩.\hfill \end{array}$$ [15]

The response functions can be read from the diagrams shown in SI Appendix, Figs. S2 and S3. The corresponding diagram rules are summarized in refs. 30–32. The calculated harmonic spectra based on Eqs. 12 and 14 with time delay $T_1=27fs$ are shown in Fig. 4A. By scanning the time delay $T_1$ between the two pulses, we can obtain the time-resolved high-order harmonic spectra. Since $P_1$ pulse is only used to excite the system and the HHG is generated from the $P_2$ pulse alone, one can use the ideal time domain approximation for the pump pulse $E_1\left(t\right)=f\left(t\right)cos\left({\omega }_1\left(t-{\tau }_1\right)\right)\delta \left(t-{\tau }_1\right)$ to speed up the numerical simulations. Note that the ideal time-domain approximation cannot be applied to the probe pulse $P_2$, since the HHG is a subcycle process. Thus the semiperturbative method allows us to achieve a significant improvement in computation time. For instance, the scan of time delay from 27 to 227 fs with 0.1-fs time step for $P^{\left(1\right)}$ and $P_1^{\left(2\right)}$ can be completed within 7 min on Intel(R) Xeon(R) CPU 2.27 GHz single-core processor. It takes about 16 h for $P_2^{\left(2\right)}$ since the ideal time-domain approximation cannot be applied to $t_2$. This can be compared with about 100 h that is required to compute a complete nonperturbative dynamics (Eq. 3). Thus, the semiperturbative approach allows to speed up the computation process without a significant loss of the computational precision.

Fig. 4.

HHG spectra for different order responses. (A) Calculated harmonic spectra based on $P^{\left(1\right)},P_2^{\left(2\right)}$ and $P_1^{\left(2\right)}$ with time delay $T_1$ = 27 fs. Direct transitions between ground states and excited states have been ignored. (B) The one-dimensional spectra obtained by taking Fourier transformation of time delay dependent ${\mathrm{8}}^{\text{th}}$-order harmonic signal calculated from $P^{\left(1\right)}$. (C and D) The same as B but for the second-order responses $P_2^{\left(2\right)}$ and $P_1^{\left(2\right)}$, respectively.

Fig. 4B shows the spectra obtained by the taking Fourier transform of the time-resolved $8^{\text{th}}$-order harmonic signal calculated from $P^{\left(1\right)}$. Fig. 4 C and D are the same as B but for the second-order responses $P_2^{\left(2\right)}$ and $P_1^{\left(2\right)}$, respectively. In Fig. 4 B and C, the four peaks correspond to ${\mathrm{\Omega }}_1$ = 1, 3, 5, and 6 eV. From Eqs. 12 and 13, it follows that the intensity of the harmonic signal is

$$\sim \sum _m|\left({\mu }_{1,m}^{c,P_2}+{\mu }_{m,1}^{c*,P_2}\right){\mu }_{1,m}^{c,P_1}{e}^{i\left(E_m-E_1\right)T_1}-c.c.{|}^2.$$ [16]

Thus, the peaks in frequency domain correspond to a combination of the bound state energies $\pm \mathrm{\Delta }{E}_{mg}\pm \mathrm{\Delta }{E}_{ng}$, where $\mathrm{\Delta }{E}_{mn}=E_m-E_n$. The energy levels are set relative to the continuum as $E_g=-5$, $E_{e_1}=-3$, $E_{e_2}=-2.5$, and $E_{e_3}=-2$ eV. Generally, there should be more than four peaks in Fig. 4B. However, the transition dipole moments between $e_1$, $e_2$, and continuum are negligible, so several peaks involving these two dark states cannot be observed. The transition dipole moment between $e_3$ and continuum state is dominant, so that the peaks involving $e_3$ correspond to ${\mathrm{\Omega }}_1=$3 $\pm$ 0 eV, 3 $\pm$ 2 eV, and 3 $\pm$ 3 eV. The harmonic intensity generated from $S_1^{\left(2\right)}$ is

$$\begin{array}{ccc}\hfill & \sim \sum _{m,n}|\left({\mu }_{g,m}^{c,P_2}+{\mu }_{m,g}^{c*,P_2}\right)\left({\mu }_{m,n}^b+{\mu }_{m,n}^{c,P_1}\right){\mu }_{n,g}^{c,P_1}{e}^{i\left(E_1-E_m\right)T_1}\hfill & \hfill \\ \hfill & -\left({\mu }_{g,m}^b+{\mu }_{g,m}^{c,P_1}\right)\left({\mu }_{m,n}^{c,P_2}+{\mu }_{n,m}^{c*,P_2}\right){\mu }_{n,g}^{c,P_1}{e}^{i\left(E_m-E_n\right)T_1}+c.c|.\hfill & \hfill \\ \hfill & \hfill & \hfill \end{array}$$ [17]

The peaks in Fourier transformed spectra should be thus observed at $\pm \mathrm{\Delta }{E}_{gm}\pm \mathrm{\Delta }{E}_{mn}$. The peaks involving state $e_1$ can be further enhanced due to the large dipole moments, which explains the appearance of the additional peaks in Fig. 4D. Since dipole moments between $e_2$ and continuum are negligible, the transitions directly involving $e_2$ cannot be observed.

Conclusion

In summary, we proposed a method to detect long-range electronic coherence with all-optical pulses. Coherent excitation is achieved by SFRS process, and coherent information is detected by the high-order harmonic spectra driven by a combination of optical pump and probe pulses. A semiperturbative model is developed to describe the pump-probe high-harmonic signals by expanding over the field–matter interactions between bound and continuum states while treating the free electron dynamics nonperturbatively. Typically, material responses in spectroscopic measurements suffer from inhomogeneous broadening since different molecules have random orientations. So far, the relevant experiments were carried out only for the aligned molecules, such as NO (11). For small molecules with a few internal degrees of freedom, optical alignment is a well-prescribed way to reduce the effect of inhomogeneous broadening. However, for the large molecules with complex internal structure or in the solution, it becomes difficult to get high-degree alignment. An alternative method used in multidimensional spectroscopy is based on the photon echo techniques, where the inhomogeneous broadening can be selectively eliminated by a proper phase matching selection (24). Usually, photon echo is observed in the four-wave mixing signal which requires four pulses. Our present approach is a first step toward the implementation of the photon echo technique and multidimensional spectroscopy in the high-harmonic regime. Experimentally, a relatively large number of peaks originate from the coherences between dense levels and different order responses, which may lead to the congested spectra. Since net absorption of different order harmonics ($m{\omega }_1$) can access information about different energy levels, the spectral congestion can be relieved by selectively detecting the wave mixing signal $\mathrm{\Omega }=m{\omega }_1+n{\omega }_2\hspace{1em}\left(k_s=m{k}_1+n{k}_2\right)$ in noncollinear geometry using pump-probe setup. It deserves further efforts to describe the details of the possible future experiments.

Data Availability.

The code calculating the results presented in this paper can be found at Open Science Framework, https://osf.io/7qnh9/. All of the other data discussed in the paper are available in the main text and SI Appendix.