Multi-wave mixing in the high harmonic regime: monitoring electronic dynamics

Shicheng Jiang; Markus Kowalewski; Konstantin E. Dorfman

doi:10.1364/OE.414619

1. Introduction

Spectroscopic detection of the coherent molecular dynamics provides an intriguing opportunity to access a unique material information, including the phase evolution of molecular eigenstates, coherence dephasing timescales, population relaxation rates and couplings to the environment. To prepare a coherent excitation, the bandwidth of the pump pulse should be broad enough to cover the energy range of the involved states. On the other hand, the oscillation periods of the polarization in the coherent superposition of electronic states scale inversely with the energy differences. It means that the time resolution of the pump-probe scheme should be high enough if the energy scale is very large [1]. To overcome these two obstacles, usually a broadband X-ray light source generated by a free-electron laser (FEL) [2] or a high-harmonic generation (HHG) [3] is implemented.

Although X-ray light sources have many advantages and have been widely applied in multidimensional spectroscopy to detect ultrafast dynamics [4], development of the all-optical spectroscopic methods is still highly desirable. The first question is how can we prepare a broadband coherent excitation by strong optical pulse? Coherent excitation of cations generated by strong-field ionization with optical pulse has been previously discussed theoretically [5–8] and realized experimentally [9–11]. As discussed in [5], since strong-field tunneling ionization happens wihtin a short time scale using a few-cycle optical pulse, the broad bandwidth in frequency domain will ensure that the states of cation can be populated coherently. As an opposite process, laser-induced recombination of electrons can also populate the neutral states coherently [12,13]. The relevant discussion about laser-induced coherent excitation of neutral states can be found in the Ref. [14]. Once the coherent superposition is prepared, the related electronic dynamics can be detected by another time-delayed strong optical probe pulse. It has been shown that a 0.1$\%$ excitation fraction can be converted into a 20$\%$ modulation of the HHG signal [15] making HHG a suitable tool to detect the coherent dynamics.

So far, most of the HHG pump-probe schemes suitable for detection of electronic coherences were limited to collinear geometries and one-dimensional spectroscopy (e.g. a single pump pulse followed by a probe pulse) [14,16–18]. In the collinear geometry, the background signal can not be eliminated. Additionally, if the target is a large biological macromolecule, the dense energy spectra with significant linewidth broadening would result in a potential spectral congestion. Nonlinear spectroscopic methods are not limited to simple pump-probe schemes or collinear geometries. In the perturbation regime, the abundance of different spectroscopic methods that utilize spatial phase matching such as transient grating [19], photon echo [20] and other multi-wave mixing techniques [21] have significant differences and benefits for approaching matter information with different detection windows. Wavemixing of HHG generated by two strong optical pulses overlapped in time domain has also been carried out [22]. Therefore, it is natural to use non-collinear multi-pulses well separated in time to detect coherent dynamics. Indeed, the semi-perturbative model developed in Ref. [14] aims to realize the strong-field multidimensional spectroscopy in non-collinear geometry. In the strong-field non-collinear geometry, same harmonics order can be generated as a result of a combination of different quanta of the input lasers.

In this paper, we will show that using sophisticated post selection techniques to filter out given combinations of various harmonic orders yields a possibility to reduce the spectral congestion. Furthermore, when the pump and the probe pulses are well separated in time, the coherences $\rho _{mn}$ between the bound states m and n (n$\neq$ m) are capable of storing the wave vectors of the pump pulse, while the population $\rho _{mm}$ is not. This means the polarization explicitly depends on $\bf {k}$ and $\bf {r}$ after interaction with a single pump pulse. The excited state population is homogeneous, e.g. independent of $\bf {k}$ and $\bf {r}$. Thus, the population dynamics initiated by the pump pulse propagating along $\bf {k_1}$ can be probed by the HHG signal co-propagating with the probe along $\bf {k_2}$. In contrast, the coherent dynamics can be detected in the phase-matched directions ($q_1\textbf {k}_{1}+q_2\textbf {k}_{2}$), where ($q_1,q_2$) represents the net absorption of $q_1$ photons of pump pulse and $q_2$ photons of probe pulse.

The paper is organized as follows. We first present the extension of the semi-perturbative model from collinear to non-collinear geometry. The following parts will present the calculated results for multi-wave mixing signals using the extended semi-perturbative model. In particular, we will show the influence of the pump pulse bandwidth; explaining the ways to reduce spectral congestion by selectively detecting the wave-mixing signal; and presenting a method to separate the dynamics of coherences and populations in the pump-probe scheme. Finally, we will conclude and discuss the future research directions. Details of the ab-initio simulation for energy levels and transition dipole moments of coumarin will be presented in the Supplementary Materials (MS).

2. Semi-perturbative model

By introducing an effective strong-field transition dipole $\mu _{mn}^c(t)$ connecting bound state $m$ and $n$ through continuum states via strong field Raman scattering (SFRS) [14], the macroscopic polarization can be expanded in perturbative series:

(1)$$P(t) \equiv P^{(0)}(t)+P^{(1)}(t)+P^{(2)}(t)+\cdots$$

Details of the semi-perturbative model and discussion regarding the applicability of such expansion can be found in Ref. [14]. Note that each perturbation order contains explicitly strong field ionization and recombination processes. The zeroth order polarization $P^{(0)}=\eta _{gg}(t)+c.c.$ represents the strong-field approximation (SFA) model including only the ground state [23], where the operator $\eta$ is defined bellow. Because the zeroth order does not describe the wave mixing signal, and the second and higher order response are much weaker than the first order, we only focus on the first order response, which can be written as

(2)$$\begin{array}{l} P^{(1)}(t) = i \int_{0}^{t} d t_{1}\left\{\sum_{m} (\eta^{H}_{gm})^{ p_{2}}\left(t, t_{1}\right) (\eta_{mg})^{p_{1}}\left(t-t_{1}, 0\right)-c . c .\right\} E_{1}(t-t_{1})\end{array}$$

where

(3)$$\eta_{mn}(t,t^{\prime})=\mu_{mn}^{c*}(t)e^{{-}i(\epsilon_m-\epsilon_n-i\gamma_{bc})t^{\prime}},$$

and

(4)$$\eta^{H}_{mn}(t,t^{\prime})=(\mu_{mn}^{c*}(t)+\mu_{nm}^{c}(t))e^{{-}i(\epsilon_m-\epsilon_n-i\gamma_{bc})t^{\prime}}.$$

The superscripts $p_i$ in Eq. (2) represent different pulses, $\gamma$ is the dephasing rate of the polarization, and $\epsilon _m$ are the energy levels of bound states, $g$ means ground state. The electric field is $E_j(t)=f_j(t)(e^{i\omega _j t+i\phi _j}+c.c.)$, where $f_j(t)$ is the $\sin^2$-like envelope function . The effective dipole moment

(5)$$\begin{array}{l} \mu_{mn}^c(t)= i \int_{0}^{t} dt^{\prime} \sqrt{\frac{-2 \pi \mathrm{i}}{\mathrm{t}-t^{\prime}-i \varepsilon}}\mu_{m, p_r} E\left(t^{\prime}\right) \mu_{p_i, n} \times e^{{-}iS(t^{\prime},t,p_{st})+i \epsilon_{n}(t-t^{\prime})}\end{array}$$

is a complex quantity responsible for the transition from the initial state $n$ to the final state $m$ through the continuum state. The action $S(t^{\prime },t,p_{st})=\int _{t^{\prime }}^{t} d t^{\prime \prime }\left [(p_{s t}-A(t^{\prime \prime })^{2} / 2\right ]$, $p_i=p_{st}-A(t^{\prime })$, $p_r=p_{st}-A(t)$, $p_{s t}=\frac {\int _{t^{\prime }}^{t} A\left (t^{\prime \prime }\right ) d t^{\prime \prime }}{t-t^{\prime }}$. Here, the saddle point approximation is applied to the integral over momentum $p$. In Eq. (3) and (4), the direct transition between bound states are neglected. The discussion about such a simplification can be found in Supplement 1. In order to recast the polarization in the form analogous to the expression in perturbative regime [see Eq. (17) in Ref. [24]], we need to modify Eq. (2) by two steps. First, using a wavelet transformation we obtain the time-frequency profile of effective transition dipole:

(6)$$\begin{aligned}\mu_{m n}^{c*}(t)\sim\sum_{q={-}\infty}^{+\infty} W_{m n}(q, t) e^{i q \omega t},\\ \mu_{mn}^{c*}(t)+\mu_{nm}^c(t)\sim\sum_{q={-}\infty}^{+\infty} W_{m n}^H(q, t) e^{i q \omega t}, \end{aligned}$$

where $W_{m n}(q, t)$ is the absolute amplitude of the wavelet transformation of the effective transition dipole moment, $\omega$ is the frequency of the laser. Second, replacing the effective transition dipole in Eq. (3) and (4) by the time-frequency profile in Eq. (6), the first-order polarization can be recast as:

(7)$$P^{(1)}(t)=i\int_{0}^{\infty} dt_{1}(J_1-c.c.)\Big[f_1(t-t_1)e^{i\omega_1(t-t_1)}+c.c.\Big],$$

where

(8)$$J_1=\sum_{m,q_2,q_1}W_{gm}^{H,p_2}(q_2,t)e^{iq_2\omega_2t} W_{mg}^{p_1}(q_1,t-t_1) e^{iq_1\omega_1(t-t_1)+i(\epsilon_m-\epsilon_g-i\gamma_{bc})t_1}.$$

The wave vector can be further introduced by replacing $q\omega _{1/2}t$ with $q\omega _{1/2}t- q\textbf {k}_{1/2}\textbf {r}$, where $\textbf{k}_{1/2}$ is the wave vector of the pump or probe laser [25]. The polarization thus reads:

(9)$$P^{(1)}(r, t)=\sum_{k_{s}, \omega_{s}} e^{(i \omega_{s}t-\textbf{ik}_{\textbf s} \cdot {\textbf r})} P^{(1)}(\textbf{k}_{s}, \omega_s,{t}),$$

where $\omega _s$ is the frequency of the radiation, and $\bf {k}_s$ is the wave vector determining the direction of the propagation. It is more convenient to write the polarization in the form of Eq. (9), because the spectroscopic observables can be directly calculated for different detection techniques [26]. If the detection angle is restricted within the intersection angle, e.g. the detection angle $\alpha \in (0,\beta )$ as shown in Fig. 1, and neglecting the fast oscillation in the integral over $t_1$, the frequencies of the detected signals are defined by $\omega _s=q_2\omega _2+(q_1\pm 1)\omega _1$, $\textbf {k}_s=q_2\textbf {k}_2+(q_1\pm 1)\textbf {k}_1$ with $q_2$ and $(q_1\pm 1)$ are integer numbers. For the time-integrated detection [26], the intensity of the signal $\omega _s$ is

(10)$$S(\textbf{k}_{\textbf{s}},\omega_s)=\int_{0}^{\infty}|P^{(1)}\left(\textbf{k}_{\textbf{s}}, \omega_s,t\right)|^2 dt$$

with

(11)$$\begin{aligned} P^{(1)}\left(\textbf{k}_{\textbf{s}}, \omega_s,t\right)=i\int_{0}^{\infty} dt_{1}\Big[\sum_{m}( W_{gm}^{H,P_2}(q_2,t)) W_{mg}^{P_1}(q_1,t-t_1)f_1(t-t_1)e^{{-}i(q_1\pm1)\omega_1t_1+i(\epsilon_m-\epsilon_g-i\gamma_{bc})t_1}\Big]\\ \end{aligned}$$

Fig. 1. Illustration of the scheme. Two noncollinear pulses with tunable time delays intersect in the HHG gas cell.. The different harmonic singal can be detected along different propagation direction because of the phase-matched condition $\textbf{k}_{\textbf{s}}=q_1\textbf{k}_{\textbf1}+q_2\textbf{k}_{\textbf{2}}$. The crossing angle $\beta \sim 12.5$ mrad. The inserted structure is coumarin. In the simulation, the laser is polarized along $z$ axis.

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3. Results and discussion

3.1 Pulse duration control

In order to separate the harmonic signals into different phase matching directions according to the conservation of momentum, a non-collinear geometry setup is needed. As shown in Fig. 1, there is a small angle between the propagation directions of pump and probe lasers. Therefore, the wavemixing harmonic signal with energy $\Omega _s=q_1\omega _1+q_2\omega _2$ can be detected in the direction $\mathbf {k}_s=q_1\mathbf {k}_1+q_2\mathbf {k}_2$, where $\omega _1$ ($\omega _2$) and $\mathbf {k}_1$ ($\mathbf {k}_2$) are the carrier frequency and wave vector of the pump (probe) laser. The angle between $\mathbf {k}_1$ and $\mathbf {k}_2$ is set to $12.5$ mrad. The intensity and wavelength of the pump laser are $2\times 10^{13}$ W/cm$^2$ and 800 nm, respectively. The corresponding parameters of the probe laser are fixed as $2\times 10^{13}$ W/cm$^2$ of peak intensity, 400 nm of central wavelength and 30 fs of pulse duration. We can obtain the time-resolved high harmonic signal by scanning the time delay $t_d$ between the two pulses. We simulate the signal for the coumarin molecule whose electronic energy levels are shown in Fig. 2(d). We have excluded the vibrational degrees of freedom in the calculations. The transition dipole moments between bound states and continuum states are calculated using Dyson orbitals [27]. The lowest three bound states of the neutral molecule and two lowest states of the ion are included in the calculation. The remaining states do not significantly contribute to the signals due to small transition dipole moments between the bound and continuum states.

It has been shown in Ref. [14], that the the energy range covered by the strong-field Raman scattering is determined by the Fourier transformation of $\sum _m f(t) \mu _{mg}^{c}(t)$. Thus, the pulse duration of the pump laser should play an important role in the coherent excitation. Figures 2(a)-(c) depict the detection angle-dependent high harmonic spectra with time-delay fixed to be 45 fs for three different values of pulse duration 5.4 fs, 10.0 fs, and 20 fs, respectively. In Fig. 2(b), the signal labeled as ($q_1,q_2$) indicates the net absorption of $q_1$ photons with energy $\omega _1$ and $q_2$ photons with energy $\omega _2$ which leads to the emission of high energy photons with energy $\Omega =q_1\omega _1+q_2\omega _2$. In order to obtain high spectral resolution, the Gaussian window $g(x)=\left (\frac {1}{\sqrt {\delta }}\right ) e^{it} e^{-t^{2} / 2 \delta ^{2}}$ with $\delta =100$ is used in the wavelet transformation. Spectral features along the line ($q_1,q_2=3,4,5$) are blurred when the pulse duration is decreased to two optical cycles. This phenomenon is identical to generating a supercontinuum harmonic spectrum using a few-cycle laser. Figure 2(d) depicts the Fourier transform of the time-delay for the harmonic signal of (4,1). The two peaks at 1.14 eV and 1.68 eV originate from the coherence between $e_1$(5.07 eV), $e_2$(6.24 eV) and $e_2$(6.24 eV), $e_3$(7.92 eV). If the duration of the pump pulse is too small, as seen in the case for 5.4 fs, this timescale is not sufficient to generate significant excitation probability. As a result, the 1D spectra depicted by a black line in Fig. 2(d) is weak. On the other hand, if the pulse duration is very large , e.g. 20 fs, the bandwidth covered by the $\mu ^c(t)$ is too narrow to generate a significant degree of coherence. In real experiments, therefore the pulse duration and intensity of the pump laser should be carefully tuned to reach a condition when both excitation probability and electronic coherence are sufficiently large.

Fig. 2. Detection angle-dependent high harmonic spectra for different durations of the pump pulse, 5.4 fs - (a), 10 fs - (b) and 20 fs - (c). Time-delay between pump and probe pulses are fixed to be 45 fs. The horizontal axis is the detection angle $\alpha$ as shown in Fig. 1. Vertical axis is the harmonic order $\Omega /\omega _1$. (d) Fourier transform spectra of the time-delay dependent harmonic signal (4,1) for different durations of pump pulse. . The inset shows the energy level diagram of the coumarin.

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3.2 Phase matching control of the electronic coherence

In Fig. 3(a), the black line corresponds to the Fourier transformation of $\sum _m f_1(t)\mu _{mg}^{c}(t)$ for 5.4 fs pulse duration, the bandwidth of which is significantly enlarged and is capable of covering the whole band of excited states. Figure 3(b) shows that, the absorption of different number of $\omega _1$ photons corresponds to the multiphoton resonance condition with different excited states. Thus, detection of the time-delay dependent harmonic signal of different orders allows to selectively analyze the dynamics between different excited states. In Fig. 3(c)-(e), we show the spectra corresponding to different combinations of pump and probe photon quanta, e.g. (3,1), (4,1) and (5,1). Since 3$\omega _1$ is near resonance with the lowest two excited states, one can observe the coherence between $e_1$ and $e_2$. Absorption of 4$\omega _1$ is near resonance with the states e$_1$, e$_2$ and e$_3$, so the coherence between $e_1$(5.07 eV), $e_2$(6.24 eV) and $e_2$(6.24 eV), $e_3$(7.92 eV) can be observed. For the signal of (5,1), the energy of 5$\omega _1$ is close to the energy of states e$_2$ and e$_3$, so one can detect the coherence between e$_2$ and e$_3$. If the system has a larger number of higher excited states, one can selectively detect the coherent dynamics by choosing an appropriate set of quanta $q_1$ and $q_2$, corresponding to the multi-photon resonance condition, which can be detected in a well defined unique phase-matching direction. Thus, the phase matching control can reduce the spectral congestion in further experiments on larger size biological macromolecules.

Fig. 3. (a) Fourier transformation of $\sum _m f_1(t)\mu _{mg}^{c}(t)$ for the pump pulse with 5.4 fs pulse duration, $2\times 10^{13}$W/cm$^2$ laser intensity and 800 nm wavelength. (b) Energy levels of the neutral coumarin. The red dashed lines represent the transition paths via SFRS by absorbing different numbers of $\omega _1$. (c)-(e) Fourier transform spectra of the time-delay dependent harmonic signal for different number of pump photons.

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3.3 Separation of the dephasing timescales

Time-resolved pump-probe HHG or fluorescence signals have demonstrated an important role of the excited states dynamics [17,28,29], where the collision induced changing of population in excited states was regarded as the main reason for the decay and recovery of time-delay-dependent optical signals. It is still unknown whether the coherences were created and whether the dephasing or rephasing of coherences play any significant role. The polarization generated by a single pump pulse can store the wavevector of the pulse, but the population density is unable to do so [30]. Note that this is different from the transient grating technique in which two pump pulses overlap in the time domain [19]. The high harmonic signal driven by the probe pulse may have the same initial and final electronic state, e.g.

(12)$$\begin{array}{l} P(t) = \sum_{m} \eta_{mm}^{p_{2}}\left(t, 0\right)\rho_{mm} +c . c . \end{array}$$

where $\rho _{mm}$ is the population density of state $m$ after the pump pulse. The high harmonic signal from Eq. (12) co-propagates with the probe pulse [31,32]. As demonstrated in Ref. [17,28,29], the population density of excited states can be decreased by collision induced ionization or increased by collision induced excitation. Thus, $\rho _{mm}$ can be regarded as a time-delay-dependent parameter. Here, the population density depending on the time-delay can be written approximately by $\rho _{mm}=\rho _{mm}^0+\rho _{mm}^{coll}(1-e^{-\Gamma t_d})$ [33], where $\rho _{mm}^0$ is the population density after the pump pulse, $\rho _{mm}^{coll}$ is the total probability of collision induced excitation, $\Gamma$ is the collision rate of excitation, and $t_d$ is the time-delay between pump and probe pulses. For simplification, both $\rho _{mm}^0$ and $\rho _{mm}^{coll}$ are set to be $1.5\%$. $1/\Gamma =100 $fs. The above parameters are set phenomenologically as they set to add a qualitative understanding of the phenomena. Note that only the collision induced excitation is included, while the collisions induced ionization from the excited states which results in decreasing of the HHG intensity is not considered. Figure 4 shows the decay of the wavemixing HHG signal due to dephasing and recovery of HHG signal due to collision induced excitation of population. These two kinds of signals are detected in different direction. Furthermore, one can also extract the coherence dephasing and population changing time scales from the contents in Fig. 4 by fitting using Eq. 10. Note that, if inhomogeneous broadening is significant in the target, pure dephasing can only be detected through the photon echo signal.

Fig. 4. Time-resolved pump-probe signal. The black line represents the decay of the wavemixing HHG singal corresponding to (4,3). The detection direction is along $4\mathbf {k}_1+3\mathbf {k}_2$. The blue line represents the recovery of HHG signal by Eq. (13) due to collision excitation. Here, the pump pulse parameters are the same as in Fig. 3, while the parameters of the probe pulse are 800 nm of wavelength, 30 fs of pulse duration, and 2$\times 10^{13}$ W/cm$^2$ of laser intensity.

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4. Conclusion

In this work, we developed a framework for multiwave mixing nonlinear HHG signals by extending the semi-perturbative model to noncollinear geometries. The benefit of a noncollinear geometry is that one can spatially separate different harmonic orders with different combinations of $(q_1,q_2)$. Since the pulse duration of the pump laser plays an important role in coherent excitation, we first investigated the effects of the pump pulse duration. It is found that the pump pulse should be neither too short nor too long in order to maximize both the excitation probability as well as the coherence across multiple electronic states. Setting the pulse duration to be 5.4 fs, we calculate the Fourier transform spectra of the time-delay dependent signals for different harmonic orders. Our results show that the coherent dynamics can be selectively analyzed by detecting different combinations of $(q_1\omega _1+q_2\omega _2)$. Depending on the choice of the pump photon number $q_1$, electronic excitation may occur in multi-photon resonance with various electronic states. This can be used to reduce the spectral congestion, which is especially crucial in the context of large biological macromolecules. Additionally, we show that the decay due to dephasing of coherence and recovery due to collision included excitation of population density can be separated in space by a non-collinear setup. In the present model, the Coulomb potential, the nuclear degrees of freedom, and the coupling between ionic states are not considered. These effects may influence the excitation probability and coherence degree of neutral excited states, which can influence the intensity of the peaks in the Fourier-transformed spectra. At the same time, the general structure, for example, the positions of the resonant peaks in the Fourier-transformed spectroscopy, will not be changed. Once the related experimental data is available, it would be important to estimate the influence of these factors on the detected signals, and improve the theoretical model accordingly.

Funding

National Natural Science Foundation of China (12074124); Zijiang Endowed Young Scholar Fund, East China Normal University; China Postdoctoral Science Foundation (2019TQ0098); Overseas Expertise Introduction Project for Discipline Innovation (B12024); Swedish Research Council, Vetenskapsrådet (VR 2018-05346).

Disclosures

The authors declare no conflicts of interest.

Supplemental document

See Supplement 1 for supporting content.

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Multi-wave mixing in the high harmonic regime: monitoring electronic dynamics

Abstract

1. Introduction

2. Semi-perturbative model

3. Results and discussion

3.1 Pulse duration control

3.2 Phase matching control of the electronic coherence

3.3 Separation of the dephasing timescales

4. Conclusion

Funding

Disclosures

Supplemental document

References

Supplementary Material (1)

Cited By

Figures (4)

Equations (12)

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