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Abstract
We theoretically study the spectral and temporal fine subpeak structures in the below-threshold harmonic (BTH) spectra of the stretched H2 and N2 molecules by solving the one-electron time-dependent Schrödinger equation (TDSE) in conjunction with the wavelet time-frequency analysis. We identify such fine subpeaks come from the Rabi-flopping between the ground state and the first excited state using the simple two-state model. We also confirm that these subpeak structures in BTH spectra are common for molecules at large internuclear distances if two molecular states are strongly coupled. Furthermore, the spacing between the adjacent subpeaks in BTH spectra can be determined approximately by analyzing the induced dipole moment in the time domain.
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1. Introduction
Owing to the potential applications as a tabletop light source covering from XUV to soft X-ray [1,2] and for the production of ultrashort attosecond pulses [3,4], high-order harmonic generation (HHG) of atoms and molecules in intense laser fields has been a topic of great interests over the past few decades [5,6]. Nowadays, HHG can provide a novel tool for probing ultrafast electronic dynamics in atoms and molecules in real-time [7–10] and for tomographically imaging molecular orbitals [11–17]. The HHG procedure can be well understood by the semiclassical three-step model [18], namely an electron is firstly ionized and then accelerated by a laser field, finally driven back towards the parent core to recombine into the ground state and simultaneously emit a single harmonic photon.
In recent years, the below-threshold harmonic (BTH) generation of atoms and molecules has received considerable theoretical and experimental attentions [19–36], this is because the below-threshold harmonics with spatially coherent and high conversion efficiency can be used to obtain the coherent vacuum-ultraviolet (VUV) light sources and more intense attosecond pulses [35–39]. For molecular systems, which have more degrees of freedom and more complex physical mechanism than atoms, some new phenomena have been discovered in molecular below-threshold harmonic (MBTH). For example, Bandrauk et al. [40] found that the strong coupling of charge resonance states results in the low-frequency harmonic-generation (HG) plateau of $\textrm {H}_{2}^{+}$ at large internuclear distances, and the efficiency exceeds that of the atomic HG plateau by about three orders of magnitude. Soifer et al. [41] investigated experimentally and theoretically the near-threshold harmonic generation of aligned molecules and found two distinct contributions related to primary ionization and excitation processes, which provides a deeper understanding of the origin of harmonics near the ionization threshold. Avanaki et al. [42] studied theoretically the above- and below-threshold high-order-harmonic generation of $\textrm {H}_{2}^{+}$ in intense elliptically polarized laser fields. Dong et al. [43] investigated theoretically the polarization properties of below-threshold harmonics from aligned molecules $\textrm {H}_{2}^{+}$ in linearly polarized laser fields. Li et al. [44] studied the ellipticity of near-threshold harmonics from aligned molecules $\textrm {H}_{2}^{+}$ with large internuclear distances numerically and analytically. Li et al. [45,46] theoretically studied the role of nuclear symmetry in below-threshold harmonic generation of molecules and the multichannel-resolved dynamics in resonance-enhanced below-threshold harmonic generation of $\textrm {H}_{2}^{+}$ molecular ions. In our previous work [47], we found spectral and temporal fine subpeak structures in the BTH of the stretched $\textrm {H}_{2}^{+}$ and attributed these fine subpeaks to the interferences in cycles of the multiphoton radiation during the dipole transition between the ground state $1\sigma _{g}$ and the first excited state $1\sigma _{u}$. So far, most of attentions have been paid to the BTH of the simplest one-electron $\textrm {H}_{2}^{+}$, and some theoretical methods were also used to uncover the physical mechanism of the MBTH, but it is still controversial and remains an open question.
Recall that semiconductors were modeled as a two-level system and interacted with constant light, a periodic oscillation of the population inversion was predicted by Rabi in the 1930s [48], so called Rabi flopping. Then atoms were also simplistically modeled as a two-level system, and the conventional Rabi-flopping behaviors were observed [49]. Avanaki et al. [50] showed a comprehensive theoretical and computational study on harmonic generation of Li atoms in one- and two-photon Rabi-flopping regimes. Frasca [51] studied the dynamics of a two-level model in a strong coupling regime through the analysis of the probability amplitudes and confirmed the Rabi flopping also exists for the strong coupling case. The main goal of this paper is to explore the fine subpeak structures in BTH spectra of the stretched $\textrm {H}_{2}$ and $\textrm {N}_{2}$ molecules driven by a linearly polarized laser field and interpret the fine structures in BTH of the stretched $\textrm {H}_{2}$ and $\textrm {N}_{2}$ as Rabi-flopping between two strongly coupled states. We first calculate single-active-electron potentials for $\textrm {H}_{2}$ and $\textrm {N}_{2}$ molecules using the density functional theory (DFT) [52,53], then study the below-threshold harmonic generation of $\textrm {H}_{2}$ and $\textrm {N}_{2}$ molecules at large internuclear distances in intense linearly laser fields by solving the one-electron time-dependent Schrödinger equation (TDSE) in conjunction with the wavelet time-frequency analysis. We identify the spectral and temporal fine subpeak structures in the below-threshold harmonic spectra of the stretched $\textrm {H}_{2}$ and $\textrm {N}_{2}$ molecules come from the Rabi-flopping between the ground state and the first excited state using the simple two-state model.
The rest of this paper is arranged as follows. In Sec. II, We describe how to solve the single-electron TDSE for linear molecules, and describe the basic equations of the two-state model. In Sec. III, we present results and discussions about the spectral and temporal fine subpeak structures in the BTH spectra of the stretched $\textrm {H}_{2}$ and $\textrm {N}_{2}$ molecules. Conclusions are given in Sec. IV.
2. Theoretical methods
The theory part is divided into two subsections. We first present how to calculate the MBTH spectra of $\textrm {H}_{2}$ and $\textrm {N}_{2}$ molecules by solving the one-electron TDSE. We then briefly describe basic equations of the two-state model for calculating the MBTH spectra.
2.1 TDSE method of linear molecules
Under the dipole approximation and the length gauge, the one-electron TDSE of linear molecules in linearly polarized laser fields can be written as (atomic units are used unless stated otherwise):
where $\hat H_{0}=-\frac {1}{2}\nabla ^{2}+V(r,\theta )$ is the unperturbed electronic Hamiltonian, and the molecular potential $V(r,\theta )$ is expanded as where $P_{l}(\cos \theta )$ is the Legendre polynomial, $l_{\textrm {max}}=100$ is used in our present calculations, the partial radial potential can be rewritten as [52,53] where the first two terms is the electrostatic potential and the last term describes the exchange-correlation interaction. The detailed description for constructing the single-active-electron potentials of linear molecules can be found in [52,53]. The time-dependent molecule-field interaction is where $E(t)$ is the electric field given by $E(t)=E_{0}e^{-2 \ln 2(t^{2}/\tau ^{2})}\sin (\omega t+\phi )$. Here, $E_{0}$ is the peak field amplitude, $\omega$ is the laser angular frequency, $\phi$ is the carrier-envelope phase (CEP) chosen as 0, and $\tau$ is the full width at half maximum (FWHM).The time-dependent wave functions are expanded in terms of B splines [54]:
When the time-dependent wave function $\Psi (\vec {r},t)$ is available, ionization probability is given by
where $\psi _{n}(\vec {r})$ is the bound-state wave function obtained by diagonalizing the field-free Hamiltonian matrix. One can also calculate the time-dependent population of bound states $\psi _{n}(\vec {r})$ using $|\langle \psi _{n}(\vec {r})|\Psi (\vec {r},t)\rangle |^{2}$.We can calculate the expectation value of the induced dipole moment both in the length and acceleration forms, which can be respectively written as:
The spectra density of the harmonic radiation can be obtained by using the Fourier transformation of the time-dependent dipole moment in the length form and acceleration form respectively, given by:
orHere, $d(\omega )$ and $a(\omega )$ are the Fourier transformations of the time-dependent dipole moment in the length and acceleration forms, respectively.
The dynamic features of the HHG process can be explored in more detail using the time-frequency analysis on the induced dipole moment with the Morlet wavelet given by
where andIn the present calculations, we choose the window width parameter $\tau =15$.
2.2 Two-state model
A widely used "diabatic basis" are the field-free-states, in this case, considering a closed two-state system where $|\phi _{1}^{D}\rangle$ and $|\phi _{2}^{D}\rangle$ represent these two bound states, we expand the time-dependent wave function as
with $C_{n}(t)=\langle \phi _{n}^{D}|\psi (t)\rangle (n=1,2)$ is the probability amplitude of state $|\phi _{n}^{D}\rangle$ and $C_{1}(t)^{2}+C_{2}(t)^{2}=1$, the basis states $|\phi _{n}^{D}\rangle$ are chosen according to the problem at hand. The evolution of the system is described by the time-dependent Schrödinger equation, where the Hamiltonian matrix $H^{D}$ is given byIn order to diagonalize $H^{D}$, we adopt an important basis set (so called adiabatic basis) [58–60] which means that the states follow the field. The diagonalized Hamiltonian is obtained by using the unitary transformation
Applying $U_{D \rightarrow A}$ to the diabatic basis states, one obtains the field-dressed, "adiabatic" states
In order to calculate the harmonic spectra, one needs to perform the Fourier transform of the time-dependent dipole moment [59,61], and its length form is
and acceleration form is3. Results and discussion
3.1 Spectral and temporal fine subpeak structures in BTH spectra of the stretched $H_{2}$ and $N_{2}$ molecules
In this paper, we assume the molecular axis and the polarization direction of laser fields are directed along the z axis and two nucleuses are located on the z axis (i.e., parallel alignment) at the positions of -a and a, respectively. That means the internuclear distance R=2a. Fig. 1 compares the calculated HHG spectra of $\textrm {H}_{2}$ at R=6 a.u. driven by a Gaussian laser pulse with the FWHM of 23 femtosecond (fs), laser intensity of $3.0\times 10^{13} \textrm {W}/\textrm {cm}^{2}$ and wavelength of 1400 nm. The HHG spectra are obtained by the Fourier transform of the induced dipole moment and acceleration, respectively. In our calculations, 120 radial B splines and 30 angular B splines, a spherical box with radius of 100 a.u., and time step of 0.1 a.u. are used. A $\textrm {cos}^{1/8}$ absorber function is used near the boundary (i.e., $\textrm {r}_{0}=60$ a.u.) to avoid unphysical reflections of the electron wave packet from the boundary [62]. As can be clearly seen that the HHG spectra from the length form agree well with that of the acceleration one which indicates the present calculations are fully convergent. Thereafter, we only present the HHG power spectrum in the length form.
Fig. 1. Comparison of the HHG power spectra of $\textrm {H}_{2}$ at the internuclear distance R=6 a.u. obtained from the length form (black solid line) and the acceleration form (red dashed line). We use a Gaussian pulse with peak intensity of $3.0\times 10^{13} \textrm {W}/\textrm {cm}^{2}$, wavelength of 1400 nm and the FWHM of 23 fs.
In Fig. 2(a), we present the calculated HHG spectra of $\textrm {H}_{2}$ in 1400 nm laser field with the FWHM of 23 fs and the peak intensity of $3.0\times 10^{13} \textrm {W}/\textrm {cm}^{2}$ at the large internuclear distance R=6 a.u.. In Fig. 2(b), we also show the HHG power spectra of $\textrm {N}_{2}$ in 1600 nm laser field with the FWHM of 27 fs and peak intensity of $2.0\times 10^{13} \textrm {W}/\textrm {cm}^{2}$ at the large internuclear distance R=6.8 a.u.. According to the well-known three-step model [18], the cutoff of the harmonic spectra is located at the energy of $I_{p}+3.17U_{p}$, where $U_{p}=E_{0}^{2}/4\omega ^{2}$ is the ponderomotive energy and $I_{p}$ is ionization potential of molecules. Here, the absolute value of orbital energies of 8.51 eV (8.19 eV) calculated by solving the time-independent Schrödinger equation agree well with the vertical ionization potentials of 8.49 eV (8.28 eV) obtained from the Gaussian code for $\textrm {H}_{2}$ ($\textrm {N}_{2}$) respectively. The estimated cutoffs are 29 and 30, which agree well with our TDSE results in Figs. 2(a) and 2(b). We are particularly interested in the fine subpeak structures in the below-threshold harmonic spectra of the stretched $\textrm {H}_{2}$ and $\textrm {N}_{2}$ molecules (see the insets of Figs. 2(a) and 2(b)).
Fig. 2. (a) The HHG power spectra of $\textrm {H}_{2}$ in a 1400 nm laser field with intensity of $3.0\times 10^{13} \textrm {W}/\textrm {cm}^{2}$ and the FWHM of 23 fs at internuclear distance R=6 a.u.; (b) The HHG power spectra of $\textrm {N}_{2}$ in a 1600 nm laser field with intensity of $2.0\times 10^{13} \textrm {W}/\textrm {cm}^{2}$ and the FWHM of 27 fs at internuclear distance R=6.8 a.u..
To demonstrate the spectral and temporal fine subpeak structures in the BTH spectra shown in Fig. 2, we perform the wavelet time-frequency analysis on the induced dipole moment of $\textrm {H}_{2}$ and $\textrm {N}_{2}$ at internuclear distances 6 a.u. and 6.8 a.u., and the wavelet time-frequency spectra are shown in Figs. 3(a) and 3(b), respectively. We find some unusual minima show up in the time-frequency spectra of the stretched $\textrm {H}_{2}$ and $\textrm {N}_{2}$ molecules along the emission of the 1st and 3rd harmonics, which correspond to the fine subpeak structures in below-threshold harmonic spectra of the stretched $\textrm {H}_{2}$ and $\textrm {N}_{2}$ molecules. These minima structures originate from the population inversion between two strongly coupled states (see Figs. 4(b) and 4(d)). In other words, the harmonic emission is suppressed when the population inversion happens. The situation is quite similar to the $\textrm {H}_{2}^{+}$ case [47,61], we confirm that these subpeak structures do exist in the BTH spectra of molecules at the large internuclear distance, and the reason why these subpeak structures are common will be further discussed in this paper.
Fig. 3. Wavelet time-frequency spectra of the dipole moment along the molecular axis (z direction). All the parameters are the same as Fig. 2.
Fig. 4. (a) Function of populations for the ground state $1\sigma _{g}$ (black thick line) and the first excited state $1\sigma _{u}$ (red thin line) with time calculated by the TDSE for $\textrm {H}_{2}$ at R=6 a.u.; (b) function of populations for the field-dressed state $1\sigma _{g}$ (black thick line) and the field-dressed state $1\sigma _{u}$ (red thin line) with time obtained from the two-state model for $\textrm {H}_{2}$ at R=6 a.u.; (c) function of populations for the ground state $3\sigma _{g}$ (black thick line) and the first excited state $3\sigma _{u}$ (red thin line) with time calculated by the TDSE for $\textrm {N}_{2}$ at R=6.8 a.u.; (d) function of populations for the field-dressed state $3\sigma _{g}$ (black thick line) and field-dressed state $3\sigma _{u}$ (red thin line) with time obtained from the two-state model for $\textrm {N}_{2}$ at R=6.8 a.u.. The laser parameters are the same as Fig. 2.
3.2 Uncovering the physical mechanism of the fine subpeak structures in the MBTH spectra with the two-state model
It is well-known that the dominant contribution to the BTH comes from the bound-bound transitions [63], so it is very important to know which bound states are involved. Figures 4(a) and 4(c) show the time-dependent populations of the ground state and the first excited state for $\textrm {H}_{2}$ and $\textrm {N}_{2}$ at internuclear distances R=6 a.u. and R=6.8 a.u., respectively. The time-dependent populations of the other states are almost 0 (not shown). Figures 4(b) and 4(d) present the time-dependent populations of the corresponding field-dressed states, and the population inversion can be more intuitively seen. In Fig. 4, one can see that the ground state and the first excited state of the stretched $\textrm {H}_{2}$ and $\textrm {N}_{2}$ molecules are strongly coupled. The present transition dipole $x_{21}$ is 3.07 (3.46), the Rabi frequency $\Omega _{R}=x_{21}E_{0}$ is 0.090 a.u. (0.083 a.u.), and the corresponding $\delta$ value is 5.5 (5.8) for the stretched $\textrm {H}_{2}$ ($\textrm {N}_{2}$). It indicates the strong coupling criteria [40] $\delta =2\Omega _{R}/\omega \gg 1$ is satisfied indeed. The ground state populates and depopulates during the laser pulse in agreement with the Rabi oscillation in the two-state system, the active electron in the ground state is firstly excited to the field-dressed first excited state by a dipole transition, then it gains energy from the field, and finally it can be recombined back to the field-dressed ground state and simultaneously emit low-energy photons (so called three-step model).
According to the time-dependent population analysis in Fig. 4, one can find that the dominant contributions to the MBTH spectra come from the Rabi-flopping between two strongly coupled molecular states at the large internuclear distance. Therefore, the two-state model can be used to interpret the generation of subpeak structures in the below-threshold harmonic spectra of the stretched $\textrm {H}_{2}$ and $\textrm {N}_{2}$. In Figs. 5(a) and 5(c), we compare the BTH spectra obtained from the TDSE simulations and the two-state model for $\textrm {H}_{2}$ molecule at R=6 a.u. and for $\textrm {N}_{2}$ molecule at R=6.8 a.u., respectively. As can be seen, the MBTH spectra calculated from the TDSE method agree well with those from the simple two-state model. It indicates that these fine subpeak structures in MBTH spectra can be attributed to Rabi-flopping between two strongly coupled molecular states.
Fig. 5. Comparison of the below-threshold harmonic spectra obtained from the TDSE simulations (black solid line) and the two-state model (red dashed line) (a) for $\textrm {H}_{2}$ molecule at R=6 a.u., (c) for $\textrm {N}_{2}$ molecule at R=6.8 a.u.; time-dependent induced dipole moments in the laser field (b) for $\textrm {H}_{2}$ molecule at R=6 a.u., (d) for $\textrm {N}_{2}$ molecule at R=6.8 a.u..
Furthermore, as can be seen in Figs. 5(a) and 5(c), the spacing between the adjacent subpeaks is about $0.2\omega$, which corresponds to $\frac {1}{14}\Omega _{R}$. The origin of these fine oscillations can be further understood by analyzing the induced dipole moment in the time domain. In Figs. 5(b) and 5(d), we can see two deep minima of the time-dependent induced dipole moment $d(t)$ when population inversions take place (see Figs. 4(b) and 4(d)). That means the $d(t)$ is separated into four parts and the second (four) part shifted from the first (third) one by about 5 optical cycle or 14 times the Rabi period. Therefore, the $d(t)$ can be approximately expanded as a sum of these four contributions,
where the second (fourth) contribution can be approximately obtained as the first (third) one shifted by $\frac {28\pi }{\Omega _{R}}$, and By performing the Fourier transform of $d(t)$, one obtains the dipole moment in the frequency domain3.3 Checking the validity of the two-state model for the MBTH
Next, we turn to check carefully the validity of the two-state model at different laser intensities and wavelengths by taking $\textrm {H}_{2}$ at R=6 a.u. as an example. Figure 6 compares the below-threshold harmonic spectra obtained from the TDSE simulations and the two-state model at wavelength of 1400 nm, the FWHM of 23 fs and laser intensities of $7.0\times 10^{13} \textrm {W}/\textrm {cm}^{2}$, $6.0\times 10^{13} \textrm {W}/\textrm {cm}^{2}$, $5.0\times 10^{13} \textrm {W}/\textrm {cm}^{2}$, and $4.0\times 10^{13} \textrm {W}/\textrm {cm}^{2}$, respectively. We can see that the agreement between the TDSE and the two-state model becomes worse as laser intensity increases due to the higher excited states and the continuum state will be involved at higher laser intensities. Figure 7 presents the time-dependent populations of the ground state $1\sigma _{g}$ and the first excited state $1\sigma _{u}$ and the time-dependent ionization probabilities obtained from the TDSE simulations for $\textrm {H}_{2}$ at R=6 a.u.. As can be clearly seen, two lowest bound states (i.e., $1\sigma _{g}$ and $1\sigma _{u}$) are coupled strongly in the rising part of laser field, then the electron is ionized as the electric field further increases (i.e., the final ionization probabilities are 67$\%$, 53$\%$, 40$\%$, and 25$\%$, respectively.), and thus the two-state model fails to work when other higher bound states or the continuum states are populated significantly.
Fig. 6. Comparison of the below-threshold harmonic spectra of $\textrm {H}_{2}$ at R=6 a.u. obtained from the TDSE (black thick line) and the two-state model (red thin line) for laser peak intensities of (a) $7.0\times 10^{13} \textrm {W}/\textrm {cm}^{2}$; (b) $6.0\times 10^{13} \textrm {W}/\textrm {cm}^{2}$; (c) $5.0\times 10^{13} \textrm {W}/\textrm {cm}^{2}$, and (d) $4.0\times 10^{13} \textrm {W}/\textrm {cm}^{2}$. The laser wavelength is 1400 nm and the FWHM is 23 fs in all cases.
Fig. 7. Function of populations for the ground state $1\sigma _{g}$ (black thick line) and the first excited state $1\sigma _{u}$ (red thin line) of $\textrm {H}_{2}$ at R=6 a.u. with time and the time-dependent ionization probabilities are also given. The laser parameters correspond to Fig. 6.
Figure 8 compares the BTH spectra of $\textrm {H}_{2}$ at R=6 a.u. for different laser wavelengths obtained from the TDSE and the two-state model. We use a laser pulse with the laser peak intensity of $3.0\times 10^{13} \textrm {W}/\textrm {cm}^{2}$, and wavelength covering from 1200 nm to 1800 nm. The maximum ionization probability is less than 17$\%$ and thus the two-state model is expected to work well in these cases. Indeed, the BTH spectra calculated by the two-state model agree well with those of TDSE simulations. In summary, the fine subpeak structures in BTH spectra of the stretched $\textrm {H}_{2}$ and $\textrm {N}_{2}$ can survive at different laser intensities and wavelengths, and the two-state model works well if tunneling ionization or excitation to higher excited states can be ignored.
Fig. 8. Comparison of the below-threshold harmonic spectra of $\textrm {H}_{2}$ at R=6 a.u. obtained from the TDSE (black thick line) and the two-state model (red thin line) for wavelengths of (a) 1800 nm; (b) 1600 nm; (c) 1400 nm and (d) 1200 nm. The corresponding FWHM is (a) 30 fs, (b)27 fs, (c) 23 fs and (d) 20 fs, respectively. The laser intensity is $3.0\times 10^{13} \textrm {W}/\textrm {cm}^{2}$ in all cases.
4. Conclusion
In this paper, we theoretically investigate the below-threshold harmonic generation of the stretched $\textrm {H}_{2}$ and $\textrm {N}_{2}$ molecules in intense linearly polarized laser fields by solving the one-electron time-dependent Schrödinger equation. We found some fine subpeak structures in the below-threshold harmonic spectra of the stretched $\textrm {H}_{2}$ and $\textrm {N}_{2}$, which is similar to those subpeaks in BTH spectra of the stretched $\textrm {H}_{2}^{+}$ [47]. Combining with the two-state model and the time-frequency analysis, we can conclude that the generation of fine subpeak structures in the BTH spectra of the stretched $\textrm {H}_{2}$ and $\textrm {N}_{2}$ results from the Rabi-flopping between two strongly coupled molecular states (i.e., the ground state and the first excited state). We believe these fine subpeak structures in BTH spectra are general when two states in atoms or molecules are strongly coupled and the Rabi-flopping takes place. In addition, the two-state model for harmonics is expected a very good approximated model unless ionization or excitation to higher bound states become considerable. We mention that the multi-state model should be used if more than two bound states are involved. The alignment dependence of these fine subpeaks in BTH spectra of molecules should be further studied by solving the three-dimensional TDSE [64,65] or the approximated two-dimensional TDSE [66–68] in the near future. Hopefully, these splitted harmonics have some applications for synthesizing coherently the VUV light sources or studying the wavelength tunable excitations.
Funding
National Natural Science Foundation of China (12164044, 11804405, 11864037, 11765018); Key-Area Research and Development Program of Guangdong Province (2019B030330001); Science and Technology Program of Guangzhou (China) (201904020024); Natural Science Foundation of Gansu Province (20JR5RA420); Key Research and Development Program of Ningxia (2019BEB04024).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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