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Abstract
We have performed a comparison study of the Rydberg-state excitation of model molecules (1πg and 1πu states) in different laser fields by the approaches of time-dependent Schrödinger equation and a fully quantum-mechanical model, and both simulations show good accordance. It is found that the peak structure of the Rydberg-state population vs laser intensity becomes pronounced for longer laser pulses due to the stronger interference effect between the subwave packets released in different optical cycles, and the locations of the intensity-dependent peaks closely satisfy the multi-photon resonant transition condition. In addition, it is demonstrated that the populations of the Rydberg states possessing the identical parity oscillate in an inverse manner with increasing laser intensity for different initial states (1πg and 1πu), and the aforementioned distinct phenomenon is attributed to the additional phase introduced by the symmetry of 1πg state with respect to that of 1πu state.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Ionization frequently takes place when atoms or molecules are exposed to a laser field, and it is the starting point for a multitude of intriguing strong-field physical phenomena, such as above-threshold ionization (ATI), high-order harmonic generation (HHG), and molecular fragmentation [1–7]. For the laser field which is equivalent to the field strength of the Coulomb interaction in ground-state atoms or molecules, the electrons tend to tunnel through the potential barrier of the combined Coulomb and laser fields. After tunneling, the electrons are driven by the time-varying laser field and most of them will eventually escape the Coulomb field of the parent ion. However, a part of the tunneled electrons do not gain sufficient energy from the laser pulse and are subsequently recaptured by the Coulomb field into the high-lying Rydberg states, which is dubbed as recapture or frustrated tunneling ionization (FTI) [8,9].
Investigations on Rydberg states of different samples subjected to intense laser fields have drawn considerable attentions recently, since high-$n$ Rydberg atoms or molecules provide quantum objects of mesoscopic size to probe the transition from the quantum to the classical world [10–13], and the Rydberg-state excitation process can also be employed to accelerate neutral particle, understand the photoelectron spectral characteristics, and reveal the mechanisms of below-threshold harmonics generation [10,14–16]. In addition to He atom [9], excited neutral particles have also been observed for other species exposed to strong laser fields in experiments. The resemblance of excited neutral fragment spectrum and the ionic fragmentation spectrum due to Coulomb explosion are observed for H$_2$ and N$_2$, and the above observation supports the mechanism that an electron is recaptured into highly excited states by one of the participating ions during the Coulomb explosion [17,18]. Wu et al. detected FTI during the dissociative multiple ionization of argon dimer in strong laser fields, and found that the Rydberg electron prefers to localize at the atomic ion with the higher charge state and confirmed the prediction that all ionized electrons only those are trapped which have tunneled near the peak electric field [19]. Recently, a comparison study has shown that the behaviors of ionization and Rydberg-state excitation of N$_2$ are similar to those of its counterpart atom Ar, while Rydberg-state excitation of O$_2$ is more significantly suppressed than the ionization with respect to Xe atom in a high laser intensity region, which partially results from the distinct molecular structure of O$_2$ [20].
In theory, the approach of time-dependent Schrödinger equation (TDSE) is usually used as a benchmark to investigate the formation of neutral atoms and molecules in strong laser pulses [9,20–24]. In Ref. [25], Rydberg-state excitation is ascribed to the continuation of ATI into the below-threshold negative energy region, and it is different from the Freeman-resonance perspective, in which the electron directly populates the Rydberg states via resonant multiphoton excitation [21,22]. The computation is formidable for TDSE simulations due to the large spatial distribution of high-lying Rydberg states, and it is difficult to extract more physical mechanisms from TDSE simulations, so semi-classical simulations are often utilized to uncover underlying dynamics of the generation of Rydberg-state atoms [9,26–29]. In Ref. [30], it is shown that Coulomb force and initial lateral momentum play a vital role in the generation of Rydberg-state atom compared semi-classical calculations with TDSE simulations. Huang et al. found that a fraction of atoms remain unionized at the end of the laser pulse if the electrons are emitted in a certain window of initial field phase and transverse velocity [31]. Recently, a fully quantum model (QM) has been proposed to investigate the Rydberg-state excitation process of atoms in intense laser fields, which well reproduces the TDSE simulations and experimental results [32]. Rydberg-state excitation is explained as a coherent recapture process along with ATI, and the well-developed peak structure in the intensity dependence of the Rydberg-state population is ascribed to the interference of wave packets released in different optical cycles during the laser pulse [32].
As mentioned above, the interference of electron wave packets emitted during different optical cycles has an important impact on the atomic Rydberg-state excitation based on the QM simulations [32], while a comparison study of the role of the laser-pulse width in the Rydberg-state excitation employed TDSE and QM calculations still lacks, and the generalization of QM calculations from atomic Rydberg-state excitation to molecular Rydberg-state excitation is in need. The intensity dependence of the Rydberg-state population exhibits peak structures in QM simulations [32], and the accurate prediction of the sites of the peaks is an intriguing problem. What’s more, the symmetry of molecular orbital has an important impact on molecular ionization [33,34], which is the starting point for molecular Rydberg-state excitation [18,20], and it is natural to expect that the orbital symmetry plays a crucial role in molecular Rydberg-state excitation. In this paper, it is found that the bandwidth of laser pulses and orbital symmetry play an important role in the Rydberg-state excitation of molecules subjected to strong laser fields based on TDSE and QM simulations. Atomic units (a.u.) are utilized unless otherwise indicated.
2. Theoretical methods
2.1 Time-dependent Schrödinger equation (TDSE)
Within the dipole approximation and the velocity gauge, the TDSE of model molecules in the presence of linear polarized laser fields is given by
The time-varying vector potential is written as $\textbf {A}(t)$=$\!E_{0}/\omega \cos ^{2}(\pi t/t_{\mathrm {max}})\sin (\omega t)\varepsilon$ with the peak electric field $E_0$, the frequency of the laser pulse $\omega$, unit vector $\varepsilon$, and -$t_{\mathrm {max}}/2\!<\!t\!<\!t_{\mathrm {max}}/2$, which is linearly polarized parallel to the molecular axis. We adopt $\xi$=$\cos \theta$ and expand the wave function by B-spline as
In the present work, the truncated radius is r$_{\mathrm {max}}$=1500 a.u., 1000 B-splines and 40 B-splines are utilized in the radial direction and in the angular direction, respectively, and the magnetic quantum number is $m$=0. The model molecules are exposed to laser pulses of frequency $\omega \!=\!0.057$ a.u. ($\lambda \!\!=\!\!800$ nm), and the time step is $\Delta t$=0.08 a.u.. Convergence of numerical calculations is reached with the above settings.
2.2 Quantum model (QM)
The Rydberg-state excitation of the molecules subjected to intense laser fields is also examined by a quantum model, and the expression of the transition amplitude is given by [32]
Throughout the paper, we will refer to Eq. (5) as the quantum model (QM). The population of the $n$th Rydberg state is calculated as $P_n=\sum _{l,m}|M_{nlm}|^2$ at laser turnoff, and the magnetic quantum number is $m$=0 in this work, since the laser field is linearly polarized along the molecular axis. If the two trajectories born at subsequent optical cycles to interfere constructively, the difference of the phases must be a multiple of 2$\pi$ [32], which gives rise to the following condition
where $\mu$ denotes the number of photons absorbed by the electron, and $U_p$ ($=E_0^2/4\omega ^2$) indicates the cycle average of the quiver energy. Equation (8) obeys the energy conservation, which can be used to predict locations of the peaks generated by the interference of electron wave packets emitted during different optical cycles, and the peaks are separated by an intensity interval of about 25TW/cm$^2$ for 800 nm laser fields [32]. More details of the quantum model can be found in our previous work [32].In the present work, the initial states are represented as an appropriate linear combination of scaled hydrogen 2$p_x$ atomic orbital as $\phi _{1\pi _u}(\textbf {r},\textbf {R})$=${\bigg [}\psi _{2p_x}(\textbf {r}-\frac {\textbf {R}}{2}) +\psi _{2p_x}(\textbf {r}+\frac {\textbf {R}}{2}){\bigg ]}/\sqrt {2[1+S_{2p_x}(R)]}$ and $\phi _{1\pi _g}(\textbf {r},\textbf {R})$=${\bigg [}\psi _{2p_x}(\textbf {r}-\frac {\textbf {R}}{2}) -\psi _{2p_x}(\textbf {r}+\frac {\textbf {R}}{2}){\bigg ]}/\sqrt {2[1-S_{2p_x}(R)]}$, where $S_{2p_x}(R)$ indicates the atomic orbital overlap integral. The 2$p_x$ orbital is expressed as $\psi _{2p_x}(\textbf {r})$=$\frac {r}{\sqrt {\pi }}\kappa ^{5/2}e^{-\kappa \textbf {r}}\sin \theta \cos \varphi$ with $\kappa$=$\sqrt {2I_p}$, and the ionization energy is 0.5 a.u. in the present work. The Fourier transform of the initial states leads to the interference factors of $\cos ({M_p}\cdot {R}/2)$ for 1$\pi _u$ state and $\sin ({M_p}\cdot {R}/2)$ for 1$\pi _g$ state with ${M_p}={P}+{A}(t')$ in Eq. (5) [39].
3. Results and discussions
Figure 1 depicts the probabilities of Rydberg state as a function of laser intensity calculated by TDSE and QM calculations for laser pulses with different durations, and the initial state is 1$\pi _g$. It is shown that the TDSE calculations are well reproduced by the QM simulations apart from a small shift of the peaks. The intensity dependence of Rydberg-state populations is a nearly straight line for 1$T$ in Fig. 1, and the intensity dependence of peak structures become pronounced with the increasing bandwidth of laser pulses due to the interference of wave packets released during different optical cycles [32]. The above distinct feature is analogous to the resonance-like enhancement of high-order above threshold ionization of atoms exposed to intense laser fields [40,41], which also shows strong dependence on the bandwidth of laser pulses. In the following discussions, the total duration of laser pulses 8$T$ is used in TDSE and QM simulations.
Figure 2 depicts the probabilities of Rydberg states with even and odd parity as a function of laser intensities for the initial states of 1$\pi _g$ and 1$\pi _u$, which are calculated by TDSE (the Rydberg states are in the energy range -0.032 a.u. $<$E$<$0) and QM (3$<n<$21), respectively. In general, the peak structure of populations for Rydberg states with even or odd parity is observed alternately as a function of laser intensity for 1$\pi _g$ and 1$\pi _u$ states obtained by TDSE simulations, which are well reproduced by the QM calculations. The peaks are separated by an intensity interval of around 25TW/cm$^2$ for Rydberg states with different parities, which corresponds to a shift of the ponderomotive energy $\Delta U_p$=$\hbar \omega$=1.5 eV (the photon energy of the laser field), and the above distinct feature is ascribed to the interference of electron wave packets emitted during different optical cycles for QM simulations [32]. Moreover, a separation of $\Delta U_p$=2$\hbar \omega$ can be clearly seen for the peaks of the populations for the Rydberg state possessing the same parity with the increasing laser intensity, which is attributed to the interference of orbits released in adjacent half cycles and captured in opposite directions for QM calculations [32]. However, a closer inspection reveals that the peak structures of populations of Rydberg states with the identical parity appear in inverse sequence for 1$\pi _g$ and 1$\pi _u$ states as a function of laser intensity obtained by TDSE calculations in Figs. 2(a) and 2(b), which is well reproduced by QM simulations in Figs. 2(c) and 2(d) apart from a small shift of the peaks.
Fig. 1. (a) and (b): Populations of Rydberg states as a function of laser intensity obtained by TDSE and QM simulations for 800-nm laser pulses with different durations, respectively, and $T=2\pi /\omega$. The initial state is 1$\pi _g$, the Rydberg-state energies are in the range -0.032 a.u. $<E<$0 for TDSE calculation, and the principal quantum number is 3$<n<$21 for QM simulations.
Fig. 2. (a) and (b): Populations of Rydberg states with even and odd parities as a function of laser intensity for the initial states of 1$\pi _g$ and 1$\pi _u$ obtained by TDSE calculations, respectively. (c) and (d): Same as (a) and (b), but calculated by QM simulations.
In Eq. (5), the electron is firstly pumped by the laser field into a continuum state, then it evolves driven by the external field, and a fraction of electrons are trapped by the parent ion into Rydberg states. In Eq. (5), the Fourier transform of the initial states of molecules with different symmetries gives rise to different interference terms, and the interference terms are $\cos (\textbf {M}\cdot \textbf {R}/2)$ and $\sin (\textbf {M}\cdot \textbf {R}/2)$ for 1$\pi _u$ and 1$\pi _g$ states ($\textbf {M}=\textbf {P}+\textbf {A}(t')$ indicates the mechanical momentum), respectively, in which the undressed modified molecular SFA is adopted [39]. To further understand the difference of intensity dependence of the Rydberg-state population for the initial stats of 1$\pi _g$ and 1$\pi _u$ states, we calculate the probability of the Rydberg state ($n$=6) for the initial states of 1$\pi _g$ and 1$\pi _u$ states by QM simulations adopting different interference terms in Fig. 3, since the population of Rydberg state $n$=6 is much larger with respect to those of other Rydberg states. In Figs. 3(a) and 3(b), the interference factors are $\sin (\textbf {M}\cdot \textbf {R}/2)$ and $\cos (\textbf {M}\cdot \textbf {R}/2)$, respectively. It is found that the population of the Rydberg state with the identical parity appears in alternate sequence as a function of laser intensity which is similar to those in Fig. 2, and the positions of the peaks are in a good accordance with the predictions of Eq. (8) in Fig. 3(a) (see the blue arrows). In Figs. 3(c) and 3(d), the interference factors are $|\sin (\textbf {M}\cdot \textbf {R}/2)|$ and $|\cos (\textbf {M}\cdot \textbf {R}/2)|$, respectively, and it is demonstrated that the peaks of the probability for the Rydberg state possessing the identical parity shows up in the same sequence as a function of laser intensity. In addition, it is shown that the intensity-dependent peak structure of Rydberg-state populations appears in the same order for the same parity in Figs. 3(b)–3(d).
Fig. 3. (a) and (b): Probabilities of the Rydberg state ($n$=6) of even and odd parities versus laser intensity for initial states of 1$\pi_g$ and 1$\pi_u$ calculated via QM with the interference terms $\sin(\textbf{M}\cdot \textbf{R}/2)$ and $\cos(\textbf{M}\cdot \textbf{R}/2)$, respectively. The blue arrows indicate the locations of the peaks predicted by Eq. (8) between $\mu$=11 and $\mu$=14 in (a). (c) and (d): Same as (a) and (b), but for 1$\pi_g$ state with the interference term $|\sin(\textbf{M}\cdot \textbf{R}/2)|$ and 1$\pi_u$ state with the interference term $|\cos(\textbf{M}\cdot \textbf{R}/2)|$, respectively.
In order to shed more physical insights into the results in Figs. 2 and 3, we calculate the contribution to the Rydberg states with different angular quantum numbers of the ionization times in the regime $\nu T<t'<(\nu +1/2)T$ and $(\nu +1/2)T<t'<(\nu +1)T$ (the integer number $\nu$ is between 0 and 7, and $T=2\pi /\omega$), and show the phase difference of the transition amplitude (Eq. 5) obtained in the above-stated two ionization time domains in Fig. 4. It is found that the difference of $\pi$ shows up for the phase difference calculated by the initial state of 1$\pi _g$ with respect to that of 1$\pi _u$ at the same laser intensity in Fig. 4(a) (the principal quantum number and the angular quantum number of the Rydberg state are $n$=6 and $l$=4, respectively), which also holds in Fig. 4(b) for the Rydberg state of $n$=6 and $l$=5. However, if the interference terms of $|\sin (\textbf {M}\cdot \textbf {R}/2)|$ and $|\cos (\textbf {M}\cdot \textbf {R}/2)|$ are used for the initial states of 1$\pi _g$ and 1$\pi _u$ in Fig. 4(c), respectively, the phase difference of the transition amplitudes for the Rydberg state ($n$=6 and $l$=4) shows nearly the same behavior vs laser intensity comparing with those in Fig. 4(a), which also holds for the Rydberg state of $n$=6 and $l$=5 obtained by the initial states of 1$\pi _g$ and 1$\pi _u$ in Fig. 4(d) (the interference factors of $|\sin (\textbf {M}\cdot \textbf {R}/2)|$ and $|\cos (\textbf {M}\cdot \textbf {R}/2)|$ are used for 1$\pi _g$ and 1$\pi _u$ states, respectively) in comparison with those in Fig. 4(b). In Fig. 4, it is demonstrated that a phase difference of $\pi$ is introduced comparing the interference factor $\sin (\textbf {M}\cdot \textbf {R}/2)$ with $\cos (\textbf {M}\cdot \textbf {R}/2)$ for the same laser intensity, while the phase difference of the transition amplitude exhibits analogous behavior as a function of laser intensity for the interference terms of $\cos (\textbf {M}\cdot \textbf {R}/2)$, $|\cos (\textbf {M}\cdot \textbf {R}/2)|$, and $|\sin (\textbf {M}\cdot \textbf {R}/2)|$.
Fig. 4. The phase difference of the transition amplitude (Eq. 5) obtained by the electron ionized in the positive and negative electric fields for Rydberg states possessing different quantum numbers ($n$=6, $l$=4 or 5) as a function of the laser intensities. (a) and (b): The initial states are 1$\pi_g$ and 1$\pi_u$, and the Rydberg states correspond to principal quantum number $n$=6 and angular quantum numbers $l$=4 or 5. (c) and (d): Same as (a) and (b), but the interference terms are $|\sin(\textbf{M}\cdot \textbf{R}/2)|$ and $|\cos(\textbf{M}\cdot \textbf{R}/2)|$ for the initial states of 1$\pi_g$ and 1$\pi_u$, respectively (see the text for more details). The horizontal green dashed lines are shown for visual convenience.
To understand the aforementioned distinct phenomena, we plot the distribution of $\cos (\textbf {M}\cdot \textbf {R}/2)$, $\sin (\textbf {M}\cdot \textbf {R}/2)$, $|\cos (\textbf {M}\cdot \textbf {R}/2)|$, and $|\sin (\textbf {M}\cdot \textbf {R}/2)|$ as a function of the ionization time $t'$ and the recapture time $t$ in Fig. 5. In general, stripes show up periodically as a function of the ionization time $t'$ and the recapture time $t$. However, a closer inspection reveals some significant differences in the distribution of $\sin (\textbf {M}\cdot \textbf {R}/2)$ comparing with those of $\cos (\textbf {M}\cdot \textbf {R}/2)$, $|\cos (\textbf {M}\cdot \textbf {R}/2)|$, and $|\sin (\textbf {M}\cdot \textbf {R}/2)|$. The values of $\cos (\textbf {M}\cdot \textbf {R}/2)$, $|\cos (\textbf {M}\cdot \textbf {R}/2)|$, and $|\sin (\textbf {M}\cdot \textbf {R}/2)|$ are no less than zero. The values of $\sin (\textbf {M}\cdot \textbf {R}/2)$ are mainly greater than zero in the ionization time domain $\nu T<t'<(\nu +1/2)T$ while the results of $\sin (\textbf {M}\cdot \textbf {R}/2)$ are mostly less than zero in the ionization time range $(\nu +1/2)T<t'<(\nu +1)T$ (the integer number $\nu$ is between 0 and 7). Consequently, a $\pi$ difference is introduced for the phase difference of the transition amplitude obtained by the interference factor $\sin (\textbf {M}\cdot \textbf {R}/2)$ comparing with $\cos (\textbf {M}\cdot \textbf {R}/2)$, $|\cos (\textbf {M}\cdot \textbf {R}/2)|$, and $|\sin (\textbf {M}\cdot \textbf {R}/2)|$ for the same laser intensity in Fig. 4, and it gives rise to the feature that the population of Rydberg states with a specific parity is out of step as a function of laser intensity for different initial states of 1$\pi _g (\cos )$ and 1$\pi _u (\sin )$ in Figs. 2 and 3.
Fig. 5. The dependence of interference terms on the ionization time $t'$ and the recapture time $t$ ($\textbf {M}=\textbf {P}+\textbf {A}(t')$). (a)-(d): The interference terms of $\cos (\textbf {M}\cdot \textbf {R}/2)$, $\sin (\textbf {M}\cdot \textbf {R}/2)$, $|\cos (\textbf {M}\cdot \textbf {R}/2)|$, and $|\sin (\textbf {M}\cdot \textbf {R}/2)|$, respectively ($T=2\pi /\omega$, and see the text for more details).
4. Conclusions
We have investigated the Rydberg-state excitation of model molecules exposed to different laser pulses based on velocity-gauge TDSE, which possess the similar ionization energy and different orbital symmetries (1$\pi _g$ and 1$\pi _u$ states), and the TDSE calculations are well reproduced by quantum model simulations. It is found that the intensity-dependent peak structure of Rydberg-state probability becomes evident with increasing laser periods, which originates from the stronger interference effect by the electronic wave packets emitted during different optical cycles of the longer laser pulse. The intensity dependence of Rydberg-state population clearly exhibits peak structures, and the sites of the peaks are in a good accordance with the prediction of the energy conservation. Furthermore, it is found that the probability of Rydberg states with the same parity is out of phase as a function of laser intensity for initial states of 1$\pi _g$ and 1$\pi _u$, which is well reproduced by the quantum model calculation. The above intriguing characteristic is ascribed to the two-center interference effect introduced by the symmetry of 1$\pi _g$ state, which gives rise to an additional phase $\pi$ in comparison with the symmetry of 1$\pi _u$ state for the same laser intensity.
Funding
National Key Research and Development Program of China (2016YFA0401100, 2019YFA0307700); National Natural Science Foundation of China (11425414, 11804405); Fundamental Research Funds for the Central Universities (18lgpy77).
Disclosures
The authors declare no conflicts of interest.
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