Influence of multiphoton resonance excitation on the above-threshold ionization of a hydrogen atom

Haiying Yuan; Haiying Yuan; Yujun Yang; Yujun Yang; Fuming Guo; Jun Wang; Zhiwen Cui; Zhiwen Cui

doi:10.1364/OE.461582

1. Introduction

With the rapid development of laser technology, the intensity of laser filed can be obtained to be close to or even higher than the intensity of the Coulomb field that electrons can feel in an atom [1]. When atoms and molecules are irradiated by ultrashort intense laser pulses, there are many nonlinear phenomena, such as above-threshold ionization (ATI), non-sequential double ionization and high-order harmonic generation (HHG) [2–11] etc. ATI is a nonlinear photoionization process in which the number of photons absorbed by the bound electron from the laser field exceeds the minimum number of photons required for its ionization threshold [12–18]. ATI can be used to measure the carrier envelope phase (CEP) of incident laser pulses [19,20] and probe the internal structure of atoms and molecules [21–23]. In order to better understand the characteristics of the photo-electron emission spectrum, people have conducted in-depth research on its generation process [24–27].

Due to the high driving laser intensity, the laser electric field can be regarded as a classical field. With the action of this electric field, the potential function of the atom is distorted to form a potential barrier, and the bound electrons have the opportunity under go the tunneling ionization. Since effective ionization can occur in each half cycle of the multiple peak distributions with energy interval of one photon frequency can be observed on the photo-electron emission spectrum as these ionized electrons interfere. The formation of these photo-electron peaks can also be explained by multiphoton transitions, that is, the electrons absorb the energy of multiple photons and jump to the continuous state for to be ionized. Under this mechanism, due to the quantum characteristics of photons, the interval between adjacent peaks in the formed photo-electron emission spectrum is also one photon energy.

Driven the action of a strong laser electric field, the energy level of the atom will move, especially the energy level near the continuum state will change greatly due to the smaller influence of the Coulomb potential. The electrons in the ground state of the system are strongly attracted by the nucleus, and the energy level is less affected by the external field. Therefore, under the action of a strong laser, the peak energy position of the observed distribution of the ionized electron is no longer multiple photons energy minus the ionization energy, but can be given by $E_{k} = n\omega -I_p-U_p$($I_P$ is the ionization energy of an atom and $U_P$ is the ponderomotive energy of the laser electric filed. In $U_{P}=E_{0}^{2} / 4 \omega ^{2}$, $E_{0}$ and $\omega$ are the peak amplitude and frequency of the laser electric filed) [28].

In the photo-electron emission spectrum, many small photo-electron peaks can be observed besides the ATI peaks generated by the interference between the ionized electrons produced by the laser pulse at different cycles. There are many causes for the generation of these small photo-electron peaks. For example, within a certain light intensity range, the peak position of the photo-electron emission spectrum from hydrogen atom irradiated by the high-frequency laser pulse has shifted and produced multiple interference peaks [29,30] as the increase of laser intensity. This phenomenon can be attributed to the Dynamic Stark effect and the interference between the wave-packets of ionized electron on the rising and falling edges of the laser pulse [31–33]. By studying the interaction of the single-photon resonant laser pulse with the hydrogen atoms, Müller $et$ $al.$ found that the photo-electron emission spectrum of hydrogen atoms multiphoton ionization showed clear multi-peak structures [34]. Tumakov $et$ $al.$ discovered that in the single-photon resonance region of Li atom, the Rabi oscillation between the ground 2s state and the excited 2p state exhibits interference structures with minima in the photo-electron emission spectrum [35]. In our previous work, it was found that when hydrogen atom was irradiated by the amplitude modulated sinusoidally phase-modulated pulse, a distinct subpeak structure was produced in the photo-electron emission spectrum and this subpeak structure was originated from the interference between the ionized electrons of different sub-pulses [36].

In addition, since the driving laser pulse is usually tens of femtoseconds, rather than a monochromatic continuous pulse. The pulse envelope effect can not only produce the main photo-electron emission peak in the photo-electron emission spectrum, but also other subpeaks with equal spacing. More generally, under the irradiation of a laser pulse with a specific light intensity, the atom can absorb several photons and then resonantly excite to Rydberg states. Because of the ponderomotive shift of the Rydberg state close to the ionization energy, the atom can be ionized by absorbing one or more additional photons. This resonant multiphoton ionization process is called Freeman resonance [37,38]. These peaks can reflect the bound state structure of the system [39–45]. In normal conditions, since the electrons are observed to transition to higher bound states, the energy movements of these bound states are close to or the same as the continuous state with the action of the laser field, the energy position of the electrons ionized from these states hardly varies with the driving light intensity [46,47]. Nevertheless, the nonresonant ATI peak shifts toward lower energy with the increase of the driving laser intensity [48]. Shao $et$ $al.$ have isolated the photo-electron emission spectra generated by the two ATI mechanisms using an intensity-resolved photo-electron emission spectrum [49]. In previous studies, 800 nm infrared laser was usually used. Recently, with the advancement of high-order harmonic generation and free electron laser technology, strong coherent light sources in the ultraviolet can be generated. In this paper, we used this kind of light source to irradiate the atom and found that the electrons in the atom have the opportunity to jump to a lower energy level via a few photons, and then are ionized from the excited state. Whether the energy of the excited state changes during this process and whether this change can be reflected in the photo-electron emission spectrum is the core of this paper.

2. Theoretical methods

In order to study the photo-electron emission process of the atom exposed by laser electric field, it is necessary to numerically solve the time-dependent Schrödinger equation (TDSE). In the coordinate space, the ionized electrons will quickly move away from the parent ion under the action of the laser electric field. The calculation of the photo-electron emission spectrum needs to collect all the ionized electrons, which requires a very large computational space. Looking at this process from the momentum space, the maximum energy that the electrons can obtain with the action of the laser electric field is limited, so the photo-electron emission spectrum can be accurately calculated with a small calculation space. For this reason, this paper simulates the photoionization process based on the time-dependent pseudo-spectrum scheme in momentum space [50–53]. In the dipole approximation and velocity gauge, the behavior of the atom in a strong laser field can be described by the following TDSE (Atomic units are used throughout this paper unless explicitly stated):

(1)$$i \frac{\partial}{\partial t} \psi({\mathbf{r}}, t)=\left[ \frac{\hat {\mathbf{p}}^{2}}{2}+ \frac{1}{c} \mathbf{A}(t) \cdot \hat {\mathbf{p}}+V( \mathbf{r})\right] \psi(\mathbf{r}, t)$$

where $V(\mathbf {r})$ refers to the Coulomb potential that the electron feels from the nucleus, ${\textbf {p}}$ is the dynamic momentum, $\mathbf {A}(t)$ represents the vector potential of the laser field, and $c$ denotes the speed of light in vacuum. Through the Fourier transform of the coordinate space wave function $\psi (\mathbf {r}, t)$, the time-dependent wave function of the momentum space can be obtained as follows:

(2)$$\Phi(\mathbf{p}, t)=\frac{1}{(2 \pi)^{3 / 2}} \int \psi(\mathbf{r}, t) \exp ({-}i \mathbf{p} \cdot r) d \mathbf{r}.$$

By applying Eq. (2) to Eq. (1), the TDSE in momentum space can be obtained:

(3)$$i \frac{\partial}{\partial t} \Phi(\mathbf{p}, t)=\left[\frac{\textbf{p}^{2}}{2}+\frac{1}{c} \mathbf{A}(t) \cdot \textbf{p}\right] \Phi(\mathbf{p}, t)+\int V\left(\mathbf{p}, \mathbf{p}^{\prime}\right) \Phi\left(\mathbf{p}^{\prime}, t\right) d \mathbf{p}^{\prime},$$

in which $V\left (\mathbf {p}, \mathbf {p}^{\prime }\right )$ (the momentum space Coulomb potential) can be defined as

(4)$$V\left(\mathbf{p}, \mathbf{p}^{\prime}\right)=\frac{1}{(2 \pi)^{3}} \int V(\mathbf{r}) \exp [i(\mathbf{p}^{\prime}-\mathbf{p}) \cdot \mathbf{r}] d \mathbf{r},$$

The hydrogen atom Coulomb potential is

(5)$$V\left(\textbf{p}, \mathbf{p}^{\prime}\right)={-}\frac{1}{2 \pi^{2}} \frac{1}{\left|\mathbf{p}-\mathbf{p}^{\prime}\right|^{2}}.$$

Assuming that the incident direction of the linearly polarized laser electric field is $Z$ axis, the system is symmetric about $Z$ axis during the evolution process, and the momentum space wave function $\Phi (\mathbf {p}, t)$ can be expanded by partial waves:

(6)$$\Phi({\textrm{p}}, t)=\frac{1}{\textrm{p}} \sum_{l=0}^{l_{\max }} \Phi_{l}(\mathbf{p}, t) Y_{l 0}(\theta, \varphi),$$

where $l_{\max }$ is the maximum of partial waves, $\Phi _{l}(\textrm {p}, t)$ refers to the radial wave function, $Y_{l m}(\theta, \varphi )$ represents a spherical harmonic function. For the Coulomb potential of the hydrogen atom in Eq. (5) , it can be changed by using the partial wave expansion into [54–56].

(7)$$V\left(\mathbf{p}, \mathbf{p}^{\prime}\right)=\frac{1}{\textrm{p} \textrm{p}^{\prime}} \sum_{l=0}^{l_{\max }} \sum_{m={-}1}^{l} V_{l}\left(\textrm{p}, \textrm{p}^{\prime}\right) \times Y_{l m}(\theta, \varphi) Y^{*}{ }_{l m}\left(\theta^{\prime}, \varphi^{\prime}\right),$$

where ${\textrm {p}}=|{\mathbf {p}}|$ and

(8)$$V_{l}\left(\textrm{p}, \textrm{p}^{\prime}\right)={-}\frac{1}{\pi} Q_{l}\left(\frac{\textrm{p}^{2}+\textrm{p}^{\prime 2}}{2 \textrm{p} \textrm{p}^{\prime}}\right),$$

in which $Q_{l}$ is the Legendre functions of the second kind. By substituting Eq. (6) and Eq. (7) into Eq. (3), the equation of the momentum space radial wave function $\Phi _{l}(\textrm {p}, t)$ can be obtained as

(9)$$\begin{aligned} i \frac{\partial}{\partial t} \Phi_{l}(\textrm {p}, t)=\frac{\textrm {p}^{2}}{2} \Phi_{l}(\mathbf{p}, t)+\int V_{l}\left(\textrm{p}, \textrm{p}^{\prime}\right) \Phi_{l}\left(\textrm {p}^{\prime}, t\right) d \textrm{p}^{\prime}+ \\ \frac{1}{c} \textrm{A}(t) \textrm{p}\left[\alpha_{l+1} \Phi_{l+1}(\textrm{p}, t)+\alpha_{l} \Phi_{l-1}(\mathbf{p}, t)\right]. \end{aligned}$$

where $\alpha _{l}=1 / \sqrt {(2 l-1)(2 l+1)}$.

Since the Legendre function of the second kind has a logarithmic singularity at $\textrm {p}=\textrm {p}^{\prime }$, it will be difficult to solve the TDSE in momentum space. The Legendre method can be adopted so as to remove this singularity [54].

The time-dependent pseudo-spectral scheme is used to solve Eq. (3) in momentum space to obtain the time-dependent wave function of the system at any time. At the end of the laser pulse, the system wave function will be projected onto the continuous scattering state. If $b_{l}(\varepsilon, t)$ is the normalized the population of the continuous state amplitude of the energy of different partial waves, the corresponding single differential scattering cross section is

(10)$$\frac{{d{P_\varepsilon }}}{{d\varepsilon }} = {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \sum_l {{{\left| {{b_l}(\varepsilon ,t)} \right|}^{2}}}.$$

The population of the bound state is calculated by projecting the bound state on the wave function of the system at the end of the laser:

(11)$$p_{n, l}=\left|\left\langle\psi_{n, l}(\mathbf{p}) \mid \Phi(\mathbf{p}, t)\right\rangle\right|^{2}.$$

3. Results and discussions

We used a linearly polarized pulse, and the laser polarization direction is along the $Z$ axis. The vector potential form of the laser electric field is $A(t)=\frac {cE_{0}}{\omega } f(t) \sin (\omega t+\phi )$ and the electric field form is $E(t)=-\frac {\partial A(t)}{c\partial t}$, in which $E_{0}$ is the peak amplitude of the electric field , $\omega =0.1436$ is the center frequency of the incident laser pulse, $f(t)=\sin ^{2}\left (\frac {\omega t}{2 \tau _{R}}\right )$ is the pulse envelope function, and $\tau _{R}$ is the laser pulse duration. As for $\tau _{R}$, we choose the incident laser pulse duration to be 30 optical cycles (o.c.). $\phi =0$ is the CEP of the incident laser pulse. Figure 1 shows the photo-electron emission spectrum of the hydrogen atom under the driving laser with the peak intensity of $I=1.62 \times 10^{14} \mathrm {~W} / \mathrm {cm}^{2}$. The peak position of the photo-electron emission spectrum is usually given by the energy position formula $E_{k}=n \omega -I_{p}-U_{p}$. It can be clearly seen from Fig. 1 that there are subpeaks next to the main peak except for the main peak (in the blue dashed box) of the photo-electron emission spectrum. The source of these subpeaks can be attributed to the pulse width effect of the laser pulse. In general, the intensity of these subpeaks should be smaller than the main peak. It is noted that the subpeak intensity of the peak of the photo-electron emission spectrum near 0 energy is beyond the intensity of the main peak, like the subpeak (in the red dashed box area) in Fig. 1. In order to understand this feature of the photo-electron emission spectrum, we systematically studied the photo-electron emission spectra in different peak intensities.

Figure 2(a) shows the variation of photo-electron emission spectrum with the peak intensity of the driving laser. In order to clearly see the change of the photo-electron emission spectrum, the low-energy part of the photo-electron emission spectrum (the white solid line box in Fig. 2(a)) is enlarged, as shown in Fig. 2(b). As the peak intensity of the driving laser increases, the peak position of the photo-electron emission spectrum moves to lower energy, and the peak energy position can be provided by $E=n \omega -I_{p}-U_{p}$. According to this formula, the increase of peak intensity means the increase of $U_P$. Therefore, the corresponding ATI peak energy gradually decreases. As shown from Fig. 2, the sideband with a smaller intensity can also be seen besides the main peak of the photo-electron emission spectrum. In addition, Fig. 2(b) shows interference structure when the ionized electron energy is around $0.04$ a.u.- $0.06$ a.u.. As the increase of the peak intensity, the maximum and minimum of the structure is moving toward with high energy in the photo-electron emission spectrum (for example, the dotted white line marked in Fig. 2(b) is enhanced, and the dotted black line is weakened). It can be seen from the photo-electron emission spectrum of the changing peak intensity that the appearance of interference is a universal law, and its mechanism needs to be analyzed in depth. The bound electrons in the atom are ionized under the action of the laser electric field, which not only satisfies the conservation of energy, but also the conservation of angular momentum. In order to explain the producing mechanism of the photo-electron emission spectrum, it is necessary to analyze it from the perspective of conservation of angular momentum. As shown in Fig. 3(a), the intensity of each partial wave emitted by the photo-electron in Fig. 1 changes with the energy of the electron. With this peak intensity, the increase in the subpeaks intensity of the emission spectrum is mainly reflected in the lower energy photo-electron emission. For this purpose, the energy range selected in this figure is 0 a.u.-0.1 a.u.. It can be seen from the figure that the partial waves including $l=0$, $l=2$ and $l=4$ account for the main contribution for the reason that the electrons need to absorb four photons so as to ionize under a laser pulse with the frequency of 0.1436 a.u.. According to the conservation of angular momentum, the ionized electrons need to absorb even-numbered photons. The weights of the partial waves near the energy of the increased subpeaks intensity are given in Fig. 3(b). It shows that the maximum proportion of the total partial waves occupied by $l=2$ is 0.7674, followed by $l=0$, the proportion of partial waves in the total is 0.1337, and the proportion of partial waves when $l=4$ is the smallest, that is 0.0199. This transition may be caused by the ground state electron absorbs three photons with the same spin and a photon of opposite spin, or first resonant transitions to an excited state and then ionizes. For purpose of analyzing the contribution of the excited state, the population of the excited state of the atom at the end of the laser is analyzed.

Fig. 1. Photo-electron emission spectrum of the hydrogen atom at the laser peak intensity with $I=1.62 \times 10^{14} \mathrm {~W} / \mathrm {cm}^{2}$.

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Fig. 2. Dependence of the photo-electron emission spectrum on the peak intensity of the driving laser for the hydrogen atom.

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Fig. 3. (a) Partial wave photo-electron spectrum of the hydrogen atom under the driving laser with peak intensity of $I=1.62 \times 10^{14} \mathrm {~W} / \mathrm {cm}^{2}$ and (b) partial wave probability (normalized) with the energy of 0.0493 a.u..

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Corresponding to the laser pulse intensity range in Fig. 2, the populations of the 2s, 2p, 3s, 3p and 4f states of the atom change with the peak intensity of the driving laser at the end of the laser, as shown in Fig. 4(a) (the calculations show that higher excited states will not have significant populations, so it is not given in the figure). From Fig. 4(a), the populations of 2s, 3s, 3p and 4f states are almost zero compared with the population of 2p state of the atom. When the peak intensity is small, the population of 2p state is also small. When the peak intensity reaches $I=0.72 \times 10^{14} \mathrm {~W} / \mathrm {cm}^{2}$, its population probability rapidly increases with the quick growth of the peak intensity, and then it exhibits an oscillating behavior as the peak intensity increases. It indicates that the excited state that contributes to ionization may mainly come from the 2p state of the atom. On this basis, the population of the total ionization and ground state of the atom are calculated as the laser intensity changes, as shown in Fig. 4(b). The total ionization rate of the atom (blue dash dot dot line) seems to increase in Fig. 4(b). The increase shows periodic peak-to-valley oscillation characteristics, and the ionization rate and the population of the 2p state are synchronized to oscillate. Both of them reach the maximum and minimum at the same time with the increase of laser intensity. They are opposite to the ground state oscillation behavior, which means the ionization rate and population of 2p state rise rapidly when the population of the ground state (black dash dot line) drops quickly. At the same time, it can be noticed that the laser electric field intensity corresponding to these oscillation peaks is the intensity of the increased subpeak intensity observed in Fig. 2. The 2p oscillation can be attributed to the few-photon resonance. Due to the strong coupling between 1s state and 2p state, under the action of the laser field, electrons have a greater probability of jumping to the 2p state, and then ionization occurs. The population of the 2p state is greatly affected by the parameters of the driving laser, showing obvious oscillation characteristics. Therefore, the resonant peak in the photo-electron emission spectrum may be attributed to the increase in the population of the 2p state resulting from the resonant transition from the ground state to the 2p state, which is then generated by the interference between the ionized electrons from the 2p state and the ionized electrons from the ground state. To further test this inference, the photo-electron emission spectrum with the same laser parameters is calculated by using strong-field approximation (SFA).

Fig. 4. (a) Variation of the populations of the hydrogen atom in the 2s, 2p, 3s, 3p and 4f states with the peak intensity of the driving laser. (b) Change of the population of the total ionization and ground state of the hydrogen atom with the peak intensity of the driving laser.

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Figure 5 exhibits the change of the photo-electron emission spectrum with the peak intensity of the driving laser by using the SFA scheme. In our calculation, the selected laser parameters are the same as those in Fig. 2, and the calculation only includes the ionization of the ground state of the atom. The photo-electron emission spectrum also shows a main ATI peak and multiple subpeaks in Fig. 5.The appearance of these sub-peaks is due to the interference between the ionizations of 1s electrons on the rising and falling edges of the driving pulse. As the peak intensity of the driving laser of increases, the energy position of the peak of the photo-electron spectrum shifts to lower energy. The predicted position can be given by $E=n \omega -I_{p}-U_{p}$. It should be pointed out that the enhancement of the subpeaks intensity of the photo-electron emission spectrum cannot be observed in Fig. 5, which indicates that the previously observed subpeaks intensity enhancement of the photo-electron emission spectrum is related to the contribution of the excited state.

Fig. 5. Change of the photo-electron emission spectrum with the peak intensity of the driving laser from SFA.

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We further investigated the photo-electron emission spectrum ionized from the excited state separately. The initial state of the atom is selected as the 2p state, and its photo-electron emission spectrum is studied with the same laser parameters. Figure 6(a) shows the variation of the photo-electron emission spectrum with the peak intensity of the driving laser under this initial state. From Fig. 6(a), unlike the photo-electron emission in the ground state of the system, as the intensity of the driving laser increases, the intensity of the photo-electron emission spectrum no longer moves to low energy, but slightly to high energy. In Fig. 6(a), with the increase of the driving laser intensity, the changing process of energy position of the enhanced subpeaks emitted by photo-electron is given in Fig. 2 (as shown by the white dotted line). It is found that the movement of this resonance enhancement peak is equal to the peak energy of the photo-electron emission spectrum whose initial state is 2p. It is noted that, as two independent channels as shown in Fig. 5 and Fig. 6(a), as the light intensity increases, the energy and intensities of ATI peaks changes continuously, and the subpeaks of ATI cannot be observed. For the TDSE calculation that includes both ionization channels, the interference characteristics of alternating strengthening and weakening can be observed (for example, the dotted black line marked in Fig. 2(b) is weakened, and the dotted white line is enhanced).

Fig. 6. (a) Variation of the photo-electron emission spectrum of the atom whose initial state is the 2p state with the peak intensity of the driving laser. (b) Variation of the photo-electron emission spectrum of the atom whose initial state is the superposition of the 1s state and the 2p state with the peak intensity of the driving laser.

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In order to more clearly analyze the causes of the subpeaks structure in the photo-electron emission spectrum, we studied the variation of the photo-electron emission spectrum generated by the atom whose initial state is the superposition of the 1s state and the 2p state (the weight of the 1s state is 0.95 and the weight of the 2p state is 0.05) with the peak intensity of the driving laser, as shown in Fig. 6(b). It can be seen that at the lower intensity, the peaks of the photo-electron emission spectrum of the ionization of the 1s state and 2p state can be observed independently. As the driving laser intensity increases, the ATI peaks ionized from the 2p state shift to higher energy, consistent with the behavior of a single 2p initial state, and the ATI peaks ionized from the 1s state shift to lower energy. When the driving laser intensity exceeds $I=0.72 \times 10^{14} \mathrm {~W} / \mathrm {cm}^{2}$, the energy coverage of electrons originating from the ionization of the 1s state and the 2p state is consistent, and a clear interference structure can be observed in Fig. 6(b). Therefore, it can be observed that the subpeaks features in the photo-electron emission spectrum is attributed to the resonant transition from the ground state to the 2p state, which increased the population of 2p state, and then produced by the interference of the ionized electrons between the 2p state and the ground state. From the view of energy, it can also be attributed to the resonance of the 2p state as an intermediate state.

For the purpose of further verifying the generation mechanism of the photo-electron emission spectrum, we analyzed the evolution of the populations of the ground state and the 2p state, as shown in Fig. 7. When the intensity of the driving laser is $I=1.62 \times 10^{14} \mathrm {~W} / \mathrm {cm}^{2}$, the population of the ground state of the atom decreases rapidly, and the population is about 0.7 at the end of the laser. It is worth noting that as time passes, the population of the initial state shows envelope oscillation behavior instead of decreasing monotonically. Around the envelope, multiple oscillating structures in each cycle can be clearly seen. For the time-dependent population of the 2p state, it exhibits a complementary behavior with the ground state. The complementarity is not only reflected in the overall envelope, but also in the multiple oscillatory structures in the period, as shown in the inset in Fig. 7. Here, in order to clearly see the complementary features, 0.8 is added to the population of the 2p state in Fig. 7. According to the characteristics of complementary, it can be reflected that under the action of the driving laser, there is a strong coupling between the 1s state and 2p state. Therefore, the electrons have the opportunity to jump to the 2p state through resonance, and then ionize, resulting in the observed photo-electron emission spectrum.

Fig. 7. Evolution of the populations of the ground state and the 2p state of the hydrogen atom with time at the laser peak intensity with $I=1.62 \times 10^{14} \mathrm {~W} / \mathrm {cm}^{2}$.

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4. Conclusion

Using the time-dependent pseudo-spectral scheme in momentum space, we study the intensity influence of driving UV laser pulses on the spectra of photo-electron of the hydrogen atom. In the photo-electron emission spectrum, there are some subpeaks with higher intensity. The enhanced energy position of the subpeaks shifts to higher energy as the peak intensity of driving laser pulse increases. Combining the SFA calculation, ionization of the excited state, superposition and time-dependent population of bound states, it is found that the appearance of the subpeak originates from the interference of the electrons ionized from the ground state and the electrons ionized from the 2p state after the resonant transition from the ground state to the 2p state. From the perspective of energy, it can also be attributed to the Freeman resonance of the 2p state as an intermediate state. The study of this phenomenon is conductive to deepen the understanding of the change of lower excited state energy under the action of strong field.

Funding

National Key Research and Development Program of China (No.2019YFA0307700); National Natural Science Foundation of China (No. 11627807, No. 11774129, No. 12074145).

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 maybe obtained from the authors upon reasonable request.

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Influence of multiphoton resonance excitation on the above-threshold ionization of a hydrogen atom

Abstract

1. Introduction

2. Theoretical methods

3. Results and discussions

4. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (7)

Equations (11)

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