1Institute of Atomic and Molecular Physics, Jilin University, Changchun 130012, China
2Institute of Applied Physics and Computational Mathematics, Beijing 100088, China
3Research Center for Advanced Optics and Photoelectronics, Department of Physics, College of Science, Shantou University, Shantou 515063, China
4Key Laboratory of Intelligent Manufacturing Technology of MOE, Shantou University, Shantou 515063, China
5Max-Born-Institut, 12489 Berlin, Germany
6Hefei National Laboratory, Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China
7Shenzhen Key Laboratory of Ultraintense Laser and Advanced Material Technology, Center for Advanced Material Diagnostic Technology, and College of Engineering Physics, Shenzhen Technology University, Shenzhen 518118, China
Conservation of parity plays a fundamental role in our understanding of various quantum processes. However, it is difficult to observe in atomic and molecular processes induced by a strong laser field due to their multiphoton character and the large number of states involved. Here we report an effect of parity in strong-field Rydberg-state excitation (RSE) by comparing the RSE probabilities of the molecule and its companion atom Ar, which has a similar ionization potential but opposite parity of its ground state. Experimentally, we observe an oscillatory structure as a function of intensity with a period of about in the ratio between the RSE yields of the two species, which can be reproduced by simulations using the time-dependent Schrödinger equation (TDSE). We analyze a quantum-mechanical model, which allows for interference of electrons captured in different spatial regions of the Rydberg-state wave function. In the intensity-dependent RSE yield, it results in peaks with alternating heights with a spacing of and at the same intensity for both species. However, due to the opposite parities of their ground states, pronounced RSE peaks in Ar correspond to less pronounced peaks in and vice versa, which leads to the period of in their ratio. Our work reveals a novel parity-related interference effect in the coherent-capture picture of the RSE process in intense laser fields.
【AIGC One Sentence Reading】:Observed parity effects in Rydberg-state excitation of N2 and Ar under intense laser fields, with oscillatory structure in yield ratio.
【AIGC Short Abstract】:Our study unveils a parity effect in Rydberg-state excitation under intense laser fields, observed through contrasting N2 and Ar. The RSE yield ratio oscillates with intensity, showing a 50 TW/cm2 period due to opposite ground-state parities. Quantum simulations support findings, revealing alternating peaks in RSE yields caused by parity-related interference.
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1. INTRODUCTION
Parity is a fundamental concept in quantum physics [1]. In the electromagnetic interaction, conservation of parity serves as a powerful fundamental principle governing the selection rules of quantum transitions in atomic and molecular systems if emission or absorption of a specific number of photons is involved. The effect of parity conservation is easy to observe for one-photon or few-photon transitions between two or very few states having definite parities. However, it is difficult to track for multiphoton transitions with many states involved, for example, for atomic ionization in strong laser fields. When atoms or molecules are subjected to intense laser fields, the electron in the ground state can be pumped to continuum states via absorption of a large number of photons [dubbed above-threshold ionization (ATI)] [2]. To understand this process and the subsequent dynamics of the photoelectron in the laser field as observed, e.g., in high-order ATI [3] and high-harmonic generation [4], quantum features such as discrete excited states and their parities are usually ignored and a semiclassical picture based on tunneling ionization is adopted [5,6], which has achieved great success and constitutes the foundation of attosecond physics [7]. Nevertheless, the fact that only odd harmonics are observed in the harmonic spectrum [4] or the carpet structure in the photoelectron spectrum in the direction perpendicular to the laser polarization [8] is the reminder that parity plays a dominant role.
As an important complement to tunneling-induced physical processes [9,10], neutral Rydberg-state excitation (RSE) of an atom or molecule in a strong laser field has attracted intense experimental and theoretical research during the last decade [11–23]; for a recent review, see Ref. [24]. Strong-field RSE can be applied to the acceleration of neutral particles [25] and the generation of coherent EUV light through free-induction decay [26]. Thus, it may lead to new applications in both fundamental and applied physics. A large number of theoretical studies, including calculations using time-dependent Schrödinger equation (TDSE) and semiclassical simulations, have been carried out on atomic RSE in strong laser fields. Nonetheless, its underlying physical mechanisms are still under debate [11,13–16,19–21,27,28].
Recently, a fully quantum-mechanical (QM) model based on -matrix theory has been proposed, which well reproduces the peaks in the intensity dependence of RSE as obtained via the TDSE [29]. The quantum model describes strong-field RSE as a coherent recapture process accompanying above-threshold ionization [29]. According to this picture, the oscillating peaks in the intensity dependence of the RSE yield are attributed to the interference of electronic wave packets liberated in different optical half cycles of the driving field and captured by the Rydberg-state maxima on opposite sides of the ion. It is at this point that the parity of the Rydberg state enters the selection rules. The results of this model are well consistent with experimental measurements [29–31]. In addition, for intensity-dependent RSE in mid-infrared laser fields an extension of this model that includes the interaction between the tunneled photoelectron and the ionic Coulomb potential well reproduces a new oscillatory structure, which can be related to the interference of the trajectories that generate the various peaks in the low-energy structure in the photoelectron spectrum [20].
Still, our understanding of the quantum picture of strong-field RSE is far from complete. Since the initial and final states of RSE are discrete bound states, it can be expected that the parity or symmetry of these states will affect the RSE process. To the best of our knowledge, this has not been addressed yet. In fact, both the TDSE and the quantum-model calculations show peaks with alternating heights in the intensity dependence of RSE [27,29]. However, this is difficult to observe in an experiment considering the unavoidable fluctuations in the laser intensity and the data acquisition. Whether and how parity plays a role in the oscillations of the intensity-dependent RSE probability has remained unknown. In this paper, we present a joint experimental and theoretical study of strong-field RSE of the molecule and its companion atom Ar, which has an ionization potential similar to the molecule but opposite parity of the ground state. We observe and compare the modulation of the intensity-dependent population of RSE in strong laser fields for these two targets. An oscillating peak structure with a period of about in the intensity dependence of the ratio is observed, which is well reproduced by both the TDSE and QM simulations. Our analysis reveals the novel parity-related interference in the coherent recapture picture of the strong-field RSE process. A schematic picture of the above RSE process is shown in Fig. 1. Atomic units are used throughout this paper unless specified otherwise.
Figure 1.Rydberg-state excitation process of the molecule and the Ar atom in intense laser fields. The valence electron escapes into the continuum through tunneling and propagates in the laser field. The electrons are captured into a certain Rydberg state coherently and interference occurs, which gives rise to peak structures in the intensity dependence of the RSE. In addition, the interference of the contributions of trajectories captured at positive and negative extrema of the Rydberg wave functions causes the peaks to alternate in heights, as indicated on the right of the figure, such that strong peaks in correspond to weak peaks in Ar and vice versa. This is due to the different parities of the ground states of and Ar (see text for more details).
In our experiments, neutral Rydberg atoms produced by the strong laser field were measured by the delayed-static-field-ionization method using a time-of-flight (ToF) mass spectrometer operated under a pulsed-electric-field mode [12,16]. Briefly, the molecular or Ar atomic beam was introduced into the reaction chamber through a pulsed valve with a stagnation pressure of 1 atm. A Ti:sapphire chirped-pulse amplified laser with a central wavelength of 800 nm and a pulse duration of 50 fs was focused by a 25 cm plane-convex lens to interact with the atoms or molecules. The direct-ionized ions ( or ) were pushed away from the detector by a pulsed electric field, and then after a delay time of 1.0 μs the remaining high-lying neutral Rydberg yields ( or ) were field-ionized by another pulsed electric field and detected by dual micro-channel plates at the end of the ToF tube of about 50 cm. The measured Rydberg states were estimated to be in the range of , according to the saddle-point model of static-field ionization [] [12,16,32]. A half-wave plate and a Glan prism were used to control the laser pulse energy. The peak intensity of the focused laser pulse was calibrated by comparing the measured saturation intensity of Ar with that calculated by the Ammosov–Delone–Krainov (ADK) model [33]. The pulse energy was simultaneously recorded, which was monitored by a fast photodiode. Both the ion signal and the pulse energy were analyzed shot-to-shot to ensure that the fluctuation of the laser intensity of each data point can be precisely controlled. In the present experiment, each data point was an averaged result of laser shots with an intensity uncertainty of .
B. Time-Dependent Schrödinger Equation
The interaction between the Ar atom or the molecule and a strong laser field can be modeled by numerically performing the three-dimensional TDSE simulations in the length gauge [12,34]: For the Ar atom, the model potential is given by with the parameters , , , , , and [35]. For the molecule, the effective potential is written as [36] where with the internuclear separation (see Fig. 2), and the parameters and are 24.25 and 1.311, respectively.
The time-dependent wave functions are expanded in terms of B-splines in the following form [34]: The order of the B-splines is and [37]. The laser field is assumed to be linearly polarized along the axis, and the magnetic quantum number is . The vector potential is given by with the unit vector , , where , , and indicate the electric field amplitude, the duration, and the central frequency of the laser pulse, respectively. The time-dependent electric field is , and the time-dependent wave functions are evolved by the Crank–Nicolson scheme [34,37]. The Rydberg-state probability is obtained by projecting the wave function onto the corresponding field-free bound states after the laser pulses are switched off, and the bound states are calculated by diagonalization of the field-free Hamiltonian matrix.
For , the internuclear separation is a.u. The initial states of and are assumed for the Ar atom and the molecule, respectively. In this work, 1200 radial and 20 angular B-splines are used, and a truncation radius a.u. A 10-cycle laser pulse with a.u. is adopted, and the time step is a.u.
C. Quantum Model
The Rydberg-state population is calculated by using our recently proposed quantum model [29]. The population of the th Rydberg state at the end of the laser pulse is written as , and its excitation amplitude has the form [29] where and indicate the ionization and capture time, respectively. The field-free initial state is represented by the wave function with the ionization energy. The Volkov states with asymptotic momentum are and the field-dressed Rydberg state is approximated by where is a field-free Rydberg state of the hydrogen atom with the principal quantum number , angular quantum number , magnetic quantum number ( in this work), and the energy . The Coulomb potential is . The 10-cycle time-dependent electric field and its vector potential have the forms and with the unit vector and the laser frequency a.u. To evaluate Eq. (5), the integration over is calculated by the saddle-point approach, and the integrations over the capture time and the ionization time are carried out numerically. More details of the above quantum model have been described in Ref. [29].
In the present work, the initial wave function of the diatomic molecule is based on LCAO theory within the Hartree–Fock–Roothaan approach [38]: with the Slater-type atomic orbitals where the vector indicates the internuclear separation, and , , and are the principal quantum number, orbital quantum number, and magnetic quantum number, respectively. The parameter is assumed for gerade () orbital symmetry. For the molecule, the highest occupied molecular orbital () is modeled by the orbitals with and [38], and the coefficients of and for each orbital can be found in Ref. [38]. For Ar (), the wave function is with and . The ionization energy for both Ar () and () is 0.57 a.u.
3. RESULTS AND DISCUSSION
In Fig. 3(a) we present the measured RSE yields of the molecule () and its companion Ar atom () as functions of the intensity of the strong irradiating 800 nm laser field. It is crucial that we are able to control the laser intensity precisely by simultaneously monitoring the pulse energy. The yields of are generally comparable to those of in the intensity range considered, in agreement with the results of our previous study [12]. We observe a modulation in the intensity-dependent RSE for and Ar. Two steps can be clearly observed at particular laser intensities as indicated by the arrows in Fig. 3(a). For example, the Rydberg yields are not very sensitive to the laser intensity around , but increase fast at about . Such a modulation structure has been investigated in atomic RSE in some recent studies [30,39] but not been observed so far in molecular RSE. The modulations of the intensity-dependent Rydberg yields of the molecule and the Ar atom are qualitatively reproduced by TDSE simulations without focal averaging, as shown in Fig. 3(b). We find that the oscillations in the intensity dependence of RSE are very similar in Ar and . Both exhibit peaks at the same intensities, separated by an interval of , which corresponds to for 800 nm laser fields ( denotes the ponderomotive energy of the laser field). The peaks alternate in height: a rather high peak is followed by a rather low peak. However, closer inspection of the TDSE results shows that the patterns for Ar and are out of phase, i.e., a high peak in Ar corresponds to a low peak in and vice versa. This out-of-phase structure is also visible in the experimental data [see the results at and in Fig. 3(a)] though only the rather high peaks can be distinguished in the low-intensity region.
Figure 3.(a) The measured yields of RSE for Ar and at 800 nm as a function of laser intensity. (b) The calculated yields as a function of laser intensity using the TDSE without focal averaging. (c) The measured (black open squares) and calculated (red filled squares) yield ratios of at 800 nm as functions of the laser intensity. In the TDSE simulation, focal averaging is included.
To further elucidate the differences between the intensity-dependent RSE of and Ar, we show in Fig. 3(c) the measured yield ratio as a function of laser intensity and, for comparison, the TDSE simulation with focal averaging included. Despite a small shift of the peaks, both the experimental measurements and the TDSE simulations show a clear oscillation of the ratio as a function of laser intensity with a period of about , twice as long as the periods for and . This intriguing oscillation of the intensity-dependent ratio is a consequence of the peak pattern of the two species being out-of-phase as mentioned above. It is worthwhile to mention that for the angular distribution of the tunneling electron always peaks in the field direction, irrespective of the alignment angle, which is very similar to atoms [40]. So, according to the recapture mechanism, the RSE probability will be analogous to the Ar atom even for different alignment angles although the ionization rate decreases with increasing alignment angle [41]. In our experiment, the molecules are randomly aligned. On the other hand, in the TDSE calculations, the molecular axis is assumed to be parallel to the polarization direction of the laser field (i.e., parallel alignment) to simplify the calculation. The qualitative agreement between the TDSE and the experimental results [see Fig. 3(c)] indicates that the main result will not be affected by the effect of alignment.
In order to trace the out-of-phase oscillating peak structures to the angular momentum of the Rydberg states and the opposite parities of the ground states of Ar and , we perform additional calculations and analysis based on the recently developed QM model [29]. Since the final Rydberg states only depend on the quantum numbers , , and and not on the target species (Ar or ), the matrix elements for Ar and only differ by the wave functions of the initial states [see Eq. (5) for details], which have opposite parities.
In Fig. 4(a), we show the calculated probabilities of RSE for the Rydberg states with as a function of laser intensity for and . In this calculation, no focal averaging is considered. The QM calculations reproduce the main features of the peak structure as obtained by the TDSE simulations, i.e., peaks separated by an interval of around at the same laser intensities for and with alternating heights. To show more clearly the differences between the intensity-dependent RSE processes of the molecule and the atom, the ratio of versus laser intensity is plotted in Fig. 4(b). Now, the peaks are separated by an intensity interval of about , which is consistent with the results of the TDSE simulations and the experimental data of Fig. 3(c).
Figure 4.(a) The excitation probability of the Rydberg states versus laser intensity calculated via the QM model for Ar and exposed to an 800 nm laser field. (b) The corresponding ratio of as a function of the laser intensity.
To shed more light on the above findings, we choose the Rydberg state of Ar and for the further analysis, since the probabilities decrease fast with increasing in the region . As shown in Fig. 5(a), the intensity-dependent excitation probabilities of and () present the same peak structure as the total RSE probabilities of [see Fig. 4(a)]. We then present in Figs. 5(b) and 5(c) the probabilities of the Rydberg states with different angular momenta () as functions of the laser intensity for Ar and , respectively (the Rydberg states with make the dominant contributions). If the initial state and the Rydberg states have specific parities, two consecutive peaks are separated by approximately , which can be readily understood from the selection rule in the multiphoton transition picture of RSE. Since the energies of the Rydberg states are close to the threshold, their Stark shifts are all approximately equal to the Stark shift of the continuum. Multiphoton resonances between Rydberg states and the initial state occur when the intensities differ by . Furthermore, the selection rule only permits even-order (odd-order) multiphoton transitions between states of the same (opposite) parity, resulting in a separation of [29]. As shown in Figs. 5(b) and 5(c), states with even and odd are occupied, but the capture probabilities for and 20 are much larger than for and 19, both for the Ar atom and the molecule in the intensity regime considered. For each angular momentum, the excitation probabilities for Ar and look much the same, except that they are shifted versus each other by the intensity corresponding to , i.e., by about . For a peak at the same intensity and for fixed , either Ar or has an excitation probability that is higher by almost an order of magnitude. For example, Rydberg states with odd are populated for the Ar atom at with comparatively low probability [Fig. 5(b)], while, for the molecule at the same intensity, the Rydberg states with even are populated with a much higher probability [Fig. 5(c)]. This is because the initial states of the Ar atom and the molecule are and states, which possess parities of and , respectively. As the laser intensity increases, the high and low peaks alternate for both the Ar atom and the molecule, but the respective patterns are shifted versus each other by . Thus, it is the different parities of the initial states of Ar and that cause the out-of-phase oscillating peak structures and the resulting oscillation in the ratio versus laser intensity as exhibited in Figs. 3(c) and 4(b).
Figure 5.(a) Intensity-dependent excitation probability of the Rydberg states obtained by QM calculations for Ar and . (b) and (c) Probability of the Rydberg states with fixed angular momenta versus laser intensity calculated by QM simulations for Ar and , respectively. (d) Radial wave functions of the Rydberg states with , , and ; (e) integrals of the wave functions depicted in panel (d) over the radius ; (f) integrals of the RSE amplitude [Eq. (5)] over the radius for the four states shown in panel (d).
To explain the parity dependence of the RSE yield for fixed angular momentum shown in Figs. 5(b) and 5(c), in Fig. 5(d) we plot for the radial wave functions for these angular momenta and in Fig. 5(e) their integrals over the radius . The radial wave functions oscillate with increasing and the number of extrema (both positive and negative) is odd for even and even for odd . Hence the integrals of the radial wave functions with odd angular momenta are small due to cancelation between positive and negative contributions. In contrast, the integrals with even are much larger. For the RSE amplitude Eq. (5), the integral over the radius also shows a similar dependence on , that is, the amplitude for even angular momentum integrated over is large, while for odd angular momentum it is much smaller, as shown in Fig. 5(f). According to the coherent-capture picture of the RSE, the total excitation amplitude is the coherent sum of the amplitudes of the photoelectron captured in different regions of the Rydberg state (see Fig. 1). Therefore, the capture events at the peaks depicted in Fig. 5(d) interfere with each other, and due to the positive and negative values of the wave function, the total RSE yield is relatively small for the states with even number of extrema and relatively large for the states with odd number of extrema, as schematically illustrated in Fig. 1. This explains the parity-dependent interference effect in the coherent capture of the tunneled electron, leading to the parity dependence of the RSE probability for given angular momentum of the Rydberg state. In practice, the Ar atom is the companion atom of the molecule, possessing a similar ionization potential but opposite initial-state parity. Therefore, for resonant intensities, Ar and can be pumped to states with even (odd) and odd (even) angular momenta, respectively, and the parities of the states reverse with increasing laser intensities. According to the above analysis, the RSE yield of a state with even angular momentum is much higher than for a state with odd angular momentum due to interference between electrons captured in different spatial regions. Therefore, their intensity-dependent peak structures are out of phase, with high peaks in one case corresponding to less pronounced peaks in the other. The ratio between these two out-of-phase structures amplifies the difference between high and low peaks, which leads to the pronounced oscillatory pattern.
4. SUMMARY
In summary, we investigate the effect of the parity of the initial state in the strong-field Rydberg-state excitation process by comparing the RSE yields of Ar and as a function of laser intensity. An oscillatory structure with a period of is observed in the yield ratio of . This can be attributed to oscillations with a period of of the peaks in the intensity dependence of the RSE yields, which are out of phase for Ar versus . The experimental result is well reproduced by both the TDSE and QM simulations. The analysis based on the QM model shows that the interference between electrons captured in different regions of the wave function of the Rydberg state leads to a dependence of the RSE probability on the parity of the angular momentum of the final Rydberg state. As a result, when different parities of the initial atomic and molecular states are considered, the RSE yields for Ar and for as functions of intensity exhibit a series of peaks located at the same intensities but with alternating heights, such that strong peaks for Ar correspond to weak peaks for and vice versa. Our work reveals a distinct parity effect in the RSE process in intense laser fields. Due to the multiphoton character and the large number of states involved, such parity effects are usually difficult to explore.
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