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Abstract
We theoretically investigated frustrated tunneling ionization (FTI) in the interaction of atoms with elliptically polarized laser pulses by a semiclassical ensemble model. Our results show that the yield of frustrated tunneling ionization events exhibits an anomalous behavior which maximizes at the nonzero ellipticity. By tracing back the initial tunneling coordinates, we show that this anomalous behavior is due to the fact that the initial transverse velocity at tunneling of the FTI events is nonzero in the linear laser pulses and it moves across zero as the ellipticity increases. The FTI yield maximizes at the ellipticity when the initial transverse momentum for being trapped is zero. Moreover, the angular momentum distribution of the FTI events and its ellipticity dependence are also explored. The anomalous behavior revealed in our work is very similar to the previously observed ellipticity dependence of the near- and below-threshold harmonics, and thus our work may uncover the mechanism of the below-threshold harmonics which is still a controversial issue.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Tunneling ionization is a fundamental process in the intense laser-atom/molecule interactions. When the electric field strength of the laser pulse becomes comparable to the binding Coulomb field inside the atom, a potential barrier is formed through which an electron can tunnel out of the atom [1,2]. The tunneled electron is then driven by the laser field and could return back to the parent ion [3], initiating various interesting processes, such as high-order above-threshold ionization [4,5], high-order harmonic generation [6–11], enhanced multiple ionization [12,13], etc. These processes have drawn considerable attention during the past decades and their applications have been widely explored. For example, the recollision of the tunneled electron could be used for attosecond pulse generation [14,15], molecular orbital tomography [16], laser-induced electron diffraction [17] and time-resolved strong-field photoelectron holography [18–23].
When atoms are exposed to the intense laser fields, it has also been observed that a large population of neutral atoms survive in Rydberg states [24–26]. The underlying mechanism for creation of these Rydberg states has attracted much interest. Two mechanisms for forming the Rydberg state have been proposed. In the multiphoton ionization region, the Rydberg states are created via AC Stark-shifted multiphoton resonant excitation during the laser pulse [24,25,27–29]. Recent experiments with coincident measurement of the correlated ion and electron provided solid evidence of this multiphoton resonance excitation for the Rydberg atoms by the UV light [30]. In the tunneling ionization regime, it was demonstrated that the Rydberg states are formed through the frustrated tunneling ionization (FTI) process [26]. In this process, the tunneled electron does not gain enough drift energy from the laser field thus it is finally recaptured by the Coulomb field of the parents atomic [26,31] or molecular ion [32–38]. Initially, it was proposed that rescattering is responsible for the recapture of the tunneled electron [26]. Subsequent studies showed that rescattering is not necessary for FTI and the electron could directly launch into the elliptical orbit and finally stays in the highly excited state after the end of the laser pulses [31,39]. This FTI model has very successfully reproduced the n-distribution (energy) of the Rydberg states [26,40,41] and has initiated a variety of studies in laser-atom interactions, such as strong-field acceleration of neutral atoms [42–44] and the survival of Rydberg states [45] and so on [46,47]. With the semiclassical model, the details of the FTI process have been revealed [26,39,48]. This FTI scenario is supported by experiments on the ellipticity dependence of the excited neutral atom yield [26,39].
There is an another comparable phenomena in the interaction of a strong laser field with atoms and molecules, known as below-threshold harmonic generation [49]. Different from the high-order harmonics wherein the underlying dynamics has been well understood, the mechanism for the below-threshold harmonics is still under debate. During the past decade, it has drawn increasing theoretical as well as experimental attention, partly due to its potential application for the coherent extreme-ultraviolet light sources [50,51]. Recently, the correspondence between the below-threshold harmonic generation and FTI has been investigated with a semiclassical model [52]. It is shown that these two processes share the same series of trajectories. By solving the time-dependent Schrödinger equation, it is shown that FTI and below-threshold harmonics have a similar carrier-envelope phase dependence on the driving pulses [52], indicating that FTI should be responsible for the below-threshold harmonic generation. Two decades ago, an intriguing phenomena in below- and near-threshold harmonics was reported [53–56]. It was found that the yield of those harmonics exhibited an anomalous dependence on the ellipticity of the driving field, i.e., the yield maximized at a nonzero ellipticity. For FTI, this anomalous behavior should also exist if it shares the same process as the below-threshold harmonic generation. However, previous studies have shown that the Rydberg atom yield decreases monotonously with the ellipticity [26,39].
In this work, we study FTI in the interaction of the atom with the elliptically polarized laser pulses. We observed an anomalous ellipticity dependence of the Rydberg atom yield, which maximizes at a nonzero ellipticity. By tracing back the initial tunneling coordinates, we show that this anomalous behavior is due to the fact that the initial transverse velocity at tunneling of the FTI events is nonzero in the linear laser pulses and it moves across zero as the ellipticity increases. The FTI yield maximizes at the ellipticity when the initial transverse momentum for being trapped is zero. The presence of this anomalous behavior depends on the laser wavelength, intensity and pulse duration. This explains why this anomalous behavior has not been observed in previous studies. Moreover, the angular momentum distribution of the FTI events and its ellipticity dependence are also explored. The anomalous behavior revealed in our work is very similar to the previously observed ellipticity dependence of the near- and below-threshold harmonics, thus our work may uncover the mechanism of the below-threshold harmonics.
2. Theoretical model
In this work, we employ the classical-trajectory Monte Carlo (CTMC) ensemble model [26,48] to study the FTI driven by elliptically polarized laser pulses. This CTMC ensemble model has been well established and has been widely used in studying FTI [26,39,48,57]. In this model, FTI process includes two steps: tunneling ionization and classical evolution of the tunneled electron in a combination of the laser field and Coulomb potential of the ion.
In the first step, i.e., tunneling ionization, the electron is launched at the tunnel exit with a zero initial longitudinal momentum and a Gaussian-like initial transverse momentum distribution. The weight of the trajectory was determined by the tunneling theory [58–60] (atomic units are used unless stated otherwise)
In our calculations, we consider FTI of Ar($I_p=0.59$ a.u.). The tunneling exit is approximately estimated by $\textbf {r}_0=-I_p/\textbf {F}(t_0)$. The ensemble is obtained by sampling four millions of classical trajectories over the tunneling ionization time and initial transverse momentum. In our calculations, we only consider the events where tunnel ionization occurs around one electric field peak of the laser pulse, $-T/4<t_0<T/4$, where T is the optical cycle of the laser field. For the long pulses, the electric fields of the laser are similar from cycle to cycle and thus the electron dynamics in different cycles should be similar. So, the results demonstrated in this paper will not change when more laser cycles for tunneling ionization are considered. We examine the energy of the electron when the laser pulse is turned off. The electrons with the final energy $E_f =v^2/2-1/r>0$ stand for the ionized ones. Those with $E_f<0$ correspond to the tunneled electrons recaptured by the ion, which are the FTI events that we focus on in this study. We estimate the main quantum number (n) of the bound electron with the Rydberg formula $E_f=-0.5/n^2$ [26]. The angular momentum is calculated as $\textbf {L}_f =\textbf {r}_f\times \textbf {P}_f$. In this study, we focus on the ellipticity dependence of yield and the angular momentum distribution of the FTI events.
3. Ellipticity dependence of the Rydberg-atom yields
Figure 1 displays the yields of the excited atoms as a function of the ellipticity of the laser pulses. Here the pulse contains 30 optical cycles (i.e., N=30). We performed the calculations at different laser wavelengths and intensities. For the 1600 nm field at the intensity of $8\times 10^{13}$ W/cm$^2$ (curve c), the yield decreases rapidly when the ellipticity increases. This result is consistent with previous theoretical and experimental results [26,39]. This ellipticity dependence is a solid evidence that FTI is responsible for the excited atoms in strong laser fields. For the wavelength of 800 nm pulse with the intensity of $1.6\times 10^{14}$ W/cm$^2$ (curve b), the yield almost does not change as the ellipticity varies from 0 to 0.2 and then decrease sharply as the ellipticity further increases. At the lower laser intensity, as displayed by the blue circles (curve a), the yield first increases and then decreases with the ellipticity. The yield curve shows a surprising peak at the ellipticity of 0.2.
Fig. 1. The yields of excited atoms as a function of the ellipticity of the laser pulses. The wavelengths and the laser intensities are specified in the legend.
In order to understand the underlying dynamics for this anomalous ellipticity dependence, we take the advantage of the CTMC model and trace back to the initial coordinates of the FTI events. In Fig. 2, we show the distributions of the tunneling ionization time and initial transverse momentum of the FTI events at different ellipticities, correspond to the anomalous curve a in Fig. 1. It can be seen that the area of the Rydberg-state atoms is crescent-shaped. The ionization time and the initial transverse momentum for these recaptured electrons are correlated. For the linear laser field ($\xi$=0), the electrons emitted around the crest electric field should possess a proper nonzero initial transverse momentum to be recaptured, and these with nearly zero initial transverse momentum are emitted before the peak of the electric field, as shown in Fig. 2(a). Here we divided the Rydberg area into three regions, AI, AII and B. Regions AI and AII correspond to the nearly vertical distribution in the initial coordinates and Region B corresponds to the nearly horizontal area. As the ellipticity increases, it is shown that the shape of the Rydberg area keeps almost unchanged while the location shifts to the right side gradually as the ellipticity changes from 0 to 0.2. Thus, the probability of the distribution in region AII and region B decrease because the required initial transverse momentum increases, while the probability in region AI increases due to the reduced initial transverse momentum requirement. The competition of the increasing probability in region AII and the decreasing probability in regions AII and B results in the different ellipticity-dependence of the three FTI yield curves in Fig. 1. As the ellipticity further increases, region AI passes through the $p_\perp =0$ plane, and thus the probability decreases in all of the regions, resulting in the rapid decreasing of the excited atom yield for ellipticity larger than 0.2.
Fig. 2. Probability distributions of the FTI events on the coordinates of the tunneling ionization time $t_0$ and initial transverse momentum $P_\perp$ at different ellipticities (from 0 to 0.4) and same laser parameters. The laser intensity (I=$8\times 10^{13}$W/cm$^2$) and wavelength (800nm) are same with the curve a of Fig. 1. The colorbar on the right represents the recapture probability. The increasing pulse ellipticity leads to the crescent-shaped Rydberg area shifting to the right gradually. The black crosses indicate the coordinate of ($t_0$=0, $P_\perp$=0). In Fig. 2(a), the dashed red line divides the Rydberg area into three regions, vertical regions AI, AII and horizontal region B.
To explore the ellipticity dependence of yield for different laser parameters shown in Fig. 1, we display the distributions of the ionization time and the initial transverse momentum at different laser wavelengths and intensities in Fig. 3. The laser parameters in Figs. 3(a)–3(c) are the same as these for the three curves in Fig. 1, respectively. As explained in Fig. 2, the location of the Rydberg area shifts toward the right side when the ellipticity increases. Thus, the yield from region AI increases while the yields in region B and AII decrease. In Fig. 3(a), regions AI and AII have the main contribution to the total Rydberg atom yield. As the ellipticity increases, region AI moves toward $p_\perp$=0 and regions AII and B move away from $p_\perp$=0, and thus the capture probability in region AI increases and those in regions AII and B decrease. The increase of the capture probability in AI is stronger than the decrease in regions AII and B, resulting in the increase of the total FTI yield. As the ellipticity further increases, the yield in region AI decreases and thus the total yield decreases, resulting in an anomalous peak in the yield curve.
Fig. 3. Probability distributions of the FTI events in the initial tunneling coordinates for the linear laser pulses($\xi$=0). The laser wavelengths and intensities are (a) 800 nm and $8\times 10^{13}$ W/cm$^2$, (b) 800 nm and $1.6\times 10^{14}$ W/cm$^2$, (c) 1600 nm and 8$\times 10^{13}$ W/cm$^2$. The colorbar on the right represents the recapture probability. The dashed black lines are the level curves of the probability of FTI events. The dashed red line divides the Rydberg area into three regions, vertical regions AI, AII and horizontal region B.
As the laser intensity increases, as shown in Fig. 3(b), the location of the region B is more close to $t_0$=0 and thus the weight of region B increases. So, region AI and region B, AII have the similar contribution to the total yield. As the ellipticity increases, the yield increases in region AI and decreases in regions B, AII. The competition of these three regions leads to total yield almost unchanged with the ellipticity between 0$-$0.2. At the longer wavelength, as shown in Fig. 3(c), the location of region B is very close to the peak of the electric field, and thus it has the dominant contribution to the total Rydberg atom yield. The yield in region B decreases monotonously with the ellipticity and thus the total yield exhibits the sharp decreasing behavior, as shown by the squares (curve c) in Fig. 1.
We mention that the behavior of the Rydberg atom yield on ellipticity in strong field ionization does not only depend on the laser intensity and wavelength [61], but also on the pulses duration [57]. We repeated our calculation at a short pulse duration with N=10 and N=4, as shown in Fig. 4. For the shorter pulse duration, the area of region B increases and thus the contribution to the total yield increase, as shown in Fig. 4(a) and Fig. 4(c). Consequently, the yield decreases with the laser ellipticity, as shown in Fig. 4(b) and Fig. 4(d). This is in consistent with previous theoretical and experimental results [26,39].
Fig. 4. (a, c) The distribution of FTI events in the initial tunneling coordinates and (b, d) the yield of FTI events for laser fields with shorter pulse duration. The laser intensity and wavelength are $8\times 10^{13}$W/cm$^2$ and 800 nm. The pulse duration in (a) (b) is N=10 and in (c) (d) is N=4.
The calculations above show that the anomalous behavior is more obvious at the nonadiabatic tunneling region. In nonadiabatic tunneling, it has been shown that the most likely transverse momentum in tunneling by the elliptical laser pulse is nonzero [62]. However, this most probable nonzero initial transverse momentum is much less than the corresponding initial transverse momentum of region AI. So, as ellipticity increases, region AI will move towards the most probable initial transverse momentum and the probability increases. Therefore, the yield of the Rydberg atoms increases and the anomalous behavior will persist when the nonadiabatic effect is considered in our calculations. Of course, the exact ellipticity for the location of the peak will change slightly due to the nonadiabatic effect.
We notice that in the below- and near-threshold harmonic generation, a similar anomalous ellipticity dependence of the yield has been experimentally observed 20 years ago [53–56]. The mechanism of the below- and near-threshold harmonic generation is still a hot and controversial topic. Recently, it was suggested that there is a close correspondence between the below-threshold harmonics and FTI [52]. Our results revealed another surprising resemblance on the ellipticity dependence between FTI and near- and below-threshold harmonics. According to the analysis above, the anomalous ellipticity dependence results from the nonzero initial transverse momentum of the responsible trajectories in the linear laser pulses, which moves towards zero momentum as the ellipticity increases. Thus, our results indicate that the emission of the below- and near-threshold harmonics in the linear laser pulse might be also related to the trajectories of the nonzero initial transverse momentum at tunneling. In previous studies, the dynamics for the near- and below-threshold harmonics have been analyzed with the one-dimensional classical trajectories model [63]. Our results indicate that the initial transverse momentum should be considered and thus the one-dimensional analysis should be inadequate. Because of the nonzero initial transverse momentum, the electron trajectory should be an ellipse even in the linear driving pulses (as will be detailedly shown in the subsection below). It implies that the near- and below-threshold harmonics should possess nonzero ellipticity even in the linear driving pulses, which could be easily verified in further experiments. Our results on FTI may help us to recover the mechanism of below-and near-threshold harmonics.
4. Angular momentum distributions
Most of the previous studies on FTI are focused on the yields and the energy distribution of Rydberg atoms. The angular momentum of the excited atoms is also a very important quantity of Rydberg atoms. The angular momentum of Rydberg states is of great importance for both fundamental and applied physics. For example, the low angular-momentum Rydberg states were used for optical quantum information manipulations [64,65] and long-lived circular Rydberg levels with high angular momentum are tools in the exploration of cavity quantum electrodynamic effects [66,67]. Here, we study the angular momentum distribution of the Rydberg atoms in FTI with the semiclassical model.
In Fig. 5, we show the angular momentum number distribution of Rydberg-state atoms at different ellipticities. Here,the final angular momentum number is calculated as $L =[\textbf {L}_f] =[\textbf {r}_f\times \textbf {P}_f$], where [x] denotes the nearest integer of x. The laser wavelength is 800 nm and the intensity is $8\times 10^{13}$ W/cm$^2$. It is shown that the angular momentum number distribution exhibits a symmetric triple-hump structure for this linear laser field ($\xi$=0). As the ellipticity increases, the right and the middle peak are suppressed and the distribution exhibits a single peak in the left side.
Fig. 5. The angular momentum number distribution of Rydberg-state atoms at different ellipticities. The laser intensity (I=$8\times 10^{13}$W/cm$^2$) and wavelength (800nm) are same with the curve a of Fig. 1. It exhibits a symmetric triple-hump structure for the linear laser field ($\xi$=0).
In order to understand the dependence of the angular momentum distribution on the ellipticity, we trace back to the initial coordinates of the FTI events. Fig. 6(a) shows the angular momentum number distribution in the coordinates of initial transverse momentum and tunneling time at $\xi$=0. Note that here we show the absolute value of L. It is shown that angular quantum number distributed exhibits stripes nearly parallel to the coordinate of tunneling time. It means that the final angular momentum is sensitive to the initial transverse velocity but is almost independent on the tunneling time. Thus, it can be expected that there was a correlation between the final angular momentum and the initial transverse momentum. In Fig. 6(b), we also show the principal quantum number (n) distribution in the initial tunneling coordinates. This distribution exhibits the crescent shape with the smaller principle quantum numbers located at the inner edge, the larger n at outer edge of the crescent. The reason for this crescent shape has been explained in a recent paper [57]. The distributions for the angular momentum and the energy in the initial tunneling coordinates are totally different and this difference results in the different ellipticity dependence. As shown in Fig. 2, the location of the distribution moves towards right with the shape unchanged as the ellipticity increases. Therefore, the weight of Rydberg state atomic in the left side of Fig. 6(a) increases and this in the center and right side decreases. Consequently, the center and right peak in Fig. 5 are suppressed as ellipticity increases. On the contrary, the energy distribution keeps almost unchanged with the laser ellipticity (not shown here) but can be controlled via the intensities, wavelengths and duration of the laser field [28,57,68–70].
Fig. 6. (a) The distribution of angular momentum number L in the coordinates of initial transverse momentum and tunneling time at $\xi$=0. (b) The distribution of principal quantum number n in the coordinates of initial transverse momentum and tunneling time at $\xi$=0. Different colors represent different numbers L and n.
Moreover, the relationship between the final angular momentum and the initial transverse momentum can be easily established using the ’guiding center’ model [71]. The classical trajectory dynamics is governed by the Hamiltonian
At the lowest order of the perturbative expansion, the electron coordinates are of the form [71] where ($\textbf {r}_g,\textbf {p}_g$) are the canonically conjugate variables of the guiding center, and $\textbf {A}(t)$ is the vector potential of the laser field. The guiding-center dynamics is governed by the averaged Hamiltonian [71] This Hamiltonian no longer depends on time, as a result of averaging. At the lowest order in the perturbative expansion, $V_{eff}(\textbf {r})=V(\textbf {r})$. Consequently, its energy $\varepsilon =\bar {H}(\textbf {r}_g,\textbf {p}_g)$ and the angular momentum of the guiding center is conserved.In Fig. 7, we show two classical trajectories of the FTI events with the initial coordinated indicated by the black cross in Figs. 2(f) and 2(g). The guiding center trajectory which averages the disturbance of the electric field is in good agreement with the CTMC trajectory after the laser field turned off. Thus we can use the guiding-center trajectory to estimate the angular momentum. Due to the conservation of angular momentum of the guiding center trajectory, we can predict the relationship between the final angular momentum and the initial transverse momentum. We obtain the initial conditions of the corresponding guiding center according Eqs. (5-6) [71], when t0=0
Fig. 7. (a) A sample classical trajectories of the FTI events (the blue solid curve) and the corresponding guiding center trajectory (the dashed cyan curve) at $\xi =0.25$. (b) The same trajectories at $\xi =0.30$. (a) and (b) have the same initial conditions, $t_0=0, P_\perp =0$. The black crosses show the initial position of classical trajectory, and the black dot marks the zero point of the coordinate.
Fig. 8. The distribution of the final angular momentum ($L_f$) versus the initial transverse momentum at tunneling ($P_\perp$). The dashed magenta line shows the analytical formula (9). The black cross marks the zero point of the coordinate. The colorbar on the right represents the recapture probability.
5. Conclusions
Using CTMC simulations we found that the yield of frustrated tunneling ionization events exhibits an anomalous behavior which maximizes at the nonzero ellipticity. This anomalous behavior is very similar to the previously observed ellipticity dependence of the near- and below-threshold harmonics. By tracing back the initial tunneling coordinates, we show that this anomalous behavior is because the initial transverse velocity at tunneling of the FTI events is nonzero in the linear laser pulses and it moves across zero as the ellipticity increases. The FTI yield maximizes at the nonzero ellipticity when the initial transverse momentum for being trapped is zero. We show that the presence of this anomalous behavior depends on the laser intensity, wavelength and pulse duration.
Moreover, the angular momentum distribution of the FTI events and its ellipticity dependence are also explored. The angular momentum distribution of Rydberg state changes from a symmetric triple-hump structure at the linear laser field to a single peak structure as the ellipticity increases. By tracing back the initial tunneling coordinates, we found that the angular quantum number does not depend on the tunneling time but is very sensitive to the initial transverse momentum at tunnel. Using guiding center theory, we obtained the correlation between the final angular momentum and the initial transverse momentum.
Funding
National Natural Science Foundation of China (NSFC) (11604108, 11622431, 11627809, 11874163); Program for HUST Academic Frontier Youth Team.
Acknowledgments
Numerical simulations presented in this paper were carried out using the High Performance Computing Center experimental testbed in SCTS/CGCL (see http://grid.hust.edu.cn/hpcc).
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