We theoretically investigate the above-threshold ionization of hydrogen atoms driven by few-cycle phase jump laser pulses. By numerically solving the three-dimensional time-dependent Schrödinger equation, we demonstrate that the phase jump plays an important role in the ionization process. The cutoff of the photoelectron energy spectrum can extend to a range of very high energy, and the yield of the photoelectrons can be dramatically enhanced by choosing proper phase jump times. Both the classical simulations and Fourier transform method are used to understand the spectra features found in our investigation.
© 2013 Optical Society of America
Above-threshold ionization (ATI)  is one of the most well-known nonlinear phenomena in atom or molecular physics. It occurs when an atom or molecule absorbs more photons than the minimum number required to ionize, with the leftover energy being converted to the kinetic energy of the released electron (the photoelectron). Most of the early works [2–6] have concentrated on the low-energy part of the spectrum, and perturbation theory has been applied to the ATI process [7, 8]. Later, owing to the significant increase of the experimental precision in detecting electron spectra, high energy electron spectrum with an extend plateau as well as rings in the electron angular emission were observed [9, 10]. These new ATI features can be understood by the simple man’s theory [11–14] which was firstly proposed to exploit a classical analysis for predicting the electron energy distribution. According to this theory, ATI could be understood as a combination of direct ionization process and indirect process (rescattering process), and the maximum electron kinetic energies of each processes are 2Up and 10Up (Up is the ponderomotive energy).
Up to now, a lot of studies both in theory and experiment have been done to exploit ATI process and its control through different ways [15–33]. The ATI spectra exhibit a characteristic left-right asymmetry with the variation of the carrier-envelope phase of the few-cycle laser pulse . Both the photoelectron energy spectra and photoelectron angular distributions are strongly dependent on the chirp rate of the linear chirped pulse since the presence of the intermediate bound states . We have also found that the cutoff energy of the ATI spectrum could dramatically extend as the nonlinear chirp enhances . Furthermore, an unexpected structure at the low energy spectra was revealed [24, 25], and attributed to the Coulomb field effects [25–27]. Recently, one type of spatially inhomogeneous field is widely used to study the high-order harmonic generation (HHG) and the ATI process [28–34]. On the other hand, some light-atom interactions have been studied by phase-jump pulses with different asymmetry degrees. For instance, complete population transfer was shown to occur for suitable jump phases [35, 37]. Qian et al. found that a phase-jump pulse may cause the breakdown of dipole blockade in strong Rydberg blockade regime . Cleary et al. demonstrated that the center-of-mass motion of a Bose-Einstein condensate can be quenched by applying a carefully timed and sized jump in the phase of the rotating field . And in our previous study of HHG process by phase-jump laser pulses , the HHG spectrum cutoff can have a dramatic extension, the intensity and the coherence of the continuum spectrum can be controlled by the phase jump time. Motivated by the above works, we will investigate the ATI process driven by phase jump pulses with different jump times in this paper.
2. Theoretical model and method
We use the three-dimensional time-dependent Schrödinger equation with dipole approximation to describe the interaction between the hydrogen atom and the laser field [The atomic units (a.u.) are used, unless otherwise mentioned].
The electric field of a laser pulse with a phase jump can be written as:Eq. (2)) can be realized by modern femtosecond pulse-shaping technology, such as described in . In all of our calculations, the peak amplitude of the laser pulse F is 0.0377 a.u. (corresponds to an intensity of I = 5 × 1013W/cm2), the frequency ω is 0.057 a.u. (corresponds to a wavelength of λ = 800 nm), and the CEP φ = 0. We use a Gaussian shaped envelope f (t) = exp(−2ln2(t/τ)2) with τ = 5fs being full width at half maximum (FWHM) of the pulse.
We assume the target atom is in the ground state (1s) before we turn on the laser pulse and this state can be obtained by employing the image-time transformation. Then, we solve Eq. (1) by using a partial-wave decomposition of the wave function method , together with the Peaceman-Rachford scheme . When the time-dependent wave function ψ(r, t) is obtained, the energy-resolved photoelectron spectra then can be calculated by using the window function technique developed by Schafer [42, 43].
3. Results and discussions
For the laser field expressed in Eq. (2), the symmetry of the electric field is broken by the phase jump, and the degree of the asymmetry is dependent on both the jump phase ϕ and jump time t0. As mentioned in our previous work , in order to obtain a larger asymmetry of the electric field, we consider the cases that the phase jump occurs at the zero points near the pulse center with a jump phase of π. In this paper two new cases are also considered which results in six phase jump cases, i.e., t0 = ±0.75T, ±0.50T and ±0.25T (where T is the optical period of the laser pulse). As a reference, the case without phase jump (ϕ = 0) is also considered.
Figure 1 displays the ATI spectra driven by laser pulses for the above mentioned six different phase jump times and the laser pulse without phase jump. It can be seen that the phase jump has significant effects on the ATI spectra. For the case without phase jump, the spectrum exhibits the typical ATI behavior, namely, two plateaus, 2Up and 10Up cutoffs. But for the cases of t0 = ±0.75T and t0 = ±0.25T, the cutoff energy of the whole spectrum extends very much, and some oscillations appear. For the cases of t0 = ±0.50T, the photoelectrons with higher energies far beyond the classical prediction 10Up energy are present in the spectrum. Moreover, the yield of the photoelectrons is enhanced for all the cases with phase jump, especially for the cases of t0 = ±0.50T in the low-energy regime.
Now we first concentrate on the cutoff extension of the whole photoelectron energy spectrum. As is stated in , electrons are born at time tb (ionization time, or born time) with zero velocity, and then they move under the influence of the laser field. For some ionization time, electrons may never return to the core (called the direct electrons). But for some other ionization time, electrons may return to the core at time tr and elastically backscatter (or rescatter) with the core, and subsequently they move in the laser field until the terminal time tf of the pulse (called the rescattered electrons). After the termination of the laser pulse, the kinetic energy of electron do not change anymore and the electron continues to fly to the detector. According to this theory, the final kinetic energy of electron can be calculated by solving the Newton equation of motion ẍ = −E (t), ignoring the effect of the ionization potential. For the direct process, the kinetic energy of an electron born at time tb is
Figure 2 shows the classical results for the kinetic energy of the photoelectron. We use the same laser parameters as in Fig. 1. For the case without phase jump, we find that the maximum kinetic energies of the electron which undergoes the direct and rescattering processes are 2Up and 10Up, respectively. This is well consistent with the quantum simulation as shown in Fig. 1. For the phase jump cases, we can observe the strong modifications of the electron energy distribution caused by phase jump. In particular for the cases of t0 = −0.75T, t0 = −0.25T, t0 = 0.25T and t0 = 0.75T, the maximum kinetic energies of the rescattered electrons extend to about 15Up, 27Up, 34Up and 30Up, respectively. These energies are approximately consistent with the quantum simulations as shown in Fig. 1.
As described in our previous study of HHG , once the electron is born, it will obtain an impulse to get away from the nucleus, and after a while it will obtain a reversed impulse to slow down and get back to the nucleus. Because of the symmetry break of the laser field caused by the static electric field or the chirp, the difference of these two impulses enlarges and thus the electron could obtain a larger kinetic energy when it returns to the nucleus. Therefore, the cutoff of the HHG spectrum could be extended. For the laser fields used in this paper, the symmetry of the laser field is broken by phase jump. For the cases of t0 = ±0.75T and t0 = ±0.25T, all the rescattered electrons which have the maximum kinetic energy (called the cutoff electrons) return to the core just at the time before half an optical period of the phase jump time t0. At this moment, the laser field value is zero, and it will change the sign, and then the electrons backscatter at the same time as shown in Figs. 2(b), 2(c), 2(f) and 2(g). This coincidence in time exactly gives a positive force to the electrons to accelerate from this moment. For the laser pulse with phase jump, this acceleration process can last a whole optical period at most, while it can only last half an optical period at most for the pulse without phase jump. As a result, the electron could obtain a larger impulse from the laser pulse with phase jump.
As to the cutoff electrons, from their backscattering time to the terminal time of the laser pulse, we mark the impulse whose direction deviates from the nucleus as A and the impulse with a reversed direction as B in Fig. 2. In order to quantitatively show the significant effect of the laser field asymmetry, we calculate the cutoff electrons’ velocity at their backscattering time, the impulse obtained from the backscattering time to the terminal time and their final velocity at the terminal time. The results are shown in Table 1. We can see that, for all the phase jump cases, their cutoff electrons have a larger final velocity than that of the case without phase jump. A further view of these results shows us that the impulse obtained from backscattering time to the terminal time have different degrees of increase compared to that of the case without phase jump. This is because of the different degrees of asymmetry caused by the different phase jump times. For the cases of t0 = ±0.75T and t0 = ±0.25T, the asymmetry changes so much that the impulse obtained is much larger than that of the case without phase jump. Thus the electron’s final velocity is much larger even though its velocity at the backscattering time is smaller. However, for the cases of t0 = ±0.50T, the symmetry doesn’t change so much and the impulse obtained is just a little bit larger than that of the case without phase jump. As a result, the final velocity is almost the same with that of the case without phase jump. In conclusion, the asymmetry of the laser field is the reason why the cutoff extends.
As we know, the photoelectrons with the same final kinetic energy may come from many electrons that are born at different times, namely, different trajectories. Interferences between these different trajectories will cause oscillations in the spectrum . In our present study, we also observed obvious oscillations in the spectra shown in Fig. 1 for the cases of t0 = ±0.75T and t0 = ±0.25T. Take the case of t0 = −0.25T for an example, we see from Fig. 2(f) that there exist many trajectories between 5Up and 25Up, and hence oscillations occur between 5Up and 25Up in the spectrum shown as the thin blue dash-dotted line in Fig. 1.
Phase jump in the laser pulse could make the frequency spectrum of laser field be changed. We do Fast Fourier Transform (FFT) for all the laser fields, and obtain the frequency spectra shown in Fig. 3. It shows that, for the laser pulses with phase jump, the intensity of the center frequency ω is weakened substantially, while the intensities of the lower and higher frequencies with a broad bandwidth are enhanced. Just as discussed in , since the few-cycle pulse has a broader spectral bandwidth than many-cycle pulse, n-photon ionization becomes n′ (n′ < n)-photon ionization, and then the ionization efficiency by few-cycle pulses will be larger than that by many-cycle pulses. Now, for all the phase jump pulses we used here, they have a much broader spectral bandwidth than the pulse without phase jump. Therefore, they have a larger ionization efficiency than that of the case without phase jump. Furthermore, we can see from Fig. 3 that, two distinct peaks appear astride the center frequency in 0.8ω and 1.2ω. And for the cases of t0 = ±0.50T, the intensity of the high frequency peak 1.2ω is stronger than that of all the other four phase jump cases. Then, the multiphoton ionization can be easier to take place for them than the other four cases. Thus the yield of the photoelectrons is much higher in the low-energy regime for t0 = ±0.50T.
Moreover, we note that the high frequency extends to an infinite position with a considerable intensity for the cases of t0 = ±0.50T. This will make the multiphoton ionization be easier and even make the single-photon ionization take place. We consider that the photoelectrons with energy far beyond the classical prediction 10Up cutoff mainly come from the multiphoton ionization or single-photon ionization of the very high energy photons.
Maybe someone has noticed that, for the cutoff electrons of the cases t0 = −0.75T and t0 = −0.25T, the laser field value at the born time is smaller than 0.6F, which is shown in Figs. 2(b) and 2(f). According to , only the solutions whose ionization time corresponds to a reasonably high laser field (|E(t)| ≥ 0.6F) have considerable contribution to the electron spectrum. Based on this theory, the cutoffs of t0 = −0.75T and t0 = −0.25T are about 10Up and 15Up, which are obviously smaller than the quantum simulation values of 15Up and 27Up. In our calculations, since the Coulomb potential was used to characterize the level structure of hydrogen atom, there exists lots of excited bound states. As was reported by , the excited bound states could play important roles in the ionization process. During the rising time of the laser pulse, the excited states could be populated through multiphoton absorption, and as the pulse approaches the peak intensity, tunneling ionization takes places very easily from there. In order to make sure whether the excited bound states play some roles in the ionization process, we introduce a short-range potential V (r) = −Z ·exp(−αr)  instead of Coulomb potential. Here Z and α are chosen to be 2 and 1.1144, which results in only the existence of the ground state. Then we calculate the ionization probability by numerically solving three-dimensional time-dependent Schrödinger equation using Coulomb potential and the short-range potential. In Table 2 we list the ionization probabilities of the hydrogen atoms for t0 = −0.75 and t0 = −0.25T. Pcoul and Psr are the ionization probability calculated for the Coulomb and short-range potential. We may approximate the contribution of the excited bound states by Pcoul − Psr. For the cases of t0 = −0.75T and t0 = −0.25, we can see that Pcoul is much larger than Psr. This exactly demonstrates the important roles of the excited bound states in the ionization process. When the excited bound states are populated through multiphoton absorption, the ionization potential could be reduced that there does not need so strong laser field to tunneling ionize. Thus, tunneling process takes place even though the laser field is less than 0.6F. Therefore, the cutoff electrons have considerable yield for t0 = −0.75T and t0 = −0.25T, which results in the cutoffs reaching to the 15Up and 27Up.
In this paper, we have theoretically investigated the ATI process of hydrogen atoms driven by few-cycle laser pulses with a π-phase jump. We found that the photoelectron energy spectrum is strongly modified by the phase jump time t0. If proper t0 is selected, the cutoff of the spectrum could be dramatically extended, as well as the yield of the photoelectrons. We found that it is the asymmetry of the laser field that results in the cutoff extension. Since the phase jump makes the frequencies both lower and higher than center frequency ω with a broad bandwidth appear, the yield of the photoelectrons is enhanced for all the cases with phase jump, especially for the cases of t0 = ±0.50T in the low-energy regime. Moreover, we also demonstrated that the excited bound states play important roles in the ionization process.
This work was supported by the National Nature Science Foundation of China (Grants No. 11074263, No. 60921004, and No. 61308029). We thank Dr. Li Zhang from Vienna University of Technology very much for her valuable comments and suggestions.
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