Role of quantum trajectory in high-order harmonic generation in the Keldysh multiphoton regime

We present a systematic study of spectral and temporal structure of high-order harmonic generation (HHG) by solving accurately the time-dependent Schrödinger equation for a hydrogen atom in the multiphoton regime where the Keldysh parameter is greater unity. Combining with a time-frequency transform and an extended semiclassical analysis, we explore the role of quantum trajectory in HHG. We find that the time-frequency spectra of the HHG plateau near cutoff exhibit a decrease in intensity associated with the shortand long-trajectories when the ionization process is pushed from the multiphoton regime into the tunneling regime. This implies that the harmonic emission spectra in the region of the HHG plateau near and before the cutoff are suppressed. To see the generality of this prediction, we also present a time-dependent density-functional theoretical study of the effect of correlated multi-electron responses on the spectral and temporal structure of the HHG plateau of the Ar atom. © 2016 Optical Society of America OCIS codes: (020.2649) Strong field laser physics; (020.4180) Multiphoton processes. References and links 1. M. Protopapas, C. H. Keitel, and P. L. Knight, “Atomic physics with super-high intensity lasers,” Rep. Prog. Phys. 60(4), 389–486 (1997). 2. F. Krausz and M. Ivanov, “Attosecond physics,” Rev. Mod. Phys. 81(1), 163–234 (2009). 3. M. Chini, X. Wang, Y. Cheng, H. Wang, Y. Wu, E. Cunningham, P. C. Li, J. Heslar, D. A. Telnov, S. I. Chu, and Z. Chang, “Coherent phase-matched VUV generation by field-controlled bound states,” Nat. Photon. 8, 437–441 (2014). 4. P. B. Corkum and F. Krausz, “Attosecond science,” Nat. Phys. 3, 381–387 (2007). 5. A. Paul, R. A. Bartels, R. Tobey, H. Green, S. Weiman, I. P. Christov, M. 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Introduction
High-order harmonic generation (HHG) has attracted much attention over past two decades [1][2][3]. HHG provides a promising method to produce the coherent extreme ultraviolet (XUV) source in the attosecond time scale [4][5][6], leading to applications such as the observation and control of the electronic dynamic behaviors [7][8][9]. The HHG spectrum is characterized by a rapid drop at low orders followed by a broad plateau where all harmonics have the similar amplitude, and finally a sharp cutoff beyond which no further harmonic emission is seen. The physical picture of the HHG can be understood by semi-classical three-step model [10,11]: (i) An electron tunnels out the potential barrier formed by the coulomb potential and the laser field. (ii) The tunneled electron is accelerated by the laser field and then bounced back to the core at a later time when the laser field changes to the opposite direction. (iii) The returning electron emits a harmonic photon and recombines to the ground states of the atom. Experiments in the HHG from atoms and molecules driven by the intense laser pulses involve multiphoton and tunneling ionization regimes. According to the Keldysh theory [12], the description of nonlinear ionization regime can be quantitatively differentiated by introducing the Keldysh γ parameter (γ = I p /2U p , where I p is the atomic ionization potential and U p is the ponderomotive energy of the laser field). Multiphoton ionization regime is typically classified by γ > 1, while tunneling ionization regime is classified by γ < 1. The physical picture of most studies [13,14] of laser-driven ionization commonly used is tunneling ionization regime, and this tunneling model agrees rather well with experimental results. However, More recently considerable attention has been paid to the intermediate domain between the tunneling and multiphoton regimes [15,16], particularly the studies of the dynamical origin of the HHG in multiphoton regime have been increased considerably [17][18][19].
In this paper, we present an ab initio study of the role of quantum trajectory in HHG of hydrogen (H) atom and many-electron argon (Ar) atom in the multiphoton regime where the Keldysh parameter is greater unity. We perform a time-frequency transform [20] of the harmonic spectra to explore the role of quantum trajectory in HHG in multiphoton regime. We find that the time-frequency spectra of the HHG plateau near cutoff exhibit a decrease in intensity associated with the short-and long-trajectories when the ionization process is pushed from the multiphoton regime into the tunneling regime. This implies that the harmonic emission in the region of the HHG plateau near cutoff is suppressed. Combining with the time-frequency spectra and an extended semiclassical approach [21] with the inclusion of atomic potential, we confirm that the low distribution of time-frequency spectra in intensity near the cutoff is associated with a competition between the multiphoton and tunneling regimes. Our result enables us to obtain a deeper understanding of the underlying mechanism and delicate electron electron dynamics of the HHG process.

Theoretical methods
The harmonic spectrum of hydrogen atom can be obtained by solving the time-dependent Schrödinger equation (TDSE) directly. For the Ar atom, we use the self-interaction-free timedependent density-functional theoretical (TDDFT) approach with proper long-range potential [22,23]. Both the TDSE and TDDFT can be solved accurately and efficiently by means of the time-dependent generalized pseudospectral method (TDGPS) [24]. It has been demonstrated that the TDGPS method is computationally more accurate and efficient than the conventional time-dependent propagation techniques using equal-spacing grid discretization. The numerical scheme of the TDGPS method consists of two essential steps: (i) The spatial coordinates are optimally discretized in a non-uniform spatial grid by the generalized pseudospectral (GPS) technique [24]. This discretization uses only a modest number of grid points, and it is characterized by denser grids near the nuclear origin and sparser grids for larger distances. (ii) A second-order split-operator technique in the energy representation is used for the time propagation of the wave function, which can be expressed as When the time-dependent wave function ψ(r,t) is determined, and then we can calculate the expectation value of the induced dipole moment d(t) [25], so the HHG power spectra can be calculated through the Fourier transform of the induced dipole moment d(t). To explore the detailed spectral and temporal structures of HHG, we perform the time-frequency analysis by means of the wavelet transform, For the harmonic emission, a natural choice of the mother wavelet is given by the Morlet wavelet: To explore the underlying mechanism of the distribution of time-frequency spectra in intensity associated with the quantum trajectories in HHG, we calculate the probability of the electrons with the corresponding return time t and return energy E by using an extended semiclassical method [21], which can be obtained from the following expression: where Γ(t ) is the time-dependent ionization rate which can be defined as a logarithmic derivative of the time-dependent population P(t ) calculated by solving TDSE, i. e., [26], t r and E r are the returning time and returning energy for given trajectories, and P(v) is the Gaussian initial velocity distribution. Each trajectory is monitored for all the approaches to the parent ion. If an electron trajectory is such that it can return to the parent ionic core at time t r with returning kinetic energy E r , the factor C t (t , Figure 1 shows the wavelet time-frequency analysis of the HHG of hydrogen atom driven by an intense 800-nm infrared laser pulse with the peak intensity (a) I=0.1I 0 (I 0 =10 14 W/cm 2 ), (b) I=0.5I 0 , and (c) I=0.8I 0 , respectively. The left pattern corresponds to the harmonic spectra. In calculation, we adopt a laser pulse with the cosine-squared shape and the duration of 20 optical cycles. In Fig. 1(a), the laser peak intensity is 0.1I 0 , and the corresponding Keldysh parameter γ is equal to 3.4. This implies the multiphoton regime is dominant. The atomic ionization energy of ground state of hydrogen atom is equal to 0.5 a.u., which coincides with the near-threshold 9th harmonic order for an intense 800-nm infrared laser pulse, so the strong resonant structure is located at the 9th harmonic order. It is well known that there are two quantum trajectories called the short trajectory and long trajectory. Typically the travel times of the long trajectories are larger than the short ones, and the long trajectory has a late harmonic emission in one optical cycle. In Fig.1(a), the harmonic spectra do not show the short-and long-trajectories behaviors. With increasing the laser intensity, the ionization process is pushing from the multiphoton regime to close to the tunneling regime. And the HHG with the higher energy will be produced and the contributions of short trajectories become strong as shown in Fig. 1(b) (γ =1.5) and Fig.  1(c) (γ =1.2). However, in Fig. 1(b), an interesting feature is that the time-frequency spectrum between harmonic 11 and harmonic 15 exhibits a weak distribution in intensity. Particularly, in Fig. 1(c), the distribution is broken and has a behavior as a function of the envelope of laser pulse. On the other hand, this broken distribution is located at harmonic 11 to harmonic 17, indicating the harmonic emission is suppressed in these broken distributions. The corresponding harmonic spectra are presented in black solid line (left pattern).

Results and discussion
To explore the role of quantum trajectories in HHG, we present an extended semiclassical analysis by using Eq. (4). The semiclassical calculation including the atomic potential is sensitive to the choice of the initial conditions. Therefore we define two kinds of the initial conditions [17,27]. The first one corresponds to the multiphoton ionization regime where initially the electrons with an initial velocity are released at the core. Here the minimum of initial velocity in the laser polarized direction z is v 0z =0 and the maximum is the height of the barrier at the time of release. Typically the initial velocities used are |v 0z |=0.2 (in a.u.) and z 0 (t )=0. The other set of initial conditions corresponds to the tunneling-ionized regime where initially the electrons with an instantaneous position are released at the core. Typically the initial instantaneous positions used are z 0 (t ) = and |v 0z |=0. To avoid unlikely tunneling distances, we have filtered out those trajectories with initial positions greater than 20 a.u. The other laser parameters used are the same as those in Fig. 1(c).
In Fig. 2(a), it is clear that the short trajectory in the initial condition of multiphoton ionization model is dominant and the maximum returning energy is located at the 17th harmonic order. On the other hand, the maximum returning energy is located at the 21th harmonic order in the initial condition of tunneling ionization model and the short trajectory is absent at the The laser parameters used are the same as those in Fig. 1(c). plateau of the HHG as shown in Fig. 2(b). In other words, with increasing the harmonic order in HHG, the contributions of the short trajectories with the dominated multiphoton ionization regime become smaller. The contributions of the short trajectories with the dominated tunneling ionization regime, on the other hand, are increased and has a strong contribution near cutoff, but the contributions are absent at low energy harmonics. The contributions of the long trajectories in both the dominated multiphoton and tunneling regime are weak. In Fig. 2(c), a comparison of semicalssical calculations between the multiphoton ionization and tunneling ionization model defined by both the initial conditions is presented. We find that the maximum returning energy with the tunneling ionization model is higher than that one of the multiphoton ionization model, and the maximum returning energy has a gap between the multiphoton ionization and tunneling ionization model. This result is in good agreement with the broken distribution of time-frequency spectra shown in Fig. 2(d). The suppression of harmonic emission is located harmonic 11 to harmonic 17. Therefore, we confirm that the competition between the multiphoton and tunneling ionization regimes leads to a weak contribution associated with the quantum trajectories in HHG near cutoff.
To check the sensitivity of the distribution of time-frequency spectra associated with the quantum trajectories when the laser wavelength is changed, we present the wavelet timefrequency analysis of the dipole moment and the harmonic spectrum of hydrogen atom in a 1600-nm and 2000-nm mid-infraed laser field as shown in Figs. 3(a) and 3(b), respectively, and the corresponding Keldysh parameter γ equal to 1.2. The other laser parameters used are the same as those in Fig. 1. It is clear that the weak distribution of time-frequency spectra in intensity occurs between the harmonic 25 and harmonic 35. The similar result of hydrogen atom driven by a 2000-nm mid-infraed laser field can be found in Fig. 3(b), the weak distribution of time-frequency appears at between harmonic 30 and harmonic 40.  Fig. 1(c). To demonstrate the effects of correlated multielectron responses on the time-frequency characteristic which associates with the spectral distribution in the multiphoton regime, we calculate the HHG spectra of Ar by solving the TDDFT. In Fig. 4(a), we present the time-dependent dipole moments of the Ar atom from individual valence electron orbitals 3s, 3p 0 , and 3p 1 . Our calculations of dipole moments are based on a numerical solution of the self-interaction-free time-dependent Kohn-Sham equations for N-electron atomic systems [22,23]. The laser parameters used are the same as those in Fig. 1(c). It is clear that the dipole moments of the 3p is higher than that of the 3s subshell due to the easier ionization. In Fig. 4(b), the wavelet timefrequency analysis of the total dipole moment shows several weak distributions between the harmonic 13 and harmonic 17, and the corresponding harmonic spectra is suppressed as shown in left pattern. In Figs. 5(a)-5(c), we show the wavelet time-frequency analysis of the individual valence electron orbitals (3s, 3p 0 , and 3p 1 ) and the corresponding harmonic spectra, and the similar results are obtained. In other words, the low distribution of the time-frequency spectrum associated with the competition between the multiphoton and tunneling ionization regime is not sensitive to the correlated multielectron responses.

Conclusion
In this paper, we have explored the dynamical origin of the low intensity of the time-frequency spectra of the HHG plateau near the cutoff which implies the harmonic emission spectra near and before the cutoff is suppressed. This phenomenon, which appears to be general, is seen for both H and Ar atoms in the multiphoton regime where the Keldysh parameter is greater unity. We found this phenomenon is associated with the electronic ionization regime in the laser field rather than with the electronic structure of the atom. To analyze the spectral and temporal structure of the HHG, we perform the wavelet time-frequency transform of the HHG and an extended semiclassical calculations. We found that the low intensity of the time-frequency spectrum of the HHG near cutoff can be attributed from the competition of ionization regime associated with the short-and long-trajectories when the ionization process is pushed from the multiphoton into the tunneling regime. The reason is that the maximum returning energy of the quantum trajectories with the multiphoton ionization and tunneling ionization regime has a gap and the contribution of the HHG near cutoff is weak in multiphoton ionization regime, leading to the suppression of higher energy harmonic emission.