Determination of Ionization and Tunneling Times in High-Order Harmonic Generation

Abstract

From the numerical solution of the time-dependent Schrödinger equation, we obtain the times of ionization and return of the laser-driven electron in high-order harmonic generation by probing the dynamics with a second harmonic field polarized orthogonal to the fundamental field and observing the harmonic emission in dependence on the two-color delay. Our retrieval method using complex-time evolution gives ionization and return times in excellent agreement with the quantum-orbit model, while a retrieval based on real-time classical dynamics can introduce substantial errors. Because of the imaginary parts, the harmonic signal polarized along the probe field is nonzero for any two-color delay. The tunneling time can be retrieved under an assumption for the return time. DOI: 10.1103/PhysRevLett.111.043901 PACS numbers: 42.65.Ky, 32.80.Rm High-order harmonic generation (HHG) from atoms or molecules exposed to strong laser pulses has been intensively investigated in the past two decades [1,2]. HHG as a highly nonlinear process provides a unique source of coherent extreme ultraviolet radiation in the form of single attosecond pulses or attosecond pulse trains, which paves the way for monitoring and controlling electronic dynamics on the attosecond time scale [3,4]. Because the HHG signal contains information about electronic dynamics and molecular structure, it is also extensively used for highharmonic spectroscopy [5][6][7][8][9][10].The key to understanding HHG is the three-step model [2]: An electron tunnels into the continuum through the potential barrier formed by the atomic potential and the laser field. Then the electron is accelerated as a free particle in the strong oscillating field. Finally, the electron may recollide with the parent ion and recombine to the initial state by emission of an extreme ultraviolet photon. The photon carries away the sum of the binding energy and the electronic kinetic energy acquired in the continuum. Classical electron dynamics in the continuum is sufficient to explain approximately the cutoff in the harmonic spectrum, but the initial tunnel ionization is a quantummechanical process that requires separate treatment. A complete quantum-mechanical description, based on the strong-field approximation, was developed to give a quantitative treatment of HHG [1]. Resulting from this approach is the quantum-orbit (QO) model, where each harmonic emission frequency is attributed to a few dominant quantum trajectories evolving in complex time [11]. The question arises how accurately reality follows these model trajectories, especially since time-resolved highharmonic spectroscopy is based on the knowledge of the electron excursion times. Another question concerns the physical meaning of the imaginary parts.With the development of time-resolved methods operating on the attosecond scale, the precise timing of electron release from an atom has been investigated for various types of ionization processes. As for single-photon ionization, isolated attosecond pulses were applied to set electr...

Attosecond chronoscopy of photoemission

“…The connections to the controversial subject of "tunneling time", the time it takes a wavepacket to penetrate a barrier has been the subject of a large number of publications, (e.g., Kolomeitsev and Voskresensky, 2013;MacColl, 1932;Hartman, 1962;Hauge and Støvneng, 1989;Landauer and Martin, 1994;Steinberg, 1995;Olkhovsky et al, 2004;Winful, 2006). Attempts to observe tunneling times on the attosecond scale for strong-field ionization of atoms (Eckle et al, 2008b;Pfeiffer et al, 2013;Shafir et al, 2012;Zhao and Lein, 2013;McDonald et al, 2013;Klaiber et al, 2013;Orlando et al, 2014) have so far been inconclusive. While outside the main scope of the present article on the time-resolved photoelectric effect, we will briefly discuss the challenge in extracting such timing information in Section IX.…”

Analysis of electron trajectories with two-color strong-field ionization

“…Fast electrons, however, are due to recollisions, and in analogy to the HHG experiments [15,18] the fast-electron yield is strongly delay dependent. The strongest enhancement for electrons above 4U p is around φ ≈ 0.96π and φ ≈ 1.96π , which represents a shift of about π/2 compared to HHG [18,21]. We show below that this is due to the dominance of the long rescattering trajectories.…”

“…The experiment showed excellent agreement with the quantumorbit (QO) model [19,20] and lesser agreement with the purely classical model. Theoretical analysis using the timedependent Schrödinger equation (TDSE) also confirmed the QO model [21,22]. Despite strong interest in the subject, only few investigations of the accompanying photoelectrons have been reported for two-color ionization in fields with orthogonal polarization [23,24], compared to fields with parallel polarization [25][26][27][28][29][30][31][32].…”

“…corresponds to 800-nm wavelength. The small relative strength = 0.1 of the second harmonic field allows us to view it as a perturbation [18,21]. The field strength is E 0 = 0.107 a.u., corresponding to an intensity of 4.0 × 10 14 W/cm 2 for the component along x.…”

Attosecond dynamics of light-induced resonant hole transfer in high-order-harmonic generation

“…If the second harmonic field is weak it will barely modify the original odd harmonics, but the symmetry breaking between half optical cycles leads to the generation of weak even harmonics. These even harmonics have been used to retrieve emission times of attosecond pulses and recombination times of the electrons [28][29][30][31]. Some ideas beyond single-active electron approximation have also been theoretically discussed to extend the HHG cutoff.…”