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Studying the universality of field induced tunnel ionization times via high-order harmonic spectroscopy

High-harmonics generation spectroscopy is a promising tool for resolving electron dynamics and structure in atomic and molecular systems. This scheme, commonly described by the strong field approximation, requires a deep insight into the basic mechanism that leads to the harmonics generation. Recently, we have demonstrated the ability to resolve the first stage of the process -- field induced tunnel ionization -- by adding a weak perturbation to the strong fundamental field. Here we generalize this approach and show that the assumptions behind the strong field approximation are valid over a wide range of tunnel ionization conditions. Performing a systematic study -- modifying the fundamental wavelength, intensity and atomic system -- we observed a good agreement with quantum path analysis over a range of Keldysh parameters. The generality of this scheme opens new perspectives in high harmonics spectroscopy, holding the potential of probing large, complex molecular systems.


I. INTRODUCTION
Strong field light matter interactions open new horizons in ultrafast physics, in the generation of coherent XUV or x-ray radiation and energetic electrons. Understanding their basic mechanism is an essential step in establishing these new directions of research. In this framework an extensively studied phenomenon is high harmonic generation (HHG). In the strong field regime the mechanism that leads to the emission of high harmonics can be described in semiclassical terms. Near the peak of the optical cycle the Coulomb barrier is lowered by the strong laser field leading to tunnel ionization. Once the electron is free, it interacts with the laser field and accumulates high kinetic energy before it recollides with the parent ion and emits a high-energy photon [1]. This semiclassical picture provides us with two important assumptions. It defines three separate steps which can be treated independently. In addition, it provides a simple link between them: there is a direct mapping between the time of ionization t 0 of the electron, its time of return t r and the corresponding emitted photon energy N ω 0 where N is the harmonic order and ω 0 is the fundamental frequency of the driving laser. These two important components of the semiclassical picture are at the heart of HHG spectroscopy. In this picture the tunneling serves as a 'pump', initiating a hole wavefunction, while the recollision serves as a 'probe', mapping the evolved wavefunction onto the HHG spectrum [2][3][4][5].
Recently we have demonstrated the ability to decouple the two important steps in the process -the ionization and recollision [6]. By adding a weak second harmonic field, orthogonally polarized with respect to the strong driving field, we induce a subcycle amplitude gate. Manipulating the two color delay shifts the gate within the optical cycle, probing a narrow range of electron trajectories according to their ionization time. A perturbative analysis enabled us to map each harmonic number to its time of ionization t 0 and time of recollision t r . Our results, measured in helium atoms, demonstrated a deviation from the semi-classical picture, in agreement with quantum path analysis [7]. Additionally, we have demonstrated the reconstruction of ionization times in both short and long trajectories in argon [8]. The extension of our approach to more complex electronic systems raises the following question: what are the limitations of the strong field assumptions? Specifically, can we generalize our results to a wider range of fundamental parameters such as the laser frequency, intensity or atomic ionization potential?
The semiclassical picture provides a simple answer: there is a unique mapping between t 0 , is the pondermotive potential and E 0 is the field amplitude (atomic units are used throughout the paper). Such mapping provides a universal scaling law, in which the cutoff photon energy scales linearly with U p . A more accurate picture is provided by the quantum mechanical description [7]. Quantum mechanically, the induced dipole moment is dictated by all quantum trajectories that contribute to this process. These trajectories are coherently integrated with a relative phase known as the semi-classical action S(p, t 0 , t r ), given by: where p is the canonical momentum of the electron, A is the vector potential of the driving electric field and I p is the ionization potential of the target atom.
In the strong field regime the main contribution to the integral comes from the stationary  In this paper we study the link between the ionization time and emitted photon energy for a range of Keldysh parameters between 0.64 and 1.23. Current laser technology enables us to achieve this range by scanning the fundamental wavelength [10]. In addition, we also modify the field intensity and vary the atomic systems, thus changing the ionization potential. To avoid contributions of multi-electron dynamics we focus the study to small, rare-gas, atomic systems. Our systematic study demonstrates the generality of the gating approach. Importantly, we confirm the validity of quantum path analysis in a regime were the basic assumptions that underly the strong field model (i.e. γ 1), reach their limit.

II. METHODS
In the following we summarize the basic mechanism of the gating measurement. Consider a two color field in orthogonal configuration: where E 0 is the amplitude of the fundamental field, is the amplitude ratio between the two colors and φ is the two colors delay. In the limit of 1 the second harmonic field acts as a weak perturbation inducing a small lateral shift of the recolliding electron. This shift reduces the recollision probability, suppressing the emitted signal. The lateral shift accumulated between t 0 and t r can be classicaly described as [6]: Here v 0y is the lateral electron velocity at the moment of ionization. The recollision condition which requires y L = 0 at t = t r defines the initial velocity v 0y (t 0 , t r ) that compensates for the field induced shift. The probability of an electron to tunnel with an initial velocity v 0y (t 0 , t r ) is centered around v 0y = 0, having a width dictated by the tunneling process. Controlling the two colors delay, φ, modulates the lateral shift and therefore v 0y . Its optimal value, φ max , for which the signal maximizes, is set at v 0y (t 0 , t r ) = 0, encoding the main parameters of the interaction: t 0 and t r .
Our systematic study was performed both in the 800 nm and the mid-IR regime. Figure   2 shows a schematic description of the experimental setup. We used a BBO crystal followed by a calcite window for generating the two color driving field, consisting of a strong HHG-generating field with perpendicular polarization relative to the weak second harmonic field. The relative phase of the two fields was controlled with a pair of glass wedges. High order harmonics were generated by focusing the two-color driving field on a pulsed gas jet.
Harmonics were detected by means of an XUV spectrometer coupled to an MCP and a CCD detector. The 800 nm experiments were driven by 30 fs pulses generated by an amplified Ti:sapphire laser system. The experiment driven by mid-IR pulses (20 fs, tunable between 1300 and 1800 nm) were performed exploiting an optical parametric amplifier (OPA) pumped by an amplified Ti:sapphire laser system (60 fs, 20 mJ, 800 nm) [11]. We performed a systematic study of the gating experiment over a range of Keldysh parameters. Figure 4 summarizes the results, displaying the two-color optimal delay φ max vs photon energy for four values of Keldysh parameter. These data were obtained from three different atoms, and with three wavelengths, as follows: (1) Argon, 800nm, γ = 1.23. The theoretical curves (solid lines) were calculated by solving the gating equation (eq. 2) for φ, setting y L = 0, v 0y = 0, based on the ionization times obtained from the quantum path analysis [7,12]. Since the peak intensity of the laser was not accurately known, U p was an additional fitting parameter in calculating the theoretical curves. Our study demonstrates a good agreement with quantum path analysis within this range of Keldysh parameters.

III. RESULTS
In the next stage, we map the information from the spectral domain to the temporal domain and reconstruct the ionization times. The reconstruction procedure is described in detail in [8]. Specifically, our experiment resolves φ max as a function of the harmonic number for each value of Keldysh parameter. Relying on the perturbative approach we calculate the lateral gate and extract the theoretical φ max (t r , t 0 ). Comparing φ max (N ω 0 ) to φ max (t r , t 0 ) and relying on the theoretical values of t r (which have been independently measured before [13,14]) we extract Re{t 0 }. Figure 5  These results display a striking universality. Although obtained with a large range of experimental parameters -varying I p , peak intensity and wavelength -they can all be described by a curve which depends only on the Keldysh parameter. Importantly, we obtained a very good fit even for a measurement with γ = 1.23. This observation demonstrates the validity of quantum path analysis of HHG in a regime where the adiabatic assumption of γ 1 fails [15].
Previous experiments have measured the attochirp in HHG generated by mid-IR lasers [14]. In this work we report the first measurement of ionization times at wavelengths longer than 800 nm. Combined with our previous results [6,8], we confirm the validity of the strong field approximation and the single-electron picture for a variety of atomic systems and a wide range of laser parameters. This is an important step for extending HHG spectroscopy to mid-IR wavelengths, a regime which holds great promise [16,17].
Generalizing the two-color gating scheme to longer wavelengths enables extending our scheme to resolve tunneling dynamics in more complex molecular systems. In these systems structural and dynamical effect are often coupled [18], adding a major obstacle to HHG molecular spectroscopy [3,19]. In addition, the tunneling process may involve nontrivial multielectron dynamics. Extending the dimensionality of the measurement by combining two-color schemes with tuneable mid-IR sources will enable us to disentangle the main degrees of freedoms, resolving attosecond scale processes, so far hidden in many HHG experiments.