The time it takes a bound electron to respond to the electromagnetic force of light sets a fundamental speed limit on the dynamic control of matter and electromagnetic signal processing. Time-integrated measurements of the nonlinear refractive index of matter indicate that the nonlinear response of bound electrons to optical fields is not instantaneous; however, a complete spectral characterization of the nonlinear susceptibility tensors--which is essential to deduce the temporal response of a medium to arbitrary driving forces using spectral measurements--has not yet been achieved. With the establishment of attosecond chronoscopy, the impulsive response of positive-energy electrons to electromagnetic fields has been explored through ionization of atoms and solids by an extreme-ultraviolet attosecond pulse or by strong near-infrared fields. However, none of the attosecond studies carried out so far have provided direct access to the nonlinear response of bound electrons. Here we demonstrate that intense optical attosecond pulses synthesized in the visible and nearby spectral ranges allow sub-femtosecond control and metrology of bound-electron dynamics. Vacuum ultraviolet spectra emanating from krypton atoms, exposed to intense waveform-controlled optical attosecond pulses, reveal a finite nonlinear response time of bound electrons of up to 115 attoseconds, which is sensitive to and controllable by the super-octave optical field. Our study could enable new spectroscopies of bound electrons in atomic, molecular or lattice potentials of solids, as well as light-based electronics operating on sub-femtosecond timescales and at petahertz rates.
The Keldysh theory of photoionization in a solid dielectric is generalized to the case of arbitrarily short driving pulses of arbitrary pulse shape. We derive a closed-form solution for the nonadiabatic ionization rate in a transparent solid with a periodic dispersion relation, which reveals ultrafast ionization dynamics within the field cycle and recovers the key results of the Keldysh theory in the appropriate limiting regimes.In his seminal 1964 paper [1], Keldysh has presented his celebrated formulas for photoionization, providing a uniform description of multiphoton and tunneling ionization. Over the next five decades, the Keldysh theory of photoionization has been pivotal to the research in laser science, providing a commonly accepted framework for a quantitative analysis of ionization in a remarkable diversity of light-matter interaction phenomena, including laser-induced breakdown [2,3], high-order harmonic [4] and terahertz [5] generation, as well as filamentation of ultrashort light pulses [6,7]. While the original Keldysh formulas were intended to describe photoionization in a continuous-wave field, several elegant approaches have been proposed [8][9][10] in the context of rapidly progressing ultrafast technologies [11] and attosecond science [12], to include the wave-packet nature of ultrashort driver pulses inducing an ultrafast ionization of gases. These approaches help identify new field-cycle-sensitive phenomena in electron tunneling [13,14] and develop novel experimental methods for all-optical detection of electron tunneling dynamics [15,16].Extension of the Keldysh model to ultrafast photoionization in solids is a standalone challenge in quantum physics. Meeting this challenge not only requires an adequate treatment of broadband driver fields, but also calls for a revision of the standard, hyperbolic model of the electron band structure adopted in the Keldysh formalism. The hyperbolic band model enables an accurate description of weak-field optical properties of solids [17,18], but fails in the strong-field regime, where effects of zone edges become significant. A Schrödinger-equation treatment with a 1D cosine-type dispersion [19,20] has been shown to partially address this problem, offering an adequate framework for the numerical analysis of an important class of ultrafast ionization effects in solids [21]. Still, in the lack of a closed-form solution for the photoionzation rate valid for ultrashort pulses of arbitrary shape, the physical intuition based on the Keldysh theory of photoionization of solids often has to be pushed beyond the range where this theory is rigorously valid, for the sake of compact semianalytical description and overall physical clarity [16,22].Here, we derive a closed-form solution for the nonadiabatic ionization rate in a transparent solid, which can be used not only to calculate the probability of ionization in the wake of the pulse and after each field cycle, but also to analyze the behavior of the ionization rate within the field cycle. Our analysis presented bel...
We show that a broadly accepted criterion of laser-induced breakdown in solids, defining the laserbreakdown threshold in terms of the laser fluence or laser intensity needed to generate a certain fraction of the critical electron density rc within the laser pulse, fails in the case of high-intensity fewcycle laser pulses. Such laser pulses can give rise to subcycle oscillations of electron density ρ with peak ρ values well above ρ c even when the total energy of the laser pulse is too low to induce a laser damage of material. The central idea of our approach is that, instead of the ρ = ρ c ratio, the laser-breakdown threshold connects to the total laser energy coupled to the electron subsystem and subsequently transferred to the crystal lattice. With this approach, as we show in this work, predictions of the physical model start to converge to the available experimental data.Laser-induced breakdown of solid materials has been a subject of in-depth research since the invention of lasers 1,2 . In the era of rapidly progressing laser sources of extremely short and broadband optical field waveforms 3,4 , understanding the regimes and scenarios of laser-induced breakdown, as well as the available parameter space for a reversible photoionization-assisted control of optical properties of solids is central for emerging petahertz optoelectronic technologies 5,6 , nonlinear-optical bioimaging 7,8 , short-pulse laser surgery 9,10 , laser micromachining 11 , and compression of high-peak-power ultrashort laser pulses in transparent solids [12][13][14] . Systematic experimental studies of optical breakdown and laser-induced damage, performed within more than five decades, have revealed distinctly different physical scenarios of optical breakdown induced by laser pulses of broadly varying intensities, fluences, and pulse widths [15][16][17] . These studies helped identify a broad range of physical processes contributing to laser-induced breakdown 18 , including field-induced and avalanche ionization, nonlinear dynamics of a laser beam, plasma effects, radiation absorption by impurity and defect states, as well as collisional dynamics, diffusion, and recombination of free carriers 9 . While the specific regime of laser-induced breakdown can depend on all the above-listed factors, ionization dynamics and the related buildup of free-carrier density always play a central role in laser breakdown, providing a mechanism whereby the laser field is coupled to a material. This fact is recognized by a broadly accepted criterion of laser-induced breakdown 9,18-25 that defines the laser breakdown threshold in terms of the laser fluence or laser intensity needed to generate a certain fraction of the critical electron density within the laser pulse. This criterion has proven to be useful in a broad range of pulse widths, offering a powerful tool for the analysis of a laser breakdown by pico-and femtosecond light pulses and helping understand a variety of related laser-matter interaction phenomena in a broad class of solid materials and syste...
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