In the article the effect of twisting of Bessel radiation on two-photon ionization of single atoms localized on the axis of the incident beam is studied. The matrix element of two-photon ionization of this type is obtained for arbitrary polarization and multipolarity of the incident radiation. The differential and integral probability of ionization of an atom over the photoemission angle is analyzed. Illustrative calculations are performed for helium and neon atoms in the simplest case of a circularly polarized field in the electric dipole approximation in the single-active-electron model.

Attosecond photoelectron interferometry based on the combination of an attosecond pulse train and a synchronized infrared field is a fundamental technique for the temporal characterization of attosecond waveforms and for the investigation of electron dynamics in the photoionization process. In this approach, the comb of extreme ultraviolet harmonics typically lies above the ionization threshold of the target under investigation, thus releasing a photoelectron by single-photon absorption. The interaction of the outgoing photoelectron with the infrared pulse results in the absorption or emission of infrared photons, thereby creating additional peaks in the photoelectron spectrum, referred to as sidebands. While, in the absence of resonances in the first ionization step, the phases imparted on the photoionization process evolve smoothly with the photon energy, the presence of intermediate resonances imprints a large additional phase on the outgoing photoelectron wave packet. In this paper, using a comb of harmonics below and above the ionization threshold of neon, we investigate the effect of intermediate bound excited states on attosecond photoelectron interferometry. We show that the phase of the oscillations of the sidebands and their angular distributions are strongly affected by such resonances. By slightly tuning the photon energies of the extreme ultraviolet harmonics, we show how the contributions of selected resonances can be enhanced or suppressed.

It is shown that under optimal conditions, the generation of the 3rd, 5th, 7th, and 9th harmonics of the short-wave, ultraviolet or vacuum ultraviolet, laser field by helium atoms is mainly due to transitions between bound states, and the maximum energy of the harmonics is achieved under conditions of their resonant multiphoton excitation. In this case, the optima for the generation of the 3rd, 5th, 7th, and 9th harmonics of the field correspond to three-, four-, five-, and six-photon resonances, and the optimal value of the peak field intensity, depending on the harmonic order, varies from 2.5*10^14 W/cm^2 to 1.2*10^15 W/cm^2. The total probability of excitation and ionization of an atom at the end of a laser pulse under corresponding conditions exceeds 1/2. With such intensity and not-too-high frequency of the laser field, the Stark effect turns out to be very significant, which allows it to be tuned in resonance with an arbitrary excited state of the atom by changing the intensity of the field without changing its frequency. It is shown that for the laser field parameters maximizing the energy of the Nth harmonic, N from 3 to 9, this harmonic dominates in the dipole acceleration spectrum. At the same time, the amplitudes of the remaining harmonics increase as the harmonic order approaches N. In particular, under the conditions maximizing the yield of the 9th harmonic, the harmonic amplitudes increase when moving from the 3rd harmonic to the 5th and then to the 7th and 9th harmonics, and the harmonics form an attosecond pulse train. At the same time, under the conditions maximizing the yield of the 3rd harmonic, its amplitude in the dipole acceleration spectrum exceeds not only the amplitudes of the other harmonics, but also the amplitude of the atomic response at the frequency of the driving field.

Учебное пособие предназначено для студентов IV курса отделения ядерной физики физического факультета МГУ им. М.В. Ломоносова, изучающих дисциплину "Теоретический практикум". Пособие содержит краткое изложение теоретического материала и большое количесво задач, позволяющих выработать у обучающих практические навыки применения техники квантовомеханических выкладок в задачах ядерной и атомной физики, а также в смежных областях. Многие задачи снабжены ответами и решениями. Пособие включает следующие главы: коммутационные соотношения и матричные элементы оператора углового момента; коэффициенты Клебша-Гордана; углы Эйлера; D-функции Вигнера; 6j-символы; 9j-символы; неприводимые тензорные операторы и теорема Вигнера-Эккарта; электромагнитное поле. Пособие может использоваться студентами физических специальностей, изучающими квантовую механику, для углубленного освоения методов квантовой теории углового момента.

In the paper we theoretically investigate the features of RABBITT (Reconstruction of Attosecond Beating By Interference of Two-photon Transitions) spectroscopy under conditions when transitions through discrete spectrum states play a significant role. Two approaches are applied in the article: the numerical solution of rate equations with continuum discretization and the perturbation theory up to the third order in amplitude. Both approaches use transition matrix elements and photoionization amplitudes obtained by the high-precision R-matrix method. Within the framework of these approaches, photoelectron spectra, the amplitude and phase of RABBITT oscillations were obtained, and the effect of the seed optical field intensity and detuning from a resonance upon excitation of discrete states was studied.

The modern Free-Electron-Lasers generate a highly intense polarized radiation which initiate a sequence of ionization and decay events. Their probability depends on the polarization of each state as function of time. Its complete accounting is limited by the fact that a state can be formed in various ways. Here we present the equivalent of rate equations for population that completely accounts polarization of radiation and formulated in terms of the statistical tensors. To illustrate our approach we theoretically consider sequential photoionization of krypton by an intense extreme ultraviolet femtosecond pulse for the photon energies below the 3d-shell excitation threshold. The calculations of the ion yields, photoelectron spectra and ionic polarization for various photon fluence are presented and role of polarization is discussed.

In the paper we theoretically investigate the features of RABBITT (Reconstruction of Attosecond Beating By Interference of Two-photon Transitions) spectroscopy under conditions when transitions through discrete spectrum states play a significant role. Two approaches are applied in the article: the numerical solution of rate equations with continuum discretization and the perturbation theory up to the third order in amplitude. Both approaches use transition matrix elements and photoionization amplitudes obtained by the high-precision R-matrix method. Within the framework of these approaches, photoelectron spectra, the amplitude and phase of RABBITT oscillations were obtained, and the effect of the seed optical field intensity and detuning from a resonance upon excitation of discrete states was studied.

In the paper we propose a method for characterizing VUV pulse(s) in a bichromatic ionization setup. The scheme is based on s-shell ionization by joint action of circularly polarized fundamental harmonic and linearly polarized second one. The advantage of the proposed approach is the existence of kinematic (geometrical) zeros of partial amplitudes which positions can be extracted with minimal number of theoretical (spectroscopic) assumptions and therefore they may serve as natural reference points in measuring the relative phase and amplitude of the harmonics. In the paper, we investigate a general possible geometry setup with more detailed consideration of the edge cases and present calculation and numerical stimulation for helium ionization as an illustrative example.

The optimal regime of three-photon resonant excitation of a helium atom via a femtosecond ultraviolet (UV) pulse was discovered and numerically studied, at which the maximum power of the third harmonic of the UV field is achieved in the spectrum of dipole acceleration (the second time derivative of the induced dipole moment) of the atom. It is shown that the optimal frequency of the UV field nearly coincides with the frequency of the three-photon transition |1s2⟩–|1s2p⟩, taking into account its shift as a result of the dynamic Stark effect, and the intensity of the UV field is dictated by the condition of maximizing the product of the populations of the |1s2⟩ and |1s2p⟩ states, averaged over the time interval during which the UV field is non-zero. For the considered UV field durations, from 10 to 100 cycles of the carrier frequency (from units to tens of femtoseconds), the optimal intensity lies in the range from 1014 W/cm2 to several units of 1014 W/cm2. It is shown that with an optimal choice of the frequency and intensity of the UV field, the dynamics of excitation of bound and continuum states, as well as the shape of the time envelope of the dipole acceleration of the atom, weakly depend on the duration of the UV field envelope; only their time scale changes significantly. In addition, under optimal conditions, the average power of the third harmonic signal in the dipole acceleration spectrum is practically independent of the duration of the UV field envelope.

In the paper we present the angular distributions of photoelectrons in ionization of neon atom by a field of several multiple frequencies. The considered setup is referred to the RABBITT (reconstruction of attosecond beating by interference of two-photon transitions) spectroscopy under condition that the field frequencies are selected in such a way that resonant transitions through discrete states play an important role. The role of the phase of the seeding infrared field on the angular distributions of photoemission is analyzed. A significant difference in the anisotropy parameters at the near-threshold sideband caused by transitions through discrete states is shown. Two methods are compared: numerical solution of the rate equations with continuum discretization and third-order perturbation theory.

A multilevel model that makes it possible to describe multiphoton processes taking into account ionization in a multielectron atom irradiated by an intense laser field is proposed. Using the He atom as an example, it is shown that this model reproduces the main regularities of multiphoton ionization of an atom by an intense high-frequency laser field.

We investigate features of electron emission from atoms in the field of the fundamental and second harmonic of a laser in conditions relevant to those realized in free-electron lasers. General expressions for the angular anisotropy and photoelectron spin polarization are obtained. These expressions are used for describing of the ionization of the valence shell of inert gas atoms and illustrated by calculations for neon. The possibility of control over the electron emission characteristics by varying the relative phase between harmonic (i.e., realization of one of the forms of the coherent control) is considered.

In coherent control schemes, pathways connecting an initial and a final state can be independently controlled by manipulating the complex amplitudes of their transition matrix elements. For paths characterized by the absorption of multiple photons, these quantities depend on the magnitude and phase between the intermediate steps, and are expected to be strongly affected by the presence of resonances. We investigate the coherent control of the photoemission process in neon using a phase-controlled two-color extreme ultraviolet pulse with frequency in proximity of an excited energy state. Using helium as a reference, we show that the presence of such a resonance in neon modifies the amplitude and phase of the asymmetric emission of photoelectrons. Theoretical simulations based on perturbation theory are in fair agreement with the experimental observations.

An extensive study of photoionization from neon excited states was performed. The R-matrix approach was applied to calculate a photoionization cross-section from the metastable 2p5(2PJf)3s[K]0,2 and dipole-allowed 2p5(2PJf)3s[K]1 states. The resonance structures and Cooper minimum accessible in photoionization from the excited states by the photons with energy below 30 eV were analyzed. The parameters of the lowest autoionizing states (AISs) of even parity were extracted by fitting of the photoionization cross-section. For the dipole-allowed states, calculations are presented for unpolarized, linearly and circularly polarized radiation.

A fundamental phenomenon of coherent control is investigated theoretically using the example of neon photoionization by the bichromatic field of a free-electron laser. A system exposed to coherent fields with commensurable frequencies loses some symmetry, which manifests itself in the angular distribution and spin polarization of the electron emission. We predict several such effects, for example, the violation of symmetry with respect to the plane perpendicular to the polarization vector of the second harmonic and the appearance of new components of spin polarization. Furthermore, we predict a very efficient control of spin polarization via manipulation of the phase between the harmonics. Experimental observation of these effects is accessible with modern free-electron lasers operating in the extreme ultraviolet wavelength regime.

Synopsis Interference effects resulting from direct single-photon and resonant two-photon ionization of neon and argon by bichromatic circularly polarized radiation were investigated in the framework of the perturbation theory. Analytical formulas are obtained for the differential cross section of the process in multielectron noble gas atoms. The possibility of coherent control of the photoelectron angular distribution and spin polarization will be discussed.

Quantum mechanically, photoionization can be fully described by the complex photoionization amplitudes that describe the transition between the ground state and the continuum state. Knowledge of the value of the phase of these amplitudes has been a central interest in photoionization studies and newly developing attosecond science, since the phase can reveal important information about phenomena such as electron correlation. We present a new attosecond-precision interferometric method of angle-resolved measurement for the phase of the photoionization amplitudes, using two phase-locked Extreme Ultraviolet pulses of frequency $\omega$ and $2\omega$, from a Free-Electron Laser. Phase differences $\Delta \tilde \eta$ between one- and two-photon ionization channels, averaged over multiple wave packets, are extracted for neon $2p$ electrons as a function of emission angle at photoelectron energies 7.9, 10.2, and 16.6 eV. $\Delta \tilde \eta$ is nearly constant for emission parallel to the electric vector but increases at 10.2 eV for emission perpendicular to the electric vector. We model our observations with both perturbation and \textit{ab initio} theory, and find excellent agreement. In the existing method for attosecond measurement, Reconstruction of Attosecond Beating By Interference of Two-photon Transitions (RABBITT), a phase difference between two-photon pathways involving absorption and emission of an infrared photon is extracted. Our method can be used for extraction of a phase difference between single-photon and two-photon pathways and provides a new tool for attosecond science, which is complementary to RABBITT.