High-harmonic generation - the emission of high-frequency radiation by the ionization and subsequent recombination of an atomic electron driven by a strong laser field - is widely understood using a quasiclassical trajectory formalism, derived from a saddle-point approximation, where each saddle corresponds to a complex-valued trajectory whose recombination contributes to the harmonic emission. However, the classification of these saddle-points into individual quantum orbits remains a high-friction part of the formalism. Here we present a scheme to classify these trajectories, based on a natural identification of the (complex) time that corresponds to the harmonic cutoff. This identification also provides a natural complex value for the cutoff energy, whose imaginary part controls the strength of quantum-path interference between the quantum orbits that meet at the cutoff. Our construction gives an efficient method to evaluate the location and brightness of the cutoff for a wide class of driver waveforms by solving a single saddle-point equation. It also allows us to explore the intricate topologies of the Riemann surfaces formed by the quantum orbits induced by nontrivial waveforms.
Starting from a time-dependent Schrödinger equation, stationary states of 3D central potentials are obtained. An imaginary-time evolution technique coupled with the minimization of energy expectation value, subject to the orthogonality constraint leads to ground and excited states. The desired diffusion equation is solved by means of a finite-difference approach to produce accurate wave functions, energies, probability densities and other expectation values. Applications in case of 3D isotropic harmonic oscillator, Morse as well the spiked harmonic oscillator are made. Comparison with literatu…
Moderate-terahertz-induced plateau expansion of high-order harmonic generation to soft X-ray region
Duy Tan University（越南科技大学）；Van Lang University（文郎大学）；Ton Duc Thang University（胡志明市铁城大学）；Ho Chi Minh City University of Education（胡志明市教育大学）；M. V. Lomonosov Moscow State University（莫斯科国立罗蒙诺索夫大学）
We study the Schrödinger operator with a potential given by the sum of the potentials for harmonic oscillator and imaginary cubic oscillator and we focus on its pseudospectral properties. A summary of known results about the operator and its spectrum is provided and the importance of examining its pseudospectrum as well is emphasized. This is achieved by employing scaling techniques and treating the operator using semiclassical methods. The existence of pseudoeigenvalues very far from the spectrum is proven, and as a consequence, the spectrum of the operator is unstable with respect to small p…
Two-Electron Effects Extend High-Harmonic Generation into the keV Regime
ICFO – Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology（ICFO-光子科学研究所，巴塞罗那科学技术研究所）；Department of Physics, Guangdong Technion - Israel Institute of Technology（广东以色列理工学院物理系）；Technion – Israel Institute of Technology（以色列理工学院）；Department of Physics and Astronomy, University College London（伦敦大学学院物理与天文学系）；Atominstitut, Technische Universität Wien（维也纳工业大学原子物理研究所）；Foundation for Research and Technology-Hellas, Institute of Electronic Structure & Laser（希腊研究与技术基金会电子结构与激光研究所）；Center for Quantum Science and Technologies (FORTH-QuTech)（量子科学与技术中心（FORTH-QuTech））；ELI-ALPS, ELI-Hu Non-Profit Ltd.（ELI-ALPS，匈牙利ELI非营利有限公司）
We present theoretical studies of high-order harmonic generation (HHG) produced by non-homogeneous fields as resulting from the illumination of plasmonic nanostructures with a short laser pulse. We show that both the inhomogeneity of the local fields and the confinement of the electron movement play an important role in the HHG process and lead to the generation of even harmonics and a significantly increased cutoff, more pronounced for the longer wavelengths cases studied. In order to understand and characterize the new HHG features we employ two different approaches: the numerical solution o…
We formulate the Born approximation for finding resonance poles in the complex plane for potential scattering problems. Using the method, we study the distribution of resonance poles for several scattering potentials. In particular, we find for an exponential potential with a cutoff that the cutoff generates an infinite series of extra resonance poles below and along the real axis, which would not exist without the cutoff. We also find for a Gaussian potential that the series of resonance poles approach the imaginary axis of the complex energy plane from left. In other words, the real parts of…
High-order harmonic generation (HHG) from molecular ion HeH$^{2+}$ in two initial states of $1sσ$ (ground state) and $2pσ$ (first excited state) is investigated theoretically, in homogeneous and plasmonic-enhanced laser fields. The electron ionization and the enhancement of HHG yield in ground and first excited states, as the initial states, is studied and a strong orientation effect for ionization of electron in $2pσ$ state is discovered, that it can be used to reach a longer lifetime for the electron in the first excited state, which is responsible for the enhancement in HHG yield, during th…
We find that a donor-doped band-gap material can enhance the overall high-order harmonic generation (HHG) efficiency by several orders of magnitude, compared with undoped and acceptor-doped materials. This significant enhancement, predicted by time-dependent density functional theory simulations, originates from the highest-occupied impurity state which has an isolated energy located within the band gap. The impurity-state HHG is rationalized by a three-step model, taking into account that the impurity-state electron tunnels into the conduction band and then moves according to its band structu…
We consider a quantum mechanical model for the high-order harmonic generation in bulk solids. The bandgap is assumed to be considerably larger than the exciting photon energy. Using dipole approximation, the dynamical equations for different initial Bloch states are decoupled in the velocity gauge. Although there is no quantum mechanical interference between the time evolution of different initial states, the complete harmonic radiation results from the interference of fields emitted by all the initial (valence band) states. In particular, this interference is shown to be responsible for the s…
We consider two ultracold particles confined in spherically symmetric harmonic trap and interacting via isotropic potential with absorbing boundary conditions at short range that models reactive scattering. First, we apply the contact pseudopotential with complex scattering length and investigate the properties of eigenergies and eigenfunctions for different trap states as a function of real and imaginary part of the scattering length. In the analyzed case the eigenenergies take complex values, and their imaginary part can be interpreted in terms of decay rates. Then we introduce the model of …
We investigate high-order harmonic generation in a solid, modeled as a multi-level system dressed by a strong infrared laser field. We show that the cutoff energies and the relative strengths of the multiple plateaus that emerge in the harmonic spectrum can be understood both qualitatively and quantitatively by considering a combination of adiabatic and diabatic processes driven by the strong field. Such a model was recently used to interpret the multiple plateaus exhibited in harmonic spectra generated by solid argon and krypton [Ndabashimiye {\it et al.}, Nature 534, 520 (2016)]. We also sho…
The quantum harmonic oscillator with parity-time ($\mathcal{PT}$) symmetry, obtained from the ordinary (Hermitian) quantum harmonic oscillator by an imaginary displacement of the spatial coordinate, provides an important and exactly-solvable model to investigate non-Hermitian extension of the Ehrenfest theorem. Here it is shown that transverse light dynamics in an optical resonator with off-axis longitudinal pumping can emulate a $\mathcal{PT}$-symmetric quantum harmonic oscillator, providing an experimentally accessible system to investigate non-Hermitian coherent state propagation.
We show that and how the Coulomb potential can be regularized and solved exactly at the imaginary couplings. The new spectrum of energies is real and bounded as expected, but its explicit form proves totally different from the usual real-coupling case.
High-harmonic generation by a laser-driven solid slab is simulated using time-dependent density functional theory. Multiple harmonic plateaus up to very high harmonic orders are observed already at surprisingly low field strengths. The full all-electron harmonic spectra are, in general, very different from those of any individual Kohn-Sham orbital. Freezing the Kohn-Sham potential instead is found to be a good approximation for the laser intensities and harmonic orders considered. The origins of the plateau cutoffs are explained in terms of band gaps that can be reached by Kohn-Sham electrons …
We study high-order harmonic generation (HHG) resulting from the illumination of plasmonic nanostructures with a short laser pulse. We show that both the inhomogeneities of the local electric field and the confinement of the electron motion play an important role in the HHG process and lead to a significant increase of the harmonic cutoff. In order to understand and characterize this feature, we combine the numerical solution of the time dependent Schroedinger equation (TDSE) with the electric fields obtained from 3D finite element simulations. We employ time-frequency analysis to extract more…
An imaginary-time nonuniform mesh method is presented and used to find the first 50 eigenstates and energies of up to five strongly interacting spinless quantum Lennard-Jones particles trapped in a one-dimensional harmonic potential. We show that the use of tailored grids reduces drastically the computational effort needed to diagonalize the Hamiltonian and results in a favorable scaling with dimensionality. Solutions to both bosonic and fermionic counterparts of this strongly interacting system are obtained, the bosonic case clustering as a Tonks-Girardeau crystal exhibiting the phenomenon of…
Quantum probing consists of suitably exploiting a simple, small, and controllable quantum system to characterize a larger and more complex system. Here, we address the estimation of the cutoff frequency of the Ohmic spectral density of a harmonic reservoir by quantum probes. To this aim, we address the use of single-qubit and two-qubit systems and different kinds of coupling with the bath of oscillators. We assess the estimation precision by the quantum Fisher information of the sole quantum probe as well as the corresponding quantum signal-to-noise ratio. We prove that, for most of the values…
In strong field laser physics it is a common practice to use the high-order harmonic cutoff to estimate the laser intensity of the pulse that generates the harmonic radiation. Based on the semiclassical arguments it is possible to find a direct relationship between the maximum value of the photon energy and the laser intensity. This approach is only valid if the electric field driving HHG is spatially homogenous. In laser-matter processes driven by plasmonics fields, the enhanced fields present a spatial dependence that strongly modifies the electron motion and consequently the laser driven ph…
We have simulated the ground states of quantum harmonic oscillators driven either by constant forces of different magnitudes or time-dependent driving forces. The expectation values of position for various combinations of mass, natural angular frequency, and the coupling constant $λ$ were calculated for both driving modes. For constant forcing, coherent states were obtained. The results for both forcing scenarios match the theoretically expected values almost exactly. For the simulations, the Metropolis algorithm was implemented on a discrete time lattice to evaluate the imaginary time path in…
Quantum confinement is studied by numerically solving time-dependent Schrödinger equation. An imaginary-time evolution technique is employed in conjunction with the minimization of an expectation value, to reach the global minimum. Excited states are obtained by imposing the orthogonality constraint with all lower states. Applications are made on three important model quantum systems, namely, harmonic, repulsive and quartic oscillators; enclosed inside an impenetrable box. The resulting diffusion equation is solved using finite-difference method. Both symmetric and asymmetric confinement are c…
Introduction Path Integrals - Introduction - Propagator - Free Particle - Path Integral Representation of Quantum Mechanics - Particle on a Ring - Particle in a Box - Driven Harmonic Oscillator - Semiclassical Approximation - Imaginary Time Path Integral Dissipative Systems - Introduction - Environment as Collection of Harmonic Oscillators - Effective Action Damped Harmonic Oscillator - Partition Function - Ground State Energy and Density of States - Position Autocorrelation Function
