Above-threshold ionization with highly-charged ions in super-strong laser fields: II. Relativistic Coulomb-corrected strong field approximation

We develop a relativistic Coulomb-corrected strong field approximation (SFA) for the investigation of spin effects at above-threshold ionization in relativistically strong laser fields with highly charged hydrogen-like ions. The Coulomb-corrected SFA is based on the relativistic eikonal-Volkov wave function describing the ionized electron laser-driven continuum dynamics disturbed by the Coulomb field of the ionic core. The SFA in different partitions of the total Hamiltonian is considered. The formalism is applied for direct ionization of a hydrogen-like system in a strong linearly polarized laser field. The differential and total ionization rates are calculated analytically. The relativistic analogue of the Perelomov-Popov-Terent'ev ionization rate is retrieved within the SFA technique. The physical relevance of the SFA in different partitions is discussed.Comment: 11 pages, 4 figure

The Wigner time delay for laser induced tunnel-ionization via the electron propagator

Recent attoclock experiments using the attsecond angular streaking technique enabled the measurement of the tunneling time delay during laser induced strong field ionization. Theoretically the tunneling time delay is commonly modelled by the Wigner time delay concept which is derived from the derivative of the electron wave function phase with respect to energy. Here, we present an alternative method for the calculation of the Wigner time delay by using the fixed energy propagator. The developed formalism is applied to the nonrelativistic as well as to the relativistic regime of the tunnel-ionization process from a zero-range potential, where in the latter regime the propagator can be given by means of the proper-time method.Comment: 5 pages, 4 figure

Above-threshold ionization with highly-charged ions in super-strong laser fields: I. Coulomb-corrected strong field approximation

Aiming at the investigation of above-threshold ionization in super-strong laser fields with highly charged ions, we develop a Coulomb-corrected strong field approximation (SFA). The influence of the Coulomb potential of the atomic core on the ionized electron dynamics in the continuum is taken into account via the eikonal approximation, treating the Coulomb potential perturbatively in the phase of the quasi-classical wave function of the continuum electron. In this paper the formalism of the Coulomb-corrected SFA for the nonrelativistic regime is discussed employing velocity and length gauge. Direct ionization of a hydrogen-like system in a strong linearly polarized laser field is considered. The relation of the results in the different gauges to the Perelomov-Popov-Terent'ev imaginary-time method is discussed.Comment: 8 pages, 3 figure

Coherent hard x-rays from attosecond pulse train-assisted harmonic generation

High-order harmonic generation from atomic systems is considered in the crossed fields of a relativistically strong infrared laser and a weak attosecond-pulse train of soft x-rays. Due to one-photon ionization by the x-ray pulse, the ionized electron obtains a starting momentum that compensates the relativistic drift which is induced by the laser magnetic field, and allows the electron to efficiently emit harmonic radiation upon recombination with the atomic core in the relativistic regime. In this way, short pulses of coherent hard x-rays of up to 40 keV energy and 10 as duration can be brought about

Phase-matched coherent hard x-rays from relativistic high-order harmonic generation

High-order harmonic generation (HHG) with relativistically strong laser pulses is considered employing electron ionization-recollisions from multiply charged ions in counterpropagating, linearly polarized attosecond pulse trains. The propagation of the harmonics through the medium and the scaling of HHG into the multi-kilo-electronvolt regime are investigated. We show that the phase mismatch caused by the free electron background can be compensated by an additional phase of the emitted harmonics specific to the considered setup which depends on the delay time between the pulse trains. This renders feasible the phase-matched emission of harmonics with photon energies of several tens of kilo-electronvolt from an underdense plasma

Under-the-barrier dynamics in laser-induced relativistic tunneling

The tunneling dynamics in relativistic strong-field ionization is investigated with the aim to develop an intuitive picture for the relativistic tunneling regime. We demonstrate that the tunneling picture applies also in the relativistic regime by introducing position dependent energy levels. The quantum dynamics in the classically forbidden region features two time scales, the typical time that characterizes the probability density's decay of the ionizing electron under the barrier (Keldysh time) and the time interval which the electron spends inside the barrier (Eisenbud-Wigner-Smith tunneling time). In the relativistic regime, an electron momentum shift as well as a spatial shift along the laser propagation direction arise during the under-the-barrier motion which are caused by the laser magnetic field induced Lorentz force. The momentum shift is proportional to the Keldysh time, while the wave-packet's spatial drift is proportional to the Eisenbud-Wigner-Smith time. The signature of the momentum shift is shown to be present in the ionization spectrum at the detector and, therefore, observable experimentally. In contrast, the signature of the Eisenbud-Wigner-Smith time delay disappears at far distances for pure quasistatic tunneling dynamics

Estimating the Price Elasticity of Demand for Water with Quasi Experimental Methods

There is a growing recognition in both the professional and popular literatures that water scarcity is a key policy issue that is especially important in arid, urban settings with the prospects for shortfalls in water availability due to the effects of climate change. Those evaluating these types of water problems usually conclude prices must be reformed so that incentives facing water users change to reflect this scarcity. Demand functions provide the basic economic relationships required to understand how water use will respond to such changes. This paper proposes a new method for estimating the price elasticity of demand that meets policy needs and can accommodate the presence of increasing block pricing structures.Water Demand Elasticity, Quasi Experiment, Climate Change, Consumer/Household Economics, Demand and Price Analysis, Environmental Economics and Policy, Resource /Energy Economics and Policy,