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Journal of Physics B Atomic Molecular and Optical Physics

Analytic formulae for occupation probabilities of atomic states in strong short laser pulses

Darko Dimitrovski, Jendrik Poloczek, J S Briggs

Published Jul 6, 2006• 9 citations
Publisher

Abstract

Explicit analytic expressions, involving elementary functions, are given for occupation probabilities of atomic states after one or a few cycles of a strong short laser pulse. These formulae are strictly valid in the asymptotic limit of large momentum transfer from the field to the atom during a half cycle of the pulse. The cases of a hydrogenic atom, of a negative ion and of multielectron atoms in the independent-electron approximation are considered. The asymptotic expressions, derived in the limit of infinite momentum transfer from the field, have also the correct limit for vanishingly small momentum transfers, so that fortuitously they are valid for arbitrary momentum transfer.

DOI
10.1088/0953-4075/39/14/009

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Explicit analytic expressions, involving elementary functions, are given for occupation probabilities of atomic states after one or a few cycles of a strong short laser pulse. These formulae are strictly valid in the asymptotic limit of large momentum transfer from the field to the atom during a half cycle of the pulse. The cases of a hydrogenic atom, of a negative ion and of multielectron atoms in the independent-electron approximation are considered. The asymptotic expressions, derived in the limit of infinite momentum transfer from the field, have also the correct limit for vanishingly small momentum transfers, so that fortuitously they are valid for arbitrary momentum transfer.
We treat the interaction of an atom with a short intense few-cycle laser pulse by the use of the Magnus expansion of the time-evolution operator. Terms of the Magnus expansion up to the third order in the pulse duration are evaluated explicitly, and expressions for the transition probability between atomic eigenstates are given. The transition operators up to this order are particularly simple, involving only momentum and position shifts arising from the laser field alone. The connection to other existing approaches is established. For a variety of generic short-pulse forms the results of the Magnus approximation are in excellent agreement with time-dependent transition probabilities obtained from accurate ab initio numerical calculations. However, the limitation of the Magnus expansion for pulses having both vanishing momentum and position shifts is demonstrated also.
Abstract We consider the interaction of atomic hydrogen, in its ground state, with an electromagnetic pulse whose duration is fixed in terms of the number of optical cycles. We study the probability of excitation of the atom in the static field limit i.e. for field frequencies going to zero. Despite the fact that the well-known Born–Fock adiabatic theorem is valid only for a system whose energy spectrum is discrete, we show that it is still possible to use this theorem to derive, in the low frequency limit, an analytical formula which gives the probability of transition to any excited state of the atom as a function of the field intensity, the carrier envelope phase and the number of optical cycles within the pulse. The results for the probability of excitation to low-lying excited states, obtained with this formula, agree with those we get by solving the time-dependent Schrödinger equation. The domain of validity is discussed in detail.
Classical, quantum-mechanical, and semiclassical expressions for the transition probability in one-dimensional Rydberg atom irradiated by short half-cycle pulse are derived and compared. The simple formulas obtained for excitation of Rydberg atom by two time delayed weak half-cycle pulses reproduce well the experimental data and the solutions of time-dependent Schr\"odinger equation. When the transferred momenta are stronger and positive, the transition probabilities exhibit fast oscillations with time delay between the pulses. The classical transition probability is constant in time. For negative transferred momenta a focusing phenomenon is observed, and there is a region in time delay, where the transition probabilities oscillate with the Kepler period.
Based on the results of a previous communication [Dimitrovski et al., Phys. Rev. Lett. 93, 083003 (2004)], we study ionization and excitation of a hydrogenic atom from the ground and first excited states in short electric field pulses of several cycles. A process of ionization and recombination which occurs periodically in time is identified, for both small and extremely large peak electric field strengths. In the limit of large electric peak fields closed-form analytic expressions for the population of the initial state after single- and few-cycle pulses are derived. These formulas, strictly valid for asymptotically large momentum transfer from the field, give excellent agreement with fully numerical calculations for all momentum transfers.
We develop an entirely quantum mechanical analytical description of scattering of atoms with angular momenta j g →j→j e =j (j is an integer) by a pulsed σ +−σ − field. In the stationary-atom approximation with exact accounting for recoil effects, we solve the problem of the change in the distribution of atoms among the internal and translational degrees of freedom initiated by a single pulse for j g =1, 2. We find in analytical form recurrence formulas that make it possible to calculate the distribution of the atoms after an arbitrary sequence of pulses has acted on the system. We show that for discrete (resonant) values of the time interval between the pulses, the action of N pulses leads to effective formation and narrowing of peaks at discrete points in momentum space and to a broadening of the envelope of these peaks. In the case of a broad initial momentum distribution we derive explicit formulas for the peaks and the envelope and study their asymptotic behavior for N≫1. Finally, in the weak-field limit we study numerically the dependence of the contrast of the scattering diagram on pulse length.
Using the Magnus expansion to the fourth order, we obtain analytic expressions for the atomic state of a two-level system driven by a laser pulse of arbitrary shape with small pulse area. We also determine the limitation of our obtained formulas due to limited range of convergence of the Magnus series. We compare our method to the recently developed method of Rostovtsev et al. (PRA 2009, 79, 063833) for several detunings. Our analysis shows that our technique based on the Magnus expansion can be used as a complementary method to the one in PRA 2009.