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Physical Review A

Time-independent theory of multiphoton ionization of an atom by an intense field

R. M. Potvliege, Robin Shakeshaft

Published Nov 1, 1988• 83 citations
Publisher

Abstract

We formulate a computationally feasible method for calculating multiphoton ionization rates for atoms exposed to intense fields in the intensity regime where perturbation theory ceases to apply. The method is based on the time-independent picture of ionization, which starts with the Floquet ansatz. The question of the gauge of the radiation field is discussed in some detail. Various expressions for the ionization amplitude are derived from a variational principle, with the radiation field expressed in the velocity gauge. We consider in particular an approximation in which the wave vector developing from the initial state is replaced by a trial vector that is the Floquet expansion truncated just below the threshold for ionization, and in which the wave vector developing into the final state is replaced by a trial vector that is just the wave vector appearing in the Kroll-Watson low-frequency approximation in scattering theory. We have applied this approximation to hydrogen and we present some results for both nonresonant and resonant multiphoton ionization. We argue that the experimentally observed resonance structure in the above-threshold peaks of the ionization signal occurs through the electron jumping from one dressed-state energy-eigenvalue curve to another.

DOI
10.1103/physreva.38.4597

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We formulate a computationally feasible method for calculating multiphoton ionization rates for atoms exposed to intense fields in the intensity regime where perturbation theory ceases to apply. The method is based on the time-independent picture of ionization, which starts with the Floquet ansatz. The question of the gauge of the radiation field is discussed in some detail. Various expressions for the ionization amplitude are derived from a variational principle, with the radiation field expressed in the velocity gauge. We consider in particular an approximation in which the wave vector developing from the initial state is replaced by a trial vector that is the Floquet expansion truncated just below the threshold for ionization, and in which the wave vector developing into the final state is replaced by a trial vector that is just the wave vector appearing in the Kroll-Watson low-frequency approximation in scattering theory. We have applied this approximation to hydrogen and we present some results for both nonresonant and resonant multiphoton ionization. We argue that the experimentally observed resonance structure in the above-threshold peaks of the ionization signal occurs through the electron jumping from one dressed-state energy-eigenvalue curve to another.
The strong-field multiphoton ionization of atoms is considered and a theoretical approach dealing nonperturbatively with the radiation field formulated. The general computational scheme is the conventional perturbation theory, but the intermediate states are dressed by the field. We present in detail a method to dress the continuum states and to study the dipole transitions within the continuum. In the high-frequency domain, the proposed procedure rapidly converges over a wide range of field intensity and offers an interesting framework for calculating ionization rates for arbitrary numbers of absorbed (above-threshold) photons and field polarization.
An approximate non-perturbative theory of multiphoton ionization of a hydrogen atom by two strong radiation fields is outlined. The special case is then considered for when one of the fields is of low intensity and high frequency and the other field is of high intensity and low frequency. A set of representative calculations has been performed for this case; that is, (a) angular distributions for given numbers of absorbed and emitted low-frequency photons and different geometries; (b) total cross-sections versus the strong-field intensity; (c) angular distributions summed over all the low-frequency photon exchanges, showing a fourfold symmetry in a geometry in which each ionization channel exhibits, instead, only a twofold symmetry; (d) the average energy gained by the ejected electrons. Calculations are performed for a strong low-frequency field of arbitrary polarization. In many respects, the present calculations are expected to complement those of single-field multiphoton ionization. It is hoped that the results of the calculations reported here may prompt systematic measurements also in this new class of processes, as the predictions are well within the state of the art of present multiphoton-ionization experiments.
High-frequency Floquet theory (HFFT) is again applied to photoionization in a linearly polarized field of the hydrogen atom in its ground state. The new feature is the systematic use of a two-pole approximation to represent both the dressed potential (with two positive charges) and the coupling responsible for one-photon ionization (with two charges of opposite sign). Within such a scheme, both the initial bound and the final continuum states can be very accurately determined. The final continuum state is obtained by deriving, from Floquet theory, an equation involving a source term. The model accounts reasonably well, even at rather low frequencies, for the stabilization of the atom in an intense field and supports the claim that HFFT represents the photoionization rate with a golden rule expression involving dressed states.
A variational procedure is developed for the calculation of the multiphoton-ionization rate of an atom interacting with a superposition of two time-dependent electric fields, one weak and the other intense and slowly varying. Trial functions are chosen that take into account the effect of both the atomic binding potential and the external fields. The variational expression contains, as its leading term, an approximation for the transition amplitude that was derived previously. The correction term is carefully analyzed here with regard to its dependence on suitably defined frequency and intensity parameters. The analysis provides firmly based estimates of the order, relative to the leading term, of the error in the approximation. A sum rule is derived which, owing to the cancellation of higher-order correction terms, allows for an accurate representation, in a relatively simple form, of the total transition rate summed over the net number of photons exchanged with the low-frequency field. A modified final-state trial function is presented which may be used, without loss of accuracy, to account for the existence of scattering resonances with energies close to that of the emergent electron. The calculational procedure is tested numerically on a one-dimensional model problem; the results are consistent with the more formal analysis of the relative order of the correction term and the accuracy of the sum rule.
Formal time-independent scattering theory is applied to multiphoton ionization of atoms in intense electromagnetic fields. The quantized-field version of the Volkov solution makes this approach possible. With the electron-photon interaction in a monochromatic photon field, it is found that, in the nonrelativistic and large-photon-number limits, the final scattering state exists only in the special case in which the ponderomotive potential per unit photon energy is an integer; otherwise the final state vanishes. In the integer case the corresponding wave function reduces to a single Volkov function, multiplied by an overlap factor. A simple interpretation of this result is given, and some other consequences of this work are discussed.