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Journal of Physics B Atomic and Molecular Physics

The time dependence of two-phonon ionisation

W A McClean, S. Swain

Published Jul 28, 1979• 13 citations
Publisher

Abstract

The authors treat a model for photoionisation in which the atom has many bound atomic levels and the laser may be either monochromatic or possess a finite bandwidth. The treatment is restricted to the case where the laser is not sufficiently intense to cause Stark splitting of the atomic energy levels. They obtain an expression for the two-photon ionisation probability as a function of time which covers both resonant and non-resonant excitation. They identify several time regimes where the probability takes on a particularly simple form. In particular they discuss the short time behaviour and the conditions under which a time-independent transition rate exists. Numerical plots of the ionisation probability against laser frequency for a wide range of pulse durations are given for the two-level model.

DOI
10.1088/0022-3700/12/14/013

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Australian Journal of Physics•Oct 1, 1995•0 citations
The authors treat a model for photoionisation in which the atom has many bound atomic levels and the laser may be either monochromatic or possess a finite bandwidth. The treatment is restricted to the case where the laser is not sufficiently intense to cause Stark splitting of the atomic energy levels. They obtain an expression for the two-photon ionisation probability as a function of time which covers both resonant and non-resonant excitation. They identify several time regimes where the probability takes on a particularly simple form. In particular they discuss the short time behaviour and the conditions under which a time-independent transition rate exists. Numerical plots of the ionisation probability against laser frequency for a wide range of pulse durations are given for the two-level model.
The author examines some simple models for multiphoton ionisation which can be constructed when the bound-free transition rates are much smaller than the bound-bound transition rates and considers how to proceed when this condition is relaxed. The author identifies a regime where the transition rate is time independent. If the laser has a sufficiently broad linewidth one can use a rate equation approach; this assertion is justified and applied to two-, three- and N-photon ionisation. If the laser is narrow-band, a density matrix method has to be used and the effects of different laser lineshapes are discussed. In three-photon ionisation the role of optical pumping effects is emphasised. Explicit expressions for the multiphoton ionisation probabilities are presented and finally some extensions of the method are briefly discussed.
A theoretical procedure is presented for the evaluation of resonant multiphoton ionisation rates for atoms interacting with intense monochromatic lasers. The essence of the procedure is to treat first the strong coupling by solution of the truncated Schrodinger equation to obtain a time-dependent superposition state. This is then used as the lowest order approximation to iterate the full time-dependent Schrodinger equation to obtain the transition amplitude to the continuum. A model-atom study of the time evolution of the ionisation probability is carried out and the approach to rate behaviour in the long-time limit is demonstrated. Ionisation rates are evaluated for a variety of resonant processes in H, Na and Cs involving one or two lasers, as a function of intensities and detunings. Comparison with experiment is still in a fragmentary stage, and some large discrepancies are found to exist.
The authors investigate two-photon ionisation using a generalised rate equation technique for a model in which the atom is assumed to have only two bound levels (the ground state and an excited state) together with a simple continuum of ionised states. Spontaneous emission is allowed from the excited to the ground state, and the laser is assumed to have a finite bandwidth due to phase diffusion. The photoelectron energy distribution is calculated; it is found that if the Rabi frequency exceeds the ionisation rate the energy spectrum is a sensitive function of the linewidth and spontaneous emission rate, whereas if the ionisation rate exceeds the Rabi frequency then the sensitivity of the energy spectrum to these parameters depends upon the initial state of the system. A quite small value of laser linewidth is found to considerably reduce the heights of peaks and increase the widths. Analytic and numerical results are presented.
A theory of two-photon ionisation of atomic systems is developed incorporating into the treatment details of the laser spectrum. A general expression for the time-integrated two-photon ionisation probability is obtained together with general expressions for the induced shifts and widths of the atomic levels. These expressions are valid near resonance, when in the ordinary perturbation treatment the energy denominators vanish. The general expressions are then evaluated for the special case of a Lorentzian laser spectrum, when simple analytic results are obtained. The treatment is not confined to the two-level atom or the rotating-wave approximation.
The theory of resonant and non-resonant three-photon ionisation of a multi-level atomic system illuminated by a not too intense source (in a sense which is defined precisely in the text) with a Lorentzian spectrum is presented. The following new results are obtained: (i) the authors find different expressions for the non-resonant ionisation probability under monochromatic and broad band excitation; (ii) in addition to the usual one- and two-photon resonances connecting the ground state to excited states, the resonance which occurs when the photon energy is equal to the difference in energy between two excited states is discussed; and (iii) the triple resonance (that is the resonance occurring when the conditions for a one-photon and two-photon resonance are simultaneously satisfied) is examined.
Perturbation theory of multiphoton ionisation due to a classical electromagnetic field is modified to allow for intermediate resonances with bound states. Complex energies, generally associated with resonances, do not enter into this formalism. For a monochromatic field of frequency u, a constant ionisation rate can then be defined unambiguously and only such continuum states are excited which correspond to the absorption of energy in integral multiples of hu. As an application, differential and total cross sections for the two-photon ionisation of hydrogen, for frequencies below the n = 3 resonance region, are obtained in closed form. Existing data for generalised cross sections, calculated numerically using the complex coordinate method, are in good agreement with the present results. Finally, the mean fractional ionisation resulting from a pulse of finite duration is estimated on the basis of the associated power spectrum. For short pulses, the time dependence of ionisation exhibits a departure from that expected of a time-independent rate.