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Journal of Physics B Atomic and Molecular Physics

Theory of resonant two-photon ionisation

W A McClean, S. Swain

Published May 28, 1978• 24 citations
Publisher

Abstract

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.

DOI
10.1088/0022-3700/11/10/010

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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.

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Use of the Green's function formalism in resonant two-photon ionisation of alkali-metal atoms. II. Two-photon ionisation of Cs near the 7P1/2-7P3/2resonances

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For pt.I see ibid., vol. 17, p.4643 (1984). The authors apply the theoretical results obtained in a previous paper in order to study the two-photon resonant ionisation of atomic caesium in the neighbourhood of the fine-structure doublet 7P 1/2 -7P 3/2 when the incident light is right-hand circularly polarised. This formalism allows one to consider these two resonant states on an equal footing and it includes the effects of the infinite summations involved in higher-order perturbation theory. The numerical results show that the two-photon resonant ionisation of caesium takes place through an anticrossing of levels, at least in the intensity range studied. The temporal behaviour of the probability exhibits two distinct temporal regimes: linear for long interaction times and cubic for the shorter ones. The intermediate regime is characterised by quantum beats. At high intensity values, it is found that the induced light shift of the resonances is time dependent. It is also shown that the contribution of the higher-order effects may explain the slight departures between the theoretical and experimental results in the off-resonant two-photon ionisation of atomic caesium.

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An unconventional perturbation treatment of the theory of two-photon absorption in a four-level atomic system close to two-photon resonance is developed on the basis of the optical Bloch equations. The results for the longtime populations of the intermediate and upper states are valid in low- and medium-intensity regimes. In both regimes the upper state population shows a lorentzian resonance centred on zero two-photon detuning with a width equal to the natural linewidth of the upper state. The intermediate state population shows a similar behaviour in the intermediate regime only, with no significant dependence on two-photon detuning in the low-intensity regime. However, in both regimes the population of the virtual intermediate state is at least comparable to that of the resonant upper state. The perturbation theory results have been checked by direct numerical calculations based on the Bloch equations. Numerical calculations confirm the presence of a high-intensity regime characterized by power shifts, broadening and saturation effects.

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Two-photon spectroscopy. III. General strong-field aspects

Rainer Salomaa

For pt.II see ibid., vol.9, p.1221 (1976). Discusses the steady-state response of a three-level atom in two-photon absorption. The applied laser fields can have arbitrary intensities (within the rotating-wave approximation (RWA)). A general expression is derived for the population excited on the uppermost energy level. Two special cases which allow a trivial Doppler averaging are treated in detail. For an off-resonant intermediate state the two-photon spectrum is a simple Lorentzian showing a power shift and modest cross power broadening. The same features are retained when several off-resonance intermediate levels are assumed. For very strong laser fields the Doppler shifts can be neglected in the first approximation. The corresponding spectra are analysed.

Journal of Physics B Atomic and Molecular Physics•Oct 21, 1977•53 citations
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.
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.
I present a perturbation theory of multiphoton ionization based on the resolvent operator. The theory is applied to a detailed study of two-photon ionization under near-resonance conditions. It is shown that a distinction must be made between linear level shifts and widths, and resonance shifts and widths which appear in the transition probability. The resonance shifts and widths exhibit saturation effects and their values tend to be much smaller than those of the linear level shifts and widths. Using numerical examples corresponding to tunable dye lasers and alkali-metal atoms, we show that the above shifts can have a dramatic effect on the ionization rate even when the laser is not in exact resonance with the intermediate level. The question of "one-step" vs "two-step" two-photon ionization is also discussed. It is shown that exact resonance does not necessarily imply a two-step process. In fact, in most cases it is quite the opposite. A quantitative criterion for this distinction is established. Finally, a distinction is made between instrumental saturation and saturation of the transition probability. The two are shown to be of different nature. Further extensions of the work are also discussed.
For pt.I see ibid., vol. 17, p.4643 (1984). The authors apply the theoretical results obtained in a previous paper in order to study the two-photon resonant ionisation of atomic caesium in the neighbourhood of the fine-structure doublet 7P 1/2 -7P 3/2 when the incident light is right-hand circularly polarised. This formalism allows one to consider these two resonant states on an equal footing and it includes the effects of the infinite summations involved in higher-order perturbation theory. The numerical results show that the two-photon resonant ionisation of caesium takes place through an anticrossing of levels, at least in the intensity range studied. The temporal behaviour of the probability exhibits two distinct temporal regimes: linear for long interaction times and cubic for the shorter ones. The intermediate regime is characterised by quantum beats. At high intensity values, it is found that the induced light shift of the resonances is time dependent. It is also shown that the contribution of the higher-order effects may explain the slight departures between the theoretical and experimental results in the off-resonant two-photon ionisation of atomic caesium.
An unconventional perturbation treatment of the theory of two-photon absorption in a four-level atomic system close to two-photon resonance is developed on the basis of the optical Bloch equations. The results for the longtime populations of the intermediate and upper states are valid in low- and medium-intensity regimes. In both regimes the upper state population shows a lorentzian resonance centred on zero two-photon detuning with a width equal to the natural linewidth of the upper state. The intermediate state population shows a similar behaviour in the intermediate regime only, with no significant dependence on two-photon detuning in the low-intensity regime. However, in both regimes the population of the virtual intermediate state is at least comparable to that of the resonant upper state. The perturbation theory results have been checked by direct numerical calculations based on the Bloch equations. Numerical calculations confirm the presence of a high-intensity regime characterized by power shifts, broadening and saturation effects.
For pt.II see ibid., vol.9, p.1221 (1976). Discusses the steady-state response of a three-level atom in two-photon absorption. The applied laser fields can have arbitrary intensities (within the rotating-wave approximation (RWA)). A general expression is derived for the population excited on the uppermost energy level. Two special cases which allow a trivial Doppler averaging are treated in detail. For an off-resonant intermediate state the two-photon spectrum is a simple Lorentzian showing a power shift and modest cross power broadening. The same features are retained when several off-resonance intermediate levels are assumed. For very strong laser fields the Doppler shifts can be neglected in the first approximation. The corresponding spectra are analysed.