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Journal of Physics B Atomic Molecular and Optical Physics

Theory of single-photon bound - free transitions: extension of the pole approximation

Abraham G. Kofman

Published Nov 28, 1997• 15 citations
Publisher

Abstract

A theory of single-photon ionization, detachment, and dissociation of quantum systems (atom, molecule, ion, etc) is presented. A unified treatment of excitation from the ground and excited states is performed, via a consideration of the rotating- and counter-rotating-wave terms on equal footing. The familiar pole approximation (PA) is extended to obtain higher-order corrections, which allows one to establish the PA validity conditions and provide a unified description of the cases of above- and below-threshold excitation. As a specific example, photodetachment of negative ions is considered. An exact long-time solution is obtained for the general case. A familiar model of photodetachment is used to validate the present PA formalism. The latter has allowed us to derive the PA validity condition, which holds for all values of the parameters, involving quite different situations. The applicability of the existing versions of the rotating-wave approximation has been checked.

DOI
10.1088/0953-4075/30/22/013

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Photoassociation: From a quasicontinuum model to coherent matter optics

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We consider the process of photoassociation (PA) that occurs when a pair of colliding atoms interacts with a photon, thereby making a transition from the two-atom continuum to a bound molecular state. Such free-bound transitions are described by quantizing the molecule's dissociation continuum, and coupling each of these quasicontinuum (QC) levels to a similarly discrete molecular level. The correct free-atom results are recovered no later than at the end of a given calculation by taking the limit as the QC spacing tends to zero. From the quasicontinuum model, we are led to adapt the few-level and second-quantization methods of photon optics to analyze free-bound transitions. In particular, we establish the PA density parameter as nearly identical to the phase-space parameter as a Bose-Einstein condensate (BEC), illustrate the near-unit efficiency of one- and two-color photoassociation of a BEC, and develop coherent atom-molecule transitions as the matter optics version of second-harmonic generated photons. ^

•Oct 1, 1999•1 citation
A theory of single-photon ionization, detachment, and dissociation of quantum systems (atom, molecule, ion, etc) is presented. A unified treatment of excitation from the ground and excited states is performed, via a consideration of the rotating- and counter-rotating-wave terms on equal footing. The familiar pole approximation (PA) is extended to obtain higher-order corrections, which allows one to establish the PA validity conditions and provide a unified description of the cases of above- and below-threshold excitation. As a specific example, photodetachment of negative ions is considered. An exact long-time solution is obtained for the general case. A familiar model of photodetachment is used to validate the present PA formalism. The latter has allowed us to derive the PA validity condition, which holds for all values of the parameters, involving quite different situations. The applicability of the existing versions of the rotating-wave approximation has been checked.
Summary form only given. We have developed a general theory of single-photon bound-free transitions, which takes into account all the states of the quantum system of interest and treats the rotating- and counterrotating-wave terms on equal footing. With the help of the smoothed-dynamics approximation (SDA), which holds until the laser field is much weaker than the intramolecular field, a closed analytical result for the Laplace transform of the initial-state amplitude has been obtained. To perform the inverse Laplace transform, the pole approximation, extended to obtain higher-order corrections, has been employed.
The multiple-linear-time-scale method is used to construct a perturbation theory for NNN-level quantum systems subjected to time-dependent perturbations. The secular and small-denominator terms which plague conventional time-dependent perturbation theory are avoided; consequently, the theory is useful for treating long-term behavior and resonant interactions. The introduction of a self-energy operator allows the shifts in energy levels to be displayed explicitly. When the perturbation is the dipole interaction with an electromagnetic field, the successive approximations yield the well-known rotating-wave approximation, corrections due to counter rotating terms, and the Bloch-Siegert frequency shift. The formalism is applicable to arbitrary pulse shapes, provided that $\ensuremath{\parallel}{H}_{1}\ensuremath{\parallel}\ensuremath{\ll}\ensuremath{\hbar}\overline{\ensuremath{\omega}}$, where $\ensuremath{\parallel}{H}_{1}\ensuremath{\parallel}$ represents the inter-action strength and $\overline{\ensuremath{\omega}}$ is a characteristic frequency of the field. Rabi oscillations induced by multiphoton resonances are automatically included, and this effect is demonstrated in the case of a square pulse with frequency approximately one half the resonance frequency for a transition between two vibrational levels of a molecule. These calculations are compared to an exact numerical solution in order to find the limits of validity of the approximation. Even at high intensities ($I\ensuremath{\sim}{10}^{14}$ W/cm2{\mathrm{cm}}^{2}cm2) the shifted resonance frequency is quite accurate; however, the approximate and exact solutions are slightly out of phase, so that the approximate solution can only be trusted for a few Rabi cycles. For much lower intensities ($I\ensuremath{\sim}{10}^{10}$ W/cm2{\mathrm{cm}}^{2}cm2), the approximate solution is valid for many thousands of Rabi cycles.
Near-threshold single-photon photodetachment processes are considered for simple model systems representing negative ions exposed to monochromatic radiation. The model systems considered feature an initial discrete state, the photoelectron continuum, and in some cases an excited or autodetaching discrete state. Population trapping through the formation of a bound dressed state and the photoelectron spectra are examined. Two photoelectron spectra are defined : a primary spectrum which results from decay while the laser interaction is on and a secondary spectrum which results from the destabilization of the bound dressed state if the interaction is turned off again. Careful attention is given to the passage of the poles of the Green's operator through the low-energy boundary of a continuum, and it is concluded that such poles move nonmonotonically through the boundary of an s-wave continuum, but monotonically through the boundary of a p-wave or higher-l-wave continuum. Two example systems, one featuring a single discrete state and an s-wave continuum and one featuring two discrete states and a p-wave continuum, are considered in detail.
A general framework for description of quantum systems in strong external fields is established. The most important application of the theory is concerned with the multiphoton ionization of atoms and molecules. The adiabatic approximation is developed. The Hamiltonian of the system is diagonalized by a time-dependent unitary transformation, which generates an interaction energy vanishing for very slow fields. A general analysis of the photoionization rates is performed. A specific model is considered: one-dimensional motion in the δ-function potential and external harmonic electric field. This model allows the exact solution for time-dependent eigenvalues and eigenfunctions. These solutions are compared with the conventional free electron approach to the ionization states.
Detailed descriptions of one- and two-photon ionization are presented. We go beyond conventional perturbation theory to investigate the dynamics of both processes for a broad range of field intensities. The general theory of unstable decaying states is applied to the problem of one-photon ionization (the photoeffect). It is found that a non-trivial dependence of the transition dipole moment on the continuum state energy can lead to corrections to exponential decay, and even oscillations in the bound state population. As examples we consider the photoionization of hydrogen and atoms with autoionizing states. Only in the absence of autoionizing states corrections to exponential decay remain inconsequential for any experimentally relevant intensity. Our description of two-photon ionization is formulated in the Heisenberg picture. Besides the usual two-level atom projection operators we introduce explicitly the operators referring to the atomic continuum. By eliminating from the equations of motion the terms related to the electromagnetic self field as well as the terms referring to the continuum, transverse and longitudinal damping corresponding to spontaneous emission and photoionization is obtained. The use of field correlation functions allows us to account for the partial coherence of the applied field. We develop a technique to extract from the equations of motion a unique two-photon ionization rate. Such a characterization of the ionization process is not only instructive because of its simplicity, it is also in many cases completely satisfactory. The resulting general expression for the two-photon ionization rate is the central result of our treatment. From it we recover several familiar results in different limiting cases. In addition, our general rate expression provides the first description of the complete hierarchy of saturation effects that one should expect. We present a number of examples where we calculate the two-photon ionization rate as a function of the detuning, and where we emphasize the saturation and field correlation effects.
We consider the process of photoassociation (PA) that occurs when a pair of colliding atoms interacts with a photon, thereby making a transition from the two-atom continuum to a bound molecular state. Such free-bound transitions are described by quantizing the molecule's dissociation continuum, and coupling each of these quasicontinuum (QC) levels to a similarly discrete molecular level. The correct free-atom results are recovered no later than at the end of a given calculation by taking the limit as the QC spacing tends to zero. From the quasicontinuum model, we are led to adapt the few-level and second-quantization methods of photon optics to analyze free-bound transitions. In particular, we establish the PA density parameter as nearly identical to the phase-space parameter as a Bose-Einstein condensate (BEC), illustrate the near-unit efficiency of one- and two-color photoassociation of a BEC, and develop coherent atom-molecule transitions as the matter optics version of second-harmonic generated photons. ^