A successive-approximation procedure for calculating multiphoton-transition amplitudes is presented. The theory, designed for use in the strong-field domain where ordinary perturbative methods are unsuitable, provides an extension of previous work along these lines by inclusion of the effects of free-free transitions, i.e., interactions with the field which take the atomic system from one continuum state to another. This approach is based on the introduction of an effective Hamiltonian which operates only in the subspace of states, degenerate or nearly degenerate, among which real transitions can take place. Contributions from virtual intermediate states are included in the structure of the effective potential. This approach provides the framework for the construction of models, defined by the approximations made for the effective potential, which build in the constraints of unitarity and which properly account for the effects of intermediate-state resonances. The present work is focussed on those new features of the problem which arise from the fact that both the electron and the residual atom to which it was originally bound can interact with the field in asymptotic states. To avoid complications associated with long-ranged Coulomb effects, the residual atom is assumed to be neutral. (The initial state is a negative ion.) An approximation procedure is described for calculating the field-modified asymptotic states appropriate for such a system. By way of an illustrative example a low-frequency approximation is given which leads to considerable simplification of the calculational procedure.
We outline a formalism and develop a computational procedure to treat the process of multiphoton ionization (MPI) of atomic targets in strong laser fields. We treat the MPI process nonperturbatively as a decay phenomenon by solving a coupled set of the integral Lippmann-Schwinger equations. As basic building blocks of the theory, we use a complete set of field-free atomic states, discrete and continuous. This approach should enable us to provide both the total and differential cross sections of MPI of atoms with one and two electrons. As an illustration, we apply the proposed procedure to a simple model of MPI from a square-well potential and to the hydrogen atom.
Analyses of the resonant multiphoton ionization of atoms require knowledge of ac Stark energy shifts and of multiphoton, bound-to-bound state, transition amplitudes. In this paper, we consider the three-photon photoionization of hydrogen atoms at frequencies that are at and surrounding the two-photon 1s to 2s resonance. AC energy shift sums of both the 1s and 2s states are calculated as a function of the laser frequency along with two-photon 1s ? 2s resonant transition amplitude sums. These quantities are calculated using an extended version of a method, which has often been employed in a variety of ways, of calculating these sums by expressing them in terms of solutions to a variety of differential equations that are derived from the different sums being evaluated. We demonstrate how exact solutions are obtained to these differential equations, which lead to exact evaluations of the corresponding sums. A variety of different cases are analysed, some involving analytic continuation, some involving real number analysis and some involving complex number analysis. A dc Stark sum calculation of the 2s state is carried out to illustrate the case where analytic continuation, pole isolation and pole subtraction are required and where the calculation can be carried out analytically; the 2s state, ac Stark shift sum calculations involve a case where no analytic continuation is required, but where the solution to the differential equation produces complex numbers owing to the finite photoionization lifetime of the 2s state. Results from these calculations are then used to calculate three-photon ionization probabilities of relevance to an analysis of the multiphoton ionization data published by Kyrala and Nichols (1991 Phys. Rev. A 44, R1450).
The construction of physical occupation and transition probabilities in non-Hermitian, effective-Hamiltonian models of coupled near-resonant discrete states which decay to a continuum is analyzed. We consider for definiteness the particular physical example of atomic multiphoton ionization. The possibility of finite duration and arbitrary modulation of this interaction, and of the subsequent ionization and/or decay, in such a system allows us to invoke "adiabatic switching" considerations in conventional
S-matrix theory. The effective Hamiltonian $\stackrel{^}{H}(t)$ is derived for both stationary and time-dependent Schr\"odinger equations. These derivations yield the same effective $\stackrel{^}{H}(t)$, and further reveal that this operator is associated with discrete-state-projected "incoming" scattering states. The Hermitian conjugate operator ${\stackrel{^}{H}}^{\ifmmode\dagger\else\textdagger\fi{}}(t)$ is similarly shown to be associated with discrete-state "outgoing" scattering states. This shows that effective-Hamiltonian theories are intrinsically
S-matrix theories. This fact, in turn, is employed to construct transition amplitudes. The possibility of resonance in the atom-field interaction requires that both the projected incoming states and projected outgoing states be employed in this construction. The projected incoming and outgoing states in the (discrete) bound space are quite conveniently described by effective time-dependent Schr\"odinger equations, supplemented by initial- and final-state boundary conditions. The fact of independent-exponential decay from each mode throughout the history of the interaction, for an arbitrary initial superposition state, in the adiabatic limit, suggests a practical construction for intermediate-time bound-state probabilities. This construction permits the formal definition of individual-state probabilities, which satisfy a generalized adiabatic theorem, and leads to the satisfying result that the total bound-state probability at all intermediate times is the sum of complex-mode probabilities. In marked contrast to the conventional norm-of-state definition of nonionization probability, this sum does not have oscillations at intermediate times, in the limit of adiabatic modulation of the interaction. Superposition-state probabilities, however, exhibit oscillations at intermediate times. The resulting non-Hermitian quantum dynamics is especially suited, and even essential, for an accurate description of near-resonance ionization. Precise matching of the time-dependent $\stackrel{^}{H}(t)$ and ${\stackrel{^}{H}}^{\ifmmode\dagger\else\textdagger\fi{}}(t)$ with bases of their instantaneous eigenstates is required by approximate unitarity in the discrete decaying space, for a consistent theory. We illustrate these considerations in a general way in the context of resolvent-operator techniques. The implementation of the theory in both its time-dependent and stationary-state formulations is presented in an Appendix, for the completely general two-state non-Hermitian Hamiltonian. The results of these formulations are found to agree, in the adiabatic limit, to all orders in the non-Hermitian interaction. The utility of direct diagonalization of the effective $\stackrel{^}{H}(t)$, advocated by Armstrong and Baker, is transparent. The overall context of interpretation of their earlier work is, however, significantly altered. Similar considerations apply for nonswitchable interactions, such as for
K-meson decays. These results represent a practical generalization of quantum mechanics to non-Hermitian systems. The utility, and the necessity, of such a generalization has considerable theoretical interest, and direct experimental implications.
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.
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.