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Physical Review A

First iteration within the high-frequency Floquet theory of laser-atom interactions

M. Marinescu, Mihai Gavrila

Published Apr 1, 1996• 85 citations
Publisher

Abstract

The high-frequency Floquet theory (HFFT) of laser-atom interactions solves the space-translated version of the Schr\"odinger equation by an iterative procedure, leading to corrections of increasing order in the inverse photon energy. The lowest-order approximation (high-frequency limit) has been often evaluated before, but its accuracy at finite frequencies has not been established. To explore this issue we have computed the corrections yielded by the first-order iteration to the energy levels, and the ionization rates of a one-dimensional atomic model with a Gauss attractive potential. We have then compared, at frequencies above the field-free ionization potential \ensuremath{\Vert}${\mathit{W}}_{0}$\ensuremath{\Vert}, the HFFT results with those of a full Floquet calculation. We show that the agreement is substantially improved by the inclusion of the HFFT corrective terms. The agreement is good at all intensities for photon energies larger than several times \ensuremath{\Vert}${\mathit{W}}_{0}$\ensuremath{\Vert}. Even when the photon energy is marginally larger than \ensuremath{\Vert}${\mathit{W}}_{0}$\ensuremath{\Vert} the discrepancies vanish at sufficiently high intensities. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physreva.53.2513

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The high-frequency Floquet theory (HFFT) of laser-atom interactions solves the space-translated version of the Schr"odinger equation by an iterative procedure, leading to corrections of increasing order in the inverse photon energy. The lowest-order approximation (high-frequency limit) has been often evaluated before, but its accuracy at finite frequencies has not been established. To explore this issue we have computed the corrections yielded by the first-order iteration to the energy levels, and the ionization rates of a one-dimensional atomic model with a Gauss attractive potential. We have then compared, at frequencies above the field-free ionization potential \ensuremath{\Vert}W0{\mathit{W}}_{0}W0​\ensuremath{\Vert}, the HFFT results with those of a full Floquet calculation. We show that the agreement is substantially improved by the inclusion of the HFFT corrective terms. The agreement is good at all intensities for photon energies larger than several times \ensuremath{\Vert}W0{\mathit{W}}_{0}W0​\ensuremath{\Vert}. Even when the photon energy is marginally larger than \ensuremath{\Vert}W0{\mathit{W}}_{0}W0​\ensuremath{\Vert} the discrepancies vanish at sufficiently high intensities. \textcopyright{} 1996 The American Physical Society.
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.
In this chapter,we shall analyze the particular case of an atom interacting with a laser pulse whose duration is sufficiently long, so that the evolution of the atom in the laser field is adiabatic. When this condition is fulfilled, the atom can be considered to interact with a monochromatic laser field. As a consequence, the Hamiltonian of the system is periodic in time, and the Floquet theory [1] can be used to solve the time-dependent Schrödinger equation (TDSE) non-perturbatively. We begin in Section 4.1 by considering the Hermitian Floquet theory. We first derive the Floquet theorem for a monochromatic, spatially homogeneous laser field and show that the solutions of the TDSE correspond to dressed states having real quasi-energies , which can be obtained by solving an infinite system of time-independent coupled equations. We then generalize the Floquet theory to multicolor laser fields and to “non-dipole” laser fields which are not spatially homogeneous. In Section 4.2, the Floquet theory is applied to study the dynamics of a model atom having M discrete levels interacting with a monochromatic laser field. In this case, the coupling between the bound and continuum atomic states is neglected.We analyze the relationship between the Floquet theory and the rotating wave approximation, and examine the perturbative limit of the Floquet theory. We also consider the population transfer between Floquet dressed states.
We have made several different types of measurements of the three-photon ionization of Li produced by 3-ps laser pulses and describe the results using a Floquet picture. Over the photon frequency range 15 000 to 15 800 ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$, Li represents a strongly coupled three-state system with the 2s ground state coupled to the 2p and 3d states by one and two photons, respectively. Energy analysis of the photoelectrons allows the measurement of the intensity dependent shift of the 2s Floquet state during the laser pulse. The shift shows a strong frequency dependence that is not predicted by first-order perturbation theory. We have also measured the total ionization spectrum over several ranges of frequency, as well as the angular distribution of the ionization and the first above-threshold ionization peak for frequencies where the ground state is near resonance with the 4s and 4d excited states. Calculations based on the Floquet Hamiltonian indicate that all of these processes may be understood in terms of a Floquet description.
A general program has been developed to calculate multiphoton processes in atomic hydrogen to arbitrarily high order in perturbation theory. This program can be used to calculate quantities in intense field multiphoton processes, such as the generalized cross section of multiphoton absorption, the a.c. Stark effect (shift and the broadening of the level due to the field interaction), and the parameters in nonlinear optics such as the nonlinear index of refraction and the nonlinear susceptibility. Effective parametrization of high-order susceptibilities has been achieved over a wide range of frequencies, utilizing concepts from quantum defect theory to express the results of lengthy calculations in a compact form that also permits extrapolation across thresholds and resonance poles. Significant progress has been made in direct numerical solution of the time-dependent Schroedinger equation. In addition, contributions have been made in: understanding the role of spatial dimensionality in the solution of model systems (e.G., the one-dimensional delta-function potential); the development of a general R-matrix code for perturbative computations of multiphoton processes in many-electron atoms; and multiphoton ionization with two commensurate laser frequencies.
In systems of ultracold atoms, pairwise interactions are resonantly enhanced by the application of an oscillating magnetic field that is parallel to the spin-quantization axis of the atoms. The resonance occurs when the frequency of the applied field is precisely tuned near the transition frequency between the scattering atoms and a diatomic molecule. The resulting cross section can be made more than two orders of magnitude larger than the cross section in the absence of the oscillating field. The low momentum resonance properties have a universal description that is independent of the atomic species. To arrive at these conclusions, we first develop a formal extension of Floquet theory to describe scattering of atoms with time-periodic, short-range interaction potentials. We then calculate the atomic scattering properties by modeling the atomic interactions with a square well potential with oscillating depth and then explicitly solving the time-dependent Schrodinger equation. We then apply the Floquet formalism to the case of atoms scattering with a contact interaction described by a time-periodic scattering length, obtaining analytic results that agree with those obtained by solving the time-dependent Schrodinger equation.
The stabilization of a one-dimensional delta potential model atom under the influence of a high frequency, short laser pulse is investigated, through both an ab initio numerical integration of the time-dependent Schrödinger equation and a Floquet calculation in the single-dressed-state approximation. Provided the intensity is not too high, the variation of the photodetachment probability is explained by adiabatic stabilization and its oscillating behaviour by that of the Floquet rate of photodetachment.