A rotating-wave approximation (RWA) for the single-photon and multiphoton resonance profiles for a two-level molecule interacting with both sinusoidal and static electric fields is derived. Included is a useful expression for the coupling between the molecule and the applied sinusoidal field which exhibits the effects of both diagonal dipole matrix elements (permanent dipole moments) and the applied static electric field. Comparisons of the RWA results with exact two-level calculations provide qualitative information concerning the validity of the RWA as a function of molecular and field parameters. These examples are also used to illustrate the interpretive nature of the RWA results for the resonance profiles and some of the effects of permanent dipoles and static fields on the profiles. Of particular interest are examples involving Boltzmann rotationally averaged spectra as a function of temperature where it is shown that minima in the spectra occurring for high temperatures can be reduced or removed completely if the temperature or applied static field is reduced.
A rotating-wave approximation (RWA) for the single-photon and multiphoton resonance profiles for a two-level molecule interacting with both sinusoidal and static electric fields is derived. Included is a useful expression for the coupling between the molecule and the applied sinusoidal field which exhibits the effects of both diagonal dipole matrix elements (permanent dipole moments) and the applied static electric field. Comparisons of the RWA results with exact two-level calculations provide qualitative information concerning the validity of the RWA as a function of molecular and field parameters. These examples are also used to illustrate the interpretive nature of the RWA results for the resonance profiles and some of the effects of permanent dipoles and static fields on the profiles. Of particular interest are examples involving Boltzmann rotationally averaged spectra as a function of temperature where it is shown that minima in the spectra occurring for high temperatures can be reduced or removed completely if the temperature or applied static field is reduced.
An analytic expression for the absorption spectra of a two-level molecule or atom is derived, within the rotating wave approximation (RWA), which includes the effects of permanent dipole moments and static electric fields. The derivation is for the interaction of the system with a plane-polarized sinusoidal electromagnetic field (EMF) in the semi-classical electric dipole approximation. The RWA resonance profile, and a series of exactly calculated two-level model spectra, are used to investigate some single- and multi-photon spectral effects due to permanent dipoles and static fields, relative to the atomic problem (no permanent dipoles). These effects include the occurrence of even as well as odd photon transitions. Permanent dipole moments can cause narrowing of the resonances, oscillatory fringes around the resonances as a function of frequency, and decreases in the molecule-EMF coupling, relative to the atomic results. Comparisons with exact two-level model spectra are used to study the validity of the RWA results and indicate that several features, such as dynamic backgrounds and shifts of the resonance frequencies from the weak EMF limits of {dollar}\\omega\\sbsp{lcub}\\rm res{rcub}{lcub}\\rm N{rcub}{dollar} = {dollar}\\Delta{dollar}E/N, N = 1,2,3, dots, are missing in the RWA spectra as the coupling increases.;Perturbative corrections to the RWA absorption spectra and the full widths at half maxima for the resonances are derived, neglecting static fields, and used to investigate and explain the effects missing in the RWA; the RWA resonance profile is a zeroth plus first order result obtainable from a time-independent Floquet Hamiltonian secular equation. The usefulness and validity of the perturbative corrections are investigated and it is concluded that these corrections are not useful computationally past second order. However, the corrections are useful in understanding some of the deficiencies of the RWA. These include the shifts of the N-photon resonance positions to low frequency, with respect to {dollar}\\Delta{dollar}E/N, that can occur for molecules with non-zero permanent dipoles; the shift is always to high frequency for atoms. A series of exact model calculations, for "giant dipole" molecules, is also used to discuss the effects of permanent dipole moments on spectra and to establish comparisons with the recent literature on the subject.
Perturbative corrections are derived for the rotating-wave approximation (RWA) for the single-photon and multiphoton resonance profiles due to the interaction between a two-level molecule, with nonzero permanent dipoles, and a sinusoidal time-dependent electric field. The derivation is carried out through the use of a Floquet secular equation that introduces the effects of the permanent dipole moments into the problem through the argument, d\ensuremath{\cdot}e^scrE/\ensuremath{\omega}, of Bessel functions contained in molecule-laser coupling parameters; here d=${\ensuremath{\mu}}_{22}$-${\ensuremath{\mu}}_{11}$, where ${\ensuremath{\mu}}_{\mathrm{ii}}$ is the permanent dipole of molecular state i, and scrE, e^, and \ensuremath{\omega} are the field strength, direction of polarization unit vector, and the circular frequency of the electromagnetic field (EMF). Expansions for the N-photon resonance profiles and the associated resonance frequencies and full widths at half maximum, in powers of the couplings between the transition and permanent dipole moments and the applied EMF, are obtained by expanding the Bessel functions in the perturbative results and are compared with those in the literature (available mostly for d=0 only). The perturbative results, and a series of exactly calculated two-level model absorption spectra, are used to discuss (1) the ability of the explicit perturbative corrections to the RWA to explain the differences between exact and RWA resonance profiles (for example, the positive or negative Bloch-Siegert shifts and dynamic backgrounds absent in the RWA), (2) the usefulness of the perturbative corrections in general, and (3) the effects of permanent dipoles on single-photon and multiphoton absorption spectra.
The interaction of two continuous wave electric fields (lasers) with a two-level molecule, that has a nonzero difference d, between the permanent dipoles of the two states involved in a multiphoton two-color transition, is studied in the rotating wave approximation and by using exact methods. Analytical rotating wave approximation results for the time-dependent populations of the molecular states and for the resonance profiles are derived and used to help discuss the effects of permanent dipole moments in the two-color absorption problem. Several examples of multiphoton, two-color, two-level resonance profiles calculated in the rotating wave approximation and by using exact techniques, are employed for illustrative purposes. They are used to help quantify the conditions for the validity of the two-color rotating wave approximation, which are considerably more restrictive than the analogous conditions for the one-color rotating wave approximation with d≠0, and to illustrate the interpretive and predictive nature of the rotating wave approximation results. For example, there are many multiphoton, two-color transitions that are available if a molecule possesses permanent dipoles (d≠0). The analytic rotating wave approximation expression for the molecule-two laser coupling can often be used to reliably estimate the field parameters of the two lasers that are required to optimize a given transition relative to others.
The phase and the rotational averaging of single and multi-photon transition probabilities for the interaction of an atom or molecule with an applied sinusoidal field is discussed. It is shown that while the correct interval for carrying out the phase average is 2π in general, a phase interval of π is satisfactory under certain specified conditions. Illustrative two-level examples indicate that if these conditions are not satisfied, the two choices of phase interval can lead to significant differences in the phase averaged temporal transition probabilities which are subsequently essentially removed upon long-time averaging. A numerical scheme for evaluating rotational averaged transition probabilities is developed and illustrated for model calculations involving two-level molecular systems. The examples are chosen to illustrate the effects of rotational averaging, both when a molecule possesses non-zero diagonal dipole matrix elements (‘permanent dipole moments’) and when it does not, on single and multi-photon spectra. Often resonance profiles are sharpened, Bloch Siegert shifts and resonance maxima reduced, and the structure due to diagonal dipole matrix elemnts eliminated, upon rotational averaging. A recently developed rotating wave or Rabi type approximation is used to explain the effects of diagonal dipole matrix elements for both the fixed orientation and rotationally averaged spectra.
One of the purposes of this paper is to develop an analytical many-level generalized rotating wave approximation (GRWA), including the effects of permanent dipoles, for the excitation of many-level molecules through the simultaneous absorption of two photons. Included are expressions for the two-photon laser–molecule coupling C, and its two components Cd and Cv corresponding to the direct permanent dipole and the virtual state excitation mechanisms, respectively, and related observables such as the time-dependent populations of the initial and final states of the excitation process and resonance profiles. This GRWA treatment also includes an energy shift parameter e, which causes shifts in the position of the resonance energy as the laser intensity increases. The effects of permanent dipoles are very different in Cv and e versus Cd. These effects have been discussed previously for Cd using analytic two-level RWA approaches. The analytical results for Cv and e obtained here are new as is their use in discussing the influence of permanent dipoles in the parts of the two-photon excitation process involving virtual states. In the absence of permanent dipoles Cd is zero whereas Cv and e are not; they equal the corresponding perturbation theory results. The GRWA and perturbative results are related by Bessel function damping functions which, for dipolar molecules, damp out the divergence of the perturbative results as the laser intensity increases. Illustrative examples are given for a two-photon excitation involving a model chromophore which has a significant virtual state contribution through both the laser–molecule coupling and the energy shift parameter.
A rotating-wave approximation (RWA) is developed to describe the evolution of a two-level system, which has permanent dipole moments, interacting with a pulsed laser. Comparisons with exact calculations for one- and two-photon excitations involving the two lowest vibrational states of the ground electronic state of HeH+ are given to illustrate the validity of the RWA formula.