Solutions of the time-dependent Schrödinger equation, in the semi-classical electric dipole approximation, are developed for the interaction of a two-level system, possessing both transition and diagonal dipole matrix elements, with applied static and sinusoidal electric fields. Exact solutions can only be obtained subject to conditions involving the orientations and strengths of the applied fields. These, and perturbative extensions, are used to discuss some of the features of multi-photon processes that depend on the existence of non-zero diagonal dipole matrix elements. Some of the results apply, in the limit of vanishing permanent moments, to the atomic two-level problem.
Solutions of the time-dependent Schrödinger equation, in the semi-classical electric dipole approximation, are developed for the interaction of a two-level system, possessing both transition and diagonal dipole matrix elements, with applied static and sinusoidal electric fields. Exact solutions can only be obtained subject to conditions involving the orientations and strengths of the applied fields. These, and perturbative extensions, are used to discuss some of the features of multi-photon processes that depend on the existence of non-zero diagonal dipole matrix elements. Some of the results apply, in the limit of vanishing permanent moments, to the atomic two-level problem.
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.
The consequences of having state-specific diagonal transition matrix elements on the interaction of a two-level system with a laser field and a static electric field are examined within the dipole and rotating-wave approximations. A finite difference between the permanent dipole moments can result in the appearance of subsaturating ancilliary fringes flanking the usual Lorentzian transition resonance. The requisite conditions for a two-level system to exhibit an oscillatory line shape are discussed. These fringes arise from the essential transparency of the system to the field at prescribed intensity-dependent frequencies.
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.
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.
A solution for the time-dependent Schrödinger equation is given for an N-level system interacting with a sinusoidal field of arbitrary amplitude , frequency v and phase δ, in the presence of an applied static field of arbitrary strength . The solution is obtained by a modification of a previous exact method of solution for and retains the essential analytic features of the exact wavefunction for the problem. The Floquet form of the solution yields (a) the location and the widths of allowed transitions and the location of forbidden transitions, by use of characteristic exponent plots, but without evaluating the spectrum, and (b) convenient expressions for the steady-state induced transition probabilities, including the effects of a uniform relaxation mechanism, that only require the solution of the wave equation over the initial period of the hamiltonian. The approach is equally applicable to Stark (or Zeeman) frequency sweep or tuning problems. The ability to incorporate the effects of static fields in the solution is important since a variety of problems (e.g. anti-crossing spectroscopy) involve the application of such fields to a level configuration interacting with a sinusoidal field. As a detailed specific example, the multi-photon steady-state induced transition probabilities and characteristic exponents are studied, as a function of frequency v, for the unperturbed and the perturbed two-level system under saturating conditions. It is also demonstrated, for the coupling strengths of interest in multi-photon calculations, that the method of solution discussed here is a viable unified alternative to dressed-atom techniques.
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.