A general and computationally efficient method for averaging both the time-dependent and the steady-state atomic or molecular state populations over the phases, ${\ensuremath{\delta}}_{1}$ and ${\ensuremath{\delta}}_{2},$ of two continuous-wave laser fields involved in an excitation process is developed based on the Floquet formalism. Explicit calculations are presented for the coherent one- and three-photon electronic excitation of a two-level model molecule in order to illustrate the importance of phase averaging in situations where the relative phase difference between the two fields is fixed. While the explicit results involve electronic excitation, they are presented in reduced form so that they can be scaled to other regions of the electromagnetic spectrum and to other field strengths. The results have important implications in situations where the relative phase difference between two intense continuous-wave laser fields is used to control the excitation process.
A general and computationally efficient method for averaging both the time-dependent and the steady-state atomic or molecular state populations over the phases, \ensuremathδ1 and \ensuremathδ2, of two continuous-wave laser fields involved in an excitation process is developed based on the Floquet formalism. Explicit calculations are presented for the coherent one- and three-photon electronic excitation of a two-level model molecule in order to illustrate the importance of phase averaging in situations where the relative phase difference between the two fields is fixed. While the explicit results involve electronic excitation, they are presented in reduced form so that they can be scaled to other regions of the electromagnetic spectrum and to other field strengths. The results have important implications in situations where the relative phase difference between two intense continuous-wave laser fields is used to control the excitation process.
The absolute laser phase dependence of the time-dependent populations of the molecular states, including the steady-state (long time) populations of the states, associated with the interaction of a molecule with a pulsed laser is investigated using illustrative two-level examples. One-photon transitions, including the effects of permanent dipoles, are discussed as a function of the pulse duration, intensity, and (absolute) laser phase, for selected laser frequencies. The effects of laser phase can be large, depending on the values of the pulse duration for a given frequency and intensity. The effects of permanent dipoles, relative to no permanent dipoles, are significant for large laser field strengths ε0. When the laser-molecule coupling parameter b=μ12ε0/E21⩾0.2, where μ12 and E21 are the transition dipole and energy difference between the ground and excited states, respectively, the dynamics of the pulse-molecule interaction are (strongly) phase dependent, independent of pulse duration, whereas the corresponding steady-state populations of the molecular states may or may not be phase-dependent depending on the pulse duration. Analytical rotating wave approximations for pulsed laser-molecule interactions are useful for interpreting the dynamics and the steady-state results as a function of field strength and pulse duration, including the effects of permanent dipole moments. The results reported in this paper are based on molecular parameters associated with an S0→S1 electronic transition in a dipolar molecule. However, they are presented in reduced form and therefore can be scaled to other regions of the electromagnetic spectrum. Short, intense pulses at or beyond the limits of current laser technology will often be required for the types of absolute laser phase effects of this paper to be appreciable for electronic excitations. The discussion, in the UV-VIS, also suffers from the use of a two-level model and from the requirement of field intensities that can be beyond the Keldysh limit. For other spectral regions, these absolute laser phase effects will be much more readily applicable.
The effects of permanent dipole moments and those due to the randomness of molecular orientation in the phase control of molecular excitation are discussed for the simultaneous one- and three-photon excitation of a two-level model molecule. In this transition scheme both transitions can occur with or without the presence of permanent dipoles and the results are contrasted to those corresponding to the one- and two-photon excitation of a two-level molecule, which requires the presence of permanent dipoles. The dependence of the temporal evolution of the excited state and the associated resonance profiles on the relative phase of the lasers is used to monitor the control of the excitation process. Analytical perturbation theory, the rotating-wave approximation, and exact Floquet results for those observables are used for this purpose. Both fixed molecule-laser configurations and situations where the absorbing molecules assume random orientations with respect to the laser beams are considered. \textcopyright{} 1996 The American Physical Society.
The spectral and dynamic effects associated with a non-zero difference, d, between the permanent dipole moments of two molecular states involved in a transition, are investigated for a variety of electromagnetic field-molecule interactions. Both analytical and exact numerical techniques are used to solve the time-dependent Schrodinger equation and to obtain the temporal populations of the molecular states and the associated absorption spectra, for molecular interactions with a Gaussian pulsed, or one or two continuous wave (CW), laser(s).;Analytical rotating wave approximation results are derived for a single CW laser interacting with a two-level d {dollar}\\ne{dollar} 0 molecule with an excited state lifetime, and for a two-level d {dollar}\\ne{dollar} 0 molecule interacting with two CW lasers. A symmetry-adapted Riemann product integral method for exact calculations of pulsed laser-molecule interactions is developed and applied to specific problems. The Floquet technique for the treatment of periodic Hamiltonians is extended to accommodate the interaction of a molecule with two CW lasers.;The one-field RWA results are used to predict and interpret the effects of permanent dipoles on the temporal behaviour of the molecular states for a decaying two-level system interacting with a CW laser, and for a non-decaying two-level system interacting with a pulsed laser. It is shown that the excited state lifetime and the pulse duration can act as internal and external probes, respectively, of the temporal effects of permanent dipoles, relative to the often-studied d = 0 problem. The two-field analytical RWA solutions are used to help discuss and predict the effects of permanent dipole moments in two-laser multi-photon transitions. Several examples of multi-photon, two-colour, two-level resonance profiles, calculated in the RWA and by using exact Floquet techniques, are used to demonstrate how particular multi-photon transitions can be optimized, and to help quantify the conditions for the validity of the two-colour RWA. The effects of neighbouring energy levels on the transition of interest, for both probe and pump-probe laser-molecule interactions, are investigated using a series of three- to seven-level model molecular systems.
A rotating-wave approximation (RWA) is developed to describe the interaction of a two-level system, which has permanent dipole moments, with two pulsed lasers. The RWA expressions for the time-dependent populations of the molecular states are applied to model laser-molecule interactions and tested by comparison with exact results. The results are used to discuss the pulsed-laser phase control of molecular excitation through the interplay of competing one- and two-photon resonances involving the effects of a nonzero difference d between the permanent dipoles of the two states involved in the transition; the competition vanishes if d=0.
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.