The assumed existence of a time-independent Kubo-type operator renders possible an analytic expression for the evolution operator of a quantized system. For a multilevel system dipole interacting with a continuous-wave laser, the realization of a suitable Kubo-type transformation through an algebraic algorithm may require that the Hamiltonian operator of the system be simplified through, say, the imposition of the rotating-wave approximation and/or the appeal to transition dipole selection rules that validate the neglect of unimportant couplings. The resultant sparse Hamiltonian operator is arbitrarily constructed on the basis of the near-resonant laser coupling of the ground state to the excited states, but without direct appeal to the magnitude of the dipole coupling strength parameters. The implementation and associated computational aspects of a resolvent method for evaluating the evolution operator is discussed and illustrated through the calculation of the transition probabilities for the excitation of an anharmonic oscillator model of the CH stretch in ${\mathrm{CD}}_{3}$H.
The assumed existence of a time-independent Kubo-type operator renders possible an analytic expression for the evolution operator of a quantized system. For a multilevel system dipole interacting with a continuous-wave laser, the realization of a suitable Kubo-type transformation through an algebraic algorithm may require that the Hamiltonian operator of the system be simplified through, say, the imposition of the rotating-wave approximation and/or the appeal to transition dipole selection rules that validate the neglect of unimportant couplings. The resultant sparse Hamiltonian operator is arbitrarily constructed on the basis of the near-resonant laser coupling of the ground state to the excited states, but without direct appeal to the magnitude of the dipole coupling strength parameters. The implementation and associated computational aspects of a resolvent method for evaluating the evolution operator is discussed and illustrated through the calculation of the transition probabilities for the excitation of an anharmonic oscillator model of the CH stretch in CD3H.
An algorithm [G. F. Thomas and W. J. Meath, J. Phys. B 16, 951 (1983)] for evaluating the evolution operator of a multilevel system dipole interacting with either a continuous-wave or Gaussian-pulsed source, based on its Riemann product integral representation in conjunction with Frazer's method of mean coefficients, is appraised and extended. Our extension allows for the inclusion of spontaneous decay or a dissociation (or ionization) channel and useful relations are derived for the eigenvalues of the evolution operator for both the continuous wave and the Gaussian pulse. For the latter we now admit an arbitrary phase in the field and we derive a simple relation for cascading the state amplitudes of the system over the duration of an arbitrary number of identical phase-coherent pulses. As part of our appraisal, improvements in the implementation of the algorithm are provided. For a two-level system, the evolution operator is given as the chronologically ordered product of readily computed 2\ifmmode\times\else\texttimes\fi{}2 matrices while for multilevel system it is conveniently evaluated using a resolvent method based on Leverrier's algorithm for computing the Bateman matrices. The asymptotic behavior of the evolution operator and the fluence in the direction of propagation for both the continuous-wave and the Gaussian-pulsed sources are explored. Applications to model two-level systems illustrate the influence of the source's phase and interaction duration on resonant and nonresonant transition probabilities, respectively. The saturation of a two-level system by a train of resonant phase-coherent Gaussian pulses is demonstrated.
An algorithm for calculating the time-resolved evolution operator of a multilevel system dipole interacting with a Gaussian-shaped laser pulse is developed through use of a compound Magnus expansion in compliance with Maricq's convergence criterion. In conjunction with the Leverrier-Bateman resolvent method, the algorithm allows the frequency spectrum for the interaction of the multilevel system with a train of identical phase-coherent Gaussian-shaped laser pulses to be readily evaluated. Furthermore, the algorithm renders possible the expeditious averaging of operator expectation values over the spectral bandwidth of the laser field. A prescription for gauging the accuracy of the algorithm is provided.
Transition probabilities between quantum states of a molecular system interacting with a low intensity time dependent field were obtained. The molecular system is modeled as a quartic anharmonic oscillator, whereas the interaction with the electric field is described by the electric-dipole approximation. We apply Lie algebraic techniques and make use of iterative Bogoliubov transformations to derive an approximate expression for the evolution operator. The methodological procedure proposed by Yuen allows us to evaluate the final time evolution operator in a closed for, using the optical properties of the two- photon coherent state of the radiation field.
The interaction is studied of a Gaussian pulsed or a continuous wave (CW) electric field with a molecule that has a nonzero difference d between the permanent dipoles of the ground and excited states involved in a transition. The unusual spectral effects associated with the transition with d ≠ 0, relative to the often studied case with d = 0, have been investigated previously in the two-level approximation and for continuous wave lasers. The observation of these effects, including markedly increased periods of excited state populations and negative Bloch-Siegart shifts, will often require the intense fields associated with pulsed rather than purely continuous wave electric fields. Transitions induced by continuous wave and pulsed laser-‘permanent dipole molecule’ interactions, for high field intensities and varying pulse durations, are investigated with model calculations carried out using exact Floquet and symmetry-adapted Riemann product integral methods. These calculations illustrate how the increased period of the excited state populations for dipolar molecules, and the associated molecule-field couplings, can be probed by sampling the time-dependent populations of the excited states using ultrashort to short pulses. The discussion illustrates the predictive and interpretive importance of the analytical expression for the molecule-EMF coupling provided by the rotating wave approximation (RWA) with d ≠ 0. The examples considered show, for example, temporal beating effects between the field and the RWA or Rabi frequencies and their dependence on d.
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.
An efficient computational algorithm for evaluating the evolution operator, and related induced transition probabilities, for a multilevel system interacting with an amplitude-modulated sinusoidal field of arbitrary strength, modulating envelope, frequency and phase, is developed. The algorithm is based on the Riemann product integral representation of the evolution operator in conjunction with Frazer's method of mean coefficients and avoids the introduction of commonly used perturbative or rotating wave approximations. Model calculations for the interaction of a two-level system with a Gaussian pulse are presented and used to discuss the dependence of time-resolved and steady-state frequency sweep spectra on some of the pulse characteristics.