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Physical Review A

Coherent dynamics of N -level atoms and molecules. II. Analytic solutions

Zofia Białynicka-Birula, Iwo Białynicki‐Birula, Joseph H. Eberly, Bruce W. Shore

Published Nov 1, 1977• 108 citations
Publisher

Abstract

We show that a class of multiphoton-absorption problems has exact solutions. In these problems an $N$-level atom or molecule absorbs energy from near-resonance monochromatic laser fields, and one asks for the time-dependent populations of the energy levels of the absorber. We find that the equal-Rabi and the harmonic-Rabi $N$-level systems studied numerically in the preceding paper are associated with Chebyshev and Hermite polynomials. Using the exact solutions we find exact expressions for the time averages of occupation probabilities in terms of sums of squares of polynomials. In addition we find the first approximate analytic expression for the long-period population oscillations observed in numerical solutions; and we show that a steady-state population inversion between the highest and lowest levels may be maintained by multiphoton pumping.

DOI
10.1103/physreva.16.2048

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arXiv:1404.1248•Mar 31, 2014•0 citations
We show that a class of multiphoton-absorption problems has exact solutions. In these problems an NNN-level atom or molecule absorbs energy from near-resonance monochromatic laser fields, and one asks for the time-dependent populations of the energy levels of the absorber. We find that the equal-Rabi and the harmonic-Rabi NNN-level systems studied numerically in the preceding paper are associated with Chebyshev and Hermite polynomials. Using the exact solutions we find exact expressions for the time averages of occupation probabilities in terms of sums of squares of polynomials. In addition we find the first approximate analytic expression for the long-period population oscillations observed in numerical solutions; and we show that a steady-state population inversion between the highest and lowest levels may be maintained by multiphoton pumping.
The multiple-linear-time-scale method is used to construct a perturbation theory for NNN-level quantum systems subjected to time-dependent perturbations. The secular and small-denominator terms which plague conventional time-dependent perturbation theory are avoided; consequently, the theory is useful for treating long-term behavior and resonant interactions. The introduction of a self-energy operator allows the shifts in energy levels to be displayed explicitly. When the perturbation is the dipole interaction with an electromagnetic field, the successive approximations yield the well-known rotating-wave approximation, corrections due to counter rotating terms, and the Bloch-Siegert frequency shift. The formalism is applicable to arbitrary pulse shapes, provided that $\ensuremath{\parallel}{H}_{1}\ensuremath{\parallel}\ensuremath{\ll}\ensuremath{\hbar}\overline{\ensuremath{\omega}}$, where $\ensuremath{\parallel}{H}_{1}\ensuremath{\parallel}$ represents the inter-action strength and $\overline{\ensuremath{\omega}}$ is a characteristic frequency of the field. Rabi oscillations induced by multiphoton resonances are automatically included, and this effect is demonstrated in the case of a square pulse with frequency approximately one half the resonance frequency for a transition between two vibrational levels of a molecule. These calculations are compared to an exact numerical solution in order to find the limits of validity of the approximation. Even at high intensities ($I\ensuremath{\sim}{10}^{14}$ W/cm2{\mathrm{cm}}^{2}cm2) the shifted resonance frequency is quite accurate; however, the approximate and exact solutions are slightly out of phase, so that the approximate solution can only be trusted for a few Rabi cycles. For much lower intensities ($I\ensuremath{\sim}{10}^{10}$ W/cm2{\mathrm{cm}}^{2}cm2), the approximate solution is valid for many thousands of Rabi cycles.
A method based on the use of integral transform and orthogonal polynomials is presented to obtain the analytic solutions of the equations for coherent dynamics of multilevel quantum systems as simple molecular models describing molecule-radiation interactions in the high-power laser field and many other processes. As an example the Jacobi polynomials have been used and exact analytic solution has been obtained for dynamics of the family of multilevel Jacobi systems with both equidistant and nonequidistant energy levels, with various dependencies of dipole transition moments on energy, with both resonant and non-resonant laser excitation. Some special cases of the solution for definite systems from the family are given.
We study numerically the solution to Schr\"odinger's equation for the radiative interaction of fictitious NNN-level model atomic and molecular systems with monochromatic radiation fields. The numerical method of solution allows us to study multiphoton transitions in NNN-level systems unhampered by the assumptions usually adopted, such as weak fields or large detunings of intermediate states, in perturbative and quasiperturbative analytic studies. We point out regularities and near regularities in the population dynamics that appear to be qualitatively different from those encountered in two- and three-level systems.
Results are reported on coherent monochromatic multiphoton excitation in many-level systems, which are representative for some of the basic mechanisms for atomic and molecular multiphoton processes. Numerical solutions are discussed that use the Floquet and quasi-resonant approximations in the framework of the URIMIR program package. The excitation schemes include direct three-photon excitation, two-photon excitation with diagonal coupling, Göppert-Mayer-type two-photon processes, multiphoton excitation with off-resonant inter-mediates, and practically irreversible coherent excitation into dense spectral structures. Several interesting phenomena are observed, such as nonlinear line shifts and broadenings of multi-photon resonances of relevance for multiphoton spectroscopy and almost constant intermediate population inversions, potentially useful for laser design. The accurate numerical results are compared with approximate solutions from perturbation theory, and with simple analytical solutions from Rabi-type formulae.
A simple procedure is described for computing steady-state excited vibrational state populations for multiphoton absorption processes in many-level systems. The procedure utilizes techniques already in the literature, i.e., expansion of the density matrix as a Fourier series in the frequency of the monochromatic radiation field, and introduction of linewidths and thermal relaxation by the addition of empirical terms to the density matrix time-development equation. Numerical computations can be carried out quickly on idealized systems, using commercially available band matrix inversion routines. A sample calculation for a 10-level, nine-photon process is presented.
A method to build various integrable models for description of coherent excitation of multilevel media by laser pulses is suggested. Distribution functions over the energy levels of quantum systems depending on the time and frequency detuning are obtained. The distributions follow from Schrödinger equation exact solutions and give the complete dynamical description of laser-excited quantum multilevel systems. Interpretation based on the Fourier spectra of the probability amplitudes of a quantum system is presented. The spectra are expressed in terms of orthonormal polynomials and their weight functions. Matrix elements of the dipole transitions between levels are equal to coefficients of the recurrence formula for the orthonormal polynomial system. Some examples are presented. The Kravchuk oscillator family as an integrable model is constructed to describe the coherent excitation dynamics of multilevel resonance media. It is based on the use of the Kravchuk orthogonal polynomials. The Kravchuk oscillator excitation dynamics is described by the binomial distribution of energy level populations and the distribution parameter depends on excitation conditions. Two known basic models in quantum physics -- the harmonic oscillator and two-level system are the special representatives of the Kravchuk oscillator family.