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Physical Review A

Two-level behavior of coherent excitation of multilevel systems

Bruce W. Shore

Published Sep 1, 1981• 34 citations
Publisher

Abstract

Previously published papers have noted the existence of two-level periodicities in idealized theoretical models of monochromatic excitation of multilevel systems. The present paper provides a theoretical derivation of such two-level dynamics from the more general $N$-level rotating-wave-approximation Schr\"odinger equation and applies the resulting formulas to different extremes of excitation in a nondegenerate system of sequential (ladderlike) excitation: first, when intermediate levels have much larger cumulative detunings than Rabi frequencies, and second, when intermediate Rabi frequencies are much larger than detunings. In each case one has simple analytic expressions for the effective two-level Rabi frequencies and detunings.

DOI
10.1103/physreva.24.1413

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Previously published papers have noted the existence of two-level periodicities in idealized theoretical models of monochromatic excitation of multilevel systems. The present paper provides a theoretical derivation of such two-level dynamics from the more general NNN-level rotating-wave-approximation Schr"odinger equation and applies the resulting formulas to different extremes of excitation in a nondegenerate system of sequential (ladderlike) excitation: first, when intermediate levels have much larger cumulative detunings than Rabi frequencies, and second, when intermediate Rabi frequencies are much larger than detunings. In each case one has simple analytic expressions for the effective two-level Rabi frequencies and detunings.
By constructing a vector model we study the coherent dynamics of a special three-level quantum system--one with equal detunings and equal Rabi frequencies. This three-level problem can be fully related to a two-level problem. We find that complete population depletion can be achieved even in the case of constant nonzero detunings when the three-level system is prepared initially in the common level of the two transitions. Complete population depletion with constant detunings can also occur in other odd-number multilevel systems. An explanation in terms of an equivalent resonant excitation is presented.
This paper describes the dynamics of multilevel excitation by coherent radiatin, in the NNN-level rotating-wave approximation, when the levels have nonzero angular momentum JJJ and consequent spatial-orientation degeneracy 2J+12J+12J+1. Observable populations average over 2J+12J+12J+1 independent sublevel excitation sequences. The phenomenon of inaccessible sublevels is noted, as is the effect of polarization. The paper illustrates how averaging over orientation can, in some cases, render illegible the Rabi oscillations and the ac Stark effect, whereas in other cases ($J\ensuremath{\rightarrow}J$, circular polarization on $J\ensuremath{\rightarrow}J\ifmmode\pm\else\textpm\fi{}1$ linear polarization) appreciable structure remains.
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.
It is shown that any pulse with a time-asymmetric envelope and a constant non-zero detuning excites a two-level quantum system with a probability of nearly one-half provided the pulse is very asymmetric, adiabatic in the slowly rising part of its envelope, sudden in the steep part and the peak Rabi frequency is much greater than the detuning. Three particular cases are considered analytically: the asymmetric hyperbolic-secant, the Bambini-Berman and the exponential pulse, which illustrate the general conclusions. The results are applied to a special three-level 'vee' system whose two transitions are driven by two asymmetric laser pulses of the same Rabi frequencies and the same detunings. Analysis shows that if such a system is prepared initially in its ground state, effective population depletion of this state can be achieved, the results being quite insensitive to variations in the Rabi frequency and the detuning.
For Rabi frequencies comparable to, or even larger than, the transition frequency of a two-level system, the regime of extreme nonlinear optics is reached. Here, we give an overview of the radiated light intensity as a function of carrier frequency of light, transition frequency, Rabi frequency, spectrometer frequency, as well as of the shape and duration of the exciting optical pulses. The graphical representations reveal an amazing complexity and beauty of the nonlinear optical response. Analytical results within the ``square-wave approximation'' qualitatively reproduce many of the intricate features of the exact numerical calculations.