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Physical Review A

Coherent dynamics of N -level atoms and molecules. III. An analytically soluble periodic case

Richard J. Cook, Bruce W. Shore

Published Aug 1, 1979• 134 citations
Publisher

Abstract

By employing an analogy with a spin-$J$ system in a constant magnetic field the authors obtain a simple soluble model for stepwise laser excitation of an $N$-level system ($N=2J+1$), including analytic treatment of Doppler detuning and of ionization loss. The solutions are periodic and hence permit complete population inversion. A simple graphical realization of the population dynamics, a generalization of the Feynman, Vernon, and Hellwarth vector model of the two-level Bloch equation, is then described.

DOI
10.1103/physreva.20.539

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By employing an analogy with a spin-JJJ system in a constant magnetic field the authors obtain a simple soluble model for stepwise laser excitation of an NNN-level system (N=2J+1N=2J+1N=2J+1), including analytic treatment of Doppler detuning and of ionization loss. The solutions are periodic and hence permit complete population inversion. A simple graphical realization of the population dynamics, a generalization of the Feynman, Vernon, and Hellwarth vector model of the two-level Bloch equation, is then described.
Laser excitation of a two-level atom is analytically studied for arbitrary pulse shape, area, and energy. The ionization mechanism is modeled as a homogeneous width $\ensuremath{\gamma}$ of the atomic excited state. A transformation of the optical Bloch equations allows the dynamics to be completely described by a single equation for the area of the pulse at time ttt, $\ensuremath{\Phi}(t)$. Analytic solutions for the total bound population for a wide variety of pulse shapes can be found in two regions: $\ensuremath{\Phi}(t)\ensuremath{\approx}sin\ensuremath{\Phi}(t)$ and $\ensuremath{\Omega}(t)>\ensuremath{\gamma}$, where $\ensuremath{\Omega}(t)$ is the resonance Rabi frequency. The generalization of the two-level atom results to the NNN-level atom is discussed.
We present a comprehensive analytical study of the optical Bloch equations for a closed two-level atom. We obtain a map in phase space of the Rabi frequency and the laser frequency detuning, discriminating the characteristics of solutions to the optical Bloch equations separated by a boundary in phase space. Outside the boundary, oscillatory behavior is shown, while simple decaying behavior is shown inside the boundary. The behavior of the solutions is also found to be dependent on the relative magnitude of the decay rates of the optical coherence and the excited state population.
The problem of achieving population inversion adiabatically in an NNN-level system using one or more laser fields whose detunings and/or amplitudes are continuously varied is studied analytically and numerically. The SU(N)\mathrm{SU}(N)SU(N) coherence vector picture is shown to suggest unexpected inversion procedures and also to give a generalized interpretation of adiabatic following. It is shown that the (${N}^{2}\ensuremath{-}1$)-dimensional SU(N)\mathrm{SU}(N)SU(N) space contains an ($N\ensuremath{-}1$)--dimensional steady-state subspace $\mathit{\ensuremath{\Gamma}}(t)$ whose orthonormal basis vectors ${\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\Gamma}}}_{1},\dots{}, {\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\Gamma}}}_{N\ensuremath{-}1}$ are given explicitly in terms of the Hamiltonian matrix elements. The motion of the system can be interpreted as a "generalized precession" of $\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}$ about $\mathit{\ensuremath{\Gamma}}$. Multilevel adiabatic following occurs when the angle $\ensuremath{\chi}(t)$ between the coherence vector $\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}$ and its projection onto $\mathit{\ensuremath{\Gamma}}$ is very small. The multiple dimension of $\mathit{\ensuremath{\Gamma}}$ is shown to provide a variety of paths for adiabatic inversion. The adiabatic solution is obtained by solving $N\ensuremath{-}1$ simple equations for the directional cosines of $\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}$ on ${\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\Gamma}}}_{i}$. The adiabatic solution and time scale and the state taken up by the atomic variable are discussed analytically and numerically for a three-level system.
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 authors demonstrate special cases for which the resonantly excited NNN-level atom displays particularly simple periodic behavior involving only two levels. The cases, which occur when first and last transitions are weak, have analytic solutions. A striking difference is noted between even-NNN and odd-NNN, related to the resonance structure of the dressed ($N\ensuremath{-}2$)-level atom.