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Physical Review A

State reconstruction of anharmonic molecular vibrations: Morse-oscillator model

Ulf Leonhardt

Published Apr 1, 1997• 26 citations
Publisher

Abstract

The quantum state of a one-dimensional molecular vibration can be reconstructed via molecular emission tomography [T. J. Dunn, I. A. Walmsley, and S. Mukamel, Phys. Rev. Lett. 74, 884 (1995)] even in the case of anharmonic potentials. We assume a Morse-oscillator model for the vibration and calculate analytically the irregular wave functions. These mathematical objects serve as ingredients of the reconstruction procedure [U. Leonhardt and M. G. Raymer, Phys. Rev. Lett. 76, 1989 (1996)]. Additionally, we show why the observation time until the revival of order is sufficient for an accurate state determination.

DOI
10.1103/physreva.55.3164

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The quantum state of a one-dimensional molecular vibration can be reconstructed via molecular emission tomography [T. J. Dunn, I. A. Walmsley, and S. Mukamel, Phys. Rev. Lett. 74, 884 (1995)] even in the case of anharmonic potentials. We assume a Morse-oscillator model for the vibration and calculate analytically the irregular wave functions. These mathematical objects serve as ingredients of the reconstruction procedure [U. Leonhardt and M. G. Raymer, Phys. Rev. Lett. 76, 1989 (1996)]. Additionally, we show why the observation time until the revival of order is sufficient for an accurate state determination.
We show how to reconstruct the quantum state of one-dimensional wave packets in the continuous part of the spectrum. We assume that the position probability distribution is measured for a sufficiently long time interval and that the potential is known, but otherwise arbitrary. Our paper fills the gap between the tomographic state reconstruction based on free evolution [J. Bertrand and P. Bertrand, Found. Phys. 17, 397 (1987)] and the state determination of moving bound states [U. Leonhardt and M. G. Raymer, Phys. Rev. Lett. 76, 1985 (1996)]. Our result may be relevant with respect to the state determination of atomic beams [Ch. Kurtsiefer, T. Pfau, and J. Mlynek, Nature (London) 386, 150 (1997)] moving in arbitrary potentials.
For the one-dimensional Morse oscillator we have built a set of creation and annihilation operators and established the corresponding coherent states. The average values of some observables in these coherent states have been calculated and their time evolution established. In order to study the strong vibrations of di-atomic molecules we use a three-dimensional Morse oscillator model and its dynamic group. For this system we have built the coherent states and calculated the dynamic modifications of the refractive index of the molecules, as the result of interaction with a laser beam.
We propose a method for reconstruction of the density matrix from measurable time-dependent (probability) distributions of physical quantities. The applicability of the method based on least-squares inversion is---compared with other methods---very universal. It can be used to reconstruct quantum states of various systems, such as harmonic and anharmonic oscillators including molecular vibrations in vibronic transitions and damped motion. It also enables one to take into account various specific features of experiments, such as limited sets of data and data smearing owing to limited resolution. To illustrate the method, we consider a Morse oscillator and give a comparison with other state-reconstruction methods suggested recently.
A method for calculating the analytical solutions of the one-dimensional Schr\"odinger equation is suggested. A general discussion of the possible forms of the potentials and wave functions that are necessary to get the analytical solution is presented. In general, the analytical solutions appear in multiplets corresponding to the quantum number n of the harmonic oscillator. As an application, known solutions for the anharmonic oscillators are critically recalculated and a few additional results are found. Analytical solutions are also found for the generalized Morse oscillators. \textcopyright{} 1996 The American Physical Society.
In this paper, we consider the one-dimensional anharmonic oscillator, which represents well the anharmonic vibrations in diatomic molecules. For the description of the associate potential we use the Morse potential, which gives a good approximation of the experimentally observed vibrational modes of molecules and hence contributes to the realistic description of the spectrum of diatomic molecules. The generalized and Gaussian coherent states are thus constructed and compared in terms of the localization of the particle in those states. We apply these results to the example of the sodium chloride molecule, 1 H 35 Cl.