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Kluwer Academic Publishers eBooks

Analysis of Quantum Spectra by Harmonic Inversion

Jörg Main, Günter Wunner, Vladimir A. Mandelshtam, Howard S. Taylor

Published Feb 2, 2006• 0 citations
Publisher

Abstract

We have shown that interesting physical phenomena can be revealed in high resolution quantum recurrence spectra by application of the harmonic inversion technique, thereby circumventing the restrictions imposed by the uncertainty principle of the conventional Fourier transform. The method allows, e.g., to test the validity of semiclassical theories, to identify hidden ghost or bits in the quantum spectra, and to observe the symmetry breaking in the spectra of the hydrogen atom in crossed magnetic and electric fields. The analysis has been demonstrated here on theoretically calculated quantum spectra but can be applied to experimental spectra as well.

DOI
10.1007/0-306-47074-8_27

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We have shown that interesting physical phenomena can be revealed in high resolution quantum recurrence spectra by application of the harmonic inversion technique, thereby circumventing the restrictions imposed by the uncertainty principle of the conventional Fourier transform. The method allows, e.g., to test the validity of semiclassical theories, to identify hidden ghost or bits in the quantum spectra, and to observe the symmetry breaking in the spectra of the hydrogen atom in crossed magnetic and electric fields. The analysis has been demonstrated here on theoretically calculated quantum spectra but can be applied to experimental spectra as well.
Harmonic inversion is introduced as a powerful tool for both the analysis of quantum spectra and semiclassical periodic orbit quantization. The method allows to circumvent the uncertainty principle of the conventional Fourier transform and to extract dynamical information from quantum spectra which has been unattainable before, such as bifurcations of orbits, the uncovering of hidden ghost orbits in complex phase space, and the direct observation of symmetry breaking effects. The method also solves the fundamental convergence problems in semiclassical periodic orbit theories -- for both the Berry-Tabor formula and Gutzwiller's trace formula -- and can therefore be applied as a novel technique for periodic orbit quantization, i.e., to calculate semiclassical eigenenergies from a finite set of classical periodic orbits. The advantage of periodic orbit quantization by harmonic inversion is the universality and wide applicability of the method, which will be demonstrated in this work for v...
Harmonic inversion has already been proven to be a powerful tool for the analysis of quantum spectra and the periodic orbit orbit quantization of chaotic systems. The harmonic inversion technique circumvents the convergence problems of the periodic orbit sum and the uncertainty principle of the usual Fourier analysis, thus yielding results of high resolution and high precision. Based on the close analogy between periodic orbit trace formulae for regular and chaotic systems the technique is generalized in this paper for the semiclassical quantization of integrable systems. Thus, harmonic inversion is shown to be a universal tool which can be applied to a wide range of physical systems. The method is further generalized in two directions: firstly, the periodic orbit quantization will be extended to include higher order ~ corrections to the periodic orbit sum. Secondly, the use of cross-correlated periodic orbit sums allows us to significantly reduce the required number of orbi...
Semiclassical spectra weighted with products of diagonal matrix elements of operators ${A}_{\ensuremath{\alpha}},$ i.e., ${g}_{\ensuremath{\alpha}{\ensuremath{\alpha}}^{\ensuremath{'}}}(E)={\ensuremath{\sum}}_{n}〈n|{A}_{\ensuremath{\alpha}}|n〉〈n|{A}_{{\ensuremath{\alpha}}^{\ensuremath{'}}}|n〉/(E\ensuremath{-}{E}_{n}),$ are obtained by harmonic inversion of a cross-correlation signal constructed of classical periodic orbits. The method provides highly resolved semiclassical spectra even in situations of nearly degenerate states, and opens the way to reducing the required signal lengths to shorter than the Heisenberg time. This implies a significant reduction of the number of orbits required for periodic orbit quantization by harmonic inversion.
The correspondence between the quantum spectra and the classical orbits of the two-dimensional elliptic billiards is investigated. In order to study systems without analytic solutions in the future, an expansion method for stationary states (EMSS) to get highly accurate numerical solutions is applied and tested. The closed-orbit theory was extended to the open orbit case. By comparing the Fourier-transformed spectra with the classical orbits, we find that the peak positions in the quantum spectra match with the lengths of the classical orbits with small numerical differences. This is another example showing that semiclassical method provides a bridge between quantum and classical mechanics.
We present a theoretical method to study atomic photoabsorption processes in a temporally periodic external field. The time development of the atomic dipole moment is propagated by numerical time propagation, and the photoabsorption cross sections is obtained directly from the Fourier transform of the autocorrelation function without summing up over the Floquet-Fourier components. As an example, we study the photoabsorption spectra of hydrogen atoms exposed in a mixture of static and periodically oscillating magnetic fields for the transition from $1s\ensuremath{\rightarrow}2p$. The atomic energy structure and photoabsorption spectra show interesting dependence on the strength of the oscillating magnetic field. Tuning the combination of the magnetic field strengths, we can control the photoabsorption process. The proposed method is general and can be applied to processes in which the number of relevant high harmonic components of the Fourier expansion is so large that the conventional Floquet method is intractable.