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Physics Letters A

Generalized Heisenberg algebra and algebraic method: The example of an infinite square-well potential

Evaldo M. F. Curado, Y. Hassouni, M. A. Rego-Monteiro, Ligia M. C. S. Rodrigues

Published Feb 21, 2008• 33 citations
Publisher
DOI
10.1016/j.physleta.2008.01.086

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We study how catlike superpositions of generalized Heisenberg algebras (GHA) nonlinear coherent states behave under dissipative decoherence. Two cases are presented: the infinite square well potential and systems whose spectra, given by infinite strictly increasing sequences of nonnegative real numbers, can be considered perturbations of the harmonic oscillator. The decoherence effect caused by the interaction of these systems with a thermal bath is analyzed: from their fidelity behavior we see that a region always exists in the parameter space where the quantum coherence is better preserved as compared to the harmonic oscillator. Moreover, we show that the qualitative behavior of GHA systems under the studied mechanism of decoherence can be inferred from the algebraic structure via the analysis of their GHA characteristic functions.
An adaptation of Kieu's hypercomputational quantum algorithm (KHQA) is presented. The method that was used was to replace the Weyl–Heisenberg algebra by other dynamical algebra of low dimension that admits infinite-dimensional irreducible representations with naturally defined generalized coherent states. We have selected the Lie algebra , because this algebra possesses the necessary characteristics to realize the hypercomputation and also because such algebra has been identified as the dynamical algebra associated with many relatively simple quantum systems. In addition to an algebraic adaptation of KHQA over the algebra , we presented an adaptation of KHQA over some concrete physical referents: the infinite square well, the infinite cylindrical well, the perturbed infinite cylindrical well, the Pöschl–Teller potentials, the Holstein–Primakoff system and the Laguerre oscillator. We conclude that it is possible to have many physical systems within condensed matter and quantum optics in which it is possible to consider an implementation of KHQA.