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1 IPNL - Institut de Physique Nucléaire de Lyon

Abstract : We introduce a one-parameter generalized oscillator algebra Ak that covers the case of the harmonic oscillator algebra and discuss its finite- and infinite-dimensional representations according to the sign of the parameter k. We define an Hamiltonian operator associated with Ak and examine the degeneracies of its spectrum. For the finite when k < 0 and the infinite when k > 0 or = 0 representations of Ak, we construct the associated phase operators and build temporally stable phase states as eigenstates of the phase operators. To overcome the difficulties related to the phase operator in the infinite-dimensional case and to avoid the degeneracy problem for the finite-dimensional case, we introduce a truncation procedure which generalizes the one used by Pegg and Barnett for the harmonic oscillator. This yields a truncated generalized oscillator algebra Ak,s, where s denotes the truncation order. We construct two types of temporally stable states for Ak,s as eigenstates of a phase operator and as eigenstates of a polynomial in the generators of Ak,s. Two applications are considered in this article. The first concerns physical realizations of Ak and Ak,s in the context of one-dimensional quantum systems with finite Morse system or infinite Poeschl-Teller system discrete spectra. The second deals with mutually unbiased bases used in quantum information.

Keywords : Gauss sums mutually unbiased bases exactly solvable potentials phase operators temporally stable phase states coherents states





Author: Mohammed Daoud - Maurice Robert Kibler -

Source: https://hal.archives-ouvertes.fr/



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