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Lettere al nuovo cimento della societa italiana di fisica/Lettere al nuovo cimento

On the nature of the spectrum of generalized oscillator phase operators

Evangelos K. Ifantis

Published Nov 1, 1971• 8 citations
Publisher
DOI
10.1007/bf02780628

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Admissible cyclic representations and an algebraic approach to quantum phase

Tugrul Hakioglu

Nonadmissible, weakly admissible and admissible cyclic representations and other algebraic properties of the generalized homographic oscillator (GHO) are studied in detail. For certain ranges of the deformation parameter, it is shown that this new deformed oscillator is a prototype cyclic oscillator endowed with a non-negative (admissible) spectrum. By changing the deformation parameter, the cyclic spectrum can be tuned to have an arbitrarily large period. It is shown that the standard harmonic oscillator is recovered at the nonadmissible infinite-period limit of the GHO. With these properties, the GHO provides a concrete example of an oscillator rich in a variety of cyclic representations. It is well known that such representations are of relevance to the proper algebraic formulation of the quantum-phase operator. Using a general scheme, it is shown that admissible cyclic algebras permit a well-defined Hermitian phase operator of which properties are studied in detail at finite periods as well as at the infinite-period limit. Fujikawa's index approach is applied to admissible cyclic representations and in particular to the phase operator in such algebras. Using the specific example of GHO it is confirmed that the infinite-period limit is distinctively singular. The connection with the Pegg-Barnett phase formalism is established in this singular limit as the period of the cyclic representations tends to infinity. The singular behaviour at this limit identifies the algebraic problems, in a concrete example, emerging in the formulation of a standard quantum harmonic-oscillator phase operator.

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Phase operators, temporally stable phase states, mutually unbiased bases and exactly solvable quantum systems

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We introduce a one-parameter-generalized oscillator algebra (that covers the case of the harmonic oscillator algebra) and discuss its finite- and infinite-dimensional representations according to the sign of the parameter κ. We define an (Hamiltonian) operator associated with and examine the degeneracies of its spectrum. For the finite (when κ < 0) and the infinite (when κ ⩾ 0) representations of , 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 , where s denotes the truncation order. We construct two types of temporally stable states for (as eigenstates of a phase operator and as eigenstates of a polynomial in the generators of ). Two applications are considered in this paper. The first concerns physical realizations of and in the context of one-dimensional quantum systems with finite (Morse system) or infinite (Pöschl–Teller system) discrete spectra. The second deals with mutually unbiased bases used in quantum information.

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Some Mathematical Properties of Oscillator Phase Operators

E. C. Lerner, Huajun Huang, G. E. Walters

A general definition of ``cosine'' and ``sine'' operators, C and S, for harmonic oscillator phase is proposed and its consequences examined. An important feature of the spectral analysis is the ``chain sequence'' condition which ensures that C and S have unit norm. The (nonunitary) operator U = C + iS is shown to be an annihilation-type operator whose spectral properties bear a remarkable analogy to those of the standard annihilation operator, although its spectrum fills the unit circle rather than the entire complex plane. Statistical properties of the eigenstates of U are discussed briefly.

Journal of Mathematical Physics•May 1, 1970•87 citations
Nonadmissible, weakly admissible and admissible cyclic representations and other algebraic properties of the generalized homographic oscillator (GHO) are studied in detail. For certain ranges of the deformation parameter, it is shown that this new deformed oscillator is a prototype cyclic oscillator endowed with a non-negative (admissible) spectrum. By changing the deformation parameter, the cyclic spectrum can be tuned to have an arbitrarily large period. It is shown that the standard harmonic oscillator is recovered at the nonadmissible infinite-period limit of the GHO. With these properties, the GHO provides a concrete example of an oscillator rich in a variety of cyclic representations. It is well known that such representations are of relevance to the proper algebraic formulation of the quantum-phase operator. Using a general scheme, it is shown that admissible cyclic algebras permit a well-defined Hermitian phase operator of which properties are studied in detail at finite periods as well as at the infinite-period limit. Fujikawa's index approach is applied to admissible cyclic representations and in particular to the phase operator in such algebras. Using the specific example of GHO it is confirmed that the infinite-period limit is distinctively singular. The connection with the Pegg-Barnett phase formalism is established in this singular limit as the period of the cyclic representations tends to infinity. The singular behaviour at this limit identifies the algebraic problems, in a concrete example, emerging in the formulation of a standard quantum harmonic-oscillator phase operator.
We introduce a one-parameter-generalized oscillator algebra (that covers the case of the harmonic oscillator algebra) and discuss its finite- and infinite-dimensional representations according to the sign of the parameter κ. We define an (Hamiltonian) operator associated with and examine the degeneracies of its spectrum. For the finite (when κ < 0) and the infinite (when κ ⩾ 0) representations of , 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 , where s denotes the truncation order. We construct two types of temporally stable states for (as eigenstates of a phase operator and as eigenstates of a polynomial in the generators of ). Two applications are considered in this paper. The first concerns physical realizations of and in the context of one-dimensional quantum systems with finite (Morse system) or infinite (Pöschl–Teller system) discrete spectra. The second deals with mutually unbiased bases used in quantum information.
A general definition of ``cosine'' and ``sine'' operators, C and S, for harmonic oscillator phase is proposed and its consequences examined. An important feature of the spectral analysis is the ``chain sequence'' condition which ensures that C and S have unit norm. The (nonunitary) operator U = C + iS is shown to be an annihilation-type operator whose spectral properties bear a remarkable analogy to those of the standard annihilation operator, although its spectrum fills the unit circle rather than the entire complex plane. Statistical properties of the eigenstates of U are discussed briefly.