q-Phase Operators of Finite Dimensional Two-Mode q-Oscillator Algebra

Mohammed Daoud A, M. Kibler

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.

L. C. Kwek, C. H. Oh

Won Sang Chung

Demosthenes Ellinas

The problem of quantization of the classical phase of a harmonic oscillator (HO) is solved here in two steps. First, polar decomposition of the step operators of the u(2) algebra is performed. Second, the method of group contraction is used through which, in the limit j→∞, â,â° is passed to for the quantized HO and its Hermitian phase operators. Also, phase states, i.e., states with sharply defined phase, are constructed and the dynamical aspects of the contraction limit between the Jaynes–Cummings model (JCM) and a finite-dimensional counterpart with increasing j parameter are studied. Finally, the old problem of the phase operators is discussed in the wider frame of rank-1 algebras and classify the previous works in this frame.

Yee Kin Tsui

T Hakioglu

Schwinger's finite ( D ) dimensional periodic Hilbert space representations are studied on the toroidal lattice with specific emphasis on the deformed oscillator subalgebras and the generalized representations of the Wigner function. These subalgebras are shown to be admissible endowed with the non-negative norm of Hilbert space vectors. Hence, they provide the desired canonical basis for the algebraic formulation of the quantum phase problem. Certain equivalence classes in the space of labels are identified within each subalgebra, and connections with area-preserving canonical transformations are examined. The generalized representations of the Wigner function are examined in the finite-dimensional cyclic Schwinger basis. These representations are shown to conform to all fundamental conditions of the generalized phase space Wigner distribution. As a specific application of the Schwinger basis, the number-phase unitary operator pair in is studied and, based on the admissibility of the underlying q -oscillator subalgebra, an algebraic approach to the unitary quantum phase operator is established. This being the focus of this work, connections with the Susskind-Glogower-Carruthers-Nieto phase operator formalism as well as standard action-angle Wigner function formalisms are examined in the infinite-period limit. The concept of continuously shifted Fock basis is introduced to facilitate the Fock space representations of the Wigner function.