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

We introduce a one-parameter-generalized oscillator algebra Aκ (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 Aκ and examine the degeneracies of its spectrum. For the finite (when κ < 0) and the infinite (when κ ≥ 0) representations of Aκ, 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 Aκ,s, where s denotes the truncation order. We construct two types of temporally stable states for Aκ,s (as eigenstates of a phase operator and as eigenstates of a polynomial in the generators of Aκ,s). Two applications are considered in this paper. The first concerns physical realizations of Aκ and Aκ,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.