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Explicit, analytic and closed expressions for boson realizations of the (m+3)-parameter nonlinearly deformed angular momentum algebra with its highest power m of polynomial function being arbitrary, which combines and generalizes Witten's two deformation schemes, are investigated in terms of the single boson and the single inversion boson, respectively. For each kind, the unitary Holstein–Primakoff-like realization, the non-unitary Dyson–Maléev-like realization and their connections are respectively discussed. Using these realizations, the corresponding representations of as well as their respective acting spaces in the Fock space are obtained.

In this paper, we introduce the deformed algebra whose number operator is expressed in terms of the product of the creation operator and annihilation operator. We give some examples for these kinds of deformed algebras. For Arik–Coon's q-oscillator algebra, we discuss, especially, the photon-added states and the photon-subtracted states and construct their associated generation functions.

$(1+1)$-dimensional Dirac oscillator with minimal uncertainty in position and maximal in momentum is investigated. To obtain energy spectrum, SUSY QM technique is applied. It is shown that the Dirac oscillator has two branches of spectrum, the first one gives the standard spectrum of the Dirac oscillator when the parameter of deformation goes to zero and the second branch does not have nondeformed limit. Maximal momentum brings an upper bound for the energy and it gives rise to the conclusion that the energy spectrum contains a finite number of eigenvalues. We also calculate partition function for the spectrum of the first type. The partition function allows us to derive thermodynamic functions of the oscillator which are obtained numerically.

In this paper, using a deformed algebra $[X,P]=i\hslash /(1-{\alpha }^2{P}^2)$ which is originated from various theories of gravity, we study thermodynamical properties of the classical and extreme relativistic gases in canonical ensembles. In this regards, we exactly calculate the modified partition function, Helmholtz free energy, internal energy, entropy, heat capacity and the thermal pressure which conclude to the familiar form of the equation of state for the ideal gas. The advantage of applying this algebra is not only considering all natural cutoffs but also its structure is similar to the other effective quantum gravity models such as polymer, Snyder and noncommutative space–time frameworks. Moreover, after obtaining some thermodynamical quantities including internal energy and entropy, we conclude at high temperature limits due to the decreasing of the number of microstates, these quantities reach to maximal bounds which do not exist in standard cases and it concludes that at the presence of gravity for both micro-canonic and canonic ensembles, the internal energy and the entropy tend to these upper bounds.

In this paper, we propose a simple scheme for generating a deformed cat state based on a dissipative amplitude interaction between an anharmonic oscillator and a special modified bosonic group. The nonclassical properties of the superposition result exhibit an interesting behavior depending on the input parameters. Finally, we discuss some prominent possibilities in creating a generalized Bell state, resulting from an approximate displacement operator under a beam splitter interaction.

The quantum dynamics of a driven single-band tight-binding model with infinite and Dirichlet boundary conditions is considered. The polynomial algebra for the above model but with periodic boundary conditions (quantum ring) is constructed. Based on analyzing the algebraic structures of Hamiltonian, the solution of the time-dependent Schrödinger equation is also obtained exactly.

Phase-locked loops (PLL) is a phase and/or frequency tracking system, widely used in communication and control systems. The sinusoidal multiplicative type PLL still remains a recurrent model, due the fact that its derivation is originated from the maximum likelihood approach. In this note, it is showed as a generalized product, called q-product, which can be used to implement the phase detector and improve some important parameters of the PLL system, as the block linearity and pull-in characteristics. Numerical examples are presented in order to illustrate the proposal.

A unified method of calculating structure functions from commutation relations of deformed single-mode oscillator algebras is presented. A natural approach to building coherent states associated to deformed algebras is then deduced.