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Sakarya University Journal of Science

A Study of q-Deformed Bosons, and Their Implications to Quantum Optics

Mustafa Şenay

Published Oct 23, 2024• 0 citations
Publisher

Abstract

In this study, we investigate three types of q-deformed boson oscillators, focusing on their mathematical frameworks and thermodynamic properties. We calculate key thermodynamic quantities, such as internal energy and entropy, as functions of the deformation parameter q. Our results reveal that these oscillators are eigenstates of specific deformed boson annihilation operators. We also analyze their unique characteristics and implications in deformed quantum optics. Furthermore, we examine the impact of q-deformation on qutrit logic gates, including cycle, self-shift, controlled cycle, controlled self-shift, Feynman, ternary Toffoli, and Fredkin gates, highlighting their altered computational properties. This research contributes to a deeper understanding of q-deformed systems and their applications in quantum computing. Overall, it opens new avenues for exploring the interplay between deformation parameters and quantum information processing.

DOI
10.16984/saufenbilder.1522962

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In this study, we investigate three types of q-deformed boson oscillators, focusing on their mathematical frameworks and thermodynamic properties. We calculate key thermodynamic quantities, such as internal energy and entropy, as functions of the deformation parameter q. Our results reveal that these oscillators are eigenstates of specific deformed boson annihilation operators. We also analyze their unique characteristics and implications in deformed quantum optics. Furthermore, we examine the impact of q-deformation on qutrit logic gates, including cycle, self-shift, controlled cycle, controlled self-shift, Feynman, ternary Toffoli, and Fredkin gates, highlighting their altered computational properties. This research contributes to a deeper understanding of q-deformed systems and their applications in quantum computing. Overall, it opens new avenues for exploring the interplay between deformation parameters and quantum information processing.
The multi-dimensional q-deformed bosonic Newton oscillator algebra with SU(d)-symmetry is considered. In this framework, we first introduce some new properties concerning the q-deformed calculus related to the algebra, and we then discuss possible consequences of applying these deformed oscillators in some quantum optical issues such as in the construction of coherent states and their effects on the photon statistics. Second, we investigate the role of q-deformation on both the energy levels and the wave functions of the bosonic Newton oscillators by constructing the q-deformed Hermite polynomials. The results obtained in this work might have some implications for studies on quantum information based technologies such as in photonic quantum computing.
The q-deformed bosonic Tamm–Dancoff oscillator algebra is considered. The coherent state of the q-deformed bosonic Tamm–Dancoff oscillator algebra is first constructed in detail. Second, the high-temperature thermostatistical properties of a gas of the Tamm–Dancoff oscillators are investigated. For high temperatures, the specific heat, the entropy and the equation of state for the system are derived in terms of the real deformation parameter q. The results obtained by the effects of Tamm–Dancoff type q-deformation show that they could be useful for further researches on understanding of mutual interactions between bosons and fermions in many-body quantum systems.
In this paper, the [Formula: see text]-deformed bosonic Tamm–Dancoff oscillator algebra is considered. First, the quantum algebraic and representative properties of these deformed bosons are analyzed in detail. The representations of the [Formula: see text]-fermion algebra of Tamm–Dancoff type are also studied. Second, the high-temperature thermostatistical properties of a gas of Tamm–Dancoff type [Formula: see text]-fermions are investigated. The fermionic distribution function and the other important thermodynamic functions such as the entropy and the specific heat are derived in terms of the real deformation parameter [Formula: see text]. Finally, the time evaluation of a two-level atom in a Tamm–Dancoff oscillator trap interacting with a single-mode traveling light field is concisely discussed.
We present a comprehensive quantum many-body theory for qqq-deformed fermions and bosons, offering a novel framework that relates particle statistics directly to effective interaction strength. Deformed by the parameters kkk and qqq, these particles exhibit statistical behaviors that interpolate between conventional bosonic and fermionic systems, enabling us to model complex interactions via statistical modifications. We develop a generalized Wick theorem and extended Feynman diagrammatics tailored to kqkqkq particles, allowing us to calculate two types of Green's functions. Explicit expressions for these Green's functions are derived in both direct and momentum spaces, providing key insights into the collective properties of qqq-fermionic and bosonic systems. Using a random-phase approximation, we estimate the dielectric function for qqq-fermion gas ($k=\ensuremath{-}1$) and analyze the Friedel oscillations, the plasmon excitations, and the energy-loss function. Our results demonstrate that the effective interaction is tuned by the value of qqq so that a noninteracting limit is obtained as $q\ensuremath{\rightarrow}0$, where the Friedel and the plasma oscillations disappear. An optimal value of qqq, namely, q∗{q}^{*}q∗, the plasma frequency, and the energy-loss function show an absolute maximum and the effective interaction changes behavior.
Within the framework of the theory of q-deformed bosons, we investigate the problem of fluctuations of a q-deformed boson system, and obtain the general result for thermodynamic fluctuations of such a system. Especially, for the system of a q-deformed boson gas, we get the expressions for the fluctuations of internal energy U and particle number N, and the correlation of the fluctuations of the two physical quantities above. Further numerical calculation shows that the fluctuations of such a q-deformed boson gas have some interesting features. We consider that this work may provide much insight into the theory of q bosons, and may also be useful for the further study of a concrete q-deformed boson gas.
A Novel qqq-Derivative Framework with Applications to qqq-Deformed Thermodynamics and Leakage Suppression in Superconducting Qubits
We propose a new qqq-derivative operator built directly from Jackson's qqq-number formulation, designed to preserve the structural properties of standard differential calculus while incorporating deformation effects. By analyzing qqq-deformed Heisenberg algebras, we demonstrate that this formulation maintains the consistency of thermodynamic quantities-such as internal energy, particle number, and specific heat-in dilute gas limits without requiring ad-hoc chain-rule modifications. Furthermore, we explore the physical implications of algebraic deformation using the Biedenharn-Macfarlane realization, showing how qqq-deformation induces intrinsic anharmonicity in quantum oscillator spectra and affects multi-level quantum systems (d≥3d \ge 3d≥3). Applying this algebraic scheme to superconducting transmon qubits, we derive analytical pulse-shaping corrections that generalize the Derivative Removal by Adiabatic Gate (DRAG) technique, offering a robust method to suppress computational leakage in ultra-fast quantum logic operations.