Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review A

Quantum phase and aq-deformed quantum oscillator

Demosthenes Ellinas

Published Mar 1, 1992• 40 citations
Publisher

Abstract

The q-deformed quantum oscillator for q=${\mathit{e}}^{\mathit{i}2\mathrm{\ensuremath{\pi}}/(\mathit{s}+1)}$ has reducible representations that admit unitary phase operators. In the classical limit s\ensuremath{\rightarrow}\ensuremath{\infty}, of zero deformation, they provide the phase angles of the ordinary quantum oscillator. The formalism is applied to the master equation of a q-deformed laser mode that is introduced and solved in the steady state and then used to evaluate the statistical moments of the cosine and sine operators.

DOI
10.1103/physreva.45.3358

Related papers

The Wigner function of aq-deformed harmonic oscillator model

E I Jafarov, S Lievens, S M Nagiyev +1 more

The phase space representation for a q -deformed model of the quantum harmonic oscillator is constructed. We have found explicit expressions for both the Wigner and Husimi distribution functions for the stationary states of the q -oscillator model under consideration. The Wigner function is expressed as a basic hypergeometric series, related to the Al-Salam-Chihara polynomials. It is shown that, in the limit case h → 0 ( q → 1), both the Wigner and Husimi distribution functions reduce correctly to their well-known non-relativistic analogues. Surprisingly, examination of both distribution functions in the q -deformed model shows that, when q ≪ 1, their behaviour in the phase space is similar to the ground state of the ordinary quantum oscillator, but with a displacement towards negative values of the momentum. We have also computed the mean values of the position and momentum using the Wigner function. Unlike the ordinary case, the mean value of the momentum is not zero and it depends on q and n . The ground-state-like behaviour of the distribution functions for excited states in the q -deformed model opens quite new perspectives for further experimental measurements of quantum systems in the phase space.

arXiv:math-ph/0702082•Journal of Physics A Mathematical and Theoretical•Apr 30, 2007•13 citations

Interpretation of the q-Deformed 1-D Quantum Harmonic Oscillator

Ahmed Shaker Mahmood, Mohammed A. Z. Habeeb

The interpretation of the q-deformed 1-D quantum harmonic oscillator is investigated for two definitions of q-deformation.This investigation is achieved by using Zaslavskii's method to obtain the Heisenberg equations of motion (quantum Liouville equations) in the undeformed phase space.These quantum Liouville equations exhibit a non-commutative geometry produce from the existence of the dilatation operator which is inherent in the q-deformation process.The classical limits of these equations are obtained by applying a special classical limiting condition to produce the classical Liouville equations of the q-deformed oscillator.These classical Liouville equations are solved by using the method of characteristics in order to obtain the classical probability distribution functions for this system.The 2-D and 3-D behaviors of these functions were then investigated using a computer visualization method.The results of the mathematical derivations together with the computer visualization method show that the classical limit of the quantum Liouville equations for the q-deformed 1-D quantum harmonic oscillator are statistical in nature where the nonlinearity parameter for the q-deformed oscillator is connected with h .This result conforms to that obtained by Ghosh et al. for the undeformed 1-D quantum harmonic oscillator.The obtained classical probability distribution functions exhibit whorl shapes that evolve with time in phase space that are similar to the shapes obtained for the 1-D classical q-deformed oscillator.These whorl shapes in phase space are similar to those introduced by Milburn for the 1-D classical anharmonic oscillator.This similarity results from the fact that the anharmonicity itself represents a kind of deformation with a frequency that is a function of amplitude.

Journal of Al-Nahrain University-Science•Sep 1, 2016•1 citation

Dynamics of aQ-analogue of the Quantum Harmonic Oscillator

Vladimír Bužek

We analyse the dynamics of the q-deformed quantum harmonic oscillator initially prepared in the q-analogue of the coherent state. Non-trivial behaviour of the mean values of the q-position operator is observed. The squeezing of the second-order moments of this operator is studied.

Journal of Modern Optics•Apr 1, 1991•44 citations

The Liouville Dynamics of the q-Deformed 1-D Classical Harmonic Oscillator

Ahmed Shaker Mahmood, Mohammed A. Z. Habeeb

The Liouville equation for the q-deformed 1-D classical harmonic oscillator is derived for two definitions of q-deformation.This derivation is achieved by using two different representations for the q-deformed Hamiltonian of this oscillator corresponding to undeformed and deformed phase spaces.The resulting Liouville equation is solved by using the method of characteristics in order to obtain the classical probability distribution function for this system.The 2-D and 3-D behaviors of this function are then investigated using a computer visualization method.The results are compared with those for the classical anharmonic oscillator.This comparison reveals that there are some similarities between these two models, where the results for the q-deformed oscillator exhibit similar whorl shapes that evolve with time as for the anharmonic oscillator.It is concluded that studying the Liouville dynamics gives more details about the physical nature of q-deformation than using the equation of motion method.It is also concluded that this result could have reflections on the interpretation of the quantized version of this q-deformed oscillator.

arXiv:1611.03682•Journal of Al-Nahrain University-Science•Mar 1, 2016•0 citations

q-PHASE OPERATOR IN A FINITE-DIMENSIONAL HILBERT SPACE

Yaping Yang, Zurong Yu

In this letter, the unitary and Hermitian phase operators for the single-mode electromagnetic field are given in the q-deformed case. A new kind of q-coherent states of a q-harmonic oscillator in a finite-dimensional Hilbert space are introduced. Some properties of these operators and q-coherent states are discussed.

Modern Physics Letters A•Feb 10, 1995•7 citations

q-Deformed Boson Oscillators and Zero Point Energy

P. Narayana Swamy

Just as for the ordinary quantum harmonic oscillators, we expect the zero-point energy to play a crucial role in the correct high temperature behavior. We accordingly reformulate the theory of the statistical distribution function for the q-deformed boson oscillators and develop an approximate theory incorporating the zero-point energy. We are then able to demonstrate that for small deformations, ln q≪1, the theory reproduces the correct limits both for very high temperatures and for very low temperatures. The deformed theory thus reduces to the undeformed theory in these extreme cases.

arXiv:quant-ph/9909069•International Journal of Modern Physics B•Dec 30, 1998•6 citations
The q-deformed quantum oscillator for q=ei2\ensuremathπ/(s+1){\mathit{e}}^{\mathit{i}2\mathrm{\ensuremath{\pi}}/(\mathit{s}+1)}ei2\ensuremathπ/(s+1) has reducible representations that admit unitary phase operators. In the classical limit s\ensuremath{\rightarrow}\ensuremath{\infty}, of zero deformation, they provide the phase angles of the ordinary quantum oscillator. The formalism is applied to the master equation of a q-deformed laser mode that is introduced and solved in the steady state and then used to evaluate the statistical moments of the cosine and sine operators.
The phase space representation for a q -deformed model of the quantum harmonic oscillator is constructed. We have found explicit expressions for both the Wigner and Husimi distribution functions for the stationary states of the q -oscillator model under consideration. The Wigner function is expressed as a basic hypergeometric series, related to the Al-Salam-Chihara polynomials. It is shown that, in the limit case h → 0 ( q → 1), both the Wigner and Husimi distribution functions reduce correctly to their well-known non-relativistic analogues. Surprisingly, examination of both distribution functions in the q -deformed model shows that, when q ≪ 1, their behaviour in the phase space is similar to the ground state of the ordinary quantum oscillator, but with a displacement towards negative values of the momentum. We have also computed the mean values of the position and momentum using the Wigner function. Unlike the ordinary case, the mean value of the momentum is not zero and it depends on q and n . The ground-state-like behaviour of the distribution functions for excited states in the q -deformed model opens quite new perspectives for further experimental measurements of quantum systems in the phase space.
The interpretation of the q-deformed 1-D quantum harmonic oscillator is investigated for two definitions of q-deformation.This investigation is achieved by using Zaslavskii's method to obtain the Heisenberg equations of motion (quantum Liouville equations) in the undeformed phase space.These quantum Liouville equations exhibit a non-commutative geometry produce from the existence of the dilatation operator which is inherent in the q-deformation process.The classical limits of these equations are obtained by applying a special classical limiting condition to produce the classical Liouville equations of the q-deformed oscillator.These classical Liouville equations are solved by using the method of characteristics in order to obtain the classical probability distribution functions for this system.The 2-D and 3-D behaviors of these functions were then investigated using a computer visualization method.The results of the mathematical derivations together with the computer visualization method show that the classical limit of the quantum Liouville equations for the q-deformed 1-D quantum harmonic oscillator are statistical in nature where the nonlinearity parameter for the q-deformed oscillator is connected with h .This result conforms to that obtained by Ghosh et al. for the undeformed 1-D quantum harmonic oscillator.The obtained classical probability distribution functions exhibit whorl shapes that evolve with time in phase space that are similar to the shapes obtained for the 1-D classical q-deformed oscillator.These whorl shapes in phase space are similar to those introduced by Milburn for the 1-D classical anharmonic oscillator.This similarity results from the fact that the anharmonicity itself represents a kind of deformation with a frequency that is a function of amplitude.
The Liouville equation for the q-deformed 1-D classical harmonic oscillator is derived for two definitions of q-deformation.This derivation is achieved by using two different representations for the q-deformed Hamiltonian of this oscillator corresponding to undeformed and deformed phase spaces.The resulting Liouville equation is solved by using the method of characteristics in order to obtain the classical probability distribution function for this system.The 2-D and 3-D behaviors of this function are then investigated using a computer visualization method.The results are compared with those for the classical anharmonic oscillator.This comparison reveals that there are some similarities between these two models, where the results for the q-deformed oscillator exhibit similar whorl shapes that evolve with time as for the anharmonic oscillator.It is concluded that studying the Liouville dynamics gives more details about the physical nature of q-deformation than using the equation of motion method.It is also concluded that this result could have reflections on the interpretation of the quantized version of this q-deformed oscillator.
Just as for the ordinary quantum harmonic oscillators, we expect the zero-point energy to play a crucial role in the correct high temperature behavior. We accordingly reformulate the theory of the statistical distribution function for the q-deformed boson oscillators and develop an approximate theory incorporating the zero-point energy. We are then able to demonstrate that for small deformations, ln q≪1, the theory reproduces the correct limits both for very high temperatures and for very low temperatures. The deformed theory thus reduces to the undeformed theory in these extreme cases.