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Physical Review A

Anharmonic and nonclassical effects of the quantum-deformed harmonic oscillator

M. Artoni, J. Zang, Joseph L. Birman

Published Apr 1, 1993• 18 citations
Publisher

Abstract

An isomorphism is realized between the (q)-deformed harmonic oscillator and a particular anharmonic oscillator model: it permits insight into the problem of the physics behind the deformation parameter q. In our model, the anharmonic coupling strength is shown to be proportional to q. The isomorphism permits us also to analyze and to interpret the appearance of various nonclassical features induced by a q deformation during the time evoluton of an su(2) coherent state.

DOI
10.1103/physreva.47.2555

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An isomorphism is realized between the (q)-deformed harmonic oscillator and a particular anharmonic oscillator model: it permits insight into the problem of the physics behind the deformation parameter q. In our model, the anharmonic coupling strength is shown to be proportional to q. The isomorphism permits us also to analyze and to interpret the appearance of various nonclassical features induced by a q deformation during the time evoluton of an su(2) coherent state.
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.
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.
Decoherence of q−q-q−Deformed Photon Added Coherent State
In this study, we explore the behavior of photon added coherent states in a deformed harmonic oscillator subjected to dissipative decoherence. We use q−q-q−deformation as our nonlinear function to model our system. By adjusting the deformation parameter, we show that q−q-q−deformed photon added coherent state (DPACS) exhibit greater nonclassicality and resilience to decoherence compared to those of a standard harmonic oscillator. Additionally, we investigate the nonclassical properties and entanglement of DPACS under decoherence induced by interaction with a dissipative photon-loss environment.
This work addresses the study of the oscillator algebra, defined by four parameters ppp, qqq, ααα, and ννν. The time-independent Schrödinger equation for the induced deformed harmonic oscillator is solved; explicit analytic expressions of the energy spectrum are given. Deformed states are built and discussed with respect to the criteria of coherent state construction. Various commutators involving annihilation and creation operators and their combinatorics are computed and analyzed. Finally, the correlation functions of matrix elements of main normal and antinormal forms, pertinent for quantum optics analysis, are computed.
We introduce a quantum antibunching effect. The quantum antibunching effect of the K eigenstates of the K th powere ( K greater than or equal to 3) of the annihilation operator in the Q-deformed non-harmonic oscillator is investigated. The physical meaning of the K states are explored. The results show that there is the quantum antibunching effect in all of those states. All of them can be generated by a linear superposition of generalized coherent states produced by the time-dependent Q-deformation non-harmonic oscillator at different instants.