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Journal of Modern Optics

Dynamics of aQ-analogue of the Quantum Harmonic Oscillator

Vladimír Bužek

Published Apr 1, 1991• 44 citations
Publisher

Abstract

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.
DOI
10.1080/09500349114550781

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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 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.