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Physical Review A

Aspects ofq-oscillator quantum mechanics

S. Chaturvedi, Venkatraman Srinivasan

Published Dec 1, 1991• 50 citations
Publisher

Abstract

We investigate some aspects of q Heisenberg algebra. We show how su(2) and su(1,1) generators can be constructed in terms of the q creation and annihilation operators. We also construct the coherent states for the q oscillator and show that they can be obtained by the action of a displacement operator on the vacuum. For the multimode case with q=0, corresponding to the infinite statistics of Greenberg [Phys. Rev. Lett. 64, 705 (1990)], we generalize our single-mode construction to obtain the corresponding coherent states. These states, which are eigenstates of the annihilation operators, interestingly, turn out to be degenerate owing to the noncommutativity of the displacement operators.

DOI
10.1103/physreva.44.8020

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We investigate some aspects of q Heisenberg algebra. We show how su(2) and su(1,1) generators can be constructed in terms of the q creation and annihilation operators. We also construct the coherent states for the q oscillator and show that they can be obtained by the action of a displacement operator on the vacuum. For the multimode case with q=0, corresponding to the infinite statistics of Greenberg [Phys. Rev. Lett. 64, 705 (1990)], we generalize our single-mode construction to obtain the corresponding coherent states. These states, which are eigenstates of the annihilation operators, interestingly, turn out to be degenerate owing to the noncommutativity of the displacement operators.
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Generalized q-boson operators associated with the simultaneous creation of several q-bosons are introduced. These operators give rise to a realization of the quantum Weyl-Heisenberg algebra and are applied to the construction of corresponding Holstein-Primakoff realizations of the quantum algebras SU q (2) and SU q (1,1). Coherent states of these algebras are defined in the various ways suggested by the equivalent definitions of the harmonic oscillator coherent states, and some of their properties are studied. Particular attention is devoted to the squeezing properties of the quadrature of the electromagnetic field in these states.
This subject of this thesis is the physical application of deformations of Lie algebras and their use in generalising some exotic quantum optical states. We begin by examining the theory of quantum groups and the q-boson algebras used in their representation theory. Following a review of the properties of conventional coherent states, we describe the extension of the theory to various deformed Heisenberg-Weyl algebras, as well as the q-deformations of su(2) and su(1,1). Using the Deformed Oscillator Algebra of Bonatsos and Daskaloyannis, we construct generalised deformed coherent states and investigate some of their quantum optical properties. We then demonstrate a resolution of unity for such states and suggest a way of investigating the geometric effects of the deformation. The formalism devised by Rembielinski et al is used to consider coherent states of the q-boson algebra over the quantum complex plane. We propose a new unitary operator which is a q-analogue of the displacement operator of conventional coherent state theory: This is used to construct q-displaced vacuum states which are eigenstates of the annihilation operator. Some quantum mechanical properties of these states are investigated and it is shown that they formally satisfy a Heisenberg-type minimum uncertainty relation. After briefly reviewing the theory of conventional squeezed states, we examine the various q-generalisations. We propose a q-analogue of the squeezed vacuum state, and use this in conjunction with the unitary q-displacement operator to construct a general q-squeezed state, parameterised by noncommuting variables.. It is shown that, like their conventional counterparts, such states satisfy the Robertson-Schrodinger Uncertainty Relation. We conclude with a brief discussion about the appearance of noncommuting variables in the states that have been considered.