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Physics Letters B

Linear dependence of deformed two-photon quadrature operators on spatial noncommutative parameter

Ying Wu

Published Jan 31, 2006• 7 citations
Publisher

Abstract

By deriving useful formulas on the variances of the deformed two-photon quadrature operators in noncommutative space, we show that the variances in general have a linear dependence on the spatial noncommutative parameter θ, significantly different from the previous θ2-dependence results. The θ-dependence instead of θ2-dependence results make the scheme of testing the space noncommutativity via measuring such variances much easier to realize experimentally, well within the reach of nowadays technology.

DOI
10.1016/j.physletb.2006.01.018

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By deriving useful formulas on the variances of the deformed two-photon quadrature operators in noncommutative space, we show that the variances in general have a linear dependence on the spatial noncommutative parameter θ, significantly different from the previous θ2-dependence results. The θ-dependence instead of θ2-dependence results make the scheme of testing the space noncommutativity via measuring such variances much easier to realize experimentally, well within the reach of nowadays technology.
Recent studies on nonperturbation aspects of noncommutative quantum mechanics explored a new type of boson commutation relations at the deformed level, described by deformed annihilation–creation operators in noncommutative space. This correlated boson commutator correlates different degrees of freedom, and shows an essential influence on dynamics. This Letter devotes to the development of formalism of deformed two-photon squeezed states in noncommutative space. General representations of deformed annihilation–creation operators and the consistency condition for the electromagnetic wave with a single mode of frequency in noncommunicative space are obtained. Two-photon squeezed states are studied. One finds that variances of the dimensionless Hermitian quadratures of the annihilation operator in one degree of freedom include variances in the other degree of freedom. Such correlations show the new feature of spatial noncommutativity and allow a deeper understanding of the correlated boson commutator.
We present analytical expressions for displacements that involve noncommutative operators. We generalize the standard displacement formula D a f(x)=f(x-a) by considering successively the displacement effect of momentum operators in noncommutative quantum mechanics [p i ,p j ]=iθ ij , d-dimensional radial momentum operators p r (d)=- iℏ(∂ r +(d−1)/2) and spin-dependent operators. The noncommutativity results in three characteristic effects: the appearance of phase factors, space-modulated displacements and staggered spin-dependent displacements.
We introduce the deformed boson operators which satisfy a deformed boson algebra in some special types of generalized noncommutative phase space. Based on the deformed boson algebra, we construct coherent state representations. We calculate the variances of the coordinate operators on the coherent states and investigate the corresponding Heisenberg uncertainty relations. It is found that there are some restriction relations of the noncommutative parameters in these special types of noncommutative phase space.
We study several classical-like properties of qqq-deformed nonlinear coherent states as well as nonclassical behaviors of qqq-deformed version of the Schr\"odinger cat states in noncommutative space. Coherent states in qqq-deformed space are found to be minimum uncertainty states together with the squeezed photon distributions unlike the ordinary systems, where the photon distributions are always Poissonian. Several advantages of utilizing cat states in noncommutative space over the standard quantum mechanical spaces have been reported here. For instance, the qqq-deformed parameter has been utilized to improve the squeezing of the quadrature beyond the ordinary case. Most importantly, the parameter provides an extra degree of freedom by which we achieve both quadrature squeezed and number squeezed cat states at the same time in a single system, which is impossible to achieve from ordinary cat states.
In two-dimensional noncommutive space for the case of both position - position and momentum - momentum noncommuting, a constraint between noncommutative parameters is investigated. The related topic of guaranteeing Bose - Einstein statistics in noncommutive space in the general case are elucidated: Bose - Einstein statistics is guaranteed by the deformed Heisenberg - Weyl algebra itself, independent of dynamics. A special character of a dynamical system is represented by a constraint between noncommutative parameters. The general feature of the constraint for any system is a direct proportionality between noncommutative parameters with a proportional coefficient depending on characteristic parameters of the system under study. The constraint for a harmonic oscillator is illustrated.