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Physical Review A

Nonclassical properties of the q -coherent and q -cat states of the Biedenharn-Macfarlane q oscillator with q>1

Hossein Fakhri, A. Hashemi

Published Jan 4, 2016• 30 citations
Publisher

Abstract

This paper has been motivated by a recent paper by Dey [Phys. Rev. D 91, 044024 (2015)] on the known Arik-Coon $q$ oscillator. We construct $q$ coherent, even and odd $q$-cat states in Fock representation for the Biedenharn-Macfarlane $q$ oscillator with $q>1$ and study their nonclassical properties. The $q$-coherent states minimize the Heisenberg uncertainty relation between the generalized position and momentum operators as well as the $x$ and $y$ components of a $q$-deformed $\text{su}(1,1)$ algebra in the Schwinger boson representation. The latter is also minimized by the even and odd $q$-cat states. We show that, contrary to the undeformed harmonic oscillator, the squeezing effect in both position and momentum operators can be exhibited by odd $q$-cat states. It is also violated by even $q$-cat states. Furthermore, it is shown that the antibunching effect and sub-Poissonian or super-Poissonian statistics can simultaneously appear by each of the even or odd $q$-cat states. Finally, a unitary Fock representation of the $q$-deformed $\text{su}(1,1)$ algebra is obtained by the $q$-deformed Bargmann-Fock realization.

DOI
10.1103/physreva.93.013802

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This paper has been motivated by a recent paper by Dey [Phys. Rev. D 91, 044024 (2015)] on the known Arik-Coon qqq oscillator. We construct qqq coherent, even and odd qqq-cat states in Fock representation for the Biedenharn-Macfarlane qqq oscillator with q>1q>1q>1 and study their nonclassical properties. The qqq-coherent states minimize the Heisenberg uncertainty relation between the generalized position and momentum operators as well as the xxx and yyy components of a qqq-deformed su(1,1)\text{su}(1,1)su(1,1) algebra in the Schwinger boson representation. The latter is also minimized by the even and odd qqq-cat states. We show that, contrary to the undeformed harmonic oscillator, the squeezing effect in both position and momentum operators can be exhibited by odd qqq-cat states. It is also violated by even qqq-cat states. Furthermore, it is shown that the antibunching effect and sub-Poissonian or super-Poissonian statistics can simultaneously appear by each of the even or odd qqq-cat states. Finally, a unitary Fock representation of the qqq-deformed su(1,1)\text{su}(1,1)su(1,1) algebra is obtained by the qqq-deformed Bargmann-Fock realization.
This work has focused on the violation of uncertainty relation, squeezing effect, photon antibunching and sub-Poissonian statistics for the Arik–Coon [Formula: see text]-oscillator coherent states associated with the noncommutative complex plane [Formula: see text]. It is shown that one has to use a generalized definition for the covariance between the operators [Formula: see text] and [Formula: see text]. For [Formula: see text], Heisenberg's inequality violation with two different behaviors related to the role of the deformation parameter [Formula: see text] on the variances of the position and momentum operators is illustrated. We conclude that both weak and strong squeezing effects are exhibited by the [Formula: see text]-coherent states. In particular, strong squeezing effect is a direct consequent of the violation of Heisenberg's inequality. Moreover, the photon antibunching and sub-Poissonian photon statistics are two features of the [Formula: see text]-coherent states which are realized simultaneously with the squeezing effects. Clearly, the three later behaviors are different from their corresponding counterparts in the Arik–Coon [Formula: see text]-oscillator coherent states associated with a commutative complex plane.
The normalized even and odd [Formula: see text]-cat states corresponding to Arik–Coon [Formula: see text]-oscillator on the noncommutative complex plane [Formula: see text] are constructed as the eigenstates of the lowering operator of a [Formula: see text]-deformed [Formula: see text] algebra with the left eigenvalues. We present the appropriate noncommutative measures in order to realize the resolution of the identity condition by the even and odd [Formula: see text]-cat states. Then, we obtain the [Formula: see text]-Bargmann–Fock realizations of the Fock representation of the [Formula: see text]-deformed [Formula: see text] algebra as well as the inner products of standard states in the [Formula: see text]-Bargmann representations of the even and odd subspaces. Also, the Euler’s formula of the [Formula: see text]-factorial and the Gaussian integrals based on the noncommutative [Formula: see text]-integration are obtained. Violation of the uncertainty relation, photon antibunching effect and sub-Poissonian photon statistics by the even and odd [Formula: see text]-cat states are considered in the cases [Formula: see text] and [Formula: see text].
The optical charge pair coherent states for a two-mode electromagnetic field corresponding to [Formula: see text]- and [Formula: see text]-deformed oscillator algebras that are both of the Arik–Coon or Biedenharn–Macfarlane type are constructed. Next, their associated even and odd cat states are formulated in terms of the charge, deformation parameters [Formula: see text] and [Formula: see text], and coherent field, in a linear combination form of two-mode number states of the deformed oscillators. The two-mode higher-order anti-bunching, squeezing, and entanglement traits of the even and odd charge cat states corresponding to the oscillators with the same deformation parameters are considered in terms of the charge and the coherent field as well as different values for deformation parameter itself. We compare the difference of these non-classical behaviors not only between the odd and even states of each two-mode oscillator but also between two-mode Arik–Coon and Biedenharn–Macfarlane oscillators. For example, we answer whether the entanglement of the two-mode cat states increases or decreases with the increase of the charge.
We present the truncated (or [Formula: see text]-deformed) para-Bose oscillator algebra of order [Formula: see text], with [Formula: see text] being a half-integer, along with its [Formula: see text]-dimensional Fock representation. The model reduces to the truncated simple harmonic oscillator and the conventional para-Bose oscillator algebras in the limiting cases where the parameters [Formula: see text] and [Formula: see text] approach zero and infinity, respectively. A Fock–Bargmann realization is constructed in such a way that the creation operator is expressed as a function of both deformation parameters [Formula: see text] and [Formula: see text]. Two-component orthogonal cat states are formulated through suitable superpositions of even and odd parity states within the Fock representation space. The corresponding positive-definite measures over the entire complex plane are derived in terms of Meijer’s [Formula: see text]-functions parameterized by [Formula: see text] and [Formula: see text]. In addition, the nonclassical properties of the even and odd cat states are analyzed, including the manifestation of sub-Poissonian statistics, photon antibunching effect, and quadrature squeezing. We further investigate how the parameters [Formula: see text] and [Formula: see text] individually influence the enhancement or suppression of entanglement in the associated quasi-Bell states.
We define Schrödinger cat states as superpositions of q-deformed Barut–Girardello su (1, 1) coherent states with an adjustable angle φ in a q-deformed Fock space. We study the statistical properties of the q-deformed Barut–Girardello su (1, 1) coherent states and Schrödinger cat states. The statistical properties of photons are always sub-Poissonian for q-deformed Barut–Girardello su (1, 1) coherent states. For Schrödinger cat states in the cases φ = 0, π /2, π , the statistical properties of photons are always sub-Poissonian if φ = π /2, and the other cases are hard to determine because they depend on the parameters q and k. Moreover, we find some interesting properties of Schrödinger cat states in the limit | z | → 0, where z is the parameter of those states. We also derive that the statistical properties of photons are sub-Poissonian in the undeformed case where π /2 ≤ φ ≤ 3 π /2.
The generalized even and odd coherent states of a q-Deformation Non-harmonic Oscillator(E-O q-NHCSs) are constructed,their Higher-Order Squeezing and Antibunching Properties are studied in detail.It is increasing the Squeezing effect in different parameter ranges.With q-deformation increasing,the squeezing effect of E q-NHCSs are weakened and those of O q-NHCSs are enhanced.The E q-NHCSs appear high-order(N≥2) antibunching properties while the O q-NHCSs appear high-order bunching properties in some ranges of the parameter q and r 2 with the q-deformation increasing.Both antibunching effects of E q-NHCSs and bunching effects of O q-NHCSs are enhanced.