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.
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.
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 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 su(1,1) algebra is obtained by the q-deformed Bargmann-Fock realization.
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.
Nonclassical features of the pointer states in the q-deformed post-selected weak measurement
We study q-deformed coherent states of the Arik-Coon harmonic oscillator as the quantum resource in the post-selected weak measurement. First, we show how the precision of weak measurement is improved significantly due to q-deformation. Next, we focus on the role of the interplay between the deformation parameter and the interaction strength on the nonclassical nature of light. In particular, we show that sub-Poissonian photon distribution as characterized by Mandel parameter, photon antibunching effect, and quadrature squeezing are greatly enhanced by q-deformation. Our results not only advance the understanding of the fundamentals of q-deformed quantum mechanics, but also raise the potential to contribute to quantum technologies.
We construct the photon-added coherent states of a noncommutative harmonic oscillator associated to a q-deformed oscillator algebra. Various nonclassical properties of the corresponding system are explored, first, by studying two different types of higher-order quadrature squeezing, namely, the Hillery type and the Hong-Mandel type, and second, by testing the sub-Poissonian nature of photon statistics in higher order with the help of the correlation function and the Mandel parameter. Also, we compare the behavior of different types of quadrature and photon number squeezing of our system with those of the ordinary harmonic oscillator by considering the same set of parameters.
Noncommutative optical squeezed vacuum states are constructed as eigenstates of an appropriate two-photon annihilation operator corresponding to the Biedenharn–Macfarlane [Formula: see text]-oscillator. We consider in details the role of noncommutativity parameter [Formula: see text] on the nonclassical behaviors including quadrature squeezing and sub-Poissonian statistics. Also, we construct the noncommutative photon-added squeezed vacuum states and consider their Hillery-type higher-order squeezing and single-mode noise band.
Quantum statistical properties of general superpositions of coherent states of the type Nα,βα+eiξβ have been studied. Formulas for fluctuations (variances) of the coordinate and momentum quadratures and fluctuations of the number of photons in these states are obtained. The amplitude Nα,βnαeiφ+eiξnβeiφ,φ=φα=φβ and phase Nα,βneiφα+eiξneiφβ,n=nα=nβ superpositions of coherent states are considered separately. In the case of amplitude superpositions, a field under certain conditions occurs in a quadrature-squeezed state. At the same time, the photon distribution in the case of amplitude superpositions is always super-Poissonian. In the case of phase superpositions, a field under certain conditions occurs in quadrature-squeezed states for small average numbers of photons. For certain parameters of coherent states, the photon statistics in phase superpositions is sub-Poissonian. Using amplitude superpositions, it is possible to generate a microscopic electromagnetic field with the average photon number below or on the order of unity, as well as mesoscopic and even macroscopic fields with arbitrary large average photon numbers, characterized by significant quadrature squeezing. Using phase superpositions, it is possible to create under certain conditions the electromagnetic field states with sub-Poissonian statistics (squeezed distribution of photon numbers) for an arbitrary large average photon number. Expressions for a ‘strict’ coordinate–momentum uncertainty relation (Cauchy inequality) are obtained and analysed in cases of general, amplitude and phase superpositions of coherent states.