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arXiv:1907.04506 · Physical review. D/Physical review. D.

Deformed Gazeau-Klauder Schrödinger cat states with modified commutation relations

Chee Leong Ching, Wei Khim Ng

Published Oct 25, 2019• 8 citations
arXivHTMLPDF

Abstract

Generalized coherent states under deformed quantum mechanics that exhibit intrinsic minimum length [Phys. Rev. D 86, 064038 (2012)] and maximum momentum [Phys. Rev. D 88, 084009 (2013)] have been well studied following the Gazeau-Klauder approach. In this paper, as an extension to the study of quantum deformation, we investigate the famous Schr\"odinger cat states (SCs) under these two classes of quantum deformation. Following the concept of generalized Gazeau-Klauder Schr\"odinger cat states (GKSCs) in [J. Phys. A 45, 244006 (2012)], we construct the deformed GKSCs for both phenomenological models that exhibit intrinsic minimum length and/or maximum momentum. All comparisons between minimum length and maximum momentum deformations are illustrated and plots are done in even and odd cat states since they are one of the most important classic statistical characteristics of SCs. Probability distribution and entropies are studied. In general, deformed cat states do not possess the original even and odd state statistical properties. Nonclassical properties of the deformed GKSCs are explored in terms of the Mandel $Q$ parameter, quadrature squeezing $(\mathrm{\ensuremath{\Delta}}{X}_{q})\ifmmode\cdot\else\textperiodcentered\fi{}(\mathrm{\ensuremath{\Delta}}{Y}_{q})$, as well as Husimi quasiprobability distribution $\mathcal{Q}$. Some of these distinguishing quantum-gravitational features may possibly be realized qualitatively and even be measured quantitatively in future experiments with the advanced development in quantum atomic and optics technology.

DOI
10.1103/physrevd.100.085018

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Generalized coherent states under deformed quantum mechanics that exhibit intrinsic minimum length [Phys. Rev. D 86, 064038 (2012)] and maximum momentum [Phys. Rev. D 88, 084009 (2013)] have been well studied following the Gazeau-Klauder approach. In this paper, as an extension to the study of quantum deformation, we investigate the famous Schr"odinger cat states (SCs) under these two classes of quantum deformation. Following the concept of generalized Gazeau-Klauder Schr"odinger cat states (GKSCs) in [J. Phys. A 45, 244006 (2012)], we construct the deformed GKSCs for both phenomenological models that exhibit intrinsic minimum length and/or maximum momentum. All comparisons between minimum length and maximum momentum deformations are illustrated and plots are done in even and odd cat states since they are one of the most important classic statistical characteristics of SCs. Probability distribution and entropies are studied. In general, deformed cat states do not possess the original even and odd state statistical properties. Nonclassical properties of the deformed GKSCs are explored in terms of the Mandel QQQ parameter, quadrature squeezing (\ensuremathΔXq)\ifmmode⋅\else\textperiodcentered\fi(\ensuremathΔYq)(\mathrm{\ensuremath{\Delta}}{X}_{q})\ifmmode\cdot\else\textperiodcentered\fi{}(\mathrm{\ensuremath{\Delta}}{Y}_{q})(\ensuremathΔXq​)\ifmmode⋅\else\textperiodcentered\fi(\ensuremathΔYq​), as well as Husimi quasiprobability distribution Q\mathcal{Q}Q. Some of these distinguishing quantum-gravitational features may possibly be realized qualitatively and even be measured quantitatively in future experiments with the advanced development in quantum atomic and optics technology.
Following the Gazeau--Klauder approach, we construct generalized coherent states (GCS) as the quantum simulator to examine the deformed quantum mechanics, which exhibits an intrinsic maximum momentum. We study deformed harmonic oscillators and compute their probability distribution and entropy of states exactly. Also, a particle in an infinite potential box is studied perturbatively. In particular, unlike usual quantum mechanics, the present deformed case increases the entropy of the Planck scale quantum optical system. Furthermore, for simplicity, we obtain the modified uncertainty principle (MUP) with the perturbative treatment up to leading order. MUP turns out to increase generally. However, for certain values of $\ensuremath{\gamma}$ (a parameter of GCS), it is possible that the MUP will vanish and hence will exhibit the classical characteristic. This is interpreted as the manifestation of the intrinsic high-momentum cutoff at lower momentum in a perturbative treatment. Although the GCS saturates the minimal uncertainty in a simultaneous measurement of physical position and momentum operators, thus constituting the squeezed states, complete coherency is impossible in quantum gravitational physics. The Mandel Q number is calculated, and it is shown that the statistics can be Poissonian and super-/sub-Poissonian depending on $\ensuremath{\gamma}$. The equation of motion is studied, and both Ehrenfest's theorem and the correspondence principle are recovered. Fractional revival times are obtained through the autocorrelation, and they indicate that the superposition of a classical-like subwave packet is natural in GCS. We also contrast our results with the string-motivated (Snyder) type of deformed quantum mechanics, which incorporates a minimum position uncertainty rather than a maximum momentum. With the advances of quantum optics technology, it might be possible to realize some of these distinguishing quantum-gravitational features within the domain of future experiments.
Generalization of single-mode Schrödinger cat states is proposed, and their construction in terms of superposition of Gazeau–Klauder coherent states is presented. A comparative analysis of selected properties of this novel states for a nonlinear Kerr oscillator and the ‘traditional’ Schrödinger cats is given with emphasis on the photon statistics and the properties of non-cyclic geometric phase. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Coherent states: mathematical and physical aspects’.
Abstract In this paper, we present coherent states à la Gazeau-Klauder for a free particle in square well potential within position-deformed Heisenberg algebra . These states satisfy the Klauder’s mathematical requirement to build coherent states. Some statistical properties such as the probability distribution, the intensity correlation function and the Mandel parameter are calculated and analyzed. We find that these states are sub-Poissonian in nature. We also construct for these coherent states, the even cat states and we evaluate its Wigner function which analyses the quasiprobability distribution of these states. We graphically demonstrate that these states exhibit nonclassical behavior.
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.
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.
Abstract In this paper, we introduce some distinct classes of entangled cat states associated to generalized displaced Fock states. For this purpose, we use the formalism of nonlinear coherent states corresponding to nonlinear oscillator algebra which yields various kinds of f-deformed entangled states. We also take a particular class of Gilmore-Perelomov-type of SU(1, 1) and a class of SU(2) coherent states. We then obtain the amount of entanglement between subsystems of the quantum states of interest by applying the measure of concurrence. Furthermore, examining some of the most important criteria, such as Mandels Q parameter, quadrature squeezing and Vogels characteristic function, we study the nonclassicality of the introduced quantum states. The numerical results show remarkable values of entanglement, sub-Poissonian statistics of the field, and squeezing indicating that the introduced states can be regarded as possible candidates for nonclassical entangled states. Afterwards, we see that the Vogel function for quantum states of interests goes beyond the value of characteristic function of the ground state, which shows the nonclassicality of the introduced states.