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Theoretical and Mathematical Physics

q-Deformed Barut–Girardello su(1, 1) coherent states and Schrödinger cat states

Yuefeng Zhao, Yan Zeng, Honggang Liu, Qi Song, Gangcheng Wang, Kang Xue

Published Dec 1, 2017• 3 citations
Publisher

Abstract

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.

DOI
10.1134/s0040577917120108

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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.
Canonical coherent states of a quantum harmonic oscillator have been generalized by requiring the conditions of normalizability, continuity in the label and resolution of the identity operator with a positive weight function. Superpositions of these states are considered in the present scenario as a generalization of the optical Schrödinger cat states. The Fock space is assumed to be canonical or finite-dimensional. The photon number distribution of these generalized Schrödinger cat states departs from the Poisson statistics in various ways for high photon numbers. For small nonvanishing values of the label, the photon number distribution is sub-Poissonian (nonclassical) or super-Poissonian, according to the interference properties. In fact, the sub- or super-Poissonian statistics is determined by the interplay between the relative phase and a critical value of the phase. The photon number distribution is uniquely sub-Poissonian for large values of the label.
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.
By introducing a generalization of the (p, q)-deformed boson oscillator algebra, we establish a two-parameter deformed oscillator algebra in an infinite-dimensional subspace of the Hilbert space of a harmonic oscillator without first finite Fock states. We construct the associated coherent states, which can be interpreted as photon-added deformed states. In addition to the mathematical characteristics, the quantum statistical properties of these states are discussed in detail analytically and numerically in the context of conventional as well as deformed quantum optics. Particularly, we find that for conventional (nondeformed) photons the states may be quadrature squeezed in both cases Q = pq < 1, Q = pq > 1 and their photon number statistics exhibits a transition from sub-Poissonian to super-Poissonian for Q < 1 whereas for Q > 1 they are always sub-Poissonian. On the other hand, for deformed photons, the states are sub-Poissonian for Q > 1 and no quadrature squeezing occurs while for Q < 1 they show super-Poissonian behavior and there is a simultaneous squeezing in both field quadratures.PACS Nos.: 42.50.Ar, 03.65.–w
The Barut-Girardello (BG) coherent states (CS) representation is extended to the noncompact algebras and in (reducible) quadratic boson realizations. The BG CS take the form of multimode ordinary Schrödinger cat states. Macroscopic superpositions of CS ( canonical CS, n = 1, 2, ...) are pointed out which are overcomplete in the N -mode Hilbert space and the relation between the canonical CS and the BG-type CS representations is established. The sets of and BG CS and their discrete superpositions contain many states studied in quantum optics (even and odd N -mode CS, pair CS) and provide an approach to quadrature squeezing, alternative to that of intelligent states. New subsets of weakly and strongly nonclassical states are pointed out and their statistical properties (first- and second-order squeezing, photon number distributions) are discussed. For specific values of the angle parameters and small amplitude of the canonical CS components, these states approach multimode Fock states with one, two or three bosons/photons. It is shown that eigenstates of a squared non-Hermitian operator (generalized cat states) can exhibit squeezing of the quadratures of A .
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.
In this Letter, we introduce four distinct classes of f-deformed photon-added nonlinear cat state. This would be achieved by recalling a nonlinear coherent states approach, as well as a particular class of Gilmore-Perelomov-type of SU(1,1) coherent state and a class of SU(2) coherent state. We then examine the role of photon addition and nonlinearity functions in the phase space structure and sub-Poissonianity of even, odd, and Yurke-Stoler nonlinear cat states. The effect of photon addition, which results in a π phase shift at the origin of the Wigner function toward negativity, interestingly enhances the nonclassicality by means of the Wigner function and Mandel parameter. Furthermore, owing to photon addition, we can observe a deformation in the Gaussian shape of the Wigner function, which may be found to be potentially useful in quantum noise reduction. Moreover, the deformation function shows a remarkable role in revealing the nonclassical behavior and can increase the depth and the domain of nonclassicality.