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arXiv:1211.6889 · Journal of Physics A Mathematical and Theoretical

Ladder operators and squeezed coherent states of a three-dimensional generalized isotonic nonlinear oscillator

V. Chithiika Ruby, S Karthiga, Murugaian Senthilvelan

Published Dec 13, 2012• 7 citations

Abstract

We explore squeezed coherent states of a three-dimensional generalized isotonic oscillator whose radial part is the newly introduced generalized isotonic oscillator whose bound state solutions have been shown to admit the recently discovered X 1 -Laguerre polynomials. We construct a complete set of squeezed coherent states of this oscillator by exploring the squeezed coherent states of the radial part and combining the latter with the squeezed coherent states of the angular part. We also prove that the three-mode squeezed coherent states resolve the identity operator. We evaluate Mandel’s Q -parameter of the obtained states and demonstrate that these states exhibit sub-Possionian and super-Possionian photon statistics. Furthermore, we illustrate the squeezing properties of these states, both in the radial and angular parts, by choosing appropriate observables in the respective parts. We also evaluate the Wigner function of these three-mode squeezed coherent states and demonstrate the squeezing property explicitly.

DOI: 10.1088/1751-8113/46/2/025305

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We explore squeezed coherent states of a three-dimensional generalized isotonic oscillator whose radial part is the newly introduced generalized isotonic oscillator whose bound state solutions have been shown to admit the recently discovered X 1 -Laguerre polynomials. We construct a complete set of squeezed coherent states of this oscillator by exploring the squeezed coherent states of the radial part and combining the latter with the squeezed coherent states of the angular part. We also prove that the three-mode squeezed coherent states resolve the identity operator. We evaluate Mandel’s Q -parameter of the obtained states and demonstrate that these states exhibit sub-Possionian and super-Possionian photon statistics. Furthermore, we illustrate the squeezing properties of these states, both in the radial and angular parts, by choosing appropriate observables in the respective parts. We also evaluate the Wigner function of these three-mode squeezed coherent states and demonstrate the squeezing property explicitly.

We construct nonlinear squeezed states of a generalized isotonic oscillator potential. We demonstrate the non-existence of a dual counterpart of nonlinear squeezed states in this system. We investigate the statistical properties exhibited by the squeezed states, in particular Mandel's parameter, second-order correlation function, photon number distributions and parameter A 3 in detail. We also examine the quadrature and amplitude-squared squeezing effects. Finally, we derive the expression for the s -parameterized quasi-probability distribution function of these states. All this information about the system is new to the literature.

In this paper, we construct nonlinear coherent states for the generalized isotonic oscillator and study their non-classical properties in detail. By transforming the deformed ladder operators suitably, which generate the quadratic algebra, we obtain Heisenberg algebra. From the algebra, we define two non-unitary and one unitary displacement type operators. While the action of one of the non-unitary type operators reproduces the original nonlinear coherent states, the other one fails to produce a new set of nonlinear coherent states (dual pair). We show that these dual states are not normalizable. For the nonlinear coherent states, we evaluate the parameter A3 and examine the non-classical nature of the states through quadratic and amplitude-squared squeezing effect. Further, we derive analytical formula for the P-function, Q-function, and the Wigner function for the nonlinear coherent states. All of them confirm the non-classicality of the nonlinear coherent states. In addition to the above, we obtain the harmonic oscillator type coherent states from the unitary displacement operator.

Abstract We construct a new class of nonlinear coherent states for the isotonic oscillator by replacing the factorial of the coefficients z n / n !

{z}^{n}/\sqrt{n!}

of the canonical coherent states by the factorial x n γ ! = x 1 γ . x 2 γ … x n γ

{x}_{n}^{\gamma }!={x}_{1}^{\gamma }.{x}_{2}^{\gamma }\dots {x}_{n}^{\gamma }

with x 0 γ = 0

{x}_{0}^{\gamma }=0

, where x n γ

{x}_{n}^{\gamma }

is a sequence of positive numbers and γ is a positive real parameter. This also leads to the construction of a Bargmann-type integral transform which will allow us to find some integral transforms for orthogonal polynomials. The statistics of our coherent states will also be considered by the calculus of one called Mandel parameter. The squeezing phenomenon was also discussed.

We generalized the squeeze and displacement operators of the one-dimensional harmonic oscillator to the three-dimensional case and based on these operators we construct the corresponding coherent and squeezed states. We have also calculated the Wigner function for the three-dimensional harmonic oscillator and from the analysis of time evolution of this function, the quantum Liouville equation is also presented. Further properties of the quantum states including Mandel's Q and quadrature squeezing parameters are discussed as well.

We used the numerical method to study nonclassical properties of the even and odd generalized coherent states and superposition states of generalized coherent states for an isotonic oscillator. The following was shown. (1) The quantum statistical properties of the even and odd generalized coherent states are very different from those of the usual even and odd coherent states, and the N th-order ( N = 2 m + 1, m = 0,1,2,...) squeezing and sub-Poisson distribution appear alternately for both the even and odd generalized coherent states in some ranges of z = |β| 2 . The weaker the isotonic oscillator potential, the narrower the ranges. (2) The superposition states of generalized coherent states may exhibit the N th-order squeezing effect too, and this kind of higher-order squeezing effect appears periodically.

We construct a new type of photon-added squeezed coherent state generated by repeatedly operating the bosonic creation operator on a new type of squeezed coherent state [Fan H Y and Xiao M 1996 Phys. Lett. A 220 81]. We find that its normalization factor is related to single-variable Hermite polynomials. Furthermore, we investigate its statistical properties, such as Mandel’s Q parameter, photon-number distribution, and Wigner function. The nonclassicality is displayed in terms of the intense oscillation of photon-number distribution and the negativity of the Wigner function.