Nonclassical properties of two families of q-coherent states in the Fock representation space of q-oscillator algebra

Abstract

We clarify that the q-generalization of the simple harmonic oscillator to the Arik–Coon one leads us to obtain two different families of q-coherent states in a Fock representation space of the system. They are eigenstates of unbounded and bounded annihilation operators associated with the Arik–Coon q-oscillator. The first family satisfies the resolution of identity condition on all the complex plane and the second one on a disc in radius $1/\sqrt{1-q}$. Their positive definite q-measures are different, but in the limit $q\rightarrow 1$ both of them convert to the measure of well-known coherent states for the simple harmonic oscillator. The first and second families of the q-coherent states are also deformed eigenstates of the bounded and unbounded annihilation operators, respectively. Thus, it is possible to study the statistical properties of both q-coherent states via both bounded and unbounded operators. The nonclassical behaviours of interest in this article are signal-to-quantum noise ratio, sub-Poissonian photon statistics, photon antibunching, quadrature squeezing effect and bipartite entanglement for the two families of the q-coherent states, as well as Hillery-type higher-order squeezing for their corresponding photon-added states.

Appendix A: q-exponential functions and q-integral representations of the q-factorial

For a fixed parameter q, the two different generalizations of exponential function are defined as [42]

$$\begin{aligned}&{E_q(x)}:=\sum _{n=0}^{\infty }\frac{q^\frac{n(n-1)}{2}}{(q;q)_n}{x^n}, \end{aligned}$$

(A1a)

$$\begin{aligned}&{e_q(x)}:=\sum _{n=0}^{\infty }\frac{x^n}{(q;q)_n}, \end{aligned}$$

(A1b)

where ${(a;q)_n:=(1-a)(1-aq)(1-aq^2)\cdots (1-aq^{n-1})}$. The convergence regions of the q-exponential functions for $0<q<1$ are $|x|<\infty$ and $|x|<1$, respectively. The relation ${e_q(x)}{E_q(-x)}=1$ and the classical limits $\lim _{q\rightarrow 1} {e_q((1-q)x)}=exp(x)$ and $\lim _{q\rightarrow 1}{E_q((1-q)x)}=exp(x)$ are readily derived from (A1a) and (A1b). Furthermore, in the case $|q|<1$, there exists an infinite product expansion as follows [46]

$$\begin{aligned} e_{q}(x)=\frac{1}{\prod _{k=0}^\infty (1-q^{k} x)}. \end{aligned}$$

(A2)

Let us assume that f(x) and g(x) are two arbitrary and continuous functions on the real line. The symmetric and asymmetric q-derivatives $\widetilde{D}_{q,x}$ and $D_{q,x}$ are defined as [42]

$$\begin{aligned}&\widetilde{D}_{q,x}f(x):=\frac{f(q x)-f(q^{-1}x)}{(q-q^{-1})x},&\end{aligned}$$

(A3a)

$$\begin{aligned}&D_{q,x}f(x)=\frac{f(x)-f(qx)}{(1-q)x},&\end{aligned}$$

(A3b)

with the following q-analogues for the Leibniz rule:

$$\begin{aligned}&\widetilde{D}_{q,x}(f(x) g(x))=(\widetilde{D}_{q,x}f(x))\,g(q^{-1}x)+f(q x)\,\widetilde{D}_{q,x}g(x),&\end{aligned}$$

(A4a)

$$\begin{aligned}&{D_{q,x}}(f(x) g(x))=({D_{q,x}}f(x))\, g(qx)+f(x)\,D_{q,x}g(x).&\end{aligned}$$

(A4b)

Being the inverse operations of the q-derivatives $\widetilde{D}_{q,x}$ and ${D_{q,x}}$, the q-integrals on the intervals $[0,\infty )$ and [0, c] with c as a positive real number are defined as [41, 46]

$$\begin{aligned}&\int _0^{\infty }f(x)\widetilde{d_q}x=(q^{-1}-q)\sum _{j=-\infty }^{\infty }q^{2j+1}f(q^{2j+1}),&\end{aligned}$$

(A5a)

$$\begin{aligned}&\int _{0}^{c}f(x)d_{q}x:=(1-q)c\sum _{j=0}^{\infty }q^j{f(q^j{c})}.&\end{aligned}$$

(A5b)

For any nonzero complex number n, a q-number as $[n]_q:=\frac{1-q^n}{1-q}$ is associated. Also, the q-factorial is defined as $[0]_q!=1$ and $[n]_q!:=[1]_q [2]_q \cdots [n]_q$ for n as a positive integer number. For $0<q<1$, the following infinite and finite integral representations are obtained by the above definitions for the q- and $q^2$-factorials, respectively [40, 41, 46, 47]

$$\begin{aligned}&\int _{0}^{\infty }\frac{x^n}{E_{q^2}((q^{-1}-q)x)}\widetilde{d}_{q}x=q^{-n^2}[n]_{q^2}!,&\end{aligned}$$

(A6a)

$$\begin{aligned}&\int _{0}^{\frac{1}{1-q}}\frac{x^n}{{e_q}((1-q)x)}d_{q}x=[n]_q!.&\end{aligned}$$

(A6b)

We call the attention of the reader to an important issue: quantum number introduced in [46] is symmetric, and to obtain integral relation (A6a), one should write it in terms of the asymmetric quantum number used in this article. Both of the q-integrals convert to the infinite integral representation of the ordinary factorial as $q\rightarrow 1$.