Deformed Boson Algebra with Parity Operator and Non-classical Properties of Coherent States

Abstract

In this paper we introduce the deformed boson algebra where the even number states as well as the odd number states are deformed. Based on the set of observables associated with the quantum state, we assume that photon obeys this deformed algebra. We construct the coherent states and investigate the non-classical properties of the coherent states such as photon statistics, Mandel parameter, and quantum fluctuations of quadrature.

Appendices

Appendix A

The Fock space for the deformed boson algebra is defined as

$$\begin{aligned} \mathcal{F} = \{ |n\rangle : n=0, 1, 2, \ldots \} \end{aligned}$$

(68)

For the deformed boson algebra including the parity operator, Fock space is decomposed into two sub Fock spaces as

$$\begin{aligned} \mathcal{F} =\mathcal{F}_e \oplus \mathcal{F}_o, \end{aligned}$$

(69)

where

$$\begin{aligned} \mathcal{F}_e= & {} \{ |n\rangle : n=2k, ~k=0, 1, 2, \ldots \} \end{aligned}$$

(70)

$$\begin{aligned} \mathcal{F}_o= & {} \{ |n\rangle : n=2k+1, \quad k=0, 1, 2, \ldots \} \end{aligned}$$

(71)

Now let us introduce the even coherent states and odd coherent states as

$$\begin{aligned} |z\rangle _e = N_e ( |z\rangle + |-z\rangle ) = 2 N_e c_0 \sum _{n=0}^{\infty } \frac{z^{2n}}{[2n]_{\nu }!} |2n\rangle \end{aligned}$$

(72)

and

$$\begin{aligned} |z\rangle _o = N_o ( |z\rangle - |-z\rangle )= 2 N_o c_0 \sum _{n=0}^{\infty } \frac{z^{2n+1}}{[2n+1]_{\nu }!} |2n+1\rangle , \end{aligned}$$

(73)

where the normalization constants are

$$\begin{aligned} N_e= & {} \frac{1}{ 2 c_0(x) \sqrt{ \cosh _{\nu } (x)}} \end{aligned}$$

(74)

$$\begin{aligned} N_o= & {} \frac{1}{ 2 c_0(x) \sqrt{ \sinh _{\nu } (x)}} \end{aligned}$$

(75)

Then, the completeness relation is given by

$$\begin{aligned} \int \int dz dz^* \left( \mu _e(x) |z\rangle _e {}_e\langle z| + \mu _o(x) |z\rangle _o {}_o\langle z| \right) =I \end{aligned}$$

(76)

Thus, if we find the weighting factors $\mu _e$ and $\mu _o$, the completeness relation is proved. From the Eq. (76), we have

$$\begin{aligned}&\int \int dz dz^* \left( \mu _e(x) |z\rangle _e {}_e\langle z| + \mu _o(x) |z\rangle _o {}_o\langle z| \right) \nonumber \\&\quad = \int \int dz dz^* \left( \mu _e ( 2N_e c_0)^2 \sum _{n=0}^{\infty } \sum _m \frac{z^{2n}(z^*)^{2m}}{\sqrt{[2n]_{\nu }! [2m]_{\nu }!}} |2n\rangle \langle 2m| \right) \nonumber \\&\qquad + \int \int dz dz^* \left( \mu _o ( 2N_o c_0)^2 \sum _{n=0}^{\infty } \sum _m \frac{z^{2n+1}(z^*)^{2m+1}}{\sqrt{[2n+1]_{\nu }! [2m+1]_{\nu }!}} |2n+1\rangle \langle 2m+1| \right) \nonumber \\&\quad = \pi \int _0^{\infty } dx \left( \mu _e ( 2N_e c_0)^2 \sum _{n=0}^{\infty } \frac{x^{2n}}{[2n]_{\nu }! } |2n\rangle \langle 2n|+ \mu _o ( 2N_o c_0)^2 \sum _{n=0}^{\infty } \frac{x^{2n+1}}{[2n+1]_{\nu }! } |2n+1\rangle \langle 2n+1| \right) \end{aligned}$$

(77)

Thus, we demand

$$\begin{aligned} \pi \int _0^{\infty } dx \mu _e ( 2N_e c_0)^2 x^{2n} = [2n]_{\nu }! \end{aligned}$$

(78)

and

$$\begin{aligned} \pi \int _0^{\infty } dx \mu _o ( 2N_o c_0)^2 x^{2n+1}= [2n+1]_{\nu }! \end{aligned}$$

(79)

Using the Eq. (26), we obtain

$$\begin{aligned} \pi \mu _e ( 2N_e c_0)^2= & {} \frac{1}{\sqrt{ 1 - \nu ^2}} e^{ - \frac{x}{\sqrt{ 1 - \nu ^2}}} \end{aligned}$$

(80)

$$\begin{aligned} \pi \mu _o ( 2N_o c_0)^2= & {} \frac{1}{1+\nu } e^{ - \frac{x}{\sqrt{ 1 - \nu ^2}}}, \end{aligned}$$

(81)

which completes completeness relation.

Appendix B

The ordinary Poisson distribution has the probability

$$\begin{aligned} P(n, x) = \frac{\lambda ^n }{n!} e^{ - \lambda }, \quad \lambda >0 \end{aligned}$$

(82)

The mean in the Poisson distribution is

$$\begin{aligned} \langle n \rangle = \lambda \end{aligned}$$

(83)

Based on the Eq. (45), the $\nu$-deformed Poisson distribution has the probability

$$\begin{aligned} P_{\nu }(n, x) = \frac{\lambda ^n}{[n]_{\nu }!} ( e_{\nu }(\lambda ))^{-1}, \quad \lambda >0 \end{aligned}$$

(84)

In this case the mean is

$$\begin{aligned} \langle n \rangle _{\nu } = \lambda \left[ \frac{ \frac{1}{1-\nu }\cosh \left( \frac{\xi }{\sqrt{1-\nu ^2}} \right) + \frac{ 1}{ \sqrt{ 1 - \nu ^2}} \sinh \left( \frac{\xi }{\sqrt{1-\nu ^2}} \right) }{\cosh \left( \frac{\xi }{\sqrt{1-\nu ^2}} \right) + \sqrt{ \frac{ 1 +\nu }{1-\nu }}\sinh \left( \frac{\xi }{\sqrt{1-\nu ^2}} \right) }\right] \end{aligned}$$

(85)

Thus, $\lambda$ is not the mean any more unless $\nu =0$. For small $\nu$, we have the mean

$$\begin{aligned} \langle n \rangle _{\nu } \approx \lambda \left( 1 + \nu e^{ - 2 \lambda } \right) \end{aligned}$$

(86)