Evolution Law of the Negative Binomial State in Laser Channel and its Photon-Number Decay Formula

Abstract

For the first time we examine how a negative binomial state (NBS), whose density operator is ${\sum }_{n=0}^{\infty }\frac {\left (n+s\right ) !} {n!s!}\gamma ^{s+1}\left (1-\gamma \right )^{n}\left \vert n\right \rangle \left \langle n\right \vert ,$ evolves in a laser channel. By using a newly derived generating function formula about Laguerre polynomial we obtain the evolution law of NBS, which turns out to be an infinite operator-sum of photon-added negative binomial state with a new negative-binomial parameter, and the photon number of NBS decays with e ^{−2(κ−g)t}, where g and κ represent the cavity gain and loss respectively. The technique of integration (summation) within an ordered product of operators is used in our discussions.

Appendix

The proof of Eq. 27.

In order to derive the new generating function formula about the Laguerre polynomials, we firstly calculate the infinite series sum

$$ \sum\limits_{n=0}^{\infty}\frac{\lambda^{n}}{n!}H_{n+m,n+s}\left(x,y\right) , $$

(37)

where

$$ H_{m,s}\left(x,y\right) =\sum\limits_{n=0}^{\min \left(m,s\right) }\frac {m!s!}{n!\left(m-n\right) !\left(s-n\right) !}\left(-1\right)^{n}x^{m-n}y^{s-n} $$

(38)

is the two-variable Hermite polynomials. Instead of directly calculating Eq. 37 we consider its operator counterpart ${\sum }_{n=0}^{\infty }\frac {\lambda ^{n}}{n!}\left (-i\right )^{m+s+2n}H_{n+m,n+s}\left (ia^{\dagger },ia\right )$. Using the operator identity [23]

$$ a^{s}a^{\dagger m}=\left(-i\right)^{m+s}\colon H_{m,s}\left(ia^{\dagger },ia\right) \colon, $$

(39)

we have

$$\begin{array}{@{}rcl@{}} && \sum\limits_{n=0}^{\infty}\frac{\lambda^{n}}{n!}\left(-i\right)^{m+s+2n} H_{n+m,n+s}\left(ia^{\dagger},ia\right) \\ && =\sum\limits_{n=0}^{\infty}\frac{\lambda^{n}}{n!}a^{n+s}\left(a^{\dagger }\right)^{n+m} \\ && =a^{s}{\vdots} e^{\lambda aa^{\dagger}}{\vdots} a^{\dagger m}. \end{array} $$

(40)

Then employing the completeness of coherent states $\left \vert z\right \rangle =\exp \left [ -\frac {\left \vert z\right \vert ^{2}}{2}+za^{\dagger }\right ] \left \vert 0\right \rangle$

$$ \int \frac{d^{2}z}{\pi}\left \vert z\right \rangle \left \langle z\right \vert =\int \frac{d^{2}z}{\pi}\colon \exp \left[ -\left \vert z\right \vert^{2}+za^{\dagger}+z^{\ast}a-a^{\dagger}a\right] \colon=1 $$

(41)

and the IWOP technique, we obtain

$$\begin{array}{@{}rcl@{}} a^{s}{\vdots} e^{\lambda aa^{\dagger}}{\vdots} a^{\dagger m} & =&\int \frac {d^{2}z}{\pi}z^{s}e^{\lambda \left \vert z\right \vert^{2}}z^{\ast m}\left \vert z\right \rangle \left \langle z\right \vert \\ & =&\int \frac{d^{2}z}{\pi}z^{s}z^{\ast m}\left \vert z\right \rangle \left \langle z\right \vert \colon e^{-\left(1-\lambda \right) \left \vert z\right \vert^{2}+za^{\dagger}+z^{\ast}a-a^{\dagger}a}\colon \\ & =&\left(-i\right)^{m+s}\left(1-\lambda \right)^{-\frac{\left(s+m\right) }{2}-1}\colon e^{\frac{\lambda a^{\dagger}a}{1-\lambda}} H_{m,s}\left(\frac{ia^{\dagger}}{\sqrt{1-\lambda}},\frac{ia}{\sqrt {1-\lambda}}\right) \colon. \end{array} $$

(42)

Comparing Eq. 40 with Eq. 42, we derive the identity of operators

$$ \sum\limits_{n=0}^{\infty}\frac{\left(-\lambda \right)^{n}}{n!}\colon{} H_{n+m,n+s}\left(ia^{\dagger},ia\right) \colon{} =\left(1{}-\lambda \right)^{-\frac{\left(s+m\right) }{2}-1}\colon e^{\frac{\lambda a^{\dagger} a}{1-\lambda}}H_{m,s}\left(\frac{ia^{\dagger}}{\sqrt{1-\lambda}},\frac {ia}{\sqrt{1-\lambda}}\right) \colon. $$

(43)

Returning to the classical case, i.e., substituting x for i a ^†, y for ia, we obtain

$$ \sum\limits_{n=0}^{\infty}\frac{\lambda^{n}}{n!}H_{n+m,n+s}\left(x,y\right) =\left(1+\lambda \right)^{-\frac{\left(s+m\right) }{2}-1}e^{\frac {\lambda xy}{1+\lambda}}H_{m,s}\left(\frac{x}{\sqrt{1+\lambda}},\frac {y}{\sqrt{1+\lambda}}\right) , $$

(44)

which is a new generating function formula about the two-variable Hermite polynomials H _{n + m, n + s} (x, y). Especially, when m = s, Eq. 44 reduces to

$$ \sum\limits_{n=0}^{\infty}\frac{\lambda^{n}}{n!}H_{n+s,n+s}\left(x,y\right) =\left(1+\lambda \right)^{-s-1}e^{\frac{\lambda xy}{1+\lambda}} H_{s,s}\left(\frac{x}{\sqrt{1+\lambda}},\frac{y}{\sqrt{1+\lambda}}\right). $$

(45)

On the other hand, comparing Eq. 11 with Eq. 38, we see

$$ L_{s}\left(xy\right) =\frac{\left(-1\right)^{s}}{s!}H_{s,s}\left(x,y\right) . $$

(46)

Then substituting (s + n)!(−1)^{s + n} L _{s + n} (z), (z = x y) for H _{n + s, n + s} (x, y) in Eq. 45, we obtain Eq. 27.