# Density matrix of the superposition of excitation on coherent states with
thermal light and its statistical properties

Li-yun Hu  Department of Physics, Shanghai Jiao Tong University, Shanghai
200030, China  Hong-yi Fan  Department of Physics, Shanghai Jiao Tong
University, Shanghai 200030, China

(5 July 2008)

###### Abstract

A beam’s density matrix that is described by the superposition of excitation
on coherent states with thermal noise (SECST) is presented, and its matrix
elements in Fock space are calculated. The maximum information transmitted by
the SECST beam is derived. It is more than that by coherent light beam and
increases as the excitation photon number increases. In addition, the
nonclassicality of density matrix is demonstrated by calculating its Wigner
function.

PACS numbers: 42.50.Dv, 03.67.Hk, 03.65.Ud

excitation on coherent states, thermal light, Wigner function

##  I Introduction

Recently, much attention has been paid to the excitation on coherent states
(ECS) 1 ; 2 ; 3 ; 4 ; 5 ; 6 . As pointed out in Refs.2 ; 3 , the single photon
ECS causes a classical-to-quantum (nonclassical) transition. The ECSs can be
considered as a generalization of coherent states 7 ; 8  and number
eigenstates. All these states can be used as signal beams in optical
communications field, in which the nonclassicality of signals plays an
important role.

However, in reality, signal beams are usually mixed with thermal noise.
Statistical properties of the superposition of (squeezed) coherent states with
thermal light (SCST) have been investigated by calculating the photon number
matrix elements $\left\langle N\right|\rho\left|M\right\rangle$ of SCST’s
density matrix 9 ; 10 . These properties are useful in quantum optics and
quantum electronics (e.g. how lasers working well above threshold, heterodyne
detection of light, etc.) 11 . Some general properties of the density matrices
which describe coherent, squeezed and number eigenstates in thermal noise are
studied in Ref.12 . It is found that the information transmitted by the
superposition of number eigenstates with thermal light (SNET) beam is less
than that by the SCST beam 13 .

In this paper, we investigate statistical properties of the superposition of
ECS with thermal light (SECST). We present the relevant density matrix in Fock
space and derive the Mandel $Q$ parameter. The SECST field can exhibit a
significant amount of super-Poissonian photon statistics (PPS) due to the
presence of thermal noise for excitation photon number $m=0;$ while for $m\neq
0$ the SECST field can present the sub-PPS when the thermal mean photon number
is less than a threshold value. In addition, the threshold value increases as
$m$ increases. We also calculate the maximum information (channel capacity)
transmitted by the SECST beam, which increases as $m$ increases. In addition,
the nonclassicality of density matrix is also presented by calculating the
Wigner function of the SECST.

Our paper is arranged as follows. In Sec. II we present the density matrix
$\rho$ that describes the SECST and calculate its matrix elements in Fock
space by using the normal ordered form of $\rho$. The PPS distributions are
discussed in Sec III. The maximum information is calculated in Sec. IV. Sec. V
is devoted to deriving the Wigner function of the SECST and discussing its
nonclassicality in details. Conclusions are summarized in the last section.

##  II Excitation on coherent states with thermal noise

Firstly, let us briefly review the excitation on coherent states (ECSs). The
ECSs, first introduced by Agarwal and Tara 1 , are the result of successive
elementary one-photon excitations of a coherent state, and is an intermediate
state in between the Fock state and the coherent state, since it exhibits the
sub-Poissonian character. Theoretically, the ECSs can be obtained by
repeatedly operating the photon creation operator $a^{{\dagger}}$ on a
coherent state, so its density operator is

$\rho_{0}=C_{\alpha,m}a^{{\dagger}m}\left|\alpha\right\rangle\left\langle\alpha\right|a^{m},$
(1)

where $C_{\alpha,m}=[m!L_{m}(-\left|\alpha\right|^{2})]^{-1}$ is the
normalization factor,
$\left|\alpha\right\rangle=\exp(-\left|\alpha\right|^{2}/2+\alpha
a^{\dagger})\left|0\right\rangle$ is the coherent state 7 ; 8 , and
$L_{m}\left(x\right)$ is the $m$th-order Laguerre polynomial.

The SECST is described by the density matrix 12

$\displaystyle\rho$
$\displaystyle=\int\frac{d^{2}z}{\pi}P\left(z\right)D\left(z\right)\rho_{0}D^{{\dagger}}\left(z\right),$
(2) $\displaystyle P\left(z\right)$
$\displaystyle=\frac{1}{\bar{n}_{t}}\exp\left[-\frac{\left|z\right|^{2}}{\bar{n}_{t}}\right],$
(3)

where $D\left(z\right)=\exp(za^{{\dagger}}-z^{\ast}a)$ is the displacement
operator, and $\bar{n}_{t}$ is the mean number of thermal photons for
$\rho_{0}\rightarrow\left|0\right\rangle\left\langle 0\right|$. We can easily
prove that $\mathtt{Tr}\rho=1,$ as it should be. In fact,

$\displaystyle\mathtt{Tr}\rho$
$\displaystyle=\int\frac{d^{2}z}{\pi}P\left(z\right)\mathtt{Tr}\left[D\left(z\right)\rho_{0}D^{{\dagger}}\left(z\right)\right]$
$\displaystyle=\int\frac{d^{2}z}{\pi}P\left(z\right)\mathtt{Tr}\left(\rho_{0}\right)$
$\displaystyle=\int\frac{d^{2}z}{\pi}P\left(z\right)=1.$ (4)

###  II.1 Normal ordering form of the SECST

For the simplicity in our later calculation, we first perform the integration
in Eq.(2) by using the technique of integration within an ordered product
(IWOP) of operators 14 ; 15 . Using the normal ordering form of the vacuum
projector $\left|0\right\rangle\left\langle
0\right|=\colon\exp(-a^{{\dagger}}a)\colon,$ we can reform Eq.(2) as the
following form

$\displaystyle\rho$
$\displaystyle=C_{\alpha,m}e^{-\left|\alpha\right|^{2}}\int\frac{d^{2}z}{\pi}P\left(z\right)D\left(z\right)\colon
a^{{\dagger}m}$ $\displaystyle\times\exp\left(\alpha
a^{{\dagger}}+\alpha^{\ast}a-a^{{\dagger}}a\right)a^{m}\colon
D^{{\dagger}}\left(z\right)$
$\displaystyle=\frac{C_{\alpha,m}}{\bar{n}_{t}}\colon\exp\left(-\left|\alpha\right|^{2}-a^{{\dagger}}a+\allowbreak
a^{{\dagger}}\alpha+a\alpha^{\ast}\right)$
$\displaystyle\times\int\frac{d^{2}z}{\pi}\exp\left[-\frac{1+\bar{n}_{t}}{\bar{n}_{t}}\left|z\right|^{2}\right]$
$\displaystyle\times\exp\left[\left(a^{{\dagger}}-\alpha^{\ast}\right)\allowbreak
z+\left(a-\alpha\right)z^{\ast}\right]$
$\displaystyle\times\left(a^{{\dagger}}-z^{\ast}\right)^{m}\left(a-z\right)^{m}\colon.$
(5)

In the last step of (5), we noticed that for any operator $f(a^{{\dagger}},a)$

$D\left(z\right)f(a^{{\dagger}},a)D^{{\dagger}}\left(z\right)=f(a^{{\dagger}}-z^{\ast},a-z).$
(6)

Making two independent variable displacements,

$a^{{\dagger}}-z^{\ast}\rightarrow\beta^{\ast},a-z\rightarrow\beta,$

(note that operators $a^{{\dagger}},a$ can be considered as C-number within
the normal order $\colon\colon$), thus Eq.(5) can be rewritten as

$\displaystyle\rho$
$\displaystyle=\frac{C_{\alpha,m}}{\bar{n}_{t}}\colon\exp\left(-\left|\alpha\right|^{2}-\frac{1}{\bar{n}_{t}}a^{{\dagger}}a\right)$
$\displaystyle\times\int\frac{d^{2}\beta}{\pi}\beta^{\ast
m}\beta^{m}\exp\left[-\lambda_{t}^{-2}\left|\beta\right|^{2}\right.$
$\displaystyle\left.+\left(\allowbreak\alpha^{\ast}+\allowbreak\frac{a^{{\dagger}}}{\bar{n}_{t}}\right)\beta+\left(\frac{a}{\bar{n}_{t}}+\alpha\right)\beta^{\ast}\right]\colon$
$\displaystyle=\frac{C_{\alpha,m}}{\bar{n}_{t}}\lambda_{t}^{2m+2}\colon\exp\left(-\left|\alpha\right|^{2}-\frac{1}{\bar{n}_{t}}a^{{\dagger}}a\right)$
$\displaystyle\times\int\frac{d^{2}\beta}{\pi}\beta^{\ast
m}\beta^{m}\exp\left[-\left|\beta\right|^{2}+A^{{\dagger}}\beta+A\beta^{\ast}\right]\colon,$
(7)

where we have set $\lambda_{t}=\sqrt{\bar{n}_{t}/(1+\bar{n}_{t})}$ and
$A=\lambda_{t}(\frac{1}{\bar{n}_{t}}a+\alpha).$ Then using the integration
expression of two-variable Hermite polynomial $H_{m,n}$ 16 ,

$\displaystyle(-1)^{n}e^{-\xi\eta}H_{m,n}\left(\xi,\eta\right)$
$\displaystyle=\int\frac{d^{2}z}{\pi}z^{n}z^{\ast
m}\exp\left[-\left|z\right|^{2}+\xi z-\eta z^{\ast}\right],$ (8)

we can put Eq.(7) into

$\displaystyle\rho$
$\displaystyle=\frac{C_{\alpha,m}}{\bar{n}_{t}}\lambda_{t}^{2m+2}\colon\left(-1\right)^{m}H_{m,m}\left(A^{{\dagger}},-A\right)$
$\displaystyle\times\exp\left[-\frac{\left(a-\alpha\right)\left(a^{{\dagger}}-\alpha^{\ast}\right)}{\bar{n}_{t}+1}\right]\colon.$
(9)

In particular, when $m=0$, corresponding to the case of superposition of
coherent state with thermal noise, Eq.(9) reduces to

$\rho=\frac{1}{\bar{n}_{t}+1}D\left(\alpha\right)\colon
e^{-\frac{a^{{\dagger}}a}{\bar{n}_{t}+1}}\colon
D^{{\dagger}}\left(\alpha\right),$ (10)

which can be directly checked by using Eqs.(2) and (3) as well as noticing
$\rho_{0}=\left|\alpha\right\rangle\left\langle\alpha\right|.$

Further employing the relation between Hermite polynomial and Laguerre
polynomial 16 ,

$H_{m,n}\left(\xi,\kappa\right)=\left\\{\begin{array}[c]{cc}n!\left(-1\right)^{n}\xi^{m-n}L_{n}^{m-n}\left(\xi\kappa\right),&m>n\\\
m!\left(-1\right)^{m}\kappa^{n-m}L_{m}^{n-m}\left(\xi\kappa\right),&m<n\end{array}\right.,$
(11)

we can see that

$\displaystyle\rho$
$\displaystyle=\frac{1}{L_{m}(-\left|\alpha\right|^{2})}\frac{\bar{n}_{t}^{m}}{(1+\bar{n}_{t})^{m+1}}\colon
L_{m}\left(-A^{{\dagger}}A\right)$
$\displaystyle\times\exp\left[-\frac{\left(a-\alpha\right)\left(a^{{\dagger}}-\alpha^{\ast}\right)}{\bar{n}_{t}+1}\right]\colon.$
(12)

Eqs.(9) and (12) are the normal ordering form of the SECST. From these it is
convenient to calculate the phase space distributions, such as Q-function,
P-representation and Wigner function.

###  II.2 The matrix elements $\left\langle N\right|\rho\left|M\right\rangle$

Now we calculate the matrix elements of $\rho$ in Eq.(2) between two number
states $\left\langle N\right|$ and $\left|M\right\rangle,$ i.e., $\left\langle
N\right|\rho\left|M\right\rangle.$ Employing the overcompleteness of coherent
states, one can express the matrix elements $\left\langle
N\right|\rho\left|M\right\rangle$ as

$\left\langle N\right|\rho\left|M\right\rangle=\int\frac{d^{2}\beta
d^{2}\gamma}{\pi^{2}}\left\langle
N\right.\left|\beta\right\rangle\left\langle\beta\right|\rho\left|\gamma\right\rangle\left\langle\gamma\right.\left|M\right\rangle,$
(13)

where the overlap between the coherent state and the number state is given by

$\left\langle\gamma\right.\left|M\right\rangle=\frac{1}{\sqrt{M!}}e^{-\left|\gamma\right|^{2}/2}\gamma^{\ast
M},$ (14)

and the matrix elements
$\left\langle\beta\right|\rho\left|\gamma\right\rangle$ can be obtained from
Eq.(9) due to $\rho^{\prime}$s normal ordering form,

$\displaystyle\left\langle\beta\right|\rho\left|\gamma\right\rangle$
$\displaystyle=\left(-1\right)^{m}\frac{C_{\alpha,m}}{\bar{n}_{t}}\lambda_{t}^{2m+2}e^{-\left|\alpha\right|^{2}/(\bar{n}_{t}+1)}$
$\displaystyle\times\frac{\partial^{2m}}{\partial\tau^{m}\partial\tau^{\prime
m}}\exp\left[-\tau\tau^{\prime}+\lambda_{t}\tau\alpha^{\ast}-\lambda_{t}\alpha\tau^{\prime}\right]$
$\displaystyle\times\exp\left\\{\left(\frac{\alpha+\bar{n}_{t}\allowbreak\gamma}{\bar{n}_{t}+1}+\frac{\lambda_{t}\tau}{\bar{n}_{t}}\right)\beta^{\ast}-\frac{1}{2}\left|\beta\right|^{2}\right.$
$\displaystyle-\left.\frac{1}{2}\left|\gamma\right|^{2}+\left(\frac{\alpha^{\ast}}{\bar{n}_{t}+1}-\frac{\lambda_{t}\tau^{\prime}}{\bar{n}_{t}}\right)\gamma\right\\}_{\tau=\tau^{\prime}=0},$
(15)

where we have used the generating function of two-variable Hermite polynomial
$H_{m,n},$

$H_{m,n}\left(x,y\right)=\left.\frac{\partial^{m+n}}{\partial t^{m}\partial
t^{\prime
n}}\exp\left[-tt^{\prime}+tx+t^{\prime}y\right]\right|_{t=t^{\prime}=0}.$ (16)

When $M=N,$ $\left\langle N\right|\rho\left|N\right\rangle$ is just the photon
number distribution of the SECST. Then combing with Eqs.(15), (13) and (14),
after a lengthy but straightforward calculation, one can get the matrix
elements $\left\langle N\right|\rho\left|M\right\rangle,$ (without loss of
generality, let $M\geqslant N$)

$\displaystyle\left\langle N\right|\rho\left|M\right\rangle$
$\displaystyle=\frac{\left(-1\right)^{N}}{\sqrt{M!N!}}\frac{\lambda_{t}^{2N}C_{\alpha,m}}{\bar{n}_{t}+1}e^{-\left|\alpha\right|^{2}}\frac{\partial^{2m}}{\partial\upsilon^{m}\partial\upsilon^{\prime
m}}$
$\displaystyle.\left\\{e^{\lambda_{t}^{2}\upsilon\upsilon^{\prime}}H_{M,N}\left(\frac{\upsilon^{\prime}}{\bar{n}_{t}+1},-\frac{\upsilon}{\bar{n}_{t}}\right)\right\\}_{\upsilon=\alpha,\upsilon^{\prime}=\alpha^{\ast}},$
(17)

where we have used the integral formula 17

$\int\frac{d^{2}\beta}{\pi}f\left(\beta^{\ast}\right)\exp\left\\{-\left|\beta\right|^{2}+\tau\beta\right\\}=f\left(\tau\right),$
(18)

and another expression of two-variable Hermite polynomial $H_{m,n},$

$H_{m,n}\left(\xi,\kappa\right)=\sum_{l=0}^{\min(m,n)}\frac{m!n!\left(-1\right)^{l}\xi^{m-l}\kappa^{n-l}}{l!\left(n-l\right)!\left(m-l\right)!}.$
(19)

In particular, when $m=0$, noticing $M\geqslant N$ and Eq.(11), Eq.(17)
reduces to

$\displaystyle\left\langle N\right|\rho\left|M\right\rangle$
$\displaystyle=\sqrt{\frac{N!}{M!}}\alpha^{\ast
M-N}\frac{\left(\bar{n}_{t}\right)^{N}}{\left(\bar{n}_{t}+1\right)^{M+1}}$
$\displaystyle\times
e^{-\left|\alpha\right|^{2}/(\bar{n}_{t}+1)}L_{N}^{M-N}\left[-\frac{\left|\alpha\right|^{2}}{\bar{n}_{t}\left(\bar{n}_{t}+1\right)}\right],$
(20)

which is just the Glauber-Lachs formula 9  when
$\bar{n}_{t}=(e^{\beta\omega}-1)^{-1}$. While for $\alpha=0,$ corresponding to
the case of superposition of number state with thermal light, using Eq.(19),
Eq.(17) becomes

$\left\langle N\right|\rho\left|M\right\rangle=\delta_{M,N}P_{N},$ (21)

where ($k_{0}=\max[0,m-N]$)

$P_{N}=\frac{m!N!}{\bar{n}_{t}+1}\sum_{k=k_{0}}^{m}\frac{1}{k!}\frac{\left(\frac{\bar{n}_{t}}{\bar{n}_{t}+1}\right)^{k+N}\left[\bar{n}_{t}\left(\bar{n}_{t}+1\right)\right]^{k-m}}{\left(k+N-m\right)!\left[(m-k)!\right]^{2}}.$
(22)

Eq.(21) is just the result of Ref. 13 .

##  III Sub-Poissonian photon statistics

To see clearly the photon statistics properties of the SECST, in this section,
we pay our attention to the variance of the photon number operator
$\left\langle\left(\Delta\hat{n}\right)^{2}\right\rangle=\left\langle\hat{n}^{2}\right\rangle-\left\langle\hat{n}\right\rangle^{2}.$
In particular, we will examine the evolution of the Mandel $Q$ parameter
defined as

$\displaystyle Q$
$\displaystyle=\frac{\left\langle\left(a^{{\dagger}}a\right)^{2}\right\rangle}{\left\langle
a^{{\dagger}}a\right\rangle}-\left\langle a^{{\dagger}}a\right\rangle$
$\displaystyle=\frac{\left\langle
a^{2}a^{{\dagger}2}\right\rangle-\left\langle
aa^{{\dagger}}\right\rangle^{2}-\left\langle
aa^{{\dagger}}\right\rangle}{\left\langle aa^{{\dagger}}\right\rangle-1},$
(23)

which measures the derivation of the variance of the photon number
distribution of the field state under consideration from the Poissonian
distribution of the coherent state. $Q=1,Q>1$ and $Q<1$ correspond to
Poissonian photon statistics (PPS), super-PPS and sub-PPS, respectively.

In order to calculate the average value in Eq.(23), we first calculate the
value of
$\left\langle\alpha\right|a^{n}a^{{\dagger}m}\left|\alpha\right\rangle.$ In
fact, using

$\left\langle\alpha\right|a^{m+n}a^{{\dagger}m+n}\left|\alpha\right\rangle=\left(m+n\right)!L_{m+n}(-\left|\alpha\right|^{2})$
(24)

and

$\int\frac{d^{2}z}{\pi}z^{n}z^{\ast
m}P\left(z\right)=\bar{n}_{t}^{m}m!\delta_{m,n},$ (25)

we can evaluate (for writing’s convenience, let $L_{m}$ denote
$L_{m}(-\left|\alpha\right|^{2})$)

$\left\langle
a^{{\dagger}}a\right\rangle=\frac{1+m}{L_{m}}L_{m+1}+\bar{n}_{t}-1,$ (26)

and

$\left\langle
a^{2}a^{{\dagger}2}\right\rangle=2\bar{n}_{t}^{2}+\frac{m+1}{L_{m}}\left[4\bar{n}_{t}L_{m+1}+\left(m+2\right)L_{m+2}\right].$
(27)

Substituting Eqs.(26) and (27) into (23) leads to

$\displaystyle Q$
$\displaystyle=\frac{\bar{n}_{t}\left(\bar{n}_{t}-1\right)L_{m}+\left(2\bar{n}_{t}-1\right)\left(m+1\right)L_{m+1}}{\left(1+m\right)L_{m+1}+\left(\bar{n}_{t}-1\right)L_{m}}$
$\displaystyle+\frac{(m+1)(m+2)L_{m+2}-\frac{\left(m+1\right)^{2}}{L_{m}}L_{m+1}^{2}}{\left(1+m\right)L_{m+1}+\left(\bar{n}_{t}-1\right)L_{m}}.$
(28)

At the zero-temperature limit $(\bar{n}_{t}\rightarrow 0)$, Eq.(28) just
reduces to Eq.(2.20) in Ref.1 .

Figure 1: (Color online) The evolution of Mandel $Q$ parameter as a function
of ($n_{t},\left|\alpha\right|$) for different values $m.$

In Fig.1, we display the parameter $Q\left(n_{t},\left|\alpha\right|\right)$
as a function of $\left(n_{t},\left|\alpha\right|\right)$ for different values
$m.$ From Fig.1, we see that, for the excitation photon number $m=0$ (see
Fig.1 (a)), $Q\left(\bar{n}_{t}=0,\left|\alpha\right|\right)=1$ corresponding
to coherent state (a PPS); while $Q\left(\bar{n}_{t}\neq
0,\left|\alpha\right|\right)>1$, i.e., the SECST field exhibits a significant
amount of super-PPS due to the presence of $\bar{n}_{t}$. From Fig.1 (b) and
(c), we see that, when $m\neq 0,$ the SECST field presents the sub-PPS when
$\bar{n}_{t}$ is less than a threshold value for a given
$\left|\alpha\right|;$ the threshold value increases as $m$ increases. For
example, when $\left|\alpha\right|=0,$ the threshold values are about 0.414
and 0.481, respectively, for $m=1$ and $m=6$.

##  IV Information transmitted by the SECST beam

According to the negentropy principle of Brillouin 18 , the maximum
information $I$ transmitted by a beam is

$I=S_{\max}-S_{act},$ (29)

in which $S_{\max}$ and $S_{act}$ represent the maximum entropy and the actual
entropy, respectively, possessed by the quantum mechanical system described by
a density matrix $\rho$. Here the maximum information $I$ is an ideal one
transmitted through an ideal optical communication system.

For the SECST system, the actual entropy is

$S_{act}=-\mathtt{Tr}\left(\rho\ln\rho\right)=-\sum_{N}\sigma_{N}\ln\sigma_{N},$
(30)

where $\rho=\sum_{N}\sigma_{N}\left|N\right\rangle\left\langle N\right|,$ and
$\sigma_{N}=\left\langle N\right|\rho\left|N\right\rangle.$ $\sigma_{N}$ can
be obtained from Eq.(17), i.e.,

$\displaystyle\sigma_{N}$
$\displaystyle=\frac{\bar{n}_{t}^{N}e^{-\left|\alpha\right|^{2}}C_{\alpha,m}}{\left(\bar{n}_{t}+1\right)^{N+1}}\frac{\partial^{2m}}{\partial\upsilon^{m}\partial\upsilon^{\prime
m}}$
$\displaystyle\times\left\\{e^{\lambda_{t}^{2}\upsilon\upsilon^{\prime}}L_{N}\left(\frac{-\upsilon\upsilon^{\prime}}{\bar{n}_{t}\left(\bar{n}_{t}+1\right)}\right)\right\\}_{\upsilon=\alpha,\upsilon^{\prime}=\alpha^{\ast}},$
(31)

which is independent of the phase of $\alpha.$ On the other hand, for a system
in thermal equilibrium, described by the density matrix $\rho_{th}$, with mean
photons number $\bar{n}_{t}$, its entropy is

$S=-\sum_{N}P_{N}\ln
P_{N}=\ln(1+\bar{n}_{t})+\bar{n}_{t}\ln\frac{\bar{n}_{t}+1}{\bar{n}_{t}},$
(32)

where $P_{N}=\bar{n}_{t}^{N}/\left(\bar{n}_{t}+1\right)^{N=1}$ obtained from
Eq.(20) under the condition $m=0,\alpha=0.$ Note that the maximum entropy of
the system is equal to the entropy of a system in thermal equilibrium, with an
equal mean number of photons. The mean photons number of the SECST is given by
Eq.(26). Therefore, using Eq.(32), we have

$\displaystyle S_{\max}$
$\displaystyle=\ln\left(\left(1+m\right)\frac{L_{m+1}}{L_{m}}+\bar{n}_{t}\right)$
$\displaystyle+\left(\left(1+m\right)\frac{L_{m+1}}{L_{m}}+\bar{n}_{t}-1\right)$
$\displaystyle\times\ln\left(\frac{\left(1+m\right)L_{m+1}+\bar{n}_{t}L_{m}}{\left(1+m\right)L_{m+1}+\left(\bar{n}_{t}-1\right)L_{m}}\right).$
(33)

Figure 2: (Color online) The maximum information
$I\left(\bar{n}_{t},\left|\alpha\right|\right)$as a function of
$\left(\bar{n}_{t},\left|\alpha\right|\right)$ for some different values (a)
$m=0,(b)$ $m=1,$(truncating the infinite sum at $N_{\max}=70$). Figure 3:
(Color online) The maximum information
$I\left(\bar{n}_{t},\left|\alpha\right|=1\right)$as a function of
$\left(\bar{n}_{t}\right)$ for some different values (a) $m=0,(b)$ $m=1,(c)$
${\small m=2}$ (truncating the infinite sum at $N_{\max}=70$).

From Eqs.(29), (30) and (33), we can calculate the maximum information
transmitted by the SECST beam. In Fig. 2, the maximum information
$I\left(\bar{n}_{t},\left|\alpha\right|\right)$ is plotted as a function of
$\left(\bar{n}_{t},\left|\alpha\right|\right)$ for some different values $m$
(truncating the infinite sum at $N_{\max}=70$). From Fig.2, we can see that,
for a given $\bar{n}_{t},$ $I\left(\bar{n}_{t},\left|\alpha\right|\right)$
increases as the value $\left|\alpha\right|$ increases; for given
$\left|\alpha\right|,$ in general,
$I\left(\bar{n}_{t},\left|\alpha\right|\right)$ grows up as the value
$\bar{n}_{t}$ increases. In order to see clearly the effect of different
parameter $m$ to $I\left(\bar{n}_{t},\left|\alpha\right|\right),$ we presented
a plot in Fig.3, from which it is obvious that
$I\left(\bar{n}_{t},\left|\alpha\right|\right)$ becomes bigger due to the
presence of $m$, and increases as $m$ increases. In other words, the maximum
information transmitted by the SECST beam is larger than that by the SCST
($m=0$). The channel of ECS can carry with more information than that of
coherent state. In Ref. 13 , Vourdas has pointed out that the coherent signals
(of known phase) can transmit more information than the number eigenvectors
signals. Thus among these three beams, the SECST beam can transmit most
information.

##  V The Wigner function of the SECST

###  V.1 The Wigner function

The Wigner function (WF) plays an important role in quantum optics, especially
the WF can be reconstructed from measurements 19 ; 20 . The WF is a powerful
tool to investigate the nonclassicality of optical fields. The presence of
negativity in the WF of optical field is a signature of its nonclassicality is
often used to describe the decoherence of quantum states. In this section,
using the normally ordered form of the SECST, we evaluate its WF. For a
single-mode system, the WF is given by 21

$W\left(\gamma,\gamma^{\ast}\right)=\frac{e^{2\left|\gamma\right|^{2}}}{\pi}\int\frac{d^{2}\beta}{\pi}\left\langle-\beta\right|\rho\left|\beta\right\rangle
e^{2\left(\beta^{\ast}\gamma-\beta\gamma^{\ast}\right)},$ (34)

where $\left|\beta\right\rangle$ is the coherent state and $\gamma=x+iy$. From
Eq.(34) it is easy to see that once the normal ordered form of $\rho$ is
known, we can conveniently obtain the WF of $\rho.$

On substituting Eq.(9) into Eq.(34) we obtain the WF of the SECST

$\displaystyle W\left(\gamma,\gamma^{\ast}\right)$
$\displaystyle=\frac{\left(\lambda_{t}^{2}A_{1}^{2}\right)^{m}C_{\alpha,m}}{\pi\left(2\bar{n}_{t}+1\right)}\exp\left\\{-\frac{2\left|\alpha-\gamma\right|^{2}}{2n_{t}+1}\right\\}$
$\displaystyle\times\left(-1\right)^{m}H_{m,m}\left(\frac{A_{2}^{\ast}}{A_{1}},-\frac{A_{2}}{A_{1}}\right)$
$\displaystyle=\frac{\left(\lambda_{t}^{2}A_{1}^{2}\right)^{m}\exp\left\\{-\frac{2\left|\alpha-\gamma\right|^{2}}{2\bar{n}_{t}+1}\right\\}}{\pi\left(2\bar{n}_{t}+1\right)L_{m}\left(-\left|\alpha\right|^{2}\right)}L_{m}\left(-\left|A_{2}\right|^{2}/A_{1}^{2}\right),$
(35)

where we have set

$\displaystyle A_{1}^{2}$
$\displaystyle=1-\frac{1}{\left(2\bar{n}_{t}+1\right)\bar{n}_{t}},$
$\displaystyle A_{2}$
$\displaystyle=\frac{\lambda_{t}\left(\bar{n}_{t}+1\right)}{\left(2\bar{n}_{t}+1\right)\bar{n}_{t}}\left(2\bar{n}_{t}\alpha-\alpha+2\gamma\right).$
(36)

Figure 4: (Color online) The evolution of the Wigner function of the SECST
with $\alpha=0.2+0.2i$ for several different values $m$ and $\bar{n}_{t}.$

Noticing that $L_{m}(-\left|\alpha\right|^{2})>0,$ and
$L_{m}[-\left|A_{2}\right|^{2}/A_{1}^{2}]>0$ when
$1-\frac{1}{\left(2\bar{n}_{t}+1\right)\bar{n}_{t}}>0,$ thus the WF of the
SECST is always positive under the condition of $\bar{n}_{t}>1/2$. In
particular, when $m=0$, Eq.(35) becomes

$W\left(\gamma,\gamma^{\ast}\right)=\frac{1}{\pi\left(2\bar{n}_{t}+1\right)}\exp\left\\{-\frac{2\left|\alpha-\gamma\right|^{2}}{2\bar{n}_{t}+1}\right\\},$
(37)

which corresponds to the thermal state with mean photon number $\bar{n}_{t}$.
While for $\alpha=0,$ $A_{2}\rightarrow
2\gamma\lambda_{t}\left(\bar{n}_{t}+1\right)/[\left(2\bar{n}_{t}+1\right)\bar{n}_{t}],$
$\left|A_{2}\right|^{2}/A_{1}^{2}\rightarrow
4\left|\gamma\right|^{2}\left(\bar{n}_{t}+1\right)/\\{\left(2\bar{n}_{t}+1\right)\left[\left(2\bar{n}_{t}+1\right)\bar{n}_{t}-1\right]\\}\equiv\xi$,
Eq. (35) yields

$W\left(\gamma,\gamma^{\ast}\right)=\frac{\left[\left(2\bar{n}_{t}+1\right)\bar{n}_{t}-1\right]^{m}}{\pi\left(2\bar{n}_{t}+1\right)^{m+1}\left(\bar{n}_{t}+1\right)^{m}}e^{-\frac{2\left|\gamma\right|^{2}}{2\bar{n}_{t}+1}}L_{m}\left(-\xi\right).$
(38)

At the zero-temperature limit, $T\rightarrow 0,\bar{n}_{t}\rightarrow 0,$
Eq.(37) reduces now into
$\frac{1}{\pi}\exp(-2\left|\alpha-\gamma\right|^{2}),$ i.e., the WF of
coherent state (a Guassian form), which can be seen from Eq.(2) yielding
$\rho=\left|\alpha\right\rangle\left\langle\alpha\right|$ under the condition
$m=0$; while Eq.(38) becomes
$\frac{1}{\pi}(-1)^{m}e^{-2\left|\gamma\right|^{2}}L_{m}(4\left|\gamma\right|^{2}),$
corresponding to the WF of number state.

Using Eq.(35), the WFs of the SECST are depicted in Fig.4 in phase space with
$\alpha=0.2+0.2i$ for several different values $m$ and $\bar{n}_{t}.$ It is
easy to see that the negative region of WF gradually disappears as $m$ and
$\bar{n}_{t}$ and increases.

###  V.2 The marginal distributions of the SECST

We now find the probability distribution of position or momentum$|$—–the
marginal distributions, by performing the WF either over the variable $y$ or
the variable $x$, respectively. Using Eqs.(35) and (36) we can derive (denote
$\gamma=x+iy,$ $\alpha=q+ip$)

$\displaystyle\mathrm{P}\left(x,\bar{n}_{t}\right)$ $\displaystyle\equiv\int
W\left(x,y\right)dy$
$\displaystyle=\frac{\left(\lambda_{t}^{2}A_{1}^{2}\right)^{m}C_{\alpha,m}}{\sqrt{2\pi\left(2\bar{n}_{t}+1\right)}}\frac{\left[m!\right]^{2}e^{-\frac{2\left(q-x\right)^{2}}{2\bar{n}_{t}+1}}}{\left(2\bar{n}_{t}-1\right)^{m}}$
$\displaystyle\times\sum_{k=0}^{m}\frac{2^{2k-m}\bar{n}_{t}^{k}}{k!\left[(m-k)!\right]^{2}}\left|H_{m-k}\left(E_{1}\right)\right|^{2},$
(39)

where $H_{m}\left(x\right)$ is single variable Hermite polynomial and
$E_{1}=\left[\left(2\bar{n}_{t}-1\right)\alpha+2x+2ip\right]/\sqrt{2\left(2\bar{n}_{t}+1\right)}$.
Eq.(39) is the marginal distribution of WF of the SECST in “$x$-direction”.

On the other hand, performing the integration over $dx$ yields the other
marginal distribution in “$y$-direction”,

$\displaystyle\mathrm{P}\left(y,\bar{n}_{t}\right)$
$\displaystyle=\frac{\left(\lambda_{t}^{2}A_{1}^{2}\right)^{m}C_{\alpha,m}}{\sqrt{2\pi\left(2\bar{n}_{t}+1\right)}}\frac{\left[m!\right]^{2}e^{-\frac{2(p-y)^{2}}{2\bar{n}_{t}+1}}}{\left(2\bar{n}_{t}-1\right)^{m}}$
$\displaystyle\times\sum_{k=0}^{m}\frac{2^{2k-m}\bar{n}_{t}^{k}}{k!\left[(m-k)!\right]^{2}}\left|H_{m-k}\left(E_{2}\right)\right|^{2},$
(40)

where
$E_{2}=i\left(2\bar{n}_{t}\alpha-\alpha+2q+2iy\right)/[\sqrt{2\left(2\bar{n}_{t}+1\right)}]$.
As expected, the two marginal distributions are both real.

##  VI Conclusions

In summary, we have investigated the photon statistics properties of the
SECST, described by the density matrix $\rho$ (2). We have calculated the
matrix elements $\left\langle N\right|\rho\left|M\right\rangle$ in Fock space
and the Mandel $Q$ parameter. It is found that the SECST field exhibits a
significant amount of super-PPS due to the presence of thermal noise
($\bar{n}_{t}$) for excitation photon number $m=0$ and that, for $m\neq 0$ and
a given $\left|\alpha\right|,$ the SECST field presents the sub-PPS when
$\bar{n}_{t}$ is less than a threshold value. In addition, the threshold value
increases as $m$ increases. We have presented the maximum information (channel
capacity) transmitted by the SECST beam. It is shown that the maximum
information transmitted increases as $m$ increases. This implies that among
the coherent signals, the eigen-number signals and the ECS in thermal light,
the last one can transmit the most information. Further, as one of the photon
statistical properties, the Wigner function and the marginal distributions of
the SECST have also been derived, from which one can clearly see the
nonclassicality. The negative region has no chance to be present when the
average photon number $\bar{n}_{t}$ of thermal noise exceeds $1/2.$ The
marginal distributions are related to the Hermite polynomial.

###### Acknowledgements.

This work is supported by the National Natural Science Foundation of China
(Grant No 10775097). L.-Y. Hu’s email address is<EMAIL_ADDRESS>or
<EMAIL_ADDRESS>

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