# Nonclassical properties of Hermite polynomial’s excitation on squeezed
vacuum and its decoherence in phase-sensitive reservoirs

Shi-You Liu1, Ya-Zhou Li1,Li-Yun Hu1,2∗,Jie-Hui Huang1,2, Xue-Xiang Xu1, and
Xiang-Yang Tao1 Corresponding author. Email<EMAIL_ADDRESS>1Center for
Quantum Science and Technology, Jiangxi Normal University, Nanchang 330022,
China 2Beijing Computational Science Research Center, Beijing, 100084, China

###### Abstract

We introduce Hermite polynomial excitation squeezed vacuum (SV)
$H_{n}(\hat{O})S\left(r\right)\left|0\right\rangle$ with $\hat{O}=\mu a+\nu
a^{\dagger}$. We investigate analytically the nonclassical properties
according to Mandel’s Q parameter, second correlation function, squeezing
effect and the negativity of Wigner function (WF). It is found that all these
nonclassicalities can be enhanced by $H_{n}(\hat{O})$operation and adjustable
parameters $\mu$and $\nu$. In particular, the optimal negative volume
$\delta_{opt}$of WF can be achieved by modulating $\mu$and $\nu$ for
$n\geqslant 2,$while $\delta$ is kept unchanged for $n=1$. Furthermore, the
decoherence effect of phase-sensitive enviornment on this state is examined.
It is shown that $\delta$ with bigger $n$diminishes more quickly than that
with lower $n$, which indicates that single-photon subtraction SV presents
more roboustness. Parameter $M$of reservoirs can be effectively used to
improve the nonclassicality.

Keywords: completeness of representation, IWOP method, single- and two-
variable Hermite polynomials

PACS: 03.65 -a. 42.50.Dv

##  I Introduction

Nonclassical light fields play a critical rule in quantum optics and quantum
information process 1 . Generation and manipulation of these states have
attracted much attention to obtain a more effective quantum processing, such
as teleportation, dense coding and quantum cloning. There are many schemes
proposed to realize this purpose. Non-Gaussian operation, say photon addition
and photon subtraction, has been widely employed to enhance the nonclassical
properties of the input states 2 ; 3 ; 4 ; 5 ; 6 ; 6a ; 6b ; 6c ; 6d ; 6e ; 6f
; 6g ; 6h . For instance, a quantum-to-classical transition has been realized
experimentally through single-photon–added coherent states of light 7 . For
any photon–addition coherent state in the dissipative channel, its
nonclassical properties are examined theoretically 8  by the analytical
expression of the Wigner function (a Laguerre–Gaussian function). As another
example, photon subtraction or addition has been used to improve entanglement
between Gaussian states and the average fidelity of quantum teleportation 9 ;
10 .

On the other hand, superposition of operators, such as $a^{\dagger}a$,
$ta^{\dagger}+ra$, $a^{2}+b^{2}$, $a^{\dagger 2}+b^{\dagger 2}$, are also
applied to generate nonclassical states 11 ; 12 ; 13 ; 13a ; 13b ; 14 . For
example, the $ta^{\dagger}+ra$ operator is uesed to realize quantum state
engineering and improve quantum entanglement or non-Gaussian entanglement
distillation or the effect of quantum teleportation 11 ; 12 . In addition,
this superposition operation is employed to enhance the degree of entanglement
of even entangled coherent state 15 . Thus it will be interesting to
investigate the different combination of elementary non-Gaussian operations to
manipulate nonclassical quantum states. As a kind of polynomials states, for
instace, the squeezed Hermoite states is found to be the minimum uncertain
states for amplitude-squared squeezing 16 . In addition, the squeezed two-
variable Hermite polynomial state is shown to be the minimum uncertain states
for amplitude-squared squeezing, which is called as the sum-frequency
squeezing states 17 .

Recently, the Hermite polynomial’s coherent state
$H_{n}\left(Q\right)\left|\alpha\right\rangle$ is introduced 18 , where
$Q=(a+a^{\dagger})/\sqrt{2}$ is the coordinate operator and
$\left|\alpha\right\rangle$=$\exp\\{\alpha
a^{\dagger}-\alpha^{\ast}a\\}\left|0\right\rangle$ is the Glauber coherent
state. Then some nonclassical properties are discussed in details. In this
paper, we shall introduce another kind of non-Gaussian state, which can be
generated by operating Hermite polynomial of superposition of coherent photon-
subtraction and addition (HPS), i.e., $H_{n}\left(\mu a+\nu
a^{\dagger}\right)$ on single-mode squeezed vacuum (SV)
$S\left(r\right)\left|0\right\rangle$. Single photon subtraction/addition SV,
Hermite polynomial’s subtraction $H_{n}\left(\mu a\right)$ and addition
$H_{n}\left(\nu a^{\dagger}\right)$ SV can be considered as special cases of
the HPS. It is interesting to notice that the HPS-SV can be generated by
superposing some photon-addition and photon-subtraction SVs. As far as we
know, there is no report in literature before.

This paper is arranged as follows. In section 2, we shall derive the
normalization factor $N_{\mu_{1},\nu_{1}}$ for the non-Gaussian states. It is
shown that $N_{\mu_{1},\nu_{1}}$ is just the Legendra polynomials, which is
needed for clearly discussing the statistical properties of the HPS-SV. In
section 3, we shall discuss nonclassical properties of the HPS-VS by
ananlytically deriving Mandel’s Q parameter, second correlation function,
photon-number distribution, squeezing effect. In section 4, the Wigner
function (WF) of the HPS-SV is obtained by using the property of Weyl ordered
operators’ invariance under similar transformations. In particular, the
nonclassical property is presented according to the negativity of the WF.
Section 5 is devoted to considering the effect of phase-sensitive reservoirs
on the HPS-SV in terms of the negativity of WF. The last section is used to
draw a conclusion.

##  II The HPS-SV and its normalization

The HPS-SV can be generated by operating Hemite polynomial operator
$H_{n}\left(\mu a+\nu a^{\dagger}\right)$ on single-mode squeezed vacuum
$S\left(r\right)\left|0\right\rangle$,

$\left|\Psi\right\rangle_{H}=N_{\mu_{1},\nu_{1}}H_{n}(\hat{O})S\left(r\right)\left|0\right\rangle,\hat{O}\equiv\mu
a+\nu a^{\dagger},$ (1)

where $N_{\mu_{1},\nu_{1}}$ is the normailzarion factor to be determined, and
$S\left(r\right)=\exp\\{\frac{r}{2}\left(a^{2}-a^{{\dagger}2}\right)\\}$ is
the squeezing operator with $r$ being squeezing parameter, and $H_{n}(x)$ is
the single-variable Hermite polynomials. $a$ and $a^{\dagger}$ are the Bose
anahilate and creation operator, respectively, satisfying communicative
relation $[a,a^{\dagger}]=1$.

In order to calculate $N_{\mu_{1},\nu_{1}}$, using the transformation relation
of single-mode squeezed operator 19 ,
$S^{\dagger}\left(r\right)aS\left(r\right)=a\cosh r-a^{{\dagger}}\sinh r$,
$S^{\dagger}\left(r\right)a^{{\dagger}}S\left(r\right)=a^{{\dagger}}\cosh
r-a\sinh r$, we can get

$S^{\dagger}\left(r\right)H_{n}(\hat{O})S\left(r\right)=H_{n}(\hat{O}_{1}),(\hat{O}_{1}\equiv\mu_{1}a+\nu_{1}a^{\dagger}),$
(2)

where $\mu_{1}=\mu\cosh r-\allowbreak\nu\sinh r$, $\nu_{1}=\nu\cosh r-\mu\sinh
r.$ Thus the factor $N_{\mu_{1},\nu_{1}}$ can be calculated according to the
normalization $1=\left\langle\Psi\right|\left.\Psi\right\rangle_{H},$ i.e.,

$N_{\mu_{1},\nu_{1}}^{-2}=\left\langle
0\right|H_{n}(\hat{O}_{1}^{\dagger})H_{n}(\hat{O}_{1})\left|0\right\rangle.$
(3)

Then further employing the generating of function of single-variable Hermite
polynomial,

$H_{n}\left(x\right)=\left.\frac{\partial^{n}}{\partial
t^{n}}e^{-t^{2}+2tx}\right|_{t=0},$ (4)

and the following operator identity 19
$e^{A+B}=e^{A}e^{B}e^{-\frac{1}{2}\left[A,B\right]}=e^{B}e^{A}e^{\frac{1}{2}\left[A,B\right]}$,
which is valid for $[A,[A,B]]=[B,[A,B]]=0$, we can put Eq.(3) into the form

$\displaystyle N_{\mu_{1},\nu_{1}}^{-2}$
$\displaystyle=\left.\frac{\partial^{2n}}{\partial\tau^{n}\partial
t^{n}}\exp\left\\{-A\left(\tau^{2}+t^{2}\right)+4\nu_{1}^{2}t\tau\right\\}\right|_{\tau,t=0}$
$\displaystyle=2^{n}n!B^{n}P_{n}\left(2\nu_{1}^{2}/B\right),$ (5)

where $A=1-2\mu_{1}\nu_{1},$ $B=\sqrt{4\nu_{1}^{4}-A^{2}}$, $P_{n}$ is the
Legendre polynomial, and we have used the new formula 20

$\displaystyle\frac{\partial^{2m}}{\partial
t^{m}\partial\tau^{m}}\left.\exp\left(-t^{2}-\tau^{2}+\frac{2x\tau
t}{\sqrt{x^{2}-1}}\right)\right|_{t,\tau=0}$
$\displaystyle=\frac{2^{m}m!}{\left(x^{2}-1\right)^{m/2}}P_{m}\left(x\right),$
(6)

in the last step of Eq.(5). Eq.(5) is just the analytical expression of the
normalization factor $N_{\mu_{1},\nu_{1}}^{-2}$.

In particular, when $(\mu,\nu)=(1,0),\allowbreak(0,1),$ leading to
$(\mu_{1},\nu_{1})=(\cosh r,-\sinh r)$, $(-\allowbreak\sinh r,\cosh r)$, and
$B=\sqrt{1-2e^{2r}}$, $\sqrt{1+2e^{-2r}}$, thus Eq.(5) reduces to

$\displaystyle N_{1,0}^{-2}$
$\displaystyle=2^{n}n!\left(\sqrt{1-2e^{2r}}\right)^{n}P_{n}\left(\frac{2\sinh^{2}r}{\sqrt{1-2e^{2r}}}\right),$
(7) $\displaystyle N_{0,1}^{-2}$
$\displaystyle=2^{n}n!\left(\sqrt{1+2e^{-2r}}\right)^{n}P_{n}\left(\frac{2\cosh^{2}r}{\sqrt{1+2e^{-2r}}}\right),$
(8)

which are just the normalization factors of Hermite subtraction and Hermite
addition squeezed vacuum, respectively. In addition, when the squeezing
parameter $r=0\ $leading to $\mu_{1}=\mu$, $\nu_{1}=\nu,$ i.e., the HPS vacuum
($\left|\Psi\right\rangle_{HPS}\rightarrow\left|\Psi\right\rangle$), we see

$\left|\Psi\right\rangle\equiv
N_{\mu,\nu}H_{n}(\hat{O})\left|0\right\rangle,\text{
}N_{\mu,\nu}^{-2}=\left\\{N_{\mu_{1},\nu_{1}}^{-2}\right\\}_{\left(\mu_{1},\nu_{1}\right)\rightarrow\left(\mu,\nu\right)}.$
(9)

In the state, we can get the average

$\left\langle
a^{{\dagger}l}a^{k}\right\rangle=\frac{N_{\mu,\nu}^{2}\nu^{2n}\left(2\lambda\right)^{l+k}\left(n!\right)^{2}}{\lambda^{2n}\left(n-l\right)!\left(n-k\right)!}F_{n-l,n-k}\left(\lambda^{2}\right),$
(10)

where $\lambda=\nu/\sqrt{1-2\mu\nu}$ and we have define a special function
whose mother function is given by

$F_{m,n}\left(\lambda^{2}\right)\equiv\left.\frac{\partial^{m}}{\partial
s^{m}}\frac{\partial^{n}}{\partial
t^{n}}e^{-t^{2}-s^{2}+4st\lambda^{2}}\right|_{s=t=0}.$ (11)

From Eq.(10) one can see that $\left\langle
a^{{\dagger}l}a^{k}\right\rangle=\left\langle
a^{{\dagger}k}a^{l}\right\rangle^{\ast}$, $\left\langle
a^{{\dagger}l}\right\rangle=\left\langle a^{l}\right\rangle^{\ast}$ and
$\left\langle a\right\rangle=0$, as expected. In particular, when $k=l,$
Eq.(10) reduces to

$\displaystyle\left\langle a^{{\dagger}l}a^{l}\right\rangle$
$\displaystyle=\frac{2^{n+l}N_{\mu,\nu}^{2}\left(n!\right)^{2}}{\left(n-l\right)!}v^{2l}K^{(n-l)/2}P_{n-l}\left(\frac{2\nu^{2}}{\sqrt{K}}\right)$
(12) $\displaystyle(K$
$\displaystyle=4\nu^{4}-\left(1-2\mu\nu\right)^{2},n\geqslant l),$

Eq.(10) shall be useful for further calculations.

##  III Nonclassical properties of the HPS-VS

In this section, we study the nonclassical properties of the HPS-VS according
to Mandel’s Q parameter, photon-number distribution, and squeezing effect.

###  III.1 Mandel’s Q parameter and second-order correlation function

We examine the sub-Poissonian photon statistics by using Mandel’s Q parameter
21 , which is defined as

$Q_{M}=\frac{\left\langle a^{{\dagger}2}a^{2}\right\rangle_{H}}{\left\langle
a^{{\dagger}}a\right\rangle_{H}}-\left\langle
a^{{\dagger}}a\right\rangle_{H}.$ (13)

Super-Poissonian, Poissonian, and sub-Poissonian statistics correspond to
$Q_{M}>0$, $Q_{M}=0,$ and $Q_{M}<0,$respectively. In order to obtain the
result (13), it will be convinient to derive some average values:
$\left\langle a^{2}\right\rangle$, $\left\langle a^{4}\right\rangle,$
$\left\langle a^{{\dagger}}a\right\rangle,$ $\left\langle
a^{{\dagger}3}a\right\rangle$ and $\left\langle
a^{{\dagger}2}a^{2}\right\rangle$ under the state $\left|\Psi\right\rangle$
(9). These averages are obtained from Eq.(10). Thus under the state
$\left|\Psi\right\rangle_{H}$, these corresponding average values
($\left\langle...\right\rangle_{H}=\left\\{\left\langle
S^{{\dagger}}...S\right\rangle\right\\}_{\mu\rightarrow\mu_{1},\nu\rightarrow\nu_{1}}$)
are given by

$\displaystyle\left\langle a^{{\dagger}}a\right\rangle_{H}$
$\displaystyle=\left\\{\left\langle a^{{\dagger}}a\right\rangle\cosh
2r+\sinh^{2}r\right.$ $\displaystyle\left.-\frac{\sinh 2r}{2}\left\langle
a^{{\dagger}2}+a^{2}\right\rangle\right\\}_{\mu\rightarrow\mu_{1},\nu\rightarrow\nu_{1}},$
(14)

and

$\displaystyle\left\langle a^{{\dagger}2}a^{2}\right\rangle_{H}$
$\displaystyle=\left\\{\frac{1}{4}\left(3\cosh 4r+1\right)\left\langle
a^{{\dagger}2}a^{2}\right\rangle+\frac{1}{4}\left\langle
a^{{\dagger}4}+a^{4}\right\rangle\sinh^{2}2r\right.$
$\displaystyle+\left(\sinh 2r-\frac{3}{4}\sinh 4r\right)\left\langle
a^{{\dagger}2}+a^{2}\right\rangle\allowbreak$
$\displaystyle-\frac{1}{2}\left\langle
a^{{\dagger}}a^{3}+a^{{\dagger}3}a\right\rangle\sinh 4r+4\left\langle
a^{{\dagger}}a\right\rangle\left(\allowbreak 3\cosh^{2}r-1\right)\sinh^{2}r$
$\displaystyle\left.+(3\cosh^{2}r-2)\sinh^{2}r\right\\}_{\mu,\nu\rightarrow\mu_{1},\nu_{1}}.$
(15)

Substituting Eq.(10) into Eq.(13), we can get the Mandel’s Q parameter. The
second-order correlation function 22  $g^{(2)}=\left\langle
a^{{\dagger}2}a^{2}\right\rangle_{H}/\left\langle
a^{{\dagger}}a\right\rangle_{H}^{2}$ can also be gotten by using Eqs.(14) and
(16). In particular, when $n=0$ (corresponding to the squeezed vacuum), the
Mandel’s Q parameter and the second-order correlation function are given by
$Q_{M}=\cosh 2r>1$, and$\;g^{(2)}=3+1/\sinh^{2}r>3$, respectively.

In order to clearly see the effects of Hermite polynomial on squeezed vacuum,
the numerical calculation results of squeezing parameter $r$ and Mandel’s Q
parameter $Q_{M}$, $g^{(2)}$ are plotted as the functions of squeezing
parameter $r$ in Figs.1, 2, respectively. From Fig.1(a) for a given
$(\mu,\nu)$=$(1,1)$, it is easy to see that the HPS-SV presents a sub-
Poissonian statistics (except for $n=0$) in a small region of parameter
$r\lesssim 0.5$ and the value of $Q_{M}$ increases with $r$. However, the
absolute value of $Q_{M}$ decreases with $n$ in this region. For a given value
of $n=2$, and several asymmetrical cases of $(\mu,\nu),$ on one hand, from
Fig.1(b) one can see that the negative feature can be enhanced by Hermite
polymials addition $H_{2}(a^{\dagger})$ operation rather than Hermite
polymials subtraction operator $H_{2}(a)$. The latter shows a similar trend to
the SV due to their similar photon-number distributions. On the other hand,
the asymmetrical coherent superposition of subtraction and addition (say,
$H_{2}(a+9a^{\dagger})$) can be more effective for improving the negative
feature of $Q_{M}$ than $H_{2}(a)$ and $H_{2}(a+a^{\dagger})$. In addition,
from Fig.2 one can get similar results for the second-order correlation
function. For instance, the HPS-SV appears antibunching effect in a small
region due to the Hermite operation (except for $n=0$).

Figure 1: (Color online) Mandel‘s $Q$ parameter $Q_{M}$ as a function of
squeezing parameter $r$ for several different values of $n$ and ($\mu,\nu$).
(a) ($\mu,\nu$)$=$($1,1$); (b) $n=2.$

Figure 2: (Color online) The second-order correlation function $g^{(2)}$ as
the function of squeezing parameter $r$ for several different values of $n$
and ($\mu,\nu$). (a) ($\mu,\nu$)$=$($1,1$); (b) $n=2.$

###  III.2 Photon-number distribution

Now, we discuss the photon-number distribution of the HPS-SV. In this field,
the photon-number distribution (PND) of finding $m$ photons is given by
$P_{H}=\left|N_{\mu_{1},\nu_{1}}\left\langle
m\right|H_{n}(\hat{O})S\left(r\right)\left|0\right\rangle\right|^{2}$.
Employing the un-normalized coherent state
$\left|\alpha\right\rangle=\exp[\alpha a^{{\dagger}}]\left|0\right\rangle$
($\left\langle 0\right.\left|\alpha\right\rangle=1$) 24 ; 25 , leading to
$\left|m\right\rangle=\frac{1}{\sqrt{m!}}\frac{\mathtt{d}^{m}}{\mathtt{d}\alpha^{m}}\left|\alpha\right\rangle\left|{}_{\alpha=0}\right.,$
$\left(\left\langle\beta\right.\left|\alpha\right\rangle=e^{\alpha\beta^{\ast}}\right)$,
and the SV 19 ; 22

$S\left(r\right)\left|0\right\rangle=\text{sech}^{1/2}r\exp\left(-\frac{1}{2}a^{{\dagger}2}\tanh
r\right)\left|0\right\rangle,$ (16)

as well as Eq.(4) ($e^{\alpha a}a^{{\dagger}}e^{-\alpha
a}=a^{{\dagger}}+\alpha$), we have

$\displaystyle\left\langle
m\right|H_{n}(\hat{O})S\left(r\right)\left|0\right\rangle$
$\displaystyle=\frac{\text{sech}^{1/2}r}{\sqrt{m!}}\left.\frac{\partial^{m}\partial^{n}}{\partial\alpha^{\ast
m}\partial t^{n}}e^{-A_{1}^{2}t^{2}-B_{1}^{2}\alpha^{\ast
2}+2t\alpha^{\ast}C_{1}}\right|_{t=\alpha^{\ast}=0},$ (17)

where we have set $A_{1}^{2}=1+2\mu^{2}\tanh r-2\mu\nu$,
$B_{1}^{2}=\frac{1}{2}\tanh r$, $C_{1}=\nu-\mu\tanh r.$ Thus the photon-number
distribution is

$\displaystyle P_{H}\left(m\right)$
$\displaystyle=N_{\mu_{1},\nu_{1}}^{2}m!\left(n!\right)^{2}\text{sech}r$
$\displaystyle\times\left|\left.\frac{\partial^{m}\partial^{n}}{\partial\alpha^{\ast
m}\partial t^{n}}e^{-A_{1}^{2}t^{2}-B_{1}^{2}\alpha^{\ast
2}+2t\alpha^{\ast}C_{1}}\right|_{t=\alpha^{\ast}=0}\right|^{2},$ (18)

which is the PND of the HPS-SV. It is easy to see that Eq.(18) just reduces to
the photon-number distribution of SV when $n=0$; while for $r=0$, (i.e., the
Hermite polynomial vacuum state $H_{n}(\hat{O})\left|0\right\rangle$), the PND
is given by

$\displaystyle P_{H}\left(m\right)$
$\displaystyle=N_{\mu,\nu}^{2}m!\left(n!\right)^{2}$
$\displaystyle\times\left|\sum_{l=0}^{[n/2]}\frac{\left(2\mu\nu-1\right)^{l}\left(2\nu\right)^{n-2l}}{l!\left(n-2l\right)!}\delta_{m,n-2l}\right|^{2}.$
(19)

In particular, when $m=n$, using the formula 20

$P_{m}\left(x\right)=x^{m}\sum_{l=0}^{[m/2]}\frac{m!}{2^{2l}l!l!(m-2l)!}\left(1-\frac{1}{x^{2}}\right)^{l},$
(20)

we have

$\displaystyle P_{H}\left(n\right)$
$\displaystyle=N_{\mu,\nu}^{2}\left(n!\right)^{2}2^{2n}\text{sech}r\left|\text{
}D_{1}^{n/2}P_{n}\left(C_{1}/\sqrt{D_{1}}\right)\right|^{2},$ (21)
$\displaystyle\left(D_{1}=C_{1}^{2}-A_{1}^{2}B_{1}^{2}\right).$

In Fig.3, the PND is poltted for different values $\left(\mu,\nu\right)$, $r$
and $n$, from which one can see that (i) by modulating the order of Hermite
polynomials, one has able to change the position of peak [see Fig.3 (a) and
(d)]; (ii) for a small squeezing (say $r=0.3$), the peak of PND is mainly
located at $n$ [see Fig. 3 (a), (b), (d)]; (iii) for a large squeezing (say
$r=0.9$), the peak moves to the small photon-number region (see Fig.3(b));
(iv) in addition, the PND can also be modulated by the parameters [see Fig.3
(a) and (c)], expecially for $n\geqslant 2$.

Figure 3: (Color online) The photon-number of the HPS-SV distribution as a
function of $m$ for several different parameters $r$, $n$ and ($\mu,\nu$). (a)
$n=2,r=0.3$, $\mu,\nu=1/\sqrt{2}$; (b) $n=2,r=0.9$, $\mu,\nu=1$/$\sqrt{2}$;
(c) $n=2,r=0.3$, $\mu=1,\nu=3$; (d) $n=3,r=0.3,$ $\mu,\nu=1$/$\sqrt{2}$.

###  III.3 Squeezing effects

In this subsection, we consider the squeezing effects of the HPS-SV,
eapecially from the Hermite polynomial operation. First, we examine the wave
function which can reflect the the squeezing effect of quantum state to some
extent. Using the natural expression of single-mode squeezing operator in the
momentum representation $\left|p\right\rangle$ 27 ,

$S\left(r\right)=\sqrt{u}\int_{-\infty}^{\infty}dp\left|up\right\rangle\left\langle
p\right|,u=e^{r},$ (22)

which leads to
$S^{{\dagger}}\left(r\right)\left|p\right\rangle=1/\sqrt{u}\left|p/u\right\rangle$,
thus the wave function can be derived as

$\displaystyle\Psi_{H}\left(p\right)$ $\displaystyle=\left\langle
p\right|\left.\Psi\right\rangle_{H}=N_{\mu,\nu}\left\langle
p\right|SS^{{\dagger}}H_{n}\left(\hat{O}\right)S\left|0\right\rangle$
$\displaystyle=\frac{N_{\mu,\nu}}{\sqrt{u}}\left\langle\frac{p}{u}\right|H_{n}\left(\hat{O}_{1}\right)\left|0\right\rangle.$
(23)

Now we calculate the matrix element $\left\langle
p\right|H_{n}(\hat{O})\left|0\right\rangle$. Using the normal ordering form of
$H_{n}(\hat{O})$ and Eq.(4), ($e^{\alpha a}a^{{\dagger}}e^{-\alpha
a}=a^{{\dagger}}+\alpha$), and noticing that 27
$\left|p\right\rangle=\pi^{-1/4}\exp\\{-\frac{1}{2}p^{2}+\sqrt{2}ipa^{{\dagger}}+\frac{1}{2}a^{{\dagger}2}\\}\left|0\right\rangle$,
we have ($\lambda=\nu/\sqrt{1-2\mu\nu}$)

$\displaystyle\left\langle
p\right|H_{n}\left(\hat{O}\right)\left|0\right\rangle$
$\displaystyle=\frac{1}{\pi^{1/4}}\left(\sqrt{1-2\lambda^{2}}\frac{\nu}{\lambda}\right)^{n}e^{-\frac{1}{2}p^{2}}H_{n}\left(\frac{-i\sqrt{2}\lambda
p}{\sqrt{1-2\lambda^{2}}}\right).$ (24)

Thus the distribution of the quadrature $p$ is given by

$\displaystyle\left|\Psi_{H}\left(p\right)\right|^{2}$
$\displaystyle=\frac{N_{\mu,\nu}^{2}}{\pi^{1/2}u}e^{-u^{-2}p^{2}}\left|1-2\mu_{1}\nu_{1}-2\nu_{1}^{2}\right|^{n}$
$\displaystyle\times\left|H_{n}\left(\frac{-i\sqrt{2}\nu_{1}p/u}{\sqrt{1-2\mu_{1}\nu_{1}-2v_{1}^{2}}}\right)\right|^{2},$
(25)

where $\left(u=e^{r}\right)$ and $\mu_{1},\nu_{1}$ are defined above. Eq.(25)
is just the Hermite-Gaussian function.

In order to clearly see the squeezing effect, we present the distribution in
Fig.4 where the distributions are plotted for different values of $n$ and
($\mu,\nu$). Different from the Gaussian distribution of squeezed state (n=0),
the HPS-SV has several different peak distributions with different $n(\neq 0)$
values. In addition, the amplitude values of peaks affected by the parameters
$\mu,\nu$ (see Fig.4(b)).

Figure 4: (Color online) The distributions of the quadrature $p$ of the HPS-SV
as a function of $p$ for several different parameters $n$ and ($\mu,\nu$) with
$r=0.2$. Red, Blue, Green and Gray lines correspond to $n=0,1,2,4$,
respectively. (a) $\mu=\nu=1/\sqrt{2};$(b) n=1,2,4 correspond to
$\left(\mu,\nu\right)=\left(0,1\right)$, $\left(1,2\right)$,
$\left(1,6\right)$, respectively.

Next, we further discuss the squeezing property of the HPS-SV by using the
standard analysis of quadrature squeezing, i.e., $\left(\triangle
Q\right)^{2}<1$ or $\left(\triangle P\right)^{2}<1$ which indicates the
squeezing or sub-Poissonian statistics. Here, we introduce a quadrature
operator $Q_{\theta}=ae^{-i\theta}+a^{\dagger}e^{i\theta}$. Thus the squeezing
can be characterized by the minimum value
$\left\langle\triangle^{2}Q_{\theta}\right\rangle$ $<1$ with respect to
$\theta$, or by the normal ordering form
$\left\langle\colon\triangle^{2}Q_{\theta}\colon\right\rangle<0$ 28 . Upon
expanding the terms of
$\left\langle\colon\triangle^{2}Q_{\theta}\colon\right\rangle$, one can
minimize its value over the whole angle $\theta$, which is given by 29
$S_{opt}=-2\left|\left\langle a^{\dagger 2}\right\rangle-\left\langle
a^{\dagger}\right\rangle^{2}\right|+2\left\langle
a^{\dagger}a\right\rangle-2\left|\left\langle
a^{\dagger}\right\rangle\right|^{2}$, then its negative value in the range
$\left[-1,\text{ }0\right)$ indicates squeezing (or nonclassical). For the
HPS-SV, $\left\langle a^{\dagger}\right\rangle=0,$ using Eq.(14), the degree
of squeezing of the HPS-SV can be obtained

$S_{HPS}=2\left\\{\left\langle
a^{\dagger}a\right\rangle_{H}-\left|\left\langle a^{\dagger
2}\right\rangle_{H}\right|\right\\}<0,$ (26)

which indicates that the negative value of $S_{HPS}$ only emerges when
$\left\langle a^{\dagger}a\right\rangle_{H}<\left|\left\langle a^{\dagger
2}\right\rangle_{H}\right|,$ where $\left\langle
a^{2}\right\rangle_{H}=\left\langle
a^{2}\cosh^{2}r+a^{{\dagger}2}\sinh^{2}r-a^{{\dagger}}a\sinh
2r\right\rangle_{\mu\rightarrow\mu_{1},\nu\rightarrow\nu_{1}}$-$\frac{1}{2}\sinh
2r$. In particular, when $n=0\ $(i.e., the case of squeezed vacuum),
$S_{HPS}=-2e^{-r}\sinh r,$ as expected.

Figure 5: (Color online) The degree of squeezing $S_{opt}$ of HPS-SV as the
function of squeezing parameter $r$ for several different values of $n$ and
($\mu,\nu$). (a) ($\mu,\nu$)$=$($1,1$); (b) $n=2.$

The degree of squeezing of the HPS-SV is shown in Fig. 5 for several different
values of parameters $n$ and ($\mu,\nu$). It is found that the degree of
squeezing of the HPS-SV increases with $r$. In Fig.5 (a) with a given
($\mu,\nu$)=(1,1), comparing with the SV (n=0), the HPS-SV can present
squeezing only when the squeezing parameter $r$ exceeds a certain threshold
value (say 0.5); the degree of squeezing can be enhanced by Hermite
polynomials superposition operation (say $n=2,4$) in a larger region of
squeezing parameter. In Fig.5 (b) with $n=2$, it is shown that (i) the Hermite
photon-subtraction operation $H_{2}\left(a\right)$ on the SV can be used to
improve the degree of squeezing in a small region ($r\lesssim 0.6$), while for
$H_{2}\left(a^{{\dagger}}\right)$ the case is not true; the coherent
superposition operation ($H_{2}\left(a+a^{{\dagger}}\right)$), rather than the
$H_{2}\left(a\right)$ or $H_{2}\left(a^{{\dagger}}\right)$ operations, can
improve the degree of squeezing in a large region. This implies that the
coherent operation $\mu a+\nu a^{{\dagger}}$ achieves better squeezing than
the mere photon-subtraction (-addition) in a large region of $r$. In addition,
the maximum degree of squeezing of the HPS-SV is $-1$.

##  IV Wigner distribution of the HPS-SV

As a kind of quasi-probability function, the Wigner function (WF) is a
powerful tool to describe the nonclassicality of optical fields, whose partial
negativity implies the highly nonclassical properties of quantum states. In
addition, the negativity is often used to present the decoherence of quantum
states. In this section, we derive the analytical expression of WF for the
HPS-SV by using the the Weyl ordered operators’ invariance under similar
transformations 30 . For a single-mode quantum system, the WF can be
calculated as $W=$tr$\left[\rho\Delta\left(\alpha\right)\right]$, where
$\Delta\left(\alpha\right)$ single-mode Wigner operator 30 ; 31 ,

$\displaystyle\Delta\left(\alpha\right)$
$\displaystyle=\frac{e^{2\left|\alpha\right|^{2}}}{\pi}{\displaystyle\int}\frac{d^{2}\beta}{\pi}\left|\beta\right\rangle\left\langle-\beta\right|e^{2\left(\alpha\beta^{\ast}-\alpha^{\ast}\beta\right)}$
$\displaystyle=\frac{1}{2}\genfrac{}{}{0.0pt}{}{\colon}{\colon}\delta\left(\alpha-a\right)\delta\left(\alpha^{\ast}-a^{{\dagger}}\right)\genfrac{}{}{0.0pt}{}{\colon}{\colon}.$
(27)

Here $\alpha=\left(q+ip\right)/\sqrt{2}$ and the symbol
$\genfrac{}{}{0.0pt}{}{\colon}{\colon}\genfrac{}{}{0.0pt}{}{\colon}{\colon}$
denotes Weyl ordering.

The merit of Weyl ordering lies in the Weyl ordered operators’ invariance
under similar transformations proved, which means
$S\genfrac{}{}{0.0pt}{}{:}{:}\left(\circ\circ\circ\right)\genfrac{}{}{0.0pt}{}{:}{:}S^{-1}=\genfrac{}{}{0.0pt}{}{:}{:}S\left(\circ\circ\circ\right)S^{-1}\genfrac{}{}{0.0pt}{}{:}{:}$,
as if the “fence” $\genfrac{}{}{0.0pt}{}{:}{:}\genfrac{}{}{0.0pt}{}{:}{:}$did
not exist, so $S$ can pass through it. Then following this invariance and the
above squeezing transform relations, we have

$S^{\dagger}\Delta\left(\alpha\right)S=\frac{1}{2}\genfrac{}{}{0.0pt}{}{\colon}{\colon}\delta\left(\bar{\alpha}-a\right)\delta\left(\bar{\alpha}^{\ast}-a^{{\dagger}}\right)\genfrac{}{}{0.0pt}{}{\colon}{\colon}=\Delta\left(\bar{\alpha}\right),$
(28)

where $\bar{\alpha}=\alpha\cosh r+\alpha^{\ast}\sinh r$. Thus the WF can be
derived

$W\left(\alpha,\alpha^{\ast}\right)=N_{\mu_{1},\nu_{1}}^{2}\left\langle
0\right|H_{n}\left(\hat{O}_{1}^{{\dagger}}\right)\Delta\left(\bar{\alpha}\right)H_{n}\left(\hat{O}_{1}\right)\left|0\right\rangle.$
(29)

Further employing Eq.(4) we can obtain

$\displaystyle W\left(\alpha,\alpha^{\ast}\right)$
$\displaystyle=\frac{e^{-2\left|\bar{\alpha}\right|^{2}}}{\pi
N_{\mu_{1},\nu_{1}}^{-2}}\frac{\partial^{2n}}{\partial\tau^{n}\partial t^{n}}$
$\displaystyle\times\left.e^{4\nu_{1}\bar{\alpha}\tau+4\nu_{1}\bar{\alpha}^{\ast}t-\allowbreak
4\nu_{1}^{2}t\tau+\left(2\mu_{1}\nu_{1}-1\right)\left(\tau^{2}+t^{2}\right)}\right|_{\tau=t=0}$
$\displaystyle=\frac{1}{\pi}{\displaystyle\sum\limits_{l=0}^{n}}\frac{\left(n!\right)^{2}\left(-\allowbreak
4\nu_{1}^{2}\right)^{l}\left(2\mu_{1}\nu_{1}-1\right)^{n-l}}{l!\left[\left(n-l\right)!\right]^{2}N_{\mu,\nu}^{-2}e^{2\left|\bar{\alpha}\right|^{2}}}$
$\displaystyle\times\left|H_{n-l}\left(\frac{2\nu_{1}\bar{\alpha}}{i\sqrt{2\mu_{1}\nu_{1}-1}}\right)\right|^{2}.$
(30)

Obviously, the WF $W\left(\alpha,\alpha^{\ast}\right)$ in Eq.(30) is a real
function and is non-Gaussian in phase space due to the presence of
$H_{n-l}\left(x\right)$. In particular, when $n=0,$ Eq.(30) just reduces to
the WF of the squeezed vacuum, $\frac{1}{\pi}e^{-2\left|\alpha\cosh
r+\alpha^{\ast}\cosh r\right|^{2}}$, as expected. In addition, when $n=1$
corresponding to the single photon-subtraction (-addition) squeezed vacuum,
($H_{0}\left(x\right)=1,H_{1}\left(x\right)=2x$)

$W\left(\alpha,\alpha^{\ast}\right)=\frac{e^{-2\left|\bar{\alpha}\right|^{2}}}{\pi}\left(4\left|\bar{\alpha}\right|^{2}-\allowbreak
1\right),$ (31)

which indicates that there is always negative region at the center of phase
space $\alpha=0$ (independent of the two parameters $\mu,\nu$). In Fig.6, the
Wigner distributions are depicted in phase space for several different
parameter values $n$, and $\left(\mu,\nu\right)$, from which it clearly see
that there are some obvious negative regions of the WF in the phase space
which is an indicator of the nonclassicality of the state. In addition, these
negative areas are modulated not only by $n$ [see Fig.6 (a)-(c)], but also by
the parameters $\left(\mu,\nu\right)$ [see Fig.6 (b) and (d)]. For instance,
there is obvious difference of WF distribution between Fig.6 (b) and (d). That
is to say, for higher order case $n\geqslant 2$, the negative area depends on
the two parameters. In order to clearly see this point, we can quanlify the
negative volume of the WF, defined by
$\delta$=$\frac{1}{2}$[$\int_{-\infty}^{\infty}dqdp\left|W(q,p)\right|$-1] 32
.

Figure 6: (Color online) Contour plot of the Wigner function after the Hermite
polynomials coherent operation on the SV with $r=0.3$ for several different n
and ($\mu,\nu$). (a) $n=1$; (b) $n=2$, $\mu,\nu=1;$ (c) $n=3$, $\mu,\nu=1;$
(d) $n=2$, $\mu=1$, $\nu=9$, where only the negative regions are colored in
blue.

In Fig. 7, the negative volume of WF as a function of $r$ or $\nu$ on applying
the Hermite coherent suposition operatopn $H_{n}(\mu a+\nu a^{\dagger})$ for
several different $n$. From Fig.7(a), one can find that the negative volume
$\delta$ increases with the order $n$ (in a certain region of $r\lesssim
0.45$) and decreases with $r$. In particular, for the case of $n=1$
(corresponding to a superposition between single-photon addition/subtraction
SV and SV), the negative volume is independent of parameters $r$ and
($\mu,\nu$) and is kept unchanged ($\delta=0.2131$). In fact, using Eq.(31)
one can calculate that the negative volume of WF with $n=1$ is
$\delta=2/\sqrt{e}-1\approx 0.2131$. In Fig.7(b), we optimize the negative
volume for different $r$ and $n,$ where $\mu$,$\nu$ are taken as
$\mu=\sqrt{1-\nu^{2}}$. From Fig.7(b), it is found that (i) when
$\mu=\nu=1/\sqrt{2}$, the negative volume $\delta$ increases with the order
$n$ for a given small $r$ and decreases with $r$ for a given $n$ (see the
vertical dotted line at the point of $\nu=1/\sqrt{2}$); (ii) for a given
parameter $r=0.1$, the negative volume $\delta$ increases with $n$ when $\nu$
exceeds a certain threshold ($\nu\approx 0.41$); (iii) for a given $n=2,$
$\delta$ decreases with $r$ when $\nu$ exceeds a certain threshold
($\nu\approx 0.45$); (iv) the negative volume $\delta$ does not monotonously
increase with $\nu;$ in particular, one can find the maximum negative volume
$\delta$ in a bigger region of $\nu$ ($\nu\geqslant 0.41$) for $n=2,3,4$. This
optimal value of $\delta$ can be achieved at neither $\nu=0$ nor $\nu=1$. For
instance, this points are $\nu\approx 0.71,0.74,0.78$ for different values of
$n=2,3,4$ and $r=0.1$. These indicate that the effects of the coherent
operation $H_{n}(\mu a+\nu a^{\dagger})$ with higher order $n\geqslant 2$ are
prominent than those of the mere photon subtraction $H_{n}(\nu a^{\dagger})$
and the addition $H_{n}(\nu a^{\dagger})$ particularly in the larger region of
parameter $\nu$, whereas the optimal operation is not the photon subtraction
or the photon addition in this region. This result is different from that in
Ref.11 .

Figure 7: (Color online) Negative volume of the Wigner function as a function
of (a) squeezing parameter $r$, and $\mu$=$\nu$=$1/\sqrt{2}$; (b) parameter
$\nu$ with $\mu=\sqrt{1-\nu^{2}}$ for several different $n=1,2,3,4$, and
$r=0.1,0.3,0.5,0.8.$

##  V Decoherence of the HPS-SV in Phase-Sensitive Reservoirs

In this section, we shall examine the time evolution of the HPS-SV at the
presence of phase-sensitive reservoirs. In the interaction picture and the
Born and Markow approximation, the time evolution of the density matrix is
governed by the master equation (ME) 33 :

$\displaystyle\frac{d}{dt}\rho\left(t\right)$
$\displaystyle=\kappa\bar{n}L\left[a^{{\dagger}}\right]\rho+\kappa\left(\bar{n}+1\right)L\left[a\right]\rho$
$\displaystyle+\kappa MD\left[a\right]\rho+\kappa
M^{\ast}D\left[a^{{\dagger}}\right]\rho,$ (32)

and

$\displaystyle L\left[O^{{\dagger}}\right]\rho$
$\displaystyle=2O^{{\dagger}}\rho O-OO^{{\dagger}}\rho-\rho OO^{{\dagger}},$
(33) $\displaystyle D\left[a\right]\rho$ $\displaystyle=2a^{{\dagger}}\rho
a^{{\dagger}}-a^{{\dagger}2}\rho-\rho a^{{\dagger}2}.$ (34)

where $\kappa$ and $\bar{n}$ are the dissipative coefficient and the average
thermal photon number of the environment, respectively. Here $M$ is the
complex correlation parameter between modes symmetrically displaced about
center frequency. In fact, the ME in Eq.(32) includes two special cases: (1)
For an uncorrelated reservoir, i.e., $M=0,$ Eq.(32) becomes the ME describing
the interaction between a system and a thermal environment at finite
temperature; (2) when $M=\bar{n}=0,$ Eq.(32) reducess to the one describing
the photon-loss channel. For an (non)ideally squeezed reservoir, the
constraint condition $\left|M\right|^{2}=\bar{n}\left(\bar{n}+1\right)$
($\left|M\right|^{2}<\bar{n}\left(\bar{n}+1\right)$) is required.

In Ref.34 , we derived the Kraus operator-sum representation of density
operator $\rho$ and the time evolution of some distibution functions by using
the thermal entangled state representation $\left\langle\eta\right|$. The
evolution of Wigner function is given by

$W\left(\alpha,t\right)=\frac{2\mu_{\infty}}{T}\int\frac{d^{2}\beta}{\pi}e^{-\frac{2\mu_{\infty}^{2}}{T}\Sigma\left(\bar{\alpha},\bar{\alpha}^{\ast}\right)}W\left(\beta,0\right),$
(35)

where $T=1-e^{-2\allowbreak\kappa t},$ $\bar{\alpha}=\alpha-\beta e^{-\kappa
t}$, and
$\mu_{\infty}=1/\sqrt{\left(2\bar{n}+1\right)^{2}-4\left|M\right|^{2}}$,
$\Sigma\left(\bar{\alpha},\bar{\alpha}^{\ast}\right)$ is defined as

$\displaystyle\Sigma\left(\bar{\alpha},\bar{\alpha}^{\ast}\right)$
$\displaystyle=\left(\begin{array}[c]{cc}\bar{\alpha}&\bar{\alpha}^{\ast}\end{array}\right)\sigma_{\infty}\left(\begin{array}[c]{c}\bar{\alpha}\\\
\bar{\alpha}^{\ast}\end{array}\right),$ (39) $\displaystyle\sigma_{\infty}$
$\displaystyle=\left(\begin{array}[c]{cc}M^{\ast}&\bar{n}+\frac{1}{2}\\\
\bar{n}+\frac{1}{2}&M\end{array}\right).$ (42)

Noting the differential expression of Wigner function Eq.(30), we can finally
obtain

$W\left(\alpha,t\right)=W_{r}\left(\alpha,t\right)F_{n}\left(\alpha,t\right),$
(43)

where $W_{r}\left(\alpha,t\right)$ is the evolution of Wigner function of
squeezed vacuum in phase sensitive resevoire, and $F_{n}\left(\alpha,t\right)$
is a non-Gaussian item due to the presence of Hermite excitation,

$\displaystyle W_{r}\left(\alpha,t\right)$
$\displaystyle=\frac{\mu_{\infty}e^{-P}}{\pi\sqrt{D}}e^{\frac{1}{TD}\left(2R_{1}R_{2}R_{2}^{\ast}-R_{3}R_{2}^{\ast}{}^{2}-R_{2}^{2}R_{3}^{\ast}\right)},$
(44) $\displaystyle F_{n}\left(\alpha,t\right)$
$\displaystyle=\sum_{l=0}^{n}\frac{\left[n!\right]^{2}\left(-G_{1}\right)^{l}\left|G_{2}\right|^{n-l}}{l!\left[\left(n-l\right)!\right]^{2}}$
$\displaystyle\times
N_{\mu,\nu}^{2}\left|H_{n-l}\left(\frac{G_{3}}{2i\sqrt{G_{2}}}\right)\right|^{2},$
(45)

and $D=R_{1}^{2}-\left|R_{3}\right|^{2},$

$\displaystyle P$
$\displaystyle=\frac{2\mu_{\infty}^{2}}{T}\left[\left(2\bar{n}+1\right)\left|\alpha\right|^{2}+M\alpha^{\ast}{}^{2}+M^{\ast}\alpha^{2}\right],$
$\displaystyle R_{1}$
$\displaystyle=\left(1+2\bar{n}\right)\mu_{\infty}^{2}e^{-2(t\kappa)}+\allowbreak
T\allowbreak\cosh 2r,$ $\displaystyle R_{2}$
$\displaystyle=\mu_{\infty}^{2}\left(\alpha^{\ast}+2\bar{n}\alpha^{\ast}+2\alpha
M^{\ast}\right)e^{-\kappa t},$ $\displaystyle R_{3}$
$\displaystyle=2\mu_{\infty}^{2}M^{\ast}e^{-2(t\kappa)}+T\sinh 2r,$ (46)

as well as

$\displaystyle G_{1}$
$\displaystyle=4\nu_{1}^{2}\left\\{1+\frac{T}{D}\left[\left(R_{3}+R_{3}^{\ast}\right)\sinh
2r-2R_{1}\cosh 2r\right]\right\\},$ $\displaystyle G_{2}$
$\displaystyle=2\mu_{1}\nu_{1}-1+\frac{2T\nu_{1}^{2}}{D}\left\\{2R_{1}\sinh
2r\right.$
$\displaystyle\left.-\allowbreak\left(R_{3}-R_{3}^{\ast}+\left(R_{3}+R_{3}^{\ast}\right)\cosh
2r\right)\right\\},$ $\displaystyle G_{3}$
$\displaystyle=\frac{4\nu_{1}}{D}\allowbreak R_{1}\left(\allowbreak
R_{2}^{\ast}\sinh r+R_{2}\cosh r\allowbreak\right)$
$\displaystyle-\frac{4\nu_{1}}{D}\allowbreak\left(R_{2}^{\ast}R_{3}\cosh
r+R_{3}^{\ast}\allowbreak R_{2}\sinh r\right).$ (47)

In partciular, at the center of phase space $\alpha=0$, we have
$R_{2}=0,P=0,G_{3}=0.$ Thus for the case of $n=1$, we can get
$W\left(\alpha,t\right)\propto-G_{1}$. Thus the existence of negative volume
of WF is determined by $G_{1}>0,$ which leads to

$\kappa t<\kappa t_{c}=\frac{1}{2}\ln\left(\mu_{\infty}+1\right),$ (48)

which is independent of squeezing parameter $r$. It is easy to see that for
any $M$ ranging from $0$ to $\bar{n}(\bar{n}+1)$,

$\frac{1}{2}\ln\left(\frac{2\bar{n}+2}{2\bar{n}+1}\right)\leqslant\frac{1}{2}\ln\left(\mu_{\infty}+1\right)\leqslant\frac{1}{2}\ln
2,$ (49)

which indicates that the characteristic time of decoherence of single-photon
added squeezed vacuum state in phase sensitive reservoirs is larger than that
in the thermal enviornment and smaller than that in photon-loss channel.

Figure 8: (Color online) The evolution of Wigner function distribution in
phase space with $n=1,$ $r=0.3$. (a) $\kappa t=0.01,$ $M=0.1,$ $\bar{n}=1;$
(b) $\kappa t=0.08,$ $M=0.1,$ $\bar{n}=1;$ (c) $\kappa t=0.08,$ $M=1,$
$\bar{n}=1;$ (d) $\kappa t=0.08,$ $M=0.1,$ $\bar{n}=1.5;$

In order to measure the degree of nonclassicality for the evoluted state, we
consider the negative area and the nagative volume in phase space. As shown in
Fig. 8, it is shown that the nagative area gradually dissapears with the
increasemenet of $\bar{n}$, $\kappa t$, while increases with parameter $M$. To
clearly see the effects of the decoherence and parameter $M$ on the
nonclassical properties, the evolutions of negative volume with time and
squeezing parameter $M$ are plotted in Fig.9 for given $n=1$, $r=0.3$ and
$\bar{n}=0.5$. From Fig.9(a) one can see that the nagative volume
monotonically diminnishes with $\kappa t$, and there is a more rapid
attenuation for a big $n$ than a small one; this leads to a smaller negative
volume for a big $n$ than a small one when $\kappa t$ ecseeds a certain value.
From this point, one can draw a conclusion that single-photon
subtraction/addition SV present a more strongger roboustness against the
reservoirs than a higher-order photon subtraction/addition which could have a
bigger negative volume at initial time. From Fig.9(b), it is shown that the
parameter $M$ can be used to enhance the nonclassicality of quantum state in a
phase-sensitive reservoirs. Specially speaking, the negative volume increases
monotonically with $M$. The optimal volume appears at the the maximum value of
$\left|M\right|^{2}=\bar{n}(\bar{n}+1)$, which inreases with $n$, as expected.

Figure 9: (Color online) Evolution of Negative volume of Wigner function as a
function of (a) $\kappa t$ with $M=0.1$; (b) $M$ with $\kappa t=0.03$ for
several different $n=1,2,3$, and $r=0.3$, $\bar{n}=0.5,$ as well as
$\mu$=$\nu$=$1/\sqrt{2}$.

##  VI Conclusions

In this paper, we have introduced a new non-Gaussian state, which is generated
by applying Hermite-polynomial excitation on squeezed vacuum. Single-photon
addition/subtraction and Hermite polynomial’s addition/subtraction can be seen
as its three special cases. Its normalized factor is found to be a Legendre
polynomial. Then we have investigated its nonclassicality according to the
Mandel’s Q parameter, second-order correlation function, photon-number
distribution, squeezing effects and the negativity of WF in phase space. It is
shown that all these nonclassical properties can be obviously improved by the
HPS operation and can be remarkably modulated by superposition parameters
$\mu$ and $\nu$. The degree of squeezing of the HPS-SV increases with $r$. In
particular, comparing with the SV, an observable improvement of squeezing
effect can be achieved by Hermite photon-subtraction operation
$H_{2}\left(a\right)$ and the coherent superposition operation
($H_{2}\left(a+a^{{\dagger}}\right)$) in a small region ($r\lesssim 0.6$) and
a large region, respectively.

In addition, the numerical calculation of negative volume $\delta$ of WF
showed that $\delta$ increases with the order $n$ for $r\lesssim 0.45$ and
decreases with $r$. It is interesting to notice that the negative volume
$\delta$ is given $\delta=2/\sqrt{e}-1\approx 0.2131$ independent of $\mu$ and
$\nu$, as well as $r$. For high-order excitation ($n\geqslant 2$), the
negative volume can be optimized by modulating parameters $\nu$
($\mu=\sqrt{1-\nu^{2}}$) and $n$ for a given $r$. It is found that the
negative volume $\delta$ may increase with $n$ and decrease with $r$ when
$\nu$ exceeds a certain threshold. In particular, the optimal value of
$\delta$ can be obtained in a bigger region of $\nu$ ($1>\nu\gtrsim 0.41$) not
at $\nu=0$ or $\nu=1$. This implies that the effects of the coherent operation
$H_{n}(\mu a+\nu a^{\dagger})$ with higher order $n\geqslant 2$ are prominent
than those of the mere photon subtraction $H_{n}(\nu a^{\dagger})$ and the
addition $H_{n}(\nu a^{\dagger})$ particularly in the larger region of
parameter $\nu$, whereas the optimal operation is not the photon subtraction
or the photon addition in this region, which is a new result.

Furthermore, we have considered the decoherence effects of the HPS-SV in
phase-sensitive reservoirs according to the analitically derived WF
distribution. It is shown that the negative area and volume diminish gradually
with the evolution of time and dissapear eventually. However, the negative
volume of the HPS-SV with higher order excitation decays more rapidly with
time, which implies that single-photon subtraction/addition SV has a more
strongger roboustness than a higher-order photon subtraction/addition although
the latter has a bigger negative volume at initial time. In addition, the
parameter $M$ describing the squeezing characteristic of reservoirs can be
effectively used to enhance the nonclassicality.

Acknowledgments: Project supported by the National Natural Science Foundation
of China (Grant No.11264018), the Natural Science Foundation of Jiangxi
Province of China (Grant No. 20132BAB212006), and the Research Foundation of
the Education Department of Jiangxi Province of China (no GJJ14274) as well as
Degree and postgraduate education teaching reform project of jiangxi
province(No. JXYJG-2013-027).

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